Subgroups — #24916529

Order 10624.

410 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128251	C₂	1	1	yes	maximal, characteristic
2	#19127780	C₂	1	2	yes	maximal, complemented
3	#19127778	C₂	1	2	yes	maximal, complemented
4	C₈×D₃₃₂	C₂	1	2	yes	maximal, complemented
5	#19127766	C₂	1	2	yes	maximal, complemented
6	#19127754	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127666	C₂	1	4	yes	maximal, complemented
8	#431×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13050590	C₂²	1	2	yes	—
10	C₂×#13050573	C₂²	1	4	yes	—
11	C₂×C₈×D₈₃	C₂²	1	4	yes	—
12	#14581792	C₄	1	1	yes	characteristic
13	#14581789	C₄	1	1	yes	characteristic
14	#14581780	C₄	1	2	yes	—
15	#14581766	C₄	1	2	yes	—
16	#14581766	C₄	1	2	yes	—
17	C₄×D₃₃₂	C₂²	1	2	yes	—
18	C₄×D₃₃₂	C₂²	1	2	yes	—
19	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂²⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
22	(C₂×D₈₃)⋊C₈	C₂²	1	4	yes	—
23	(C₂×D₈₃)⋊C₈	C₂²	1	4	yes	—
24	C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
25	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
28	C₂²⋊C₈×C₈₃	C₂²	1	2	yes	—
29	C₄×C₈×C₈₃	C₂²	1	2	yes	—
30	#13050590	—	4	2	no	—
31	#13050590	—	4	2	no	—
32	C₂×Dic₁₆₆	C₂×C₄	1	2	yes	—
33	C₂×D₃₃₂	C₂×C₄	1	2	yes	—
34	C₂×D₃₃₂	C₂×C₄	1	2	yes	—
35	C₂×D₃₃₂	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊D₈₃	C₂×C₄	1	4	yes	—
37	C₂×C₄×D₈₃	C₂³	1	4	yes	—
38	#13050573	—	4	4	no	—
39	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
40	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
41	C₈×D₈₃	—	4	4	no	—
42	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
44	(C₂×D₈₃)⋊C₄	C₂×C₄	1	4	yes	—
45	(C₂×D₈₃)⋊C₄	C₂×C₄	1	4	yes	—
46	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₈₃	C₂³	1	2	yes	—
50	C₂×C₈×C₈₃	C₂³	1	2	yes	—
51	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₈₃	C₂³	1	1	yes	characteristic
54	C₄²×C₈₃	C₂³	1	1	yes	characteristic
55	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×D₈₃	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₈₃	—	2	2	no	—
58	C₂×C₄×C₈₃	—	2	2	no	—
59	C₂×C₄×C₈₃	—	2	1	no	—
60	C₂×C₄×C₈₃	—	2	1	no	—
61	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
67	C₂²⋊D₈₃	—	4	4	no	—
68	C₂×Dic₈₃	C₂²×C₄	1	4	yes	—
69	D₃₃₂	—	2	4	no	—
70	D₃₃₂	—	2	4	no	—
71	D₃₃₂	—	4	2	no	—
72	C₄×D₈₃	—	2	4	no	—
73	C₄×D₈₃	—	2	4	no	—
74	Dic₁₆₆	—	4	2	no	—
75	C₈×C₈₃	—	2	2	no	—
76	C₈×C₈₃	—	2	2	no	—
77	C₈₃⋊C₈	—	2	2	no	—
78	C₈₃⋊C₈	—	2	2	no	—
79	C₂²×C₈₃	—	2	1	no	—
80	C₂²×C₈₃	—	2	1	no	—
81	C₂²×C₈₃	—	2	1	no	—
82	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
83	C₂×D₈₃	—	2	4	no	—
84	C₂×D₈₃	—	2	4	no	—
85	C₂×D₈₃	—	2	4	no	—
86	C₄×C₈₃	—	2	1	no	—
87	C₄×C₈₃	—	2	1	no	—
88	C₄×C₈₃	—	2	1	no	—
89	C₄×C₈₃	#131	1	2	yes	—
90	C₄×C₈₃	#131	1	2	yes	—
91	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
92	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
93	Dic₈₃	—	2	4	no	—
94	C₂×C₈₃	—	4	1	no	—
95	C₂×C₈₃	#518	1	1	yes	characteristic
96	C₂×C₈₃	C₂×#131	1	1	yes	characteristic
97	C₂×C₈₃	C₂×#131	1	1	yes	characteristic
98	D₈₃	—	4	4	no	—
99	#2979	—	83	1	no	maximal, Sylow
