Subgroups — #89662

Order 448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6456	C₂	1	1	yes	maximal, characteristic
2	#6451	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	#6395	C₂	1	1	yes	maximal, characteristic
5	Dic₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#6383	C₂	1	1	yes	maximal, characteristic
7	#6381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₂₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₇	—	2	1	no	—
10	C₂×Dic₁₄	—	2	1	no	—
11	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	(C₂×D₇)⋊C₄	—	2	1	no	—
15	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	Dic₇⋊C₄	—	2	1	no	—
17	C₄×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#474	—	7	1	no	maximal, Sylow
20	C₂²×D₇	—	2	1	no	—
21	C₂×C₄×C₇	—	2	1	no	—
22	C₂×C₄×C₇	—	2	1	no	—
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂×Dic₇	—	2	1	no	—
25	C₂×Dic₇	D₄	1	1	yes	characteristic
26	C₂×Dic₇	D₄	1	1	yes	characteristic
27	D₂₈	—	2	1	no	—
28	D₂₈	Q₈	1	2	yes	complemented
29	C₄×D₇	—	4	2	no	—
30	Dic₁₄	—	4	2	no	—
31	C₇×C₈	—	2	1	no	—
32	C₇⋊C₈	—	2	1	no	—
33	C₄⋊Q₈	—	7	1	no	—
34	C₄×D₄	—	7	1	no	—
35	#107	—	7	1	no	—
36	#107	D₇	1	1	yes	characteristic, complemented
37	C₄⋊C₈	—	7	1	no	—
38	#102	—	7	1	no	—
39	#102	—	7	1	no	—
40	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂×D₇	—	2	2	no	—
42	C₂×D₇	—	4	1	no	—
43	C₂×D₇	—	4	1	no	—
44	C₄×C₇	—	4	1	no	—
45	C₄×C₇	—	4	1	no	—
46	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
47	C₄×C₇	#41	1	1	yes	characteristic
48	Dic₇	—	2	1	no	—
49	Dic₇	—	4	1	no	—
50	Dic₇	D₈	1	2	yes	—
51	C₂×Q₈	—	14	1	no	—
52	C₂×D₄	—	7	1	no	—
53	C₂²×C₄	—	14	1	no	—
54	C₂×C₈	—	7	1	no	—
55	C₂×C₈	C₂×D₇	1	1	yes	characteristic
56	C₄⋊C₄	—	7	1	no	—
57	C₄⋊C₄	—	14	1	no	—
58	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
60	C₂²⋊C₄	—	14	1	no	—
61	C₄²	—	7	1	no	—
62	C₂×C₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂×C₇	C₂×D₈	1	1	yes	characteristic
64	C₂×C₇	#137	1	1	yes	characteristic
65	D₇	—	4	2	no	—
66	C₂³	—	14	1	no	—
67	Q₈	—	28	2	no	—
68	D₄	—	7	2	no	—
69	D₄	—	14	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	7	1	no	—
73	C₂×C₄	—	7	1	no	—
74	C₂×C₄	—	14	1	no	—
75	C₂×C₄	—	28	2	no	—
76	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	14	1	no	—
79	C₇	#474	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	14	2	no	—
81	C₂²	—	28	1	no	—
82	C₂²	—	28	1	no	—
83	C₂²	D₄×D₇	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	7	2	no	—
87	C₄	—	14	1	no	—
88	C₄	—	28	1	no	—
89	C₄	#1157	1	1	yes	characteristic
90	C₄	Q₈×D₇	1	1	yes	characteristic
91	C₂	—	28	2	no	—
92	C₂	#6459	1	1	yes	characteristic
93	C₂	D₇×D₈	1	1	yes	characteristic
94	C₂	#6481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#107	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×D₂₈	C₂²
hypercenter	#107	D₇
hypocenter	C₇	#474
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
2. #6451	C₂	semidirect
3. C₇×#107	C₂	semidirect
5. Dic₇⋊C₈	C₂	semidirect
7. #6381	C₂	semidirect
28. D₂₈	Q₈	semidirect
36. #107	D₇	semidirect
79. C₇	#474	semidirect

Maximal subgroups

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

subgroup	index
#6456	2
#6451	2
C₇×#107	2
#6395	2
Dic₇⋊C₈	2
#6383	2
#6381	2
#474	7