Subgroups — C₃₁×#62245 #21519004

Order 7936.

2422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3496	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3334	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3294	C₂	1	8	yes	maximal, complemented
4	D₄×D₈×C₃₁	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3245	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3186	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3169	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#466	C₂	1	4	yes	maximal, complemented
9	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×H(5,𝔽₂)	—	2	2	no	—
11	C₂²×D₈×C₃₁	C₂²	1	4	yes	complemented
12	C₂³×C₈×C₃₁	C₂²	1	2	yes	complemented
13	C₃₁×#547	C₂²	1	8	yes	complemented
14	C₃₁×#546	—	2	8	no	—
15	D₄²×C₃₁	C₂²	1	8	yes	complemented
16	C₃₁×#537	—	2	2	no	—
17	C₃₁×#534	—	2	4	no	—
18	C₃₁×#532	C₂²	1	4	yes	complemented
19	C₂×C₃₁×#121	C₂²	1	8	yes	complemented
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#500	C₂²	1	2	yes	complemented
22	C₃₁×#493	C₂²	1	2	yes	complemented
23	C₃₁×#486	C₂²	1	4	yes	complemented
24	C₃₁×#480	C₂²	1	16	yes	complemented
25	C₃₁×#474	C₂²	1	8	yes	complemented
26	C₃₁×#466	C₂²	1	8	yes	complemented
27	C₃₁×#466	C₂²	1	4	yes	complemented
28	C₃₁×#466	C₂²	1	4	yes	complemented
29	C₃₁×#466	C₂²	1	8	yes	complemented
30	C₃₁×#466	C₂²	1	8	yes	complemented
31	C₃₁×#459	C₂²	1	8	yes	complemented
32	C₃₁×#447	C₂²	1	16	yes	complemented
33	C₄×D₈×C₃₁	C₂²	1	4	yes	complemented
34	C₄×D₈×C₃₁	C₂²	1	4	yes	complemented
35	C₈×D₄×C₃₁	C₂²	1	4	yes	complemented
36	C₈×D₄×C₃₁	C₂²	1	4	yes	complemented
37	C₂×C₃₁×#107	C₂²	1	4	yes	complemented
38	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₃₁×#102	C₂²	1	8	yes	complemented
40	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
41	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×H(5,𝔽₂)	—	4	16	no	—
43	C₂×#41×C₃₁	—	2	8	no	—
44	C₂×#41×C₃₁	—	2	4	no	—
45	C₂²×D₄×C₃₁	—	2	8	no	—
46	C₂²×D₄×C₃₁	—	2	8	no	—
47	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
49	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
50	C₂×D₈×C₃₁	—	4	8	no	—
51	C₂×D₈×C₃₁	C₂³	1	4	yes	—
52	C₂×D₈×C₃₁	C₂³	1	4	yes	—
53	C₂×D₈×C₃₁	C₂³	1	8	yes	—
54	C₂²×C₈×C₃₁	—	4	4	no	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
60	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
61	C₃₁×#127	C₂³	1	4	yes	—
62	C₃₁×#124	—	2	4	no	—
63	C₃₁×#123	—	2	16	no	—
64	C₃₁×C₂²⋊Q₈	—	2	8	no	—
65	C₃₁×#121	—	2	8	no	—
66	C₃₁×#121	—	2	8	no	—
67	C₃₁×#121	—	2	4	no	—
68	C₃₁×#121	—	2	8	no	—
69	C₃₁×#121	C₂³	1	16	yes	—
70	C₃₁×#121	C₂³	1	8	yes	—
71	C₃₁×#120	—	2	16	no	—
72	C₄×D₄×C₃₁	—	2	4	no	—
73	C₄×D₄×C₃₁	D₄	1	4	yes	—
74	C₄×D₄×C₃₁	C₂³	1	8	yes	—
75	C₄×D₄×C₃₁	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
79	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₃₁×#107	C₂³	1	4	yes	—
82	C₃₁×#107	C₂³	1	8	yes	—
83	C₃₁×#107	C₂³	1	4	yes	—
84	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
85	C₃₁×#102	C₂³	1	8	yes	—
86	C₃₁×#102	C₂³	1	16	yes	—
87	C₃₁×#102	C₂³	1	8	yes	—
88	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
89	C₄×C₈×C₃₁	C₂³	1	2	yes	—
90	C₄×C₈×C₃₁	C₂³	1	2	yes	—
91	C₂⁴×C₃₁	—	2	8	no	—
92	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
93	#41×C₃₁	—	4	32	no	—
