Subgroups — C₃₁×#13264 #21470023

Order 7936.

386 conjugacy classes of subgroups.

The subgroup lattice has 192 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	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1374	C₂	1	4	yes	maximal
5	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
8	C₈×Q₈×C₃₁	—	2	2	no	—
9	C₈×Q₈×C₃₁	—	2	2	no	—
10	C₂×C₃₁×C₄⋊C₈	C₄	1	2	yes	—
11	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×C₄⋊C₁₆	—	2	2	no	—
15	C₃₁×#346	—	2	2	no	—
16	C₂²×Q₈×C₃₁	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
20	C₄×Q₈×C₃₁	—	2	2	no	—
21	C₄×Q₈×C₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₃₁	C₈	1	2	yes	—
24	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₃₁	—	2	2	no	—
27	C₂×C₁₆×C₃₁	—	2	2	no	—
28	C₂×C₁₆×C₃₁	—	2	2	no	—
29	C₂×C₁₆×C₃₁	—	2	4	no	—
30	C₂×C₁₆×C₃₁	Q₈	1	4	yes	complemented
31	C₃₁×C₄⋊C₈	—	2	4	no	—
32	C₃₁×C₄⋊C₈	—	2	2	no	—
33	C₄×C₈×C₃₁	—	2	2	no	—
34	C₄×C₈×C₃₁	—	2	2	no	—
35	C₄×C₈×C₃₁	D₄	1	2	yes	—
36	C₄×C₈×C₃₁	D₄	1	2	yes	—
37	C₂×Q₈×C₃₁	—	2	2	no	—
38	C₂×Q₈×C₃₁	—	2	4	no	—
39	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₃₁	—	2	2	no	—
45	C₂×C₈×C₃₁	—	2	2	no	—
46	C₂×C₈×C₃₁	—	2	4	no	—
47	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₃₁	#41	1	2	yes	—
52	C₄⋊C₄×C₃₁	—	2	4	no	—
53	C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₄²×C₃₁	—	2	2	no	—
55	C₄²×C₃₁	—	2	2	no	—
56	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₃₁	M₁₆	1	2	yes	—
58	C₁₆×C₃₁	—	2	4	no	—
59	C₁₆×C₃₁	—	4	2	no	—
60	#13264	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
62	Q₈×C₃₁	—	2	8	no	—
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	—	2	2	no	—
65	C₂×C₄×C₃₁	—	2	2	no	—
66	C₂×C₄×C₃₁	—	2	2	no	—
67	C₂×C₄×C₃₁	—	2	2	no	—
68	C₂×C₄×C₃₁	—	2	4	no	—
69	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
70	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
71	C₂×C₄×C₃₁	M₃₂	1	4	yes	—
72	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
74	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₃₁	#131	1	2	yes	—
76	C₈×C₃₁	—	2	2	no	—
77	C₈×C₃₁	—	4	2	no	—
78	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
79	C₈×C₃₁	#124	1	1	yes	characteristic
80	C₈×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
82	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
83	C₂×#346	C₂×C₃₁	1	1	yes	characteristic
84	#1374	C₂×C₃₁	1	4	yes	—
85	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
86	C₂²×C₃₁	C₈×Q₈	1	2	yes	—
87	C₂²×C₃₁	C₂×M₃₂	1	2	yes	—
88	C₂²×C₃₁	#504	1	2	yes	—
89	C₄×C₃₁	—	2	2	no	—
90	C₄×C₃₁	—	2	4	no	—
91	C₄×C₃₁	—	2	2	no	—
92	C₄×C₃₁	—	4	2	no	—
93	C₄×C₃₁	#391	1	1	yes	characteristic
94	C₄×C₃₁	#408	1	1	yes	characteristic
95	C₄×C₃₁	#446	1	2	yes	—
96	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
97	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
98	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
99	C₈×Q₈	—	2	2	no	—
100	C₈×Q₈	—	2	2	no	—
101	C₂×C₄⋊C₈	C₄×C₃₁	1	2	yes	—
102	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
105	C₄⋊C₁₆	—	2	2	no	—
106	#346	—	2	2	no	—
107	C₂×C₃₁	#1984	1	1	yes	characteristic
108	C₂×C₃₁	#2112	1	1	yes	characteristic
109	C₂×C₃₁	#2113	1	1	yes	characteristic
110	C₂×C₃₁	#2183	1	4	yes	—
111	C₂²×Q₈	C₈×C₃₁	1	1	yes	characteristic
112	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
113	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
115	C₄×Q₈	—	2	2	no	—
116	C₄×Q₈	—	2	2	no	—
117	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₈×C₃₁	1	2	yes	—
119	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
120	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	4	no	—
125	C₂×C₁₆	Q₈×C₃₁	1	4	yes	complemented
126	C₄⋊C₈	—	2	4	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	D₄×C₃₁	1	2	yes	—
131	C₄×C₈	D₄×C₃₁	1	2	yes	—
132	C₃₁	#13264	1	1	yes	characteristic, direct product factor, Sylow
133	C₂×Q₈	—	2	2	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
136	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
139	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
144	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
145	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
146	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
147	C₂×C₈	#41×C₃₁	1	2	yes	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
153	C₄²	M₁₆×C₃₁	1	2	yes	—
154	C₁₆	—	2	4	no	—
155	C₁₆	—	4	2	no	—
156	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
157	Q₈	—	2	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
165	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
166	C₂×C₄	C₃₁×M₃₂	1	4	yes	—
167	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
168	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
169	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
170	C₂×C₄	C₃₁×#131	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	4	2	no	—
173	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
174	C₈	C₃₁×#124	1	1	yes	characteristic
175	C₈	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
176	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
177	C₂²	C₈×Q₈×C₃₁	1	2	yes	—
178	C₂²	C₂×C₃₁×M₃₂	1	2	yes	—
179	C₂²	C₃₁×#504	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	4	no	—
182	C₄	—	2	2	no	—
183	C₄	—	4	2	no	—
184	C₄	C₃₁×#391	1	1	yes	characteristic
185	C₄	C₃₁×#408	1	1	yes	characteristic
186	C₄	C₃₁×#446	1	2	yes	—
187	C₂	C₃₁×#1984	1	1	yes	characteristic
188	C₂	C₃₁×#2112	1	1	yes	characteristic
189	C₂	C₃₁×#2113	1	1	yes	characteristic
190	C₂	C₃₁×#2183	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
30. C₂×C₁₆×C₃₁	Q₈	semidirect
60. #13264	C₃₁	direct (N × H)
125. C₂×C₁₆	Q₈×C₃₁	semidirect
132. C₃₁	#13264	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
C₂×C₃₁×C₄⋊C₁₆	2
C₂×C₃₁×#346	2
C₃₁×#1374	2
#13264	31

Subgroups — C31×#13264 #21470023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13264 #21470023