Subgroups — #89324

Order 448.

114 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	C₂²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₇	—	2	1	no	—
5	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂²×Dic₇	—	2	1	no	—
7	C₇×C₄⋊C₄	—	2	1	no	—
8	C₇×C₄⋊C₄	C₄	1	2	yes	complemented
9	C₄⋊Dic₇	—	2	1	no	—
10	C₄⋊Dic₇	C₄	1	2	yes	complemented
11	C₂×C₇⋊C₈	—	2	1	no	—
12	C₂×C₇⋊C₈	—	2	1	no	—
13	C₂×C₇⋊C₈	C₄	1	2	yes	complemented
14	#340	—	7	1	no	maximal, Sylow
15	C₂³×C₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₇	—	2	2	no	—
17	C₂×C₄×C₇	—	4	1	no	—
18	C₂×C₄×C₇	—	4	1	no	—
19	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₇	Q₈	1	1	yes	characteristic
25	C₂×Dic₇	—	2	2	no	—
26	C₂×Dic₇	—	4	1	no	—
27	C₂×Dic₇	—	4	1	no	—
28	C₇⋊C₈	—	2	2	no	—
29	C₂²×C₈	—	7	1	no	—
30	C₂×C₄⋊C₄	—	7	1	no	—
31	C₂×C₄⋊C₄	D₇	1	1	yes	characteristic
32	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₇	D₈	1	1	yes	characteristic
36	C₂²×C₇	SD₁₆	1	1	yes	characteristic
37	C₂²×C₇	SD₁₆	1	1	yes	characteristic
38	C₂²×C₇	Q₁₆	1	1	yes	characteristic
39	C₄×C₇	—	4	2	no	—
40	C₄×C₇	C₄²	1	1	yes	characteristic
41	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
44	Dic₇	—	4	2	no	—
45	C₂²×C₄	—	2	1	no	—
46	C₂²×C₄	—	14	1	no	—
47	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
48	C₂×C₈	—	7	2	no	—
49	C₂×C₈	—	14	1	no	—
50	C₂×C₈	—	14	1	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	7	2	no	—
53	C₄⋊C₄	—	14	1	no	—
54	C₄⋊C₄	Dic₇	1	2	yes	complemented
55	C₂×C₇	#95	1	1	yes	characteristic
56	C₂×C₇	#102	1	1	yes	characteristic
57	C₂×C₇	#102	1	1	yes	characteristic
58	C₂×C₇	#103	1	1	yes	characteristic
59	C₂×C₇	#103	1	1	yes	characteristic
60	C₂×C₇	#106	1	1	yes	characteristic
61	C₂×C₇	#107	1	1	yes	characteristic
62	C₂³	C₂²⋊D₇	1	1	yes	characteristic
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	4	1	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	14	2	no	—
67	C₂×C₄	—	28	1	no	—
68	C₂×C₄	—	28	1	no	—
69	C₂×C₄	Dic₁₄	1	1	yes	characteristic
70	C₂×C₄	C₄×D₇	1	1	yes	characteristic
71	C₂×C₄	C₄×D₇	1	1	yes	characteristic
72	C₂×C₄	D₂₈	1	1	yes	characteristic
73	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
75	C₈	—	14	2	no	—
76	C₇	#340	1	1	yes	characteristic, complemented, Sylow
77	C₂²	Dic₇⋊C₄	1	1	yes	characteristic
78	C₂²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
79	C₂²	#1141	1	1	yes	characteristic
80	C₂²	#1142	1	1	yes	characteristic
81	C₂²	#1143	1	1	yes	characteristic
82	C₂²	#1144	1	1	yes	characteristic
83	C₂²	C₂²⋊Dic₇	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	28	2	no	—
86	C₄	C₄×Dic₇	1	1	yes	characteristic
87	C₄	Dic₇⋊C₄	1	1	yes	characteristic
88	C₄	C₄⋊Dic₇	1	1	yes	characteristic
89	C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
90	C₂	#6381	1	1	yes	characteristic
91	C₂	#6382	1	1	yes	characteristic
92	C₂	#6383	1	1	yes	characteristic
93	C₂	#6384	1	1	yes	characteristic
94	C₂	C₇⋊#95	1	1	yes	characteristic
95	C₂	#6406	1	1	yes	characteristic
96	C₂	#6409	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₇
commutator	C₄×C₇	C₄²
Frattini	C₂²×C₄	C₂×D₇
Fitting	C₂×C₇×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₇	D₄
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	C₂×C₄⋊C₄	D₇
hypocenter	C₇	#340
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
8. C₇×C₄⋊C₄	C₄	semidirect
10. C₄⋊Dic₇	C₄	semidirect
13. C₂×C₇⋊C₈	C₄	semidirect
54. C₄⋊C₄	Dic₇	semidirect
76. C₇	#340	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
C₂²×C₇⋊C₈	2
#340	7