Subgroups — C₂×#125×D₈₃ #24920713

Order 10624.

1192 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14581784	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14581783	C₂	1	2	yes	maximal, complemented
3	C₂×#14581782	C₂	1	4	yes	maximal, complemented
4	#125×D₈₃	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₄⋊C₄×D₈₃	C₂	1	4	yes	maximal
6	C₂×C₄⋊C₄×D₈₃	C₂	1	2	yes	maximal
7	C₂×C₄²×D₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#125×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₈₃	C₂²	1	2	yes	—
10	C₂²×C₄×D₈₃	C₂²	1	4	yes	—
11	C₂²×C₄×D₈₃	C₂²	1	1	yes	characteristic
12	#14581784	C₂²	1	8	yes	complemented
13	#14581783	C₂²	1	16	yes	complemented
14	#14581782	C₂²	1	32	yes	complemented
15	C₂×C₄⋊Dic₈₃	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
17	C₂×Dic₈₃⋊C₄	C₂²	1	4	yes	—
18	C₂×Dic₈₃⋊C₄	C₂²	1	4	yes	—
19	C₂×Dic₈₃⋊C₄	C₂²	1	4	yes	—
20	C₄⋊C₄×D₈₃	C₂²	1	32	yes	—
21	C₄⋊C₄×D₈₃	C₂²	1	16	yes	—
22	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
23	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
24	C₄²×D₈₃	C₂²	1	8	yes	—
25	#125×C₈₃	C₂²	1	8	yes	complemented
26	C₂×C₄⋊C₄×C₈₃	C₂²	1	4	yes	—
27	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
28	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
29	C₂³×D₈₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₈₃	C₂³	1	2	yes	—
31	C₂²×Dic₈₃	C₂³	1	4	yes	—
32	C₂²×Dic₈₃	C₂³	1	1	yes	characteristic
33	C₂×C₄×D₈₃	—	2	8	no	—
34	C₂×C₄×D₈₃	—	2	16	no	—
35	C₂×C₄×D₈₃	Q₈	1	8	yes	—
36	C₂×C₄×D₈₃	C₂³	1	8	yes	—
37	C₂×C₄×D₈₃	C₂³	1	16	yes	—
38	C₂×C₄×D₈₃	C₂³	1	4	yes	—
39	C₄⋊Dic₈₃	C₂³	1	16	yes	—
40	C₄⋊Dic₈₃	C₂³	1	8	yes	—
41	Dic₈₃⋊C₄	C₂³	1	16	yes	—
42	Dic₈₃⋊C₄	C₂³	1	16	yes	—
43	Dic₈₃⋊C₄	C₂³	1	16	yes	—
44	C₄×Dic₈₃	C₂³	1	8	yes	—
45	C₄×Dic₈₃	C₂³	1	4	yes	—
46	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
48	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₈₃	C₂³	1	16	yes	—
50	C₄⋊C₄×C₈₃	C₂³	1	8	yes	—
51	C₄²×C₈₃	C₂³	1	4	yes	—
52	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
53	C₂²×D₈₃	C₂×Q₈	1	4	yes	—
54	C₂²×D₈₃	#41	1	8	yes	—
55	C₂²×D₈₃	C₂⁴	1	2	yes	—
56	C₂×C₄×C₈₃	—	2	4	no	—
57	C₂×C₄×C₈₃	—	2	8	no	—
