Group #19099274 — subgroups

Subgroups — #19099274

Order 5056.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14579152	C₂	1	2	yes	maximal, complemented
2	#14579141	C₂	1	2	yes	maximal, complemented
3	#14579137	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₃₁₆	C₂	1	1	yes	maximal, characteristic
6	#14579123	C₂	1	2	yes	maximal
7	C₄⋊C₄×D₇₉	C₂	1	2	yes	maximal, complemented
8	C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₁₅₈	C₂²	1	1	yes	characteristic
10	C₂×D₃₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₇₉	C₂²	1	2	yes	—
12	C₂×C₄×D₇₉	C₂²	1	4	yes	—
13	C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₇₉	C₂²	1	4	yes	—
16	Dic₇₉⋊C₄	C₂²	1	4	yes	—
17	Dic₇₉⋊C₄	C₂²	1	2	yes	—
18	(C₂×D₇₉)⋊C₄	C₂²	1	4	yes	—
19	(C₂×D₇₉)⋊C₄	C₂²	1	2	yes	—
20	C₄×Dic₇₉	C₂²	1	2	yes	—
21	C₂×Q₈×C₇₉	C₂²	1	2	yes	—
22	C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
23	C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
24	C₄²×C₇₉	C₂²	1	1	yes	characteristic
25	C₂²×D₇₉	C₂³	1	2	yes	—
26	Q₈×C₇₉	—	4	2	no	—
27	C₂×C₄×C₇₉	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₇₉	C₂³	1	4	yes	—
29	C₂×C₄×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₇₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₇₉	C₂³	1	2	yes	—
32	C₂×Dic₇₉	C₂³	1	2	yes	—
33	C₂×Dic₇₉	C₂³	1	2	yes	—
34	D₃₁₆	Q₈	1	4	yes	complemented
35	C₄×D₇₉	—	2	4	no	—
36	C₄×D₇₉	—	4	4	no	—
37	Dic₁₅₈	—	2	2	no	—
38	C₂²×C₇₉	C₂⁴	1	1	yes	characteristic
39	C₂×D₇₉	—	2	2	no	—
40	C₂×D₇₉	—	2	2	no	—
41	C₂×D₇₉	C₂×Q₈	1	4	yes	—
42	C₄×C₇₉	—	2	1	no	—
43	C₄×C₇₉	—	2	4	no	—
44	C₄×C₇₉	C₂×Q₈	1	2	yes	—
45	C₄×C₇₉	#41	1	2	yes	—
46	Dic₇₉	—	2	2	no	—
47	Dic₇₉	—	2	2	no	—
48	Dic₇₉	—	2	2	no	—
49	C₂×C₇₉	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₇₉	C₂×#41	1	1	yes	characteristic
51	C₂×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
52	D₇₉	—	2	4	no	—
53	C₇₉	#554	1	1	yes	characteristic, complemented, Sylow
54	#554	—	79	1	no	maximal, Sylow
55	C₄⋊Q₈	D₇₉	1	1	yes	characteristic, complemented
56	#125	—	79	2	no	—
57	C₂²⋊Q₈	—	79	4	no	—
58	C₂²⋊Q₈	—	79	2	no	—
59	C₄×Q₈	—	79	1	no	—
60	C₄×D₄	—	79	2	no	—
61	C₄×D₄	—	79	1	no	—
62	C₂×C₄⋊C₄	—	79	2	no	—
63	C₂×Q₈	—	79	1	no	—
64	C₂×Q₈	C₂×D₇₉	1	2	yes	—
65	C₂×D₄	—	79	1	no	—
66	C₂²×C₄	—	79	4	no	—
67	C₂²×C₄	—	79	2	no	—
68	C₄⋊C₄	—	79	4	no	—
69	C₄⋊C₄	—	79	2	no	—
70	C₄⋊C₄	—	79	1	no	—
71	C₄⋊C₄	—	79	1	no	—
72	C₄⋊C₄	—	79	4	no	—
73	C₄⋊C₄	C₂×D₇₉	1	2	yes	—
74	C₄⋊C₄	C₂×D₇₉	1	2	yes	—
75	C₂²⋊C₄	—	79	4	no	—
76	C₂²⋊C₄	—	79	2	no	—
77	C₄²	—	79	2	no	—
78	C₄²	C₂×D₇₉	1	1	yes	characteristic
79	C₂³	—	79	2	no	—
80	Q₈	—	4	2	no	—
81	Q₈	—	158	2	no	—
82	D₄	—	79	4	no	—
83	C₂×C₄	—	316	4	no	—
84	C₂×C₄	—	79	2	no	—
85	C₂×C₄	—	79	2	no	—
86	C₂×C₄	—	79	2	no	—
87	C₂×C₄	—	158	4	no	—
88	C₂×C₄	C₂²×D₇₉	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₇₉	1	4	yes	—
90	C₂×C₄	C₂²×D₇₉	1	1	yes	characteristic
91	C₂×C₄	C₂²×D₇₉	1	1	yes	characteristic
92	C₂²	—	79	4	no	—
93	C₂²	—	158	2	no	—
94	C₂²	—	158	2	no	—
95	C₂²	C₂³×D₇₉	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	2	1	no	—
98	C₄	—	158	2	no	—
99	C₄	—	158	2	no	—
100	C₄	—	158	2	no	—
101	C₄	Q₈×D₇₉	1	2	yes	—
102	C₄	#11929625	1	2	yes	—
103	C₂	—	158	4	no	—
104	C₂	C₂×Q₈×D₇₉	1	1	yes	characteristic
105	C₂	C₂×#11929625	1	1	yes	characteristic
106	C₂	#14579206	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14579152

