Subgroups — C₂×#11716070 #13096567

Order 1568.

660 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₇²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716070	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₇²	C₂²	1	1	yes	characteristic
6	C₂²×C₇⋊D₇	—	2	2	no	—
7	C₂²×C₇⋊D₇	—	2	2	no	—
8	C₂²×C₇⋊D₇	C₄	1	4	yes	complemented
9	C₂²×C₇⋊D₇	C₂²	1	2	yes	—
10	C₂×C₄×C₇²	—	2	2	no	—
11	C₂×C₄×C₇²	C₂²	1	2	yes	complemented
12	C₂×C₇⋊Dic₇	—	2	2	no	—
13	C₂×C₇⋊Dic₇	C₂²	1	2	yes	complemented
14	C₂×(C₂×D₇)⋊C₄	—	7	8	no	maximal
15	C₂²×C₇²	C₂×C₄	1	2	yes	—
16	C₂²×C₇²	D₄	1	2	yes	—
17	C₂²×C₇²	D₄	1	2	yes	—
18	C₂²×C₇²	C₂³	1	1	yes	characteristic
19	C₂×C₇⋊D₇	—	2	8	no	—
20	C₂×C₇⋊D₇	—	2	2	no	—
21	C₂×C₇⋊D₇	—	2	2	no	—
22	C₂×C₇⋊D₇	C₂×C₄	1	4	yes	complemented
23	C₄×C₇²	—	2	2	no	—
24	C₇⋊Dic₇	—	2	2	no	—
25	C₂³×D₇	—	7	8	no	—
26	C₂²×C₄×C₇	D₇	1	8	yes	complemented
27	C₂²×Dic₇	—	7	8	no	—
28	(C₂×D₇)⋊C₄	—	7	32	no	—
29	C₂×C₇²	C₂²⋊C₄	1	4	yes	complemented
30	C₂×C₇²	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
32	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
33	C₇⋊D₇	—	2	4	no	—
34	C₂³×C₇	C₂×D₇	1	8	yes	—
35	C₂²×D₇	—	7	32	no	—
36	C₂²×D₇	—	7	16	no	—
37	C₂²×D₇	—	14	16	no	—
38	C₂²×D₇	—	14	16	no	—
39	C₂×C₄×C₇	—	2	16	no	—
40	C₂×C₄×C₇	C₂×D₇	1	16	yes	complemented
41	C₂×Dic₇	—	7	16	no	—
42	C₂×Dic₇	—	14	16	no	—
43	C₇²	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₂²⋊C₄	—	49	1	no	Sylow
45	C₂²×C₇	C₄×D₇	1	16	yes	—
46	C₂²×C₇	D₂₈	1	16	yes	—
47	C₂²×C₇	C₂²⋊D₇	1	16	yes	—
48	C₂²×C₇	C₂²×D₇	1	8	yes	—
49	C₂×D₇	—	7	32	no	—
50	C₂×D₇	—	14	64	no	—
51	C₂×D₇	—	14	16	no	—
52	C₂×D₇	—	14	16	no	—
53	C₄×C₇	—	2	16	no	—
54	Dic₇	—	14	16	no	—
55	C₂⁴	—	49	1	no	—
56	C₂²×C₄	—	49	1	no	—
57	C₂²×C₄	C₇⋊D₇	1	1	yes	characteristic, complemented
58	C₂²⋊C₄	—	49	4	no	—
59	C₂×C₇	(C₂×D₇)⋊C₄	1	32	yes	complemented
60	C₂×C₇	C₂×C₄×D₇	1	8	yes	—
61	C₂×C₇	C₂×D₂₈	1	8	yes	—
62	C₂×C₇	C₂×C₂²⋊D₇	1	8	yes	—
63	D₇	—	14	32	no	—
64	C₂³	—	49	4	no	—
65	C₂³	—	49	2	no	—
66	C₂³	—	98	2	no	—
67	C₂³	—	98	2	no	—
68	C₂³	C₂×C₇⋊D₇	1	1	yes	characteristic
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	49	2	no	—
71	C₂×C₄	—	98	2	no	—
72	C₂×C₄	C₂×C₇⋊D₇	1	2	yes	complemented
73	C₇	C₂×(C₂×D₇)⋊C₄	1	8	yes	complemented
74	C₂²	—	49	4	no	—
75	C₂²	—	98	2	no	—
76	C₂²	—	98	2	no	—
77	C₂²	—	98	8	no	—
78	C₂²	C₄×C₇⋊D₇	1	2	yes	—
79	C₂²	C₄⋊(C₇⋊D₇)	1	2	yes	—
80	C₂²	C₂²⋊(C₇⋊D₇)	1	2	yes	—
81	C₂²	C₂²×C₇⋊D₇	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	98	2	no	—
84	C₂	—	98	4	no	—
85	C₂	#11716070	1	4	yes	direct product factor
86	C₂	C₂×C₄×C₇⋊D₇	1	1	yes	characteristic
87	C₂	C₂×C₄⋊(C₇⋊D₇)	1	1	yes	characteristic
88	C₂	C₂×C₂²⋊(C₇⋊D₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₇⋊D₇
commutator	C₂×C₇²	C₂²×C₄
Frattini	C₂²	C₂²×C₇⋊D₇
Fitting	C₂²×C₄×C₇²	C₂
socle	C₂³×C₇²	C₂²
prime radical	C₂³×C₇⋊D₇	C₂
cosocle	C₂²×C₇²	C₂³
hypercenter	C₂²×C₄	C₇⋊D₇
hypocenter	C₇²	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₇²	C₂	semidirect
3. C₂²×C₇⋊Dic₇	C₂	semidirect
4. #11716070	C₂	direct (N × H)
8. C₂²×C₇⋊D₇	C₄	semidirect
11. C₂×C₄×C₇²	C₂²	semidirect
13. C₂×C₇⋊Dic₇	C₂²	semidirect
22. C₂×C₇⋊D₇	C₂×C₄	semidirect
26. C₂²×C₄×C₇	D₇	semidirect
29. C₂×C₇²	C₂²⋊C₄	semidirect
40. C₂×C₄×C₇	C₂×D₇	semidirect
43. C₇²	C₂×C₂²⋊C₄	semidirect
57. C₂²×C₄	C₇⋊D₇	semidirect
59. C₂×C₇	(C₂×D₇)⋊C₄	semidirect
72. C₂×C₄	C₂×C₇⋊D₇	semidirect
73. C₇	C₂×(C₂×D₇)⋊C₄	semidirect
85. C₂	#11716070	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇⋊D₇	2
C₂²×C₄×C₇²	2
C₂²×C₇⋊Dic₇	2
#11716070	2
C₂×(C₂×D₇)⋊C₄	7

Subgroups — C2×#11716070 #13096567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11716070 #13096567