Finite Group Databasebeta

Subgroups — #34209188

Order 20224.

160 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 53 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	#23765681	C₂	1	1	yes	maximal, characteristic, complemented
2	#23765281	C₂	1	1	yes	maximal, characteristic
3	#23761606	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761605	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761597	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761587	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1418	C₂	1	1	yes	maximal, characteristic
8	#19098967	—	2	1	no	—
9	#19098680	—	2	1	no	—
10	#19098397	C₂²	1	1	yes	characteristic
11	#19098393	C₂²	1	1	yes	characteristic
12	#19098391	C₂²	1	1	yes	characteristic
13	#19098390	—	2	1	no	—
14	#19098370	—	2	1	no	—
15	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
16	#508×C₇₉	C₂²	1	1	yes	characteristic
17	M₆₄×C₇₉	C₂²	1	1	yes	characteristic
18	#367×C₇₉	C₂²	1	1	yes	characteristic
19	#14579168	—	4	1	no	—
20	#14579160	—	4	1	no	—
21	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
22	#14579097	—	2	1	no	—
23	#14579095	—	2	1	no	—
24	#14579094	—	4	1	no	—
25	#14579088	—	4	1	no	—
26	C₈×Dic₇₉	D₄	1	1	yes	characteristic
27	C₇₉⋊C₃₂	—	2	1	no	—
28	#135×C₇₉	—	2	1	no	—
29	Q₃₂×C₇₉	Q₈	1	1	yes	characteristic, complemented
30	SD₃₂×C₇₉	—	2	1	no	—
31	D₁₆×C₇₉	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
33	#107×C₇₉	—	2	1	no	—
34	C₃₂×C₇₉	—	2	1	no	—
35	C₂×Dic₁₅₈	—	8	1	no	—
36	#11929608	—	4	1	no	—
37	#11929608	—	8	1	no	—
38	C₂×C₇₉⋊C₈	—	8	1	no	—
39	C₂×C₇₉⋊C₈	D₈	1	1	yes	characteristic
40	C₄⋊Dic₇₉	—	4	1	no	—
41	Dic₇₉⋊C₄	—	8	1	no	—
42	C₄×Dic₇₉	D₈	1	1	yes	characteristic
43	C₇₉⋊C₁₆	—	2	1	no	—
44	#41×C₇₉	—	4	1	no	—
45	Q₁₆×C₇₉	—	2	1	no	—
46	SD₁₆×C₇₉	—	4	1	no	—
47	D₈×C₇₉	—	2	1	no	—
48	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₄⋊C₄×C₇₉	—	4	1	no	—
50	C₁₆×C₇₉	C₂×Q₈	1	1	yes	characteristic
51	C₁₆×C₇₉	#41	1	1	yes	characteristic
52	Q₈×C₇₉	—	4	1	no	—
53	D₄×C₇₉	—	4	1	no	—
54	D₄×C₇₉	—	8	1	no	—
55	C₂×C₄×C₇₉	—	8	1	no	—
56	C₂×C₄×C₇₉	—	8	1	no	—
57	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₇₉	—	2	1	no	—
59	C₂×Dic₇₉	—	8	1	no	—
60	Dic₁₅₈	—	16	2	no	—
61	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₇₉	#137	1	1	yes	characteristic
63	C₇₉⋊C₈	—	8	1	no	—
64	C₇₉⋊C₈	SD₃₂	1	1	yes	characteristic
65	C₇₉⋊C₈	SD₃₂	1	1	yes	characteristic
66	C₂²×C₇₉	—	8	1	no	—
67	C₂²×C₇₉	#509	1	1	yes	characteristic
68	C₄×C₇₉	—	8	1	no	—
69	C₄×C₇₉	—	16	1	no	—
70	C₄×C₇₉	#474	1	1	yes	characteristic
71	C₄×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
72	Dic₇₉	—	2	2	no	—
73	Dic₇₉	—	16	1	no	—
74	#13710	—	79	1	no	maximal, Sylow
75	C₂×C₇₉	—	2	1	no	—
76	C₂×C₇₉	—	16	1	no	—
77	C₂×C₇₉	#2226	1	1	yes	characteristic