94	#41×C₃₁	—	4	16	no	—
95	C₂×Q₈×C₃₁	—	2	4	no	—
96	C₂×D₄×C₃₁	—	2	16	no	—
97	C₂×D₄×C₃₁	—	2	8	no	—
98	C₂×D₄×C₃₁	—	2	4	no	—
99	C₂×D₄×C₃₁	—	2	4	no	—
100	C₂×D₄×C₃₁	—	2	8	no	—
101	C₂×D₄×C₃₁	—	4	8	no	—
102	C₂×D₄×C₃₁	—	4	8	no	—
103	C₂×D₄×C₃₁	—	4	32	no	—
104	C₂×D₄×C₃₁	—	4	32	no	—
105	C₂×D₄×C₃₁	D₈	1	8	yes	complemented
106	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
107	C₂×D₄×C₃₁	C₂⁴	1	8	yes	—
108	C₂²×C₄×C₃₁	—	2	8	no	—
109	C₂²×C₄×C₃₁	—	2	8	no	—
110	C₂²×C₄×C₃₁	—	4	2	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
117	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
118	D₈×C₃₁	—	2	8	no	—
119	D₈×C₃₁	—	4	8	no	—
120	D₈×C₃₁	—	4	4	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	8	no	—
123	C₂×C₈×C₃₁	—	4	4	no	—
124	C₂×C₈×C₃₁	—	4	2	no	—
125	C₂×C₈×C₃₁	—	4	1	no	—
126	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
127	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
128	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
129	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
130	C₄⋊C₄×C₃₁	—	2	8	no	—
131	C₄⋊C₄×C₃₁	—	2	4	no	—
132	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
133	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
134	C₂²⋊C₄×C₃₁	—	2	8	no	—
135	C₂²⋊C₄×C₃₁	—	2	16	no	—
136	C₂²⋊C₄×C₃₁	—	2	8	no	—
137	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
138	C₄²×C₃₁	C₂×D₄	1	2	yes	—
139	C₄²×C₃₁	C₂⁴	1	2	yes	—
140	#62245	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	8	no	—
142	C₂³×C₃₁	—	4	2	no	—
143	C₂³×C₃₁	—	4	2	no	—
144	C₂³×C₃₁	—	4	16	no	—
145	C₂³×C₃₁	—	8	8	no	—
146	C₂³×C₃₁	—	8	8	no	—
147	C₂³×C₃₁	C₂×D₈	1	8	yes	complemented
148	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
149	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₃₁	—	4	8	no	—
151	D₄×C₃₁	—	2	8	no	—
152	D₄×C₃₁	—	2	8	no	—
153	D₄×C₃₁	—	4	32	no	—
154	D₄×C₃₁	—	4	16	no	—
155	D₄×C₃₁	—	4	8	no	—
156	D₄×C₃₁	—	8	4	no	—
157	D₄×C₃₁	—	8	8	no	—
158	D₄×C₃₁	C₂×D₈	1	16	yes	complemented
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	4	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	4	no	—
163	C₂×C₄×C₃₁	—	2	8	no	—
164	C₂×C₄×C₃₁	—	4	4	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	16	no	—
168	C₂×C₄×C₃₁	—	4	16	no	—
169	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	complemented
170	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
172	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
173	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
174	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
175	C₈×C₃₁	—	2	1	no	—
176	C₈×C₃₁	—	2	4	no	—
177	C₈×C₃₁	—	2	2	no	—
178	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	complemented
179	#3496	C₂×C₃₁	1	2	yes	complemented
180	#3334	C₂×C₃₁	1	4	yes	complemented
181	#3294	C₂×C₃₁	1	8	yes	complemented
182	D₄×D₈	C₂×C₃₁	1	4	yes	complemented
183	#3245	C₂×C₃₁	1	2	yes	complemented
184	#3186	C₂×C₃₁	1	4	yes	complemented
185	#3169	C₂×C₃₁	1	2	yes	complemented
186	C₂×#466	C₂×C₃₁	1	4	yes	complemented
187	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
188	C₂²×C₃₁	—	2	1	no	—
189	C₂²×C₃₁	—	2	4	no	—
190	C₂²×C₃₁	—	2	1	no	—
191	C₂²×C₃₁	—	2	4	no	—
192	C₂²×C₃₁	—	4	8	no	—
193	C₂²×C₃₁	—	4	8	no	—
194	C₂²×C₃₁	—	4	8	no	—
195	C₂²×C₃₁	—	4	8	no	—
