Subgroups — C₃₁×#33492 #21490251

Order 7936.

564 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2169	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2168	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2164	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2113	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2112	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#525	C₄	1	1	yes	characteristic
10	C₂×C₃₁×M₃₂	C₂²	1	4	yes	—
11	C₂×C₃₁×M₃₂	C₂²	1	2	yes	complemented
12	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	complemented
13	C₃₁×#435	C₄	1	4	yes	—
14	C₈×D₄×C₃₁	C₂²	1	4	yes	—
15	C₃₁×#424	C₄	1	1	yes	characteristic
16	C₃₁×#408	C₄	1	1	yes	characteristic
17	C₃₁×#407	C₄	1	1	yes	characteristic
18	C₃₁×#405	C₂²	1	1	yes	characteristic
19	C₃₁×C₄⋊C₁₆	C₂²	1	4	yes	complemented
20	C₃₁×C₂²⋊C₁₆	C₂²	1	4	yes	—
21	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₃₁×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
24	C₃₁×#346	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
26	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₃₁×#131	—	4	2	no	—
29	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₃₁	—	4	2	no	—
33	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₃₁×#123	C₈	1	2	yes	—
39	C₃₁×C₂²⋊Q₈	C₈	1	2	yes	—
40	C₃₁×#121	C₈	1	2	yes	—
41	C₃₁×#120	C₈	1	2	yes	—
42	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
43	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
44	C₃₁×M₃₂	—	4	4	no	—
45	C₃₁×M₃₂	—	4	2	no	—
46	C₂×C₁₆×C₃₁	—	4	2	no	—
47	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
48	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
49	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
50	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
51	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
52	C₃₁×#97	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
55	#41×C₃₁	—	4	1	no	—
56	#41×C₃₁	—	4	1	no	—
57	C₂×Q₈×C₃₁	C₂×C₈	1	1	yes	characteristic
58	C₂×D₄×C₃₁	C₂×C₈	1	4	yes	—
59	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₃₁	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₃₁	—	4	1	no	—
69	M₁₆×C₃₁	—	4	2	no	—
70	M₁₆×C₃₁	—	4	1	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	4	1	no	—
76	C₂×C₈×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
83	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
84	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
85	C₂²⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
86	C₂²⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
87	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
88	C₁₆×C₃₁	—	2	4	no	—
89	C₁₆×C₃₁	—	2	4	no	—
90	#33492	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	4	1	no	—
92	C₂³×C₃₁	—	4	2	no	—
93	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
94	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
95	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
96	Q₈×C₃₁	—	4	1	no	—
97	D₄×C₃₁	—	4	4	no	—
98	D₄×C₃₁	—	4	1	no	—
99	D₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	4	1	no	—
107	C₂×C₄×C₃₁	—	4	2	no	—
108	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
113	C₈×C₃₁	—	2	2	no	—
114	C₈×C₃₁	—	2	1	no	—
115	C₈×C₃₁	—	2	2	no	—
116	C₈×C₃₁	—	2	2	no	—
117	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
118	#2929	C₂×C₃₁	1	1	yes	characteristic
119	#2169	C₂×C₃₁	1	4	yes	complemented
120	#2168	C₂×C₃₁	1	4	yes	complemented
121	#2164	C₂×C₃₁	1	2	yes	complemented
122	#2113	C₂×C₃₁	1	2	yes	complemented
123	#2112	C₂×C₃₁	1	2	yes	complemented
124	C₂²×C₃₁	—	2	2	no	—
125	C₂²×C₃₁	—	2	4	no	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	2	1	no	—
129	C₂²×C₃₁	—	2	2	no	—
130	C₂²×C₃₁	—	2	2	no	—
131	C₂²×C₃₁	—	4	4	no	—
132	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₃₁	—	2	4	no	—
134	C₄×C₃₁	—	2	2	no	—
135	C₄×C₃₁	—	2	1	no	—
136	C₄×C₃₁	—	2	2	no	—
137	C₄×C₃₁	#518	1	1	yes	characteristic
138	C₄×C₃₁	#568	1	1	yes	characteristic
139	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
140	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
141	#525	C₄×C₃₁	1	1	yes	characteristic
142	C₂×M₃₂	C₂²×C₃₁	1	4	yes	—
143	C₂×M₃₂	C₂²×C₃₁	1	2	yes	complemented
144	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	complemented
145	#435	C₄×C₃₁	1	4	yes	—
146	C₈×D₄	C₂²×C₃₁	1	4	yes	—
147	#424	C₄×C₃₁	1	1	yes	characteristic
148	#408	C₄×C₃₁	1	1	yes	characteristic
149	#407	C₄×C₃₁	1	1	yes	characteristic
150	#405	C₂²×C₃₁	1	1	yes	characteristic
151	C₄⋊C₁₆	C₂²×C₃₁	1	4	yes	complemented
152	C₂²⋊C₁₆	C₂²×C₃₁	1	4	yes	—
153	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
154	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
