Subgroups — C₂×C₃₁×#2947 #21516919

Order 7936.

2730 conjugacy classes of subgroups.

The subgroup lattice has 578 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×C₃₁×#520	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#519	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2947	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×#416	C₂	1	1	yes	maximal, characteristic
11	C₂²×C₃₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#405	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	complemented
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	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#520	—	2	4	no	—
22	C₃₁×#520	C₂²	1	8	yes	—
23	C₃₁×#519	—	2	4	no	—
24	C₃₁×#519	C₂²	1	8	yes	complemented
25	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
26	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
27	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₃₁	—	2	2	no	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
32	C₂×C₃₁×#117	—	2	1	no	—
33	C₂×C₃₁×#117	—	2	2	no	—
34	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
36	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
37	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#441	C₂²	1	16	yes	complemented
39	C₃₁×#440	C₂²	1	16	yes	complemented
40	C₄×Q₁₆×C₃₁	C₂²	1	16	yes	complemented
41	C₄×SD₁₆×C₃₁	C₂²	1	16	yes	complemented
42	C₃₁×#428	C₂²	1	8	yes	complemented
43	C₂×C₃₁×#107	C₂²	1	2	yes	—
44	C₂×C₃₁×#106	C₂²	1	2	yes	—
45	C₃₁×#416	C₂²	1	8	yes	—
46	C₂×C₃₁×#103	C₂²	1	2	yes	—
47	C₂×C₃₁×#103	C₂²	1	2	yes	—
48	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
49	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×#103	C₂²	1	8	yes	complemented
51	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
53	C₃₁×#405	C₂²	1	8	yes	complemented
54	C₂×C₃₁×#97	C₂²	1	2	yes	—
55	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
56	C₂×#41×C₃₁	—	2	2	no	—
57	C₂×#41×C₃₁	—	2	4	no	—
58	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
59	C₂×#41×C₃₁	C₂³	1	4	yes	—
60	C₂²×Q₈×C₃₁	—	2	1	no	—
61	C₂²×Q₈×C₃₁	—	2	2	no	—
62	C₂²×Q₈×C₃₁	—	2	4	no	—
63	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
64	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
65	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
67	C₂²×D₄×C₃₁	—	2	1	no	—
68	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₃₁	—	2	1	no	—
70	C₂³×C₄×C₃₁	—	2	1	no	—
71	C₂³×C₄×C₃₁	—	2	1	no	—
72	C₂³×C₄×C₃₁	—	2	2	no	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#137	C₂×C₄	1	64	yes	complemented
75	C₂×Q₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
76	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
77	C₂×SD₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
78	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
79	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
80	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
81	C₂²×C₈×C₃₁	—	2	4	no	—
82	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
85	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
86	C₄×Q₈×C₃₁	—	2	8	no	—
87	C₄×Q₈×C₃₁	—	2	8	no	—
88	C₄×Q₈×C₃₁	—	2	4	no	—
89	C₄×Q₈×C₃₁	—	2	4	no	—
90	C₄×Q₈×C₃₁	—	2	4	no	—
91	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
92	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
93	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
94	C₄×D₄×C₃₁	—	2	16	no	—
95	C₄×D₄×C₃₁	—	2	4	no	—
96	C₄×D₄×C₃₁	C₂³	1	8	yes	—
97	C₃₁×#117	—	2	2	no	—
98	C₃₁×#117	—	2	8	no	—
99	C₃₁×#117	—	2	16	no	—
100	C₃₁×#117	D₄	1	4	yes	—
101	C₃₁×#117	C₂³	1	4	yes	—
