Subgroups — C₂×C₃₁×#3529 #21519450

Order 7936.

7418 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3529	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₃₁×#563	C₂	1	3	yes	maximal, complemented
3	C₂×C₃₁×#562	C₂	1	6	yes	maximal, complemented
4	C₂×C₃₁×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#560	C₂	1	6	yes	maximal, complemented
6	C₂×C₃₁×#532	C₂	1	3	yes	maximal, complemented
7	C₂×C₃₁×#529	C₂	1	6	yes	maximal, complemented
8	C₂×C₃₁×#525	C₂	1	6	yes	maximal, complemented
9	C₂²×#41×C₃₁	C₂²	1	3	yes	—
10	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic, complemented
11	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	complemented
12	C₃₁×#563	C₂²	1	48	yes	complemented
13	C₃₁×#562	C₂²	1	96	yes	complemented
14	C₃₁×H(3,𝔽₄)	C₂²	1	16	yes	complemented
15	C₃₁×#560	C₂²	1	96	yes	complemented
16	C₃₁×#532	C₂²	1	48	yes	complemented
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	complemented
18	C₂×C₃₁×#127	C₂²	1	3	yes	—
19	C₃₁×#529	C₂²	1	96	yes	complemented
20	C₂×C₃₁×#126	C₂²	1	12	yes	complemented
21	C₂×C₃₁×#125	C₂²	1	6	yes	complemented
22	C₂×C₃₁×#124	C₂²	1	6	yes	—
23	C₂×C₃₁×#124	C₂²	1	3	yes	—
24	C₂×C₃₁×#124	C₂²	1	3	yes	complemented
25	C₃₁×#525	C₂²	1	96	yes	complemented
26	C₂×C₃₁×#123	C₂²	1	6	yes	—
27	C₂×C₃₁×#123	C₂²	1	6	yes	complemented
28	C₂×C₃₁×#123	C₂²	1	12	yes	complemented
29	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	12	yes	complemented
30	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	6	yes	complemented
31	C₂×C₃₁×#121	C₂²	1	12	yes	—
32	C₂×C₃₁×#121	C₂²	1	6	yes	—
33	C₂×C₃₁×#121	C₂²	1	12	yes	complemented
34	C₂×C₃₁×#120	C₂²	1	6	yes	—
35	C₂×C₃₁×#120	C₂²	1	6	yes	complemented
36	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
37	C₂×C₄×D₄×C₃₁	C₂²	1	6	yes	—
38	C₂×C₄×D₄×C₃₁	C₂²	1	12	yes	complemented
39	C₂×C₃₁×#117	C₂²	1	6	yes	complemented
40	C₂²×C₄²×C₃₁	C₂²	1	3	yes	complemented
41	C₂⁵×C₃₁	C₂³	1	2	yes	—
42	C₂×#41×C₃₁	—	4	12	no	—
43	C₂×#41×C₃₁	—	4	4	no	—
44	C₂×#41×C₃₁	C₂³	1	24	yes	—
45	C₂×#41×C₃₁	C₂³	1	8	yes	complemented
46	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
48	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
50	C₂²×D₄×C₃₁	C₂³	1	12	yes	—
51	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
52	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
53	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
54	C₂³×C₄×C₃₁	—	4	8	no	—
55	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
56	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
59	C₂³×C₄×C₃₁	C₂³	1	16	yes	complemented
60	C₃₁×C₄⋊Q₈	C₂³	1	24	yes	complemented
