Subgroups — C₂×C₃₁×#2311 #21490802

Order 7936.

5822 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2311	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃₁×#391	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#390	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#386	C₂	1	4	yes	maximal, complemented
8	C₂×C₃₁×#380	C₂	1	4	yes	maximal, complemented
9	C₂²×#41×C₃₁	—	2	8	no	—
10	C₂²×#41×C₃₁	—	2	8	no	—
11	C₂²×#41×C₃₁	C₄	1	16	yes	complemented
12	C₂²×#41×C₃₁	C₂²	1	8	yes	—
13	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
18	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#117	—	2	8	no	—
20	C₂×C₃₁×#117	C₂²	1	8	yes	complemented
21	C₂²×C₄⋊C₄×C₃₁	—	2	8	no	—
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₃₁	C₂²	1	8	yes	complemented
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
28	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#391	C₂²	1	16	yes	complemented
30	C₃₁×#390	C₂²	1	16	yes	complemented
31	C₃₁×#386	C₂²	1	32	yes	complemented
32	C₃₁×#380	C₂²	1	32	yes	complemented
33	C₂×C₃₁×#95	C₂²	1	8	yes	—
34	C₂⁵×C₃₁	C₂³	1	2	yes	—
35	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
36	C₂×#41×C₃₁	—	2	128	no	—
37	C₂×#41×C₃₁	—	2	32	no	—
38	C₂×#41×C₃₁	—	2	32	no	—
39	C₂×#41×C₃₁	C₂×C₄	1	64	yes	complemented
40	C₂²×Q₈×C₃₁	—	2	4	no	—
41	C₂²×Q₈×C₃₁	—	2	4	no	—
42	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
43	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
44	C₂²×D₄×C₃₁	—	2	8	no	—
45	C₂²×D₄×C₃₁	—	2	8	no	—
46	C₂²×D₄×C₃₁	—	2	4	no	—
47	C₂²×D₄×C₃₁	—	2	4	no	—
48	C₂²×D₄×C₃₁	C₂×C₄	1	16	yes	—
49	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
50	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
51	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
52	C₂³×C₄×C₃₁	—	2	8	no	—
53	C₂³×C₄×C₃₁	—	2	8	no	—
54	C₂³×C₄×C₃₁	—	2	8	no	—
55	C₂³×C₄×C₃₁	—	4	8	no	—
56	C₂³×C₄×C₃₁	C₂×C₄	1	8	yes	—
57	C₂³×C₄×C₃₁	C₂×C₄	1	16	yes	complemented
58	C₂³×C₄×C₃₁	D₄	1	8	yes	—
59	C₂³×C₄×C₃₁	D₄	1	8	yes	—
60	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#117	—	2	32	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	32	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	4	16	no	—
78	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₃₁	C₂³	1	32	yes	—
80	C₂×C₄²×C₃₁	—	2	8	no	—
81	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
82	C₃₁×#95	C₂³	1	32	yes	—
83	C₂⁴×C₃₁	—	2	4	no	—
84	C₂⁴×C₃₁	—	2	4	no	—
85	C₂⁴×C₃₁	—	2	4	no	—
86	C₂⁴×C₃₁	—	2	4	no	—
87	C₂⁴×C₃₁	—	2	8	no	—
88	C₂⁴×C₃₁	—	2	8	no	—
89	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
90	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
91	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
92	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
93	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
94	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
95	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
96	#41×C₃₁	—	2	256	no	—
97	C₂×Q₈×C₃₁	—	2	32	no	—
98	C₂×Q₈×C₃₁	—	2	8	no	—
99	C₂×Q₈×C₃₁	—	2	8	no	—
100	C₂×Q₈×C₃₁	C₂²×C₄	1	16	yes	complemented
101	C₂×D₄×C₃₁	—	2	64	no	—
102	C₂×D₄×C₃₁	—	2	16	no	—
103	C₂×D₄×C₃₁	—	2	16	no	—
104	C₂×D₄×C₃₁	—	2	32	no	—
105	C₂×D₄×C₃₁	—	2	8	no	—
106	C₂×D₄×C₃₁	—	2	8	no	—
107	C₂×D₄×C₃₁	C₂²×C₄	1	32	yes	—
108	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	complemented
109	C₂²×C₄×C₃₁	—	2	4	no	—
110	C₂²×C₄×C₃₁	—	2	4	no	—
111	C₂²×C₄×C₃₁	—	2	64	no	—
112	C₂²×C₄×C₃₁	—	2	16	no	—
113	C₂²×C₄×C₃₁	—	2	16	no	—
114	C₂²×C₄×C₃₁	—	2	8	no	—
115	C₂²×C₄×C₃₁	—	2	16	no	—
116	C₂²×C₄×C₃₁	—	2	8	no	—
117	C₂²×C₄×C₃₁	—	2	8	no	—
118	C₂²×C₄×C₃₁	—	2	16	no	—
119	C₂²×C₄×C₃₁	—	2	8	no	—
120	C₂²×C₄×C₃₁	—	4	16	no	—
121	C₂²×C₄×C₃₁	—	4	32	no	—
122	C₂²×C₄×C₃₁	C₂²⋊C₄	1	64	yes	complemented
