Subgroups — #24922174

Order 10624.

1424 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128537	C₂	1	1	yes	maximal, characteristic
2	#19128468	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128467	C₂	1	6	yes	maximal, complemented
4	#19128466	C₂	1	8	yes	maximal, complemented
5	C₂×#14581798	C₂	1	6	yes	maximal, complemented
6	#131×D₈₃	C₂	1	8	yes	maximal, complemented
7	C₂×#131×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14581840	C₄	1	2	yes	—
9	#14581839	C₄	1	4	yes	—
10	#14581838	C₄	1	4	yes	—
11	#14581837	C₄	1	6	yes	—
12	C₂×#13050590	C₂²	1	6	yes	—
13	#41×D₈₃	C₂²	1	8	yes	—
14	#14581800	C₂²	1	24	yes	complemented
15	#14581799	C₂²	1	8	yes	complemented
16	#14581798	C₂²	1	12	yes	complemented
17	#14581798	C₂²	1	4	yes	complemented
18	#14581798	C₂²	1	12	yes	complemented
19	#14581798	C₂²	1	12	yes	complemented
20	C₂×#13050574	C₂²	1	6	yes	complemented
21	C₂×#13050573	C₂²	1	12	yes	complemented
22	M₁₆×D₈₃	C₂²	1	24	yes	complemented
23	C₂×C₈×D₈₃	C₂²	1	12	yes	complemented
24	C₂×#41×C₈₃	C₂²	1	1	yes	characteristic
25	#131×C₈₃	C₂²	1	4	yes	complemented
26	#131×C₈₃	C₂²	1	4	yes	complemented
27	C₂×M₁₆×C₈₃	C₂²	1	3	yes	complemented
28	C₂²×C₈×C₈₃	C₂²	1	3	yes	complemented
29	#13050591	C₂×C₄	1	8	yes	—
30	#13050590	C₂×C₄	1	12	yes	—
31	#13050590	C₂×C₄	1	12	yes	—
32	#13050590	C₂³	1	4	yes	—
33	#13050590	C₂³	1	12	yes	—
34	#13050589	C₂×C₄	1	24	yes	—
35	C₂×Dic₁₆₆	C₂×C₄	1	6	yes	—
36	Q₈×D₈₃	C₂×C₄	1	8	yes	—
37	C₂×D₃₃₂	C₂×C₄	1	6	yes	—
38	C₂×C₂²⋊D₈₃	C₂×C₄	1	12	yes	—
39	D₄×D₈₃	C₂×C₄	1	24	yes	—
40	C₂×C₄×D₈₃	C₂³	1	12	yes	—
41	#13050574	C₂³	1	12	yes	—
42	#13050574	C₂³	1	4	yes	—
43	#13050573	C₂³	1	24	yes	—
44	#13050573	C₂³	1	8	yes	—
45	C₂×C₈₃⋊C₈	C₂³	1	12	yes	—
46	C₈×D₈₃	C₂³	1	8	yes	—
47	C₈×D₈₃	C₂³	1	24	yes	—
48	#41×C₈₃	C₂×C₄	1	4	yes	—
49	#41×C₈₃	C₂³	1	4	yes	—
50	C₂×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
51	C₂×D₄×C₈₃	C₂×C₄	1	3	yes	—
52	C₂²×C₄×C₈₃	C₂³	1	3	yes	—
53	M₁₆×C₈₃	C₂³	1	12	yes	—
54	C₂×C₈×C₈₃	C₂³	1	6	yes	—
55	C₂×C₈×C₈₃	C₂³	1	6	yes	—
56	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₈₃	C₂³	1	3	yes	—
58	C₂³×C₈₃	C₂²×C₄	1	3	yes	—
59	C₂²×D₈₃	C₂²×C₄	1	12	yes	—
60	Q₈×C₈₃	C₂²×C₄	1	4	yes	—
61	D₄×C₈₃	C₂²×C₄	1	12	yes	—
62	C₂×C₄×C₈₃	C₂²×C₄	1	3	yes	—
63	C₂×C₄×C₈₃	C₂²×C₄	1	6	yes	—
64	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