61	C₃₁×#127	C₂³	1	24	yes	—
62	C₃₁×#126	C₂³	1	96	yes	complemented
63	C₃₁×#125	C₂³	1	48	yes	complemented
64	C₃₁×#124	C₂³	1	48	yes	—
65	C₃₁×#124	C₂³	1	24	yes	—
66	C₃₁×#124	C₂³	1	24	yes	complemented
67	C₃₁×#123	C₂³	1	48	yes	—
68	C₃₁×#123	C₂³	1	48	yes	complemented
69	C₃₁×#123	C₂³	1	96	yes	complemented
70	C₃₁×C₂²⋊Q₈	C₂³	1	96	yes	complemented
71	C₃₁×C₂²⋊Q₈	C₂³	1	48	yes	complemented
72	C₃₁×#121	C₂³	1	96	yes	—
73	C₃₁×#121	C₂³	1	48	yes	—
74	C₃₁×#121	C₂³	1	96	yes	complemented
75	C₃₁×#120	C₂³	1	48	yes	—
76	C₃₁×#120	C₂³	1	48	yes	complemented
77	C₄×Q₈×C₃₁	C₂³	1	16	yes	complemented
78	C₄×D₄×C₃₁	C₂³	1	48	yes	—
79	C₄×D₄×C₃₁	C₂³	1	96	yes	complemented
80	C₃₁×#117	C₂³	1	48	yes	complemented
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
85	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
86	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
87	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
88	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
89	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
90	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
91	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
92	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
93	C₂×C₄²×C₃₁	C₂³	1	24	yes	complemented
94	C₂⁴×C₃₁	—	4	4	no	—
95	C₂⁴×C₃₁	—	4	8	no	—
96	C₂⁴×C₃₁	C₂⁴	1	6	yes	—
97	C₂⁴×C₃₁	C₂⁴	1	3	yes	—
98	C₂⁴×C₃₁	C₂⁴	1	8	yes	—
99	#41×C₃₁	—	4	48	no	—
100	#41×C₃₁	—	4	16	no	—
101	C₂×Q₈×C₃₁	—	4	4	no	—
102	C₂×Q₈×C₃₁	—	4	6	no	—
103	C₂×Q₈×C₃₁	—	4	2	no	—
104	C₂×Q₈×C₃₁	C₂⁴	1	8	yes	—
105	C₂×Q₈×C₃₁	C₂⁴	1	12	yes	—
106	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
107	C₂×D₄×C₃₁	—	4	24	no	—
108	C₂×D₄×C₃₁	—	4	12	no	—
109	C₂×D₄×C₃₁	—	4	12	no	—
110	C₂×D₄×C₃₁	—	4	6	no	—
111	C₂×D₄×C₃₁	—	4	6	no	—
112	C₂×D₄×C₃₁	C₂⁴	1	24	yes	—
113	C₂×D₄×C₃₁	C₂⁴	1	48	yes	—
114	C₂×D₄×C₃₁	C₂⁴	1	12	yes	—
115	C₂×D₄×C₃₁	C₂⁴	1	24	yes	—
116	C₂×D₄×C₃₁	C₂⁴	1	12	yes	—
117	C₂²×C₄×C₃₁	—	2	24	no	—
118	C₂²×C₄×C₃₁	—	2	12	no	—
119	C₂²×C₄×C₃₁	—	4	32	no	—
120	C₂²×C₄×C₃₁	—	4	12	no	—
121	C₂²×C₄×C₃₁	—	4	12	no	—
122	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
123	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
124	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
125	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
126	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
127	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
128	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
129	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
130	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
