Subgroups — C₇×#5812 #13061525

Order 1344.

532 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇×Q₈×Dic₃	C₂	1	8	yes	maximal, complemented
3	C₇×#926	C₂	1	6	yes	maximal
4	C₇×#519	—	3	1	no	maximal, Hall
5	C₂×C₃×C₇×Q₈	C₄	1	8	yes	complemented
6	C₂×C₃×C₇×Q₈	C₂²	1	4	yes	—
7	C₂²×C₃×C₄×C₇	C₂²	1	3	yes	—
8	C₇×#217	C₂²	1	4	yes	—
9	C₇×C₄⋊Dic₃	C₂²	1	12	yes	—
10	C₄×C₇×Dic₃	C₂²	1	12	yes	—
11	C₂²×C₇×Q₈	D₃	1	1	yes	characteristic
12	C₄×C₇×Q₈	—	3	8	no	—
13	C₇×#117	—	3	6	no	—
14	#5812	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
16	C₃×C₇×Q₈	C₂×C₄	1	16	yes	—
17	C₂×C₃×C₄×C₇	C₂×C₄	1	12	yes	—
18	C₂×C₃×C₄×C₇	C₂³	1	6	yes	—
19	C₂×C₇×Dic₃	C₂³	1	8	yes	—
20	C₂×C₇×Q₈	Dic₃	1	8	yes	complemented
21	C₂×C₇×Q₈	C₂×D₃	1	4	yes	—
22	C₂²×C₄×C₇	C₂×D₃	1	3	yes	—
23	C₇×C₄⋊C₄	—	3	12	no	—
24	C₇×C₂²⋊C₄	—	3	4	no	—
25	C₄²×C₇	—	3	12	no	—
26	C₂²×C₃×Q₈	C₂×C₇	1	1	yes	characteristic
27	Q₈×Dic₃	C₂×C₇	1	8	yes	complemented
28	#926	C₂×C₇	1	6	yes	—
29	C₂²×C₃×C₇	—	2	2	no	—
30	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
31	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
32	C₃×C₄×C₇	C₂²×C₄	1	12	yes	—
33	C₇×Dic₃	—	2	8	no	—
34	#519	—	3	1	no	Sylow
35	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
36	C₇×Q₈	C₂×Dic₃	1	16	yes	—
37	C₂×C₄×C₇	—	3	8	no	—
38	C₂×C₄×C₇	C₂×Dic₃	1	12	yes	—
39	C₂×C₄×C₇	C₂²×D₃	1	6	yes	—
40	C₂×C₃×Q₈	C₄×C₇	1	8	yes	complemented
41	C₂×C₃×Q₈	C₂²×C₇	1	4	yes	—
42	C₂²×C₃×C₄	C₂²×C₇	1	3	yes	—
43	#217	C₂²×C₇	1	4	yes	—
44	C₄⋊Dic₃	C₂²×C₇	1	12	yes	—
45	C₄×Dic₃	C₂²×C₇	1	12	yes	—
46	C₂×C₃×C₇	—	2	2	no	—
47	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
48	C₂×C₃×C₇	2₋¹⁺⁴	1	2	yes	—
49	C₂²×Q₈	D₃×C₇	1	1	yes	characteristic
50	C₄×Q₈	—	3	8	no	—
51	#117	—	3	6	no	—
52	C₂²×C₇	—	2	2	no	—
53	C₂²×C₇	C₂²×Dic₃	1	2	yes	—
54	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
55	C₄×C₇	—	6	8	no	—
56	C₄×C₇	C₂²×Dic₃	1	12	yes	—
57	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
58	C₃×Q₈	C₂×C₄×C₇	1	16	yes	—
59	C₂×C₃×C₄	C₂×C₄×C₇	1	12	yes	—
60	C₂×C₃×C₄	C₂³×C₇	1	6	yes	—
61	C₂×Dic₃	C₂³×C₇	1	8	yes	—
62	C₃×C₇	#519	1	1	yes	characteristic, complemented, Hall
63	C₂×Q₈	C₇×Dic₃	1	8	yes	complemented
64	C₂×Q₈	C₂×D₃×C₇	1	4	yes	—
65	C₂²×C₄	C₂×D₃×C₇	1	3	yes	—
66	C₄⋊C₄	—	3	12	no	—
67	C₂²⋊C₄	—	3	4	no	—
68	C₄²	—	3	12	no	—
69	C₂×C₇	—	2	2	no	—
70	C₂×C₇	#1007	1	2	yes	—
71	C₂×C₇	C₂³×Dic₃	1	1	yes	characteristic
72	C₂²×C₃	—	2	2	no	—
73	C₂²×C₃	C₂²×C₄×C₇	1	2	yes	—
74	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
75	C₃×C₄	C₂²×C₄×C₇	1	12	yes	—
76	Dic₃	—	2	8	no	—
77	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
78	Q₈	C₂×C₇×Dic₃	1	16	yes	—
79	C₂×C₄	—	3	8	no	—
80	C₂×C₄	C₂×C₇×Dic₃	1	12	yes	—
81	C₂×C₄	C₂²×D₃×C₇	1	6	yes	—
82	C₇	#5812	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×C₃	—	2	2	no	—
84	C₂×C₃	C₂³×C₄×C₇	1	1	yes	characteristic
85	C₂×C₃	C₇×2₋¹⁺⁴	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	C₂²×C₇×Dic₃	1	2	yes	—
88	C₂²	C₂³×D₃×C₇	1	1	yes	characteristic
89	C₄	—	6	8	no	—
90	C₄	C₂²×C₇×Dic₃	1	12	yes	—
91	C₃	C₇×#519	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	2	2	no	—
93	C₂	C₇×#1007	1	2	yes	—
94	C₂	C₂³×C₇×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄×C₇
Frattini	C₂²	C₂³×D₃×C₇
Fitting	C₂²×C₃×C₇×Q₈	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂³×C₃×C₇	C₂³
cosocle	C₂²×C₃	C₂⁴×C₇
hypercenter	C₂²×C₇×Q₈	D₃
hypocenter	C₃	C₇×#519
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₇×Q₈

Complements

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

N	H ≅ G/N	type
2. C₇×Q₈×Dic₃	C₂	semidirect
5. C₂×C₃×C₇×Q₈	C₄	semidirect
14. #5812	C₇	direct (N × H)
20. C₂×C₇×Q₈	Dic₃	semidirect
27. Q₈×Dic₃	C₂×C₇	semidirect
40. C₂×C₃×Q₈	C₄×C₇	semidirect
62. C₃×C₇	#519	semidirect
63. C₂×Q₈	C₇×Dic₃	semidirect
82. C₇	#5812	direct (N × H)
91. C₃	C₇×#519	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₇×Q₈	2
C₇×Q₈×Dic₃	2
C₇×#926	2
C₇×#519	3
#5812	7

Subgroups — C7×#5812 #13061525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5812 #13061525