Subgroups — #90265

Order 448.

294 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	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#6456	C₂	1	3	yes	maximal, complemented
5	D₇×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
6	C₄×D₂₈	C₂	1	3	yes	maximal, complemented
7	#6435	C₂	1	3	yes	maximal, complemented
8	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
9	#1161	C₄	1	8	yes	complemented
10	C₂×D₂₈	C₂²	1	3	yes	—
11	C₂×C₄×D₇	C₂²	1	6	yes	—
12	C₂×C₄×D₇	C₂²	1	3	yes	—
13	C₇×C₄⋊C₄	C₂²	1	3	yes	—
14	C₄²×C₇	C₂²	1	3	yes	—
15	(C₂×D₇)⋊C₄	C₂²	1	3	yes	—
16	(C₂×D₇)⋊C₄	C₂²	1	3	yes	—
17	C₄⋊Dic₇	C₂²	1	3	yes	—
18	Dic₇⋊C₄	C₂²	1	1	yes	characteristic
19	Dic₇⋊C₄	C₂²	1	3	yes	—
20	C₄×Dic₇	C₂²	1	3	yes	—
21	#520	—	7	1	no	maximal, Sylow
22	C₂²×D₇	C₂³	1	3	yes	—
23	C₇×Q₈	C₂×C₄	1	4	yes	complemented
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₇	C₂³	1	3	yes	—
26	C₂×C₄×C₇	C₂³	1	3	yes	—
27	C₂×Dic₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₇	C₂³	1	3	yes	—
29	C₂×Dic₇	C₂³	1	1	yes	characteristic
30	D₂₈	C₂×C₄	1	12	yes	complemented
31	C₄×D₇	—	4	6	no	—
32	C₄×D₇	C₂×C₄	1	12	yes	complemented
33	C₂×#41	—	7	1	no	—
34	C₄×Q₈	—	7	1	no	—
35	C₄×Q₈	D₇	1	1	yes	characteristic, complemented
36	C₄×D₄	—	7	3	no	—
37	C₄×D₄	—	7	3	no	—
38	#117	—	7	3	no	—
39	C₂×C₄⋊C₄	—	7	3	no	—
40	C₂²×C₇	C₂⁴	1	1	yes	characteristic
41	C₂×D₇	—	2	3	no	—
42	C₂×D₇	—	2	3	no	—
43	C₂×D₇	C₂²×C₄	1	6	yes	—
44	C₄×C₇	—	2	3	no	—
45	C₄×C₇	—	2	1	no	—
46	C₄×C₇	C₂²×C₄	1	6	yes	—
47	Dic₇	—	2	3	no	—
48	Dic₇	—	2	1	no	—
49	Dic₇	C₂²×C₄	1	2	yes	—
50	#41	—	7	8	no	—
51	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
52	C₂×D₄	—	7	3	no	—
53	C₂²×C₄	—	7	3	no	—
54	C₂²×C₄	—	7	6	no	—
55	C₄⋊C₄	—	7	3	no	—
56	C₄⋊C₄	—	7	1	no	—
57	C₄⋊C₄	—	7	3	no	—
58	C₄⋊C₄	C₂×D₇	1	3	yes	—
59	C₂²⋊C₄	—	7	3	no	—
60	C₂²⋊C₄	—	7	3	no	—
61	C₄²	—	7	3	no	—
62	C₄²	C₂×D₇	1	3	yes	—
63	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
66	D₇	—	2	6	no	—
67	C₂³	—	7	3	no	—
68	Q₈	C₄×D₇	1	4	yes	complemented
69	D₄	—	7	12	no	—
70	C₂×C₄	—	7	1	no	—
71	C₂×C₄	—	7	12	no	—
72	C₂×C₄	—	7	1	no	—
73	C₂×C₄	—	7	3	no	—
74	C₂×C₄	—	28	6	no	—
75	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₇	1	3	yes	—
77	C₂×C₄	C₂²×D₇	1	3	yes	—
78	C₇	#520	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	7	6	no	—
80	C₂²	—	14	3	no	—
81	C₂²	—	14	3	no	—
82	C₂²	C₂³×D₇	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	3	no	—
85	C₄	—	7	2	no	—
86	C₄	—	14	1	no	—
87	C₄	—	14	3	no	—
88	C₄	C₂×C₄×D₇	1	6	yes	—
89	C₂	—	14	6	no	—
90	C₂	C₂²×C₄×D₇	1	1	yes	characteristic
91	C₂	#6551	1	1	yes	characteristic
92	C₂	#6553	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator	C₂×C₇	C₂³×C₄
Fitting	C₄×C₇×Q₈	C₂
socle, cosocle	C₂²×C₇	C₂⁴
prime radical	C₂×#1161	C₂
hypercenter	C₄×Q₈	D₇
hypocenter	C₇	#520
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×C₇×Q₈	C₂	semidirect
3. Q₈×Dic₇	C₂	semidirect
4. #6456	C₂	semidirect
5. D₇×C₄⋊C₄	C₂	semidirect
6. C₄×D₂₈	C₂	semidirect
7. #6435	C₂	semidirect
9. #1161	C₄	semidirect
23. C₇×Q₈	C₂×C₄	semidirect
30. D₂₈	C₂×C₄	semidirect
32. C₄×D₇	C₂×C₄	semidirect
35. C₄×Q₈	D₇	semidirect
68. Q₈	C₄×D₇	semidirect
78. C₇	#520	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
C₄×C₇×Q₈	2
Q₈×Dic₇	2
#6456	2
D₇×C₄⋊C₄	2
C₄×D₂₈	2
#6435	2
#520	7