Subgroups — C₃₁×#29356 #21486115

Order 7936.

798 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2858	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2073	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2025	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1988	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1982	C₂	1	2	yes	maximal
7	C₃₁×#1936	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#124	—	2	2	no	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×#102	C₂²	1	4	yes	—
15	C₂×C₃₁×#102	C₂²	1	4	yes	—
16	C₂×C₃₁×#97	C₂²	1	2	yes	—
17	C₃₁×#390	—	2	2	no	—
18	C₃₁×#390	C₂²	1	2	yes	—
19	C₃₁×#390	C₂²	1	2	yes	—
20	C₃₁×#389	C₂²	1	2	yes	—
21	C₃₁×#387	—	2	2	no	—
22	C₃₁×#382	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
24	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₄³×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#340	C₂²	1	4	yes	—
27	C₃₁×#340	C₂²	1	4	yes	—
28	C₃₁×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₃₁	—	2	1	no	—
30	C₂²×D₄×C₃₁	—	2	1	no	—
31	C₂²×D₄×C₃₁	—	2	1	no	—
32	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
34	C₃₁×C₄⋊Q₈	—	2	4	no	—
35	C₃₁×#127	—	2	4	no	—
36	C₃₁×#125	—	2	4	no	—
37	C₃₁×#124	—	4	8	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
45	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
46	C₂×C₄²×C₃₁	—	2	2	no	—
47	C₂×C₄²×C₃₁	D₄	1	2	yes	—
48	C₂×C₄²×C₃₁	D₄	1	2	yes	—
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×C₄⋊C₈	—	2	4	no	—
53	C₃₁×#102	—	2	8	no	—
54	C₃₁×#102	—	4	4	no	—
55	C₃₁×#97	—	4	2	no	—
56	C₃₁×#95	—	2	2	no	—
57	C₃₁×#95	—	2	2	no	—
58	C₃₁×#95	—	2	2	no	—
59	C₂⁴×C₃₁	—	2	2	no	—
60	C₂×Q₈×C₃₁	—	4	2	no	—
61	C₂×Q₈×C₃₁	—	4	4	no	—
62	C₂×D₄×C₃₁	—	2	4	no	—
63	C₂×D₄×C₃₁	—	2	4	no	—
64	C₂×D₄×C₃₁	—	2	2	no	—
65	C₂×D₄×C₃₁	—	2	2	no	—
66	C₂×D₄×C₃₁	—	4	4	no	—
67	C₂×D₄×C₃₁	—	4	2	no	—
68	C₂×D₄×C₃₁	—	4	2	no	—
69	C₂×D₄×C₃₁	—	4	4	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	2	2	no	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₄⋊C₄×C₃₁	—	2	8	no	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	4	4	no	—
87	C₄⋊C₄×C₃₁	—	4	2	no	—
88	C₄⋊C₄×C₃₁	#41	1	4	yes	—
89	C₂²⋊C₄×C₃₁	—	4	8	no	—
90	C₂²⋊C₄×C₃₁	—	4	8	no	—
91	C₄²×C₃₁	—	2	4	no	—
92	C₄²×C₃₁	—	2	2	no	—
93	C₄²×C₃₁	—	2	1	no	—
94	C₄²×C₃₁	—	2	1	no	—
95	C₄²×C₃₁	—	4	2	no	—
96	C₄²×C₃₁	—	4	4	no	—
97	C₄²×C₃₁	#41	1	4	yes	—
98	#29356	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₃₁	—	4	2	no	—
100	C₂³×C₃₁	—	4	2	no	—
101	C₂³×C₃₁	—	4	2	no	—
102	C₂³×C₃₁	—	4	4	no	—
103	C₂³×C₃₁	—	4	4	no	—
104	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₃₁	—	8	4	no	—
106	D₄×C₃₁	—	4	4	no	—
107	D₄×C₃₁	—	4	4	no	—
108	D₄×C₃₁	—	8	4	no	—
109	D₄×C₃₁	—	8	4	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	4	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	4	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	#124	1	2	yes	—
126	C₂×C₄×C₃₁	#124	1	2	yes	—
127	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
128	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
129	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
130	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
132	C₈×C₃₁	—	4	4	no	—
133	#2858	C₂×C₃₁	1	1	yes	characteristic
134	C₄×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
135	#2073	C₂×C₃₁	1	2	yes	complemented
136	#2025	C₂×C₃₁	1	2	yes	complemented
137	#1988	C₂×C₃₁	1	2	yes	complemented
138	#1982	C₂×C₃₁	1	2	yes	—
139	#1936	C₂×C₃₁	1	4	yes	complemented
140	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
141	C₂²×C₃₁	—	8	2	no	—
142	C₂²×C₃₁	—	8	4	no	—
143	C₂²×C₃₁	—	8	2	no	—
144	C₂²×C₃₁	—	8	2	no	—
145	C₂²×C₃₁	—	8	4	no	—
146	C₂²×C₃₁	#525	1	1	yes	characteristic
147	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
148	C₂²×C₃₁	#537	1	1	yes	characteristic
149	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
150	C₂²×C₃₁	#575	1	2	yes	—
151	C₄×C₃₁	—	2	2	no	—
152	C₄×C₃₁	—	4	2	no	—
153	C₄×C₃₁	—	4	4	no	—
154	C₄×C₃₁	—	4	2	no	—
155	C₄×C₃₁	—	8	2	no	—
156	C₄×C₃₁	—	8	2	no	—
157	C₄×C₃₁	#490	1	4	yes	—
