Subgroups — C₃₁×#29139 #21485898

Order 7936.

978 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2023	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1957	C₂	1	2	yes	maximal
5	C₃₁×#1834	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1802	C₂	1	4	yes	maximal, complemented
7	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₃₁	—	2	1	no	—
13	C₂×C₄×D₄×C₃₁	—	2	1	no	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#102	C₂²	1	8	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#390	—	2	1	no	—
19	C₃₁×#390	—	2	1	no	—
20	C₃₁×#390	C₂²	1	2	yes	—
21	C₃₁×#384	—	2	1	no	—
22	C₃₁×#384	—	2	1	no	—
23	C₃₁×#384	C₂²	1	2	yes	—
24	C₃₁×#384	C₂²	1	1	yes	characteristic
25	C₃₁×#384	C₂²	1	2	yes	—
26	C₄³×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#340	C₂²	1	8	yes	—
28	C₃₁×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₃₁	—	2	2	no	—
30	C₂²×D₄×C₃₁	—	2	1	no	—
31	C₂²×D₄×C₃₁	—	2	1	no	—
32	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
33	C₂³×C₄×C₃₁	—	2	2	no	—
34	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
35	C₃₁×C₄⋊Q₈	—	4	4	no	—
36	C₃₁×C₄⋊Q₈	—	4	4	no	—
37	C₃₁×C₄⋊Q₈	—	4	4	no	—
38	C₃₁×#127	—	2	2	no	—
39	C₃₁×#127	Q₈	1	4	yes	complemented
40	C₄×D₄×C₃₁	—	2	8	no	—
41	C₄×D₄×C₃₁	—	4	4	no	—
42	C₄×D₄×C₃₁	—	4	4	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
52	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×C₄⋊C₈	—	2	4	no	—
63	C₃₁×#102	—	4	8	no	—
64	C₄×C₈×C₃₁	—	2	4	no	—
65	C₂⁴×C₃₁	—	2	2	no	—
66	C₂×Q₈×C₃₁	—	4	4	no	—
67	C₂×Q₈×C₃₁	—	4	4	no	—
68	C₂×Q₈×C₃₁	—	4	4	no	—
69	C₂×D₄×C₃₁	—	2	4	no	—
70	C₂×D₄×C₃₁	—	2	4	no	—
71	C₂×D₄×C₃₁	—	2	4	no	—
72	C₂×D₄×C₃₁	—	2	2	no	—
73	C₂×D₄×C₃₁	—	2	4	no	—
74	C₂×D₄×C₃₁	—	4	2	no	—
75	C₂×D₄×C₃₁	—	4	2	no	—
76	C₂×D₄×C₃₁	—	4	2	no	—
77	C₂×D₄×C₃₁	—	4	2	no	—
78	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	2	4	no	—
81	C₂²×C₄×C₃₁	—	4	4	no	—
82	C₂²×C₄×C₃₁	—	4	4	no	—
83	C₂²×C₄×C₃₁	—	4	4	no	—
84	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₃₁	—	2	4	no	—
92	C₂×C₈×C₃₁	—	2	4	no	—
93	C₂×C₈×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	—	2	8	no	—
96	C₄⋊C₄×C₃₁	—	4	4	no	—
97	C₄⋊C₄×C₃₁	—	4	4	no	—
98	C₄⋊C₄×C₃₁	—	4	2	no	—
99	C₄⋊C₄×C₃₁	—	4	4	no	—
100	C₄⋊C₄×C₃₁	—	4	4	no	—
101	C₄⋊C₄×C₃₁	—	4	2	no	—
102	C₄⋊C₄×C₃₁	—	4	4	no	—
103	C₂²⋊C₄×C₃₁	—	4	4	no	—
104	C₂²⋊C₄×C₃₁	—	4	4	no	—
105	C₂²⋊C₄×C₃₁	—	4	4	no	—
106	C₄²×C₃₁	—	2	2	no	—
107	C₄²×C₃₁	—	2	2	no	—
108	C₄²×C₃₁	—	2	2	no	—
109	C₄²×C₃₁	D₈	1	4	yes	—
110	C₄²×C₃₁	SD₁₆	1	4	yes	—
111	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
112	C₄²×C₃₁	#41	1	2	yes	—
113	C₄²×C₃₁	#41	1	4	yes	—
114	#29139	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	—	2	4	no	—
116	C₂³×C₃₁	—	4	2	no	—
117	C₂³×C₃₁	—	4	2	no	—
118	C₂³×C₃₁	—	4	2	no	—
119	C₂³×C₃₁	—	4	2	no	—
120	C₂³×C₃₁	—	4	2	no	—
121	C₂³×C₃₁	—	4	2	no	—
122	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
123	Q₈×C₃₁	—	8	8	no	—
124	D₄×C₃₁	—	2	8	no	—
125	D₄×C₃₁	—	4	4	no	—
126	D₄×C₃₁	—	4	8	no	—
127	D₄×C₃₁	—	4	8	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	2	1	no	—
130	C₂×C₄×C₃₁	—	2	1	no	—
131	C₂×C₄×C₃₁	—	2	1	no	—
132	C₂×C₄×C₃₁	—	2	1	no	—
133	C₂×C₄×C₃₁	—	2	1	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	4	4	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	4	4	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	4	no	—
140	C₂×C₄×C₃₁	—	8	8	no	—
141	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
142	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₃₁	#124	1	2	yes	—
144	C₂×C₄×C₃₁	#124	1	2	yes	—
145	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
149	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
150	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
151	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