131	C₄⋊C₄×C₃₁	C₂⁴	1	48	yes	—
132	C₄⋊C₄×C₃₁	C₂⁴	1	24	yes	—
133	C₄⋊C₄×C₃₁	C₂⁴	1	48	yes	—
134	C₄⋊C₄×C₃₁	C₂⁴	1	24	yes	—
135	C₂²⋊C₄×C₃₁	C₂⁴	1	24	yes	—
136	C₂²⋊C₄×C₃₁	C₂⁴	1	24	yes	—
137	C₂²⋊C₄×C₃₁	C₂⁴	1	48	yes	—
138	C₂²⋊C₄×C₃₁	C₂⁴	1	48	yes	—
139	C₂²⋊C₄×C₃₁	C₂⁴	1	48	yes	—
140	C₄²×C₃₁	C₂⁴	1	24	yes	—
141	C₄²×C₃₁	C₂⁴	1	12	yes	—
142	C₄²×C₃₁	C₂⁴	1	12	yes	—
143	C₂×#3529	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	12	no	—
145	C₂³×C₃₁	—	2	6	no	—
146	C₂³×C₃₁	—	2	12	no	—
147	C₂³×C₃₁	—	2	24	no	—
148	C₂³×C₃₁	—	4	8	no	—
149	C₂³×C₃₁	—	4	16	no	—
150	C₂³×C₃₁	—	4	32	no	—
151	C₂³×C₃₁	C₂⁵	1	12	yes	—
152	C₂³×C₃₁	C₂⁵	1	6	yes	—
153	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
154	Q₈×C₃₁	—	4	8	no	—
155	Q₈×C₃₁	—	4	12	no	—
156	Q₈×C₃₁	—	4	4	no	—
157	D₄×C₃₁	—	4	48	no	—
158	D₄×C₃₁	—	4	24	no	—
159	D₄×C₃₁	—	4	24	no	—
160	D₄×C₃₁	—	4	12	no	—
161	D₄×C₃₁	—	4	12	no	—
162	C₂×C₄×C₃₁	—	2	12	no	—
163	C₂×C₄×C₃₁	—	2	6	no	—
164	C₂×C₄×C₃₁	—	2	12	no	—
165	C₂×C₄×C₃₁	—	2	12	no	—
166	C₂×C₄×C₃₁	—	2	48	no	—
167	C₂×C₄×C₃₁	—	2	24	no	—
168	C₂×C₄×C₃₁	—	4	24	no	—
169	C₂×C₄×C₃₁	—	4	24	no	—
170	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
171	C₂×C₄×C₃₁	C₂⁵	1	12	yes	—
172	C₂×C₄×C₃₁	C₂⁵	1	12	yes	—
173	C₂×C₄×C₃₁	C₂⁵	1	6	yes	—
174	C₂×C₄×C₃₁	C₂⁵	1	12	yes	—
175	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
176	#3529	C₂×C₃₁	1	32	yes	direct product factor
177	C₂×#563	C₂×C₃₁	1	3	yes	complemented
178	C₂×#562	C₂×C₃₁	1	6	yes	complemented
179	C₂×H(3,𝔽₄)	C₂×C₃₁	1	1	yes	characteristic, complemented
180	C₂×#560	C₂×C₃₁	1	6	yes	complemented
181	C₂×#532	C₂×C₃₁	1	3	yes	complemented
182	C₂×#529	C₂×C₃₁	1	6	yes	complemented
183	C₂×#525	C₂×C₃₁	1	6	yes	complemented
184	C₂²×C₃₁	—	2	12	no	—
185	C₂²×C₃₁	—	2	6	no	—
186	C₂²×C₃₁	—	2	12	no	—
187	C₂²×C₃₁	—	2	24	no	—
188	C₂²×C₃₁	—	4	32	no	—
189	C₂²×C₃₁	—	4	12	no	—
190	C₂²×C₃₁	—	4	24	no	—
191	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
192	C₂²×C₃₁	#585	1	4	yes	—
193	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
194	C₄×C₃₁	—	2	12	no	—
195	C₄×C₃₁	—	2	6	no	—
196	C₄×C₃₁	—	2	12	no	—
197	C₄×C₃₁	—	2	12	no	—
198	C₄×C₃₁	C₂×H(5,𝔽₂)	1	4	yes	—
199	C₂²×#41	C₂²×C₃₁	1	3	yes	—
200	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic, complemented
201	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	complemented
202	#563	C₂²×C₃₁	1	48	yes	complemented
203	#562	C₂²×C₃₁	1	96	yes	complemented
204	H(3,𝔽₄)	C₂²×C₃₁	1	16	yes	complemented
205	#560	C₂²×C₃₁	1	96	yes	complemented
206	#532	C₂²×C₃₁	1	48	yes	complemented
207	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	complemented
