Finite Group Databasebeta

Subgroups — #24909430

Order 10624.

210 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 47 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	#19128514	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128139	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14581735	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127972	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127507	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127486	C₂	1	1	yes	maximal, characteristic, complemented
7	#362×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14581840	—	2	1	no	—
9	#14581824	—	2	2	no	—
10	#14581823	—	2	1	no	—
11	#14581820	—	2	1	no	—
12	#14581815	—	2	1	no	—
13	#14581814	C₂²	1	1	yes	characteristic
14	Q₁₆×D₈₃	—	2	1	no	—
15	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
16	#14581798	C₂²	1	1	yes	characteristic
17	#14581737	—	2	1	no	—
18	#14581735	—	2	2	no	—
19	D₁₃₂₈	—	2	1	no	—
20	C₂×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	#14581708	—	2	1	no	—
22	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
23	M₃₂×C₈₃	C₂²	1	1	yes	characteristic
24	#108×C₈₃	C₂²	1	1	yes	characteristic
25	#13050591	—	2	2	no	—
26	#13050591	—	4	1	no	—
27	#13050590	—	2	1	no	—
28	#13050590	—	4	2	no	—
29	#13050590	D₄	1	1	yes	characteristic
30	Q₈×D₈₃	—	2	2	no	—
31	Q₈×D₈₃	—	4	1	no	—
32	C₂×D₃₃₂	—	2	1	no	—
33	Dic₃₃₂	D₄	1	1	yes	characteristic
34	#13050580	—	2	2	no	—
35	#13050579	—	2	1	no	—
36	#13050579	—	4	1	no	—
37	#13050577	—	2	2	no	—
38	D₆₆₄	—	2	1	no	—
39	D₆₆₄	—	2	1	no	—
40	D₆₆₄	—	4	1	no	—
41	D₆₆₄	D₄	1	1	yes	characteristic
42	#13050574	D₄	1	1	yes	characteristic
43	#13050573	—	2	1	no	—
44	C₈×D₈₃	—	2	1	no	—
45	C₈₃⋊C₁₆	—	2	1	no	—
46	C₂×Q₈×C₈₃	—	2	1	no	—
47	Q₁₆×C₈₃	—	2	1	no	—
48	Q₁₆×C₈₃	D₄	1	2	yes	—
49	M₁₆×C₈₃	—	2	1	no	—
50	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
51	C₁₆×C₈₃	—	2	1	no	—
52	C₂²×D₈₃	—	4	1	no	—
53	Q₈×C₈₃	—	2	2	no	—
54	Q₈×C₈₃	—	4	1	no	—
55	C₂×C₄×C₈₃	—	4	1	no	—
56	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₂²⋊D₈₃	—	2	1	no	—
58	C₂²⋊D₈₃	—	4	1	no	—
59	D₃₃₂	—	2	1	no	—
60	D₃₃₂	—	4	2	no	—
61	D₃₃₂	—	4	1	no	—
62	D₃₃₂	—	4	1	no	—
63	D₃₃₂	D₈	1	1	yes	characteristic
64	C₄×D₈₃	—	2	1	no	—
65	C₄×D₈₃	—	4	2	no	—
66	C₄×D₈₃	—	4	2	no	—
67	C₄×D₈₃	—	4	1	no	—
68	Dic₁₆₆	—	2	1	no	—
69	Dic₁₆₆	—	4	2	no	—
70	Dic₁₆₆	D₈	1	1	yes	characteristic
71	C₈×C₈₃	—	4	1	no	—
72	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
73	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₈₃⋊C₈	—	2	1	no	—
75	C₂²×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×D₈₃	—	2	1	no	—
77	C₂×D₈₃	—	4	1	no	—
78	C₂×D₈₃	—	8	1	no	—
79	C₂×D₈₃	—	16	1	no	—
80	C₄×C₈₃	—	4	2	no	—
81	C₄×C₈₃	#120	1	1	yes	characteristic
82	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
83	Dic₈₃	—	2	1	no	—
84	Dic₈₃	—	4	1	no	—
85	C₂×C₈₃	—	2	1	no	—
86	C₂×C₈₃	#447	1	1	yes	characteristic
87	D₈₃	—	4	1	no	—
88	D₈₃	—	8	1	no	—
89	D₈₃	—	16	1	no	—
90	#2193	—	83	1	no	maximal, Sylow