152	C₂×C₄×C₃₁	#137	1	2	yes	—
153	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
155	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
157	C₈×C₃₁	—	4	4	no	—
158	#2868	C₂×C₃₁	1	1	yes	characteristic, complemented
159	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
160	#2023	C₂×C₃₁	1	4	yes	complemented
161	#1957	C₂×C₃₁	1	2	yes	—
162	#1834	C₂×C₃₁	1	2	yes	complemented
163	#1802	C₂×C₃₁	1	4	yes	complemented
164	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
165	C₂²×C₃₁	—	4	4	no	—
166	C₂²×C₃₁	—	4	4	no	—
167	C₂²×C₃₁	—	4	4	no	—
168	C₂²×C₃₁	—	8	2	no	—
169	C₂²×C₃₁	—	8	2	no	—
170	C₂²×C₃₁	—	8	2	no	—
171	C₂²×C₃₁	—	8	2	no	—
172	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
173	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
174	C₂²×C₃₁	#535	1	1	yes	characteristic
175	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
176	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
177	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
179	C₄×C₃₁	—	2	2	no	—
180	C₄×C₃₁	—	2	2	no	—
181	C₄×C₃₁	—	2	2	no	—
182	C₄×C₃₁	—	8	2	no	—
183	C₄×C₃₁	—	8	4	no	—
184	C₄×C₃₁	#474	1	4	yes	—
185	C₄×C₃₁	#476	1	4	yes	—
186	C₄×C₃₁	#486	1	4	yes	—
187	C₄×C₃₁	#554	1	2	yes	—
188	C₄×C₃₁	#555	1	2	yes	—
189	C₂×C₄⋊Q₈	—	2	1	no	—
190	C₂×C₄⋊Q₈	—	2	1	no	—
191	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
192	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
193	C₂×C₄×D₄	—	2	1	no	—
194	C₂×C₄×D₄	—	2	1	no	—
195	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
196	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
197	C₂×#102	C₂²×C₃₁	1	8	yes	—
198	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
199	#390	—	2	1	no	—
200	#390	—	2	1	no	—
201	#390	C₂²×C₃₁	1	2	yes	—
202	#384	—	2	1	no	—
203	#384	—	2	1	no	—
204	#384	C₂²×C₃₁	1	1	yes	characteristic
205	#384	C₂²×C₃₁	1	2	yes	—
206	#384	C₂²×C₃₁	1	2	yes	—
207	C₄³	C₂²×C₃₁	1	1	yes	characteristic
208	#340	C₂²×C₃₁	1	8	yes	—
209	#336	C₂²×C₃₁	1	2	yes	—
210	C₂×C₃₁	—	8	4	no	—
211	C₂×C₃₁	#2563	1	1	yes	characteristic
212	C₂×C₃₁	C₂×#474	1	1	yes	characteristic
213	C₂×C₃₁	C₂×#476	1	1	yes	characteristic
214	C₂×C₃₁	C₂×#486	1	1	yes	characteristic
215	C₂×C₃₁	#3141	1	1	yes	characteristic
216	C₂×C₃₁	#3205	1	1	yes	characteristic
217	C₂×C₃₁	#3206	1	1	yes	characteristic
218	C₂²×Q₈	—	2	2	no	—
219	C₂²×D₄	—	2	1	no	—
220	C₂²×D₄	—	2	1	no	—
221	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
222	C₂³×C₄	—	2	2	no	—
223	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
224	C₄⋊Q₈	—	4	4	no	—
225	C₄⋊Q₈	—	4	4	no	—
226	C₄⋊Q₈	—	4	4	no	—
227	#127	—	2	2	no	—
228	#127	Q₈×C₃₁	1	4	yes	complemented
229	C₄×D₄	—	2	8	no	—
230	C₄×D₄	—	4	4	no	—
231	C₄×D₄	—	4	4	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	—	2	2	no	—
236	C₂×C₄⋊C₄	—	2	2	no	—
237	C₂×C₄⋊C₄	—	2	2	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
240	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
241	C₂×C₂²⋊C₄	—	2	2	no	—
242	C₂×C₂²⋊C₄	—	2	2	no	—
243	C₂×C₂²⋊C₄	—	2	2	no	—
244	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
246	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
249	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
250	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
251	C₄⋊C₈	—	2	4	no	—
252	#102	—	4	8	no	—
253	C₄×C₈	—	2	4	no	—
254	C₃₁	#29139	1	1	yes	characteristic, direct product factor, Sylow
255	C₂⁴	—	2	2	no	—
256	C₂×Q₈	—	4	4	no	—
257	C₂×Q₈	—	4	4	no	—
258	C₂×Q₈	—	4	4	no	—
259	C₂×D₄	—	2	4	no	—
260	C₂×D₄	—	2	4	no	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	4	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	4	4	no	—
272	C₂²×C₄	—	4	4	no	—
273	C₂²×C₄	—	4	4	no	—
274	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
