Subgroups — C₃₁×#11606 #21468365

Order 7936.

330 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	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₃₁×Q₈⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₃₁×#533	C₄	1	1	yes	characteristic
6	C₂×C₃₁×M₃₂	—	2	1	no	—
7	C₂²×C₁₆×C₃₁	—	2	1	no	—
8	C₃₁×#446	—	2	1	no	—
9	C₃₁×#446	C₄	1	2	yes	—
10	C₈×Q₈×C₃₁	—	2	1	no	—
11	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#424	—	2	1	no	—
13	C₃₁×#423	C₄	1	1	yes	characteristic
14	C₃₁×#405	—	2	1	no	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×C₄⋊C₁₆	—	2	1	no	—
17	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₃₁×#346	—	2	1	no	—
19	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
20	C₂×M₁₆×C₃₁	—	2	1	no	—
21	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
23	C₃₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
24	C₃₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
25	C₃₁×#125	—	2	1	no	—
26	C₃₁×C₂²⋊Q₈	—	2	1	no	—
27	C₃₁×C₂²⋊Q₈	C₈	1	2	yes	—
28	C₄×Q₈×C₃₁	—	2	1	no	—
29	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₃₁×#117	—	2	1	no	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₃₁×M₃₂	—	2	4	no	—
33	C₂×C₁₆×C₃₁	—	2	2	no	—
34	C₂×C₁₆×C₃₁	—	2	2	no	—
35	C₂×C₁₆×C₃₁	—	4	2	no	—
36	C₃₁×C₄⋊C₈	—	2	2	no	—
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	—	2	1	no	—
39	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₃₁×#97	—	2	2	no	—
41	C₄×C₈×C₃₁	—	2	2	no	—
42	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₃₁	—	2	1	no	—
47	C₂×Q₈×C₃₁	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
51	M₁₆×C₃₁	—	4	2	no	—
52	C₂×C₈×C₃₁	—	2	2	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
57	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
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	1	no	—
61	C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
66	C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₄²×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
70	C₄²×C₃₁	M₁₆	1	1	yes	characteristic
71	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₁₆×C₃₁	—	2	4	no	—
73	C₁₆×C₃₁	—	4	2	no	—
74	#11606	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
76	Q₈×C₃₁	—	2	4	no	—
77	Q₈×C₃₁	—	4	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	4	1	no	—
82	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₃₁	#103	1	2	yes	—
86	C₂×C₄×C₃₁	#104	1	2	yes	—
87	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
90	C₈×C₃₁	—	2	1	no	—
91	C₈×C₃₁	—	2	1	no	—
92	C₈×C₃₁	—	4	2	no	—
93	C₈×C₃₁	#103	1	2	yes	—
94	C₈×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
95	C₈×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
96	#2960	C₂×C₃₁	1	1	yes	characteristic
97	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
98	C₄×M₃₂	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	Q₈⋊C₈	1	2	yes	—
103	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄×C₃₁	—	2	1	no	—
105	C₄×C₃₁	—	2	1	no	—
106	C₄×C₃₁	—	2	1	no	—
107	C₄×C₃₁	—	4	2	no	—
108	C₄×C₃₁	—	4	2	no	—
109	C₄×C₃₁	Q₈⋊C₈	1	2	yes	—
110	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
111	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
112	C₄×C₃₁	#504	1	2	yes	—
113	#533	C₄×C₃₁	1	1	yes	characteristic
114	C₂×M₃₂	—	2	1	no	—
115	C₂²×C₁₆	—	2	1	no	—
116	#446	—	2	1	no	—
117	#446	C₄×C₃₁	1	2	yes	—
118	C₈×Q₈	—	2	1	no	—
119	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
120	#424	—	2	1	no	—
121	#423	C₄×C₃₁	1	1	yes	characteristic
122	#405	—	2	1	no	—
123	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
124	C₄⋊C₁₆	—	2	1	no	—
125	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
126	#346	—	2	1	no	—
127	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
128	C₂×C₃₁	—	2	2	no	—
129	C₂×C₃₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
130	C₂×C₃₁	#2113	1	1	yes	characteristic
131	C₂×C₃₁	#2117	1	1	yes	characteristic
132	C₂×M₁₆	—	2	1	no	—
133	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
134	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
135	C₄⋊Q₈	C₈×C₃₁	1	1	yes	characteristic
136	C₄⋊Q₈	C₈×C₃₁	1	1	yes	characteristic
137	#125	—	2	1	no	—
138	C₂²⋊Q₈	—	2	1	no	—
139	C₂²⋊Q₈	C₈×C₃₁	1	2	yes	—
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
142	#117	—	2	1	no	—
143	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
144	M₃₂	—	2	4	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	4	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
152	#97	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
155	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
156	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
158	C₃₁	#11606	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	C₂×C₈×C₃₁	1	2	yes	—
161	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
163	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
164	M₁₆	—	4	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
168	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
169	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
170	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
171	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
182	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
183	C₄²	M₁₆×C₃₁	1	1	yes	characteristic
184	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
185	C₁₆	—	2	4	no	—
186	C₁₆	—	4	2	no	—
187	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
188	Q₈	—	2	4	no	—
189	Q₈	—	4	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
195	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
196	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
197	C₂×C₄	C₃₁×#103	1	2	yes	—
198	C₂×C₄	C₃₁×#104	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
201	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	2	no	—
205	C₈	C₃₁×#103	1	2	yes	—
206	C₈	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
207	C₈	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₃₁×Q₈⋊C₈	1	2	yes	—
211	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	C₃₁×Q₈⋊C₈	1	2	yes	—
218	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
219	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
220	C₄	C₃₁×#504	1	2	yes	—
221	C₂	—	2	2	no	—
222	C₂	C₂×C₃₁×Q₈⋊C₈	1	1	yes	characteristic
223	C₂	C₃₁×#2113	1	1	yes	characteristic
224	C₂	C₃₁×#2117	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
74. #11606	C₃₁	direct (N × H)
99. Q₈⋊C₁₆	C₂×C₃₁	semidirect
158. C₃₁	#11606	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×#2960	2
C₂×C₃₁×C₄⋊C₁₆	2
C₄×C₃₁×M₃₂	2
C₃₁×Q₈⋊C₁₆	2
#11606	31

Subgroups — C31×#11606 #21468365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11606 #21468365