Subgroups — C₂×C₃₁×#3101 #21517287

Order 7936.

2570 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3101	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#483	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#476	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#461	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#450	C₂	1	2	yes	maximal
8	C₂×C₄×SD₁₆×C₃₁	C₂	1	4	yes	maximal, complemented
9	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₃₁	—	2	1	no	—
12	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	—
13	C₃₁×#533	—	2	4	no	—
14	C₃₁×#533	C₂²	1	8	yes	complemented
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃₁×#125	—	2	1	no	—
18	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
21	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
22	C₂×C₄×D₄×C₃₁	—	2	1	no	—
23	C₂×C₄×D₄×C₃₁	—	2	4	no	—
24	C₂×C₄×D₄×C₃₁	—	2	1	no	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₃₁×#117	—	2	1	no	—
29	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
31	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#483	C₂²	1	16	yes	complemented
33	C₃₁×#476	C₂²	1	16	yes	complemented
34	C₃₁×#461	C₂²	1	16	yes	complemented
35	C₃₁×#450	C₂²	1	16	yes	—
36	C₄×SD₁₆×C₃₁	C₂²	1	32	yes	complemented
37	C₃₁×#431	C₂²	1	8	yes	complemented
38	C₂×C₃₁×#106	C₂²	1	4	yes	complemented
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
40	C₂×C₃₁×#103	C₂²	1	4	yes	complemented
41	C₂×C₃₁×#102	C₂²	1	4	yes	—
42	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
44	C₂⁵×C₃₁	—	2	1	no	—
45	C₂²×Q₈×C₃₁	—	2	1	no	—
46	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×D₄×C₃₁	—	2	8	no	—
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	2	no	—
50	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
51	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
52	C₂³×C₄×C₃₁	—	2	1	no	—
53	C₂³×C₄×C₃₁	—	2	2	no	—
54	C₂³×C₄×C₃₁	—	2	8	no	—
55	C₂³×C₄×C₃₁	—	2	2	no	—
56	C₂³×C₄×C₃₁	—	4	8	no	—
57	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×SD₁₆×C₃₁	—	2	16	no	—
61	C₂×SD₁₆×C₃₁	C₂³	1	16	yes	—
62	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
63	C₃₁×C₄⋊Q₈	—	2	2	no	—
64	C₃₁×C₄⋊Q₈	—	2	2	no	—
65	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	complemented
66	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	complemented
67	C₃₁×#125	—	2	8	no	—
68	C₃₁×C₂²⋊Q₈	—	2	8	no	—
69	C₃₁×C₂²⋊Q₈	—	2	4	no	—
70	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
71	C₄×Q₈×C₃₁	—	2	8	no	—
72	C₄×Q₈×C₃₁	—	2	4	no	—
73	C₄×Q₈×C₃₁	C₂³	1	8	yes	complemented
74	C₄×D₄×C₃₁	—	2	8	no	—
75	C₄×D₄×C₃₁	—	2	4	no	—
76	C₄×D₄×C₃₁	—	2	8	no	—
77	C₄×D₄×C₃₁	—	2	4	no	—
78	C₄×D₄×C₃₁	C₂³	1	8	yes	—
79	C₄×D₄×C₃₁	C₂³	1	8	yes	—
80	C₃₁×#117	—	2	8	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
95	C₂×C₄²×C₃₁	—	2	2	no	—
96	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₃₁	D₄	1	4	yes	—
98	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
101	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
102	C₃₁×#106	C₂³	1	16	yes	complemented
103	C₃₁×C₄⋊C₈	C₂³	1	8	yes	complemented
104	C₃₁×#103	C₂³	1	16	yes	complemented
105	C₃₁×#102	C₂³	1	16	yes	—
106	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
107	C₄×C₈×C₃₁	C₂³	1	8	yes	complemented
108	C₂⁴×C₃₁	—	2	8	no	—
109	C₂⁴×C₃₁	—	2	4	no	—
110	C₂⁴×C₃₁	—	2	2	no	—
111	C₂⁴×C₃₁	—	4	4	no	—
112	C₂⁴×C₃₁	—	4	4	no	—
