Subgroups — C₇₉×#2991 #23760613

Order 10112.

394 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈×C₇₉	C₂	1	2	yes	maximal, complemented
3	#436×C₇₉	C₂	1	6	yes	maximal, complemented
4	#431×C₇₉	C₂	1	3	yes	maximal, complemented
5	#423×C₇₉	C₂	1	3	yes	maximal, complemented
6	C₂×#41×C₇₉	C₄	1	1	yes	characteristic
7	C₂×M₁₆×C₇₉	C₂²	1	6	yes	—
8	C₄×Q₈×C₇₉	C₄	1	1	yes	characteristic
9	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
10	C₄×D₄×C₇₉	C₄	1	3	yes	—
11	C₄×D₄×C₇₉	C₂²	1	3	yes	—
12	#117×C₇₉	C₄	1	3	yes	—
13	C₂×C₄²×C₇₉	C₂²	1	3	yes	—
14	C₄⋊C₈×C₇₉	C₂²	1	3	yes	—
15	C₄⋊C₈×C₇₉	C₂²	1	6	yes	—
16	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²⋊C₈×C₇₉	C₂²	1	6	yes	—
18	C₄×C₈×C₇₉	C₂²	1	6	yes	—
19	#41×C₇₉	—	2	4	no	—
20	C₂×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₇₉	C₂×C₄	1	3	yes	—
22	C₂²×C₄×C₇₉	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₇₉	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₇₉	C₂³	1	3	yes	—
25	M₁₆×C₇₉	—	4	6	no	—
26	C₂×C₈×C₇₉	C₂³	1	2	yes	—
27	C₂×C₈×C₇₉	C₂³	1	6	yes	—
28	C₄⋊C₄×C₇₉	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₇₉	C₂×C₄	1	3	yes	—
30	C₂²⋊C₄×C₇₉	C₂×C₄	1	6	yes	—
31	C₄²×C₇₉	C₂×C₄	1	3	yes	—
32	C₄²×C₇₉	C₂×C₄	1	3	yes	—
33	C₄²×C₇₉	C₂³	1	3	yes	—
34	C₄²×C₇₉	C₂³	1	1	yes	characteristic
35	C₂³×C₇₉	C₂²×C₄	1	3	yes	—
36	Q₈×C₇₉	M₁₆	1	4	yes	complemented
37	D₄×C₇₉	—	2	6	no	—
38	C₂×C₄×C₇₉	—	2	6	no	—
39	C₂×C₄×C₇₉	—	2	6	no	—
40	C₂×C₄×C₇₉	—	2	3	no	—
41	C₂×C₄×C₇₉	—	2	3	no	—
42	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₇₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₇₉	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
48	C₈×C₇₉	—	2	6	no	—
49	C₈×C₇₉	—	2	2	no	—
50	C₂²×C₇₉	—	2	3	no	—
51	C₂²×C₇₉	—	2	3	no	—
52	C₂²×C₇₉	—	2	3	no	—
53	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₇₉	—	2	3	no	—
55	C₄×C₇₉	—	2	1	no	—
56	C₄×C₇₉	—	2	3	no	—
57	C₄×C₇₉	C₂×M₁₆	1	6	yes	—
58	C₄×C₇₉	#131	1	2	yes	—
59	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
60	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
61	C₂×C₇₉	—	4	3	no	—
62	C₂×C₇₉	#520	1	1	yes	characteristic
63	C₂×C₇₉	C₂²×M₁₆	1	1	yes	characteristic
64	C₂×C₇₉	C₂×#131	1	1	yes	characteristic
65	#2991	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₇₉	#2991	1	1	yes	characteristic, direct product factor, Sylow
67	C₄×#41	C₂×C₇₉	1	1	yes	characteristic
68	C₈×Q₈	C₂×C₇₉	1	2	yes	complemented
69	#436	C₂×C₇₉	1	6	yes	complemented
70	#431	C₂×C₇₉	1	3	yes	complemented
71	#423	C₂×C₇₉	1	3	yes	complemented
72	C₂×#41	C₄×C₇₉	1	1	yes	characteristic