58	C₂×C₄×C₈₃	C₂×Q₈	1	4	yes	—
59	C₂×C₄×C₈₃	C₂⁴	1	4	yes	—
60	C₂×C₄×C₈₃	C₂⁴	1	8	yes	—
61	C₂×C₄×C₈₃	C₂⁴	1	2	yes	—
62	C₂×Dic₈₃	—	2	4	no	—
63	C₂×Dic₈₃	—	2	8	no	—
64	C₂×Dic₈₃	C₂×Q₈	1	4	yes	—
65	C₂×Dic₈₃	C₂⁴	1	4	yes	—
66	C₂×Dic₈₃	C₂⁴	1	8	yes	—
67	C₂×Dic₈₃	C₂⁴	1	2	yes	—
68	C₄×D₈₃	—	2	16	no	—
69	C₄×D₈₃	—	2	32	no	—
70	C₄×D₈₃	C₂×Q₈	1	16	yes	—
71	C₂²×C₈₃	C₂²×Q₈	1	2	yes	—
72	C₂²×C₈₃	C₂×#41	1	4	yes	—
73	C₂²×C₈₃	C₂⁵	1	1	yes	characteristic
74	C₂×D₈₃	#125	1	16	yes	direct product factor
75	C₂×D₈₃	C₂²×Q₈	1	4	yes	—
76	C₂×D₈₃	C₂×#41	1	8	yes	—
77	C₄×C₈₃	—	2	4	no	—
78	C₄×C₈₃	—	2	8	no	—
79	C₄×C₈₃	C₂²×Q₈	1	4	yes	—
80	Dic₈₃	—	2	4	no	—
81	Dic₈₃	—	2	8	no	—
82	Dic₈₃	C₂²×Q₈	1	4	yes	—
83	C₂×C₈₃	C₂×#125	1	4	yes	complemented
84	C₂×C₈₃	C₂³×Q₈	1	1	yes	characteristic
85	C₂×C₈₃	C₂²×#41	1	2	yes	—
86	D₈₃	C₂×#125	1	8	yes	direct product factor
87	C₂²×#125	—	83	1	no	maximal, Sylow
88	C₈₃	C₂²×#125	1	1	yes	characteristic, complemented, Sylow
89	C₂×#125	—	83	16	no	—
90	C₂×#125	—	83	4	no	—
91	C₂×#125	—	83	2	no	—
92	C₂×#125	—	83	1	no	—
93	C₂×#125	D₈₃	1	1	yes	characteristic, direct product factor
94	C₂²×C₄⋊C₄	—	83	2	no	—
95	C₂²×C₄⋊C₄	—	83	4	no	—
96	C₂²×C₄²	—	83	1	no	—
97	C₂³×C₄	—	83	1	no	—
98	C₂³×C₄	—	83	4	no	—
99	C₂³×C₄	—	83	2	no	—
100	#125	—	83	32	no	—
101	#125	—	83	16	no	—
102	#125	—	83	8	no	—
103	#125	C₂×D₈₃	1	8	yes	direct product factor
104	C₂×C₄⋊C₄	—	83	4	no	—
105	C₂×C₄⋊C₄	—	83	4	no	—
106	C₂×C₄⋊C₄	—	83	4	no	—
107	C₂×C₄⋊C₄	—	83	4	no	—
108	C₂×C₄⋊C₄	—	83	2	no	—
109	C₂×C₄⋊C₄	—	83	16	no	—
110	C₂×C₄⋊C₄	—	83	32	no	—
111	C₂×C₄⋊C₄	C₂×D₈₃	1	4	yes	—
112	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
113	C₂×C₄²	—	83	2	no	—
114	C₂×C₄²	—	83	1	no	—
115	C₂×C₄²	—	83	8	no	—
116	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
117	C₂⁴	—	83	1	no	—
118	C₂²×C₄	—	83	1	no	—
119	C₂²×C₄	—	83	8	no	—
120	C₂²×C₄	—	83	4	no	—
121	C₂²×C₄	—	83	2	no	—
122	C₂²×C₄	—	83	4	no	—
123	C₂²×C₄	—	83	8	no	—
124	C₂²×C₄	—	83	16	no	—
125	C₂²×C₄	—	166	16	no	—
126	C₂²×C₄	—	166	8	no	—
127	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
128	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
129	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
