Subgroups — C₈×#108×C₇₁ #31829164

Order 18176.

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×#108×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1564	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₇₁	C₂	1	2	yes	maximal
4	C₂×C₈²×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1278	C₂	1	2	yes	maximal
6	#431×C₇₁	C₂²	1	2	yes	—
7	C₂×#108×C₇₁	C₄	1	2	yes	—
8	C₄×M₁₆×C₇₁	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#341×C₇₁	C₄	1	2	yes	—
13	#335×C₇₁	C₄	1	2	yes	—
14	#334×C₇₁	C₄	1	2	yes	—
15	#322×C₇₁	—	2	2	no	—
16	C₈²×C₇₁	—	2	2	no	—
17	C₈²×C₇₁	C₄	1	2	yes	—
18	C₈²×C₇₁	C₄	1	2	yes	—
19	C₂×M₁₆×C₇₁	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₇₁	—	2	1	no	—
21	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
24	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
25	#108×C₇₁	C₈	1	8	yes	direct product factor
26	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
27	C₂²⋊C₈×C₇₁	—	2	2	no	—
28	#97×C₇₁	—	2	2	no	—
29	C₄×C₈×C₇₁	—	2	2	no	—
30	C₄×C₈×C₇₁	—	2	2	no	—
31	C₄×C₈×C₇₁	—	2	1	no	—
32	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
39	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
43	M₁₆×C₇₁	—	2	4	no	—
44	M₁₆×C₇₁	C₂×C₈	1	8	yes	—
45	C₂×C₈×C₇₁	—	2	4	no	—
46	C₂×C₈×C₇₁	—	2	2	no	—
47	C₂×C₈×C₇₁	—	2	4	no	—
48	C₂×C₈×C₇₁	—	2	2	no	—
49	C₂×C₈×C₇₁	—	2	4	no	—
50	C₂×C₈×C₇₁	—	2	4	no	—
51	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
58	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₇₁	#41	1	1	yes	characteristic
62	C₂³×C₇₁	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₇₁	—	2	2	no	—
64	C₂×C₄×C₇₁	—	2	2	no	—
65	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₇₁	#131	1	2	yes	—
73	C₈×C₇₁	—	2	4	no	—
74	C₈×C₇₁	—	2	1	no	—
75	C₈×C₇₁	—	2	1	no	—
76	C₈×C₇₁	—	2	2	no	—
77	C₈×C₇₁	—	2	8	no	—
78	C₈×C₇₁	—	4	4	no	—
79	C₈×C₇₁	C₄×C₈	1	2	yes	—
80	C₈×C₇₁	C₄×C₈	1	2	yes	—
81	C₈×C₇₁	#108	1	4	yes	direct product factor
82	C₂²×C₇₁	—	2	1	no	—
83	C₂²×C₇₁	—	2	1	no	—
84	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₇₁	C₈×Q₈	1	2	yes	—
86	C₄×C₇₁	—	2	1	no	—
87	C₄×C₇₁	—	2	1	no	—
88	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
89	C₄×C₇₁	#405	1	1	yes	characteristic
90	C₄×C₇₁	C₂×#108	1	2	yes	—
91	C₄×C₇₁	C₈×D₄	1	2	yes	—
92	C₄×C₇₁	#443	1	2	yes	—
93	C₈×#108	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×C₇₁	—	2	2	no	—
95	C₂×C₇₁	C₈×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₇₁	#1772	1	1	yes	characteristic
97	C₂×C₇₁	C₄×#108	1	1	yes	characteristic
98	C₄×#108	C₂×C₇₁	1	1	yes	characteristic
99	#1564	C₂×C₇₁	1	1	yes	characteristic
100	C₈×M₁₆	C₂×C₇₁	1	2	yes	—
101	C₂×C₈²	C₂×C₇₁	1	1	yes	characteristic
102	#1278	C₂×C₇₁	1	2	yes	—
103	C₇₁	C₈×#108	1	1	yes	characteristic, direct product factor, Sylow
104	#431	C₂²×C₇₁	1	2	yes	—
105	C₂×#108	C₄×C₇₁	1	2	yes	—
106	C₄×M₁₆	C₂²×C₇₁	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
110	#341	C₄×C₇₁	1	2	yes	—
111	#335	C₄×C₇₁	1	2	yes	—
112	#334	C₄×C₇₁	1	2	yes	—
113	#322	—	2	2	no	—
114	C₈²	—	2	2	no	—
115	C₈²	C₄×C₇₁	1	2	yes	—
116	C₈²	C₄×C₇₁	1	2	yes	—
117	C₂×M₁₆	C₂×C₄×C₇₁	1	4	yes	—