102	C₃₁×#117	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
109	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
118	C₂×C₄²×C₃₁	—	2	2	no	—
119	C₂×C₄²×C₃₁	—	2	2	no	—
120	C₂×C₄²×C₃₁	—	2	2	no	—
121	C₂×C₄²×C₃₁	—	2	1	no	—
122	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
123	C₃₁×#107	C₂³	1	8	yes	—
124	C₃₁×#106	C₂³	1	8	yes	—
125	C₃₁×#103	C₂³	1	8	yes	—
126	C₃₁×#103	C₂³	1	8	yes	—
127	C₃₁×#103	C₂³	1	4	yes	—
128	C₃₁×#103	C₂³	1	4	yes	—
129	C₃₁×#102	C₂³	1	4	yes	—
130	C₃₁×#102	C₂³	1	4	yes	—
131	C₃₁×#97	C₂³	1	8	yes	—
132	C₄×C₈×C₃₁	C₂³	1	8	yes	—
133	C₂⁴×C₃₁	—	2	1	no	—
134	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
135	#41×C₃₁	—	2	16	no	—
136	#41×C₃₁	—	2	8	no	—
137	#41×C₃₁	C₂²×C₄	1	16	yes	—
138	C₂×Q₈×C₃₁	—	2	8	no	—
139	C₂×Q₈×C₃₁	—	2	16	no	—
140	C₂×Q₈×C₃₁	—	2	4	no	—
141	C₂×Q₈×C₃₁	—	2	2	no	—
142	C₂×Q₈×C₃₁	—	2	1	no	—
143	C₂×Q₈×C₃₁	—	2	4	no	—
144	C₂×Q₈×C₃₁	—	2	8	no	—
145	C₂×Q₈×C₃₁	—	2	2	no	—
146	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
147	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
148	C₂×Q₈×C₃₁	C₂²×C₄	1	16	yes	complemented
149	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
150	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
151	C₂×D₄×C₃₁	—	2	8	no	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	2	1	no	—
154	C₂×D₄×C₃₁	—	2	2	no	—
155	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
156	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₃₁	—	2	8	no	—
158	C₂²×C₄×C₃₁	—	2	4	no	—
159	C₂²×C₄×C₃₁	—	2	1	no	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	8	no	—
162	C₂²×C₄×C₃₁	—	2	4	no	—
163	C₂²×C₄×C₃₁	—	2	1	no	—
164	C₂²×C₄×C₃₁	—	2	1	no	—
165	C₂²×C₄×C₃₁	—	2	2	no	—
166	C₂²×C₄×C₃₁	—	2	4	no	—
167	C₂²×C₄×C₃₁	—	2	8	no	—
168	C₂²×C₄×C₃₁	—	8	4	no	—
169	C₂²×C₄×C₃₁	—	8	2	no	—
170	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	#41	1	4	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
178	Q₁₆×C₃₁	C₂²×C₄	1	32	yes	complemented
179	SD₁₆×C₃₁	C₂²×C₄	1	32	yes	complemented
180	M₁₆×C₃₁	C₂²×C₄	1	16	yes	complemented
181	C₂×C₈×C₃₁	—	2	2	no	—
182	C₂×C₈×C₃₁	—	2	2	no	—
183	C₂×C₈×C₃₁	—	2	8	no	—
184	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
185	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
186	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
187	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₃₁	—	2	4	no	—
189	C₄⋊C₄×C₃₁	—	2	8	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	—	2	8	no	—
192	C₄⋊C₄×C₃₁	—	2	8	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	2	1	no	—
195	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
196	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
197	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
198	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
199	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
200	C₂²⋊C₄×C₃₁	—	2	4	no	—
201	C₂²⋊C₄×C₃₁	—	2	8	no	—
202	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
203	C₄²×C₃₁	—	2	8	no	—
204	C₄²×C₃₁	—	2	8	no	—
205	C₄²×C₃₁	—	2	8	no	—
206	C₄²×C₃₁	—	2	4	no	—
207	C₄²×C₃₁	C₂×D₄	1	4	yes	—
208	C₄²×C₃₁	C₂⁴	1	4	yes	—
209	C₂×#2947	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂³×C₃₁	—	2	2	no	—
211	C₂³×C₃₁	—	2	2	no	—
212	C₂³×C₃₁	—	2	4	no	—
213	C₂³×C₃₁	—	2	2	no	—
214	C₂³×C₃₁	—	4	2	no	—
215	C₂³×C₃₁	—	4	2	no	—
216	C₂³×C₃₁	C₄×D₄	1	4	yes	—
217	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
218	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	Q₈×C₃₁	—	2	8	no	—
220	Q₈×C₃₁	—	2	16	no	—
221	Q₈×C₃₁	—	2	4	no	—
222	Q₈×C₃₁	—	2	4	no	—
223	Q₈×C₃₁	—	2	2	no	—
