Subgroups — C₇₁×#12909 #31833781

Order 18176.

314 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#368×C₇₁	C₂	1	4	yes	maximal
2	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1848	C₂	1	2	yes	maximal
4	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
5	C₂×#108×C₇₁	—	2	2	no	—
6	C₂×#108×C₇₁	C₂²	1	4	yes	—
7	#423×C₇₁	—	2	2	no	—
8	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
9	#368×C₇₁	—	2	8	no	—
10	C₄×C₁₆×C₇₁	C₄	1	2	yes	—
11	C₄×C₁₆×C₇₁	C₄	1	2	yes	—
12	C₂×M₁₆×C₇₁	—	2	4	no	—
13	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
15	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
17	C₂×C₁₆×C₇₁	D₄	1	4	yes	—
18	C₂×C₁₆×C₇₁	Q₈	1	4	yes	—
19	#108×C₇₁	—	2	8	no	—
20	#108×C₇₁	—	4	4	no	—
21	C₄⋊C₈×C₇₁	—	4	4	no	—
22	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
27	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
28	M₁₆×C₇₁	—	4	4	no	—
29	M₁₆×C₇₁	—	4	4	no	—
30	C₂×C₈×C₇₁	—	4	4	no	—
31	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₁	D₈	1	2	yes	—
34	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
35	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
37	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
38	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₇₁	D₈	1	2	yes	—
42	C₁₆×C₇₁	C₄⋊C₄	1	8	yes	—
43	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₁	—	2	4	no	—
45	C₂×C₄×C₇₁	#107	1	2	yes	—
46	C₂×C₄×C₇₁	#107	1	2	yes	—
47	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
53	C₈×C₇₁	—	8	4	no	—
54	C₈×C₇₁	#107	1	2	yes	—
55	C₈×C₇₁	#107	1	2	yes	—
56	C₈×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
57	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
58	C₂²×C₇₁	—	2	2	no	—
59	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
60	C₂²×C₇₁	#494	1	1	yes	characteristic
61	C₂²×C₇₁	#500	1	1	yes	characteristic
62	C₄×C₇₁	—	2	2	no	—
63	C₄×C₇₁	#368	1	4	yes	complemented
64	C₄×C₇₁	#384	1	1	yes	characteristic
65	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
66	C₄×C₇₁	#493	1	1	yes	characteristic
67	C₄×C₇₁	#500	1	1	yes	characteristic
68	#12909	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₇₁	—	2	1	no	—
70	C₂×C₇₁	—	2	1	no	—
71	C₂×C₇₁	#1845	1	1	yes	characteristic
72	C₂×C₇₁	C₂×#368	1	2	yes	—
73	C₂×#368	C₂×C₇₁	1	4	yes	—
74	C₂×C₄×C₁₆	C₂×C₇₁	1	1	yes	characteristic
75	#1848	C₂×C₇₁	1	2	yes	—
76	C₇₁	#12909	1	1	yes	characteristic, direct product factor, Sylow
77	C₂²×C₁₆	C₂²×C₇₁	1	2	yes	—
78	C₂×#108	—	2	2	no	—
79	C₂×#108	C₂²×C₇₁	1	4	yes	—
80	#423	—	2	2	no	—
81	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
82	#368	—	2	8	no	—
83	C₄×C₁₆	C₄×C₇₁	1	2	yes	—
84	C₄×C₁₆	C₄×C₇₁	1	2	yes	—
85	C₂×M₁₆	—	2	4	no	—
86	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
87	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
88	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
89	C₂×C₁₆	C₂×C₄×C₇₁	1	4	yes	—
90	C₂×C₁₆	D₄×C₇₁	1	4	yes	—
91	C₂×C₁₆	Q₈×C₇₁	1	4	yes	—
92	#108	—	2	8	no	—
93	#108	—	4	4	no	—
94	C₄⋊C₈	—	4	4	no	—
95	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
96	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
97	C₄×C₈	Q₈×C₇₁	1	1	yes	characteristic
98	C₄×C₈	Q₈×C₇₁	1	1	yes	characteristic
99	C₂²×C₄	Q₁₆×C₇₁	1	2	yes	—
100	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
101	M₁₆	—	4	4	no	—
102	M₁₆	—	4	4	no	—
103	C₂×C₈	—	4	4	no	—
104	C₂×C₈	C₄⋊C₄×C₇₁	1	1	yes	characteristic
105	C₂×C₈	C₄⋊C₄×C₇₁	1	1	yes	characteristic
106	C₂×C₈	D₈×C₇₁	1	2	yes	—
107	C₂×C₈	Q₁₆×C₇₁	1	2	yes	—
108	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
109	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
110	C₂×C₈	C₂×Q₈×C₇₁	1	2	yes	—
111	C₂×C₈	C₂×Q₈×C₇₁	1	1	yes	characteristic
112	C₄²	C₄⋊C₄×C₇₁	1	1	yes	characteristic
113	C₄²	C₄⋊C₄×C₇₁	1	1	yes	characteristic
114	C₄²	D₈×C₇₁	1	2	yes	—
115	C₁₆	C₄⋊C₄×C₇₁	1	8	yes	—
116	C₂³	C₂×Q₁₆×C₇₁	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	#107×C₇₁	1	2	yes	—
119	C₂×C₄	#107×C₇₁	1	2	yes	—
120	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
121	C₂×C₄	#127×C₇₁	1	1	yes	characteristic
122	C₂×C₄	C₄⋊Q₈×C₇₁	1	1	yes	characteristic
123	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
124	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
125	C₂×C₄	C₂×Q₁₆×C₇₁	1	1	yes	characteristic
126	C₈	—	8	4	no	—
127	C₈	#107×C₇₁	1	2	yes	—
128	C₈	#107×C₇₁	1	2	yes	—
129	C₈	C₂×C₄⋊C₄×C₇₁	1	2	yes	—
130	C₈	C₄⋊Q₈×C₇₁	1	2	yes	—
131	C₂²	—	2	2	no	—
132	C₂²	C₂×#107×C₇₁	1	1	yes	characteristic
133	C₂²	#494×C₇₁	1	1	yes	characteristic
134	C₂²	#500×C₇₁	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	#368×C₇₁	1	4	yes	complemented
137	C₄	#384×C₇₁	1	1	yes	characteristic
138	C₄	C₂×#107×C₇₁	1	1	yes	characteristic
139	C₄	#493×C₇₁	1	1	yes	characteristic
140	C₄	#500×C₇₁	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	C₇₁×#1845	1	1	yes	characteristic
144	C₂	C₂×#368×C₇₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇₁
Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
63. C₄×C₇₁	#368	semidirect
68. #12909	C₇₁	direct (N × H)
76. C₇₁	#12909	direct (N × H)
136. C₄	#368×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#368×C₇₁	2
C₂×C₄×C₁₆×C₇₁	2
C₇₁×#1848	2
#12909	71

Subgroups — C71×#12909 #31833781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#12909 #31833781