Subgroups — #89612

Order 448.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6435	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₇)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	(C₂×D₇)⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#6389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₄²×C₇	C₂²	1	1	yes	characteristic
12	(C₂×D₇)⋊C₄	C₄	1	1	yes	characteristic
13	(C₂×D₇)⋊C₄	C₄	1	1	yes	characteristic
14	Dic₇⋊C₄	C₄	1	1	yes	characteristic
15	Dic₇⋊C₄	C₄	1	1	yes	characteristic
16	C₄×Dic₇	C₂²	1	1	yes	characteristic
17	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#433	—	7	1	no	maximal, Sylow
20	C₂²×D₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
27	C₄×D₇	—	2	1	no	—
28	C₄×D₇	—	2	1	no	—
29	C₇×C₈	—	2	1	no	—
30	C₇×C₈	—	2	1	no	—
31	C₇⋊C₈	—	2	1	no	—
32	C₇⋊C₈	—	2	1	no	—
33	#117	—	7	1	no	—
34	C₄⋊C₈	—	7	1	no	—
35	C₄⋊C₈	D₇	1	1	yes	characteristic, complemented
36	C₂²⋊C₈	—	7	1	no	—
37	C₂²⋊C₈	—	7	1	no	—
38	#97	—	7	1	no	—
39	C₄×C₈	—	7	1	no	—
40	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×D₇	—	2	1	no	—
42	C₂×D₇	—	2	1	no	—
43	C₂×D₇	—	2	1	no	—
44	C₄×C₇	—	2	1	no	—
45	C₄×C₇	—	2	1	no	—
46	C₄×C₇	#41	1	1	yes	characteristic
47	C₄×C₇	#41	1	1	yes	characteristic
48	Dic₇	—	2	1	no	—
49	Dic₇	—	2	1	no	—
50	Dic₇	—	2	1	no	—
51	C₂²×C₄	—	7	1	no	—
52	C₂×C₈	—	7	1	no	—
53	C₂×C₈	—	7	1	no	—
54	C₂×C₈	C₂×D₇	1	1	yes	characteristic
55	C₂×C₈	C₂×D₇	1	1	yes	characteristic
56	C₄⋊C₄	—	7	1	no	—
57	C₄⋊C₄	—	7	1	no	—
58	C₂²⋊C₄	—	7	1	no	—
59	C₂²⋊C₄	—	7	1	no	—
60	C₄²	—	7	1	no	—
61	C₄²	C₂×D₇	1	1	yes	characteristic
62	C₂×C₇	#117	1	1	yes	characteristic
63	C₂×C₇	#131	1	1	yes	characteristic
64	C₂×C₇	#131	1	1	yes	characteristic
65	D₇	—	4	1	no	—
66	C₂³	—	7	1	no	—
67	C₂×C₄	—	7	1	no	—
68	C₂×C₄	—	7	1	no	—
69	C₂×C₄	—	7	1	no	—
70	C₂×C₄	—	14	1	no	—
71	C₂×C₄	—	14	1	no	—
72	C₂×C₄	C₄×D₇	1	1	yes	characteristic
73	C₂×C₄	C₄×D₇	1	1	yes	characteristic
74	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
75	C₈	—	2	1	no	—
76	C₈	—	2	1	no	—
77	C₈	—	14	1	no	—
78	C₈	—	14	1	no	—
79	C₇	#433	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	14	1	no	—
81	C₂²	—	14	1	no	—
82	C₂²	—	14	1	no	—
83	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	14	1	no	—
87	C₄	—	14	1	no	—
88	C₄	—	14	1	no	—
89	C₄	#1159	1	1	yes	characteristic
90	C₄	#1161	1	1	yes	characteristic
91	C₂	—	28	1	no	—
92	C₂	#6455	1	1	yes	characteristic
93	C₂	#6464	1	1	yes	characteristic
94	C₂	#6470	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₄
Fitting	C₇×C₄⋊C₈	C₂
prime radical	C₂²×D₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₄⋊C₈	D₇
hypocenter	C₇	#433
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₇×C₄⋊C₈	C₂	semidirect
5. #6389	C₂	semidirect
6. C₈×Dic₇	C₂	semidirect
7. C₄⋊(C₇⋊C₈)	C₂	semidirect
35. C₄⋊C₈	D₇	semidirect
79. C₇	#433	semidirect

Maximal subgroups

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

subgroup	index
#6435	2
C₇×C₄⋊C₈	2
(C₂×D₇)⋊C₈	2
#6389	2
C₈×Dic₇	2
C₄⋊(C₇⋊C₈)	2
#433	7