Subgroups — C₃₁×#23027 #21479786

Order 7936.

1050 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2363	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1875	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#121	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₃₁×#117	C₂²	1	2	yes	—
17	C₃₁×#466	C₄	1	8	yes	complemented
18	C₄×D₈×C₃₁	—	2	4	no	—
19	C₃₁×#427	—	2	4	no	—
20	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#106	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#390	—	2	2	no	—
31	C₃₁×#390	C₂²	1	2	yes	—
32	C₃₁×#389	C₂²	1	2	yes	—
33	C₃₁×#388	—	2	2	no	—
34	C₃₁×#386	—	2	2	no	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
37	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
44	C₂³×C₄×C₃₁	—	2	2	no	—
45	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×D₈×C₃₁	—	2	2	no	—
47	C₂×D₈×C₃₁	—	2	2	no	—
48	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×#121	—	2	4	no	—
57	C₃₁×#121	C₂×C₄	1	8	yes	complemented
58	C₄×D₄×C₃₁	—	2	4	no	—
59	C₄×D₄×C₃₁	—	4	2	no	—
60	C₃₁×#117	—	2	4	no	—
61	C₃₁×#117	—	4	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	—	2	2	no	—
74	C₂×C₄²×C₃₁	—	2	2	no	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#107	—	2	2	no	—
78	C₃₁×#107	C₂×C₄	1	4	yes	complemented
79	C₃₁×#107	D₄	1	4	yes	complemented
80	C₃₁×#106	—	2	4	no	—
81	C₃₁×#102	—	2	4	no	—
82	C₃₁×#102	—	4	2	no	—
83	C₃₁×#102	C₂×C₄	1	8	yes	complemented
84	C₃₁×C₂²⋊C₈	—	2	2	no	—
85	C₄×C₈×C₃₁	—	2	2	no	—
86	C₄×C₈×C₃₁	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	D₄	1	1	yes	characteristic
90	C₃₁×#95	D₄	1	1	yes	characteristic
91	C₂⁴×C₃₁	—	2	2	no	—
92	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×D₄×C₃₁	—	2	8	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	8	2	no	—
100	C₂×D₄×C₃₁	—	8	2	no	—
101	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	1	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	4	4	no	—
110	C₂²×C₄×C₃₁	—	4	4	no	—
111	C₂²×C₄×C₃₁	—	8	2	no	—
112	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	#41	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	D₈×C₃₁	—	2	8	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	8	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₃₁	#41	1	2	yes	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	8	2	no	—
139	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
140	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
141	C₂²⋊C₄×C₃₁	—	2	8	no	—
142	C₂²⋊C₄×C₃₁	—	2	2	no	—
143	C₂²⋊C₄×C₃₁	—	2	2	no	—
144	C₂²⋊C₄×C₃₁	—	4	4	no	—
145	C₂²⋊C₄×C₃₁	—	4	4	no	—
146	C₂²⋊C₄×C₃₁	—	4	4	no	—
147	C₂²⋊C₄×C₃₁	—	8	2	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	2	1	no	—
152	C₄²×C₃₁	—	4	4	no	—
153	C₄²×C₃₁	—	8	2	no	—
154	#23027	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₃₁	—	2	1	no	—
156	C₂³×C₃₁	—	2	1	no	—
157	C₂³×C₃₁	—	2	1	no	—
158	C₂³×C₃₁	—	2	1	no	—
159	C₂³×C₃₁	—	2	1	no	—
160	C₂³×C₃₁	—	2	1	no	—
161	C₂³×C₃₁	—	2	4	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	4	2	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	4	2	no	—
166	C₂³×C₃₁	—	4	2	no	—
167	C₂³×C₃₁	—	4	2	no	—
168	C₂³×C₃₁	C₄×D₄	1	2	yes	—
169	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
170	D₄×C₃₁	—	2	4	no	—