158	C₄×C₃₁	#556	1	2	yes	—
159	C₄×C₃₁	#559	1	2	yes	—
160	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×#124	—	2	2	no	—
164	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
165	C₂×#102	C₂²×C₃₁	1	4	yes	—
166	C₂×#102	C₂²×C₃₁	1	4	yes	—
167	C₂×#97	C₂²×C₃₁	1	2	yes	—
168	#390	—	2	2	no	—
169	#390	C₂²×C₃₁	1	2	yes	—
170	#390	C₂²×C₃₁	1	2	yes	—
171	#389	C₂²×C₃₁	1	2	yes	—
172	#387	—	2	2	no	—
173	#382	C₂²×C₃₁	1	1	yes	characteristic
174	C₄×C₄⋊C₄	—	2	2	no	—
175	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
176	C₄³	C₂²×C₃₁	1	1	yes	characteristic
177	#340	C₂²×C₃₁	1	4	yes	—
178	#340	C₂²×C₃₁	1	4	yes	—
179	#336	C₂²×C₃₁	1	2	yes	—
180	C₂×C₃₁	—	16	2	no	—
181	C₂×C₃₁	#2592	1	1	yes	characteristic
182	C₂×C₃₁	#3115	1	2	yes	—
183	C₂×C₃₁	C₂×#490	1	1	yes	characteristic
184	C₂×C₃₁	#3140	1	1	yes	characteristic
185	C₂×C₃₁	#3251	1	2	yes	—
186	C₂²×Q₈	—	2	1	no	—
187	C₂²×D₄	—	2	1	no	—
188	C₂²×D₄	—	2	1	no	—
189	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
190	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
191	C₄⋊Q₈	—	2	4	no	—
192	#127	—	2	4	no	—
193	#125	—	2	4	no	—
194	#124	—	4	8	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
201	C₂×C₂²⋊C₄	—	2	4	no	—
202	C₂×C₂²⋊C₄	—	2	4	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	D₄×C₃₁	1	2	yes	—
205	C₂×C₄²	D₄×C₃₁	1	2	yes	—
206	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
209	C₄⋊C₈	—	2	4	no	—
210	#102	—	2	8	no	—
211	#102	—	4	4	no	—
212	#97	—	4	2	no	—
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	C₃₁	#29356	1	1	yes	characteristic, direct product factor, Sylow
217	C₂⁴	—	2	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	4	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	#41×C₃₁	1	4	yes	—
247	C₂²⋊C₄	—	4	8	no	—
248	C₂²⋊C₄	—	4	8	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	4	4	no	—
255	C₄²	#41×C₃₁	1	4	yes	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	4	4	no	—
261	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
262	Q₈	—	8	4	no	—
263	D₄	—	4	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	8	4	no	—
266	D₄	—	8	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₃₁×#124	1	2	yes	—
283	C₂×C₄	C₃₁×#124	1	2	yes	—
284	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
285	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
286	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
287	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
288	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
289	C₈	—	4	4	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	8	4	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	4	no	—
295	C₂²	C₃₁×#525	1	1	yes	characteristic
296	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
297	C₂²	C₃₁×#537	1	1	yes	characteristic
298	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
299	C₂²	C₃₁×#575	1	2	yes	—
300	C₄	—	2	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	4	no	—
303	C₄	—	4	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	C₃₁×#490	1	4	yes	—
307	C₄	C₃₁×#556	1	2	yes	—
308	C₄	C₃₁×#559	1	2	yes	—
309	C₂	—	16	2	no	—
310	C₂	C₃₁×#2592	1	1	yes	characteristic
311	C₂	C₃₁×#3115	1	2	yes	—
312	C₂	C₂×C₃₁×#490	1	1	yes	characteristic
313	C₂	C₃₁×#3140	1	1	yes	characteristic
314	C₂	C₃₁×#3251	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 ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₃₁×#125	C₂	semidirect
3. C₃₁×#2073	C₂	semidirect
4. C₃₁×#2025	C₂	semidirect
5. C₃₁×#1988	C₂	semidirect
7. C₃₁×#1936	C₂	semidirect
8. C₃₁×#1767	C₂	semidirect
98. #29356	C₃₁	direct (N × H)
134. C₄×#125	C₂×C₃₁	semidirect
135. #2073	C₂×C₃₁	semidirect
136. #2025	C₂×C₃₁	semidirect
137. #1988	C₂×C₃₁	semidirect
139. #1936	C₂×C₃₁	semidirect
140. #1767	C₂×C₃₁	semidirect
216. C₃₁	#29356	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×#2858	2
C₄×C₃₁×#125	2
C₃₁×#2073	2
C₃₁×#2025	2
C₃₁×#1988	2
C₃₁×#1982	2
C₃₁×#1936	2
C₃₁×#1767	2
#29356	31

Subgroups — C31×#29356 #21486115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29356 #21486115