Subgroups — #89609

Order 448.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1131	C₂	1	2	yes	maximal
2	C₄²×D₇	C₂	1	1	yes	maximal, characteristic
3	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₇	C₄	1	1	yes	characteristic
7	C₂×C₄×D₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₈	C₂²	1	2	yes	—
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₄×Dic₇	C₄	1	1	yes	characteristic
12	C₄×Dic₇	C₄	1	1	yes	characteristic
13	C₄×Dic₇	C₂²	1	1	yes	characteristic
14	C₂×C₇⋊C₈	C₂²	1	2	yes	—
15	#1131	—	2	4	no	—
16	#423	—	7	1	no	maximal, Sylow
17	C₂²×D₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
24	C₄×D₇	—	2	2	no	—
25	C₄×D₇	—	2	2	no	—
26	C₄×D₇	D₄	1	2	yes	—
27	C₄×D₇	Q₈	1	2	yes	—
28	C₇×C₈	—	2	2	no	—
29	C₇⋊C₈	—	2	2	no	—
30	C₂×M₁₆	—	7	2	no	—
31	C₂×C₄²	—	7	1	no	—
32	C₄⋊C₈	—	7	2	no	—
33	C₄⋊C₈	—	7	1	no	—
34	C₄⋊C₈	D₇	1	1	yes	characteristic, complemented
35	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×D₇	—	2	1	no	—
37	C₂×D₇	—	2	1	no	—
38	C₂×D₇	C₄⋊C₄	1	2	yes	—
39	C₄×C₇	—	2	1	no	—
40	C₄×C₇	M₁₆	1	2	yes	complemented
41	C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
43	Dic₇	—	2	1	no	—
44	Dic₇	C₄⋊C₄	1	2	yes	—
45	Dic₇	M₁₆	1	2	yes	complemented
46	C₂²×C₄	—	7	1	no	—
47	C₂²×C₄	—	7	1	no	—
48	C₂²×C₄	—	7	1	no	—
49	M₁₆	—	14	4	no	—
50	C₂×C₈	—	7	2	no	—
51	C₂×C₈	C₂×D₇	1	2	yes	—
52	C₄²	—	7	1	no	—
53	C₄²	—	7	1	no	—
54	C₄²	—	7	1	no	—
55	C₄²	C₂×D₇	1	1	yes	characteristic
56	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₇	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₇	C₂×M₁₆	1	1	yes	characteristic
59	D₇	—	2	2	no	—
60	C₂³	—	7	1	no	—
61	C₂×C₄	—	7	1	no	—
62	C₂×C₄	—	7	1	no	—
63	C₂×C₄	—	7	2	no	—
64	C₂×C₄	—	7	2	no	—
65	C₂×C₄	—	7	1	no	—
66	C₂×C₄	—	14	2	no	—
67	C₂×C₄	—	14	2	no	—
68	C₂×C₄	C₄×D₇	1	1	yes	characteristic
69	C₂×C₄	C₄×D₇	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₈	—	14	2	no	—
73	C₇	#423	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	7	2	no	—
75	C₂²	—	14	1	no	—
76	C₂²	—	14	1	no	—
77	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	7	2	no	—
80	C₄	—	7	2	no	—
81	C₄	—	14	1	no	—
82	C₄	#1131	1	2	yes	complemented
83	C₄	D₄×D₇	1	1	yes	characteristic
84	C₄	Q₈×D₇	1	1	yes	characteristic
85	C₂	—	14	2	no	—
86	C₂	D₇×C₄⋊C₄	1	1	yes	characteristic
87	C₂	C₂×#1131	1	1	yes	characteristic
88	C₂	D₇×M₁₆	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	#423
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
3. C₇×C₄⋊C₈	C₂	semidirect
4. Dic₇⋊C₈	C₂	semidirect
5. C₄⋊(C₇⋊C₈)	C₂	semidirect
34. C₄⋊C₈	D₇	semidirect
40. C₄×C₇	M₁₆	semidirect
45. Dic₇	M₁₆	semidirect
73. C₇	#423	semidirect
82. C₄	#1131	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1131	2
C₄²×D₇	2
C₇×C₄⋊C₈	2
Dic₇⋊C₈	2
C₄⋊(C₇⋊C₈)	2
#423	7