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
120	C₂²×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
121	C₂²×C₈	Q₈×C₇₁	1	2	yes	—
122	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
123	#108	C₈×C₇₁	1	8	yes	direct product factor
124	C₄⋊C₈	C₂×C₄×C₇₁	1	4	yes	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
131	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
132	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
133	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
134	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
135	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
136	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
137	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
138	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
139	C₂²×C₄	C₂×Q₈×C₇₁	1	1	yes	characteristic
140	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
141	M₁₆	—	2	4	no	—
142	M₁₆	C₂×C₈×C₇₁	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×C₇₁	1	1	yes	characteristic
150	C₂×C₈	C₄²×C₇₁	1	1	yes	characteristic
151	C₂×C₈	C₄²×C₇₁	1	1	yes	characteristic
152	C₂×C₈	C₄²×C₇₁	1	1	yes	characteristic
153	C₂×C₈	C₄⋊C₄×C₇₁	1	2	yes	—
154	C₂×C₈	C₄⋊C₄×C₇₁	1	2	yes	—
155	C₂×C₈	C₂×C₈×C₇₁	1	4	yes	—
156	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
157	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
158	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
159	C₄²	#41×C₇₁	1	1	yes	characteristic
160	C₂³	C₄×Q₈×C₇₁	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄²×C₇₁	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
165	C₂×C₄	#117×C₇₁	1	1	yes	characteristic
166	C₂×C₄	C₄×D₄×C₇₁	1	1	yes	characteristic
167	C₂×C₄	C₄×D₄×C₇₁	1	1	yes	characteristic
168	C₂×C₄	C₄×Q₈×C₇₁	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₈×C₇₁	1	2	yes	—
170	C₂×C₄	#131×C₇₁	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×C₇₁	1	2	yes	—
178	C₈	C₄×C₈×C₇₁	1	2	yes	—
179	C₈	#108×C₇₁	1	4	yes	direct product factor
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄×C₄⋊C₄×C₇₁	1	1	yes	characteristic
183	C₂²	C₈×Q₈×C₇₁	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×C₄×C₈×C₇₁	1	1	yes	characteristic
187	C₄	#405×C₇₁	1	1	yes	characteristic
188	C₄	C₂×#108×C₇₁	1	2	yes	—
189	C₄	C₈×D₄×C₇₁	1	2	yes	—
190	C₄	#443×C₇₁	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	C₈×C₄⋊C₄×C₇₁	1	1	yes	characteristic
193	C₂	C₇₁×#1772	1	1	yes	characteristic
194	C₂	C₄×#108×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈×C₇₁	D₄
commutator	C₄	C₂×C₄×C₈×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
socle	C₂²×C₇₁	C₄×C₄⋊C₄
prime radical	C₂³×C₇₁	C₄×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
25. #108×C₇₁	C₈	direct (N × H)
81. C₈×C₇₁	#108	direct (N × H)
93. C₈×#108	C₇₁	direct (N × H)
103. C₇₁	C₈×#108	direct (N × H)
123. #108	C₈×C₇₁	direct (N × H)
179. C₈	#108×C₇₁	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×#108×C₇₁	2
C₇₁×#1564	2
C₈×M₁₆×C₇₁	2
C₂×C₈²×C₇₁	2
C₇₁×#1278	2
C₈×#108	71

Subgroups — C8×#108×C71 #31829164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#108×C₇₁ #31829164