Finite Group Databasebeta

Subgroups — #34209164

Order 20224.

164 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 59 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	#23765674	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#23761600	C₂	1	1	yes	maximal, characteristic
4	#23761599	C₂	1	1	yes	maximal, characteristic
5	Dic₇₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	#23761581	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1416	C₂	1	1	yes	maximal, characteristic
8	#19098967	—	2	1	no	—
9	Q₁₆×Dic₇₉	—	2	1	no	—
10	C₂×C₇₉⋊C₃₂	C₂²	1	1	yes	characteristic
11	#19098389	C₂²	1	1	yes	characteristic
12	#19098389	C₂²	1	1	yes	characteristic
13	#19098388	—	2	1	no	—
14	#19098367	—	2	1	no	—
15	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
18	#366×C₇₉	C₂²	1	1	yes	characteristic
19	#14579160	—	4	1	no	—
20	Q₈×Dic₇₉	—	4	1	no	—
21	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
22	#14579095	—	2	1	no	—
23	#14579095	—	2	1	no	—
24	#14579094	—	4	1	no	—
25	#14579085	—	4	1	no	—
26	C₈×Dic₇₉	D₄	1	1	yes	characteristic
27	C₇₉⋊C₃₂	—	2	1	no	—
28	C₂×Q₁₆×C₇₉	—	2	1	no	—
29	Q₃₂×C₇₉	—	2	1	no	—
30	Q₃₂×C₇₉	Q₈	1	2	yes	complemented
31	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
32	#107×C₇₉	—	2	1	no	—
33	C₃₂×C₇₉	—	2	1	no	—
34	C₂×Dic₁₅₈	—	8	1	no	—
35	C₂×C₇₉⋊C₈	D₈	1	1	yes	characteristic
36	C₄⋊Dic₇₉	—	4	1	no	—
37	C₄⋊Dic₇₉	—	4	1	no	—
38	C₄⋊Dic₇₉	—	8	1	no	—
39	Dic₇₉⋊C₄	—	8	1	no	—
40	C₄×Dic₇₉	—	8	1	no	—
41	C₄×Dic₇₉	D₈	1	1	yes	characteristic
42	C₇₉⋊C₁₆	—	2	1	no	—
43	C₂×Q₈×C₇₉	—	4	1	no	—
44	Q₁₆×C₇₉	—	2	2	no	—
45	Q₁₆×C₇₉	—	4	1	no	—
46	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₇₉	—	4	1	no	—
48	C₁₆×C₇₉	C₂×Q₈	1	1	yes	characteristic
49	C₁₆×C₇₉	#41	1	1	yes	characteristic
50	Q₈×C₇₉	—	4	2	no	—
51	Q₈×C₇₉	—	8	1	no	—
52	C₂×C₄×C₇₉	—	8	1	no	—
53	C₂×C₄×C₇₉	—	8	1	no	—
54	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₇₉	—	8	1	no	—
56	C₂×Dic₇₉	—	8	1	no	—
57	C₂×Dic₇₉	D₁₆	1	1	yes	characteristic
58	C₂×Dic₇₉	D₁₆	1	1	yes	characteristic
59	Dic₁₅₈	—	16	2	no	—
60	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₈×C₇₉	#137	1	1	yes	characteristic
62	C₇₉⋊C₈	—	2	1	no	—
63	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₇₉	—	8	2	no	—
65	C₄×C₇₉	—	16	1	no	—
66	C₄×C₇₉	#474	1	1	yes	characteristic
67	C₄×C₇₉	#510	1	1	yes	characteristic
68	Dic₇₉	—	2	1	no	—
69	Dic₇₉	—	16	1	no	—
70	Dic₇₉	—	16	1	no	—
71	Dic₇₉	Q₆₄	1	2	yes	—
72	#13706	—	79	1	no	maximal, Sylow
73	C₂×C₇₉	#2224	1	1	yes	characteristic
74	C₂×C₇₉	C₂×Q₆₄	1	1	yes	characteristic
