Subgroups — C₈×C₃₁×#108 #21465051

Order 7936.

374 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₃₁	C₂	1	2	yes	maximal
4	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1278	C₂	1	2	yes	maximal
6	C₃₁×#431	C₂²	1	2	yes	—
7	C₂×C₃₁×#108	C₄	1	2	yes	—
8	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
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	C₃₁×#341	C₄	1	2	yes	—
13	C₃₁×#335	C₄	1	2	yes	—
14	C₃₁×#334	C₄	1	2	yes	—
15	C₃₁×#322	—	2	2	no	—
16	C₈²×C₃₁	—	2	2	no	—
17	C₈²×C₃₁	C₄	1	2	yes	—
18	C₈²×C₃₁	C₄	1	2	yes	—
19	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₃₁	—	2	1	no	—
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	Q₈	1	2	yes	—
24	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
25	C₃₁×#108	C₈	1	8	yes	direct product factor
26	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
27	C₃₁×C₂²⋊C₈	—	2	2	no	—
28	C₃₁×#97	—	2	2	no	—
29	C₄×C₈×C₃₁	—	2	2	no	—
30	C₄×C₈×C₃₁	—	2	2	no	—
31	C₄×C₈×C₃₁	—	2	1	no	—
32	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
43	M₁₆×C₃₁	—	2	4	no	—
44	M₁₆×C₃₁	C₂×C₈	1	8	yes	—
45	C₂×C₈×C₃₁	—	2	4	no	—
46	C₂×C₈×C₃₁	—	2	2	no	—
47	C₂×C₈×C₃₁	—	2	4	no	—
48	C₂×C₈×C₃₁	—	2	2	no	—
49	C₂×C₈×C₃₁	—	2	4	no	—
50	C₂×C₈×C₃₁	—	2	4	no	—
51	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
58	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₃₁	#41	1	1	yes	characteristic
62	C₈×#108	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₃₁	C₄×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₃₁	—	2	2	no	—
65	C₂×C₄×C₃₁	—	2	2	no	—
66	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₃₁	#131	1	2	yes	—
74	C₈×C₃₁	—	2	4	no	—
75	C₈×C₃₁	—	2	1	no	—
76	C₈×C₃₁	—	2	1	no	—
77	C₈×C₃₁	—	2	2	no	—
78	C₈×C₃₁	—	2	8	no	—
79	C₈×C₃₁	—	4	4	no	—
80	C₈×C₃₁	C₄×C₈	1	2	yes	—
81	C₈×C₃₁	C₄×C₈	1	2	yes	—
82	C₈×C₃₁	#108	1	4	yes	direct product factor
83	C₄×#108	C₂×C₃₁	1	1	yes	characteristic
84	#1564	C₂×C₃₁	1	1	yes	characteristic
85	C₈×M₁₆	C₂×C₃₁	1	2	yes	—
86	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic
87	#1278	C₂×C₃₁	1	2	yes	—
88	C₂²×C₃₁	—	2	1	no	—
89	C₂²×C₃₁	—	2	1	no	—
90	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₃₁	C₈×Q₈	1	2	yes	—
92	C₄×C₃₁	—	2	1	no	—
93	C₄×C₃₁	—	2	1	no	—
94	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
95	C₄×C₃₁	#405	1	1	yes	characteristic
96	C₄×C₃₁	C₂×#108	1	2	yes	—
97	C₄×C₃₁	C₈×D₄	1	2	yes	—
98	C₄×C₃₁	#443	1	2	yes	—
99	#431	C₂²×C₃₁	1	2	yes	—
100	C₂×#108	C₄×C₃₁	1	2	yes	—
101	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
105	#341	C₄×C₃₁	1	2	yes	—
106	#335	C₄×C₃₁	1	2	yes	—
107	#334	C₄×C₃₁	1	2	yes	—
108	#322	—	2	2	no	—
109	C₈²	—	2	2	no	—
110	C₈²	C₄×C₃₁	1	2	yes	—
111	C₈²	C₄×C₃₁	1	2	yes	—
112	C₂×C₃₁	—	2	2	no	—
113	C₂×C₃₁	C₈×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₃₁	#1772	1	1	yes	characteristic
115	C₂×C₃₁	C₄×#108	1	1	yes	characteristic
116	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
120	C₂²×C₈	Q₈×C₃₁	1	2	yes	—
121	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
122	#108	C₈×C₃₁	1	8	yes	direct product factor
123	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
124	C₂²⋊C₈	—	2	2	no	—
125	#97	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
130	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
131	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
132	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
133	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
134	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
135	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
136	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
137	C₃₁	C₈×#108	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
139	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
140	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
141	M₁₆	—	2	4	no	—
142	M₁₆	C₂×C₈×C₃₁	1	8	yes	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
150	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
151	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
152	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
153	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
154	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
155	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
156	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
157	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
158	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
159	C₄²	#41×C₃₁	1	1	yes	characteristic
160	C₂³	C₄×Q₈×C₃₁	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
166	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
167	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
168	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
170	C₂×C₄	C₃₁×#131	1	2	yes	—
171	C₈	—	2	4	no	—
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	2	2	no	—
175	C₈	—	2	8	no	—
176	C₈	—	4	4	no	—
177	C₈	C₄×C₈×C₃₁	1	2	yes	—
178	C₈	C₄×C₈×C₃₁	1	2	yes	—
179	C₈	C₃₁×#108	1	4	yes	direct product factor
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
183	C₂²	C₈×Q₈×C₃₁	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
187	C₄	C₃₁×#405	1	1	yes	characteristic
188	C₄	C₂×C₃₁×#108	1	2	yes	—
189	C₄	C₈×D₄×C₃₁	1	2	yes	—
190	C₄	C₃₁×#443	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	C₈×C₄⋊C₄×C₃₁	1	1	yes	characteristic
193	C₂	C₃₁×#1772	1	1	yes	characteristic
194	C₂	C₄×C₃₁×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈×C₃₁	D₄
commutator	C₄	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×C₄⋊C₄
prime radical	C₂³×C₃₁	C₄×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₃₁×#108	C₈	direct (N × H)
62. C₈×#108	C₃₁	direct (N × H)
82. C₈×C₃₁	#108	direct (N × H)
122. #108	C₈×C₃₁	direct (N × H)
137. C₃₁	C₈×#108	direct (N × H)
179. C₈	C₃₁×#108	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×#108	2
C₃₁×#1564	2
C₈×M₁₆×C₃₁	2
C₂×C₈²×C₃₁	2
C₃₁×#1278	2
C₈×#108	31

Subgroups — C8×C31×#108 #21465051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₃₁×#108 #21465051