Subgroups — C₃₁×#2404 #16842337

Order 3968.

1056 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#120	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#393	C₂	1	2	yes	maximal, complemented
5	C₃₁×#392	C₂	1	2	yes	maximal, complemented
6	C₃₁×#386	C₂	1	4	yes	maximal, complemented
7	C₃₁×#386	C₂	1	2	yes	maximal, complemented
8	C₃₁×#381	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×#41×C₃₁	—	2	8	no	—
11	C₂×#41×C₃₁	—	2	4	no	—
12	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄×C₃₁	C₂²	1	4	yes	—
14	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
16	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₃₁	C₂²	1	2	yes	complemented
19	C₃₁×#120	—	2	8	no	—
20	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
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	4	yes	—
25	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
26	C₃₁×#95	C₂²	1	2	yes	complemented
27	C₃₁×#95	C₂²	1	2	yes	complemented
28	C₃₁×#95	C₂²	1	4	yes	complemented
29	C₂⁴×C₃₁	—	2	2	no	—
30	C₂⁴×C₃₁	—	2	4	no	—
31	C₂⁴×C₃₁	—	2	4	no	—
32	C₂⁴×C₃₁	—	4	2	no	—
33	C₂⁴×C₃₁	C₂³	1	2	yes	—
34	C₂⁴×C₃₁	C₂³	1	2	yes	—
35	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
36	#41×C₃₁	—	2	32	no	—
37	C₂×Q₈×C₃₁	—	2	4	no	—
38	C₂×Q₈×C₃₁	—	2	2	no	—
39	C₂×D₄×C₃₁	—	2	4	no	—
40	C₂×D₄×C₃₁	—	2	4	no	—
41	C₂×D₄×C₃₁	—	2	4	no	—
42	C₂×D₄×C₃₁	—	2	8	no	—
43	C₂×D₄×C₃₁	—	2	8	no	—
44	C₂×D₄×C₃₁	—	2	2	no	—
45	C₂×D₄×C₃₁	—	4	4	no	—
46	C₂×D₄×C₃₁	D₄	1	8	yes	complemented
47	C₂²×C₄×C₃₁	—	2	4	no	—
48	C₂²×C₄×C₃₁	—	2	4	no	—
49	C₂²×C₄×C₃₁	—	2	4	no	—
50	C₂²×C₄×C₃₁	—	2	4	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	4	2	no	—
53	C₂²×C₄×C₃₁	D₄	1	4	yes	complemented
54	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
55	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
59	C₂²⋊C₄×C₃₁	—	2	4	no	—
60	C₂²⋊C₄×C₃₁	—	2	4	no	—
61	C₂²⋊C₄×C₃₁	—	2	4	no	—
62	C₂²⋊C₄×C₃₁	—	2	8	no	—
63	C₂²⋊C₄×C₃₁	—	4	4	no	—
64	C₂³×C₃₁	—	2	2	no	—
65	C₂³×C₃₁	—	2	2	no	—
66	C₂³×C₃₁	—	2	8	no	—
67	C₂³×C₃₁	—	2	2	no	—
68	C₂³×C₃₁	—	2	4	no	—
69	C₂³×C₃₁	—	2	2	no	—
70	C₂³×C₃₁	—	2	2	no	—
71	C₂³×C₃₁	—	2	4	no	—
72	C₂³×C₃₁	—	2	2	no	—
73	C₂³×C₃₁	—	2	4	no	—
74	C₂³×C₃₁	—	2	4	no	—
75	C₂³×C₃₁	—	4	16	no	—
76	C₂³×C₃₁	—	4	4	no	—
77	C₂³×C₃₁	—	4	4	no	—
78	C₂³×C₃₁	C₂×D₄	1	4	yes	—
79	C₂³×C₃₁	C₂×D₄	1	4	yes	—
80	C₂³×C₃₁	C₂×D₄	1	2	yes	—
81	C₂³×C₃₁	#41	1	4	yes	complemented
82	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
83	Q₈×C₃₁	—	2	8	no	—
84	D₄×C₃₁	—	2	8	no	—
85	D₄×C₃₁	—	2	16	no	—
86	D₄×C₃₁	—	4	16	no	—
87	C₂×C₄×C₃₁	—	2	16	no	—
88	C₂×C₄×C₃₁	—	2	8	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	4	no	—
95	C₂×C₄×C₃₁	—	2	4	no	—
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	4	4	no	—
99	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
101	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