113	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂×Q₈×C₃₁	—	2	2	no	—
115	C₂×Q₈×C₃₁	—	2	4	no	—
116	C₂×Q₈×C₃₁	—	2	8	no	—
117	C₂×Q₈×C₃₁	—	4	4	no	—
118	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
119	C₂×D₄×C₃₁	—	2	8	no	—
120	C₂×D₄×C₃₁	—	2	16	no	—
121	C₂×D₄×C₃₁	—	2	16	no	—
122	C₂×D₄×C₃₁	—	2	4	no	—
123	C₂×D₄×C₃₁	—	2	2	no	—
124	C₂×D₄×C₃₁	—	2	4	no	—
125	C₂×D₄×C₃₁	#41	1	8	yes	—
126	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
127	C₂²×C₄×C₃₁	—	2	4	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	8	no	—
133	C₂²×C₄×C₃₁	—	2	8	no	—
134	C₂²×C₄×C₃₁	—	4	2	no	—
135	C₂²×C₄×C₃₁	—	4	4	no	—
136	C₂²×C₄×C₃₁	—	4	8	no	—
137	C₂²×C₄×C₃₁	—	4	8	no	—
138	C₂²×C₄×C₃₁	—	4	32	no	—
139	C₂²×C₄×C₃₁	SD₁₆	1	8	yes	complemented
140	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
147	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
150	SD₁₆×C₃₁	—	2	32	no	—
151	C₂×C₈×C₃₁	—	2	8	no	—
152	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	4	no	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	2	8	no	—
159	C₄⋊C₄×C₃₁	—	2	8	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
162	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
163	C₂²⋊C₄×C₃₁	—	2	8	no	—
164	C₂²⋊C₄×C₃₁	—	2	8	no	—
165	C₂²⋊C₄×C₃₁	—	2	8	no	—
166	C₄²×C₃₁	—	2	8	no	—
167	C₄²×C₃₁	C₂×D₄	1	4	yes	—
168	C₄²×C₃₁	C₂×D₄	1	2	yes	—
169	C₄²×C₃₁	C₂×D₄	1	4	yes	—
170	C₄²×C₃₁	C₂×D₄	1	2	yes	—
171	C₄²×C₃₁	C₂⁴	1	2	yes	—
172	C₄²×C₃₁	C₂⁴	1	2	yes	—
173	C₂×#3101	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₃₁	—	2	2	no	—
175	C₂³×C₃₁	—	2	2	no	—
176	C₂³×C₃₁	—	2	16	no	—
177	C₂³×C₃₁	—	2	8	no	—
178	C₂³×C₃₁	—	2	4	no	—
179	C₂³×C₃₁	—	4	32	no	—
180	C₂³×C₃₁	—	4	8	no	—
181	C₂³×C₃₁	—	4	8	no	—
182	C₂³×C₃₁	—	4	4	no	—
183	C₂³×C₃₁	—	4	4	no	—
184	C₂³×C₃₁	C₂×SD₁₆	1	4	yes	—
185	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
186	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
187	Q₈×C₃₁	—	2	8	no	—
188	Q₈×C₃₁	—	4	8	no	—
189	Q₈×C₃₁	—	4	4	no	—
190	D₄×C₃₁	—	2	16	no	—
191	D₄×C₃₁	—	2	8	no	—
192	D₄×C₃₁	—	2	4	no	—
193	D₄×C₃₁	C₂×#41	1	8	yes	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	2	4	no	—
199	C₂×C₄×C₃₁	—	2	4	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	4	4	no	—
203	C₂×C₄×C₃₁	—	4	4	no	—
204	C₂×C₄×C₃₁	—	4	8	no	—
205	C₂×C₄×C₃₁	—	4	4	no	—
206	C₂×C₄×C₃₁	—	4	4	no	—
207	C₂×C₄×C₃₁	—	4	16	no	—
208	C₂×C₄×C₃₁	—	4	16	no	—
209	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
210	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
211	C₂×C₄×C₃₁	C₂×SD₁₆	1	16	yes	complemented
212	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
219	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
220	C₈×C₃₁	—	2	8	no	—
221	C₂×#533	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
223	#3101	C₂×C₃₁	1	16	yes	direct product factor
224	C₂×#483	C₂×C₃₁	1	2	yes	complemented
225	C₂×#476	C₂×C₃₁	1	2	yes	complemented
226	C₂×#461	C₂×C₃₁	1	2	yes	complemented
227	C₂×#450	C₂×C₃₁	1	2	yes	—
228	C₂×C₄×SD₁₆	C₂×C₃₁	1	4	yes	complemented
229	C₂×#431	C₂×C₃₁	1	1	yes	characteristic, complemented
230	C₂²×C₃₁	—	2	8	no	—
231	C₂²×C₃₁	—	2	2	no	—
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	2	8	no	—
234	C₂²×C₃₁	—	4	32	no	—
235	C₂²×C₃₁	—	4	16	no	—
236	C₂²×C₃₁	—	4	4	no	—
237	C₂²×C₃₁	—	4	4	no	—
238	C₂²×C₃₁	#525	1	2	yes	—
239	C₂²×C₃₁	C₂²×SD₁₆	1	4	yes	—
240	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
