Subgroups — C₇×#5398 #13057118

Order 1344.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×A₄⋊Q₈	C₂	1	2	yes	maximal
2	C₇×C₈×A₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#467	—	3	1	no	maximal, Hall
4	C₄×C₇×A₄	C₂²	1	1	yes	characteristic
5	C₇×SL(2,ℤ/4)	—	2	2	no	—
6	C₇×Dic₁₂	—	4	1	no	maximal
7	C₂×C₇×Q₁₆	—	3	1	no	—
8	C₂²×C₇×C₈	D₃	1	1	yes	characteristic
9	C₇×C₂²⋊Q₈	—	3	2	no	—
10	C₇×#107	—	3	1	no	—
11	C₇×#103	—	3	2	no	—
12	#5398	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₇×A₄	D₄	1	1	yes	characteristic
14	C₇×Dic₆	—	4	2	no	—
15	C₃×C₇×C₈	—	4	1	no	—
16	C₂×C₇×Q₈	—	3	2	no	—
17	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
18	C₇×Q₁₆	—	6	1	no	—
19	C₇×Q₁₆	—	6	1	no	—
20	C₂×C₇×C₈	—	3	1	no	—
21	C₂×C₇×C₈	—	3	1	no	—
22	C₇×C₄⋊C₄	—	3	2	no	—
23	C₇×C₄⋊C₄	—	6	2	no	—
24	C₇×C₂²⋊C₄	—	6	2	no	—
25	A₄⋊Q₈	C₂×C₇	1	2	yes	—
26	C₈×A₄	C₂×C₇	1	1	yes	characteristic
27	C₇×A₄	Q₁₆	1	1	yes	characteristic, complemented
28	C₃×C₄×C₇	—	4	1	no	—
29	C₇×Dic₃	—	8	2	no	—
30	#467	—	3	1	no	Sylow
31	C₂³×C₇	D₁₂	1	1	yes	characteristic
32	C₇×Q₈	—	6	2	no	—
33	C₇×Q₈	—	6	2	no	—
34	C₂×C₄×C₇	—	3	1	no	—
35	C₂×C₄×C₇	—	3	1	no	—
36	C₂×C₄×C₇	—	6	2	no	—
37	C₂×C₄×C₇	—	6	2	no	—
38	C₇×C₈	—	3	1	no	—
39	C₇×C₈	S₄	1	1	yes	characteristic
40	C₄×A₄	C₂²×C₇	1	1	yes	characteristic
41	SL(2,ℤ/4)	—	2	2	no	—
42	Dic₁₂	—	4	1	no	—
43	C₂×C₃×C₇	—	4	1	no	—
44	C₂×Q₁₆	—	3	1	no	—
45	C₂²×C₈	D₃×C₇	1	1	yes	characteristic
46	C₂²⋊Q₈	—	3	2	no	—
47	#107	—	3	1	no	—
48	#103	—	3	2	no	—
49	C₂²×C₇	—	3	1	no	—
50	C₂²×C₇	—	3	1	no	—
51	C₂²×C₇	Dic₁₂	1	1	yes	characteristic, complemented
52	C₄×C₇	—	3	1	no	—
53	C₄×C₇	—	12	2	no	—
54	C₄×C₇	—	12	2	no	—
55	C₄×C₇	C₂×S₄	1	1	yes	characteristic
56	C₂×A₄	C₇×D₄	1	1	yes	characteristic
57	Dic₆	—	4	2	no	—
58	C₃×C₈	—	4	1	no	—
59	C₃×C₇	—	4	1	no	Hall
60	C₂×Q₈	—	3	2	no	—
61	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
62	Q₁₆	—	6	1	no	—
63	Q₁₆	—	6	1	no	—
64	C₂×C₈	—	3	1	no	—
65	C₂×C₈	—	3	1	no	—
66	C₄⋊C₄	—	3	2	no	—
67	C₄⋊C₄	—	6	2	no	—
68	C₂²⋊C₄	—	6	2	no	—
69	C₂×C₇	—	3	1	no	—
70	C₂×C₇	—	3	1	no	—
71	C₂×C₇	C₄⋊S₄	1	1	yes	characteristic
72	A₄	C₇×Q₁₆	1	1	yes	characteristic, complemented
73	C₃×C₄	—	4	1	no	—
74	Dic₃	—	8	2	no	—
75	C₂³	C₇×D₁₂	1	1	yes	characteristic
76	Q₈	—	6	2	no	—
77	Q₈	—	6	2	no	—
78	C₂×C₄	—	3	1	no	—
79	C₂×C₄	—	3	1	no	—
80	C₂×C₄	—	6	2	no	—
81	C₂×C₄	—	6	2	no	—
82	C₈	—	3	1	no	—
83	C₈	C₇×S₄	1	1	yes	characteristic
84	C₇	#5398	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×C₃	—	4	1	no	—
86	C₂²	—	3	1	no	—
87	C₂²	—	3	1	no	—
88	C₂²	C₇×Dic₁₂	1	1	yes	characteristic, complemented
89	C₄	—	3	1	no	—
90	C₄	—	12	2	no	—
91	C₄	—	12	2	no	—
92	C₄	C₂×C₇×S₄	1	1	yes	characteristic
93	C₃	—	4	1	no	Sylow
94	C₂	—	3	1	no	—
95	C₂	—	3	1	no	—
96	C₂	C₇×C₄⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	C₄⋊S₄
commutator, cosocle	C₄×A₄	C₂²×C₇
Frattini	C₄	C₂×C₇×S₄
Fitting	C₂²×C₇×C₈	D₃
socle	C₂³×C₇	D₁₂
prime radical	C₂×C₇×A₄	D₄
hypercenter	C₇×C₈	S₄
hypocenter	A₄	C₇×Q₁₆
LCS	G ▷ C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈

Complements

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

N	H ≅ G/N	type
12. #5398	C₇	direct (N × H)
27. C₇×A₄	Q₁₆	semidirect
51. C₂²×C₇	Dic₁₂	semidirect
72. A₄	C₇×Q₁₆	semidirect
84. C₇	#5398	direct (N × H)
88. C₂²	C₇×Dic₁₂	semidirect

Maximal subgroups

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

subgroup	index
C₇×A₄⋊Q₈	2
C₇×C₈×A₄	2
C₇×#467	3
C₇×Dic₁₂	4
#5398	7

Subgroups — C7×#5398 #13057118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5398 #13057118