Subgroups — C₃₁×#2697 #16842630

Order 3968.

680 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#120	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#397	C₂	1	2	yes	maximal, complemented
4	C₃₁×#396	C₂	1	2	yes	maximal, complemented
5	C₃₁×#394	C₂	1	2	yes	maximal, complemented
6	C₃₁×#390	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#388	C₂	1	2	yes	maximal, complemented
8	C₃₁×#386	C₂	1	2	yes	maximal, complemented
9	C₃₁×#379	C₂	1	2	yes	maximal, complemented
10	C₂⁵×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
12	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#123	—	2	4	no	—
15	C₃₁×#120	—	2	4	no	—
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
19	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
29	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#95	C₂²	1	2	yes	complemented
31	C₃₁×#95	C₂²	1	2	yes	complemented
32	C₃₁×#95	C₂²	1	2	yes	complemented
33	C₂⁴×C₃₁	—	2	2	no	—
34	C₂⁴×C₃₁	—	2	4	no	—
35	C₂⁴×C₃₁	—	4	2	no	—
36	C₂⁴×C₃₁	—	4	1	no	—
37	C₂⁴×C₃₁	—	4	1	no	—
38	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
39	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
40	C₂⁴×C₃₁	C₂³	1	2	yes	—
41	C₂×D₄×C₃₁	—	2	4	no	—
42	C₂×D₄×C₃₁	—	2	4	no	—
43	C₂×D₄×C₃₁	—	4	1	no	—
44	C₂×D₄×C₃₁	—	4	1	no	—
45	C₂×D₄×C₃₁	—	4	2	no	—
46	C₂×D₄×C₃₁	D₄	1	4	yes	—
47	C₂²×C₄×C₃₁	—	2	2	no	—
48	C₂²×C₄×C₃₁	—	4	1	no	—
49	C₂²×C₄×C₃₁	—	4	1	no	—
50	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₃₁	—	2	4	no	—
57	C₄⋊C₄×C₃₁	—	4	1	no	—
58	C₄⋊C₄×C₃₁	—	4	2	no	—
59	C₂²⋊C₄×C₃₁	—	2	2	no	—
60	C₂²⋊C₄×C₃₁	—	2	4	no	—
61	C₂²⋊C₄×C₃₁	—	2	2	no	—
62	C₂²⋊C₄×C₃₁	—	2	4	no	—
63	C₂²⋊C₄×C₃₁	—	2	4	no	—
64	C₂²⋊C₄×C₃₁	—	4	2	no	—
65	C₂²⋊C₄×C₃₁	—	4	2	no	—
66	C₂²⋊C₄×C₃₁	—	4	2	no	—
67	C₂²⋊C₄×C₃₁	—	4	2	no	—
68	C₂²⋊C₄×C₃₁	—	4	2	no	—
69	C₄²×C₃₁	—	4	1	no	—
70	C₂³×C₃₁	—	2	2	no	—
71	C₂³×C₃₁	—	2	2	no	—
72	C₂³×C₃₁	—	2	2	no	—
73	C₂³×C₃₁	—	2	2	no	—
74	C₂³×C₃₁	—	2	2	no	—
75	C₂³×C₃₁	—	2	2	no	—
76	C₂³×C₃₁	—	2	2	no	—
77	C₂³×C₃₁	—	2	2	no	—
78	C₂³×C₃₁	—	2	2	no	—
79	C₂³×C₃₁	—	2	1	no	—
80	C₂³×C₃₁	—	2	1	no	—
81	C₂³×C₃₁	—	2	1	no	—
82	C₂³×C₃₁	—	2	1	no	—
83	C₂³×C₃₁	—	2	2	no	—
84	C₂³×C₃₁	—	4	8	no	—
85	C₂³×C₃₁	—	4	4	no	—
86	C₂³×C₃₁	—	4	4	no	—
87	C₂³×C₃₁	—	4	8	no	—
88	C₂³×C₃₁	—	4	4	no	—
89	C₂³×C₃₁	C₂×D₄	1	2	yes	—
90	C₂³×C₃₁	C₂×D₄	1	2	yes	—
91	C₂³×C₃₁	#41	1	4	yes	—
92	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
93	D₄×C₃₁	—	4	4	no	—
94	D₄×C₃₁	—	4	8	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
114	#2697	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₃₁	—	2	2	no	—
116	C₂²×C₃₁	—	2	2	no	—
117	C₂²×C₃₁	—	2	4	no	—
118	C₂²×C₃₁	—	2	4	no	—
119	C₂²×C₃₁	—	2	4	no	—
120	C₂²×C₃₁	—	2	4	no	—
121	C₂²×C₃₁	—	2	4	no	—
122	C₂²×C₃₁	—	4	16	no	—
123	C₂²×C₃₁	—	4	2	no	—
124	C₂²×C₃₁	—	4	2	no	—
125	C₂²×C₃₁	—	4	2	no	—
126	C₂²×C₃₁	—	4	2	no	—
127	C₂²×C₃₁	—	4	2	no	—
128	C₂²×C₃₁	—	4	1	no	—
129	C₂²×C₃₁	—	4	1	no	—
130	C₂²×C₃₁	—	4	1	no	—
131	C₂²×C₃₁	—	4	1	no	—
132	C₂²×C₃₁	—	4	2	no	—
133	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
134	C₂²×C₃₁	C₂×#41	1	2	yes	—
135	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
138	C₄×C₃₁	—	2	2	no	—
139	C₄×C₃₁	—	4	2	no	—
140	C₄×C₃₁	—	4	2	no	—
141	C₄×C₃₁	—	4	2	no	—
142	C₄×C₃₁	—	4	2	no	—
143	C₄×C₃₁	—	4	2	no	—
144	C₂×#123	C₂×C₃₁	1	1	yes	characteristic, complemented
145	C₂×#120	C₂×C₃₁	1	1	yes	characteristic
146	#397	C₂×C₃₁	1	2	yes	complemented
147	#396	C₂×C₃₁	1	2	yes	complemented
148	#394	C₂×C₃₁	1	2	yes	complemented
149	#390	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#388	C₂×C₃₁	1	2	yes	complemented
151	#386	C₂×C₃₁	1	2	yes	complemented
152	#379	C₂×C₃₁	1	2	yes	complemented
153	C₂×C₃₁	—	4	2	no	—
154	C₂×C₃₁	—	4	2	no	—
155	C₂×C₃₁	—	4	4	no	—
156	C₂×C₃₁	#534	1	1	yes	characteristic
157	C₂×C₃₁	#538	1	2	yes	—
158	C₂×C₃₁	#546	1	2	yes	—
159	C₂×C₃₁	#551	1	1	yes	characteristic
160	C₂×C₃₁	#560	1	1	yes	characteristic
161	C₂⁵	C₂²×C₃₁	1	1	yes	characteristic
162	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
163	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
164	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
165	#123	—	2	4	no	—