123	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
124	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
125	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
126	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
127	C₂²×C₄×C₃₁	C₂²×C₄	1	32	yes	complemented
128	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
130	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
132	C₂²×C₄×C₃₁	C₂×D₄	1	16	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	16	yes	—
134	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
135	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
136	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
137	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
139	C₄⋊C₄×C₃₁	—	2	16	no	—
140	C₄⋊C₄×C₃₁	—	2	16	no	—
141	C₄⋊C₄×C₃₁	—	2	16	no	—
142	C₂²⋊C₄×C₃₁	—	2	16	no	—
143	C₂²⋊C₄×C₃₁	—	4	32	no	—
144	C₄²×C₃₁	—	2	16	no	—
145	C₂×#2311	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₃₁	—	2	16	no	—
147	C₂³×C₃₁	—	2	8	no	—
148	C₂³×C₃₁	—	2	8	no	—
149	C₂³×C₃₁	—	2	16	no	—
150	C₂³×C₃₁	—	2	4	no	—
151	C₂³×C₃₁	—	2	4	no	—
152	C₂³×C₃₁	—	2	16	no	—
153	C₂³×C₃₁	—	2	32	no	—
154	C₂³×C₃₁	—	2	4	no	—
155	C₂³×C₃₁	—	2	4	no	—
156	C₂³×C₃₁	—	2	8	no	—
157	C₂³×C₃₁	—	2	8	no	—
158	C₂³×C₃₁	—	4	8	no	—
159	C₂³×C₃₁	—	4	8	no	—
160	C₂³×C₃₁	—	4	16	no	—
161	C₂³×C₃₁	C₂×C₂²⋊C₄	1	16	yes	—
162	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
163	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
164	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
165	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
166	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
167	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
168	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
169	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
170	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
171	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
172	Q₈×C₃₁	—	2	32	no	—
173	D₄×C₃₁	—	2	64	no	—
174	D₄×C₃₁	—	2	32	no	—
175	C₂×C₄×C₃₁	—	2	16	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	2	64	no	—
179	C₂×C₄×C₃₁	—	2	8	no	—
180	C₂×C₄×C₃₁	—	2	8	no	—
181	C₂×C₄×C₃₁	—	2	8	no	—
182	C₂×C₄×C₃₁	—	2	16	no	—
183	C₂×C₄×C₃₁	—	2	8	no	—
184	C₂×C₄×C₃₁	—	4	32	no	—
185	C₂×C₄×C₃₁	—	4	16	no	—
186	C₂×C₄×C₃₁	—	4	16	no	—
187	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	16	yes	—
188	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	16	yes	—
189	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	64	yes	complemented
190	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
191	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
192	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
193	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
194	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
195	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
196	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
197	C₂³×#41	C₂×C₃₁	1	1	yes	characteristic
198	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂³×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#2311	C₂×C₃₁	1	16	yes	direct product factor
201	C₂×#391	C₂×C₃₁	1	2	yes	complemented
202	C₂×#390	C₂×C₃₁	1	2	yes	complemented
203	C₂×#386	C₂×C₃₁	1	4	yes	complemented
204	C₂×#380	C₂×C₃₁	1	4	yes	complemented
205	C₂²×C₃₁	—	2	32	no	—
206	C₂²×C₃₁	—	2	8	no	—
207	C₂²×C₃₁	—	2	4	no	—
208	C₂²×C₃₁	—	2	4	no	—
209	C₂²×C₃₁	—	2	8	no	—
210	C₂²×C₃₁	—	2	8	no	—
211	C₂²×C₃₁	—	2	16	no	—
212	C₂²×C₃₁	—	4	32	no	—
213	C₂²×C₃₁	—	4	4	no	—
214	C₂²×C₃₁	—	4	4	no	—
215	C₂²×C₃₁	—	4	8	no	—
216	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	8	yes	—
217	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	4	yes	—
218	C₂²×C₃₁	#520	1	8	yes	—
219	C₂²×C₃₁	#535	1	4	yes	—
