Subgroups — C₈×C₃₁×#106 #21465039

Order 7936.

446 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal
3	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1279	C₂	1	2	yes	maximal
6	C₂×C₃₁×#106	C₄	1	2	yes	—
7	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₃₁	—	2	2	no	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#340	C₄	1	2	yes	—
14	C₃₁×#336	—	2	2	no	—
15	C₃₁×#334	C₄	1	4	yes	complemented
16	C₈²×C₃₁	C₄	1	4	yes	complemented
17	C₂²×C₈×C₃₁	—	2	1	no	—
18	C₂²×C₈×C₃₁	—	2	2	no	—
19	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₃₁	—	2	2	no	—
25	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
26	C₃₁×#106	C₈	1	8	yes	direct product factor
27	C₃₁×C₄⋊C₈	—	2	2	no	—
28	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₃₁	—	2	1	no	—
30	C₄×C₈×C₃₁	—	4	4	no	—
31	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₃₁	Q₈	1	2	yes	—
36	C₃₁×#95	—	2	2	no	—
37	C₂²×C₄×C₃₁	—	2	4	no	—
38	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
41	C₂×C₈×C₃₁	—	2	4	no	—
42	C₂×C₈×C₃₁	—	2	2	no	—
43	C₂×C₈×C₃₁	—	2	2	no	—
44	C₂×C₈×C₃₁	—	2	2	no	—
45	C₂×C₈×C₃₁	—	2	4	no	—
46	C₂×C₈×C₃₁	—	4	2	no	—
47	C₂×C₈×C₃₁	—	4	2	no	—
48	C₂×C₈×C₃₁	C₄²	1	2	yes	—
49	C₂×C₈×C₃₁	C₄²	1	2	yes	—
50	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
53	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
54	C₄⋊C₄×C₃₁	—	2	4	no	—
55	C₄⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
56	C₄²×C₃₁	—	4	2	no	—
57	C₄²×C₃₁	—	4	2	no	—
58	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₃₁	#41	1	1	yes	characteristic
62	C₈×#106	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₃₁	—	2	2	no	—
65	C₂×C₄×C₃₁	—	2	8	no	—
66	C₂×C₄×C₃₁	—	4	4	no	—
67	C₂×C₄×C₃₁	—	4	4	no	—
68	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₃₁	#131	1	2	yes	—
76	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₃₁	#135	1	2	yes	—
78	C₈×C₃₁	—	2	4	no	—
79	C₈×C₃₁	—	2	2	no	—
80	C₈×C₃₁	—	2	2	no	—
81	C₈×C₃₁	—	4	4	no	—
82	C₈×C₃₁	C₄×C₈	1	4	yes	complemented
83	C₈×C₃₁	#106	1	4	yes	direct product factor
84	C₄×#106	C₂×C₃₁	1	1	yes	characteristic
85	C₈×C₄⋊C₄	C₂×C₃₁	1	2	yes	—
86	C₂×#334	C₂×C₃₁	1	1	yes	characteristic
87	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic
88	#1279	C₂×C₃₁	1	2	yes	—
89	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₃₁	C₈×D₄	1	2	yes	—
91	C₂²×C₃₁	C₄×SD₁₆	1	2	yes	—
92	C₂²×C₃₁	#443	1	2	yes	—
93	C₄×C₃₁	—	2	1	no	—
94	C₄×C₃₁	—	2	1	no	—
95	C₄×C₃₁	—	4	8	no	—
96	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
97	C₄×C₃₁	#405	1	1	yes	characteristic
98	C₄×C₃₁	C₂×#106	1	1	yes	characteristic
99	C₄×C₃₁	#427	1	1	yes	characteristic
100	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
101	C₄×C₃₁	C₈×Q₈	1	2	yes	—
102	C₂×#106	C₄×C₃₁	1	2	yes	—
103	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
104	C₂×C₄×C₈	—	2	2	no	—
105	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
108	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
109	#340	C₄×C₃₁	1	2	yes	—
110	#336	—	2	2	no	—
111	#334	C₄×C₃₁	1	4	yes	complemented
112	C₈²	C₄×C₃₁	1	4	yes	complemented
113	C₂×C₃₁	C₈×SD₁₆	1	4	yes	—
114	C₂×C₃₁	C₈×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₃₁	#1772	1	1	yes	characteristic
116	C₂×C₃₁	C₄×#106	1	1	yes	characteristic
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
120	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
121	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
122	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
123	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
124	C₂×C₄²	—	2	2	no	—
125	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
126	#106	C₈×C₃₁	1	8	yes	direct product factor
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	4	4	no	—
131	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
132	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
133	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
134	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
135	C₄×C₈	Q₈×C₃₁	1	2	yes	—
136	#95	—	2	2	no	—
137	C₃₁	C₈×#106	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
141	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	C₄²×C₃₁	1	2	yes	—
150	C₂×C₈	C₄²×C₃₁	1	2	yes	—
151	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
152	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
153	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
154	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
157	C₄²	—	4	2	no	—
158	C₄²	—	4	2	no	—
159	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
160	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
161	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
162	C₄²	#41×C₃₁	1	1	yes	characteristic
163	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
170	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
171	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
172	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
173	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
175	C₂×C₄	C₃₁×#131	1	2	yes	—
176	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
177	C₂×C₄	C₃₁×#135	1	2	yes	—
178	C₈	—	2	4	no	—
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	—	4	4	no	—
182	C₈	C₄×C₈×C₃₁	1	4	yes	complemented
183	C₈	C₃₁×#106	1	4	yes	direct product factor
184	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
185	C₂²	C₈×D₄×C₃₁	1	2	yes	—
186	C₂²	C₄×SD₁₆×C₃₁	1	2	yes	—
187	C₂²	C₃₁×#443	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	4	8	no	—
191	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
192	C₄	C₃₁×#405	1	1	yes	characteristic
193	C₄	C₂×C₃₁×#106	1	1	yes	characteristic
194	C₄	C₃₁×#427	1	1	yes	characteristic
195	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
196	C₄	C₈×Q₈×C₃₁	1	2	yes	—
197	C₂	C₈×SD₁₆×C₃₁	1	4	yes	—
198	C₂	C₈×C₄⋊C₄×C₃₁	1	1	yes	characteristic
199	C₂	C₃₁×#1772	1	1	yes	characteristic
200	C₂	C₄×C₃₁×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₃₁	D₄
commutator	C₄	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₃₁×#334	C₄	semidirect
16. C₈²×C₃₁	C₄	semidirect
26. C₃₁×#106	C₈	direct (N × H)
62. C₈×#106	C₃₁	direct (N × H)
82. C₈×C₃₁	C₄×C₈	semidirect
83. C₈×C₃₁	#106	direct (N × H)
111. #334	C₄×C₃₁	semidirect
112. C₈²	C₄×C₃₁	semidirect
126. #106	C₈×C₃₁	direct (N × H)
137. C₃₁	C₈×#106	direct (N × H)
182. C₈	C₄×C₈×C₃₁	semidirect
183. C₈	C₃₁×#106	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#106	2
C₈×C₄⋊C₄×C₃₁	2
C₂×C₃₁×#334	2
C₂×C₈²×C₃₁	2
C₃₁×#1279	2
C₈×#106	31

Subgroups — C8×C31×#106 #21465039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₃₁×#106 #21465039