224	Q₈×C₃₁	C₂³×C₄	1	8	yes	—
225	Q₈×C₃₁	C₂³×C₄	1	4	yes	—
226	D₄×C₃₁	—	2	16	no	—
227	D₄×C₃₁	—	2	4	no	—
228	D₄×C₃₁	—	2	2	no	—
229	D₄×C₃₁	C₂³×C₄	1	4	yes	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	2	8	no	—
232	C₂×C₄×C₃₁	—	2	16	no	—
233	C₂×C₄×C₃₁	—	2	4	no	—
234	C₂×C₄×C₃₁	—	2	4	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	4	no	—
237	C₂×C₄×C₃₁	—	2	16	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	4	2	no	—
242	C₂×C₄×C₃₁	—	4	4	no	—
243	C₂×C₄×C₃₁	—	4	2	no	—
244	C₂×C₄×C₃₁	—	8	8	no	—
245	C₂×C₄×C₃₁	—	8	4	no	—
246	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
247	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
248	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
249	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
250	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
251	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
252	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
255	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
256	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
257	C₈×C₃₁	—	2	2	no	—
258	C₈×C₃₁	—	2	2	no	—
259	C₈×C₃₁	C₂³×C₄	1	8	yes	—
260	C₂²×#137	C₂×C₃₁	1	1	yes	characteristic
261	C₂×#520	C₂×C₃₁	1	1	yes	characteristic
262	C₂×#519	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#2947	C₂×C₃₁	1	16	yes	direct product factor
264	C₂×#441	C₂×C₃₁	1	2	yes	complemented
265	C₂×#440	C₂×C₃₁	1	2	yes	complemented
266	C₂×C₄×Q₁₆	C₂×C₃₁	1	2	yes	complemented
267	C₂×C₄×SD₁₆	C₂×C₃₁	1	2	yes	complemented
268	C₂×#428	C₂×C₃₁	1	1	yes	characteristic, complemented
269	C₂×#416	C₂×C₃₁	1	1	yes	characteristic
270	C₂²×#103	C₂×C₃₁	1	1	yes	characteristic, complemented
271	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₂²×C₃₁	—	2	8	no	—
273	C₂²×C₃₁	—	2	2	no	—
274	C₂²×C₃₁	—	2	2	no	—
275	C₂²×C₃₁	—	2	4	no	—
276	C₂²×C₃₁	—	4	8	no	—
277	C₂²×C₃₁	—	4	2	no	—
278	C₂²×C₃₁	—	4	2	no	—
279	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
280	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
281	C₂²×C₃₁	#577	1	2	yes	—
282	C₂²×C₃₁	#578	1	2	yes	—
283	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
284	C₄×C₃₁	—	2	4	no	—
285	C₄×C₃₁	—	2	8	no	—
286	C₄×C₃₁	—	2	4	no	—
287	C₄×C₃₁	—	2	4	no	—
288	C₄×C₃₁	—	4	2	no	—
289	C₄×C₃₁	—	4	4	no	—
290	C₄×C₃₁	—	4	2	no	—
291	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
292	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
293	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
294	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
295	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
296	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×#137	C₄×C₃₁	1	16	yes	complemented
298	C₂×#137	C₂²×C₃₁	1	8	yes	—
299	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
300	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
301	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
302	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
303	#520	—	2	4	no	—
304	#520	C₂²×C₃₁	1	8	yes	—
305	#519	—	2	4	no	—
306	#519	C₂²×C₃₁	1	8	yes	complemented
307	C₂×C₄×Q₈	—	2	1	no	—
308	C₂×C₄×Q₈	—	2	1	no	—
309	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
310	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
311	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
312	C₂×C₄×D₄	—	2	2	no	—
313	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
314	C₂×#117	—	2	1	no	—
315	C₂×#117	—	2	2	no	—
316	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
318	C₂²×C₄⋊C₄	—	2	1	no	—
319	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