102	#2404	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₃₁	—	2	2	no	—
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	—	2	8	no	—
106	C₂²×C₃₁	—	2	4	no	—
107	C₂²×C₃₁	—	2	4	no	—
108	C₂²×C₃₁	—	2	8	no	—
109	C₂²×C₃₁	—	2	8	no	—
110	C₂²×C₃₁	—	4	16	no	—
111	C₂²×C₃₁	—	4	2	no	—
112	C₂²×C₃₁	—	4	2	no	—
113	C₂²×C₃₁	—	4	2	no	—
114	C₂²×C₃₁	—	4	2	no	—
115	C₂²×C₃₁	—	4	4	no	—
116	C₂²×C₃₁	—	4	4	no	—
117	C₂²×C₃₁	#120	1	4	yes	—
118	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₃₁	C₂²×D₄	1	2	yes	—
120	C₂²×C₃₁	C₂×#41	1	2	yes	—
121	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
122	C₄×C₃₁	—	2	2	no	—
123	C₄×C₃₁	—	2	4	no	—
124	C₄×C₃₁	—	2	2	no	—
125	C₄×C₃₁	—	4	2	no	—
126	C₄×C₃₁	—	4	4	no	—
127	C₂²×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂×#120	C₂×C₃₁	1	2	yes	complemented
129	C₂²×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
130	#393	C₂×C₃₁	1	2	yes	complemented
131	#392	C₂×C₃₁	1	2	yes	complemented
132	#386	C₂×C₃₁	1	4	yes	complemented
133	#386	C₂×C₃₁	1	2	yes	complemented
134	#381	C₂×C₃₁	1	1	yes	characteristic, complemented
135	C₂×C₃₁	—	2	4	no	—
136	C₂×C₃₁	—	4	4	no	—
137	C₂×C₃₁	—	4	4	no	—
138	C₂×C₃₁	C₂×#120	1	1	yes	characteristic
139	C₂×C₃₁	#525	1	1	yes	characteristic
140	C₂×C₃₁	#535	1	1	yes	characteristic
141	C₂×C₃₁	#546	1	4	yes	—
142	C₂⁵	C₂²×C₃₁	1	1	yes	characteristic
143	C₂×#41	—	2	8	no	—
144	C₂×#41	—	2	4	no	—
145	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
146	C₂²×D₄	C₂²×C₃₁	1	4	yes	—
147	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
148	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
149	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
150	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
151	C₂³×C₄	C₂²×C₃₁	1	2	yes	complemented
152	#120	—	2	8	no	—
153	C₂×C₂²⋊C₄	—	2	4	no	—
154	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
156	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
159	#95	C₂²×C₃₁	1	2	yes	complemented
160	#95	C₂²×C₃₁	1	2	yes	complemented
161	#95	C₂²×C₃₁	1	4	yes	complemented
162	C₃₁	#2404	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁴	—	2	2	no	—
164	C₂⁴	—	2	4	no	—
165	C₂⁴	—	2	4	no	—
166	C₂⁴	—	4	2	no	—
167	C₂⁴	C₂³×C₃₁	1	2	yes	—
168	C₂⁴	C₂³×C₃₁	1	2	yes	—
169	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
170	#41	—	2	32	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	2	8	no	—
177	C₂×D₄	—	2	8	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	4	4	no	—
180	C₂×D₄	D₄×C₃₁	1	8	yes	complemented
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	D₄×C₃₁	1	4	yes	complemented
188	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
189	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
190	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	2	8	no	—
