Subgroups — #89748

Order 448.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₇⋊C₄	C₂	1	4	yes	maximal
4	C₂×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×C₇	C₂²	1	2	yes	—
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₇	C₂²	1	2	yes	—
8	C₂²×Dic₇	C₂²	1	2	yes	—
9	C₇×C₄⋊C₄	—	2	2	no	—
10	C₄⋊Dic₇	—	2	2	no	—
11	Dic₇⋊C₄	—	2	8	no	—
12	C₄×Dic₇	C₄	1	4	yes	complemented
13	#384	—	7	1	no	maximal, Sylow
14	C₂³×C₇	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₇	—	2	4	no	—
16	C₂×C₄×C₇	—	2	2	no	—
17	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₇	D₄	1	2	yes	—
19	C₂×C₄×C₇	Q₈	1	2	yes	—
20	C₂×Dic₇	—	2	4	no	—
21	C₂×Dic₇	—	2	2	no	—
22	C₂×Dic₇	C₂×C₄	1	4	yes	—
23	C₂×Dic₇	D₄	1	4	yes	—
24	C₂×Dic₇	Q₈	1	4	yes	—
25	C₂×C₄⋊C₄	—	7	4	no	—
26	C₂×C₄⋊C₄	—	7	1	no	—
27	C₂×C₄⋊C₄	D₇	1	1	yes	characteristic
28	C₂×C₄²	—	7	1	no	—
29	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₇	C₂×D₄	1	2	yes	—
31	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₇	C₂×Q₈	1	2	yes	—
34	C₄×C₇	—	4	2	no	—
35	C₄×C₇	C₄⋊C₄	1	4	yes	complemented
36	Dic₇	—	4	2	no	—
37	Dic₇	C₄⋊C₄	1	8	yes	complemented
38	C₂²×C₄	—	7	2	no	—
39	C₂²×C₄	—	7	2	no	—
40	C₂²×C₄	C₂×D₇	1	2	yes	—
41	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
42	C₄⋊C₄	—	2	2	no	—
43	C₄⋊C₄	—	14	8	no	—
44	C₄⋊C₄	—	14	2	no	—
45	C₄²	—	7	4	no	—
46	C₂×C₇	C₂×C₄⋊C₄	1	2	yes	—
47	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₇	#127	1	1	yes	characteristic
49	C₂×C₇	C₄⋊Q₈	1	2	yes	—
50	C₂×C₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂³	C₂²×D₇	1	1	yes	characteristic
52	C₂×C₄	—	2	4	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	7	4	no	—
55	C₂×C₄	—	7	4	no	—
56	C₂×C₄	—	7	4	no	—
57	C₂×C₄	—	14	4	no	—
58	C₂×C₄	—	14	2	no	—
59	C₂×C₄	Dic₁₄	1	2	yes	—
60	C₂×C₄	C₄×D₇	1	2	yes	—
61	C₂×C₄	C₂²⋊D₇	1	2	yes	—
62	C₇	#384	1	1	yes	characteristic, complemented, Sylow
63	C₂²	C₂×Dic₁₄	1	1	yes	characteristic
64	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
65	C₂²	D₄×D₇	1	2	yes	—
66	C₂²	Q₈×D₇	1	2	yes	—
67	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
68	C₄	—	4	2	no	—
69	C₄	—	7	8	no	—
70	C₄	—	28	2	no	—
71	C₄	Dic₇⋊C₄	1	4	yes	complemented
72	C₂	#6451	1	2	yes	—
73	C₂	D₇×C₄⋊C₄	1	2	yes	—
74	C₂	C₂×Dic₇⋊C₄	1	1	yes	characteristic
75	C₂	#6503	1	1	yes	characteristic
76	C₂	#6507	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₄
Fitting	C₂×C₇×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₇
hypocenter	C₇	#384
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. C₄×Dic₇	C₄	semidirect
35. C₄×C₇	C₄⋊C₄	semidirect
37. Dic₇	C₄⋊C₄	semidirect
62. C₇	#384	semidirect
71. C₄	Dic₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊C₄	2
C₂×C₄⋊Dic₇	2
C₂×Dic₇⋊C₄	2
C₂×C₄×Dic₇	2
#384	7