320	#441	C₂²×C₃₁	1	16	yes	complemented
321	#440	C₂²×C₃₁	1	16	yes	complemented
322	C₄×Q₁₆	C₂²×C₃₁	1	16	yes	complemented
323	C₄×SD₁₆	C₂²×C₃₁	1	16	yes	complemented
324	#428	C₂²×C₃₁	1	8	yes	complemented
325	C₂×#107	C₂²×C₃₁	1	2	yes	—
326	C₂×#106	C₂²×C₃₁	1	2	yes	—
327	#416	C₂²×C₃₁	1	8	yes	—
328	C₂×#103	C₂²×C₃₁	1	2	yes	—
329	C₂×#103	C₂²×C₃₁	1	2	yes	—
330	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×#103	C₂²×C₃₁	1	8	yes	complemented
333	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
334	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
335	#405	C₂²×C₃₁	1	8	yes	complemented
336	C₂×#97	C₂²×C₃₁	1	2	yes	—
337	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
338	C₂×C₃₁	—	2	4	no	—
339	C₂×C₃₁	—	4	4	no	—
340	C₂×C₃₁	#2947	1	4	yes	direct product factor
341	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
342	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
343	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
344	C₂×#41	—	2	2	no	—
345	C₂×#41	—	2	4	no	—
346	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
347	C₂×#41	C₂³×C₃₁	1	4	yes	—
348	C₂²×Q₈	—	2	1	no	—
349	C₂²×Q₈	—	2	2	no	—
350	C₂²×Q₈	—	2	4	no	—
351	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
352	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
353	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
354	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
355	C₂²×D₄	—	2	1	no	—
356	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
357	C₂³×C₄	—	2	1	no	—
358	C₂³×C₄	—	2	1	no	—
359	C₂³×C₄	—	2	1	no	—
360	C₂³×C₄	—	2	2	no	—
361	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
362	#137	C₂×C₄×C₃₁	1	64	yes	complemented
363	C₂×Q₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
364	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
365	C₂×SD₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
366	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
367	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
368	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
369	C₂²×C₈	—	2	4	no	—
370	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
371	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
372	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
373	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
374	C₄×Q₈	—	2	8	no	—
375	C₄×Q₈	—	2	8	no	—
376	C₄×Q₈	—	2	4	no	—
377	C₄×Q₈	—	2	4	no	—
378	C₄×Q₈	—	2	4	no	—
379	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
380	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
381	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
382	C₄×D₄	—	2	16	no	—
383	C₄×D₄	—	2	4	no	—
384	C₄×D₄	C₂³×C₃₁	1	8	yes	—
385	#117	—	2	2	no	—
386	#117	—	2	8	no	—
387	#117	—	2	16	no	—
388	#117	D₄×C₃₁	1	4	yes	—
389	#117	C₂³×C₃₁	1	4	yes	—
390	#117	C₂³×C₃₁	1	4	yes	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	2	no	—
393	C₂×C₄⋊C₄	—	2	2	no	—
394	C₂×C₄⋊C₄	—	2	2	no	—
395	C₂×C₄⋊C₄	—	2	2	no	—
396	C₂×C₄⋊C₄	—	2	8	no	—
397	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
398	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
401	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
403	C₂×C₂²⋊C₄	—	2	1	no	—
404	C₂×C₂²⋊C₄	—	2	2	no	—
405	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
406	C₂×C₄²	—	2	2	no	—
407	C₂×C₄²	—	2	2	no	—
408	C₂×C₄²	—	2	2	no	—
409	C₂×C₄²	—	2	1	no	—
410	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
411	#107	C₂³×C₃₁	1	8	yes	—
412	#106	C₂³×C₃₁	1	8	yes	—
413	#103	C₂³×C₃₁	1	8	yes	—
414	#103	C₂³×C₃₁	1	8	yes	—
415	#103	C₂³×C₃₁	1	4	yes	—
