Group #19099356 — subgroups

Subgroups — C₂×#14579192 #19099356

Order 5056.

258 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂×#11929624	C₂	1	1	yes	maximal, characteristic
2	#14579192	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11929614	C₂	1	2	yes	maximal, complemented
4	C₂×#11929611	C₂	1	2	yes	maximal, complemented
5	C₂×#11929608	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#11929624	—	2	2	no	—
8	#11929624	C₂²	1	4	yes	—
9	C₂×Dic₁₅₈	C₂²	1	1	yes	characteristic
10	C₂×D₃₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₇₉	—	2	1	no	—
12	C₂×C₄×D₇₉	—	2	1	no	—
13	#11929614	C₂²	1	8	yes	complemented
14	#11929611	C₂²	1	8	yes	complemented
15	#11929608	C₂²	1	4	yes	complemented
16	C₂×C₇₉⋊C₈	C₂²	1	2	yes	—
17	C₂×Q₈×C₇₉	—	2	1	no	—
18	C₂×Q₈×C₇₉	—	2	2	no	—
19	C₂×Q₈×C₇₉	C₂²	1	2	yes	—
20	C₂×Q₈×C₇₉	C₂²	1	4	yes	complemented
21	C₂²×C₄×C₇₉	—	2	1	no	—
22	C₂²×C₄×C₇₉	C₂²	1	1	yes	characteristic
23	C₂³×C₇₉	D₄	1	1	yes	characteristic
24	C₂²×D₇₉	—	2	1	no	—
25	Q₈×C₇₉	—	2	2	no	—
26	Q₈×C₇₉	—	2	4	no	—
27	Q₈×C₇₉	C₂³	1	4	yes	—
28	C₂×C₄×C₇₉	—	2	2	no	—
29	C₂×C₄×C₇₉	—	2	4	no	—
30	C₂×C₄×C₇₉	D₄	1	1	yes	characteristic
31	C₂×C₄×C₇₉	D₄	1	2	yes	—
32	C₂×C₄×C₇₉	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₇₉	C₂³	1	2	yes	—
34	C₂²⋊D₇₉	—	2	4	no	—
35	C₂×Dic₇₉	—	2	1	no	—
36	D₃₁₆	—	2	1	no	—
37	D₃₁₆	C₂³	1	2	yes	—
38	C₄×D₇₉	—	2	4	no	—
39	Dic₁₅₈	—	2	1	no	—
40	Dic₁₅₈	C₂³	1	2	yes	—
41	C₇₉⋊C₈	C₂³	1	4	yes	—
42	C₂²×C₇₉	—	2	2	no	—
43	C₂²×C₇₉	C₂×D₄	1	2	yes	—
44	C₂²×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×D₇₉	—	2	2	no	—
46	C₂×D₇₉	—	4	2	no	—
47	C₄×C₇₉	—	2	4	no	—
48	C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₄×C₇₉	C₂×D₄	1	2	yes	—
50	C₄×C₇₉	C₂⁴	1	1	yes	characteristic
51	Dic₇₉	—	2	2	no	—
52	C₂×C₇₉	—	2	2	no	—
53	C₂×C₇₉	#137	1	2	yes	complemented
54	C₂×C₇₉	C₂²×D₄	1	1	yes	characteristic
55	D₇₉	—	4	2	no	—
56	C₇₉	C₂×#137	1	1	yes	characteristic, complemented, Sylow
57	C₂×#137	—	79	1	no	maximal, Sylow
58	C₂×#41	—	79	1	no	—
59	C₂²×Q₈	D₇₉	1	1	yes	characteristic, complemented
60	#137	—	79	8	no	—
61	C₂×Q₁₆	—	79	2	no	—
62	C₂×SD₁₆	—	79	2	no	—
63	C₂×M₁₆	—	79	1	no	—
64	#41	—	79	4	no	—
65	#41	—	158	2	no	—
66	C₂×Q₈	—	2	2	no	—
67	C₂×Q₈	—	2	1	no	—
68	C₂×Q₈	—	79	1	no	—
69	C₂×Q₈	C₂×D₇₉	1	2	yes	—
70	C₂×Q₈	C₂×D₇₉	1	4	yes	complemented
71	C₂×D₄	—	79	1	no	—
72	C₂×D₄	—	158	1	no	—
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	—	158	1	no	—
75	C₂²×C₄	C₂×D₇₉	1	1	yes	characteristic
76	Q₁₆	—	79	8	no	—
77	SD₁₆	—	79	8	no	—
78	M₁₆	—	79	4	no	—
79	C₂×C₈	—	79	2	no	—
80	C₂³	—	158	1	no	—
81	C₂³	C₂²⋊D₇₉	1	1	yes	characteristic
82	Q₈	—	2	2	no	—
83	Q₈	—	2	4	no	—
84	Q₈	—	79	2	no	—
85	Q₈	—	158	1	no	—
86	Q₈	C₂²×D₇₉	1	4	yes	—
87	D₄	—	79	2	no	—
88	D₄	—	158	4	no	—
89	D₄	—	158	1	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	158	1	no	—
93	C₂×C₄	—	158	4	no	—
94	C₂×C₄	C₂²⋊D₇₉	1	2	yes	—
95	C₂×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₇₉	1	2	yes	—
97	C₂×C₄	C₂²×D₇₉	1	1	yes	characteristic
98	C₈	—	79	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	316	2	no	—
101	C₂²	—	158	2	no	—
102	C₂²	C₂×C₂²⋊D₇₉	1	2	yes	—
103	C₂²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
104	C₄	—	2	4	no	—
105	C₄	—	158	2	no	—
106	C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
107	C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
108	C₄	C₂³×D₇₉	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	—	316	2	no	—
111	C₂	#14579192	1	2	yes	direct product factor
112	C₂	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11929624

