Subgroups — C₃₁×#32628 #21489387

Order 7936.

772 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2548	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2546	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2041	C₂	1	2	yes	maximal
5	C₃₁×#2023	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1934	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1918	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#124	—	2	1	no	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#102	C₂²	1	2	yes	—
18	C₂×C₃₁×#102	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#398	—	2	1	no	—
23	C₃₁×#398	—	2	1	no	—
24	C₃₁×#394	—	2	1	no	—
25	C₃₁×#394	—	2	1	no	—
26	C₃₁×#391	—	2	1	no	—
27	C₃₁×#390	—	2	1	no	—
28	C₃₁×#390	C₂²	1	2	yes	—
29	C₃₁×#390	C₂²	1	2	yes	—
30	C₃₁×#389	C₂²	1	2	yes	—
31	C₃₁×#389	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₂²×Q₈×C₃₁	—	2	1	no	—
40	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
41	C₂²×D₄×C₃₁	—	2	1	no	—
42	C₂²×D₄×C₃₁	—	2	1	no	—
43	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₃₁×C₄⋊Q₈	—	4	1	no	—
47	C₃₁×C₄⋊Q₈	—	4	1	no	—
48	C₃₁×C₄⋊Q₈	—	4	4	no	—
49	C₃₁×#127	—	4	1	no	—
50	C₃₁×#127	—	4	1	no	—
51	C₃₁×#124	—	4	4	no	—
52	C₄×Q₈×C₃₁	—	4	1	no	—
53	C₄×Q₈×C₃₁	—	4	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	—	2	1	no	—
68	C₂×C₄²×C₃₁	—	2	1	no	—
69	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×#107	—	2	4	no	—
73	C₃₁×C₄⋊C₈	—	4	1	no	—
74	C₃₁×#102	—	2	4	no	—
75	C₃₁×#102	—	2	4	no	—
76	C₃₁×#102	—	4	2	no	—
77	C₃₁×#102	—	4	2	no	—
78	C₄×C₈×C₃₁	—	4	1	no	—
79	C₃₁×#95	—	2	1	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₃₁×#95	—	2	1	no	—
83	C₂⁴×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	4	1	no	—
85	C₂×Q₈×C₃₁	—	4	1	no	—
86	C₂×Q₈×C₃₁	—	4	2	no	—
87	C₂×Q₈×C₃₁	—	4	2	no	—
88	C₂×Q₈×C₃₁	—	4	2	no	—
89	C₂×Q₈×C₃₁	D₈	1	4	yes	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	4	2	no	—
97	C₂×D₄×C₃₁	—	4	2	no	—
98	C₂×D₄×C₃₁	—	4	2	no	—
99	C₂×D₄×C₃₁	—	4	2	no	—
100	C₂×D₄×C₃₁	—	8	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	4	1	no	—
123	C₄⋊C₄×C₃₁	—	4	1	no	—
124	C₄⋊C₄×C₃₁	—	4	1	no	—
125	C₄⋊C₄×C₃₁	—	4	2	no	—
126	C₄⋊C₄×C₃₁	—	4	4	no	—
127	C₄⋊C₄×C₃₁	—	4	4	no	—
128	C₄⋊C₄×C₃₁	#41	1	4	yes	—
129	C₂²⋊C₄×C₃₁	—	4	4	no	—
130	C₂²⋊C₄×C₃₁	—	4	4	no	—
131	C₂²⋊C₄×C₃₁	—	8	2	no	—
132	C₂²⋊C₄×C₃₁	—	8	2	no	—
133	C₄²×C₃₁	—	2	1	no	—
134	C₄²×C₃₁	—	2	1	no	—
135	C₄²×C₃₁	—	2	1	no	—
136	C₄²×C₃₁	—	2	1	no	—
137	C₄²×C₃₁	—	2	1	no	—
138	C₄²×C₃₁	—	2	1	no	—
139	C₄²×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	—	4	2	no	—
141	#32628	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂³×C₃₁	—	4	2	no	—
143	C₂³×C₃₁	—	4	2	no	—
144	C₂³×C₃₁	—	4	2	no	—
145	C₂³×C₃₁	—	4	2	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	C₂³×C₃₁	—	4	2	no	—
148	C₂³×C₃₁	—	4	2	no	—
149	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₃₁	—	4	4	no	—
151	Q₈×C₃₁	—	8	4	no	—
152	D₄×C₃₁	—	4	2	no	—
153	D₄×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	4	2	no	—
155	D₄×C₃₁	—	4	2	no	—
156	D₄×C₃₁	—	8	4	no	—
157	D₄×C₃₁	—	8	4	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	4	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
180	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
181	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
