Subgroups — C₇₁×#7465 #31828337

Order 18176.

202 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₂×C₇₁	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
4	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
6	C₂×C₃₂×C₇₁	—	2	2	no	—
7	C₂×C₃₂×C₇₁	—	2	2	no	—
8	C₂×C₃₂×C₇₁	C₄	1	4	yes	complemented
9	C₄×C₁₆×C₇₁	—	2	2	no	—
10	C₂²×C₈×C₇₁	C₈	1	1	yes	characteristic
11	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄²×C₇₁	C₈	1	1	yes	characteristic
13	C₂×C₁₆×C₇₁	—	2	1	no	—
14	C₂×C₁₆×C₇₁	—	2	1	no	—
15	C₂×C₁₆×C₇₁	C₈	1	2	yes	—
16	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
19	C₂×C₁₆×C₇₁	Q₈	1	1	yes	characteristic
20	C₄×C₈×C₇₁	—	2	1	no	—
21	C₄×C₈×C₇₁	—	2	1	no	—
22	C₃₂×C₇₁	—	2	4	no	—
23	C₂²×C₄×C₇₁	C₁₆	1	2	yes	—
24	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
25	C₂×C₈×C₇₁	—	2	1	no	—
26	C₂×C₈×C₇₁	—	2	1	no	—
27	C₂×C₈×C₇₁	C₁₆	1	2	yes	—
28	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
34	C₄²×C₇₁	—	2	1	no	—
35	C₄²×C₇₁	—	2	1	no	—
36	C₁₆×C₇₁	—	2	2	no	—
37	C₁₆×C₇₁	C₂²⋊C₄	1	2	yes	—
38	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
39	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₄×C₇₁	—	2	2	no	—
41	C₂×C₄×C₇₁	—	2	2	no	—
42	C₂×C₄×C₇₁	C₃₂	1	4	yes	complemented
43	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₁	M₃₂	1	1	yes	characteristic
48	C₂×C₄×C₇₁	M₃₂	1	1	yes	characteristic
49	C₈×C₇₁	—	2	2	no	—
50	C₈×C₇₁	#95	1	1	yes	characteristic
51	C₈×C₇₁	#97	1	1	yes	characteristic
52	C₈×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
53	C₈×C₇₁	C₄⋊C₈	1	1	yes	characteristic
54	C₂²×C₇₁	C₄×C₁₆	1	1	yes	characteristic
55	C₂²×C₇₁	C₂²⋊C₁₆	1	1	yes	characteristic
56	C₂²×C₇₁	C₄⋊C₁₆	1	1	yes	characteristic
57	C₂²×C₇₁	C₂×C₃₂	1	2	yes	—
58	C₂²×C₇₁	M₆₄	1	2	yes	—
59	C₄×C₇₁	—	2	4	no	—
60	C₄×C₇₁	#336	1	1	yes	characteristic
61	C₄×C₇₁	#346	1	1	yes	characteristic
62	C₄×C₇₁	C₂²⋊C₁₆	1	1	yes	characteristic
63	C₄×C₇₁	C₄⋊C₁₆	1	1	yes	characteristic
64	#7465	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₇₁	#1374	1	1	yes	characteristic
66	C₂×C₇₁	C₄×C₃₂	1	1	yes	characteristic
67	C₂×C₇₁	#1397	1	1	yes	characteristic
68	C₂×C₇₁	C₂²⋊C₃₂	1	2	yes	—
69	C₂×C₇₁	C₄⋊C₃₂	1	2	yes	—
70	C₂²×C₃₂	C₂×C₇₁	1	2	yes	—
71	C₂×C₄×C₁₆	C₂×C₇₁	1	1	yes	characteristic
72	C₇₁	#7465	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×C₁₆	C₄×C₇₁	1	1	yes	characteristic
74	C₂²×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
75	C₂×C₄×C₈	C₄×C₇₁	1	1	yes	characteristic
76	C₂×C₃₂	—	2	2	no	—
77	C₂×C₃₂	—	2	2	no	—
78	C₂×C₃₂	C₄×C₇₁	1	4	yes	complemented
79	C₄×C₁₆	—	2	2	no	—
80	C₂²×C₈	C₈×C₇₁	1	1	yes	characteristic