197	C₂²⋊C₄	—	4	4	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	8	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	4	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	4	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	4	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	4	16	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	4	4	no	—
212	C₂³	C₂×D₄×C₃₁	1	4	yes	—
213	C₂³	C₂×D₄×C₃₁	1	4	yes	—
214	C₂³	C₂×D₄×C₃₁	1	2	yes	—
215	C₂³	#41×C₃₁	1	4	yes	complemented
216	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
217	Q₈	—	2	8	no	—
218	D₄	—	2	8	no	—
219	D₄	—	2	16	no	—
220	D₄	—	4	16	no	—
221	C₂×C₄	—	2	16	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
234	C₂×C₄	C₂×D₄×C₃₁	1	4	yes	—
235	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	4	16	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	4	no	—
249	C₂²	—	4	4	no	—
250	C₂²	C₃₁×#120	1	4	yes	—
251	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
252	C₂²	C₂²×D₄×C₃₁	1	2	yes	—
253	C₂²	C₂×#41×C₃₁	1	2	yes	—
254	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	4	no	—
260	C₂	—	2	4	no	—
261	C₂	—	4	4	no	—
262	C₂	—	4	4	no	—
263	C₂	C₂×C₃₁×#120	1	1	yes	characteristic
264	C₂	C₃₁×#525	1	1	yes	characteristic
265	C₂	C₃₁×#535	1	1	yes	characteristic
266	C₂	C₃₁×#546	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
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. C₂²×#41×C₃₁	C₂	semidirect
2. C₂×C₃₁×#120	C₂	semidirect
3. C₂²×C₂²⋊C₄×C₃₁	C₂	semidirect
4. C₃₁×#393	C₂	semidirect
5. C₃₁×#392	C₂	semidirect
6. C₃₁×#386	C₂	semidirect
7. C₃₁×#386	C₂	semidirect
8. C₃₁×#381	C₂	semidirect
12. C₂²×Q₈×C₃₁	C₂²	semidirect
15. C₂²×D₄×C₃₁	C₂²	semidirect
18. C₂³×C₄×C₃₁	C₂²	semidirect
25. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
26. C₃₁×#95	C₂²	semidirect
27. C₃₁×#95	C₂²	semidirect
28. C₃₁×#95	C₂²	semidirect
46. C₂×D₄×C₃₁	D₄	semidirect
53. C₂²×C₄×C₃₁	D₄	semidirect
81. C₂³×C₃₁	#41	semidirect
102. #2404	C₃₁	direct (N × H)
127. C₂²×#41	C₂×C₃₁	semidirect
128. C₂×#120	C₂×C₃₁	semidirect
129. C₂²×C₂²⋊C₄	C₂×C₃₁	semidirect
130. #393	C₂×C₃₁	semidirect
131. #392	C₂×C₃₁	semidirect
132. #386	C₂×C₃₁	semidirect
133. #386	C₂×C₃₁	semidirect
134. #381	C₂×C₃₁	semidirect
145. C₂²×Q₈	C₂²×C₃₁	semidirect
148. C₂²×D₄	C₂²×C₃₁	semidirect
151. C₂³×C₄	C₂²×C₃₁	semidirect
158. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
159. #95	C₂²×C₃₁	semidirect
160. #95	C₂²×C₃₁	semidirect
161. #95	C₂²×C₃₁	semidirect
162. C₃₁	#2404	direct (N × H)
180. C₂×D₄	D₄×C₃₁	semidirect
187. C₂²×C₄	D₄×C₃₁	semidirect
215. C₂³	#41×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#41×C₃₁	2
C₂×C₃₁×#120	2
C₂²×C₂²⋊C₄×C₃₁	2
C₃₁×#393	2
C₃₁×#392	2
C₃₁×#386	2
C₃₁×#381	2
#2404	31

Subgroups — C31×#2404 #16842337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2404 #16842337