Subgroups — C₃₁×#33765 #21490524

Order 7936.

616 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3058	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2234	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2218	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2215	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2184	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2113	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₃₁	C₂²	1	2	yes	complemented
8	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
9	C₃₁×#535	—	2	2	no	—
10	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
11	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
12	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#469	—	2	2	no	—
15	C₃₁×#469	—	2	2	no	—
16	C₃₁×#466	C₂²	1	4	yes	—
17	C₃₁×#466	C₂²	1	4	yes	complemented
18	C₃₁×#427	C₂²	1	2	yes	complemented
19	C₃₁×#418	—	2	2	no	—
20	C₃₁×#367	C₂²	1	2	yes	complemented
21	C₃₁×#366	C₂²	1	2	yes	complemented
22	C₃₁×#357	C₂²	1	4	yes	complemented
23	C₃₁×#357	C₂²	1	4	yes	complemented
24	C₃₁×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
25	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₃₁	—	2	2	no	—
27	C₂×D₈×C₃₁	—	2	2	no	—
28	C₂×D₈×C₃₁	—	4	2	no	—
29	C₂×D₈×C₃₁	—	4	2	no	—
30	C₂×D₈×C₃₁	C₂³	1	4	yes	—
31	C₃₁×#131	—	4	1	no	—
32	C₃₁×#131	—	4	1	no	—
33	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
34	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×#127	—	4	2	no	—
38	C₃₁×#124	—	4	2	no	—
39	C₃₁×#121	—	2	4	no	—
40	C₃₁×#121	—	2	4	no	—
41	C₃₁×#120	—	4	2	no	—
42	C₃₁×#120	—	4	2	no	—
43	C₃₁×#117	—	2	2	no	—
44	C₃₁×D₁₆	—	4	2	no	—
45	C₃₁×D₁₆	—	4	2	no	—
46	C₃₁×M₃₂	—	4	1	no	—
47	C₂×C₁₆×C₃₁	—	4	1	no	—
48	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
49	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#107	C₂³	1	4	yes	—
51	C₃₁×#106	—	2	2	no	—
52	C₃₁×#102	—	2	4	no	—
53	C₃₁×#102	—	2	4	no	—
54	C₂⁴×C₃₁	—	4	2	no	—
55	#41×C₃₁	—	4	1	no	—
56	#41×C₃₁	—	4	1	no	—
57	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
58	C₂×D₄×C₃₁	—	2	4	no	—
59	C₂×D₄×C₃₁	—	4	4	no	—
60	C₂×D₄×C₃₁	—	4	4	no	—
61	C₂×D₄×C₃₁	—	4	4	no	—
62	C₂×D₄×C₃₁	—	4	4	no	—
63	C₂×D₄×C₃₁	—	4	2	no	—
64	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
65	C₂×D₄×C₃₁	D₈	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	D₈×C₃₁	—	2	4	no	—
69	D₈×C₃₁	—	2	4	no	—
70	D₈×C₃₁	—	4	4	no	—
71	M₁₆×C₃₁	—	4	2	no	—
72	M₁₆×C₃₁	—	4	1	no	—
73	C₂×C₈×C₃₁	—	2	1	no	—
74	C₂×C₈×C₃₁	—	2	1	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₂²⋊C₄×C₃₁	—	4	2	no	—
82	C₂²⋊C₄×C₃₁	—	4	4	no	—
83	C₂²⋊C₄×C₃₁	—	4	4	no	—
84	C₄²×C₃₁	—	4	2	no	—
85	C₁₆×C₃₁	—	2	2	no	—
86	C₁₆×C₃₁	—	2	2	no	—
87	#33765	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	4	4	no	—
89	C₂³×C₃₁	—	8	4	no	—
90	C₂³×C₃₁	—	16	4	no	—
91	C₂³×C₃₁	C₂×D₈	1	2	yes	—
92	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₃₁	—	4	1	no	—
94	D₄×C₃₁	—	4	2	no	—
95	D₄×C₃₁	—	4	1	no	—
96	D₄×C₃₁	—	4	4	no	—
97	D₄×C₃₁	—	4	4	no	—
98	D₄×C₃₁	—	8	4	no	—
99	D₄×C₃₁	—	16	4	no	—
100	D₄×C₃₁	—	16	4	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₃₁	—	2	1	no	—
110	C₈×C₃₁	—	2	2	no	—
111	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
112	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
113	#3058	C₂×C₃₁	1	2	yes	complemented
114	#2234	C₂×C₃₁	1	4	yes	complemented
115	#2218	C₂×C₃₁	1	2	yes	complemented
116	#2215	C₂×C₃₁	1	2	yes	complemented
117	#2184	C₂×C₃₁	1	4	yes	complemented
118	#2113	C₂×C₃₁	1	1	yes	characteristic, complemented
119	C₂²×C₃₁	—	2	2	no	—
120	C₂²×C₃₁	—	2	4	no	—
121	C₂²×C₃₁	—	2	2	no	—
122	C₂²×C₃₁	—	2	1	no	—
123	C₂²×C₃₁	—	8	4	no	—
124	C₂²×C₃₁	—	8	4	no	—
125	C₂²×C₃₁	—	8	4	no	—
126	C₂²×C₃₁	—	16	8	no	—
127	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
128	C₄×C₃₁	—	2	2	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	8	4	no	—
131	C₄×C₃₁	#534	1	1	yes	characteristic
132	C₄×C₃₁	#575	1	1	yes	characteristic
133	C₂²×D₈	C₂²×C₃₁	1	2	yes	complemented
134	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
135	#535	—	2	2	no	—
136	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
137	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
138	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic, complemented
139	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
140	#469	—	2	2	no	—