220	C₂²×C₃₁	#536	1	4	yes	—
221	C₂²×C₃₁	#537	1	8	yes	—
222	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
223	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
224	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
225	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
226	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
227	C₄×C₃₁	—	2	8	no	—
228	C₄×C₃₁	—	4	8	no	—
229	C₄×C₃₁	—	4	8	no	—
230	C₄×C₃₁	C₂²×C₂²⋊C₄	1	8	yes	—
231	C₄×C₃₁	C₂²×C₂²⋊C₄	1	8	yes	—
232	C₂²×#41	—	2	8	no	—
233	C₂²×#41	—	2	8	no	—
234	C₂²×#41	C₄×C₃₁	1	16	yes	complemented
235	C₂²×#41	C₂²×C₃₁	1	8	yes	—
236	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
238	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
239	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
240	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
241	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×#117	—	2	8	no	—
243	C₂×#117	C₂²×C₃₁	1	8	yes	complemented
244	C₂²×C₄⋊C₄	—	2	8	no	—
245	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
246	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
247	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
248	C₂²×C₄⋊C₄	C₂²×C₃₁	1	8	yes	complemented
249	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
250	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
251	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
252	#391	C₂²×C₃₁	1	16	yes	complemented
253	#390	C₂²×C₃₁	1	16	yes	complemented
254	#386	C₂²×C₃₁	1	32	yes	complemented
255	#380	C₂²×C₃₁	1	32	yes	complemented
256	C₂×#95	C₂²×C₃₁	1	8	yes	—
257	C₂×C₃₁	—	2	8	no	—
258	C₂×C₃₁	—	4	8	no	—
259	C₂×C₃₁	#2311	1	8	yes	direct product factor
260	C₂×C₃₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₃₁	C₂×#520	1	2	yes	—
262	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
263	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
264	C₂×C₃₁	C₂×#537	1	2	yes	—
265	C₂⁵	C₂³×C₃₁	1	2	yes	—
266	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
267	C₂×#41	—	2	128	no	—
268	C₂×#41	—	2	32	no	—
269	C₂×#41	—	2	32	no	—
270	C₂×#41	C₂×C₄×C₃₁	1	64	yes	complemented
271	C₂²×Q₈	—	2	4	no	—
272	C₂²×Q₈	—	2	4	no	—
273	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
274	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
275	C₂²×D₄	—	2	8	no	—
276	C₂²×D₄	—	2	8	no	—
277	C₂²×D₄	—	2	4	no	—
278	C₂²×D₄	—	2	4	no	—
279	C₂²×D₄	C₂×C₄×C₃₁	1	16	yes	—
280	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
281	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
282	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
283	C₂³×C₄	—	2	8	no	—
284	C₂³×C₄	—	2	8	no	—
285	C₂³×C₄	—	2	8	no	—
286	C₂³×C₄	—	4	8	no	—
287	C₂³×C₄	C₂×C₄×C₃₁	1	8	yes	—
288	C₂³×C₄	C₂×C₄×C₃₁	1	16	yes	complemented
289	C₂³×C₄	D₄×C₃₁	1	8	yes	—
290	C₂³×C₄	D₄×C₃₁	1	8	yes	—
291	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
292	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
293	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
294	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
295	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
296	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
297	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
298	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
299	#117	—	2	32	no	—
300	C₂×C₄⋊C₄	—	2	8	no	—
301	C₂×C₄⋊C₄	—	2	8	no	—
302	C₂×C₄⋊C₄	—	2	32	no	—
303	C₂×C₄⋊C₄	—	2	8	no	—
304	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
306	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
307	C₂×C₂²⋊C₄	—	2	8	no	—
308	C₂×C₂²⋊C₄	—	4	16	no	—
309	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₃₁	1	32	yes	—
311	C₂×C₄²	—	2	8	no	—
312	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
313	#95	C₂³×C₃₁	1	32	yes	—
