Group #23765799 — subgroups

Subgroups — #384×D₇₉ #23765799

Order 10112.

888 conjugacy classes of subgroups; only the 365 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 365 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×C₄⋊C₄×D₇₉	C₂	1	6	yes	maximal
2	C₂×C₄²×D₇₉	C₂	1	1	yes	maximal, characteristic
3	#19098433	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098432	C₂	1	6	yes	maximal, complemented
5	#384×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₇₉	C₂²	1	4	yes	—
7	C₂²×C₄×D₇₉	C₂²	1	3	yes	—
8	C₂×C₄⋊Dic₇₉	C₂²	1	6	yes	—
9	C₂×Dic₇₉⋊C₄	C₂²	1	12	yes	—
10	C₄⋊C₄×D₇₉	—	2	24	no	—
11	C₂×C₄×Dic₇₉	C₂²	1	3	yes	—
12	C₄²×D₇₉	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₄×C₇₉	C₂²	1	6	yes	—
14	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
15	C₂³×D₇₉	C₂³	1	1	yes	characteristic
16	C₂²×Dic₇₉	C₂³	1	4	yes	—
17	C₂²×Dic₇₉	C₂³	1	3	yes	—
18	C₂×C₄×D₇₉	—	2	24	no	—
19	C₂×C₄×D₇₉	C₂×C₄	1	12	yes	—
20	C₂×C₄×D₇₉	D₄	1	12	yes	—
21	C₂×C₄×D₇₉	Q₈	1	12	yes	—
22	C₄⋊Dic₇₉	—	2	12	no	—
23	Dic₇₉⋊C₄	—	2	24	no	—
24	C₄×Dic₇₉	C₂×C₄	1	12	yes	complemented
25	C₂²×C₄×C₇₉	C₂³	1	4	yes	—
26	C₂²×C₄×C₇₉	C₂³	1	3	yes	—
27	C₄⋊C₄×C₇₉	—	2	12	no	—
28	C₄²×C₇₉	C₂×C₄	1	4	yes	complemented
29	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
30	C₂²×D₇₉	C₂²×C₄	1	2	yes	—
31	C₂²×D₇₉	C₂×D₄	1	6	yes	—
32	C₂²×D₇₉	C₂×Q₈	1	6	yes	—
33	C₂×C₄×C₇₉	—	2	12	no	—
34	C₂×C₄×C₇₉	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₇₉	C₂×D₄	1	6	yes	—
36	C₂×C₄×C₇₉	C₂×Q₈	1	6	yes	—
37	C₂×Dic₇₉	—	2	12	no	—
38	C₂×Dic₇₉	C₂²×C₄	1	6	yes	—
39	C₂×Dic₇₉	C₂×D₄	1	6	yes	—
40	C₂×Dic₇₉	C₂×Q₈	1	6	yes	—
41	C₄×D₇₉	—	4	16	no	—
42	C₄×D₇₉	C₄⋊C₄	1	48	yes	complemented
43	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
44	C₂²×C₇₉	C₂²×D₄	1	3	yes	—
45	C₂²×C₇₉	C₂²×Q₈	1	3	yes	—
46	C₂×D₇₉	C₂×C₄⋊C₄	1	12	yes	—
47	C₂×D₇₉	#127	1	4	yes	—
48	C₂×D₇₉	C₄⋊Q₈	1	12	yes	—
49	C₄×C₇₉	—	4	4	no	—