171	D₄×C₃₁	—	2	4	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	4	2	no	—
174	D₄×C₃₁	—	8	8	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	4	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	4	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	8	8	no	—
201	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
202	C₂×C₄×C₃₁	#135	1	2	yes	—
203	C₂×C₄×C₃₁	#135	1	2	yes	—
204	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
209	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
210	C₈×C₃₁	—	2	2	no	—
211	C₈×C₃₁	—	4	1	no	—
212	C₈×C₃₁	—	4	1	no	—
213	C₈×C₃₁	C₄×D₄	1	4	yes	—
214	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
215	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
216	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#2363	C₂×C₃₁	1	2	yes	complemented
218	#1970	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#1942	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#1926	C₂×C₃₁	1	2	yes	complemented
221	#1875	C₂×C₃₁	1	2	yes	complemented
222	#1864	C₂×C₃₁	1	2	yes	complemented
223	C₄×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
224	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
225	C₂²×C₃₁	—	2	2	no	—
226	C₂²×C₃₁	—	2	2	no	—
227	C₂²×C₃₁	—	2	2	no	—
228	C₂²×C₃₁	—	4	1	no	—
229	C₂²×C₃₁	—	4	1	no	—
230	C₂²×C₃₁	—	4	1	no	—
231	C₂²×C₃₁	—	4	1	no	—
232	C₂²×C₃₁	—	4	4	no	—
233	C₂²×C₃₁	—	4	4	no	—
234	C₂²×C₃₁	—	4	4	no	—
235	C₂²×C₃₁	—	8	2	no	—
236	C₂²×C₃₁	—	8	2	no	—
237	C₂²×C₃₁	—	8	2	no	—
238	C₂²×C₃₁	—	8	2	no	—
239	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
240	C₂²×C₃₁	D₄²	1	1	yes	characteristic
241	C₂²×C₃₁	#547	1	1	yes	characteristic
242	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
243	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
244	C₂²×C₃₁	#576	1	1	yes	characteristic
245	C₂²×C₃₁	#576	1	1	yes	characteristic
246	C₄×C₃₁	—	2	2	no	—
247	C₄×C₃₁	—	2	2	no	—
248	C₄×C₃₁	—	2	2	no	—
249	C₄×C₃₁	—	4	1	no	—
250	C₄×C₃₁	—	4	4	no	—
251	C₄×C₃₁	—	4	4	no	—
252	C₄×C₃₁	—	4	1	no	—
253	C₄×C₃₁	—	8	2	no	—
254	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
255	C₄×C₃₁	#520	1	1	yes	characteristic
256	C₄×C₃₁	#548	1	1	yes	characteristic
257	C₄×C₃₁	#553	1	1	yes	characteristic
258	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
259	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#121	C₂²×C₃₁	1	2	yes	—
261	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
262	C₂×#117	C₂²×C₃₁	1	2	yes	—
263	#466	C₄×C₃₁	1	8	yes	complemented
264	C₄×D₈	—	2	4	no	—
265	#427	—	2	4	no	—
266	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#106	C₂²×C₃₁	1	2	yes	—
270	C₂×#102	C₂²×C₃₁	1	2	yes	—
271	C₂×#102	C₂²×C₃₁	1	2	yes	—
272	C₂×#102	C₂²×C₃₁	1	2	yes	—
273	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
276	#390	—	2	2	no	—
277	#390	C₂²×C₃₁	1	2	yes	—
278	#389	C₂²×C₃₁	1	2	yes	—
279	#388	—	2	2	no	—
280	#386	—	2	2	no	—
281	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
282	C₄×C₂²⋊C₄	—	2	2	no	—
283	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
284	#340	C₂²×C₃₁	1	2	yes	—
285	#340	C₂²×C₃₁	1	2	yes	—
286	#336	C₂²×C₃₁	1	1	yes	characteristic
287	#336	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×C₃₁	—	4	2	no	—
289	C₂×C₃₁	—	8	4	no	—