C₂

yes

maximal, characteristic

#14579192

C₂

yes

maximal, direct product factor

C₂×#11929614

C₂

yes

maximal, complemented

C₂×#11929611

C₂

yes

maximal, complemented

C₂×#11929608

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₇₉

C₂

yes

maximal, characteristic, complemented

#11929624

—

#11929624

C₂²

yes

—

C₂×Dic₁₅₈

C₂²

yes

characteristic

C₂×D₃₁₆

C₂²

yes

characteristic

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

#11929614

C₂²

yes

complemented

#11929611

C₂²

yes

complemented

#11929608

C₂²

yes

complemented

C₂×C₇₉⋊C₈

C₂²

yes

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂²

yes

—

C₂×Q₈×C₇₉

C₂²

yes

complemented

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²

yes

characteristic

C₂³×C₇₉

D₄

yes

characteristic

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

C₂³

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₄

yes

characteristic

C₂×C₄×C₇₉

D₄

yes

—

C₂×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₄×C₇₉

C₂³

yes

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

C₂³

yes

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

C₂³

yes

—

C₇₉⋊C₈

C₂³

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₄

yes

—

C₂²×C₇₉

C₂×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×D₄

yes

characteristic

C₄×C₇₉

C₂×D₄

yes

—

C₄×C₇₉

C₂⁴

yes

characteristic

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#137

yes

complemented

C₂×C₇₉

C₂²×D₄

yes

characteristic

D₇₉

—

C₇₉

C₂×#137

yes

characteristic, complemented, Sylow

C₂×#137

—

maximal, Sylow

C₂×#41

—

C₂²×Q₈

D₇₉

yes

characteristic, complemented

#137

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×M₁₆

—

#41

—

#41

—

158

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₇₉

yes

—

C₂×Q₈

C₂×D₇₉

yes

complemented

C₂×D₄

—

C₂×D₄

—

158

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂×D₇₉

yes

characteristic

Q₁₆

—

SD₁₆

—

M₁₆

—

C₂×C₈

—

C₂³

—

158

—

C₂³

C₂²⋊D₇₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

158

—

Q₈

C₂²×D₇₉

yes

—

D₄

—

D₄

—

158

—

D₄

—

158

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

158

—

C₂×C₄

—

158

—

C₂×C₄

C₂²⋊D₇₉

yes

—

C₂×C₄

C₂²⋊D₇₉

yes

characteristic

C₂×C₄

C₂²×D₇₉

yes

—

C₂×C₄

C₂²×D₇₉

yes

characteristic

C₈

—

C₂²

—

100

C₂²

—

316

—

101

C₂²

—

158

—

102

C₂²

C₂×C₂²⋊D₇₉

yes

—

103

C₂²

C₂×C₂²⋊D₇₉

yes

characteristic

104

C₄

—

105

C₄

—

158

—

106

C₄

C₂×C₂²⋊D₇₉

yes

—

107

C₄

C₂×C₂²⋊D₇₉

yes

characteristic

108

C₄

C₂³×D₇₉

yes

characteristic

109

C₂

—

110

C₂

—

316

—

111

C₂

#14579192

yes

direct product factor

112

C₂

C₂²×C₂²⋊D₇₉

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂⁴
Frattini	C₄	C₂³×D₇₉
Fitting	C₂²×Q₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₄
hypercenter	C₂²×Q₈	D₇₉
hypocenter	C₇₉	C₂×#137
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₇₉

commutator

C₄×C₇₉

C₂⁴

Frattini

C₄

C₂³×D₇₉

Fitting

C₂²×Q₈×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₄

hypercenter

C₂²×Q₈

D₇₉

hypocenter

C₇₉

C₂×#137

LCS

G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #14579192	C₂	direct (N × H)
3. C₂×#11929614	C₂	semidirect
4. C₂×#11929611	C₂	semidirect
5. C₂×#11929608	C₂	semidirect
6. C₂²×Q₈×C₇₉	C₂	semidirect
13. #11929614	C₂²	semidirect
14. #11929611	C₂²	semidirect
15. #11929608	C₂²	semidirect
20. C₂×Q₈×C₇₉	C₂²	semidirect
53. C₂×C₇₉	#137	semidirect
56. C₇₉	C₂×#137	semidirect
59. C₂²×Q₈	D₇₉	semidirect
70. C₂×Q₈	C₂×D₇₉	semidirect
111. C₂	#14579192	direct (N × H)

H ≅ G/N

type

2. #14579192

C₂

direct (N × H)

3. C₂×#11929614

C₂

semidirect

4. C₂×#11929611

C₂

semidirect

5. C₂×#11929608

C₂

semidirect

6. C₂²×Q₈×C₇₉

C₂

semidirect

13. #11929614

C₂²

semidirect

14. #11929611

C₂²

semidirect

15. #11929608

C₂²

semidirect

20. C₂×Q₈×C₇₉

C₂²

semidirect

53. C₂×C₇₉

#137

semidirect

56. C₇₉

C₂×#137

semidirect

59. C₂²×Q₈

D₇₉

semidirect

70. C₂×Q₈

C₂×D₇₉

semidirect

111. C₂

#14579192

direct (N × H)

Subgroups — C₂×#14579192 #19099356

Conjugacy classes of subgroups

Complements

Subgroups — C2×#14579192 #19099356

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14579192 #19099356