Subgroups — C₃₁×#21938 #21478697

Order 7936.

1266 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#380	C₂	1	2	yes	maximal
3	C₃₁×#1853	C₂	1	8	yes	maximal, complemented
4	C₃₁×#1807	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
8	C₂²×C₂²⋊C₄×C₃₁	—	2	2	no	—
9	C₂×C₃₁×#106	C₂²	1	8	yes	complemented
10	C₂×C₃₁×C₂²⋊C₈	C₄	1	8	yes	complemented
11	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
12	C₃₁×#380	—	2	2	no	—
13	C₃₁×#380	—	2	2	no	—
14	C₃₁×#380	C₂²	1	4	yes	—
15	C₃₁×#380	C₂²	1	4	yes	—
16	C₂×C₃₁×#95	—	2	2	no	—
17	C₃₁×#340	C₂²	1	8	yes	complemented
18	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₃₁	—	2	4	no	—
20	C₂³×C₄×C₃₁	—	4	4	no	—
21	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₃₁	D₄	1	2	yes	—
23	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₃₁	Q₈	1	4	yes	—
25	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
26	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	—	4	4	no	—
28	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
29	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
30	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₃₁	—	4	8	no	—
32	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
33	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
35	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
36	C₃₁×#106	—	4	8	no	—
37	C₃₁×C₂²⋊C₈	C₂×C₄	1	16	yes	complemented
38	C₃₁×#95	—	2	4	no	—
39	C₃₁×#95	—	2	4	no	—
40	C₂⁴×C₃₁	—	4	2	no	—
41	C₂⁴×C₃₁	—	4	2	no	—
42	C₂⁴×C₃₁	C₄⋊C₄	1	4	yes	—
43	C₂⁴×C₃₁	SD₁₆	1	8	yes	—
44	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₃₁	—	2	8	no	—
47	C₂²×C₄×C₃₁	—	2	4	no	—
48	C₂²×C₄×C₃₁	—	2	8	no	—
49	C₂²×C₄×C₃₁	—	4	8	no	—
50	C₂²×C₄×C₃₁	—	4	8	no	—
51	C₂²×C₄×C₃₁	—	4	2	no	—
52	C₂²×C₄×C₃₁	—	4	4	no	—
53	C₂²×C₄×C₃₁	—	4	8	no	—
54	C₂²×C₄×C₃₁	—	4	8	no	—
55	C₂²×C₄×C₃₁	—	4	8	no	—
56	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
57	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₃₁	—	2	8	no	—
66	C₂×C₈×C₃₁	—	4	8	no	—
67	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
68	C₄⋊C₄×C₃₁	—	2	16	no	—
69	C₄⋊C₄×C₃₁	—	4	8	no	—
70	C₂²⋊C₄×C₃₁	—	4	8	no	—
71	C₂²⋊C₄×C₃₁	—	4	8	no	—
72	#21938	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₃₁	—	2	4	no	—
74	C₂³×C₃₁	—	2	2	no	—
75	C₂³×C₃₁	—	2	4	no	—
76	C₂³×C₃₁	—	2	2	no	—
77	C₂³×C₃₁	—	4	4	no	—
78	C₂³×C₃₁	—	4	4	no	—
79	C₂³×C₃₁	—	4	16	no	—
80	C₂³×C₃₁	#106	1	16	yes	complemented
81	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
82	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
83	C₂³×C₃₁	C₂×SD₁₆	1	4	yes	—
84	C₂³×C₃₁	C₂×SD₁₆	1	4	yes	—
85	C₂³×C₃₁	C₂×SD₁₆	1	4	yes	—
86	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	4	no	—
90	C₂×C₄×C₃₁	—	2	8	no	—
91	C₂×C₄×C₃₁	—	2	8	no	—
92	C₂×C₄×C₃₁	—	2	4	no	—
93	C₂×C₄×C₃₁	—	4	4	no	—
94	C₂×C₄×C₃₁	—	4	8	no	—
95	C₂×C₄×C₃₁	—	4	8	no	—
96	C₂×C₄×C₃₁	—	4	8	no	—
97	C₂×C₄×C₃₁	—	4	8	no	—
98	C₂×C₄×C₃₁	—	8	16	no	—
99	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
104	C₈×C₃₁	—	2	8	no	—
105	C₂²×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
106	C₂×#380	C₂×C₃₁	1	2	yes	—
107	#1853	C₂×C₃₁	1	8	yes	complemented
108	#1807	C₂×C₃₁	1	4	yes	complemented
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	—	2	4	no	—
111	C₂²×C₃₁	—	2	4	no	—
112	C₂²×C₃₁	—	2	2	no	—
113	C₂²×C₃₁	—	2	8	no	—
114	C₂²×C₃₁	—	2	8	no	—
115	C₂²×C₃₁	—	2	8	no	—
116	C₂²×C₃₁	—	4	16	no	—
117	C₂²×C₃₁	C₂×#106	1	4	yes	—
118	C₂²×C₃₁	C₂×#106	1	8	yes	—
119	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₃₁	#534	1	2	yes	—
121	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
122	C₂²×C₃₁	#575	1	2	yes	—
123	C₄×C₃₁	—	2	4	no	—
124	C₄×C₃₁	—	2	2	no	—
125	C₄×C₃₁	—	8	8	no	—
126	C₄×C₃₁	#518	1	2	yes	—
127	C₄×C₃₁	#543	1	2	yes	—
128	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
129	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
130	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
131	C₂²×C₂²⋊C₄	—	2	2	no	—
132	C₂×#106	C₂²×C₃₁	1	8	yes	complemented
133	C₂×C₂²⋊C₈	C₄×C₃₁	1	8	yes	complemented
134	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
135	#380	—	2	2	no	—
136	#380	—	2	2	no	—
137	#380	C₂²×C₃₁	1	4	yes	—
138	#380	C₂²×C₃₁	1	4	yes	—
139	C₂×#95	—	2	2	no	—
140	#340	C₂²×C₃₁	1	8	yes	complemented
141	C₂×C₃₁	—	2	8	no	—
