Subgroups — Dic₇×#564 #13516518

Order 1792.

1800 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#564	C₂	1	1	yes	maximal, characteristic
2	#11740472	C₂	1	15	yes	maximal
3	Dic₇×#125	C₂	1	15	yes	maximal
4	C₇×#564	C₄	1	16	yes	complemented
5	C₂×C₇×#125	C₂²	1	15	yes	—
6	#89757	C₂²	1	30	yes	—
7	#89753	C₂²	1	60	yes	—
8	C₄⋊C₄×Dic₇	C₂²	1	30	yes	—
9	#89708	C₂²	1	15	yes	—
10	C₄²×Dic₇	C₂²	1	5	yes	—
11	C₄×#564	—	7	1	no	maximal, Sylow
12	C₇×#125	C₂×C₄	1	120	yes	—
13	C₂×C₇×C₄⋊C₄	C₂³	1	30	yes	—
14	C₂×C₄²×C₇	C₂³	1	5	yes	—
15	C₂×C₄⋊Dic₇	C₂³	1	30	yes	—
16	C₂×C₄×Dic₇	C₂³	1	15	yes	—
17	C₂×C₄×Dic₇	C₂³	1	15	yes	—
18	C₇⋊#95	C₂³	1	60	yes	—
19	C₂×#564	D₇	1	1	yes	characteristic
20	#2382	—	7	15	no	—
21	C₄×#125	—	7	15	no	—
22	C₂²×C₄×C₇	C₂⁴	1	15	yes	—
23	C₂²×Dic₇	C₂⁴	1	15	yes	—
24	C₂²×Dic₇	C₂⁴	1	1	yes	characteristic
25	C₇×C₄⋊C₄	C₂²×C₄	1	120	yes	—
26	C₄²×C₇	C₂²×C₄	1	20	yes	—
27	C₄⋊Dic₇	—	4	30	no	—
28	C₄×Dic₇	—	2	30	no	—
29	C₄×Dic₇	—	4	15	no	—
30	#564	Dic₇	1	16	yes	direct product factor
31	C₂×#125	C₂×D₇	1	15	yes	—
32	#389	—	7	30	no	—
33	#387	—	7	60	no	—
34	#382	—	7	15	no	—
35	C₄×C₄⋊C₄	—	7	30	no	—
36	C₄³	—	7	5	no	—
37	C₂³×C₇	C₂⁵	1	1	yes	characteristic
38	C₂×C₄×C₇	—	2	15	no	—
39	C₂×C₄×C₇	—	2	15	no	—
40	C₂×C₄×C₇	C₂³×C₄	1	30	yes	—
41	C₂×Dic₇	—	2	15	no	—
42	C₂×Dic₇	—	2	30	no	—
43	C₂×Dic₇	2₋¹⁺⁴	1	6	yes	—
44	#125	C₂×Dic₇	1	120	yes	—
45	C₂×C₄⋊C₄	—	7	30	no	—
46	C₂×C₄⋊C₄	C₂²×D₇	1	30	yes	—
47	C₂×C₄²	—	7	15	no	—
48	C₂×C₄²	—	7	15	no	—
49	C₂×C₄²	C₂²×D₇	1	5	yes	—
50	#95	—	7	60	no	—
51	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
52	C₂²×C₇	C₂×2₋¹⁺⁴	1	3	yes	—
53	C₂²×C₇	#585	1	3	yes	—
54	C₄×C₇	—	2	30	no	—
55	Dic₇	—	4	15	no	—
56	Dic₇	#564	1	4	yes	direct product factor
57	C₂²×C₄	—	7	1	no	—
58	C₂²×C₄	—	7	15	no	—
59	C₂²×C₄	C₂³×D₇	1	15	yes	—
60	C₄⋊C₄	—	28	30	no	—
61	C₄⋊C₄	C₂²×Dic₇	1	120	yes	—
62	C₄²	—	14	30	no	—
63	C₄²	—	28	15	no	—
64	C₄²	C₂²×Dic₇	1	20	yes	—
65	C₂×C₇	C₄×2₋¹⁺⁴	1	3	yes	—
66	C₂×C₇	C₂×#564	1	1	yes	characteristic
67	C₂×C₇	#3531	1	3	yes	—
68	C₂³	C₂⁴×D₇	1	1	yes	characteristic
69	C₂×C₄	—	2	15	no	—
70	C₂×C₄	—	2	15	no	—
71	C₂×C₄	—	7	6	no	—
72	C₂×C₄	—	14	15	no	—
73	C₂×C₄	—	14	30	no	—
74	C₂×C₄	C₂³×Dic₇	1	30	yes	—
75	C₇	C₄×#564	1	1	yes	characteristic, complemented, Sylow
76	C₂²	D₇×2₋¹⁺⁴	1	3	yes	—
77	C₂²	#90620	1	3	yes	—
78	C₂²	C₂⁴×Dic₇	1	1	yes	characteristic
79	C₄	—	2	30	no	—
80	C₄	—	7	4	no	—
81	C₄	—	28	15	no	—
82	C₂	#11744207	1	3	yes	—
83	C₂	D₇×#564	1	1	yes	characteristic
84	C₂	Dic₇×2₋¹⁺⁴	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₇
commutator	C₂²×C₇	C₂⁴×C₄
Fitting	C₂×C₇×#564	C₂
socle, prime radical, cosocle	C₂³×C₇	C₂⁵
hypercenter	C₂×#564	D₇
hypocenter	C₇	C₄×#564
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×#564

Complements

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

N	H ≅ G/N	type
4. C₇×#564	C₄	semidirect
30. #564	Dic₇	direct (N × H)
56. Dic₇	#564	direct (N × H)
75. C₇	C₄×#564	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#564	2
#11740472	2
Dic₇×#125	2
C₄×#564	7

Subgroups — Dic7×#564 #13516518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₇×#564 #13516518