208	C₂×#127	C₂²×C₃₁	1	3	yes	—
209	#529	C₂²×C₃₁	1	96	yes	complemented
210	C₂×#126	C₂²×C₃₁	1	12	yes	complemented
211	C₂×#125	C₂²×C₃₁	1	6	yes	complemented
212	C₂×#124	C₂²×C₃₁	1	6	yes	—
213	C₂×#124	C₂²×C₃₁	1	3	yes	—
214	C₂×#124	C₂²×C₃₁	1	3	yes	complemented
215	#525	C₂²×C₃₁	1	96	yes	complemented
216	C₂×#123	C₂²×C₃₁	1	6	yes	—
217	C₂×#123	C₂²×C₃₁	1	6	yes	complemented
218	C₂×#123	C₂²×C₃₁	1	12	yes	complemented
219	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	12	yes	complemented
220	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	6	yes	complemented
221	C₂×#121	C₂²×C₃₁	1	12	yes	—
222	C₂×#121	C₂²×C₃₁	1	6	yes	—
223	C₂×#121	C₂²×C₃₁	1	12	yes	complemented
224	C₂×#120	C₂²×C₃₁	1	6	yes	—
225	C₂×#120	C₂²×C₃₁	1	6	yes	complemented
226	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
227	C₂×C₄×D₄	C₂²×C₃₁	1	6	yes	—
228	C₂×C₄×D₄	C₂²×C₃₁	1	12	yes	complemented
229	C₂×#117	C₂²×C₃₁	1	6	yes	complemented
230	C₂²×C₄²	C₂²×C₃₁	1	3	yes	complemented
231	C₂×C₃₁	—	4	6	no	—
232	C₂×C₃₁	—	4	12	no	—
233	C₂×C₃₁	#3529	1	4	yes	direct product factor
234	C₂×C₃₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
235	C₂×C₃₁	C₂×#585	1	2	yes	—
236	C₂⁵	C₂³×C₃₁	1	2	yes	—
237	C₂×#41	—	4	12	no	—
238	C₂×#41	—	4	4	no	—
239	C₂×#41	C₂³×C₃₁	1	24	yes	—
240	C₂×#41	C₂³×C₃₁	1	8	yes	complemented
241	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
242	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
243	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
244	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
245	C₂²×D₄	C₂³×C₃₁	1	12	yes	—
246	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
247	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
248	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
249	C₂³×C₄	—	4	8	no	—
250	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
251	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
252	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
253	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
254	C₂³×C₄	C₂³×C₃₁	1	16	yes	complemented
255	C₄⋊Q₈	C₂³×C₃₁	1	24	yes	complemented
256	#127	C₂³×C₃₁	1	24	yes	—
257	#126	C₂³×C₃₁	1	96	yes	complemented
258	#125	C₂³×C₃₁	1	48	yes	complemented
259	#124	C₂³×C₃₁	1	48	yes	—
260	#124	C₂³×C₃₁	1	24	yes	—
261	#124	C₂³×C₃₁	1	24	yes	complemented
262	#123	C₂³×C₃₁	1	48	yes	—
263	#123	C₂³×C₃₁	1	48	yes	complemented
264	#123	C₂³×C₃₁	1	96	yes	complemented
265	C₂²⋊Q₈	C₂³×C₃₁	1	96	yes	complemented
266	C₂²⋊Q₈	C₂³×C₃₁	1	48	yes	complemented
267	#121	C₂³×C₃₁	1	96	yes	—
268	#121	C₂³×C₃₁	1	48	yes	—
