Finite Group Databasebeta

Subgroups — #24906998

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	#19128049	C₂	1	1	yes	maximal, characteristic, complemented
2	#19127896	C₂	1	1	yes	maximal, characteristic
3	#19127779	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127453	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127450	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127444	C₂	1	1	yes	maximal, characteristic, complemented
7	#476×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13050575	—	2	1	no	—
9	C₂×#13050574	—	2	1	no	—
10	#14581789	C₂²	1	1	yes	characteristic
11	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
13	#14581712	—	2	1	no	—
14	#14581712	—	2	1	no	—
15	#14581712	—	2	1	no	—
16	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
17	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
20	#106×C₈₃	—	2	1	no	—
21	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
22	#102×C₈₃	—	2	1	no	—
23	C₂×D₃₃₂	—	2	1	no	—
24	C₂×D₃₃₂	—	2	1	no	—
25	C₂×D₃₃₂	—	2	1	no	—
26	#13050575	—	4	2	no	—
27	#13050574	—	4	2	no	—
28	C₂×C₈₃⋊C₈	—	2	1	no	—
29	C₂×C₈₃⋊C₈	—	2	1	no	—
30	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
31	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
32	C₂×Q₈×C₈₃	—	2	1	no	—
33	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₃	—	2	1	no	—
35	C₂×C₈×C₈₃	—	2	1	no	—
36	C₄⋊C₄×C₈₃	—	2	1	no	—
37	C₄⋊C₄×C₈₃	—	2	1	no	—
38	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₈₃	—	2	1	no	—
40	C₄²×C₈₃	C₂³	1	1	yes	characteristic
41	C₂³×C₈₃	—	2	1	no	—
42	C₂²×D₈₃	—	4	1	no	—
43	Q₈×C₈₃	—	8	1	no	—
44	D₄×C₈₃	—	2	1	no	—
45	D₄×C₈₃	—	2	1	no	—
46	C₂×C₄×C₈₃	—	2	1	no	—
47	C₂×C₄×C₈₃	—	4	1	no	—
48	C₂×C₄×C₈₃	—	4	1	no	—
49	C₂×C₄×C₈₃	—	4	1	no	—
50	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
53	D₃₃₂	—	4	1	no	—
54	D₃₃₂	—	4	1	no	—
55	D₃₃₂	—	4	1	no	—
56	D₃₃₂	—	4	1	no	—
57	D₃₃₂	—	8	1	no	—
58	C₈×C₈₃	—	4	1	no	—
59	C₈₃⋊C₈	—	2	1	no	—
60	C₈₃⋊C₈	—	2	1	no	—
61	C₈₃⋊C₈	—	4	1	no	—
62	C₈₃⋊C₈	—	4	1	no	—
63	C₂²×C₈₃	—	4	1	no	—
64	C₂²×C₈₃	—	4	1	no	—
65	C₂²×C₈₃	—	4	1	no	—
66	C₂²×C₈₃	#121	1	1	yes	characteristic
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	—	4	1	no	—
72	C₄×C₈₃	—	8	1	no	—
73	C₄×C₈₃	#135	1	1	yes	characteristic
74	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₈₃	#137	1	1	yes	characteristic
77	C₂×C₈₃	—	8	1	no	—
78	C₂×C₈₃	#463	1	1	yes	characteristic
79	C₂×C₈₃	#468	1	1	yes	characteristic
80	C₂×C₈₃	#472	1	1	yes	characteristic
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	D₈₃	1	1	yes	characteristic, complemented
86	#459	—	83	1	no	—
87	#436	—	83	1	no	—
88	#335	—	83	1	no	—
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	C₂×D₈₃	1	1	yes	characteristic
94	#127	—	83	1	no	—
95	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
96	#106	—	2	1	no	—
97	C₄⋊C₈	—	83	1	no	—
98	C₄⋊C₈	—	83	1	no	—
99	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
100	#102	—	2	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	—	83	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×D₄	—	166	1	no	—
108	C₂×D₄	—	166	1	no	—
109	C₂×D₄	—	166	1	no	—
110	C₂×D₄	C₂²⋊D₈₃	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	D₈	—	332	2	no	—
113	M₁₆	—	332	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	83	1	no	—
116	C₂×C₈	—	83	1	no	—
117	C₂×C₈	—	166	1	no	—
118	C₂×C₈	—	166	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	C₂²×D₈₃	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	—	332	1	no	—
126	Q₈	—	8	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	2	1	no	—
129	D₄	—	332	1	no	—
130	D₄	—	332	1	no	—
131	D₄	—	332	1	no	—
132	D₄	—	332	1	no	—
133	D₄	—	664	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
140	C₂×C₄	#13050589	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	332	1	no	—
143	C₈	—	332	1	no	—
144	C₈	—	166	1	no	—
145	C₈	—	166	1	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	664	1	no	—
150	C₂²	—	664	1	no	—
151	C₂²	—	664	1	no	—
152	C₂²	#14581764	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	8	1	no	—
156	C₄	#14581813	1	1	yes	characteristic
157	C₄	#14581817	1	1	yes	characteristic
158	C₄	#14581819	1	1	yes	characteristic
159	C₄	#14581823	1	1	yes	characteristic
160	C₂	—	8	1	no	—
161	C₂	—	1328	1	no	—
162	C₂	#19127918	1	1	yes	characteristic
163	C₂	#19127949	1	1	yes	characteristic
164	C₂	#19127969	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14581764
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#476×C₈₃	C₂
socle	C₂²×C₈₃	#121
prime radical	#19127896	C₂
hypercenter	#476	D₈₃
hypocenter	C₈₃	#1697
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #19128049	C₂	semidirect
3. #19127779	C₂	semidirect
4. #19127453	C₂	semidirect
5. #19127450	C₂	semidirect
6. #19127444	C₂	semidirect
7. #476×C₈₃	C₂	semidirect
83. C₈₃	#1697	semidirect
85. #476	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128049	2
#19127896	2
#19127779	2
#19127453	2
#19127450	2
#19127444	2
#476×C₈₃	2
#1697	83