280	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	2	4	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	2	8	no	—
286	C₄⋊C₄	—	4	4	no	—
287	C₄⋊C₄	—	4	4	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	4	no	—
290	C₄⋊C₄	—	4	4	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	4	4	no	—
293	C₂²⋊C₄	—	4	4	no	—
294	C₂²⋊C₄	—	4	4	no	—
295	C₂²⋊C₄	—	4	4	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	D₈×C₃₁	1	4	yes	—
300	C₄²	SD₁₆×C₃₁	1	4	yes	—
301	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
302	C₄²	#41×C₃₁	1	2	yes	—
303	C₄²	#41×C₃₁	1	4	yes	—
304	C₂³	—	2	4	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	4	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	4	2	no	—
310	C₂³	—	4	2	no	—
311	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
312	Q₈	—	8	8	no	—
313	D₄	—	2	8	no	—
314	D₄	—	4	8	no	—
315	D₄	—	4	8	no	—
316	D₄	—	4	4	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	4	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	8	8	no	—
330	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
331	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
332	C₂×C₄	C₃₁×#124	1	2	yes	—
333	C₂×C₄	C₃₁×#124	1	2	yes	—
334	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
335	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
336	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
337	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
338	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
339	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
340	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
341	C₂×C₄	C₃₁×#137	1	2	yes	—
342	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
343	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
344	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
345	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
346	C₈	—	4	4	no	—
347	C₂²	—	4	4	no	—
348	C₂²	—	4	4	no	—
349	C₂²	—	4	4	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	8	2	no	—
353	C₂²	—	8	2	no	—
354	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
355	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
356	C₂²	C₃₁×#535	1	1	yes	characteristic
357	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
358	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
359	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
360	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	2	2	no	—
364	C₄	—	8	2	no	—
365	C₄	—	8	4	no	—
366	C₄	C₃₁×#474	1	4	yes	—
367	C₄	C₃₁×#476	1	4	yes	—
368	C₄	C₃₁×#486	1	4	yes	—
369	C₄	C₃₁×#554	1	2	yes	—
370	C₄	C₃₁×#555	1	2	yes	—
371	C₂	—	8	4	no	—
372	C₂	C₃₁×#2563	1	1	yes	characteristic
373	C₂	C₂×C₃₁×#474	1	1	yes	characteristic
374	C₂	C₂×C₃₁×#476	1	1	yes	characteristic
375	C₂	C₂×C₃₁×#486	1	1	yes	characteristic
376	C₂	C₃₁×#3141	1	1	yes	characteristic
377	C₂	C₃₁×#3205	1	1	yes	characteristic
378	C₂	C₃₁×#3206	1	1	yes	characteristic

	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 ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2868	C₂	semidirect
3. C₃₁×#2023	C₂	semidirect
5. C₃₁×#1834	C₂	semidirect
6. C₃₁×#1802	C₂	semidirect
7. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
39. C₃₁×#127	Q₈	semidirect
114. #29139	C₃₁	direct (N × H)
158. #2868	C₂×C₃₁	semidirect
160. #2023	C₂×C₃₁	semidirect
162. #1834	C₂×C₃₁	semidirect
163. #1802	C₂×C₃₁	semidirect
164. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
228. #127	Q₈×C₃₁	semidirect
254. C₃₁	#29139	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×#2868	2
C₄×C₃₁×#127	2
C₃₁×#2023	2
C₃₁×#1957	2
C₃₁×#1834	2
C₃₁×#1802	2
C₄×C₃₁×C₄⋊C₈	2
#29139	31

Subgroups — C31×#29139 #21485898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29139 #21485898