142	C₂×C₃₁	—	2	4	no	—
143	C₂×C₃₁	#2315	1	1	yes	characteristic
144	C₂×C₃₁	C₂²×#106	1	1	yes	characteristic
145	C₂×C₃₁	#2911	1	1	yes	characteristic
146	C₂×C₃₁	#3187	1	4	yes	—
147	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
148	C₂³×C₄	—	2	4	no	—
149	C₂³×C₄	—	4	4	no	—
150	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
151	C₂³×C₄	D₄×C₃₁	1	2	yes	—
152	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
153	C₂³×C₄	Q₈×C₃₁	1	4	yes	—
154	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
155	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
156	C₂²×C₈	—	4	4	no	—
157	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
158	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	4	8	no	—
161	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
162	C₂×C₂²⋊C₄	—	2	4	no	—
163	C₂×C₂²⋊C₄	—	2	4	no	—
164	C₂×C₂²⋊C₄	—	4	8	no	—
165	#106	—	4	8	no	—
166	C₂²⋊C₈	C₂×C₄×C₃₁	1	16	yes	complemented
167	#95	—	2	4	no	—
168	#95	—	2	4	no	—
169	C₃₁	#21938	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁴	—	4	2	no	—
171	C₂⁴	—	4	2	no	—
172	C₂⁴	C₄⋊C₄×C₃₁	1	4	yes	—
173	C₂⁴	SD₁₆×C₃₁	1	8	yes	—
174	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
175	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	4	8	no	—
180	C₂²×C₄	—	4	8	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	—	4	8	no	—
184	C₂²×C₄	—	4	8	no	—
185	C₂²×C₄	—	4	8	no	—
186	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
187	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
188	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
189	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
192	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
193	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
194	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
195	C₂×C₈	—	2	8	no	—
196	C₂×C₈	—	4	8	no	—
197	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
198	C₄⋊C₄	—	2	16	no	—
199	C₄⋊C₄	—	4	8	no	—
200	C₂²⋊C₄	—	4	8	no	—
201	C₂²⋊C₄	—	4	8	no	—
202	C₂³	—	2	4	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	4	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	4	4	no	—
207	C₂³	—	4	4	no	—
208	C₂³	—	4	16	no	—
209	C₂³	C₃₁×#106	1	16	yes	complemented
210	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
211	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
212	C₂³	C₂×SD₁₆×C₃₁	1	4	yes	—
213	C₂³	C₂×SD₁₆×C₃₁	1	4	yes	—
214	C₂³	C₂×SD₁₆×C₃₁	1	4	yes	—
215	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	—	8	16	no	—
228	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
230	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
231	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
232	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
233	C₈	—	2	8	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	4	16	no	—
242	C₂²	C₂×C₃₁×#106	1	4	yes	—
243	C₂²	C₂×C₃₁×#106	1	8	yes	—
244	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
245	C₂²	C₃₁×#534	1	2	yes	—
246	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
247	C₂²	C₃₁×#575	1	2	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	2	no	—
250	C₄	—	8	8	no	—
251	C₄	C₃₁×#518	1	2	yes	—
252	C₄	C₃₁×#543	1	2	yes	—
253	C₂	—	2	8	no	—
254	C₂	—	2	4	no	—
255	C₂	C₃₁×#2315	1	1	yes	characteristic
256	C₂	C₂²×C₃₁×#106	1	1	yes	characteristic
257	C₂	C₃₁×#2911	1	1	yes	characteristic
258	C₂	C₃₁×#3187	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁵×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1853	C₂	semidirect
4. C₃₁×#1807	C₂	semidirect
9. C₂×C₃₁×#106	C₂²	semidirect
10. C₂×C₃₁×C₂²⋊C₈	C₄	semidirect
17. C₃₁×#340	C₂²	semidirect
28. C₂²×C₈×C₃₁	C₂×C₄	semidirect
37. C₃₁×C₂²⋊C₈	C₂×C₄	semidirect
72. #21938	C₃₁	direct (N × H)
80. C₂³×C₃₁	#106	semidirect
107. #1853	C₂×C₃₁	semidirect
108. #1807	C₂×C₃₁	semidirect
132. C₂×#106	C₂²×C₃₁	semidirect
133. C₂×C₂²⋊C₈	C₄×C₃₁	semidirect
140. #340	C₂²×C₃₁	semidirect
157. C₂²×C₈	C₂×C₄×C₃₁	semidirect
166. C₂²⋊C₈	C₂×C₄×C₃₁	semidirect
169. C₃₁	#21938	direct (N × H)
209. C₂³	C₃₁×#106	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×C₂²⋊C₈	2
C₂×C₃₁×#380	2
C₃₁×#1853	2
C₃₁×#1807	2
#21938	31

Subgroups — C31×#21938 #21478697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21938 #21478697