314	C₃₁	C₂×#2311	1	1	yes	characteristic, direct product factor, Sylow
315	C₂⁴	—	2	4	no	—
316	C₂⁴	—	2	4	no	—
317	C₂⁴	—	2	4	no	—
318	C₂⁴	—	2	4	no	—
319	C₂⁴	—	2	8	no	—
320	C₂⁴	—	2	8	no	—
321	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
322	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
323	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
324	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
325	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
326	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
327	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
328	#41	—	2	256	no	—
329	C₂×Q₈	—	2	32	no	—
330	C₂×Q₈	—	2	8	no	—
331	C₂×Q₈	—	2	8	no	—
332	C₂×Q₈	C₂²×C₄×C₃₁	1	16	yes	complemented
333	C₂×D₄	—	2	64	no	—
334	C₂×D₄	—	2	16	no	—
335	C₂×D₄	—	2	16	no	—
336	C₂×D₄	—	2	32	no	—
337	C₂×D₄	—	2	8	no	—
338	C₂×D₄	—	2	8	no	—
339	C₂×D₄	C₂²×C₄×C₃₁	1	32	yes	—
340	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	complemented
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	64	no	—
344	C₂²×C₄	—	2	16	no	—
345	C₂²×C₄	—	2	16	no	—
346	C₂²×C₄	—	2	8	no	—
347	C₂²×C₄	—	2	16	no	—
348	C₂²×C₄	—	2	8	no	—
349	C₂²×C₄	—	2	8	no	—
350	C₂²×C₄	—	2	16	no	—
351	C₂²×C₄	—	2	8	no	—
352	C₂²×C₄	—	4	16	no	—
353	C₂²×C₄	—	4	32	no	—
354	C₂²×C₄	C₂²⋊C₄×C₃₁	1	64	yes	complemented
355	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
356	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
357	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
358	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
359	C₂²×C₄	C₂²×C₄×C₃₁	1	32	yes	complemented
360	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
361	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
362	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
363	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
364	C₂²×C₄	C₂×D₄×C₃₁	1	16	yes	—
365	C₂²×C₄	C₂×D₄×C₃₁	1	16	yes	—
366	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
367	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
368	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
369	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
370	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
371	C₄⋊C₄	—	2	16	no	—
372	C₄⋊C₄	—	2	16	no	—
373	C₄⋊C₄	—	2	16	no	—
374	C₂²⋊C₄	—	2	16	no	—
375	C₂²⋊C₄	—	4	32	no	—
376	C₄²	—	2	16	no	—
377	C₂³	—	2	16	no	—
378	C₂³	—	2	8	no	—
379	C₂³	—	2	8	no	—
380	C₂³	—	2	16	no	—
381	C₂³	—	2	4	no	—
382	C₂³	—	2	4	no	—
383	C₂³	—	2	16	no	—
384	C₂³	—	2	32	no	—
385	C₂³	—	2	4	no	—
386	C₂³	—	2	4	no	—
387	C₂³	—	2	8	no	—
388	C₂³	—	2	8	no	—
389	C₂³	—	4	8	no	—
390	C₂³	—	4	8	no	—
391	C₂³	—	4	16	no	—
392	C₂³	C₂×C₂²⋊C₄×C₃₁	1	16	yes	—
393	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
394	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
395	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
396	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
397	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
398	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
399	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
400	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
401	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
402	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
403	Q₈	—	2	32	no	—
404	D₄	—	2	64	no	—
405	D₄	—	2	32	no	—
406	C₂×C₄	—	2	16	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	64	no	—
410	C₂×C₄	—	2	8	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	8	no	—
413	C₂×C₄	—	2	16	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	4	32	no	—
416	C₂×C₄	—	4	16	no	—
417	C₂×C₄	—	4	16	no	—
418	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	16	yes	—
419	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	16	yes	—