81	C₂²×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
82	C₂×C₄²	C₈×C₇₁	1	1	yes	characteristic
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	C₈×C₇₁	1	2	yes	—
86	C₂×C₁₆	C₂×C₄×C₇₁	1	2	yes	—
87	C₂×C₁₆	D₄×C₇₁	1	1	yes	characteristic
88	C₂×C₁₆	D₄×C₇₁	1	2	yes	—
89	C₂×C₁₆	Q₈×C₇₁	1	1	yes	characteristic
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₃₂	—	2	4	no	—
93	C₂²×C₄	C₁₆×C₇₁	1	2	yes	—
94	C₂²×C₄	C₂×C₈×C₇₁	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	C₁₆×C₇₁	1	2	yes	—
98	C₂×C₈	C₄²×C₇₁	1	1	yes	characteristic
99	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
100	C₂×C₈	C₄⋊C₄×C₇₁	1	1	yes	characteristic
101	C₂×C₈	C₂×C₈×C₇₁	1	1	yes	characteristic
102	C₂×C₈	M₁₆×C₇₁	1	1	yes	characteristic
103	C₂×C₈	M₁₆×C₇₁	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	—	2	1	no	—
106	C₁₆	—	2	2	no	—
107	C₁₆	C₂²⋊C₄×C₇₁	1	2	yes	—
108	C₁₆	C₄⋊C₄×C₇₁	1	2	yes	—
109	C₂³	C₂×C₁₆×C₇₁	1	1	yes	characteristic
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	C₃₂×C₇₁	1	4	yes	complemented
113	C₂×C₄	C₄×C₈×C₇₁	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊C₈×C₇₁	1	1	yes	characteristic
115	C₂×C₄	C₄⋊C₈×C₇₁	1	1	yes	characteristic
116	C₂×C₄	C₂×C₁₆×C₇₁	1	1	yes	characteristic
117	C₂×C₄	M₃₂×C₇₁	1	1	yes	characteristic
118	C₂×C₄	M₃₂×C₇₁	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	#95×C₇₁	1	1	yes	characteristic
121	C₈	#97×C₇₁	1	1	yes	characteristic
122	C₈	C₂²⋊C₈×C₇₁	1	1	yes	characteristic
123	C₈	C₄⋊C₈×C₇₁	1	1	yes	characteristic
124	C₂²	C₄×C₁₆×C₇₁	1	1	yes	characteristic
125	C₂²	C₂²⋊C₁₆×C₇₁	1	1	yes	characteristic
126	C₂²	C₄⋊C₁₆×C₇₁	1	1	yes	characteristic
127	C₂²	C₂×C₃₂×C₇₁	1	2	yes	—
128	C₂²	M₆₄×C₇₁	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	#336×C₇₁	1	1	yes	characteristic
131	C₄	#346×C₇₁	1	1	yes	characteristic
132	C₄	C₂²⋊C₁₆×C₇₁	1	1	yes	characteristic
133	C₄	C₄⋊C₁₆×C₇₁	1	1	yes	characteristic
134	C₂	C₇₁×#1374	1	1	yes	characteristic
135	C₂	C₄×C₃₂×C₇₁	1	1	yes	characteristic
136	C₂	C₇₁×#1397	1	1	yes	characteristic
137	C₂	C₇₁×C₂²⋊C₃₂	1	2	yes	—
138	C₂	C₇₁×C₄⋊C₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆×C₇₁	C₂²
commutator	C₂	C₄×C₃₂×C₇₁
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇₁
socle, prime radical	C₂³×C₇₁	C₂×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂×C₃₂×C₇₁	C₄	semidirect
42. C₂×C₄×C₇₁	C₃₂	semidirect
64. #7465	C₇₁	direct (N × H)
72. C₇₁	#7465	direct (N × H)
78. C₂×C₃₂	C₄×C₇₁	semidirect
112. C₂×C₄	C₃₂×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₂×C₇₁	2
C₂×C₄×C₁₆×C₇₁	2
#7465	71

Subgroups — C71×#7465 #31828337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7465 #31828337