Subgroups — C₃₁×#11639 #21468398

Order 7936.

334 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic
4	C₃₁×Q₈⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
6	C₈×Q₈×C₃₁	—	2	1	no	—
7	C₈×Q₈×C₃₁	—	2	1	no	—
8	C₈×Q₈×C₃₁	C₄	1	2	yes	—
9	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
11	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₃₁	—	2	1	no	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×C₄⋊C₁₆	—	2	1	no	—
15	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
17	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
18	C₃₁×#346	—	2	1	no	—
19	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
20	C₂²×Q₈×C₃₁	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	—	2	1	no	—
22	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₃₁	—	2	1	no	—
25	C₄×Q₈×C₃₁	—	2	1	no	—
26	C₄×Q₈×C₃₁	C₈	1	2	yes	—
27	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	—	2	1	no	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₃₁	—	2	2	no	—
33	C₂×C₁₆×C₃₁	—	2	2	no	—
34	C₃₁×C₄⋊C₈	—	2	2	no	—
35	C₃₁×C₄⋊C₈	—	2	2	no	—
36	C₃₁×C₄⋊C₈	—	2	1	no	—
37	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₃₁	—	2	2	no	—
39	C₄×C₈×C₃₁	—	2	2	no	—
40	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
42	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₃₁	—	2	4	no	—
45	C₂×Q₈×C₃₁	—	2	1	no	—
46	C₂×Q₈×C₃₁	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₃₁	—	2	1	no	—
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
52	C₂×C₈×C₃₁	—	2	2	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₂×C₈×C₃₁	—	4	1	no	—
55	C₂×C₈×C₃₁	—	4	2	no	—
56	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
64	C₄²×C₃₁	—	2	2	no	—
65	C₄²×C₃₁	—	2	2	no	—
66	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₃₁	M₁₆	1	1	yes	characteristic
69	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₁₆×C₃₁	—	4	2	no	—
71	C₁₆×C₃₁	—	4	2	no	—
72	#11639	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
74	Q₈×C₃₁	—	2	4	no	—
75	Q₈×C₃₁	—	2	2	no	—
76	Q₈×C₃₁	M₃₂	1	4	yes	—
77	C₂×C₄×C₃₁	—	2	4	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	4	2	no	—
85	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
91	C₈×C₃₁	—	2	1	no	—
92	C₈×C₃₁	—	2	1	no	—
93	C₈×C₃₁	—	2	1	no	—
94	C₈×C₃₁	—	4	2	no	—
95	C₈×C₃₁	#137	1	2	yes	—
96	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
97	#2163	C₂×C₃₁	1	1	yes	characteristic
98	#2162	C₂×C₃₁	1	1	yes	characteristic
99	Q₈⋊C₁₆	C₂×C₃₁	1	4	yes	complemented
100	C₂²×C₃₁	—	2	1	no	—
101	C₂²×C₃₁	—	2	1	no	—
102	C₂²×C₃₁	#350	1	2	yes	—
103	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄×C₃₁	—	2	4	no	—
105	C₄×C₃₁	—	2	1	no	—
106	C₄×C₃₁	—	2	1	no	—
107	C₄×C₃₁	—	2	1	no	—
108	C₄×C₃₁	—	2	1	no	—
109	C₄×C₃₁	—	4	2	no	—
110	C₄×C₃₁	#419	1	1	yes	characteristic
111	C₄×C₃₁	#421	1	1	yes	characteristic
112	C₄×C₃₁	C₂×M₃₂	1	2	yes	—
113	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
114	C₈×Q₈	—	2	1	no	—
115	C₈×Q₈	—	2	1	no	—
116	C₈×Q₈	C₄×C₃₁	1	2	yes	—
117	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
118	C₂×C₄⋊C₈	—	2	1	no	—
119	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
120	C₂×C₄×C₈	—	2	1	no	—
121	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
122	C₄⋊C₁₆	—	2	1	no	—
123	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
124	C₂²⋊C₁₆	—	2	1	no	—
125	C₂²⋊C₁₆	—	2	1	no	—
126	#346	—	2	1	no	—
127	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
128	C₂×C₃₁	—	2	2	no	—
129	C₂×C₃₁	#1479	1	1	yes	characteristic
130	C₂×C₃₁	#2112	1	1	yes	characteristic
131	C₂×C₃₁	C₂×#350	1	1	yes	characteristic
132	C₂²×Q₈	C₈×C₃₁	1	1	yes	characteristic
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
135	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	C₈×C₃₁	1	2	yes	—
139	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	1	no	—
149	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
153	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
154	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
155	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
156	C₃₁	#11639	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	C₂×C₈×C₃₁	1	2	yes	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
164	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
171	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
180	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
181	C₄²	M₁₆×C₃₁	1	1	yes	characteristic
182	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
183	C₁₆	—	4	2	no	—
184	C₁₆	—	4	2	no	—
185	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
186	Q₈	—	2	4	no	—
187	Q₈	—	2	2	no	—
188	Q₈	C₃₁×M₃₂	1	4	yes	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
199	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
201	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
202	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	4	2	no	—
207	C₈	C₃₁×#137	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₃₁×#350	1	2	yes	—
211	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	4	2	no	—
218	C₄	C₃₁×#419	1	1	yes	characteristic
219	C₄	C₃₁×#421	1	1	yes	characteristic
220	C₄	C₂×C₃₁×M₃₂	1	2	yes	—
221	C₂	—	2	2	no	—
222	C₂	C₃₁×#1479	1	1	yes	characteristic
223	C₂	C₃₁×#2112	1	1	yes	characteristic
224	C₂	C₂×C₃₁×#350	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₃₁	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×Q₈⋊C₁₆	C₂	semidirect
72. #11639	C₃₁	direct (N × H)
99. Q₈⋊C₁₆	C₂×C₃₁	semidirect
156. C₃₁	#11639	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×C₈×Q₈×C₃₁	2
C₃₁×#2163	2
C₃₁×#2162	2
C₃₁×Q₈⋊C₁₆	2
#11639	31

Subgroups — C31×#11639 #21468398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11639 #21468398