75	C₂×C₇₉	#2263	1	1	yes	characteristic
76	#2252	—	79	1	no	—
77	C₄×Q₃₂	—	79	1	no	—
78	#1423	—	79	1	no	—
79	#1423	—	79	1	no	—
80	C₄⋊C₃₂	—	79	1	no	—
81	#1416	—	79	1	no	—
82	#1416	D₇₉	1	1	yes	characteristic
83	C₇₉	#13706	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic
85	#500	—	158	1	no	—
86	C₄×Q₁₆	—	158	1	no	—
87	C₂×C₃₂	—	79	1	no	—
88	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
89	#366	—	79	1	no	—
90	#366	—	79	1	no	—
91	#366	—	158	1	no	—
92	#366	C₂×D₇₉	1	1	yes	characteristic
93	#358	—	158	1	no	—
94	C₄×C₁₆	—	79	1	no	—
95	C₂×Q₁₆	—	2	1	no	—
96	C₄⋊Q₈	—	316	1	no	—
97	C₄×Q₈	—	316	1	no	—
98	Q₃₂	—	2	1	no	—
99	Q₃₂	Dic₁₅₈	1	2	yes	complemented
100	C₂×C₁₆	—	79	1	no	—
101	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
102	#107	—	2	1	no	—
103	#107	—	316	1	no	—
104	#107	—	158	1	no	—
105	#107	—	158	1	no	—
106	#103	—	316	1	no	—
107	C₄×C₈	—	79	1	no	—
108	C₃₂	—	2	1	no	—
109	C₃₂	—	158	1	no	—
110	C₂×Q₈	—	4	1	no	—
111	C₂×Q₈	—	632	1	no	—
112	Q₁₆	—	2	2	no	—
113	Q₁₆	—	4	1	no	—
114	C₂×C₈	—	79	1	no	—
115	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	316	1	no	—
118	C₄⋊C₄	—	316	1	no	—
119	C₄⋊C₄	—	632	1	no	—
120	C₄⋊C₄	—	632	1	no	—
121	C₄²	—	79	1	no	—
122	C₄²	—	632	1	no	—
123	C₁₆	—	158	1	no	—
124	C₁₆	C₂×Dic₁₅₈	1	1	yes	characteristic
125	C₁₆	#11929623	1	1	yes	characteristic
126	Q₈	—	4	2	no	—
127	Q₈	—	8	1	no	—
128	Q₈	—	1264	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	79	1	no	—
132	C₂×C₄	—	79	1	no	—
133	C₂×C₄	—	632	1	no	—
134	C₂×C₄	—	632	1	no	—
135	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
136	C₈	—	158	1	no	—
137	C₈	#14579138	1	1	yes	characteristic
138	C₈	#14579190	1	1	yes	characteristic
139	C₂²	D₁₆×D₇₉	1	1	yes	characteristic
140	C₄	—	8	2	no	—
141	C₄	—	16	1	no	—
142	C₄	—	79	2	no	—
143	C₄	—	158	1	no	—
144	C₄	—	1264	1	no	—
145	C₄	—	1264	1	no	—
146	C₄	#19098841	1	1	yes	characteristic
147	C₄	#19099002	1	1	yes	characteristic
148	C₂	#23765535	1	1	yes	characteristic
149	C₂	Q₆₄×D₇₉	1	1	yes	characteristic
150	C₂	#23765714	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₉
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×#1416	C₂
socle, prime radical	C₂²×C₇₉	C₂×D₁₆
hypercenter	#1416	D₇₉
hypocenter	C₇₉	#13706
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

Complements

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

N	H ≅ G/N	type
30. Q₃₂×C₇₉	Q₈	semidirect
83. C₇₉	#13706	semidirect
99. Q₃₂	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23765674	2
Q₃₂×Dic₇₉	2
#23761600	2
#23761599	2
Dic₇₉⋊C₃₂	2
#23761581	2
C₇₉×#1416	2
#13706	79