Subgroups — C₂×C₄×C₃₁×#137 #21516916

Order 7936.

2810 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#137	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃₁×#137	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃₁×#441	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#440	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×Q₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#417	C₂	1	1	yes	maximal, characteristic
11	C₂×C₃₁×#419	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#137	C₄	1	16	yes	direct product factor
16	C₂×C₃₁×#137	C₂²	1	8	yes	—
17	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
18	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
19	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₄×#41×C₃₁	—	2	4	no	—
22	C₄×#41×C₃₁	C₂²	1	8	yes	—
23	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
24	C₂×C₄×Q₈×C₃₁	—	2	4	no	—
25	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
26	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₃₁	C₂²	1	8	yes	complemented
31	C₂×C₄×D₄×C₃₁	—	2	1	no	—
32	C₂×C₄×D₄×C₃₁	—	2	1	no	—
33	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#117	—	2	1	no	—
36	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
37	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
38	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₂²×C₄²×C₃₁	—	2	1	no	—
40	C₂²×C₄²×C₃₁	—	2	1	no	—
41	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#441	C₂²	1	16	yes	complemented
43	C₃₁×#440	C₂²	1	16	yes	complemented
44	C₄×Q₁₆×C₃₁	C₂²	1	16	yes	complemented
45	C₄×SD₁₆×C₃₁	C₂²	1	16	yes	complemented
46	C₃₁×#428	C₂²	1	8	yes	complemented
47	C₂×C₃₁×#107	C₂²	1	2	yes	—
48	C₂×C₃₁×#106	C₂²	1	2	yes	—
49	C₃₁×#419	C₂²	1	8	yes	complemented
50	C₃₁×#417	C₂²	1	8	yes	—
51	C₂×C₃₁×#103	C₂²	1	2	yes	—
52	C₂×C₃₁×#103	C₂²	1	2	yes	—
53	C₂×C₃₁×#103	C₂²	1	2	yes	—
54	C₂×C₃₁×#102	C₂²	1	2	yes	—
55	C₄×M₁₆×C₃₁	C₂²	1	8	yes	complemented
56	C₂×C₃₁×#97	C₂²	1	2	yes	—
57	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
58	C₂×#41×C₃₁	—	2	2	no	—
59	C₂×#41×C₃₁	—	2	4	no	—
60	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
61	C₂×#41×C₃₁	C₂³	1	4	yes	—
62	C₂²×Q₈×C₃₁	—	2	1	no	—
63	C₂²×Q₈×C₃₁	—	2	2	no	—
64	C₂²×Q₈×C₃₁	—	2	4	no	—
65	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
66	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
67	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
69	C₂²×D₄×C₃₁	—	2	1	no	—
70	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₃₁	—	2	1	no	—
72	C₂³×C₄×C₃₁	—	2	1	no	—
73	C₂³×C₄×C₃₁	—	2	1	no	—
74	C₂³×C₄×C₃₁	—	2	1	no	—
75	C₂³×C₄×C₃₁	—	2	1	no	—
76	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
77	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
78	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#137	C₂×C₄	1	64	yes	direct product factor
80	C₂×Q₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
81	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
82	C₂×SD₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
83	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
84	C₂×M₁₆×C₃₁	—	2	4	no	—
85	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
86	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
87	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
88	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
89	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
90	C₄×Q₈×C₃₁	—	2	4	no	—
91	C₄×Q₈×C₃₁	—	2	4	no	—
92	C₄×Q₈×C₃₁	—	2	8	no	—
93	C₄×Q₈×C₃₁	—	2	8	no	—
94	C₄×Q₈×C₃₁	—	2	2	no	—
95	C₄×Q₈×C₃₁	—	2	2	no	—
96	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