65	C₂×C₄×C₈₃	C₂⁴	1	6	yes	—
66	C₂²⋊D₈₃	C₂²×C₄	1	24	yes	—
67	C₂²⋊D₈₃	C₂²×C₄	1	8	yes	—
68	C₂×Dic₈₃	C₂²×C₄	1	12	yes	—
69	D₃₃₂	C₂²×C₄	1	12	yes	—
70	D₃₃₂	C₂²×C₄	1	4	yes	—
71	C₄×D₈₃	C₂²×C₄	1	24	yes	—
72	C₄×D₈₃	C₂⁴	1	8	yes	—
73	Dic₁₆₆	C₂²×C₄	1	12	yes	—
74	Dic₁₆₆	C₂²×C₄	1	4	yes	—
75	C₈×C₈₃	C₂⁴	1	1	yes	characteristic
76	C₈×C₈₃	C₂⁴	1	6	yes	—
77	C₈×C₈₃	C₂⁴	1	1	yes	characteristic
78	C₈₃⋊C₈	C₂⁴	1	8	yes	—
79	C₂²×C₈₃	—	4	3	no	—
80	C₂²×C₈₃	C₂³×C₄	1	6	yes	—
81	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×D₈₃	—	4	12	no	—
83	C₂×D₈₃	C₂³×C₄	1	8	yes	—
84	C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
85	C₄×C₈₃	C₂³×C₄	1	6	yes	—
86	C₄×C₈₃	C₂⁵	1	1	yes	characteristic
87	Dic₈₃	C₂³×C₄	1	8	yes	—
88	C₂×C₈₃	—	2	6	no	—
89	C₂×C₈₃	—	2	1	no	—
90	C₂×C₈₃	C₂⁴×C₄	1	1	yes	characteristic
91	D₈₃	—	2	8	no	—
92	#3573	—	83	1	no	maximal, Sylow
93	C₈₃	#3573	1	1	yes	characteristic, complemented, Sylow
94	#585	—	83	1	no	—
95	#568	—	83	1	no	—
96	#568	—	83	8	no	—
97	#568	—	83	6	no	—
98	C₂×#131	—	83	8	no	—
99	C₂×#131	—	83	6	no	—
100	C₂×#131	D₈₃	1	1	yes	characteristic, complemented
101	2₋¹⁺⁴	—	83	4	no	—
102	2₋¹⁺⁴	—	83	2	no	—
103	H(5,𝔽₂)	—	83	4	no	—
104	H(5,𝔽₂)	—	83	6	no	—
105	C₂×#41	—	83	8	no	—
106	C₂×#41	—	83	6	no	—
107	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
108	#131	—	83	8	no	—
109	#131	—	83	24	no	—
110	#131	—	83	12	no	—
111	#131	—	83	4	no	—
112	#131	—	83	12	no	—
113	#131	—	83	12	no	—
114	#131	C₂×D₈₃	1	4	yes	complemented
115	#131	C₂×D₈₃	1	4	yes	complemented
116	C₂×M₁₆	—	83	6	no	—
117	C₂×M₁₆	—	83	24	no	—
118	C₂×M₁₆	—	83	12	no	—
119	C₂×M₁₆	C₂×D₈₃	1	3	yes	complemented
120	C₂²×C₈	—	83	12	no	—
121	C₂²×C₈	C₂×D₈₃	1	3	yes	complemented
122	#41	—	83	8	no	—
123	#41	—	83	24	no	—
124	#41	—	83	12	no	—
125	#41	—	83	12	no	—
126	#41	—	83	12	no	—
127	#41	—	83	4	no	—
128	#41	C₄×D₈₃	1	4	yes	—
129	#41	C₂²×D₈₃	1	4	yes	—
130	C₂×Q₈	—	83	8	no	—
131	C₂×Q₈	—	83	6	no	—
132	C₂×Q₈	C₄×D₈₃	1	1	yes	characteristic
133	C₂×D₄	—	83	12	no	—
134	C₂×D₄	—	83	24	no	—
135	C₂×D₄	—	83	6	no	—
136	C₂×D₄	C₄×D₈₃	1	3	yes	—
137	C₂²×C₄	—	83	12	no	—
138	C₂²×C₄	C₂²×D₈₃	1	3	yes	—
139	M₁₆	—	83	12	no	—
140	M₁₆	—	83	4	no	—
141	M₁₆	—	83	24	no	—
142	M₁₆	—	83	8	no	—
143	M₁₆	C₂²×D₈₃	1	12	yes	—
144	C₂×C₈	—	83	12	no	—
145	C₂×C₈	—	83	24	no	—
