Subgroups — #89523

Order 448.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1161	C₂	1	1	yes	maximal, characteristic
2	D₇×M₁₆	C₂	1	2	yes	maximal, complemented
3	C₇×#101	C₂	1	1	yes	maximal, characteristic, complemented
4	#6410	C₂	1	1	yes	maximal, characteristic, complemented
5	#6397	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
7	#1161	—	2	2	no	—
8	#1161	—	2	2	no	—
9	C₂×D₂₈	C₄	1	2	yes	—
10	C₂×D₂₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₇	C₄	1	2	yes	—
12	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
13	C₇×M₁₆	C₂²	1	2	yes	—
14	#1136	C₂²	1	2	yes	—
15	#1131	—	2	2	no	—
16	C₈×D₇	—	2	2	no	—
17	#413	—	7	1	no	maximal, Sylow
18	C₂²×D₇	C₂×C₄	1	2	yes	—
19	C₂²×D₇	C₂×C₄	1	1	yes	characteristic
20	C₇×Q₈	—	2	2	no	—
21	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₇	C₂×C₄	1	1	yes	characteristic
24	D₂₈	—	2	4	no	—
25	D₂₈	—	2	2	no	—
26	C₄×D₇	—	2	2	no	—
27	C₄×D₇	—	2	2	no	—
28	C₄×D₇	D₄	1	2	yes	—
29	C₄×D₇	D₄	1	2	yes	—
30	C₇×C₈	—	2	2	no	—
31	C₇⋊C₈	—	2	2	no	—
32	C₂×#41	—	7	1	no	—
33	C₂×M₁₆	—	7	2	no	—
34	#101	—	7	1	no	—
35	#101	D₇	1	1	yes	characteristic, complemented
36	#100	—	7	2	no	—
37	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×D₇	—	2	2	no	—
39	C₂×D₇	—	4	1	no	—
40	C₂×D₇	—	4	2	no	—
41	C₂×D₇	C₂²⋊C₄	1	2	yes	—
42	C₄×C₇	—	2	2	no	—
43	C₄×C₇	C₂×D₄	1	2	yes	—
44	Dic₇	C₂²⋊C₄	1	1	yes	characteristic
45	Dic₇	C₂²⋊C₄	1	1	yes	characteristic
46	#41	—	14	2	no	—
47	#41	—	14	2	no	—
48	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
49	C₂×D₄	—	7	2	no	—
50	C₂×D₄	—	7	1	no	—
51	C₂²×C₄	—	7	1	no	—
52	C₂²×C₄	—	7	2	no	—
53	M₁₆	—	7	2	no	—
54	M₁₆	—	14	2	no	—
55	M₁₆	C₂×D₇	1	2	yes	—
56	C₂×C₈	—	14	2	no	—
57	C₂×C₇	—	2	1	no	—
58	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	D₇	—	2	2	no	—
60	D₇	—	4	2	no	—
61	C₂³	—	7	1	no	—
62	C₂³	—	7	2	no	—
63	Q₈	—	2	2	no	—
64	D₄	—	14	4	no	—
65	D₄	—	14	2	no	—
66	C₂×C₄	—	7	1	no	—
67	C₂×C₄	—	7	2	no	—
68	C₂×C₄	—	7	2	no	—
69	C₂×C₄	—	14	2	no	—
70	C₂×C₄	—	14	2	no	—
71	C₂×C₄	C₄×D₇	1	2	yes	—
72	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	—	14	2	no	—
75	C₇	#413	1	1	yes	characteristic, complemented, Sylow
76	C₂²	—	7	2	no	—
77	C₂²	—	14	2	no	—
78	C₂²	—	28	1	no	—
79	C₂²	—	28	2	no	—
80	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	7	1	no	—
83	C₄	—	7	1	no	—
84	C₄	D₄×D₇	1	2	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	14	2	no	—
87	C₂	—	28	2	no	—
88	C₂	D₇×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₇×C₂²⋊C₄
commutator	C₂²×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#101	C₂
socle	C₂×C₇	C₂×C₂²⋊C₄
prime radical	C₂×#1161	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#101	D₇
hypocenter	C₇	#413
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

Complements

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

N	H ≅ G/N	type
2. D₇×M₁₆	C₂	semidirect
3. C₇×#101	C₂	semidirect
4. #6410	C₂	semidirect
5. #6397	C₂	semidirect
35. #101	D₇	semidirect
75. C₇	#413	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1161	2
D₇×M₁₆	2
C₇×#101	2
#6410	2
#6397	2
#413	7