97	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
98	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
99	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
100	C₄×D₄×C₃₁	—	2	8	no	—
101	C₄×D₄×C₃₁	—	2	8	no	—
102	C₄×D₄×C₃₁	—	2	2	no	—
103	C₄×D₄×C₃₁	—	2	2	no	—
104	C₄×D₄×C₃₁	C₂³	1	4	yes	—
105	C₄×D₄×C₃₁	C₂³	1	4	yes	—
106	C₃₁×#117	—	2	2	no	—
107	C₃₁×#117	—	2	8	no	—
108	C₃₁×#117	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
112	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
113	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
115	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
121	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
122	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
123	C₂×C₄²×C₃₁	—	2	1	no	—
124	C₂×C₄²×C₃₁	—	2	1	no	—
125	C₂×C₄²×C₃₁	—	2	2	no	—
126	C₂×C₄²×C₃₁	—	2	8	no	—
127	C₂×C₄²×C₃₁	—	2	1	no	—
128	C₂×C₄²×C₃₁	—	2	2	no	—
129	C₂×C₄²×C₃₁	—	2	8	no	—
130	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
131	C₂×C₄²×C₃₁	D₄	1	4	yes	—
132	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
133	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
135	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
136	C₃₁×#107	C₂³	1	8	yes	—
137	C₃₁×#106	C₂³	1	8	yes	—
138	C₃₁×#103	C₂³	1	8	yes	—
139	C₃₁×#103	C₂³	1	8	yes	—
140	C₃₁×#103	C₂³	1	8	yes	—
141	C₃₁×#102	C₂³	1	8	yes	—
142	C₃₁×#97	C₂³	1	8	yes	—
143	C₄×C₈×C₃₁	C₂³	1	8	yes	—
144	C₂⁴×C₃₁	—	2	1	no	—
145	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
146	#41×C₃₁	—	2	16	no	—
147	#41×C₃₁	—	2	8	no	—
148	#41×C₃₁	C₂²×C₄	1	16	yes	—
149	C₂×Q₈×C₃₁	—	2	8	no	—
150	C₂×Q₈×C₃₁	—	2	16	no	—
151	C₂×Q₈×C₃₁	—	2	4	no	—
152	C₂×Q₈×C₃₁	—	2	2	no	—
153	C₂×Q₈×C₃₁	—	2	1	no	—
154	C₂×Q₈×C₃₁	—	2	4	no	—
155	C₂×Q₈×C₃₁	—	2	8	no	—
156	C₂×Q₈×C₃₁	—	2	2	no	—
157	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
158	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
159	C₂×Q₈×C₃₁	C₂²×C₄	1	16	yes	complemented
160	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
161	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
162	C₂×D₄×C₃₁	—	2	8	no	—
163	C₂×D₄×C₃₁	—	2	4	no	—
164	C₂×D₄×C₃₁	—	2	1	no	—
165	C₂×D₄×C₃₁	—	2	2	no	—
166	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
167	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₃₁	—	2	8	no	—
169	C₂²×C₄×C₃₁	—	2	1	no	—
170	C₂²×C₄×C₃₁	—	2	2	no	—
171	C₂²×C₄×C₃₁	—	2	4	no	—
172	C₂²×C₄×C₃₁	—	2	8	no	—
173	C₂²×C₄×C₃₁	—	2	4	no	—
174	C₂²×C₄×C₃₁	—	2	4	no	—
175	C₂²×C₄×C₃₁	—	2	1	no	—
176	C₂²×C₄×C₃₁	—	2	1	no	—
177	C₂²×C₄×C₃₁	—	2	4	no	—
178	C₂²×C₄×C₃₁	—	2	2	no	—
179	C₂²×C₄×C₃₁	—	2	4	no	—
180	C₂²×C₄×C₃₁	—	2	4	no	—
181	C₂²×C₄×C₃₁	—	4	2	no	—
182	C₂²×C₄×C₃₁	—	4	4	no	—
183	C₂²×C₄×C₃₁	—	4	4	no	—
184	C₂²×C₄×C₃₁	—	4	4	no	—
185	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
186	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
188	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
191	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
193	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₃₁	#41	1	4	yes	—
195	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
196	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
197	Q₁₆×C₃₁	C₂²×C₄	1	32	yes	complemented
198	SD₁₆×C₃₁	C₂²×C₄	1	32	yes	complemented
199	M₁₆×C₃₁	—	2	8	no	—
200	M₁₆×C₃₁	C₂²×C₄	1	16	yes	complemented
201	C₂×C₈×C₃₁	—	2	4	no	—
202	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
203	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
204	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
205	C₄⋊C₄×C₃₁	—	2	8	no	—