50	C₄×C₇₉	C₂×C₄⋊C₄	1	12	yes	complemented
51	Dic₇₉	—	4	4	no	—
52	Dic₇₉	C₂×C₄⋊C₄	1	12	yes	complemented
53	C₂×C₇₉	C₂²×C₄⋊C₄	1	3	yes	—
54	C₂×C₇₉	C₂×#127	1	1	yes	characteristic
55	C₂×C₇₉	C₂×C₄⋊Q₈	1	3	yes	—
56	D₇₉	#384	1	8	yes	direct product factor
57	C₂×#384	—	79	1	no	maximal, Sylow
58	C₇₉	C₂×#384	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₄⋊C₄	—	79	6	no	—
60	C₂²×C₄²	—	79	1	no	—
61	#384	—	79	6	no	—
62	#384	—	79	1	no	—
63	#384	D₇₉	1	1	yes	characteristic, direct product factor
64	C₂³×C₄	—	79	3	no	—
65	C₂³×C₄	—	79	4	no	—
66	C₂×C₄⋊C₄	—	79	12	no	—
67	C₂×C₄⋊C₄	—	79	6	no	—
68	C₂×C₄⋊C₄	—	158	24	no	—
69	C₂×C₄⋊C₄	C₂×D₇₉	1	6	yes	—
70	C₂×C₄²	—	79	3	no	—
71	C₂×C₄²	—	79	8	no	—
72	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
73	C₂⁴	—	79	1	no	—
74	C₂²×C₄	—	79	3	no	—
75	C₂²×C₄	—	79	12	no	—
76	C₂²×C₄	—	79	12	no	—
77	C₂²×C₄	—	79	12	no	—
78	C₂²×C₄	—	79	4	no	—
79	C₂²×C₄	—	158	24	no	—
80	C₂²×C₄	C₂²×D₇₉	1	4	yes	—
81	C₂²×C₄	C₂²×D₇₉	1	3	yes	—
82	C₄⋊C₄	—	2	12	no	—
83	C₄⋊C₄	—	158	24	no	—
84	C₄⋊C₄	—	158	12	no	—
85	C₄²	—	79	12	no	—
86	C₄²	C₄×D₇₉	1	4	yes	complemented
87	C₂³	—	79	6	no	—
88	C₂³	—	79	2	no	—
89	C₂³	—	79	6	no	—
90	C₂³	C₂³×D₇₉	1	1	yes	characteristic
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	316	16	no	—
93	C₂×C₄	—	79	6	no	—
94	C₂×C₄	—	79	6	no	—
95	C₂×C₄	—	79	6	no	—
96	C₂×C₄	—	79	48	no	—
97	C₂×C₄	—	158	12	no	—
98	C₂×C₄	C₂×C₄×D₇₉	1	6	yes	—
99	C₂×C₄	D₄×D₇₉	1	6	yes	—
100	C₂×C₄	Q₈×D₇₉	1	6	yes	—
101	C₂²	—	79	12	no	—
102	C₂²	—	79	4	no	—
103	C₂²	—	79	12	no	—
104	C₂²	C₂²×C₄×D₇₉	1	1	yes	characteristic
105	C₂²	C₂×D₄×D₇₉	1	3	yes	—
106	C₂²	C₂×Q₈×D₇₉	1	3	yes	—
107	C₄	—	4	4	no	—
108	C₄	—	316	4	no	—
109	C₄	—	79	12	no	—
110	C₄	C₄⋊C₄×D₇₉	1	12	yes	complemented
111	C₂	—	79	8	no	—
112	C₂	C₂×C₄⋊C₄×D₇₉	1	3	yes	—
113	C₂	#127×D₇₉	1	1	yes	characteristic
114	C₂	C₄⋊Q₈×D₇₉	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₄×D₇₉