166	#120	—	2	4	no	—
167	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
168	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
169	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
170	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
171	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
177	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
178	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
179	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
180	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic, complemented
181	#95	C₂²×C₃₁	1	2	yes	complemented
182	#95	C₂²×C₃₁	1	2	yes	complemented
183	#95	C₂²×C₃₁	1	2	yes	complemented
184	C₃₁	#2697	1	1	yes	characteristic, direct product factor, Sylow
185	C₂⁴	—	2	2	no	—
186	C₂⁴	—	2	4	no	—
187	C₂⁴	—	4	2	no	—
188	C₂⁴	—	4	1	no	—
189	C₂⁴	—	4	1	no	—
190	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
191	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
192	C₂⁴	C₂³×C₃₁	1	2	yes	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	D₄×C₃₁	1	4	yes	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
203	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
204	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
205	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
206	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
207	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	—	4	1	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	8	no	—
237	C₂³	—	4	4	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	4	8	no	—
240	C₂³	—	4	4	no	—
241	C₂³	C₂×D₄×C₃₁	1	2	yes	—
242	C₂³	C₂×D₄×C₃₁	1	2	yes	—
243	C₂³	#41×C₃₁	1	4	yes	—
244	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
245	D₄	—	4	4	no	—
246	D₄	—	4	8	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	16	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
285	C₂²	C₂×#41×C₃₁	1	2	yes	—
286	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
287	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
288	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	4	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	4	2	no	—
295	C₂	—	4	2	no	—
296	C₂	—	4	4	no	—
297	C₂	—	4	2	no	—
298	C₂	C₃₁×#534	1	1	yes	characteristic
299	C₂	C₃₁×#538	1	2	yes	—
300	C₂	C₃₁×#546	1	2	yes	—
301	C₂	C₃₁×#551	1	1	yes	characteristic
302	C₂	C₃₁×#560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#120	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#123	C₂	semidirect
3. C₃₁×#397	C₂	semidirect
4. C₃₁×#396	C₂	semidirect
5. C₃₁×#394	C₂	semidirect
6. C₃₁×#390	C₂	semidirect
7. C₃₁×#388	C₂	semidirect
8. C₃₁×#386	C₂	semidirect
9. C₃₁×#379	C₂	semidirect
13. C₂³×C₄×C₃₁	C₂²	semidirect
16. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
17. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
18. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
27. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
28. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
29. C₂×C₄²×C₃₁	C₂²	semidirect
30. C₃₁×#95	C₂²	semidirect
31. C₃₁×#95	C₂²	semidirect
32. C₃₁×#95	C₂²	semidirect
114. #2697	C₃₁	direct (N × H)
144. C₂×#123	C₂×C₃₁	semidirect
146. #397	C₂×C₃₁	semidirect
147. #396	C₂×C₃₁	semidirect
148. #394	C₂×C₃₁	semidirect
149. #390	C₂×C₃₁	semidirect
150. #388	C₂×C₃₁	semidirect
151. #386	C₂×C₃₁	semidirect
152. #379	C₂×C₃₁	semidirect
164. C₂³×C₄	C₂²×C₃₁	semidirect
167. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
168. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
169. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
178. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
179. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
180. C₂×C₄²	C₂²×C₃₁	semidirect
181. #95	C₂²×C₃₁	semidirect
182. #95	C₂²×C₃₁	semidirect
183. #95	C₂²×C₃₁	semidirect
184. C₃₁	#2697	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×#123	2
C₂×C₃₁×#120	2
C₃₁×#397	2
C₃₁×#396	2
C₃₁×#394	2
C₃₁×#390	2
C₃₁×#388	2
C₃₁×#386	2
C₃₁×#379	2
#2697	31

Subgroups — C31×#2697 #16842630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2697 #16842630