Finite Group Databasebeta

Subgroups — #37508583

Order 22912.

140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 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	#26038790	C₂	1	1	yes	maximal, characteristic
2	#26038735	C₂	1	1	yes	maximal, characteristic
3	#26038725	C₂	1	1	yes	maximal, characteristic
4	#26038569	C₂	1	1	yes	maximal, characteristic
5	#26038187	C₂	1	1	yes	maximal, characteristic
6	#26038184	C₂	1	1	yes	maximal, characteristic
7	#334×C₁₇₉	C₂	1	1	yes	maximal, characteristic
8	#19153113	C₂²	1	1	yes	characteristic
9	#19153113	C₂²	1	1	yes	characteristic
10	C₄×Dic₃₅₈	C₂²	1	1	yes	characteristic
11	#19153047	—	2	1	no	—
12	#19153045	—	2	1	no	—
13	C₄⋊(C₁₇₉⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₁₇₉⋊C₈	—	2	1	no	—
15	#19153038	—	2	1	no	—
16	#19153038	—	2	1	no	—
17	#19153038	—	2	1	no	—
18	#19153038	—	2	1	no	—
19	#19153023	—	2	1	no	—
20	C₄⋊C₈×C₁₇₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₇₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₅₈	—	2	1	no	—
24	C₂×Dic₃₅₈	—	2	1	no	—
25	C₂×Dic₃₅₈	D₄	1	1	yes	characteristic
26	C₂×C₁₇₉⋊C₈	—	2	1	no	—
27	C₄⋊Dic₁₇₉	—	2	1	no	—
28	C₄⋊Dic₁₇₉	—	2	1	no	—
29	C₄⋊Dic₁₇₉	—	2	1	no	—
30	C₄⋊Dic₁₇₉	—	2	1	no	—
31	C₄⋊Dic₁₇₉	D₄	1	1	yes	characteristic
32	Dic₁₇₉⋊C₄	—	2	1	no	—
33	C₄×Dic₁₇₉	—	2	1	no	—
34	C₂×C₈×C₁₇₉	—	2	1	no	—
35	C₂×C₈×C₁₇₉	—	2	1	no	—
36	C₂×C₈×C₁₇₉	D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₇₉	D₄	1	1	yes	characteristic
38	C₄²×C₁₇₉	C₂³	1	1	yes	characteristic
39	Dic₃₅₈	—	2	1	no	—
40	Dic₃₅₈	—	2	1	no	—
41	Dic₃₅₈	—	4	1	no	—
42	Dic₃₅₈	—	4	1	no	—
43	Dic₃₅₈	—	8	1	no	—
44	C₂×Dic₁₇₉	—	2	1	no	—
45	C₂×Dic₁₇₉	—	2	1	no	—
46	C₂×Dic₁₇₉	—	4	1	no	—
47	C₂×Dic₁₇₉	—	4	1	no	—
48	C₁₇₉⋊C₈	—	4	1	no	—
49	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
52	C₈×C₁₇₉	—	2	1	no	—
53	C₈×C₁₇₉	—	2	1	no	—
54	C₈×C₁₇₉	—	4	1	no	—
55	C₈×C₁₇₉	—	4	1	no	—
56	C₂²×C₁₇₉	#121	1	1	yes	characteristic
57	C₄×C₁₇₉	—	2	1	no	—
58	C₄×C₁₇₉	#135	1	1	yes	characteristic
59	C₄×C₁₇₉	#137	1	1	yes	characteristic
60	C₄×C₁₇₉	#137	1	1	yes	characteristic
61	C₄×C₁₇₉	#137	1	1	yes	characteristic
62	Dic₁₇₉	—	4	1	no	—
63	Dic₁₇₉	—	4	1	no	—
64	Dic₁₇₉	—	8	1	no	—
65	Dic₁₇₉	—	8	1	no	—
66	C₂×C₁₇₉	#464	1	1	yes	characteristic
67	C₂×C₁₇₉	#470	1	1	yes	characteristic
68	C₂×C₁₇₉	#471	1	1	yes	characteristic
69	C₁₇₉	#1704	1	1	yes	characteristic, complemented, Sylow
70	#1704	—	179	1	no	maximal, Sylow
71	#487	—	179	1	no	—
72	#477	—	179	1	no	—
73	#475	—	179	1	no	—
74	#446	—	179	1	no	—
75	#334	D₁₇₉	1	1	yes	characteristic
76	#333	—	179	1	no	—
77	#332	—	179	1	no	—
78	C₄⋊Q₈	—	179	1	no	—
79	C₄⋊Q₈	—	179	1	no	—
80	C₄×Q₈	—	179	1	no	—
81	#107	—	358	1	no	—
82	#106	—	358	1	no	—
83	C₄⋊C₈	—	358	1	no	—
84	C₄⋊C₈	—	179	1	no	—
85	C₄⋊C₈	C₂×D₁₇₉	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₁₇₉	1	1	yes	characteristic
87	#103	—	358	1	no	—
88	#103	—	358	1	no	—
89	#103	—	358	1	no	—
90	#103	—	358	1	no	—
91	#97	—	358	1	no	—
92	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
93	C₂×Q₈	—	358	1	no	—
94	C₂×Q₈	—	358	1	no	—
95	C₂×Q₈	—	179	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	358	1	no	—
99	C₂×C₈	D₇₁₆	1	1	yes	characteristic
100	C₂×C₈	D₇₁₆	1	1	yes	characteristic
101	C₄⋊C₄	—	358	1	no	—
102	C₄⋊C₄	—	358	1	no	—
103	C₄⋊C₄	—	358	1	no	—
104	C₄⋊C₄	—	358	1	no	—
105	C₄⋊C₄	—	358	1	no	—
106	C₄⋊C₄	—	179	1	no	—
107	C₄²	—	358	1	no	—
108	C₄²	C₂²×D₁₇₉	1	1	yes	characteristic
109	Q₈	—	358	1	no	—
110	Q₈	—	358	1	no	—
111	Q₈	—	1432	1	no	—
112	Q₈	—	716	1	no	—
113	Q₈	—	716	1	no	—
114	C₂×C₄	—	358	1	no	—
115	C₂×C₄	—	358	1	no	—
116	C₂×C₄	—	716	1	no	—
117	C₂×C₄	—	716	1	no	—
118	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
119	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
120	C₂×C₄	#15668260	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	716	1	no	—
126	C₂²	#19153088	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	1432	1	no	—
129	C₄	—	1432	1	no	—
130	C₄	—	716	1	no	—
131	C₄	—	716	1	no	—
132	C₄	#19153135	1	1	yes	characteristic
133	C₄	#19153142	1	1	yes	characteristic
134	C₄	#19153142	1	1	yes	characteristic
135	C₄	#19153146	1	1	yes	characteristic
136	C₂	#26038662	1	1	yes	characteristic
137	C₂	#26038698	1	1	yes	characteristic
138	C₂	#26038703	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19153088
commutator, cosocle	C₄²×C₁₇₉	C₂³
Frattini	C₄²	C₂²×D₁₇₉
Fitting	#334×C₁₇₉	C₂
socle, prime radical	C₂²×C₁₇₉	#121
hypercenter	#334	D₁₇₉
hypocenter	C₁₇₉	#1704
LCS	G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

Complements

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

N	H ≅ G/N	type
69. C₁₇₉	#1704	semidirect

Maximal subgroups

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

subgroup	index
#26038790	2
#26038735	2
#26038725	2
#26038569	2
#26038187	2
#26038184	2
#334×C₁₇₉	2
#1704	179