241	C₂²×C₃₁	#575	1	2	yes	—
242	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
243	C₄×C₃₁	—	2	2	no	—
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	4	4	no	—
247	C₄×C₃₁	—	4	4	no	—
248	C₄×C₃₁	—	4	4	no	—
249	C₄×C₃₁	#525	1	2	yes	—
250	C₄×C₃₁	C₂²×SD₁₆	1	4	yes	—
251	C₄×C₃₁	C₂²×SD₁₆	1	4	yes	—
252	C₄×C₃₁	C₂²×#41	1	2	yes	—
253	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
254	C₂⁴×C₄	—	2	1	no	—
255	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	—
256	#533	—	2	4	no	—
257	#533	C₂²×C₃₁	1	8	yes	complemented
258	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
259	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
260	C₂×#125	—	2	1	no	—
261	C₂×C₂²⋊Q₈	—	2	1	no	—
262	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
263	C₂×C₄×Q₈	—	2	1	no	—
264	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
265	C₂×C₄×D₄	—	2	1	no	—
266	C₂×C₄×D₄	—	2	1	no	—
267	C₂×C₄×D₄	—	2	4	no	—
268	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
269	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
270	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
271	C₂×#117	—	2	1	no	—
272	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
273	C₂²×C₂²⋊C₄	—	2	1	no	—
274	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
275	#483	C₂²×C₃₁	1	16	yes	complemented
276	#476	C₂²×C₃₁	1	16	yes	complemented
277	#461	C₂²×C₃₁	1	16	yes	complemented
278	#450	C₂²×C₃₁	1	16	yes	—
279	C₄×SD₁₆	C₂²×C₃₁	1	32	yes	complemented
280	#431	C₂²×C₃₁	1	8	yes	complemented
281	C₂×#106	C₂²×C₃₁	1	4	yes	complemented
282	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
283	C₂×#103	C₂²×C₃₁	1	4	yes	complemented
284	C₂×#102	C₂²×C₃₁	1	4	yes	—
285	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
286	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
287	C₂×C₃₁	—	2	4	no	—
288	C₂×C₃₁	—	4	8	no	—
289	C₂×C₃₁	#3101	1	4	yes	direct product factor
290	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
291	C₂×C₃₁	C₂³×SD₁₆	1	1	yes	characteristic
292	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
293	C₂⁵	—	2	1	no	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
296	C₂²×D₄	—	2	8	no	—
297	C₂²×D₄	—	2	1	no	—
298	C₂²×D₄	—	2	2	no	—
299	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
300	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
301	C₂³×C₄	—	2	1	no	—
302	C₂³×C₄	—	2	2	no	—
303	C₂³×C₄	—	2	2	no	—
304	C₂³×C₄	—	2	8	no	—
305	C₂³×C₄	—	4	8	no	—
306	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
307	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
308	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×SD₁₆	—	2	16	no	—
310	C₂×SD₁₆	C₂³×C₃₁	1	16	yes	—
311	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	—	2	2	no	—
314	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	complemented
315	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	complemented
316	#125	—	2	8	no	—
317	C₂²⋊Q₈	—	2	8	no	—
318	C₂²⋊Q₈	—	2	4	no	—
319	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
320	C₄×Q₈	—	2	8	no	—
321	C₄×Q₈	—	2	4	no	—
322	C₄×Q₈	C₂³×C₃₁	1	8	yes	complemented
323	C₄×D₄	—	2	8	no	—
324	C₄×D₄	—	2	4	no	—
325	C₄×D₄	—	2	8	no	—
326	C₄×D₄	—	2	4	no	—
327	C₄×D₄	C₂³×C₃₁	1	8	yes	—
328	C₄×D₄	C₂³×C₃₁	1	8	yes	—
329	#117	—	2	8	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
338	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