206	C₄⋊C₄×C₃₁	—	2	8	no	—
207	C₄⋊C₄×C₃₁	—	2	4	no	—
208	C₄⋊C₄×C₃₁	—	2	2	no	—
209	C₄⋊C₄×C₃₁	—	2	4	no	—
210	C₄⋊C₄×C₃₁	—	2	4	no	—
211	C₄⋊C₄×C₃₁	—	2	1	no	—
212	C₄⋊C₄×C₃₁	—	2	1	no	—
213	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
214	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
215	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
216	C₂²⋊C₄×C₃₁	—	2	4	no	—
217	C₂²⋊C₄×C₃₁	—	2	4	no	—
218	C₂²⋊C₄×C₃₁	—	2	4	no	—
219	C₄²×C₃₁	—	2	4	no	—
220	C₄²×C₃₁	—	2	4	no	—
221	C₄²×C₃₁	—	2	4	no	—
222	C₄²×C₃₁	—	2	8	no	—
223	C₄²×C₃₁	—	2	8	no	—
224	C₄²×C₃₁	C₂×D₄	1	4	yes	—
225	C₄²×C₃₁	C₂×D₄	1	4	yes	—
226	C₄²×C₃₁	C₂×D₄	1	2	yes	—
227	C₄²×C₃₁	C₂×D₄	1	2	yes	—
228	C₄²×C₃₁	C₂⁴	1	2	yes	—
229	C₄²×C₃₁	C₂⁴	1	2	yes	—
230	C₂×C₄×#137	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂³×C₃₁	—	2	2	no	—
232	C₂³×C₃₁	—	2	2	no	—
233	C₂³×C₃₁	—	2	4	no	—
234	C₂³×C₃₁	—	2	2	no	—
235	C₂³×C₃₁	—	4	2	no	—
236	C₂³×C₃₁	—	4	2	no	—
237	C₂³×C₃₁	C₄×D₄	1	4	yes	—
238	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
239	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
240	Q₈×C₃₁	—	2	8	no	—
241	Q₈×C₃₁	—	2	16	no	—
242	Q₈×C₃₁	—	2	4	no	—
243	Q₈×C₃₁	—	2	4	no	—
244	Q₈×C₃₁	—	2	2	no	—
245	Q₈×C₃₁	C₂³×C₄	1	8	yes	—
246	Q₈×C₃₁	C₂³×C₄	1	4	yes	—
247	D₄×C₃₁	—	2	16	no	—
248	D₄×C₃₁	—	2	4	no	—
249	D₄×C₃₁	—	2	2	no	—
250	D₄×C₃₁	C₂³×C₄	1	4	yes	—
251	C₂×C₄×C₃₁	—	2	4	no	—
252	C₂×C₄×C₃₁	—	2	8	no	—
253	C₂×C₄×C₃₁	—	2	16	no	—
254	C₂×C₄×C₃₁	—	2	2	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	2	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	4	no	—
259	C₂×C₄×C₃₁	—	2	16	no	—
260	C₂×C₄×C₃₁	—	2	8	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	2	no	—
263	C₂×C₄×C₃₁	—	2	4	no	—
264	C₂×C₄×C₃₁	—	4	4	no	—
265	C₂×C₄×C₃₁	—	4	8	no	—
266	C₂×C₄×C₃₁	—	4	8	no	—
267	C₂×C₄×C₃₁	—	4	2	no	—
268	C₂×C₄×C₃₁	—	4	2	no	—
269	C₂×C₄×C₃₁	—	4	4	no	—
270	C₂×C₄×C₃₁	—	4	8	no	—
271	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
272	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
273	C₂×C₄×C₃₁	#137	1	4	yes	direct product factor
274	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
275	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
276	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
277	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
278	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
280	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
282	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
283	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
284	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
285	C₈×C₃₁	—	2	4	no	—
286	C₈×C₃₁	C₂³×C₄	1	8	yes	—
287	C₂²×#137	C₂×C₃₁	1	1	yes	characteristic
288	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
289	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
290	C₄×#137	C₂×C₃₁	1	16	yes	direct product factor
291	C₂×#441	C₂×C₃₁	1	2	yes	complemented
292	C₂×#440	C₂×C₃₁	1	2	yes	complemented
293	C₂×C₄×Q₁₆	C₂×C₃₁	1	2	yes	complemented
294	C₂×C₄×SD₁₆	C₂×C₃₁	1	2	yes	complemented
295	C₂×#428	C₂×C₃₁	1	1	yes	characteristic, complemented
296	C₂×#417	C₂×C₃₁	1	1	yes	characteristic
297	C₂×#419	C₂×C₃₁	1	1	yes	characteristic, complemented
298	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
299	C₂²×C₃₁	—	2	8	no	—
300	C₂²×C₃₁	—	2	2	no	—
301	C₂²×C₃₁	—	2	2	no	—
302	C₂²×C₃₁	—	2	4	no	—
303	C₂²×C₃₁	—	4	8	no	—
304	C₂²×C₃₁	—	4	2	no	—
305	C₂²×C₃₁	—	4	2	no	—
306	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
307	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
308	C₂²×C₃₁	C₂×#137	1	2	yes	—
309	C₂²×C₃₁	#575	1	2	yes	—