269	#121	C₂³×C₃₁	1	96	yes	complemented
270	#120	C₂³×C₃₁	1	48	yes	—
271	#120	C₂³×C₃₁	1	48	yes	complemented
272	C₄×Q₈	C₂³×C₃₁	1	16	yes	complemented
273	C₄×D₄	C₂³×C₃₁	1	48	yes	—
274	C₄×D₄	C₂³×C₃₁	1	96	yes	complemented
275	#117	C₂³×C₃₁	1	48	yes	complemented
276	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
277	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
278	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
279	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
280	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
281	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
282	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
283	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
284	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
285	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
286	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
287	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
288	C₂×C₄²	C₂³×C₃₁	1	24	yes	complemented
289	C₃₁	C₂×#3529	1	1	yes	characteristic, direct product factor, Sylow
290	C₂⁴	—	4	4	no	—
291	C₂⁴	—	4	8	no	—
292	C₂⁴	C₂⁴×C₃₁	1	6	yes	—
293	C₂⁴	C₂⁴×C₃₁	1	3	yes	—
294	C₂⁴	C₂⁴×C₃₁	1	8	yes	—
295	#41	—	4	48	no	—
296	#41	—	4	16	no	—
297	C₂×Q₈	—	4	4	no	—
298	C₂×Q₈	—	4	6	no	—
299	C₂×Q₈	—	4	2	no	—
300	C₂×Q₈	C₂⁴×C₃₁	1	8	yes	—
301	C₂×Q₈	C₂⁴×C₃₁	1	12	yes	—
302	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
303	C₂×D₄	—	4	24	no	—
304	C₂×D₄	—	4	12	no	—
305	C₂×D₄	—	4	12	no	—
306	C₂×D₄	—	4	6	no	—
307	C₂×D₄	—	4	6	no	—
308	C₂×D₄	C₂⁴×C₃₁	1	24	yes	—
309	C₂×D₄	C₂⁴×C₃₁	1	48	yes	—
310	C₂×D₄	C₂⁴×C₃₁	1	12	yes	—
311	C₂×D₄	C₂⁴×C₃₁	1	24	yes	—
312	C₂×D₄	C₂⁴×C₃₁	1	12	yes	—
313	C₂²×C₄	—	2	24	no	—
314	C₂²×C₄	—	2	12	no	—
315	C₂²×C₄	—	4	32	no	—
316	C₂²×C₄	—	4	12	no	—
317	C₂²×C₄	—	4	12	no	—
318	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
319	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
320	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
321	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
322	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
323	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
324	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
325	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
326	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
327	C₄⋊C₄	C₂⁴×C₃₁	1	48	yes	—
328	C₄⋊C₄	C₂⁴×C₃₁	1	24	yes	—
329	C₄⋊C₄	C₂⁴×C₃₁	1	48	yes	—
330	C₄⋊C₄	C₂⁴×C₃₁	1	24	yes	—
331	C₂²⋊C₄	C₂⁴×C₃₁	1	24	yes	—
332	C₂²⋊C₄	C₂⁴×C₃₁	1	24	yes	—
333	C₂²⋊C₄	C₂⁴×C₃₁	1	48	yes	—
334	C₂²⋊C₄	C₂⁴×C₃₁	1	48	yes	—