339	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₂²⋊C₄	—	2	2	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	8	no	—
344	C₂×C₄²	—	2	2	no	—
345	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
346	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
347	C₂×C₄²	D₄×C₃₁	1	4	yes	—
348	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
350	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
351	#106	C₂³×C₃₁	1	16	yes	complemented
352	C₄⋊C₈	C₂³×C₃₁	1	8	yes	complemented
353	#103	C₂³×C₃₁	1	16	yes	complemented
354	#102	C₂³×C₃₁	1	16	yes	—
355	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
356	C₄×C₈	C₂³×C₃₁	1	8	yes	complemented
357	C₃₁	C₂×#3101	1	1	yes	characteristic, direct product factor, Sylow
358	C₂⁴	—	2	8	no	—
359	C₂⁴	—	2	4	no	—
360	C₂⁴	—	2	2	no	—
361	C₂⁴	—	4	4	no	—
362	C₂⁴	—	4	4	no	—
363	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	4	no	—
366	C₂×Q₈	—	2	8	no	—
367	C₂×Q₈	—	4	4	no	—
368	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
369	C₂×D₄	—	2	8	no	—
370	C₂×D₄	—	2	16	no	—
371	C₂×D₄	—	2	16	no	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	4	no	—
375	C₂×D₄	#41×C₃₁	1	8	yes	—
376	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	8	no	—
383	C₂²×C₄	—	2	8	no	—
384	C₂²×C₄	—	4	2	no	—
385	C₂²×C₄	—	4	4	no	—
386	C₂²×C₄	—	4	8	no	—
387	C₂²×C₄	—	4	8	no	—
388	C₂²×C₄	—	4	32	no	—
389	C₂²×C₄	SD₁₆×C₃₁	1	8	yes	complemented
390	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
392	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
395	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
398	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
399	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
400	SD₁₆	—	2	32	no	—
401	C₂×C₈	—	2	8	no	—
402	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
412	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
413	C₂²⋊C₄	—	2	8	no	—
414	C₂²⋊C₄	—	2	8	no	—
415	C₂²⋊C₄	—	2	8	no	—
416	C₄²	—	2	8	no	—
417	C₄²	C₂×D₄×C₃₁	1	4	yes	—
418	C₄²	C₂×D₄×C₃₁	1	2	yes	—
419	C₄²	C₂×D₄×C₃₁	1	4	yes	—
420	C₄²	C₂×D₄×C₃₁	1	2	yes	—
421	C₄²	C₂⁴×C₃₁	1	2	yes	—
422	C₄²	C₂⁴×C₃₁	1	2	yes	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	16	no	—
426	C₂³	—	2	8	no	—
427	C₂³	—	2	4	no	—
428	C₂³	—	4	32	no	—
429	C₂³	—	4	8	no	—
430	C₂³	—	4	8	no	—
431	C₂³	—	4	4	no	—
432	C₂³	—	4	4	no	—
433	C₂³	C₂×SD₁₆×C₃₁	1	4	yes	—
434	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
435	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
436	Q₈	—	2	8	no	—
437	Q₈	—	4	8	no	—
438	Q₈	—	4	4	no	—
439	D₄	—	2	16	no	—
440	D₄	—	2	8	no	—
441	D₄	—	2	4	no	—
442	D₄	C₂×#41×C₃₁	1	8	yes	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	4	no	—
453	C₂×C₄	—	4	8	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	4	16	no	—
457	C₂×C₄	—	4	16	no	—
458	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
459	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
460	C₂×C₄	C₂×SD₁₆×C₃₁	1	16	yes	complemented
461	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
464	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
465	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
466	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
467	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
468	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
469	C₈	—	2	8	no	—