130	C₄⋊C₄	—	83	16	no	—
131	C₄⋊C₄	—	83	16	no	—
132	C₄⋊C₄	—	83	16	no	—
133	C₄⋊C₄	—	83	16	no	—
134	C₄⋊C₄	—	83	8	no	—
135	C₄⋊C₄	C₂²×D₈₃	1	16	yes	—
136	C₄⋊C₄	C₂²×D₈₃	1	8	yes	—
137	C₄²	—	83	8	no	—
138	C₄²	—	83	4	no	—
139	C₄²	C₂²×D₈₃	1	4	yes	—
140	C₂³	—	83	4	no	—
141	C₂³	—	83	8	no	—
142	C₂³	—	83	2	no	—
143	C₂³	C₂³×D₈₃	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	83	4	no	—
147	C₂×C₄	—	83	2	no	—
148	C₂×C₄	—	83	16	no	—
149	C₂×C₄	—	83	4	no	—
150	C₂×C₄	—	83	8	no	—
151	C₂×C₄	—	166	16	no	—
152	C₂×C₄	—	166	32	no	—
153	C₂×C₄	—	166	4	no	—
154	C₂×C₄	—	166	8	no	—
155	C₂×C₄	Q₈×D₈₃	1	4	yes	—
156	C₂×C₄	C₂³×D₈₃	1	4	yes	—
157	C₂×C₄	C₂³×D₈₃	1	8	yes	—
158	C₂×C₄	C₂³×D₈₃	1	2	yes	—
159	C₂²	—	83	16	no	—
160	C₂²	—	83	4	no	—
161	C₂²	—	83	8	no	—
162	C₂²	C₂×Q₈×D₈₃	1	2	yes	—
163	C₂²	#41×D₈₃	1	4	yes	—
164	C₂²	C₂⁴×D₈₃	1	1	yes	characteristic
165	C₄	—	2	4	no	—
166	C₄	—	2	8	no	—
167	C₄	—	83	4	no	—
168	C₄	—	166	4	no	—
169	C₄	—	166	8	no	—
170	C₄	C₂×Q₈×D₈₃	1	4	yes	—
171	C₂	—	83	8	no	—
172	C₂	#125×D₈₃	1	4	yes	direct product factor
173	C₂	C₂²×Q₈×D₈₃	1	1	yes	characteristic
174	C₂	C₂×#41×D₈₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₈₃
commutator, cosocle	C₂²×C₈₃	C₂⁵
Frattini	C₂²	C₂⁴×D₈₃
Fitting	C₂×#125×C₈₃	C₂
socle	C₂³×C₈₃	C₂⁴
prime radical	C₂³×D₈₃	C₂³
hypercenter	C₂×#125	D₈₃
hypocenter	C₈₃	C₂²×#125
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. C₂×#14581784	C₂	semidirect
2. C₂×#14581783	C₂	semidirect
3. C₂×#14581782	C₂	semidirect
4. #125×D₈₃	C₂	direct (N × H)
8. C₂×#125×C₈₃	C₂	semidirect
12. #14581784	C₂²	semidirect
13. #14581783	C₂²	semidirect
14. #14581782	C₂²	semidirect
25. #125×C₈₃	C₂²	semidirect
74. C₂×D₈₃	#125	direct (N × H)
83. C₂×C₈₃	C₂×#125	semidirect
86. D₈₃	C₂×#125	direct (N × H)
88. C₈₃	C₂²×#125	semidirect
93. C₂×#125	D₈₃	direct (N × H)
103. #125	C₂×D₈₃	direct (N × H)
172. C₂	#125×D₈₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#14581784	2
C₂×#14581783	2
C₂×#14581782	2
#125×D₈₃	2
C₂×C₄⋊C₄×D₈₃	2
C₂×C₄²×D₈₃	2
C₂×#125×C₈₃	2
C₂²×#125	83

Subgroups — C2×#125×D83 #24920713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#125×D₈₃ #24920713