335	C₂²⋊C₄	C₂⁴×C₃₁	1	48	yes	—
336	C₄²	C₂⁴×C₃₁	1	24	yes	—
337	C₄²	C₂⁴×C₃₁	1	12	yes	—
338	C₄²	C₂⁴×C₃₁	1	12	yes	—
339	C₂³	—	2	12	no	—
340	C₂³	—	2	6	no	—
341	C₂³	—	2	12	no	—
342	C₂³	—	2	24	no	—
343	C₂³	—	4	8	no	—
344	C₂³	—	4	16	no	—
345	C₂³	—	4	32	no	—
346	C₂³	C₂⁵×C₃₁	1	12	yes	—
347	C₂³	C₂⁵×C₃₁	1	6	yes	—
348	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
349	Q₈	—	4	8	no	—
350	Q₈	—	4	12	no	—
351	Q₈	—	4	4	no	—
352	D₄	—	4	48	no	—
353	D₄	—	4	24	no	—
354	D₄	—	4	24	no	—
355	D₄	—	4	12	no	—
356	D₄	—	4	12	no	—
357	C₂×C₄	—	2	12	no	—
358	C₂×C₄	—	2	6	no	—
359	C₂×C₄	—	2	12	no	—
360	C₂×C₄	—	2	12	no	—
361	C₂×C₄	—	2	48	no	—
362	C₂×C₄	—	2	24	no	—
363	C₂×C₄	—	4	24	no	—
364	C₂×C₄	—	4	24	no	—
365	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
366	C₂×C₄	C₂⁵×C₃₁	1	12	yes	—
367	C₂×C₄	C₂⁵×C₃₁	1	12	yes	—
368	C₂×C₄	C₂⁵×C₃₁	1	6	yes	—
369	C₂×C₄	C₂⁵×C₃₁	1	12	yes	—
370	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
371	C₂²	—	2	12	no	—
372	C₂²	—	2	6	no	—
373	C₂²	—	2	12	no	—
374	C₂²	—	2	24	no	—
375	C₂²	—	4	32	no	—
376	C₂²	—	4	24	no	—
377	C₂²	—	4	12	no	—
378	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
379	C₂²	C₃₁×#585	1	4	yes	—
380	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
381	C₄	—	2	12	no	—
382	C₄	—	2	6	no	—
383	C₄	—	2	12	no	—
384	C₄	—	2	12	no	—
385	C₄	C₂×C₃₁×H(5,𝔽₂)	1	4	yes	—
386	C₂	—	4	12	no	—
387	C₂	—	4	6	no	—
388	C₂	C₃₁×#3529	1	4	yes	direct product factor
389	C₂	C₂²×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
390	C₂	C₂×C₃₁×#585	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃₁
socle	C₂³×C₃₁	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3529	C₂	direct (N × H)
2. C₂×C₃₁×#563	C₂	semidirect
3. C₂×C₃₁×#562	C₂	semidirect
4. C₂×C₃₁×H(3,𝔽₄)	C₂	semidirect
5. C₂×C₃₁×#560	C₂	semidirect
6. C₂×C₃₁×#532	C₂	semidirect
7. C₂×C₃₁×#529	C₂	semidirect
8. C₂×C₃₁×#525	C₂	semidirect
10. C₂²×#41×C₃₁	C₂²	semidirect
11. C₂⁴×C₄×C₃₁	C₂²	semidirect
12. C₃₁×#563	C₂²	semidirect
13. C₃₁×#562	C₂²	semidirect
14. C₃₁×H(3,𝔽₄)	C₂²	semidirect
15. C₃₁×#560	C₂²	semidirect
16. C₃₁×#532	C₂²	semidirect
17. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
19. C₃₁×#529	C₂²	semidirect
20. C₂×C₃₁×#126	C₂²	semidirect
21. C₂×C₃₁×#125	C₂²	semidirect
24. C₂×C₃₁×#124	C₂²	semidirect
25. C₃₁×#525	C₂²	semidirect
27. C₂×C₃₁×#123	C₂²	semidirect
28. C₂×C₃₁×#123	C₂²	semidirect
29. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
30. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
33. C₂×C₃₁×#121	C₂²	semidirect
35. C₂×C₃₁×#120	C₂²	semidirect
36. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
