Finite Group Databasebeta

Subgroups — #23765474

Order 10112.

170 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

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

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19098998	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098938	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098680	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098383	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098377	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098376	C₂	1	1	yes	maximal, characteristic, complemented
7	#509×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
9	C₂×#11929609	—	2	1	no	—
10	#14579168	—	2	1	no	—
11	#14579157	—	2	1	no	—
12	#14579104	—	2	1	no	—
13	#14579102	—	2	1	no	—
14	#14579101	C₂²	1	1	yes	characteristic
15	#14579097	C₂²	1	1	yes	characteristic
16	#14579088	—	2	1	no	—
17	C₈×Dic₇₉	C₂²	1	1	yes	characteristic
18	#135×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
20	SD₃₂×C₇₉	—	2	1	no	—
21	D₁₆×C₇₉	—	2	1	no	—
22	M₃₂×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×D₃₁₆	—	2	1	no	—
24	C₂×C₂²⋊D₇₉	—	4	2	no	—
25	D₆₃₂	—	2	1	no	—
26	D₆₃₂	—	2	1	no	—
27	#11929609	—	4	1	no	—
28	#11929609	—	4	1	no	—
29	#11929608	—	2	1	no	—
30	#11929608	—	4	1	no	—
31	C₂×C₇₉⋊C₈	—	4	1	no	—
32	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
33	C₄×Dic₇₉	D₄	1	1	yes	characteristic
34	C₇₉⋊C₁₆	—	2	1	no	—
35	#41×C₇₉	—	2	1	no	—
36	C₂×D₄×C₇₉	—	2	1	no	—
37	Q₁₆×C₇₉	D₄	1	1	yes	characteristic, complemented
38	SD₁₆×C₇₉	—	2	1	no	—
39	D₈×C₇₉	—	2	1	no	—
40	D₈×C₇₉	—	2	1	no	—
41	D₈×C₇₉	D₄	1	1	yes	characteristic, complemented
42	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
43	C₁₆×C₇₉	—	2	1	no	—
44	C₂³×C₇₉	—	4	1	no	—
45	C₂²×D₇₉	—	4	1	no	—
46	Q₈×C₇₉	—	2	1	no	—
47	D₄×C₇₉	—	2	1	no	—
48	D₄×C₇₉	—	4	1	no	—
49	D₄×C₇₉	—	4	1	no	—
50	D₄×C₇₉	—	4	1	no	—
51	C₂×C₄×C₇₉	—	4	1	no	—
52	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂²⋊D₇₉	—	8	4	no	—
54	C₂×Dic₇₉	—	2	1	no	—
55	D₃₁₆	—	4	1	no	—
56	D₃₁₆	—	4	1	no	—
57	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₈×C₇₉	#41	1	1	yes	characteristic
59	C₇₉⋊C₈	—	4	1	no	—
60	C₇₉⋊C₈	D₈	1	1	yes	characteristic
61	C₇₉⋊C₈	D₈	1	1	yes	characteristic
62	C₂²×C₇₉	—	4	1	no	—
63	C₂²×C₇₉	—	8	2	no	—
64	C₂²×C₇₉	—	8	1	no	—
65	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂×D₇₉	—	8	2	no	—
67	C₂×D₇₉	—	8	1	no	—
68	C₄×C₇₉	—	4	1	no	—
69	C₄×C₇₉	#121	1	1	yes	characteristic
70	C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
71	Dic₇₉	—	2	2	no	—
72	C₂×C₇₉	—	2	1	no	—
73	C₂×C₇₉	—	8	1	no	—
74	C₂×C₇₉	—	16	1	no	—
75	C₂×C₇₉	#459	1	1	yes	characteristic
76	D₇₉	—	16	1	no	—
77	#2213	—	79	1	no	maximal, Sylow
78	C₇₉	#2213	1	1	yes	characteristic, complemented, Sylow
79	#509	—	79	1	no	—
80	#509	D₇₉	1	1	yes	characteristic, complemented
81	#493	—	79	1	no	—
82	#443	—	79	1	no	—
83	#364	—	79	1	no	—
84	#361	—	79	1	no	—
85	#361	—	79	1	no	—
86	#135	C₂×D₇₉	1	1	yes	characteristic
87	C₂×D₈	—	79	1	no	—
88	C₂×D₈	—	158	1	no	—
89	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
90	#131	—	158	1	no	—
91	#127	—	158	1	no	—
92	SD₃₂	—	2	1	no	—
93	SD₃₂	—	158	1	no	—
94	D₁₆	—	2	1	no	—
95	D₁₆	—	158	1	no	—
96	M₃₂	—	79	1	no	—
97	M₃₂	C₂×D₇₉	1	1	yes	characteristic
98	#108	—	79	1	no	—
99	#104	—	158	1	no	—
100	C₄×C₈	—	79	1	no	—
101	#41	—	2	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	316	2	no	—
104	C₂×D₄	—	158	1	no	—
105	Q₁₆	C₂²⋊D₇₉	1	1	yes	characteristic, complemented
106	SD₁₆	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	2	1	no	—
109	D₈	—	316	1	no	—
110	D₈	—	316	1	no	—
111	D₈	—	158	1	no	—
112	D₈	—	158	1	no	—
113	D₈	C₂²⋊D₇₉	1	1	yes	characteristic, complemented
114	M₁₆	—	316	1	no	—
115	M₁₆	—	158	1	no	—
116	C₂×C₈	—	316	1	no	—
117	C₂×C₈	—	79	1	no	—
118	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
119	C₄²	—	79	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	158	1	no	—
122	C₂³	—	4	1	no	—
123	C₂³	—	316	1	no	—
124	Q₈	—	2	1	no	—
125	D₄	—	2	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	316	1	no	—
130	D₄	—	316	1	no	—
131	D₄	—	632	4	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	158	1	no	—
134	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
135	C₈	—	316	1	no	—
136	C₈	—	79	1	no	—
137	C₈	—	79	1	no	—
138	C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
139	C₈	#11929623	1	1	yes	characteristic
140	C₂²	—	4	1	no	—
141	C₂²	—	8	2	no	—
142	C₂²	—	8	1	no	—
143	C₂²	—	632	2	no	—
144	C₂²	—	632	1	no	—
145	C₂²	#14579185	1	1	yes	characteristic
146	C₄	—	4	1	no	—
147	C₄	—	158	2	no	—
148	C₄	#14579133	1	1	yes	characteristic
149	C₄	D₈×D₇₉	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	8	1	no	—
152	C₂	—	16	1	no	—
153	C₂	—	1264	1	no	—
154	C₂	#19098756	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19098756
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#509×C₇₉	C₂
socle	C₂×C₇₉	#459
hypercenter	#509	D₇₉
hypocenter	C₇₉	#2213
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #509

Complements

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

N	H ≅ G/N	type
1. #19098998	C₂	semidirect
2. #19098938	C₂	semidirect
3. #19098680	C₂	semidirect
4. #19098383	C₂	semidirect
5. #19098377	C₂	semidirect
6. #19098376	C₂	semidirect
7. #509×C₇₉	C₂	semidirect
37. Q₁₆×C₇₉	D₄	semidirect
41. D₈×C₇₉	D₄	semidirect
78. C₇₉	#2213	semidirect
80. #509	D₇₉	semidirect
105. Q₁₆	C₂²⋊D₇₉	semidirect
113. D₈	C₂²⋊D₇₉	semidirect