Finite Group Databasebeta

Subgroups — #24906997

Order 10624.

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

normal subgroups only

This group has 170 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	#19128048	C₂	1	1	yes	maximal, characteristic, complemented
2	#19127992	C₂	1	1	yes	maximal, characteristic, complemented
3	#19127897	C₂	1	1	yes	maximal, characteristic
4	#19127780	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127449	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127444	C₂	1	1	yes	maximal, characteristic, complemented
7	#335×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₆₆₄	—	2	1	no	—
9	C₂×#13050573	—	2	1	no	—
10	#14581792	C₂²	1	1	yes	characteristic
11	#14581789	C₂²	1	1	yes	characteristic
12	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
13	#14581724	—	2	1	no	—
14	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
15	#14581714	—	2	1	no	—
16	#14581714	—	2	1	no	—
17	#14581714	—	2	1	no	—
18	#14581714	—	2	1	no	—
19	(C₂×D₈₃)⋊C₈	—	2	1	no	—
20	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₆₆	—	2	1	no	—
24	C₂×D₃₃₂	—	2	1	no	—
25	C₂×D₃₃₂	—	2	1	no	—
26	C₂×D₃₃₂	—	2	1	no	—
27	C₂×D₃₃₂	D₄	1	1	yes	characteristic
28	C₂×C₄×D₈₃	—	2	1	no	—
29	D₆₆₄	—	4	2	no	—
30	#13050573	—	4	2	no	—
31	C₂×C₈₃⋊C₈	—	2	1	no	—
32	C₄⋊Dic₈₃	—	2	1	no	—
33	C₄⋊Dic₈₃	—	2	1	no	—
34	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
35	(C₂×D₈₃)⋊C₄	—	2	1	no	—
36	C₂×C₈×C₈₃	—	2	1	no	—
37	C₂×C₈×C₈₃	—	2	1	no	—
38	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
40	C₄²×C₈₃	C₂³	1	1	yes	characteristic
41	C₂²×D₈₃	—	2	1	no	—
42	C₂²×D₈₃	—	4	1	no	—
43	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
46	C₂×Dic₈₃	—	2	1	no	—
47	C₂×Dic₈₃	—	4	1	no	—
48	D₃₃₂	—	2	1	no	—
49	D₃₃₂	—	2	1	no	—
50	D₃₃₂	—	4	1	no	—
51	D₃₃₂	—	4	1	no	—
52	D₃₃₂	—	4	1	no	—
53	D₃₃₂	—	4	1	no	—
54	D₃₃₂	—	8	1	no	—
55	C₄×D₈₃	—	4	1	no	—
56	C₄×D₈₃	—	4	1	no	—
57	Dic₁₆₆	—	8	1	no	—
58	C₈×C₈₃	—	2	1	no	—
59	C₈×C₈₃	—	2	1	no	—
60	C₈×C₈₃	—	4	1	no	—
61	C₈×C₈₃	—	4	1	no	—
62	C₈₃⋊C₈	—	4	1	no	—
63	C₂²×C₈₃	#121	1	1	yes	characteristic
64	C₂×D₈₃	—	4	1	no	—
65	C₂×D₈₃	—	4	1	no	—
66	C₂×D₈₃	—	4	1	no	—
67	C₂×D₈₃	—	8	1	no	—
68	C₂×D₈₃	—	8	1	no	—
69	C₂×D₈₃	—	8	1	no	—
70	C₄×C₈₃	—	2	1	no	—
71	C₄×C₈₃	#135	1	1	yes	characteristic
72	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₈₃	#137	1	1	yes	characteristic
75	Dic₈₃	—	4	1	no	—
76	Dic₈₃	—	8	1	no	—
77	C₂×C₈₃	#463	1	1	yes	characteristic
78	C₂×C₈₃	#468	1	1	yes	characteristic
79	C₂×C₈₃	#472	1	1	yes	characteristic
80	D₈₃	—	8	1	no	—
81	D₈₃	—	16	1	no	—
82	#1697	—	83	1	no	maximal, Sylow
83	C₈₃	#1697	1	1	yes	characteristic, complemented, Sylow
84	#486	—	83	1	no	—
85	#476	—	83	1	no	—
86	#459	—	83	1	no	—