310	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
311	C₄×C₃₁	—	2	4	no	—
312	C₄×C₃₁	—	2	8	no	—
313	C₄×C₃₁	—	2	2	no	—
314	C₄×C₃₁	—	2	2	no	—
315	C₄×C₃₁	—	2	2	no	—
316	C₄×C₃₁	—	4	2	no	—
317	C₄×C₃₁	—	4	2	no	—
318	C₄×C₃₁	—	4	4	no	—
319	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
320	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
321	C₄×C₃₁	C₂×#137	1	4	yes	direct product factor
322	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
323	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
324	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
325	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
326	C₂×#137	C₄×C₃₁	1	16	yes	direct product factor
327	C₂×#137	C₂²×C₃₁	1	8	yes	—
328	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
329	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
330	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
331	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
332	C₄×#41	—	2	4	no	—
333	C₄×#41	C₂²×C₃₁	1	8	yes	—
334	C₂×C₄×Q₈	—	2	1	no	—
335	C₂×C₄×Q₈	—	2	1	no	—
336	C₂×C₄×Q₈	—	2	4	no	—
337	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
338	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
339	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
340	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
341	C₂×C₄×Q₈	C₂²×C₃₁	1	8	yes	complemented
342	C₂×C₄×D₄	—	2	1	no	—
343	C₂×C₄×D₄	—	2	1	no	—
344	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
345	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
346	C₂×#117	—	2	1	no	—
347	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
348	C₂²×C₄⋊C₄	—	2	1	no	—
349	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
350	C₂²×C₄²	—	2	1	no	—
351	C₂²×C₄²	—	2	1	no	—
352	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
353	#441	C₂²×C₃₁	1	16	yes	complemented
354	#440	C₂²×C₃₁	1	16	yes	complemented
355	C₄×Q₁₆	C₂²×C₃₁	1	16	yes	complemented
356	C₄×SD₁₆	C₂²×C₃₁	1	16	yes	complemented
357	#428	C₂²×C₃₁	1	8	yes	complemented
358	C₂×#107	C₂²×C₃₁	1	2	yes	—
359	C₂×#106	C₂²×C₃₁	1	2	yes	—
360	#419	C₂²×C₃₁	1	8	yes	complemented
361	#417	C₂²×C₃₁	1	8	yes	—
362	C₂×#103	C₂²×C₃₁	1	2	yes	—
363	C₂×#103	C₂²×C₃₁	1	2	yes	—
364	C₂×#103	C₂²×C₃₁	1	2	yes	—
365	C₂×#102	C₂²×C₃₁	1	2	yes	—
366	C₄×M₁₆	C₂²×C₃₁	1	8	yes	complemented
367	C₂×#97	C₂²×C₃₁	1	2	yes	—
368	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
369	C₂×C₃₁	—	2	4	no	—
370	C₂×C₃₁	—	4	4	no	—
371	C₂×C₃₁	C₄×#137	1	4	yes	direct product factor
372	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
373	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
374	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
375	C₂×#41	—	2	2	no	—
376	C₂×#41	—	2	4	no	—
377	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
378	C₂×#41	C₂³×C₃₁	1	4	yes	—
379	C₂²×Q₈	—	2	1	no	—
380	C₂²×Q₈	—	2	2	no	—
381	C₂²×Q₈	—	2	4	no	—
382	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
383	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
384	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
385	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
386	C₂²×D₄	—	2	1	no	—
387	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
388	C₂³×C₄	—	2	1	no	—
389	C₂³×C₄	—	2	1	no	—
390	C₂³×C₄	—	2	1	no	—
391	C₂³×C₄	—	2	1	no	—
392	C₂³×C₄	—	2	1	no	—
393	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
394	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
395	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
396	#137	C₂×C₄×C₃₁	1	64	yes	direct product factor