182	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
183	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
186	C₈×C₃₁	—	4	2	no	—
187	C₈×C₃₁	—	4	2	no	—
188	#2548	C₂×C₃₁	1	1	yes	characteristic
189	#2546	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#2067	C₂×C₃₁	1	2	yes	complemented
191	#2041	C₂×C₃₁	1	2	yes	—
192	#2023	C₂×C₃₁	1	2	yes	complemented
193	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1934	C₂×C₃₁	1	2	yes	complemented
195	#1925	C₂×C₃₁	1	2	yes	complemented
196	#1918	C₂×C₃₁	1	2	yes	complemented
197	C₂²×C₃₁	—	8	2	no	—
198	C₂²×C₃₁	—	8	2	no	—
199	C₂²×C₃₁	—	8	2	no	—
200	C₂²×C₃₁	—	8	2	no	—
201	C₂²×C₃₁	—	8	2	no	—
202	C₂²×C₃₁	—	8	2	no	—
203	C₂²×C₃₁	—	8	2	no	—
204	C₂²×C₃₁	#534	1	1	yes	characteristic
205	C₂²×C₃₁	#548	1	2	yes	—
206	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
207	C₂²×C₃₁	#575	1	1	yes	characteristic
208	C₂²×C₃₁	#577	1	2	yes	—
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	—	2	2	no	—
211	C₄×C₃₁	—	2	2	no	—
212	C₄×C₃₁	—	4	4	no	—
213	C₄×C₃₁	—	4	1	no	—
214	C₄×C₃₁	—	4	1	no	—
215	C₄×C₃₁	—	4	1	no	—
216	C₄×C₃₁	—	8	2	no	—
217	C₄×C₃₁	—	8	2	no	—
218	C₄×C₃₁	#542	1	2	yes	—
219	C₄×C₃₁	#555	1	1	yes	characteristic
220	C₄×C₃₁	#563	1	1	yes	characteristic
221	C₂×C₄⋊Q₈	—	2	1	no	—
222	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×#124	—	2	1	no	—
225	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×#107	C₂²×C₃₁	1	2	yes	—
227	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×#102	C₂²×C₃₁	1	2	yes	—
229	C₂×#102	C₂²×C₃₁	1	2	yes	—
230	C₂×#102	C₂²×C₃₁	1	2	yes	—
231	C₂×#102	C₂²×C₃₁	1	2	yes	—
232	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
233	#398	—	2	1	no	—
234	#398	—	2	1	no	—
235	#394	—	2	1	no	—
236	#394	—	2	1	no	—
237	#391	—	2	1	no	—
238	#390	—	2	1	no	—
239	#390	C₂²×C₃₁	1	2	yes	—
240	#390	C₂²×C₃₁	1	2	yes	—
241	#389	C₂²×C₃₁	1	2	yes	—
242	#389	C₂²×C₃₁	1	2	yes	—
243	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
244	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
245	#340	C₂²×C₃₁	1	2	yes	—
246	#340	C₂²×C₃₁	1	2	yes	—
247	#340	C₂²×C₃₁	1	2	yes	—
248	#336	C₂²×C₃₁	1	2	yes	—
249	#336	C₂²×C₃₁	1	2	yes	—
250	C₂×C₃₁	—	16	2	no	—
251	C₂×C₃₁	#2731	1	1	yes	characteristic
252	C₂×C₃₁	#3186	1	1	yes	characteristic
253	C₂×C₃₁	#3200	1	1	yes	characteristic
254	C₂×C₃₁	#3358	1	2	yes	—
255	C₂×C₃₁	#3371	1	2	yes	—
256	C₂²×Q₈	—	2	1	no	—
257	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
258	C₂²×D₄	—	2	1	no	—
259	C₂²×D₄	—	2	1	no	—
260	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
261	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
262	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
263	C₄⋊Q₈	—	4	1	no	—
264	C₄⋊Q₈	—	4	1	no	—
265	C₄⋊Q₈	—	4	4	no	—
266	#127	—	4	1	no	—
267	#127	—	4	1	no	—
268	#124	—	4	4	no	—
269	C₄×Q₈	—	4	1	no	—
270	C₄×Q₈	—	4	1	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	2	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
277	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
278	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₂²⋊C₄	—	2	2	no	—
282	C₂×C₂²⋊C₄	—	2	2	no	—
283	C₂×C₂²⋊C₄	—	2	2	no	—
284	C₂×C₄²	—	2	1	no	—
285	C₂×C₄²	—	2	1	no	—
286	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
287	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
289	#107	—	2	4	no	—
290	C₄⋊C₈	—	4	1	no	—