416	#103	C₂³×C₃₁	1	4	yes	—
417	#102	C₂³×C₃₁	1	4	yes	—
418	#102	C₂³×C₃₁	1	4	yes	—
419	#97	C₂³×C₃₁	1	8	yes	—
420	C₄×C₈	C₂³×C₃₁	1	8	yes	—
421	C₃₁	C₂×#2947	1	1	yes	characteristic, direct product factor, Sylow
422	C₂⁴	—	2	1	no	—
423	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
424	#41	—	2	16	no	—
425	#41	—	2	8	no	—
426	#41	C₂²×C₄×C₃₁	1	16	yes	—
427	C₂×Q₈	—	2	8	no	—
428	C₂×Q₈	—	2	16	no	—
429	C₂×Q₈	—	2	4	no	—
430	C₂×Q₈	—	2	2	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	4	no	—
433	C₂×Q₈	—	2	8	no	—
434	C₂×Q₈	—	2	2	no	—
435	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
436	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
437	C₂×Q₈	C₂²×C₄×C₃₁	1	16	yes	complemented
438	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
439	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
440	C₂×D₄	—	2	8	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
445	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
446	C₂²×C₄	—	2	8	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	8	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	8	no	—
457	C₂²×C₄	—	8	4	no	—
458	C₂²×C₄	—	8	2	no	—
459	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
460	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
461	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
462	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
464	C₂²×C₄	#41×C₃₁	1	4	yes	—
465	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
467	Q₁₆	C₂²×C₄×C₃₁	1	32	yes	complemented
468	SD₁₆	C₂²×C₄×C₃₁	1	32	yes	complemented
469	M₁₆	C₂²×C₄×C₃₁	1	16	yes	complemented
470	C₂×C₈	—	2	2	no	—
471	C₂×C₈	—	2	2	no	—
472	C₂×C₈	—	2	8	no	—
473	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
474	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
475	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
476	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	8	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	—	2	8	no	—
481	C₄⋊C₄	—	2	8	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
485	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
486	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
487	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
488	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
492	C₄²	—	2	8	no	—
493	C₄²	—	2	8	no	—
494	C₄²	—	2	8	no	—
495	C₄²	—	2	4	no	—
496	C₄²	C₂×D₄×C₃₁	1	4	yes	—
497	C₄²	C₂⁴×C₃₁	1	4	yes	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	4	2	no	—
503	C₂³	—	4	2	no	—
504	C₂³	C₄×D₄×C₃₁	1	4	yes	—
505	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
506	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
507	Q₈	—	2	8	no	—
508	Q₈	—	2	16	no	—
509	Q₈	—	2	4	no	—
510	Q₈	—	2	4	no	—
511	Q₈	—	2	2	no	—
512	Q₈	C₂³×C₄×C₃₁	1	8	yes	—
513	Q₈	C₂³×C₄×C₃₁	1	4	yes	—
514	D₄	—	2	16	no	—
515	D₄	—	2	4	no	—
516	D₄	—	2	2	no	—
517	D₄	C₂³×C₄×C₃₁	1	4	yes	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	8	no	—
520	C₂×C₄	—	2	16	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	16	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	4	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	8	8	no	—
533	C₂×C₄	—	8	4	no	—
534	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
535	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
536	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
537	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
538	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
539	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
540	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