397	C₂×Q₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
398	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
399	C₂×SD₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
400	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
401	C₂×M₁₆	—	2	4	no	—
402	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
403	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
404	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
405	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
406	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
407	C₄×Q₈	—	2	4	no	—
408	C₄×Q₈	—	2	4	no	—
409	C₄×Q₈	—	2	8	no	—
410	C₄×Q₈	—	2	8	no	—
411	C₄×Q₈	—	2	2	no	—
412	C₄×Q₈	—	2	2	no	—
413	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
414	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
415	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
416	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
417	C₄×D₄	—	2	8	no	—
418	C₄×D₄	—	2	8	no	—
419	C₄×D₄	—	2	2	no	—
420	C₄×D₄	—	2	2	no	—
421	C₄×D₄	C₂³×C₃₁	1	4	yes	—
422	C₄×D₄	C₂³×C₃₁	1	4	yes	—
423	#117	—	2	2	no	—
424	#117	—	2	8	no	—
425	#117	C₂³×C₃₁	1	4	yes	—
426	C₂×C₄⋊C₄	—	2	2	no	—
427	C₂×C₄⋊C₄	—	2	2	no	—
428	C₂×C₄⋊C₄	—	2	8	no	—
429	C₂×C₄⋊C₄	—	2	1	no	—
430	C₂×C₄⋊C₄	—	2	2	no	—
431	C₂×C₄⋊C₄	—	2	1	no	—
432	C₂×C₄⋊C₄	—	2	1	no	—
433	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
434	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
435	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
436	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	—	2	1	no	—
438	C₂×C₂²⋊C₄	—	2	1	no	—
439	C₂×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₄²	—	2	1	no	—
441	C₂×C₄²	—	2	1	no	—
442	C₂×C₄²	—	2	1	no	—
443	C₂×C₄²	—	2	2	no	—
444	C₂×C₄²	—	2	2	no	—
445	C₂×C₄²	—	2	8	no	—
446	C₂×C₄²	—	2	8	no	—
447	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
448	C₂×C₄²	D₄×C₃₁	1	4	yes	—
449	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
450	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
451	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
452	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
453	#107	C₂³×C₃₁	1	8	yes	—
454	#106	C₂³×C₃₁	1	8	yes	—
455	#103	C₂³×C₃₁	1	8	yes	—
456	#103	C₂³×C₃₁	1	8	yes	—
457	#103	C₂³×C₃₁	1	8	yes	—
458	#102	C₂³×C₃₁	1	8	yes	—
459	#97	C₂³×C₃₁	1	8	yes	—
460	C₄×C₈	C₂³×C₃₁	1	8	yes	—
461	C₃₁	C₂×C₄×#137	1	1	yes	characteristic, direct product factor, Sylow
462	C₂⁴	—	2	1	no	—
463	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
464	#41	—	2	16	no	—
465	#41	—	2	8	no	—
466	#41	C₂²×C₄×C₃₁	1	16	yes	—
467	C₂×Q₈	—	2	8	no	—
468	C₂×Q₈	—	2	16	no	—
469	C₂×Q₈	—	2	4	no	—
470	C₂×Q₈	—	2	2	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	4	no	—
473	C₂×Q₈	—	2	8	no	—
474	C₂×Q₈	—	2	2	no	—
475	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
476	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
477	C₂×Q₈	C₂²×C₄×C₃₁	1	16	yes	complemented
478	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
479	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
480	C₂×D₄	—	2	8	no	—
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	2	2	no	—
484	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
485	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
486	C₂²×C₄	—	2	8	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	4	no	—
490	C₂²×C₄	—	2	8	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	4	no	—
496	C₂²×C₄	—	2	2	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	2	4	no	—
499	C₂²×C₄	—	4	2	no	—
500	C₂²×C₄	—	4	4	no	—
501	C₂²×C₄	—	4	4	no	—
502	C₂²×C₄	—	4	4	no	—