290	C₂×C₃₁	#2361	1	1	yes	characteristic
291	C₂×C₃₁	C₄×#135	1	1	yes	characteristic
292	C₂×C₃₁	#2948	1	1	yes	characteristic
293	C₂×C₃₁	#3280	1	1	yes	characteristic
294	C₂×C₃₁	#3287	1	1	yes	characteristic
295	C₂×C₃₁	#3339	1	1	yes	characteristic
296	C₂×C₃₁	#3341	1	1	yes	characteristic
297	C₂²×D₄	—	2	2	no	—
298	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
299	C₂³×C₄	—	2	2	no	—
300	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
301	C₂×D₈	—	2	2	no	—
302	C₂×D₈	—	2	2	no	—
303	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
304	C₂²×C₈	—	2	2	no	—
305	C₂²×C₈	—	2	2	no	—
306	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
307	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
308	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
310	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
311	#121	—	2	4	no	—
312	#121	C₂×C₄×C₃₁	1	8	yes	complemented
313	C₄×D₄	—	2	4	no	—
314	C₄×D₄	—	4	2	no	—
315	#117	—	2	4	no	—
316	#117	—	4	2	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
319	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
320	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
321	C₂×C₂²⋊C₄	—	2	2	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₂²⋊C₄	—	2	2	no	—
324	C₂×C₂²⋊C₄	—	2	2	no	—
325	C₂×C₂²⋊C₄	—	2	2	no	—
326	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄²	—	2	2	no	—
329	C₂×C₄²	—	2	2	no	—
330	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
332	#107	—	2	2	no	—
333	#107	C₂×C₄×C₃₁	1	4	yes	complemented
334	#107	D₄×C₃₁	1	4	yes	complemented
335	#106	—	2	4	no	—
336	#102	—	2	4	no	—
337	#102	—	4	2	no	—
338	#102	C₂×C₄×C₃₁	1	8	yes	complemented
339	C₂²⋊C₈	—	2	2	no	—
340	C₄×C₈	—	2	2	no	—
341	C₄×C₈	—	2	2	no	—
342	#95	—	2	2	no	—
343	#95	—	2	2	no	—
344	#95	D₄×C₃₁	1	1	yes	characteristic
345	#95	D₄×C₃₁	1	1	yes	characteristic
346	C₃₁	#23027	1	1	yes	characteristic, direct product factor, Sylow
347	C₂⁴	—	2	2	no	—
348	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₂×D₄	—	2	8	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	8	2	no	—
356	C₂×D₄	—	8	2	no	—
357	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	4	4	no	—
366	C₂²×C₄	—	4	4	no	—
367	C₂²×C₄	—	8	2	no	—
368	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
369	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	#41×C₃₁	1	2	yes	—
375	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
376	D₈	—	2	8	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	8	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
387	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
388	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
389	C₂×C₈	#41×C₃₁	1	2	yes	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	8	2	no	—
395	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
396	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
397	C₂²⋊C₄	—	2	8	no	—
398	C₂²⋊C₄	—	2	2	no	—
399	C₂²⋊C₄	—	2	2	no	—
400	C₂²⋊C₄	—	4	4	no	—
401	C₂²⋊C₄	—	4	4	no	—
402	C₂²⋊C₄	—	4	4	no	—
403	C₂²⋊C₄	—	8	2	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	4	4	no	—
409	C₄²	—	8	2	no	—
410	C₂³	—	2	1	no	—
411	C₂³	—	2	1	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	—	2	1	no	—
416	C₂³	—	2	4	no	—