141	#469	—	2	2	no	—
142	#466	C₂²×C₃₁	1	4	yes	—
143	#466	C₂²×C₃₁	1	4	yes	complemented
144	#427	C₂²×C₃₁	1	2	yes	complemented
145	#418	—	2	2	no	—
146	#367	C₂²×C₃₁	1	2	yes	complemented
147	#366	C₂²×C₃₁	1	2	yes	complemented
148	#357	C₂²×C₃₁	1	4	yes	complemented
149	#357	C₂²×C₃₁	1	4	yes	complemented
150	C₂²⋊C₁₆	C₂²×C₃₁	1	4	yes	complemented
151	C₂×C₃₁	—	4	1	no	—
152	C₂×C₃₁	—	4	2	no	—
153	C₂×C₃₁	—	16	4	no	—
154	C₂×C₃₁	#3186	1	1	yes	characteristic
155	C₂×C₃₁	#3415	1	2	yes	—
156	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
157	C₂²×D₄	—	2	2	no	—
158	C₂×D₈	—	2	2	no	—
159	C₂×D₈	—	4	2	no	—
160	C₂×D₈	—	4	2	no	—
161	C₂×D₈	C₂³×C₃₁	1	4	yes	—
162	#131	—	4	1	no	—
163	#131	—	4	1	no	—
164	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
165	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
167	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
168	#127	—	4	2	no	—
169	#124	—	4	2	no	—
170	#121	—	2	4	no	—
171	#121	—	2	4	no	—
172	#120	—	4	2	no	—
173	#120	—	4	2	no	—
174	#117	—	2	2	no	—
175	D₁₆	—	4	2	no	—
176	D₁₆	—	4	2	no	—
177	M₃₂	—	4	1	no	—
178	C₂×C₁₆	—	4	1	no	—
179	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
180	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
181	#107	C₂³×C₃₁	1	4	yes	—
182	#106	—	2	2	no	—
183	#102	—	2	4	no	—
184	#102	—	2	4	no	—
185	C₃₁	#33765	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	—	4	2	no	—
187	#41	—	4	1	no	—
188	#41	—	4	1	no	—
189	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂×D₄	—	4	4	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
197	C₂×D₄	D₈×C₃₁	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
199	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
200	D₈	—	2	4	no	—
201	D₈	—	2	4	no	—
202	D₈	—	4	4	no	—
203	M₁₆	—	4	2	no	—
204	M₁₆	—	4	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
211	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
212	C₄⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₄²	—	4	2	no	—
217	C₁₆	—	2	2	no	—
218	C₁₆	—	2	2	no	—
219	C₂³	—	4	4	no	—
220	C₂³	—	8	4	no	—
221	C₂³	—	16	4	no	—
222	C₂³	C₂×D₈×C₃₁	1	2	yes	—
223	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
224	Q₈	—	4	1	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	1	no	—
227	D₄	—	4	4	no	—
228	D₄	—	4	4	no	—
229	D₄	—	8	4	no	—
230	D₄	—	16	4	no	—
231	D₄	—	16	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
238	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
240	C₈	—	2	1	no	—
241	C₈	—	2	2	no	—
242	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
243	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	8	4	no	—
249	C₂²	—	8	4	no	—
250	C₂²	—	8	4	no	—
251	C₂²	—	16	8	no	—
252	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	1	no	—
255	C₄	—	8	4	no	—
256	C₄	C₃₁×#534	1	1	yes	characteristic
257	C₄	C₃₁×#575	1	1	yes	characteristic
258	C₂	—	4	2	no	—
259	C₂	—	4	1	no	—
260	C₂	—	16	4	no	—
261	C₂	C₃₁×#3186	1	1	yes	characteristic
262	C₂	C₃₁×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×#41×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3058	C₂	semidirect
2. C₃₁×#2234	C₂	semidirect
3. C₃₁×#2218	C₂	semidirect
4. C₃₁×#2215	C₂	semidirect
5. C₃₁×#2184	C₂	semidirect
6. C₃₁×#2113	C₂	semidirect
7. C₂²×D₈×C₃₁	C₂²	semidirect
12. C₂×C₃₁×M₃₂	C₂²	semidirect
13. C₂²×C₁₆×C₃₁	C₂²	semidirect
17. C₃₁×#466	C₂²	semidirect
18. C₃₁×#427	C₂²	semidirect
20. C₃₁×#367	C₂²	semidirect
21. C₃₁×#366	C₂²	semidirect
22. C₃₁×#357	C₂²	semidirect
23. C₃₁×#357	C₂²	semidirect
24. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
87. #33765	C₃₁	direct (N × H)
113. #3058	C₂×C₃₁	semidirect
114. #2234	C₂×C₃₁	semidirect
115. #2218	C₂×C₃₁	semidirect
116. #2215	C₂×C₃₁	semidirect
117. #2184	C₂×C₃₁	semidirect
118. #2113	C₂×C₃₁	semidirect
133. C₂²×D₈	C₂²×C₃₁	semidirect
138. C₂×M₃₂	C₂²×C₃₁	semidirect
139. C₂²×C₁₆	C₂²×C₃₁	semidirect
143. #466	C₂²×C₃₁	semidirect
144. #427	C₂²×C₃₁	semidirect
146. #367	C₂²×C₃₁	semidirect
147. #366	C₂²×C₃₁	semidirect
148. #357	C₂²×C₃₁	semidirect
149. #357	C₂²×C₃₁	semidirect
150. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
185. C₃₁	#33765	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×#3058	2
C₃₁×#2234	2
C₃₁×#2218	2
C₃₁×#2215	2
C₃₁×#2184	2
C₃₁×#2113	2
#33765	31

Subgroups — C31×#33765 #21490524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33765 #21490524