503	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
504	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
505	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
506	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
507	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
508	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
509	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
510	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
511	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
512	C₂²×C₄	#41×C₃₁	1	4	yes	—
513	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
514	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
515	Q₁₆	C₂²×C₄×C₃₁	1	32	yes	complemented
516	SD₁₆	C₂²×C₄×C₃₁	1	32	yes	complemented
517	M₁₆	—	2	8	no	—
518	M₁₆	C₂²×C₄×C₃₁	1	16	yes	complemented
519	C₂×C₈	—	2	4	no	—
520	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
521	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
522	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
523	C₄⋊C₄	—	2	8	no	—
524	C₄⋊C₄	—	2	8	no	—
525	C₄⋊C₄	—	2	4	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	2	4	no	—
528	C₄⋊C₄	—	2	4	no	—
529	C₄⋊C₄	—	2	1	no	—
530	C₄⋊C₄	—	2	1	no	—
531	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
532	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
533	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₄²	—	2	4	no	—
538	C₄²	—	2	4	no	—
539	C₄²	—	2	4	no	—
540	C₄²	—	2	8	no	—
541	C₄²	—	2	8	no	—
542	C₄²	C₂×D₄×C₃₁	1	4	yes	—
543	C₄²	C₂×D₄×C₃₁	1	4	yes	—
544	C₄²	C₂×D₄×C₃₁	1	2	yes	—
545	C₄²	C₂×D₄×C₃₁	1	2	yes	—
546	C₄²	C₂⁴×C₃₁	1	2	yes	—
547	C₄²	C₂⁴×C₃₁	1	2	yes	—
548	C₂³	—	2	2	no	—
549	C₂³	—	2	2	no	—
550	C₂³	—	2	4	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	4	2	no	—
553	C₂³	—	4	2	no	—
554	C₂³	C₄×D₄×C₃₁	1	4	yes	—
555	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
556	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
557	Q₈	—	2	8	no	—
558	Q₈	—	2	16	no	—
559	Q₈	—	2	4	no	—
560	Q₈	—	2	4	no	—
561	Q₈	—	2	2	no	—
562	Q₈	C₂³×C₄×C₃₁	1	8	yes	—
563	Q₈	C₂³×C₄×C₃₁	1	4	yes	—
564	D₄	—	2	16	no	—
565	D₄	—	2	4	no	—
566	D₄	—	2	2	no	—
567	D₄	C₂³×C₄×C₃₁	1	4	yes	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	8	no	—
570	C₂×C₄	—	2	16	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	16	no	—
577	C₂×C₄	—	2	8	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	4	4	no	—
582	C₂×C₄	—	4	8	no	—
583	C₂×C₄	—	4	8	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	4	no	—
587	C₂×C₄	—	4	8	no	—
588	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
589	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
590	C₂×C₄	C₃₁×#137	1	4	yes	direct product factor
591	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
592	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
593	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
594	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
595	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
596	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
597	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
598	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
599	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
600	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
601	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
602	C₈	—	2	4	no	—
603	C₈	C₂³×C₄×C₃₁	1	8	yes	—
604	C₂²	—	2	8	no	—
605	C₂²	—	2	2	no	—
606	C₂²	—	2	2	no	—
607	C₂²	—	2	4	no	—
608	C₂²	—	4	8	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
612	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