420	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	64	yes	complemented
421	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
422	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
423	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
424	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
425	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
426	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
427	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
428	C₂²	—	2	32	no	—
429	C₂²	—	2	8	no	—
430	C₂²	—	2	4	no	—
431	C₂²	—	2	4	no	—
432	C₂²	—	2	8	no	—
433	C₂²	—	2	8	no	—
434	C₂²	—	2	16	no	—
435	C₂²	—	4	32	no	—
436	C₂²	—	4	4	no	—
437	C₂²	—	4	4	no	—
438	C₂²	—	4	8	no	—
439	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	8	yes	—
440	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	4	yes	—
441	C₂²	C₃₁×#520	1	8	yes	—
442	C₂²	C₃₁×#535	1	4	yes	—
443	C₂²	C₃₁×#536	1	4	yes	—
444	C₂²	C₃₁×#537	1	8	yes	—
445	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
446	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
447	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
448	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
449	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
450	C₄	—	2	8	no	—
451	C₄	—	4	8	no	—
452	C₄	—	4	8	no	—
453	C₄	C₂²×C₂²⋊C₄×C₃₁	1	8	yes	—
454	C₄	C₂²×C₂²⋊C₄×C₃₁	1	8	yes	—
455	C₂	—	2	8	no	—
456	C₂	—	4	8	no	—
457	C₂	C₃₁×#2311	1	8	yes	direct product factor
458	C₂	C₂³×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
459	C₂	C₂×C₃₁×#520	1	2	yes	—
460	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
461	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
462	C₂	C₂×C₃₁×#537	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#117	C₂	semidirect
3. C₂³×C₄⋊C₄×C₃₁	C₂	semidirect
4. C₃₁×#2311	C₂	direct (N × H)
5. C₂×C₃₁×#391	C₂	semidirect
6. C₂×C₃₁×#390	C₂	semidirect
7. C₂×C₃₁×#386	C₂	semidirect
8. C₂×C₃₁×#380	C₂	semidirect
11. C₂²×#41×C₃₁	C₄	semidirect
20. C₂×C₃₁×#117	C₂²	semidirect
25. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
29. C₃₁×#391	C₂²	semidirect
30. C₃₁×#390	C₂²	semidirect
31. C₃₁×#386	C₂²	semidirect
32. C₃₁×#380	C₂²	semidirect
39. C₂×#41×C₃₁	C₂×C₄	semidirect
42. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
49. C₂²×D₄×C₃₁	C₂×C₄	semidirect
57. C₂³×C₄×C₃₁	C₂×C₄	semidirect
100. C₂×Q₈×C₃₁	C₂²×C₄	semidirect
108. C₂×D₄×C₃₁	C₂²×C₄	semidirect
122. C₂²×C₄×C₃₁	C₂²⋊C₄	semidirect
127. C₂²×C₄×C₃₁	C₂²×C₄	semidirect
145. C₂×#2311	C₃₁	direct (N × H)
189. C₂×C₄×C₃₁	C₂×C₂²⋊C₄	semidirect
198. C₂²×#117	C₂×C₃₁	semidirect
199. C₂³×C₄⋊C₄	C₂×C₃₁	semidirect
200. #2311	C₂×C₃₁	direct (N × H)
201. C₂×#391	C₂×C₃₁	semidirect
202. C₂×#390	C₂×C₃₁	semidirect
203. C₂×#386	C₂×C₃₁	semidirect
204. C₂×#380	C₂×C₃₁	semidirect
234. C₂²×#41	C₄×C₃₁	semidirect
243. C₂×#117	C₂²×C₃₁	semidirect
248. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
252. #391	C₂²×C₃₁	semidirect
253. #390	C₂²×C₃₁	semidirect
254. #386	C₂²×C₃₁	semidirect
255. #380	C₂²×C₃₁	semidirect
259. C₂×C₃₁	#2311	direct (N × H)
270. C₂×#41	C₂×C₄×C₃₁	semidirect
273. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
280. C₂²×D₄	C₂×C₄×C₃₁	semidirect
288. C₂³×C₄	C₂×C₄×C₃₁	semidirect
314. C₃₁	C₂×#2311	direct (N × H)
332. C₂×Q₈	C₂²×C₄×C₃₁	semidirect
340. C₂×D₄	C₂²×C₄×C₃₁	semidirect
354. C₂²×C₄	C₂²⋊C₄×C₃₁	semidirect
359. C₂²×C₄	C₂²×C₄×C₃₁	semidirect
420. C₂×C₄	C₂×C₂²⋊C₄×C₃₁	semidirect
457. C₂	C₃₁×#2311	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₃₁	2
C₂²×C₃₁×#117	2
C₂³×C₄⋊C₄×C₃₁	2
C₃₁×#2311	2
C₂×C₃₁×#391	2
C₂×C₃₁×#390	2
C₂×C₃₁×#386	2
C₂×C₃₁×#380	2
C₂×#2311	31

Subgroups — C2×C31×#2311 #21490802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2311 #21490802