Subgroups — #90412

Order 448.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₇	C₂	1	2	yes	maximal
3	#6507	C₂	1	2	yes	maximal
4	#6453	C₂	1	2	yes	maximal
5	#6452	C₂	1	4	yes	maximal
6	#6450	C₂	1	2	yes	maximal
7	C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic
9	C₂×C₇×Q₈	C₂²	1	2	yes	—
10	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₄²×C₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₇	C₂²	1	4	yes	—
18	Dic₇⋊C₄	C₂²	1	2	yes	—
19	Dic₇⋊C₄	C₂²	1	2	yes	—
20	Dic₇⋊C₄	C₂²	1	4	yes	—
21	Dic₇⋊C₄	C₂²	1	4	yes	—
22	C₄×Dic₇	C₂²	1	2	yes	—
23	C₄×Dic₇	C₂²	1	2	yes	—
24	C₄×Dic₇	C₂²	1	4	yes	—
25	#557	—	7	1	no	maximal, Sylow
26	C₇×Q₈	—	2	4	no	—
27	C₂×C₄×C₇	C₂³	1	4	yes	—
28	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂×Dic₇	C₂³	1	2	yes	—
32	C₂×Dic₇	C₂³	1	2	yes	—
33	C₂×Dic₇	C₂³	1	2	yes	—
34	C₂×Dic₇	C₂³	1	2	yes	—
35	Dic₁₄	—	2	2	no	—
36	Dic₁₄	Q₈	1	4	yes	complemented
37	C₄⋊Q₈	—	7	2	no	—
38	C₄⋊Q₈	D₇	1	1	yes	characteristic
39	#125	—	7	4	no	—
40	#125	—	7	2	no	—
41	C₄×Q₈	—	7	2	no	—
42	C₄×Q₈	—	7	2	no	—
43	C₄×Q₈	—	7	1	no	—
44	C₄×Q₈	—	7	1	no	—
45	C₂²×C₇	C₂⁴	1	1	yes	characteristic
46	C₄×C₇	—	2	4	no	—
47	C₄×C₇	—	2	1	no	—
48	C₄×C₇	C₂×Q₈	1	2	yes	—
49	C₄×C₇	#41	1	2	yes	—
50	Dic₇	—	2	2	no	—
51	Dic₇	—	2	2	no	—
52	Dic₇	—	2	2	no	—
53	Dic₇	C₂×Q₈	1	4	yes	—
54	C₂×Q₈	—	7	1	no	—
55	C₂×Q₈	—	7	1	no	—
56	C₂×Q₈	C₂×D₇	1	2	yes	—
57	C₄⋊C₄	—	7	2	no	—
58	C₄⋊C₄	—	7	2	no	—
59	C₄⋊C₄	—	7	4	no	—
60	C₄⋊C₄	—	7	4	no	—
61	C₄⋊C₄	—	7	1	no	—
62	C₄⋊C₄	—	7	4	no	—
63	C₄⋊C₄	—	7	1	no	—
64	C₄⋊C₄	C₂×D₇	1	2	yes	—
65	C₄⋊C₄	C₂×D₇	1	2	yes	—
66	C₄²	—	7	4	no	—
67	C₄²	—	7	2	no	—
68	C₄²	—	7	2	no	—
69	C₄²	C₂×D₇	1	1	yes	characteristic
70	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₇	C₂×#41	1	1	yes	characteristic
72	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
73	Q₈	—	2	4	no	—
74	Q₈	—	7	4	no	—
75	Q₈	—	14	2	no	—
76	C₂×C₄	—	7	2	no	—
77	C₂×C₄	—	7	2	no	—
78	C₂×C₄	—	7	2	no	—
79	C₂×C₄	—	7	2	no	—
80	C₂×C₄	C₂²×D₇	1	4	yes	—
81	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
83	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
84	C₇	#557	1	1	yes	characteristic, complemented, Sylow
85	C₂²	C₂³×D₇	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	2	1	no	—
88	C₄	—	7	4	no	—
89	C₄	—	14	2	no	—
90	C₄	—	14	2	no	—
91	C₄	—	14	2	no	—
92	C₄	#1159	1	2	yes	—
93	C₄	Q₈×D₇	1	2	yes	—
94	C₂	C₂×#1159	1	1	yes	characteristic
95	C₂	C₂×Q₈×D₇	1	1	yes	characteristic
96	C₂	#6551	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	#557
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
36. Dic₁₄	Q₈	semidirect
84. C₇	#557	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₄⋊Q₈	2
Q₈×Dic₇	2
#6507	2
#6453	2
#6452	2
#6450	2
C₄×Dic₁₄	2
#557	7