78	#2254	—	79	1	no	—
79	#2178	—	79	1	no	—
80	#1426	—	79	1	no	—
81	#1426	—	79	1	no	—
82	#1422	—	79	1	no	—
83	#1418	—	79	1	no	—
84	#1418	D₇₉	1	1	yes	characteristic
85	C₇₉	#13710	1	1	yes	characteristic, complemented, Sylow
86	#508	C₂×D₇₉	1	1	yes	characteristic
87	#500	—	158	1	no	—
88	#443	—	158	1	no	—
89	M₆₄	—	79	1	no	—
90	M₆₄	C₂×D₇₉	1	1	yes	characteristic
91	#368	—	79	1	no	—
92	#367	—	79	1	no	—
93	#367	—	158	1	no	—
94	#367	C₂×D₇₉	1	1	yes	characteristic
95	#359	—	158	1	no	—
96	C₄×C₁₆	—	79	1	no	—
97	#135	—	2	1	no	—
98	#131	—	316	1	no	—
99	C₄⋊Q₈	—	316	1	no	—
100	Q₃₂	Dic₁₅₈	1	1	yes	characteristic, complemented
101	SD₃₂	—	2	1	no	—
102	D₁₆	Dic₁₅₈	1	1	yes	characteristic, complemented
103	C₂×C₁₆	—	79	1	no	—
104	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
105	#108	—	158	1	no	—
106	#107	—	2	1	no	—
107	#107	—	316	1	no	—
108	#107	—	158	1	no	—
109	#104	—	316	1	no	—
110	C₄×C₈	—	79	1	no	—
111	C₃₂	—	2	1	no	—
112	C₃₂	—	158	1	no	—
113	#41	—	4	1	no	—
114	C₂×Q₈	—	632	1	no	—
115	Q₁₆	—	2	1	no	—
116	SD₁₆	—	4	1	no	—
117	D₈	—	2	1	no	—
118	M₁₆	—	316	1	no	—
119	M₁₆	—	632	1	no	—
120	C₂×C₈	—	79	1	no	—
121	C₂×C₈	—	632	1	no	—
122	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
123	C₄⋊C₄	—	4	1	no	—
124	C₄⋊C₄	—	316	1	no	—
125	C₄⋊C₄	—	632	1	no	—
126	C₄²	—	79	1	no	—
127	C₁₆	—	158	1	no	—
128	C₁₆	C₂×Dic₁₅₈	1	1	yes	characteristic
129	C₁₆	#11929623	1	1	yes	characteristic
130	Q₈	—	4	1	no	—
131	Q₈	—	1264	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	—	632	1	no	—
137	C₂×C₄	—	158	1	no	—
138	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
139	C₈	—	79	1	no	—
140	C₈	—	79	1	no	—
141	C₈	—	632	1	no	—
142	C₈	#14579138	1	1	yes	characteristic
143	C₈	#14579190	1	1	yes	characteristic
144	C₂²	—	8	1	no	—
145	C₂²	#19098997	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	16	1	no	—
148	C₄	—	158	2	no	—
149	C₄	—	1264	1	no	—
150	C₄	#19098841	1	1	yes	characteristic
151	C₄	SD₃₂×D₇₉	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	16	1	no	—
154	C₂	#23765545	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23765545
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×#1418	C₂
socle	C₂×C₇₉	#2226
prime radical	#508×C₇₉	C₂²
hypercenter	#1418	D₇₉
hypocenter	C₇₉	#13710
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

Complements

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

N	H ≅ G/N	type
1. #23765681	C₂	semidirect
3. #23761606	C₂	semidirect
4. #23761605	C₂	semidirect
5. #23761597	C₂	semidirect
29. Q₃₂×C₇₉	Q₈	semidirect
31. D₁₆×C₇₉	Q₈	semidirect
85. C₇₉	#13710	semidirect
100. Q₃₂	Dic₁₅₈	semidirect
102. D₁₆	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23765681	2
#23765281	2
#23761606	2
#23761605	2
#23761597	2
#23761587	2
C₇₉×#1418	2
#13710	79