613	C₂²	C₂×C₃₁×#137	1	2	yes	—
614	C₂²	C₃₁×#575	1	2	yes	—
615	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
616	C₄	—	2	4	no	—
617	C₄	—	2	8	no	—
618	C₄	—	2	2	no	—
619	C₄	—	2	2	no	—
620	C₄	—	2	2	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	4	no	—
624	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
625	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
626	C₄	C₂×C₃₁×#137	1	4	yes	direct product factor
627	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
628	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
629	C₂	—	2	4	no	—
630	C₂	—	4	4	no	—
631	C₂	C₄×C₃₁×#137	1	4	yes	direct product factor
632	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
633	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic
634	C₂	C₂×C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
4. C₄×C₃₁×#137	C₂	direct (N × H)
5. C₂×C₃₁×#441	C₂	semidirect
6. C₂×C₃₁×#440	C₂	semidirect
7. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
8. C₂×C₄×SD₁₆×C₃₁	C₂	semidirect
9. C₂×C₃₁×#428	C₂	semidirect
11. C₂×C₃₁×#419	C₂	semidirect
12. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
15. C₂×C₃₁×#137	C₄	direct (N × H)
30. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
42. C₃₁×#441	C₂²	semidirect
43. C₃₁×#440	C₂²	semidirect
44. C₄×Q₁₆×C₃₁	C₂²	semidirect
45. C₄×SD₁₆×C₃₁	C₂²	semidirect
46. C₃₁×#428	C₂²	semidirect
49. C₃₁×#419	C₂²	semidirect
55. C₄×M₁₆×C₃₁	C₂²	semidirect
65. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
79. C₃₁×#137	C₂×C₄	direct (N × H)
80. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
82. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
85. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
159. C₂×Q₈×C₃₁	C₂²×C₄	semidirect
197. Q₁₆×C₃₁	C₂²×C₄	semidirect
198. SD₁₆×C₃₁	C₂²×C₄	semidirect
200. M₁₆×C₃₁	C₂²×C₄	semidirect
230. C₂×C₄×#137	C₃₁	direct (N × H)
273. C₂×C₄×C₃₁	#137	direct (N × H)
289. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
290. C₄×#137	C₂×C₃₁	direct (N × H)
291. C₂×#441	C₂×C₃₁	semidirect
292. C₂×#440	C₂×C₃₁	semidirect
293. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
294. C₂×C₄×SD₁₆	C₂×C₃₁	semidirect
295. C₂×#428	C₂×C₃₁	semidirect
297. C₂×#419	C₂×C₃₁	semidirect
298. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
321. C₄×C₃₁	C₂×#137	direct (N × H)
326. C₂×#137	C₄×C₃₁	direct (N × H)
341. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
353. #441	C₂²×C₃₁	semidirect
354. #440	C₂²×C₃₁	semidirect
355. C₄×Q₁₆	C₂²×C₃₁	semidirect
356. C₄×SD₁₆	C₂²×C₃₁	semidirect
357. #428	C₂²×C₃₁	semidirect
360. #419	C₂²×C₃₁	semidirect
366. C₄×M₁₆	C₂²×C₃₁	semidirect
371. C₂×C₃₁	C₄×#137	direct (N × H)
382. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
396. #137	C₂×C₄×C₃₁	direct (N × H)
397. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
399. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
402. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
461. C₃₁	C₂×C₄×#137	direct (N × H)
477. C₂×Q₈	C₂²×C₄×C₃₁	semidirect
515. Q₁₆	C₂²×C₄×C₃₁	semidirect
516. SD₁₆	C₂²×C₄×C₃₁	semidirect
518. M₁₆	C₂²×C₄×C₃₁	semidirect
590. C₂×C₄	C₃₁×#137	direct (N × H)
626. C₄	C₂×C₃₁×#137	direct (N × H)
631. C₂	C₄×C₃₁×#137	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃₁×#137	2
C₂×C₄×#41×C₃₁	2
C₂²×C₄×Q₈×C₃₁	2
C₄×C₃₁×#137	2
C₂×C₃₁×#441	2
C₂×C₃₁×#440	2
C₂×C₄×Q₁₆×C₃₁	2
C₂×C₄×SD₁₆×C₃₁	2
C₂×C₃₁×#428	2
C₂×C₃₁×#417	2
C₂×C₃₁×#419	2
C₂×C₄×M₁₆×C₃₁	2
C₂×C₄×#137	31

Subgroups — C2×C4×C31×#137 #21516916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₃₁×#137 #21516916