541	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
542	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
543	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
544	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
545	C₈	—	2	2	no	—
546	C₈	—	2	2	no	—
547	C₈	C₂³×C₄×C₃₁	1	8	yes	—
548	C₂²	—	2	8	no	—
549	C₂²	—	2	2	no	—
550	C₂²	—	2	2	no	—
551	C₂²	—	2	4	no	—
552	C₂²	—	4	8	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	2	no	—
555	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
556	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
557	C₂²	C₃₁×#577	1	2	yes	—
558	C₂²	C₃₁×#578	1	2	yes	—
559	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
560	C₄	—	2	4	no	—
561	C₄	—	2	8	no	—
562	C₄	—	2	4	no	—
563	C₄	—	2	4	no	—
564	C₄	—	4	4	no	—
565	C₄	—	4	2	no	—
566	C₄	—	4	2	no	—
567	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
568	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
569	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
570	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
571	C₂	—	2	4	no	—
572	C₂	—	4	4	no	—
573	C₂	C₃₁×#2947	1	4	yes	direct product factor
574	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
575	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
576	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₁×#519	C₂	semidirect
4. C₃₁×#2947	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. C₂²×C₃₁×#103	C₂	semidirect
12. C₂×C₃₁×#405	C₂	semidirect
15. C₂×C₃₁×#137	C₄	semidirect
24. C₃₁×#519	C₂²	semidirect
38. C₃₁×#441	C₂²	semidirect
39. C₃₁×#440	C₂²	semidirect
40. C₄×Q₁₆×C₃₁	C₂²	semidirect
41. C₄×SD₁₆×C₃₁	C₂²	semidirect
42. C₃₁×#428	C₂²	semidirect
50. C₂×C₃₁×#103	C₂²	semidirect
53. C₃₁×#405	C₂²	semidirect
63. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
74. C₃₁×#137	C₂×C₄	semidirect
75. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
77. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
79. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
148. C₂×Q₈×C₃₁	C₂²×C₄	semidirect
178. Q₁₆×C₃₁	C₂²×C₄	semidirect
179. SD₁₆×C₃₁	C₂²×C₄	semidirect
180. M₁₆×C₃₁	C₂²×C₄	semidirect
209. C₂×#2947	C₃₁	direct (N × H)
262. C₂×#519	C₂×C₃₁	semidirect
263. #2947	C₂×C₃₁	direct (N × H)
264. C₂×#441	C₂×C₃₁	semidirect
265. C₂×#440	C₂×C₃₁	semidirect
266. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
267. C₂×C₄×SD₁₆	C₂×C₃₁	semidirect
268. C₂×#428	C₂×C₃₁	semidirect
270. C₂²×#103	C₂×C₃₁	semidirect
271. C₂×#405	C₂×C₃₁	semidirect
297. C₂×#137	C₄×C₃₁	semidirect
306. #519	C₂²×C₃₁	semidirect
320. #441	C₂²×C₃₁	semidirect
321. #440	C₂²×C₃₁	semidirect
322. C₄×Q₁₆	C₂²×C₃₁	semidirect
323. C₄×SD₁₆	C₂²×C₃₁	semidirect
324. #428	C₂²×C₃₁	semidirect
332. C₂×#103	C₂²×C₃₁	semidirect
335. #405	C₂²×C₃₁	semidirect
340. C₂×C₃₁	#2947	direct (N × H)
351. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
362. #137	C₂×C₄×C₃₁	semidirect
363. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
365. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
367. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
421. C₃₁	C₂×#2947	direct (N × H)
437. C₂×Q₈	C₂²×C₄×C₃₁	semidirect
467. Q₁₆	C₂²×C₄×C₃₁	semidirect
468. SD₁₆	C₂²×C₄×C₃₁	semidirect
469. M₁₆	C₂²×C₄×C₃₁	semidirect
573. C₂	C₃₁×#2947	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×C₃₁×#520	2
C₂×C₃₁×#519	2
C₃₁×#2947	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×#416	2
C₂²×C₃₁×#103	2
C₂×C₃₁×#405	2
C₂×#2947	31

Subgroups — C2×C31×#2947 #21516919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2947 #21516919