196	C₂²×C₃₁	—	16	16	no	—
197	C₂²×C₃₁	C₂²×D₈	1	8	yes	—
198	C₂²×C₃₁	C₂²×D₈	1	2	yes	—
199	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
200	C₄×C₃₁	—	2	2	no	—
201	C₄×C₃₁	—	2	1	no	—
202	C₄×C₃₁	—	2	4	no	—
203	C₄×C₃₁	—	4	8	no	—
204	C₄×C₃₁	C₂²×D₈	1	4	yes	—
205	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
206	C₄×C₃₁	#585	1	1	yes	characteristic
207	C₂×H(5,𝔽₂)	—	2	2	no	—
208	C₂²×D₈	C₂²×C₃₁	1	4	yes	complemented
209	C₂³×C₈	C₂²×C₃₁	1	2	yes	complemented
210	#547	C₂²×C₃₁	1	8	yes	complemented
211	#546	—	2	8	no	—
212	D₄²	C₂²×C₃₁	1	8	yes	complemented
213	#537	—	2	2	no	—
214	#534	—	2	4	no	—
215	#532	C₂²×C₃₁	1	4	yes	complemented
216	C₂×#121	C₂²×C₃₁	1	8	yes	complemented
217	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
218	#500	C₂²×C₃₁	1	2	yes	complemented
219	#493	C₂²×C₃₁	1	2	yes	complemented
220	#486	C₂²×C₃₁	1	4	yes	complemented
221	#480	C₂²×C₃₁	1	16	yes	complemented
222	#474	C₂²×C₃₁	1	8	yes	complemented
223	#466	C₂²×C₃₁	1	8	yes	complemented
224	#466	C₂²×C₃₁	1	4	yes	complemented
225	#466	C₂²×C₃₁	1	4	yes	complemented
226	#466	C₂²×C₃₁	1	8	yes	complemented
227	#466	C₂²×C₃₁	1	8	yes	complemented
228	#459	C₂²×C₃₁	1	8	yes	complemented
229	#447	C₂²×C₃₁	1	16	yes	complemented
230	C₄×D₈	C₂²×C₃₁	1	4	yes	complemented
231	C₄×D₈	C₂²×C₃₁	1	4	yes	complemented
232	C₈×D₄	C₂²×C₃₁	1	4	yes	complemented
233	C₈×D₄	C₂²×C₃₁	1	4	yes	complemented
234	C₂×#107	C₂²×C₃₁	1	4	yes	complemented
235	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
236	C₂×#102	C₂²×C₃₁	1	8	yes	complemented
237	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	complemented
238	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
239	C₂×C₃₁	—	2	2	no	—
240	C₂×C₃₁	—	2	8	no	—
241	C₂×C₃₁	—	8	8	no	—
242	C₂×C₃₁	#3496	1	1	yes	characteristic
243	C₂×C₃₁	C₂³×D₈	1	1	yes	characteristic
244	C₂×C₃₁	#3584	1	1	yes	characteristic
245	H(5,𝔽₂)	—	4	16	no	—
246	C₂×#41	—	2	8	no	—
247	C₂×#41	—	2	4	no	—
248	C₂²×D₄	—	2	8	no	—
249	C₂²×D₄	—	2	8	no	—
250	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
251	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
252	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
253	C₂×D₈	—	4	8	no	—
254	C₂×D₈	C₂³×C₃₁	1	4	yes	—
255	C₂×D₈	C₂³×C₃₁	1	8	yes	—
256	C₂×D₈	C₂³×C₃₁	1	4	yes	—
257	C₂²×C₈	—	4	4	no	—
258	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
260	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
261	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
262	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
263	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
264	#127	C₂³×C₃₁	1	4	yes	—
265	#124	—	2	4	no	—
266	#123	—	2	16	no	—
267	C₂²⋊Q₈	—	2	8	no	—
268	#121	—	2	8	no	—
269	#121	—	2	8	no	—
270	#121	—	2	4	no	—
271	#121	—	2	8	no	—
272	#121	C₂³×C₃₁	1	8	yes	—
273	#121	C₂³×C₃₁	1	16	yes	—
274	#120	—	2	16	no	—
275	C₄×D₄	—	2	4	no	—
276	C₄×D₄	D₄×C₃₁	1	4	yes	—
277	C₄×D₄	C₂³×C₃₁	1	8	yes	—
278	C₄×D₄	C₂³×C₃₁	1	4	yes	—
279	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
281	C₂×C₂²⋊C₄	—	2	8	no	—
282	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
283	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
284	#107	C₂³×C₃₁	1	4	yes	—
285	#107	C₂³×C₃₁	1	8	yes	—
286	#107	C₂³×C₃₁	1	4	yes	—