C₂

yes

maximal, complemented

#14579141

C₂

yes

maximal, complemented

#14579137

C₂

yes

maximal, complemented

C₄×Dic₁₅₈

C₂

yes

maximal, characteristic, complemented

C₄×D₃₁₆

C₂

yes

maximal, characteristic

#14579123

C₂

yes

maximal

C₄⋊C₄×D₇₉

C₂

yes

maximal, complemented

C₄⋊Q₈×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₅₈

C₂²

yes

characteristic

C₂×D₃₁₆

C₂²

yes

characteristic

C₂×C₄×D₇₉

C₂²

yes

—

C₂×C₄×D₇₉

C₂²

yes

—

C₄⋊Dic₇₉

C₂²

yes

characteristic

C₄⋊Dic₇₉

C₂²

yes

characteristic

C₄⋊Dic₇₉

C₂²

yes

—

Dic₇₉⋊C₄

C₂²

yes

—

Dic₇₉⋊C₄

C₂²

yes

—

(C₂×D₇₉)⋊C₄

C₂²

yes

—

(C₂×D₇₉)⋊C₄

C₂²

yes

—

C₄×Dic₇₉

C₂²

yes

—

C₂×Q₈×C₇₉

C₂²

yes

—

C₄⋊C₄×C₇₉

C₂²

yes

—

C₄⋊C₄×C₇₉

C₂²

yes

—

C₄²×C₇₉

C₂²

yes

characteristic

C₂²×D₇₉

C₂³

yes

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₄×C₇₉

C₂³

yes

—

C₂×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₄×C₇₉

C₂³

yes

characteristic

C₂×Dic₇₉

C₂³

yes

—

C₂×Dic₇₉

C₂³

yes

—

C₂×Dic₇₉

C₂³

yes

—

D₃₁₆

Q₈

yes

complemented

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

C₂²×C₇₉

C₂⁴

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

C₂×Q₈

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×Q₈

yes

—

C₄×C₇₉

#41

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

C₂²×Q₈

yes

characteristic

C₂×C₇₉

C₂×#41

yes

characteristic

C₂×C₇₉

H(5,𝔽₂)

yes

characteristic

D₇₉

—

C₇₉

#554

yes

characteristic, complemented, Sylow

#554

—

maximal, Sylow

C₄⋊Q₈

D₇₉

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×Q₈

C₂×D₇₉

yes

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇₉

yes

—

C₄⋊C₄

C₂×D₇₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₇₉

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

158

—

D₄

—

C₂×C₄

—

316

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

158

—

C₂×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₄

C₂²×D₇₉

yes

—

C₂×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₄

C₂²×D₇₉

yes

characteristic

C₂²

—

C₂²

—

158

—

C₂²

—

158

—

C₂²

C₂³×D₇₉

yes

characteristic

C₄

—

C₄

—

C₄

—

158

—

C₄

—

158

—

100

C₄

—

158

—

101

C₄

Q₈×D₇₉

yes

—

102

C₄

#11929625

yes

—

103

C₂

—

158

—

104

C₂

C₂×Q₈×D₇₉

yes

characteristic

105

C₂

C₂×#11929625

yes

characteristic

106

C₂

#14579206

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇₉
commutator, socle	C₂²×C₇₉	C₂⁴
Fitting	C₄⋊Q₈×C₇₉	C₂
hypercenter	C₄⋊Q₈	D₇₉
hypocenter	C₇₉	#554
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇₉

commutator, socle

C₂²×C₇₉

C₂⁴

Fitting

C₄⋊Q₈×C₇₉

C₂

hypercenter

C₄⋊Q₈

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #14579152	C₂	semidirect
2. #14579141	C₂	semidirect
3. #14579137	C₂	semidirect
4. C₄×Dic₁₅₈	C₂	semidirect
7. C₄⋊C₄×D₇₉	C₂	semidirect
8. C₄⋊Q₈×C₇₉	C₂	semidirect
34. D₃₁₆	Q₈	semidirect
53. C₇₉	#554	semidirect
55. C₄⋊Q₈	D₇₉	semidirect

H ≅ G/N

type

1. #14579152

C₂

semidirect

2. #14579141

C₂

semidirect

3. #14579137

C₂

semidirect

4. C₄×Dic₁₅₈

C₂

semidirect

7. C₄⋊C₄×D₇₉

C₂

semidirect

8. C₄⋊Q₈×C₇₉

C₂

semidirect

34. D₃₁₆

Q₈

semidirect

53. C₇₉

#554

semidirect

55. C₄⋊Q₈

D₇₉

semidirect

Subgroups — #19099274

Conjugacy classes of subgroups

Complements