C₂

yes

maximal

C₂×C₄²×D₇₉

C₂

yes

maximal, characteristic

#19098433

C₂

yes

maximal, characteristic, complemented

#19098432

C₂

yes

maximal, complemented

#384×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₇₉

C₂²

yes

—

C₂²×C₄×D₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

C₂²

yes

—

C₄⋊C₄×D₇₉

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₄²×D₇₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂³×D₇₉

C₂³

yes

characteristic

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

C₂×C₄

yes

—

C₂×C₄×D₇₉

D₄

yes

—

C₂×C₄×D₇₉

Q₈

yes

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

C₂×C₄

yes

complemented

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

complemented

C₂³×C₇₉

C₂⁴

yes

characteristic

C₂²×D₇₉

C₂²×C₄

yes

—

C₂²×D₇₉

C₂×D₄

yes

—

C₂²×D₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂²×C₄

yes

—

C₂×Dic₇₉

C₂×D₄

yes

—

C₂×Dic₇₉

C₂×Q₈

yes

—

C₄×D₇₉

—

C₄×D₇₉

C₄⋊C₄

yes

complemented

C₂²×C₇₉

C₂³×C₄

yes

characteristic

C₂²×C₇₉

C₂²×D₄

yes

—

C₂²×C₇₉

C₂²×Q₈

yes

—

C₂×D₇₉

C₂×C₄⋊C₄

yes

—

C₂×D₇₉

#127

yes

—

C₂×D₇₉

C₄⋊Q₈

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

complemented

Dic₇₉

—

Dic₇₉

C₂×C₄⋊C₄

yes

complemented

C₂×C₇₉

C₂²×C₄⋊C₄

yes

—

C₂×C₇₉

C₂×#127

yes

characteristic

C₂×C₇₉

C₂×C₄⋊Q₈

yes

—

D₇₉

#384

yes

direct product factor

C₂×#384

—

maximal, Sylow

C₇₉

C₂×#384

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

—

C₂²×C₄²

—

#384

—

#384

—

#384

D₇₉

yes

characteristic, direct product factor

C₂³×C₄

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

158

—

C₄⋊C₄

—

158

—

C₄²

—

C₄²

C₄×D₇₉

yes

complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₇₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

316

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

158

—

C₂×C₄

C₂×C₄×D₇₉

yes

—

C₂×C₄

D₄×D₇₉

yes

—

100

C₂×C₄

Q₈×D₇₉

yes

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂²×C₄×D₇₉

yes

characteristic

105

C₂²

C₂×D₄×D₇₉

yes

—

106

C₂²

C₂×Q₈×D₇₉

yes

—

107

C₄

—

108

C₄

—

316

—

109

C₄

—

110

C₄

C₄⋊C₄×D₇₉

yes

complemented

111

C₂

—

112

C₂

C₂×C₄⋊C₄×D₇₉

yes

—

113

C₂

#127×D₇₉

yes

characteristic

114

C₂

C₄⋊Q₈×D₇₉

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂³×C₄
Fitting	#384×C₇₉	C₂
socle, cosocle	C₂³×C₇₉	C₂⁴
prime radical	C₂³×D₇₉	C₂³
hypercenter	#384	D₇₉
hypocenter	C₇₉	C₂×#384
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator

C₂²×C₇₉

C₂³×C₄

Fitting

#384×C₇₉

C₂

socle, cosocle

C₂³×C₇₉

C₂⁴

prime radical

C₂³×D₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
3. #19098433	C₂	semidirect
4. #19098432	C₂	semidirect
5. #384×C₇₉	C₂	semidirect
12. C₄²×D₇₉	C₄	semidirect
24. C₄×Dic₇₉	C₂×C₄	semidirect
28. C₄²×C₇₉	C₂×C₄	semidirect
42. C₄×D₇₉	C₄⋊C₄	semidirect
50. C₄×C₇₉	C₂×C₄⋊C₄	semidirect
52. Dic₇₉	C₂×C₄⋊C₄	semidirect
56. D₇₉	#384	direct (N × H)
58. C₇₉	C₂×#384	semidirect
63. #384	D₇₉	direct (N × H)
86. C₄²	C₄×D₇₉	semidirect
110. C₄	C₄⋊C₄×D₇₉	semidirect

H ≅ G/N

type

3. #19098433

C₂

semidirect

4. #19098432

C₂

semidirect

5. #384×C₇₉

C₂

semidirect

12. C₄²×D₇₉

C₄

semidirect

24. C₄×Dic₇₉

C₂×C₄

semidirect

28. C₄²×C₇₉

C₂×C₄

semidirect

42. C₄×D₇₉

C₄⋊C₄

semidirect

50. C₄×C₇₉

C₂×C₄⋊C₄

semidirect

52. Dic₇₉

C₂×C₄⋊C₄

semidirect

56. D₇₉

#384

direct (N × H)

58. C₇₉

C₂×#384

semidirect

63. #384

D₇₉

direct (N × H)

86. C₄²

C₄×D₇₉

semidirect

110. C₄

C₄⋊C₄×D₇₉

semidirect

Subgroups — #384×D₇₉ #23765799

Conjugacy classes of subgroups

Complements

Subgroups — #384×D79 #23765799

Conjugacy classes of subgroups

Complements

Subgroups — #384×D₇₉ #23765799