287	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
288	#102	C₂³×C₃₁	1	8	yes	—
289	#102	C₂³×C₃₁	1	16	yes	—
290	#102	C₂³×C₃₁	1	8	yes	—
291	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
292	C₄×C₈	C₂³×C₃₁	1	2	yes	—
293	C₄×C₈	C₂³×C₃₁	1	2	yes	—
294	C₃₁	#62245	1	1	yes	characteristic, direct product factor, Sylow
295	C₂⁴	—	2	8	no	—
296	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
297	#41	—	4	32	no	—
298	#41	—	4	16	no	—
299	C₂×Q₈	—	2	4	no	—
300	C₂×D₄	—	2	16	no	—
301	C₂×D₄	—	2	8	no	—
302	C₂×D₄	—	2	4	no	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	8	no	—
305	C₂×D₄	—	4	8	no	—
306	C₂×D₄	—	4	8	no	—
307	C₂×D₄	—	4	32	no	—
308	C₂×D₄	—	4	32	no	—
309	C₂×D₄	D₈×C₃₁	1	8	yes	complemented
310	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
311	C₂×D₄	C₂⁴×C₃₁	1	8	yes	—
312	C₂²×C₄	—	2	8	no	—
313	C₂²×C₄	—	2	8	no	—
314	C₂²×C₄	—	4	2	no	—
315	C₂²×C₄	—	4	2	no	—
316	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
318	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
321	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
322	D₈	—	2	8	no	—
323	D₈	—	4	8	no	—
324	D₈	—	4	4	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	8	no	—
327	C₂×C₈	—	4	4	no	—
328	C₂×C₈	—	4	2	no	—
329	C₂×C₈	—	4	1	no	—
330	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
331	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
332	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
333	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
334	C₄⋊C₄	—	2	8	no	—
335	C₄⋊C₄	—	2	4	no	—
336	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
337	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
338	C₂²⋊C₄	—	2	8	no	—
339	C₂²⋊C₄	—	2	16	no	—
340	C₂²⋊C₄	—	2	8	no	—
341	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
342	C₄²	C₂×D₄×C₃₁	1	2	yes	—
343	C₄²	C₂⁴×C₃₁	1	2	yes	—
344	C₂³	—	2	8	no	—
345	C₂³	—	4	2	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	4	16	no	—
348	C₂³	—	8	8	no	—
349	C₂³	—	8	8	no	—
350	C₂³	C₂×D₈×C₃₁	1	8	yes	complemented
351	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
352	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
353	Q₈	—	4	8	no	—
354	D₄	—	2	8	no	—
355	D₄	—	2	8	no	—
356	D₄	—	4	32	no	—
357	D₄	—	4	16	no	—
358	D₄	—	4	8	no	—
359	D₄	—	8	4	no	—
360	D₄	—	8	8	no	—
361	D₄	C₂×D₈×C₃₁	1	16	yes	complemented
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	8	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	16	no	—
371	C₂×C₄	—	4	16	no	—
372	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	complemented
373	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
375	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
376	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
377	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	2	4	no	—
380	C₈	—	2	2	no	—
381	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	complemented
382	C₂²	—	2	1	no	—
383	C₂²	—	2	4	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	4	no	—
386	C₂²	—	4	8	no	—
387	C₂²	—	4	8	no	—
388	C₂²	—	4	8	no	—
389	C₂²	—	4	8	no	—
390	C₂²	—	16	16	no	—
391	C₂²	C₂²×D₈×C₃₁	1	8	yes	—
392	C₂²	C₂²×D₈×C₃₁	1	2	yes	—
393	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
394	C₄	—	2	1	no	—
395	C₄	—	2	4	no	—
396	C₄	—	2	2	no	—
