Finite Group Databasebeta

Subgroups — #24909438

Order 10624.

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

normal subgroups only

This group has 206 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	#19128515	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581738	C₂	1	1	yes	maximal, characteristic, complemented
3	#19127973	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127518	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127508	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127486	C₂	1	1	yes	maximal, characteristic, complemented
7	#510×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14581839	—	2	1	no	—
9	#14581823	—	2	1	no	—
10	#14581822	—	2	1	no	—
11	#14581821	—	2	2	no	—
12	#14581811	—	2	1	no	—
13	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
14	Q₁₆×D₈₃	—	2	1	no	—
15	#14581798	C₂²	1	1	yes	characteristic
16	#14581738	—	2	2	no	—
17	C₂×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	#14581728	C₂²	1	1	yes	characteristic
19	#14581711	—	2	1	no	—
20	#135×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
22	Q₃₂×C₈₃	—	2	1	no	—
23	SD₃₂×C₈₃	—	2	1	no	—
24	M₃₂×C₈₃	C₂²	1	1	yes	characteristic
25	#13050590	—	4	1	no	—
26	#13050590	D₄	1	1	yes	characteristic
27	#13050589	—	2	2	no	—
28	#13050589	—	4	2	no	—
29	C₂×Dic₁₆₆	—	2	1	no	—
30	C₂×Dic₁₆₆	—	4	1	no	—
31	Q₈×D₈₃	—	2	2	no	—
32	Dic₃₃₂	—	2	1	no	—
33	Dic₃₃₂	D₄	1	2	yes	—
34	#13050580	—	2	2	no	—
35	#13050580	—	4	1	no	—
36	#13050578	—	2	2	no	—
37	#13050577	—	4	1	no	—
38	#13050574	—	2	1	no	—
39	#13050574	D₄	1	1	yes	characteristic
40	#13050573	—	2	1	no	—
41	C₈×D₈₃	—	2	1	no	—
42	C₈₃⋊C₁₆	—	2	1	no	—
43	#41×C₈₃	—	2	1	no	—
44	C₂×Q₈×C₈₃	—	2	1	no	—
45	Q₁₆×C₈₃	—	2	1	no	—
46	Q₁₆×C₈₃	—	2	1	no	—
47	Q₁₆×C₈₃	D₄	1	1	yes	characteristic
48	SD₁₆×C₈₃	—	2	1	no	—
49	D₈×C₈₃	D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
51	C₁₆×C₈₃	—	2	1	no	—
52	Q₈×C₈₃	—	2	1	no	—
53	Q₈×C₈₃	—	4	1	no	—
54	Q₈×C₈₃	—	4	1	no	—
55	D₄×C₈₃	—	2	1	no	—
56	D₄×C₈₃	—	4	1	no	—
57	C₂×C₄×C₈₃	—	4	1	no	—
58	C₂×C₄×C₈₃	—	4	1	no	—
59	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₈₃	—	2	1	no	—
61	C₂²⋊D₈₃	—	4	2	no	—
62	C₂×Dic₈₃	—	4	2	no	—
63	C₂×Dic₈₃	—	4	1	no	—
64	D₃₃₂	D₈	1	1	yes	characteristic
65	C₄×D₈₃	—	2	1	no	—
66	C₄×D₈₃	—	4	2	no	—
67	Dic₁₆₆	—	2	2	no	—
68	Dic₁₆₆	—	4	2	no	—
69	Dic₁₆₆	—	4	1	no	—
70	Dic₁₆₆	D₈	1	1	yes	characteristic
71	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
72	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
73	C₈₃⋊C₈	—	2	1	no	—
74	C₈₃⋊C₈	—	4	1	no	—
75	C₂²×C₈₃	—	4	1	no	—
76	C₂²×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₈₃	—	2	1	no	—
78	C₄×C₈₃	—	4	1	no	—
79	C₄×C₈₃	—	8	1	no	—
80	C₄×C₈₃	#120	1	1	yes	characteristic
81	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
82	Dic₈₃	—	2	1	no	—
83	Dic₈₃	—	4	2	no	—
84	C₂×C₈₃	—	2	1	no	—
85	C₂×C₈₃	—	8	1	no	—
86	C₂×C₈₃	#447	1	1	yes	characteristic
87	D₈₃	—	4	1	no	—
88	#2195	—	83	1	no	maximal, Sylow