73	C₂×M₁₆	C₂²×C₇₉	1	6	yes	—
74	C₄×Q₈	C₄×C₇₉	1	1	yes	characteristic
75	C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
76	C₄×D₄	C₄×C₇₉	1	3	yes	—
77	C₄×D₄	C₂²×C₇₉	1	3	yes	—
78	#117	C₄×C₇₉	1	3	yes	—
79	C₂×C₄²	C₂²×C₇₉	1	3	yes	—
80	C₄⋊C₈	C₂²×C₇₉	1	3	yes	—
81	C₄⋊C₈	C₂²×C₇₉	1	6	yes	—
82	C₄⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
83	C₂²⋊C₈	C₂²×C₇₉	1	6	yes	—
84	C₄×C₈	C₂²×C₇₉	1	6	yes	—
85	#41	—	2	4	no	—
86	C₂×Q₈	C₂×C₄×C₇₉	1	1	yes	characteristic
87	C₂×D₄	C₂×C₄×C₇₉	1	3	yes	—
88	C₂²×C₄	C₂×C₄×C₇₉	1	3	yes	—
89	C₂²×C₄	C₂×C₄×C₇₉	1	3	yes	—
90	C₂²×C₄	C₂³×C₇₉	1	3	yes	—
91	M₁₆	—	4	6	no	—
92	C₂×C₈	C₂³×C₇₉	1	2	yes	—
93	C₂×C₈	C₂³×C₇₉	1	6	yes	—
94	C₄⋊C₄	C₂×C₄×C₇₉	1	3	yes	—
95	C₄⋊C₄	C₂×C₄×C₇₉	1	3	yes	—
96	C₂²⋊C₄	C₂×C₄×C₇₉	1	6	yes	—
97	C₄²	C₂×C₄×C₇₉	1	3	yes	—
98	C₄²	C₂×C₄×C₇₉	1	3	yes	—
99	C₄²	C₂³×C₇₉	1	3	yes	—
100	C₄²	C₂³×C₇₉	1	1	yes	characteristic
101	C₂³	C₂²×C₄×C₇₉	1	3	yes	—
102	Q₈	M₁₆×C₇₉	1	4	yes	complemented
103	D₄	—	2	6	no	—
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	2	6	no	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	C₂²×C₄×C₇₉	1	1	yes	characteristic
109	C₂×C₄	C₂²×C₄×C₇₉	1	3	yes	—
110	C₂×C₄	C₂²×C₄×C₇₉	1	3	yes	—
111	C₂×C₄	C₂²×C₄×C₇₉	1	3	yes	—
112	C₂×C₄	C₂²×C₄×C₇₉	1	1	yes	characteristic
113	C₂×C₄	C₂⁴×C₇₉	1	1	yes	characteristic
114	C₈	—	2	6	no	—
115	C₈	—	2	2	no	—
116	C₂²	—	2	3	no	—
117	C₂²	—	2	3	no	—
118	C₂²	—	2	3	no	—
119	C₂²	C₂³×C₄×C₇₉	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	2	1	no	—
122	C₄	—	2	3	no	—
123	C₄	C₂×M₁₆×C₇₉	1	6	yes	—
124	C₄	#131×C₇₉	1	2	yes	—
125	C₄	H(5,𝔽₂)×C₇₉	1	1	yes	characteristic
126	C₄	2₋¹⁺⁴×C₇₉	1	1	yes	characteristic
127	C₂	—	4	3	no	—
128	C₂	#520×C₇₉	1	1	yes	characteristic
129	C₂	C₂²×M₁₆×C₇₉	1	1	yes	characteristic
130	C₂	C₂×#131×C₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₇₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₇₉
socle	C₂²×C₇₉	C₂³×C₄
prime radical	C₂×#41×C₇₉	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×Q₈×C₇₉	C₂	semidirect
3. #436×C₇₉	C₂	semidirect
4. #431×C₇₉	C₂	semidirect
5. #423×C₇₉	C₂	semidirect
36. Q₈×C₇₉	M₁₆	semidirect
65. #2991	C₇₉	direct (N × H)
66. C₇₉	#2991	direct (N × H)
68. C₈×Q₈	C₂×C₇₉	semidirect
69. #436	C₂×C₇₉	semidirect
70. #431	C₂×C₇₉	semidirect
71. #423	C₂×C₇₉	semidirect
102. Q₈	M₁₆×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#41×C₇₉	2
C₈×Q₈×C₇₉	2
#436×C₇₉	2
#431×C₇₉	2
#423×C₇₉	2
#2991	79

Subgroups — C79×#2991 #23760613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#2991 #23760613