397	C₄	—	4	8	no	—
398	C₄	C₂²×D₈×C₃₁	1	4	yes	—
399	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
400	C₄	C₃₁×#585	1	1	yes	characteristic
401	C₂	—	2	2	no	—
402	C₂	—	2	8	no	—
403	C₂	—	8	8	no	—
404	C₂	C₃₁×#3496	1	1	yes	characteristic
405	C₂	C₂³×D₈×C₃₁	1	1	yes	characteristic
406	C₂	C₃₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3496	C₂	semidirect
2. C₃₁×#3334	C₂	semidirect
3. C₃₁×#3294	C₂	semidirect
4. D₄×D₈×C₃₁	C₂	semidirect
5. C₃₁×#3245	C₂	semidirect
6. C₃₁×#3186	C₂	semidirect
7. C₃₁×#3169	C₂	semidirect
8. C₂×C₃₁×#466	C₂	semidirect
9. C₂×C₈×D₄×C₃₁	C₂	semidirect
11. C₂²×D₈×C₃₁	C₂²	semidirect
12. C₂³×C₈×C₃₁	C₂²	semidirect
13. C₃₁×#547	C₂²	semidirect
15. D₄²×C₃₁	C₂²	semidirect
18. C₃₁×#532	C₂²	semidirect
19. C₂×C₃₁×#121	C₂²	semidirect
21. C₃₁×#500	C₂²	semidirect
22. C₃₁×#493	C₂²	semidirect
23. C₃₁×#486	C₂²	semidirect
24. C₃₁×#480	C₂²	semidirect
25. C₃₁×#474	C₂²	semidirect
26. C₃₁×#466	C₂²	semidirect
27. C₃₁×#466	C₂²	semidirect
28. C₃₁×#466	C₂²	semidirect
29. C₃₁×#466	C₂²	semidirect
30. C₃₁×#466	C₂²	semidirect
31. C₃₁×#459	C₂²	semidirect
32. C₃₁×#447	C₂²	semidirect
33. C₄×D₈×C₃₁	C₂²	semidirect
34. C₄×D₈×C₃₁	C₂²	semidirect
35. C₈×D₄×C₃₁	C₂²	semidirect
36. C₈×D₄×C₃₁	C₂²	semidirect
37. C₂×C₃₁×#107	C₂²	semidirect
38. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
39. C₂×C₃₁×#102	C₂²	semidirect
40. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
41. C₂×C₄×C₈×C₃₁	C₂²	semidirect
105. C₂×D₄×C₃₁	D₈	semidirect
140. #62245	C₃₁	direct (N × H)
147. C₂³×C₃₁	C₂×D₈	semidirect
158. D₄×C₃₁	C₂×D₈	semidirect
169. C₂×C₄×C₃₁	C₂×D₈	semidirect
178. C₈×C₃₁	H(5,𝔽₂)	semidirect
179. #3496	C₂×C₃₁	semidirect
180. #3334	C₂×C₃₁	semidirect
181. #3294	C₂×C₃₁	semidirect
182. D₄×D₈	C₂×C₃₁	semidirect
183. #3245	C₂×C₃₁	semidirect
184. #3186	C₂×C₃₁	semidirect
185. #3169	C₂×C₃₁	semidirect
186. C₂×#466	C₂×C₃₁	semidirect
187. C₂×C₈×D₄	C₂×C₃₁	semidirect
208. C₂²×D₈	C₂²×C₃₁	semidirect
209. C₂³×C₈	C₂²×C₃₁	semidirect
210. #547	C₂²×C₃₁	semidirect
212. D₄²	C₂²×C₃₁	semidirect
215. #532	C₂²×C₃₁	semidirect
216. C₂×#121	C₂²×C₃₁	semidirect
218. #500	C₂²×C₃₁	semidirect
219. #493	C₂²×C₃₁	semidirect
220. #486	C₂²×C₃₁	semidirect
221. #480	C₂²×C₃₁	semidirect
222. #474	C₂²×C₃₁	semidirect
223. #466	C₂²×C₃₁	semidirect
224. #466	C₂²×C₃₁	semidirect
225. #466	C₂²×C₃₁	semidirect
226. #466	C₂²×C₃₁	semidirect
227. #466	C₂²×C₃₁	semidirect
228. #459	C₂²×C₃₁	semidirect
229. #447	C₂²×C₃₁	semidirect
230. C₄×D₈	C₂²×C₃₁	semidirect
231. C₄×D₈	C₂²×C₃₁	semidirect
232. C₈×D₄	C₂²×C₃₁	semidirect
233. C₈×D₄	C₂²×C₃₁	semidirect
234. C₂×#107	C₂²×C₃₁	semidirect
235. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
236. C₂×#102	C₂²×C₃₁	semidirect
237. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
238. C₂×C₄×C₈	C₂²×C₃₁	semidirect
294. C₃₁	#62245	direct (N × H)
309. C₂×D₄	D₈×C₃₁	semidirect
350. C₂³	C₂×D₈×C₃₁	semidirect
361. D₄	C₂×D₈×C₃₁	semidirect
372. C₂×C₄	C₂×D₈×C₃₁	semidirect
381. C₈	C₃₁×H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3496	2
C₃₁×#3334	2
C₃₁×#3294	2
D₄×D₈×C₃₁	2
C₃₁×#3245	2
C₃₁×#3186	2
C₃₁×#3169	2
C₂×C₃₁×#466	2
C₂×C₈×D₄×C₃₁	2
#62245	31

Subgroups — C31×#62245 #21519004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62245 #21519004