91	C₈₃	#2193	1	1	yes	characteristic, complemented, Sylow
92	#578	—	83	1	no	—
93	#509	—	83	1	no	—
94	C₂×SD₃₂	—	83	1	no	—
95	#472	—	83	1	no	—
96	#362	D₈₃	1	1	yes	characteristic, complemented
97	#359	—	83	1	no	—
98	#350	—	83	1	no	—
99	2₋¹⁺⁴	—	166	1	no	—
100	#137	—	166	1	no	—
101	#137	—	166	2	no	—
102	AGL(1,ℤ/8)	—	166	1	no	—
103	#135	—	83	1	no	—
104	#135	—	166	1	no	—
105	C₂×Q₁₆	—	166	1	no	—
106	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
107	C₂×D₈	—	83	1	no	—
108	#131	—	83	1	no	—
109	SD₃₂	—	166	1	no	—
110	SD₃₂	—	166	2	no	—
111	D₁₆	—	166	1	no	—
112	M₃₂	C₂×D₈₃	1	1	yes	characteristic
113	C₂×C₁₆	—	83	1	no	—
114	#108	C₂×D₈₃	1	1	yes	characteristic
115	#100	—	166	1	no	—
116	#41	—	332	2	no	—
117	#41	—	332	1	no	—
118	#41	—	83	1	no	—
119	#41	—	166	2	no	—
120	#41	—	166	1	no	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×Q₈	—	332	1	no	—
123	C₂×Q₈	—	166	2	no	—
124	C₂×D₄	—	166	1	no	—
125	Q₁₆	—	2	1	no	—
126	Q₁₆	—	83	1	no	—
127	Q₁₆	—	166	2	no	—
128	Q₁₆	D₃₃₂	1	2	yes	—
129	SD₁₆	—	332	1	no	—
130	SD₁₆	—	166	2	no	—
131	SD₁₆	—	166	1	no	—
132	D₈	—	332	1	no	—
133	D₈	—	83	1	no	—
134	D₈	—	166	1	no	—
135	D₈	—	166	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	83	1	no	—
138	M₁₆	—	166	1	no	—
139	C₂×C₈	—	166	1	no	—
140	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
141	C₁₆	—	2	1	no	—
142	C₁₆	—	166	1	no	—
143	C₂³	—	332	1	no	—
144	Q₈	—	2	2	no	—
145	Q₈	—	4	1	no	—
146	Q₈	—	332	2	no	—
147	Q₈	—	83	1	no	—
148	Q₈	—	166	1	no	—
149	D₄	—	332	2	no	—
150	D₄	—	332	1	no	—
151	D₄	—	332	1	no	—
152	D₄	—	332	1	no	—
153	D₄	—	83	1	no	—
154	D₄	—	166	1	no	—
155	D₄	—	166	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	332	2	no	—
158	C₂×C₄	—	332	2	no	—
159	C₂×C₄	—	332	1	no	—
160	C₂×C₄	—	166	1	no	—
161	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
162	C₈	—	4	1	no	—
163	C₈	—	166	1	no	—
164	C₈	D₄×D₈₃	1	1	yes	characteristic
165	C₈	C₂×D₃₃₂	1	1	yes	characteristic
166	C₂²	—	1328	1	no	—
167	C₂²	—	332	1	no	—
168	C₂²	—	664	1	no	—
169	C₂²	—	166	1	no	—
170	C₂²	#14581819	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	332	1	no	—
173	C₄	—	166	1	no	—
174	C₄	#14581762	1	1	yes	characteristic
175	C₄	D₈×D₈₃	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	1328	1	no	—
178	C₂	—	332	1	no	—
179	C₂	—	664	1	no	—
180	C₂	#19127836	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19127836
commutator, cosocle	C₂×C₈×C₈₃	C₂³
Frattini	C₂×C₈	C₂²×D₈₃
Fitting	#362×C₈₃	C₂
socle	C₂×C₈₃	#447
hypercenter	#362	D₈₃
hypocenter	C₈₃	#2193
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #362

Complements

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

N	H ≅ G/N	type
1. #19128514	C₂	semidirect
2. #19128139	C₂	semidirect
3. C₂×#14581735	C₂	semidirect
4. #19127972	C₂	semidirect
5. #19127507	C₂	semidirect
6. #19127486	C₂	semidirect
7. #362×C₈₃	C₂	semidirect
91. C₈₃	#2193	semidirect
96. #362	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128514	2
#19128139	2
C₂×#14581735	2
#19127972	2
#19127507	2
#19127486	2
#362×C₈₃	2
#2193	83