470	C₂²	—	2	8	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	8	no	—
474	C₂²	—	4	32	no	—
475	C₂²	—	4	16	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	4	4	no	—
478	C₂²	C₃₁×#525	1	2	yes	—
479	C₂²	C₂²×SD₁₆×C₃₁	1	4	yes	—
480	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
481	C₂²	C₃₁×#575	1	2	yes	—
482	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
483	C₄	—	2	2	no	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	4	4	no	—
487	C₄	—	4	4	no	—
488	C₄	—	4	4	no	—
489	C₄	C₃₁×#525	1	2	yes	—
490	C₄	C₂²×SD₁₆×C₃₁	1	4	yes	—
491	C₄	C₂²×SD₁₆×C₃₁	1	4	yes	—
492	C₄	C₂²×#41×C₃₁	1	2	yes	—
493	C₂	—	2	4	no	—
494	C₂	—	4	8	no	—
495	C₂	C₃₁×#3101	1	4	yes	direct product factor
496	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
497	C₂	C₂³×SD₁₆×C₃₁	1	1	yes	characteristic
498	C₂	C₂×C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂×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
1. C₂×C₃₁×#533	C₂	semidirect
3. C₃₁×#3101	C₂	direct (N × H)
4. C₂×C₃₁×#483	C₂	semidirect
5. C₂×C₃₁×#476	C₂	semidirect
6. C₂×C₃₁×#461	C₂	semidirect
8. C₂×C₄×SD₁₆×C₃₁	C₂	semidirect
9. C₂×C₃₁×#431	C₂	semidirect
14. C₃₁×#533	C₂²	semidirect
15. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
16. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
21. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
32. C₃₁×#483	C₂²	semidirect
33. C₃₁×#476	C₂²	semidirect
34. C₃₁×#461	C₂²	semidirect
36. C₄×SD₁₆×C₃₁	C₂²	semidirect
37. C₃₁×#431	C₂²	semidirect
38. C₂×C₃₁×#106	C₂²	semidirect
39. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
40. C₂×C₃₁×#103	C₂²	semidirect
43. C₂×C₄×C₈×C₃₁	C₂²	semidirect
65. C₃₁×C₄⋊Q₈	C₂³	semidirect
66. C₃₁×C₄⋊Q₈	C₂³	semidirect
73. C₄×Q₈×C₃₁	C₂³	semidirect
102. C₃₁×#106	C₂³	semidirect
103. C₃₁×C₄⋊C₈	C₂³	semidirect
104. C₃₁×#103	C₂³	semidirect
107. C₄×C₈×C₃₁	C₂³	semidirect
139. C₂²×C₄×C₃₁	SD₁₆	semidirect
173. C₂×#3101	C₃₁	direct (N × H)
211. C₂×C₄×C₃₁	C₂×SD₁₆	semidirect
221. C₂×#533	C₂×C₃₁	semidirect
223. #3101	C₂×C₃₁	direct (N × H)
224. C₂×#483	C₂×C₃₁	semidirect
225. C₂×#476	C₂×C₃₁	semidirect
226. C₂×#461	C₂×C₃₁	semidirect
228. C₂×C₄×SD₁₆	C₂×C₃₁	semidirect
229. C₂×#431	C₂×C₃₁	semidirect
257. #533	C₂²×C₃₁	semidirect
258. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
259. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
264. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
275. #483	C₂²×C₃₁	semidirect
276. #476	C₂²×C₃₁	semidirect
277. #461	C₂²×C₃₁	semidirect
279. C₄×SD₁₆	C₂²×C₃₁	semidirect
280. #431	C₂²×C₃₁	semidirect
281. C₂×#106	C₂²×C₃₁	semidirect
282. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
283. C₂×#103	C₂²×C₃₁	semidirect
286. C₂×C₄×C₈	C₂²×C₃₁	semidirect
289. C₂×C₃₁	#3101	direct (N × H)
314. C₄⋊Q₈	C₂³×C₃₁	semidirect
315. C₄⋊Q₈	C₂³×C₃₁	semidirect
322. C₄×Q₈	C₂³×C₃₁	semidirect
351. #106	C₂³×C₃₁	semidirect
352. C₄⋊C₈	C₂³×C₃₁	semidirect
353. #103	C₂³×C₃₁	semidirect
356. C₄×C₈	C₂³×C₃₁	semidirect
357. C₃₁	C₂×#3101	direct (N × H)
389. C₂²×C₄	SD₁₆×C₃₁	semidirect
460. C₂×C₄	C₂×SD₁₆×C₃₁	semidirect
495. C₂	C₃₁×#3101	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#533	2
C₂²×C₄×D₄×C₃₁	2
C₃₁×#3101	2
C₂×C₃₁×#483	2
C₂×C₃₁×#476	2
C₂×C₃₁×#461	2
C₂×C₃₁×#450	2
C₂×C₄×SD₁₆×C₃₁	2
C₂×C₃₁×#431	2
C₂×#3101	31

Subgroups — C2×C31×#3101 #21517287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3101 #21517287