38. C₂×C₄×D₄×C₃₁	C₂²	semidirect
39. C₂×C₃₁×#117	C₂²	semidirect
40. C₂²×C₄²×C₃₁	C₂²	semidirect
45. C₂×#41×C₃₁	C₂³	semidirect
59. C₂³×C₄×C₃₁	C₂³	semidirect
60. C₃₁×C₄⋊Q₈	C₂³	semidirect
62. C₃₁×#126	C₂³	semidirect
63. C₃₁×#125	C₂³	semidirect
66. C₃₁×#124	C₂³	semidirect
68. C₃₁×#123	C₂³	semidirect
69. C₃₁×#123	C₂³	semidirect
70. C₃₁×C₂²⋊Q₈	C₂³	semidirect
71. C₃₁×C₂²⋊Q₈	C₂³	semidirect
74. C₃₁×#121	C₂³	semidirect
76. C₃₁×#120	C₂³	semidirect
77. C₄×Q₈×C₃₁	C₂³	semidirect
79. C₄×D₄×C₃₁	C₂³	semidirect
80. C₃₁×#117	C₂³	semidirect
93. C₂×C₄²×C₃₁	C₂³	semidirect
143. C₂×#3529	C₃₁	direct (N × H)
176. #3529	C₂×C₃₁	direct (N × H)
177. C₂×#563	C₂×C₃₁	semidirect
178. C₂×#562	C₂×C₃₁	semidirect
179. C₂×H(3,𝔽₄)	C₂×C₃₁	semidirect
180. C₂×#560	C₂×C₃₁	semidirect
181. C₂×#532	C₂×C₃₁	semidirect
182. C₂×#529	C₂×C₃₁	semidirect
183. C₂×#525	C₂×C₃₁	semidirect
200. C₂²×#41	C₂²×C₃₁	semidirect
201. C₂⁴×C₄	C₂²×C₃₁	semidirect
202. #563	C₂²×C₃₁	semidirect
203. #562	C₂²×C₃₁	semidirect
204. H(3,𝔽₄)	C₂²×C₃₁	semidirect
205. #560	C₂²×C₃₁	semidirect
206. #532	C₂²×C₃₁	semidirect
207. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
209. #529	C₂²×C₃₁	semidirect
210. C₂×#126	C₂²×C₃₁	semidirect
211. C₂×#125	C₂²×C₃₁	semidirect
214. C₂×#124	C₂²×C₃₁	semidirect
215. #525	C₂²×C₃₁	semidirect
217. C₂×#123	C₂²×C₃₁	semidirect
218. C₂×#123	C₂²×C₃₁	semidirect
219. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
220. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
223. C₂×#121	C₂²×C₃₁	semidirect
225. C₂×#120	C₂²×C₃₁	semidirect
226. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
228. C₂×C₄×D₄	C₂²×C₃₁	semidirect
229. C₂×#117	C₂²×C₃₁	semidirect
230. C₂²×C₄²	C₂²×C₃₁	semidirect
233. C₂×C₃₁	#3529	direct (N × H)
240. C₂×#41	C₂³×C₃₁	semidirect
254. C₂³×C₄	C₂³×C₃₁	semidirect
255. C₄⋊Q₈	C₂³×C₃₁	semidirect
257. #126	C₂³×C₃₁	semidirect
258. #125	C₂³×C₃₁	semidirect
261. #124	C₂³×C₃₁	semidirect
263. #123	C₂³×C₃₁	semidirect
264. #123	C₂³×C₃₁	semidirect
265. C₂²⋊Q₈	C₂³×C₃₁	semidirect
266. C₂²⋊Q₈	C₂³×C₃₁	semidirect
269. #121	C₂³×C₃₁	semidirect
271. #120	C₂³×C₃₁	semidirect
272. C₄×Q₈	C₂³×C₃₁	semidirect
274. C₄×D₄	C₂³×C₃₁	semidirect
275. #117	C₂³×C₃₁	semidirect
288. C₂×C₄²	C₂³×C₃₁	semidirect
289. C₃₁	C₂×#3529	direct (N × H)
388. C₂	C₃₁×#3529	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3529	2
C₂×C₃₁×#563	2
C₂×C₃₁×#562	2
C₂×C₃₁×H(3,𝔽₄)	2
C₂×C₃₁×#560	2
C₂×C₃₁×#532	2
C₂×C₃₁×#529	2
C₂×C₃₁×#525	2
C₂×#3529	31

Subgroups — C2×C31×#3529 #21519450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3529 #21519450