87	#436	—	83	1	no	—
88	#335	D₈₃	1	1	yes	characteristic, complemented
89	#333	—	83	1	no	—
90	#331	—	83	1	no	—
91	C₂×D₈	—	166	1	no	—
92	C₂×M₁₆	—	166	1	no	—
93	C₄⋊Q₈	—	83	1	no	—
94	#127	—	83	1	no	—
95	C₄×D₄	—	83	1	no	—
96	#106	—	166	1	no	—
97	C₄⋊C₈	—	83	1	no	—
98	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
99	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
100	#102	—	166	1	no	—
101	#102	—	166	1	no	—
102	#102	—	166	1	no	—
103	#102	—	166	1	no	—
104	C₂²⋊C₈	—	166	1	no	—
105	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
106	C₂×Q₈	—	166	1	no	—
107	C₂×D₄	—	83	1	no	—
108	C₂×D₄	—	166	1	no	—
109	C₂×D₄	—	166	1	no	—
110	C₂×D₄	—	166	1	no	—
111	C₂²×C₄	—	166	1	no	—
112	D₈	—	332	2	no	—
113	M₁₆	—	332	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	166	1	no	—
117	C₂×C₈	D₃₃₂	1	1	yes	characteristic
118	C₂×C₈	D₃₃₂	1	1	yes	characteristic
119	C₄⋊C₄	—	83	1	no	—
120	C₄⋊C₄	—	166	1	no	—
121	C₄⋊C₄	—	166	1	no	—
122	C₂²⋊C₄	—	166	1	no	—
123	C₄²	C₂²×D₈₃	1	1	yes	characteristic
124	C₂³	—	332	1	no	—
125	C₂³	—	166	1	no	—
126	Q₈	—	664	1	no	—
127	D₄	—	332	1	no	—
128	D₄	—	332	1	no	—
129	D₄	—	332	1	no	—
130	D₄	—	332	1	no	—
131	D₄	—	664	1	no	—
132	D₄	—	166	1	no	—
133	D₄	—	166	1	no	—
134	C₂×C₄	—	332	1	no	—
135	C₂×C₄	—	332	1	no	—
136	C₂×C₄	—	332	1	no	—
137	C₂×C₄	—	166	1	no	—
138	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
139	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
140	C₂×C₄	#13050591	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	4	1	no	—
144	C₈	—	4	1	no	—
145	C₈	—	332	1	no	—
146	C₂²	—	332	1	no	—
147	C₂²	—	332	1	no	—
148	C₂²	—	332	1	no	—
149	C₂²	—	664	1	no	—
150	C₂²	—	664	1	no	—
151	C₂²	—	664	1	no	—
152	C₂²	#14581767	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	332	1	no	—
155	C₄	—	664	1	no	—
156	C₄	#14581814	1	1	yes	characteristic
157	C₄	#14581816	1	1	yes	characteristic
158	C₄	#14581820	1	1	yes	characteristic
159	C₄	#14581824	1	1	yes	characteristic
160	C₂	—	1328	1	no	—
161	C₂	—	664	1	no	—
162	C₂	#19127917	1	1	yes	characteristic
163	C₂	#19127948	1	1	yes	characteristic
164	C₂	#19127972	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14581767
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#335×C₈₃	C₂
socle	C₂²×C₈₃	#121
prime radical	#19127897	C₂
hypercenter	#335	D₈₃
hypocenter	C₈₃	#1697
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
1. #19128048	C₂	semidirect
2. #19127992	C₂	semidirect
4. #19127780	C₂	semidirect
5. #19127449	C₂	semidirect
6. #19127444	C₂	semidirect
7. #335×C₈₃	C₂	semidirect
83. C₈₃	#1697	semidirect
88. #335	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128048	2
#19127992	2
#19127897	2
#19127780	2
#19127449	2
#19127444	2
#335×C₈₃	2
#1697	83