100	C₈₃	#2979	1	1	yes	characteristic, complemented, Sylow
101	#532	—	83	1	no	—
102	#436	—	83	2	no	—
103	#436	—	83	2	no	—
104	C₈×D₄	—	83	2	no	—
105	C₈×D₄	—	83	2	no	—
106	#431	—	83	1	no	—
107	#431	D₈₃	1	1	yes	characteristic, complemented
108	#408	—	83	4	no	—
109	C₂×#41	—	83	2	no	—
110	C₂×M₁₆	—	83	4	no	—
111	C₂²×C₈	—	83	4	no	—
112	C₄⋊Q₈	—	83	1	no	—
113	#127	—	83	1	no	—
114	#124	—	83	2	no	—
115	#121	—	83	2	no	—
116	#121	—	83	2	no	—
117	C₄×D₄	—	83	2	no	—
118	C₄×D₄	—	83	2	no	—
119	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
120	C₄⋊C₈	—	83	1	no	—
121	C₄⋊C₈	—	83	1	no	—
122	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
123	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
124	C₂²⋊C₈	—	83	2	no	—
125	C₂²⋊C₈	—	83	4	no	—
126	C₂²⋊C₈	—	83	4	no	—
127	C₂²⋊C₈	C₂×D₈₃	1	2	yes	—
128	C₄×C₈	—	83	2	no	—
129	C₄×C₈	C₂×D₈₃	1	2	yes	—
130	#41	—	332	2	no	—
131	#41	—	332	2	no	—
132	C₂×Q₈	—	83	2	no	—
133	C₂×D₄	—	83	4	no	—
134	C₂×D₄	—	83	2	no	—
135	C₂×D₄	—	83	2	no	—
136	C₂×D₄	—	83	2	no	—
137	C₂²×C₄	—	83	4	no	—
138	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
139	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
141	M₁₆	—	332	4	no	—
142	C₂×C₈	—	332	4	no	—
143	C₂×C₈	—	83	2	no	—
144	C₂×C₈	—	83	2	no	—
145	C₂×C₈	C₂²×D₈₃	1	2	yes	—
146	C₂×C₈	C₂²×D₈₃	1	2	yes	—
147	C₄⋊C₄	—	83	2	no	—
148	C₄⋊C₄	—	83	2	no	—
149	C₂²⋊C₄	—	83	4	no	—
150	C₂²⋊C₄	—	83	4	no	—
151	C₄²	C₄×D₈₃	1	1	yes	characteristic
152	C₄²	C₄×D₈₃	1	1	yes	characteristic
153	C₄²	C₂²×D₈₃	1	1	yes	characteristic
154	C₄²	C₂²×D₈₃	1	1	yes	characteristic
155	C₂³	—	83	4	no	—
156	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
157	Q₈	—	332	2	no	—
158	D₄	—	332	2	no	—
159	D₄	—	332	4	no	—
160	D₄	—	166	4	no	—
161	D₄	—	166	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	83	4	no	—
167	C₂×C₄	—	166	4	no	—
168	C₂×C₄	—	166	4	no	—
169	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
174	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	—	166	2	no	—
178	C₈	—	166	2	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	166	4	no	—
183	C₂²	—	166	4	no	—
184	C₂²	—	166	4	no	—
185	C₂²	C₂²×C₄×D₈₃	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	166	4	no	—
190	C₄	#14581798	1	2	yes	—
191	C₄	#14581800	1	2	yes	—
192	C₄	#14581838	1	1	yes	characteristic
193	C₄	#14581838	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	—	332	4	no	—
196	C₂	#19128175	1	1	yes	characteristic
197	C₂	C₂×#14581798	1	1	yes	characteristic
198	C₂	C₂×#14581800	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Fitting	#431×C₈₃	C₂
prime radical	#19128251	C₂
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	#431	D₈₃
hypocenter	C₈₃	#2979
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #19127780	C₂	semidirect
3. #19127778	C₂	semidirect
4. C₈×D₃₃₂	C₂	semidirect
5. #19127766	C₂	semidirect
6. #19127754	C₂	semidirect
7. #19127666	C₂	semidirect
8. #431×C₈₃	C₂	semidirect
100. C₈₃	#2979	semidirect
107. #431	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128251	2
#19127780	2
#19127778	2
C₈×D₃₃₂	2
#19127766	2
#19127754	2
#19127666	2
#431×C₈₃	2
#2979	83