89	C₈₃	#2195	1	1	yes	characteristic, complemented, Sylow
90	#578	—	83	1	no	—
91	#510	D₈₃	1	1	yes	characteristic, complemented
92	C₂×Q₃₂	—	83	1	no	—
93	#473	—	83	1	no	—
94	#362	—	83	1	no	—
95	#359	—	83	1	no	—
96	#350	—	83	1	no	—
97	2₋¹⁺⁴	—	166	1	no	—
98	#137	—	166	1	no	—
99	#137	—	166	2	no	—
100	#137	—	166	1	no	—
101	#135	—	166	1	no	—
102	#135	C₂×D₈₃	1	1	yes	characteristic
103	C₂×Q₁₆	—	83	1	no	—
104	C₂×Q₁₆	—	166	1	no	—
105	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
106	#131	—	83	1	no	—
107	Q₃₂	—	2	1	no	—
108	Q₃₂	—	166	2	no	—
109	SD₃₂	—	2	1	no	—
110	M₃₂	C₂×D₈₃	1	1	yes	characteristic
111	C₂×C₁₆	—	83	1	no	—
112	#108	—	83	1	no	—
113	#101	—	166	1	no	—
114	#41	—	2	1	no	—
115	#41	—	332	1	no	—
116	#41	—	332	2	no	—
117	#41	—	83	1	no	—
118	#41	—	166	2	no	—
119	C₂×Q₈	—	2	1	no	—
120	C₂×Q₈	—	332	1	no	—
121	C₂×Q₈	—	166	2	no	—
122	C₂×Q₈	—	166	1	no	—
123	Q₁₆	—	2	1	no	—
124	Q₁₆	—	2	1	no	—
125	Q₁₆	—	332	1	no	—
126	Q₁₆	—	83	2	no	—
127	Q₁₆	—	166	2	no	—
128	Q₁₆	—	166	1	no	—
129	Q₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
130	SD₁₆	—	2	1	no	—
131	SD₁₆	—	332	1	no	—
132	SD₁₆	—	166	2	no	—
133	D₈	C₂²⋊D₈₃	1	1	yes	characteristic
134	M₁₆	—	83	1	no	—
135	M₁₆	—	166	1	no	—
136	M₁₆	—	166	1	no	—
137	C₂×C₈	—	166	1	no	—
138	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
139	C₁₆	—	2	1	no	—
140	C₁₆	—	166	1	no	—
141	Q₈	—	2	1	no	—
142	Q₈	—	4	1	no	—
143	Q₈	—	4	1	no	—
144	Q₈	—	332	2	no	—
145	Q₈	—	332	1	no	—
146	Q₈	—	83	1	no	—
147	Q₈	—	166	2	no	—
148	D₄	—	2	1	no	—
149	D₄	—	4	1	no	—
150	D₄	—	332	2	no	—
151	D₄	—	83	1	no	—
152	D₄	—	166	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	332	2	no	—
156	C₂×C₄	—	332	1	no	—
157	C₂×C₄	—	332	2	no	—
158	C₂×C₄	—	166	1	no	—
159	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
160	C₈	—	332	1	no	—
161	C₈	—	166	1	no	—
162	C₈	D₄×D₈₃	1	1	yes	characteristic
163	C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
164	C₂²	—	4	1	no	—
165	C₂²	—	166	1	no	—
166	C₂²	#14581816	1	1	yes	characteristic
167	C₄	—	4	1	no	—
168	C₄	—	8	1	no	—
169	C₄	—	332	2	no	—
170	C₄	—	166	1	no	—
171	C₄	#14581760	1	1	yes	characteristic
172	C₄	D₈×D₈₃	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	8	1	no	—
175	C₂	—	332	1	no	—
176	C₂	#19127833	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19127833
commutator, cosocle	C₂×C₈×C₈₃	C₂³
Frattini	C₂×C₈	C₂²×D₈₃
Fitting	#510×C₈₃	C₂
socle	C₂×C₈₃	#447
prime radical	#19128515	C₂
hypercenter	#510	D₈₃
hypocenter	C₈₃	#2195
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #510

Complements

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

N	H ≅ G/N	type
2. C₂×#14581738	C₂	semidirect
3. #19127973	C₂	semidirect
4. #19127518	C₂	semidirect
5. #19127508	C₂	semidirect
6. #19127486	C₂	semidirect
7. #510×C₈₃	C₂	semidirect
89. C₈₃	#2195	semidirect
91. #510	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128515	2
C₂×#14581738	2
#19127973	2
#19127518	2
#19127508	2
#19127486	2
#510×C₈₃	2
#2195	83