Subgroups — #89774

Order 448.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1157	C₂	1	1	yes	maximal, characteristic
2	C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6384	C₂	1	2	yes	maximal, complemented
5	#6383	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₇	—	2	1	no	—
8	#1157	—	2	2	no	—
9	#1157	C₄	1	4	yes	complemented
10	C₂×D₂₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₇	—	2	1	no	—
12	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
13	C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₇×C₂²⋊C₄	—	2	1	no	—
15	C₄²×C₇	—	2	1	no	—
16	C₂×C₇⋊C₈	—	2	2	no	—
17	C₂×C₇⋊C₈	C₂²	1	2	yes	—
18	#416	—	7	1	no	maximal, Sylow
19	C₂³×C₇	D₄	1	1	yes	characteristic
20	C₂²×D₇	—	2	1	no	—
21	C₂×C₄×C₇	—	2	2	no	—
22	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₇	D₄	1	2	yes	—
24	C₂×C₄×C₇	D₄	1	1	yes	characteristic
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂²⋊D₇	—	2	4	no	—
27	C₂×Dic₇	—	2	1	no	—
28	D₂₈	—	2	1	no	—
29	D₂₈	C₂×C₄	1	2	yes	—
30	C₄×D₇	—	2	4	no	—
31	Dic₁₄	—	2	1	no	—
32	Dic₁₄	C₂×C₄	1	2	yes	—
33	C₇⋊C₈	—	2	2	no	—
34	C₂×#41	—	7	1	no	—
35	C₂²×C₈	—	7	1	no	—
36	#117	D₇	1	1	yes	characteristic, complemented
37	#103	—	7	2	no	—
38	#102	—	7	2	no	—
39	C₂²×C₇	—	2	2	no	—
40	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
42	C₂×D₇	—	2	2	no	—
43	C₂×D₇	—	4	2	no	—
44	C₄×C₇	—	4	2	no	—
45	C₄×C₇	C₂²⋊C₄	1	2	yes	—
46	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	Dic₇	—	2	2	no	—
49	#41	—	7	4	no	—
50	#41	—	14	2	no	—
51	C₂×Q₈	—	7	1	no	—
52	C₂×D₄	—	7	1	no	—
53	C₂×D₄	—	14	1	no	—
54	C₂²×C₄	—	14	1	no	—
55	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
56	C₂×C₈	—	7	2	no	—
57	C₂×C₈	—	14	2	no	—
58	C₄⋊C₄	C₂×D₇	1	2	yes	—
59	C₂²⋊C₄	—	2	1	no	—
60	C₄²	—	2	1	no	—
61	C₂×C₇	—	2	2	no	—
62	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₇	#135	1	2	yes	—
64	D₇	—	4	2	no	—
65	C₂³	—	14	1	no	—
66	C₂³	C₂²⋊D₇	1	1	yes	characteristic
67	Q₈	—	7	2	no	—
68	Q₈	—	14	1	no	—
69	D₄	—	7	2	no	—
70	D₄	—	14	4	no	—
71	D₄	—	14	1	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	14	1	no	—
74	C₂×C₄	—	14	4	no	—
75	C₂×C₄	C₄×D₇	1	2	yes	—
76	C₂×C₄	D₂₈	1	2	yes	—
77	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
78	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
79	C₈	—	14	2	no	—
80	C₇	#416	1	1	yes	characteristic, complemented, Sylow
81	C₂²	—	2	2	no	—
82	C₂²	—	14	2	no	—
83	C₂²	—	28	2	no	—
84	C₂²	(C₂×D₇)⋊C₄	1	2	yes	—
85	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
86	C₄	—	4	2	no	—
87	C₄	—	14	2	no	—
88	C₄	(C₂×D₇)⋊C₄	1	2	yes	—
89	C₄	C₂×C₄×D₇	1	1	yes	characteristic
90	C₄	C₂×D₂₈	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	28	2	no	—
93	C₂	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
94	C₂	#6513	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#117	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×#1157	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#117	D₇
hypocenter	C₇	#416
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₇×#117	C₂	semidirect
3. C₂²×C₇⋊C₈	C₂	semidirect
4. #6384	C₂	semidirect
5. #6383	C₂	semidirect
9. #1157	C₄	semidirect
36. #117	D₇	semidirect
80. C₇	#416	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1157	2
C₇×#117	2
C₂²×C₇⋊C₈	2
#6384	2
#6383	2
#416	7