291	#102	—	2	4	no	—
292	#102	—	2	4	no	—
293	#102	—	4	2	no	—
294	#102	—	4	2	no	—
295	C₄×C₈	—	4	1	no	—
296	#95	—	2	1	no	—
297	#95	—	2	1	no	—
298	#95	—	2	2	no	—
299	#95	—	2	2	no	—
300	C₃₁	#32628	1	1	yes	characteristic, direct product factor, Sylow
301	C₂⁴	—	2	2	no	—
302	C₂×Q₈	—	4	1	no	—
303	C₂×Q₈	—	4	1	no	—
304	C₂×Q₈	—	4	2	no	—
305	C₂×Q₈	—	4	2	no	—
306	C₂×Q₈	—	4	2	no	—
307	C₂×Q₈	D₈×C₃₁	1	4	yes	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	4	2	no	—
315	C₂×D₄	—	4	2	no	—
316	C₂×D₄	—	4	2	no	—
317	C₂×D₄	—	4	2	no	—
318	C₂×D₄	—	8	2	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₄⋊C₄	—	4	1	no	—
342	C₄⋊C₄	—	4	1	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	—	4	4	no	—
345	C₄⋊C₄	—	4	4	no	—
346	C₄⋊C₄	#41×C₃₁	1	4	yes	—
347	C₂²⋊C₄	—	4	4	no	—
348	C₂²⋊C₄	—	4	4	no	—
349	C₂²⋊C₄	—	8	2	no	—
350	C₂²⋊C₄	—	8	2	no	—
351	C₄²	—	2	1	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	2	1	no	—
357	C₄²	—	4	2	no	—
358	C₄²	—	4	2	no	—
359	C₂³	—	4	2	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	4	2	no	—
362	C₂³	—	4	2	no	—
363	C₂³	—	4	2	no	—
364	C₂³	—	4	2	no	—
365	C₂³	—	4	2	no	—
366	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
367	Q₈	—	4	4	no	—
368	Q₈	—	8	4	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	8	4	no	—
374	D₄	—	8	4	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
397	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
398	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
399	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
400	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
401	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
402	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
403	C₈	—	4	2	no	—
404	C₈	—	4	2	no	—
405	C₂²	—	8	2	no	—
406	C₂²	—	8	2	no	—
407	C₂²	—	8	2	no	—
408	C₂²	—	8	2	no	—
409	C₂²	—	8	2	no	—
410	C₂²	—	8	2	no	—
411	C₂²	—	8	2	no	—
412	C₂²	C₃₁×#534	1	1	yes	characteristic
413	C₂²	C₃₁×#548	1	2	yes	—
414	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
415	C₂²	C₃₁×#575	1	1	yes	characteristic
416	C₂²	C₃₁×#577	1	2	yes	—
417	C₄	—	2	2	no	—
418	C₄	—	2	2	no	—
419	C₄	—	2	2	no	—
420	C₄	—	4	4	no	—
421	C₄	—	4	1	no	—
422	C₄	—	4	1	no	—
423	C₄	—	4	1	no	—
424	C₄	—	8	2	no	—
425	C₄	—	8	2	no	—
426	C₄	C₃₁×#542	1	2	yes	—
427	C₄	C₃₁×#555	1	1	yes	characteristic
428	C₄	C₃₁×#563	1	1	yes	characteristic
429	C₂	—	16	2	no	—
430	C₂	C₃₁×#2731	1	1	yes	characteristic
431	C₂	C₃₁×#3186	1	1	yes	characteristic
432	C₂	C₃₁×#3200	1	1	yes	characteristic
433	C₂	C₃₁×#3358	1	2	yes	—
434	C₂	C₃₁×#3371	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2546	C₂	semidirect
3. C₃₁×#2067	C₂	semidirect
5. C₃₁×#2023	C₂	semidirect
6. C₃₁×#1984	C₂	semidirect
7. C₃₁×#1934	C₂	semidirect
8. C₃₁×#1925	C₂	semidirect
9. C₃₁×#1918	C₂	semidirect
141. #32628	C₃₁	direct (N × H)
189. #2546	C₂×C₃₁	semidirect
190. #2067	C₂×C₃₁	semidirect
192. #2023	C₂×C₃₁	semidirect
193. #1984	C₂×C₃₁	semidirect
194. #1934	C₂×C₃₁	semidirect
195. #1925	C₂×C₃₁	semidirect
196. #1918	C₂×C₃₁	semidirect
300. C₃₁	#32628	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×#2548	2
C₃₁×#2546	2
C₃₁×#2067	2
C₃₁×#2041	2
C₃₁×#2023	2
C₃₁×#1984	2
C₃₁×#1934	2
C₃₁×#1925	2
C₃₁×#1918	2
#32628	31

Subgroups — C31×#32628 #21489387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32628 #21489387