146	C₂×C₈	—	83	8	no	—
147	C₂×C₈	C₂²×D₈₃	1	6	yes	—
148	C₂×C₈	C₂²×D₈₃	1	3	yes	—
149	C₂×C₈	C₂²×D₈₃	1	6	yes	—
150	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
151	C₂³	—	83	12	no	—
152	C₂³	C₂×C₄×D₈₃	1	3	yes	—
153	Q₈	—	83	12	no	—
154	Q₈	—	83	4	no	—
155	Q₈	C₂×C₄×D₈₃	1	4	yes	—
156	D₄	—	83	24	no	—
157	D₄	—	83	8	no	—
158	D₄	—	83	12	no	—
159	D₄	—	83	4	no	—
160	D₄	C₂×C₄×D₈₃	1	12	yes	—
161	C₂×C₄	—	83	12	no	—
162	C₂×C₄	—	83	24	no	—
163	C₂×C₄	—	83	8	no	—
164	C₂×C₄	C₂×C₄×D₈₃	1	6	yes	—
165	C₂×C₄	C₂×C₄×D₈₃	1	3	yes	—
166	C₂×C₄	C₂³×D₈₃	1	6	yes	—
167	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
168	C₈	—	83	8	no	—
169	C₈	C₂³×D₈₃	1	6	yes	—
170	C₈	C₂³×D₈₃	1	1	yes	characteristic
171	C₈	C₂³×D₈₃	1	1	yes	characteristic
172	C₂²	—	4	3	no	—
173	C₂²	—	332	12	no	—
174	C₂²	—	83	8	no	—
175	C₂²	C₂²×C₄×D₈₃	1	6	yes	—
176	C₂²	C₂²×C₄×D₈₃	1	1	yes	characteristic
177	C₄	—	83	8	no	—
178	C₄	C₂²×C₄×D₈₃	1	6	yes	—
179	C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
180	C₄	C₂⁴×D₈₃	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	6	no	—
183	C₂	—	166	8	no	—
184	C₂	C₂³×C₄×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂³×D₈₃
commutator, socle	C₂×C₈₃	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₈₃
Fitting	C₂×#131×C₈₃	C₂
prime radical	#19128537	C₂
cosocle	C₄×C₈₃	C₂⁵
hypercenter	C₂×#131	D₈₃
hypocenter	C₈₃	#3573
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #19128468	C₂	semidirect
3. #19128467	C₂	semidirect
4. #19128466	C₂	semidirect
5. C₂×#14581798	C₂	semidirect
6. #131×D₈₃	C₂	semidirect
7. C₂×#131×C₈₃	C₂	semidirect
14. #14581800	C₂²	semidirect
15. #14581799	C₂²	semidirect
16. #14581798	C₂²	semidirect
17. #14581798	C₂²	semidirect
18. #14581798	C₂²	semidirect
19. #14581798	C₂²	semidirect
20. C₂×#13050574	C₂²	semidirect
21. C₂×#13050573	C₂²	semidirect
22. M₁₆×D₈₃	C₂²	semidirect
23. C₂×C₈×D₈₃	C₂²	semidirect
25. #131×C₈₃	C₂²	semidirect
26. #131×C₈₃	C₂²	semidirect
27. C₂×M₁₆×C₈₃	C₂²	semidirect
28. C₂²×C₈×C₈₃	C₂²	semidirect
93. C₈₃	#3573	semidirect
100. C₂×#131	D₈₃	semidirect
114. #131	C₂×D₈₃	semidirect
115. #131	C₂×D₈₃	semidirect
119. C₂×M₁₆	C₂×D₈₃	semidirect
121. C₂²×C₈	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128537	2
#19128468	2
#19128467	2
#19128466	2
C₂×#14581798	2
#131×D₈₃	2
C₂×#131×C₈₃	2
#3573	83