Subgroups — D₄×C₇⋊Dic₇ #13096571

Order 1568.

470 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇²×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇⋊Dic₇	C₂	1	2	yes	maximal, complemented
3	C₂²⋊(C₇⋊Dic₇)	C₂	1	2	yes	maximal, complemented
4	C₄⋊(C₇⋊Dic₇)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₇⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₇²	C₂²	1	2	yes	—
7	C₇²×D₄	C₄	1	4	yes	complemented
8	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
9	C₂×C₇⋊Dic₇	—	2	4	no	—
10	C₂×C₇⋊Dic₇	C₂²	1	2	yes	—
11	C₂×C₇⋊Dic₇	C₂²	1	1	yes	characteristic
12	C₂×C₇⋊Dic₇	C₂²	1	1	yes	characteristic
13	D₄×Dic₇	—	7	8	no	maximal
14	C₂²×C₇²	—	2	2	no	—
15	C₂²×C₇²	—	2	2	no	—
16	C₂²×C₇²	C₂×C₄	1	4	yes	complemented
17	C₂²×C₇²	C₂³	1	1	yes	characteristic
18	C₄×C₇²	C₂×C₄	1	2	yes	complemented
19	C₇⋊Dic₇	—	2	2	no	—
20	C₇⋊Dic₇	—	2	1	no	—
21	C₇⋊Dic₇	D₄	1	2	yes	direct product factor
22	C₂×C₇×D₄	D₇	1	8	yes	—
23	C₂²×Dic₇	—	7	16	no	—
24	C₂²⋊Dic₇	—	7	16	no	—
25	C₄⋊Dic₇	—	7	8	no	—
26	C₄×Dic₇	—	7	8	no	—
27	C₂×C₇²	—	2	4	no	—
28	C₂×C₇²	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
30	C₂×C₇²	#41	1	1	yes	characteristic
31	C₂³×C₇	C₂×D₇	1	16	yes	—
32	C₇×D₄	Dic₇	1	32	yes	complemented
33	C₂×C₄×C₇	C₂×D₇	1	8	yes	—
34	C₂×Dic₇	—	7	8	no	—
35	C₂×Dic₇	—	7	8	no	—
36	C₂×Dic₇	—	7	16	no	—
37	C₂×Dic₇	—	14	32	no	—
38	C₇²	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
39	C₄×D₄	—	49	1	no	Sylow
40	C₂²×C₇	—	2	16	no	—
41	C₂²×C₇	—	2	16	no	—
42	C₂²×C₇	C₂×Dic₇	1	32	yes	complemented
43	C₂²×C₇	C₂²×D₇	1	8	yes	—
44	C₄×C₇	C₂×Dic₇	1	16	yes	complemented
45	Dic₇	—	7	16	no	—
46	Dic₇	—	14	16	no	—
47	Dic₇	—	14	8	no	—
48	C₂×D₄	C₇⋊D₇	1	1	yes	characteristic
49	C₂²×C₄	—	49	2	no	—
50	C₄⋊C₄	—	49	1	no	—
51	C₂²⋊C₄	—	49	2	no	—
52	C₄²	—	49	1	no	—
53	C₂×C₇	—	2	32	no	—
54	C₂×C₇	D₄×D₇	1	8	yes	—
55	C₂×C₇	#1159	1	8	yes	—
56	C₂×C₇	C₂²×Dic₇	1	8	yes	—
57	C₂³	C₂×C₇⋊D₇	1	2	yes	—
58	D₄	C₇⋊Dic₇	1	4	yes	direct product factor
59	C₂×C₄	—	49	1	no	—
60	C₂×C₄	—	49	1	no	—
61	C₂×C₄	—	49	2	no	—
62	C₂×C₄	—	98	4	no	—
63	C₂×C₄	C₂×C₇⋊D₇	1	1	yes	characteristic
64	C₇	D₄×Dic₇	1	8	yes	complemented
65	C₂²	—	2	2	no	—
66	C₂²	—	2	2	no	—
67	C₂²	C₂×C₇⋊Dic₇	1	4	yes	complemented
68	C₂²	C₂²×C₇⋊D₇	1	1	yes	characteristic
69	C₄	—	49	2	no	—
70	C₄	—	98	2	no	—
71	C₄	—	98	1	no	—
72	C₄	C₂×C₇⋊Dic₇	1	2	yes	complemented
73	C₂	—	2	4	no	—
74	C₂	D₄×C₇⋊D₇	1	1	yes	characteristic
75	C₂	#11716131	1	1	yes	characteristic
76	C₂	C₂²×C₇⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₇⋊D₇
commutator	C₂×C₇²	C₂²×C₄
Fitting, prime radical	C₂×C₇²×D₄	C₂
socle, cosocle	C₂²×C₇²	C₂³
hypercenter	C₂×D₄	C₇⋊D₇
hypocenter	C₇²	C₄×D₄
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇⋊Dic₇	C₂	semidirect
3. C₂²⋊(C₇⋊Dic₇)	C₂	semidirect
4. C₄⋊(C₇⋊Dic₇)	C₂	semidirect
5. C₄×C₇⋊Dic₇	C₂	semidirect
7. C₇²×D₄	C₄	semidirect
16. C₂²×C₇²	C₂×C₄	semidirect
18. C₄×C₇²	C₂×C₄	semidirect
21. C₇⋊Dic₇	D₄	direct (N × H)
32. C₇×D₄	Dic₇	semidirect
38. C₇²	C₄×D₄	semidirect
42. C₂²×C₇	C₂×Dic₇	semidirect
44. C₄×C₇	C₂×Dic₇	semidirect
58. D₄	C₇⋊Dic₇	direct (N × H)
64. C₇	D₄×Dic₇	semidirect
67. C₂²	C₂×C₇⋊Dic₇	semidirect
72. C₄	C₂×C₇⋊Dic₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇²×D₄	2
C₂²×C₇⋊Dic₇	2
C₂²⋊(C₇⋊Dic₇)	2
C₄⋊(C₇⋊Dic₇)	2
C₄×C₇⋊Dic₇	2
D₄×Dic₇	7

Subgroups — D4×C7⋊Dic7 #13096571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₇⋊Dic₇ #13096571