417	C₂³	—	4	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	C₄×D₄×C₃₁	1	2	yes	—
424	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
425	D₄	—	2	4	no	—
426	D₄	—	2	4	no	—
427	D₄	—	4	2	no	—
428	D₄	—	4	2	no	—
429	D₄	—	8	8	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	8	8	no	—
456	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
457	C₂×C₄	C₃₁×#135	1	2	yes	—
458	C₂×C₄	C₃₁×#135	1	2	yes	—
459	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
463	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
464	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
465	C₈	—	2	2	no	—
466	C₈	—	4	1	no	—
467	C₈	—	4	1	no	—
468	C₈	C₄×D₄×C₃₁	1	4	yes	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	8	2	no	—
480	C₂²	—	8	2	no	—
481	C₂²	—	8	2	no	—
482	C₂²	—	8	2	no	—
483	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
484	C₂²	D₄²×C₃₁	1	1	yes	characteristic
485	C₂²	C₃₁×#547	1	1	yes	characteristic
486	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
487	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
488	C₂²	C₃₁×#576	1	1	yes	characteristic
489	C₂²	C₃₁×#576	1	1	yes	characteristic
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	4	1	no	—
494	C₄	—	4	4	no	—
495	C₄	—	4	4	no	—
496	C₄	—	4	1	no	—
497	C₄	—	8	2	no	—
498	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
499	C₄	C₃₁×#520	1	1	yes	characteristic
500	C₄	C₃₁×#548	1	1	yes	characteristic
501	C₄	C₃₁×#553	1	1	yes	characteristic
502	C₂	—	4	2	no	—
503	C₂	—	8	4	no	—
504	C₂	C₃₁×#2361	1	1	yes	characteristic
505	C₂	C₄×C₃₁×#135	1	1	yes	characteristic
506	C₂	C₃₁×#2948	1	1	yes	characteristic
507	C₂	C₃₁×#3280	1	1	yes	characteristic
508	C₂	C₃₁×#3287	1	1	yes	characteristic
509	C₂	C₃₁×#3339	1	1	yes	characteristic
510	C₂	C₃₁×#3341	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₂²⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₈×C₃₁	C₂	semidirect
3. C₂×C₃₁×#427	C₂	semidirect
4. C₃₁×#2363	C₂	semidirect
5. C₃₁×#1970	C₂	semidirect
6. C₃₁×#1942	C₂	semidirect
7. C₃₁×#1926	C₂	semidirect
8. C₃₁×#1875	C₂	semidirect
9. C₃₁×#1864	C₂	semidirect
10. C₄×C₃₁×#107	C₂	semidirect
11. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
17. C₃₁×#466	C₄	semidirect
51. C₂²×C₈×C₃₁	C₂×C₄	semidirect
57. C₃₁×#121	C₂×C₄	semidirect
78. C₃₁×#107	C₂×C₄	semidirect
79. C₃₁×#107	D₄	semidirect
83. C₃₁×#102	C₂×C₄	semidirect
154. #23027	C₃₁	direct (N × H)
215. C₂×C₄×D₈	C₂×C₃₁	semidirect
216. C₂×#427	C₂×C₃₁	semidirect
217. #2363	C₂×C₃₁	semidirect
218. #1970	C₂×C₃₁	semidirect
219. #1942	C₂×C₃₁	semidirect
220. #1926	C₂×C₃₁	semidirect
221. #1875	C₂×C₃₁	semidirect
222. #1864	C₂×C₃₁	semidirect
223. C₄×#107	C₂×C₃₁	semidirect
224. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
263. #466	C₄×C₃₁	semidirect
306. C₂²×C₈	C₂×C₄×C₃₁	semidirect
312. #121	C₂×C₄×C₃₁	semidirect
333. #107	C₂×C₄×C₃₁	semidirect
334. #107	D₄×C₃₁	semidirect
338. #102	C₂×C₄×C₃₁	semidirect
346. C₃₁	#23027	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×#466	2
C₂×C₄×D₈×C₃₁	2
C₂×C₃₁×#427	2
C₃₁×#2363	2
C₃₁×#1970	2
C₃₁×#1942	2
C₃₁×#1926	2
C₃₁×#1875	2
C₃₁×#1864	2
C₄×C₃₁×#107	2
C₈×C₂²⋊C₄×C₃₁	2
#23027	31

Subgroups — C31×#23027 #21479786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23027 #21479786