155	C₂²⋊C₁₆	C₂²×C₃₁	1	4	yes	complemented
156	#346	C₂²×C₃₁	1	2	yes	complemented
157	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
158	C₂×C₃₁	—	4	2	no	—
159	C₂×C₃₁	—	4	2	no	—
160	C₂×C₃₁	—	4	1	no	—
161	C₂×C₃₁	#2972	1	1	yes	characteristic
162	C₂×C₃₁	#3407	1	2	yes	—
163	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
164	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
165	#131	—	4	2	no	—
166	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
167	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
168	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
169	C₂²×C₈	—	4	2	no	—
170	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
171	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
172	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
174	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
175	#123	C₈×C₃₁	1	2	yes	—
176	C₂²⋊Q₈	C₈×C₃₁	1	2	yes	—
177	#121	C₈×C₃₁	1	2	yes	—
178	#120	C₈×C₃₁	1	2	yes	—
179	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
180	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
181	M₃₂	—	4	4	no	—
182	M₃₂	—	4	2	no	—
183	C₂×C₁₆	—	4	2	no	—
184	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
185	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
186	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
187	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
188	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
189	#97	C₂×C₄×C₃₁	1	2	yes	—
190	C₄×C₈	C₂³×C₃₁	1	2	yes	—
191	C₃₁	#33492	1	1	yes	characteristic, direct product factor, Sylow
192	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
193	#41	—	4	1	no	—
194	#41	—	4	1	no	—
195	C₂×Q₈	C₂×C₈×C₃₁	1	1	yes	characteristic
196	C₂×D₄	C₂×C₈×C₃₁	1	4	yes	—
197	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
198	C₂×D₄	C₂×C₈×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
202	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
206	M₁₆	—	4	1	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	1	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
217	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
218	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
219	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
220	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
221	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
222	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
223	C₂²⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
224	C₂²⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
225	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
226	C₁₆	—	2	4	no	—
227	C₁₆	—	2	4	no	—
228	C₂³	—	4	1	no	—
229	C₂³	—	4	2	no	—
230	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
231	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
232	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
233	Q₈	—	4	1	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
246	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
247	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
248	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	1	no	—
252	C₈	—	2	2	no	—
253	C₈	—	2	2	no	—
254	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	4	4	no	—
263	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	C₃₁×#518	1	1	yes	characteristic
269	C₄	C₃₁×#568	1	1	yes	characteristic
270	C₂	—	4	2	no	—
271	C₂	—	4	1	no	—
272	C₂	—	4	2	no	—
273	C₂	C₃₁×#2972	1	1	yes	characteristic
274	C₂	C₃₁×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂³×C₈
prime radical	C₃₁×#525	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2169	C₂	semidirect
3. C₃₁×#2168	C₂	semidirect
4. C₃₁×#2164	C₂	semidirect
5. C₃₁×#2113	C₂	semidirect
6. C₃₁×#2112	C₂	semidirect
11. C₂×C₃₁×M₃₂	C₂²	semidirect
12. C₂²×C₁₆×C₃₁	C₂²	semidirect
19. C₃₁×C₄⋊C₁₆	C₂²	semidirect
23. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
24. C₃₁×#346	C₂²	semidirect
25. C₄×C₁₆×C₃₁	C₂²	semidirect
90. #33492	C₃₁	direct (N × H)
119. #2169	C₂×C₃₁	semidirect
120. #2168	C₂×C₃₁	semidirect
121. #2164	C₂×C₃₁	semidirect
122. #2113	C₂×C₃₁	semidirect
123. #2112	C₂×C₃₁	semidirect
143. C₂×M₃₂	C₂²×C₃₁	semidirect
144. C₂²×C₁₆	C₂²×C₃₁	semidirect
151. C₄⋊C₁₆	C₂²×C₃₁	semidirect
155. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
156. #346	C₂²×C₃₁	semidirect
157. C₄×C₁₆	C₂²×C₃₁	semidirect
191. C₃₁	#33492	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×#2929	2
C₃₁×#2169	2
C₃₁×#2168	2
C₃₁×#2164	2
C₃₁×#2113	2
C₃₁×#2112	2
#33492	31

Subgroups — C31×#33492 #21490251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33492 #21490251