Finite Group Databasebeta

Subgroups — #24908927

Order 10624.

184 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 61 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	#19127750	C₂	1	1	yes	maximal, characteristic
2	#19127710	C₂	1	1	yes	maximal, characteristic
3	C₂×#14581711	C₂	1	1	yes	maximal, characteristic
4	#19127472	C₂	1	1	yes	maximal, characteristic
5	#19127471	C₂	1	1	yes	maximal, characteristic
6	#19127457	C₂	1	1	yes	maximal, characteristic
7	#419×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050574	C₂²	1	1	yes	characteristic
9	C₂×#13050574	C₂²	1	1	yes	characteristic
10	C₂×#13050574	C₂²	1	1	yes	characteristic
11	#14581751	C₂²	1	1	yes	characteristic
12	#14581728	—	2	2	no	—
13	#14581716	—	2	2	no	—
14	#14581711	—	2	2	no	—
15	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
17	#117×C₈₃	C₂²	1	1	yes	characteristic
18	#103×C₈₃	—	2	2	no	—
19	#13050574	—	2	1	no	—
20	#13050574	—	2	1	no	—
21	#13050574	—	2	1	no	—
22	#13050574	—	2	1	no	—
23	#13050574	—	2	1	no	—
24	#13050574	—	2	1	no	—
25	C₂×C₈₃⋊C₈	—	2	1	no	—
26	C₂×C₈₃⋊C₈	—	2	1	no	—
27	C₂×C₈₃⋊C₈	—	2	1	no	—
28	C₄⋊Dic₈₃	—	2	1	no	—
29	C₂²⋊Dic₈₃	—	2	1	no	—
30	C₄×Dic₈₃	—	2	1	no	—
31	C₂×Q₈×C₈₃	—	2	1	no	—
32	C₂×Q₈×C₈₃	—	2	1	no	—
33	C₂×Q₈×C₈₃	—	2	1	no	—
34	C₂×Q₈×C₈₃	—	2	1	no	—
35	C₂×Q₈×C₈₃	—	2	1	no	—
36	C₂×Q₈×C₈₃	D₄	1	2	yes	—
37	C₂²×C₄×C₈₃	—	2	1	no	—
38	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
39	M₁₆×C₈₃	—	2	1	no	—
40	M₁₆×C₈₃	—	2	1	no	—
41	C₂×C₈×C₈₃	D₄	1	2	yes	—
42	C₄⋊C₄×C₈₃	—	2	1	no	—
43	C₂²⋊C₄×C₈₃	—	2	1	no	—
44	C₄²×C₈₃	—	2	1	no	—
45	C₂³×C₈₃	#41	1	1	yes	characteristic
46	Q₈×C₈₃	—	2	2	no	—
47	Q₈×C₈₃	—	2	2	no	—
48	Q₈×C₈₃	—	4	1	no	—
49	Q₈×C₈₃	—	4	1	no	—
50	C₂×C₄×C₈₃	—	2	2	no	—
51	C₂×C₄×C₈₃	—	4	1	no	—
52	C₂×C₄×C₈₃	—	4	1	no	—
53	C₂×C₄×C₈₃	—	4	1	no	—
54	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
57	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
58	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
59	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
60	C₂×Dic₈₃	—	4	1	no	—
61	C₈×C₈₃	—	2	2	no	—
62	C₈₃⋊C₈	—	4	1	no	—
63	C₈₃⋊C₈	—	4	1	no	—
64	C₈₃⋊C₈	—	4	1	no	—
65	C₂²×C₈₃	—	4	1	no	—
66	C₂²×C₈₃	#123	1	1	yes	characteristic
67	C₂²×C₈₃	#124	1	1	yes	characteristic
68	C₂²×C₈₃	#126	1	1	yes	characteristic
69	C₄×C₈₃	—	4	2	no	—
70	C₄×C₈₃	—	8	1	no	—
71	C₄×C₈₃	#121	1	1	yes	characteristic
72	C₄×C₈₃	#123	1	1	yes	characteristic
73	C₄×C₈₃	#123	1	1	yes	characteristic
74	C₄×C₈₃	#126	1	1	yes	characteristic
75	Dic₈₃	—	8	1	no	—
76	C₂×C₈₃	—	2	1	no	—
77	C₂×C₈₃	—	2	1	no	—
78	C₂×C₈₃	—	2	1	no	—
79	C₂×C₈₃	#397	1	1	yes	characteristic
80	#2082	—	83	1	no	maximal, Sylow
81	C₈₃	#2082	1	1	yes	characteristic, complemented, Sylow
82	#430	—	83	1	no	—
83	#419	—	83	1	no	—
84	#419	D₈₃	1	1	yes	characteristic
85	C₂×#101	—	83	1	no	—
86	#344	—	83	1	no	—
87	#344	—	83	1	no	—
88	H(3,ℤ/4)	—	83	1	no	—
89	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
90	C₂×M₁₆	—	83	1	no	—
91	C₂×M₁₆	—	83	1	no	—
92	C₂×M₁₆	—	83	1	no	—
93	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
94	#117	—	83	1	no	—
95	#117	C₂×D₈₃	1	1	yes	characteristic
96	#108	—	166	2	no	—
97	#103	—	2	2	no	—
98	#103	—	166	2	no	—
99	#101	—	166	2	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	D₃₃₂	1	2	yes	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	—	2	1	no	—
110	M₁₆	—	166	1	no	—
111	M₁₆	—	166	1	no	—
112	M₁₆	—	166	1	no	—
113	M₁₆	—	166	1	no	—
114	M₁₆	—	166	1	no	—
115	M₁₆	—	166	1	no	—
116	C₂×C₈	—	166	1	no	—
117	C₂×C₈	—	166	1	no	—
118	C₂×C₈	—	166	1	no	—
119	C₂×C₈	C₂²⋊D₈₃	1	2	yes	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	166	1	no	—
122	C₂²⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	166	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	166	1	no	—
126	C₂³	#13050589	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	332	1	no	—
136	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
137	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
138	C₂×C₄	#13050589	1	1	yes	characteristic
139	C₂×C₄	#13050589	1	1	yes	characteristic
140	C₂×C₄	#13050589	1	1	yes	characteristic
141	C₂×C₄	#13050590	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	332	1	no	—
144	C₈	—	332	1	no	—
145	C₈	—	332	1	no	—
146	C₂²	—	4	1	no	—
147	C₂²	#14581776	1	1	yes	characteristic
148	C₂²	#14581779	1	1	yes	characteristic
149	C₂²	#14581785	1	1	yes	characteristic
150	C₄	—	4	2	no	—
151	C₄	—	8	1	no	—
152	C₄	—	664	1	no	—
153	C₄	#14581766	1	1	yes	characteristic
154	C₄	#14581773	1	1	yes	characteristic
155	C₄	#14581776	1	1	yes	characteristic
156	C₄	#14581785	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#19127626	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19127626
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#419×C₈₃	C₂
socle	C₂×C₈₃	#397
prime radical	C₂³×C₈₃	#41
hypercenter	#419	D₈₃
hypocenter	C₈₃	#2082
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #419

Complements

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

N	H ≅ G/N	type
81. C₈₃	#2082	semidirect

Maximal subgroups

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

subgroup	index
#19127750	2
#19127710	2
C₂×#14581711	2
#19127472	2
#19127471	2
#19127457	2
#419×C₈₃	2
#2082	83