Subgroups — #24918102

Order 10624.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128322	C₂	1	2	yes	maximal, complemented
2	#19128106	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128085	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128061	C₂	1	2	yes	maximal, complemented
5	#19127947	C₂	1	2	yes	maximal, complemented
6	#19127942	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127937	C₂	1	1	yes	maximal, characteristic
8	#19127803	C₂	1	2	yes	maximal, complemented
9	C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₆₆₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
13	C₂×#13050579	C₂²	1	2	yes	—
14	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
15	#14581792	C₂²	1	2	yes	—
16	#14581787	—	2	2	no	—
17	#14581780	C₂²	1	2	yes	—
18	#14581778	—	2	2	no	—
19	#14581774	—	2	2	no	—
20	#14581771	C₂²	1	2	yes	—
21	#14581770	—	2	2	no	—
22	#14581766	C₂²	1	2	yes	—
23	C₄×Dic₁₆₆	C₂²	1	2	yes	—
24	C₄×D₃₃₂	C₂²	1	2	yes	—
25	#14581726	C₂²	1	1	yes	characteristic
26	#14581726	C₂²	1	1	yes	characteristic
27	#14581726	C₂²	1	2	yes	—
28	#14581717	C₂²	1	2	yes	—
29	#14581717	C₂²	1	2	yes	—
30	#14581714	C₂²	1	2	yes	—
31	#14581714	C₂²	1	2	yes	—
32	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
34	#117×C₈₃	C₂²	1	1	yes	characteristic
35	#97×C₈₃	C₂²	1	1	yes	characteristic
36	#97×C₈₃	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
39	C₂×Dic₁₆₆	—	2	2	no	—
40	C₂×Dic₁₆₆	C₂³	1	2	yes	—
41	C₂×D₃₃₂	C₂³	1	2	yes	—
42	C₂×C₂²⋊D₈₃	—	2	2	no	—
43	C₂×C₄×D₈₃	—	2	2	no	—
44	Dic₃₃₂	—	2	2	no	—
45	#13050579	—	2	4	no	—
46	D₆₆₄	—	2	2	no	—
47	C₄⋊Dic₈₃	—	2	2	no	—
48	C₄⋊Dic₈₃	C₂³	1	2	yes	—
49	C₄⋊Dic₈₃	C₂³	1	2	yes	—
50	Dic₈₃⋊C₄	—	2	2	no	—
51	Dic₈₃⋊C₄	—	2	2	no	—
52	C₂²⋊Dic₈₃	—	2	2	no	—
53	(C₂×D₈₃)⋊C₄	—	2	2	no	—
54	(C₂×D₈₃)⋊C₄	—	2	2	no	—
55	(C₂×D₈₃)⋊C₄	—	2	2	no	—
56	(C₂×D₈₃)⋊C₄	—	2	2	no	—
57	C₄×Dic₈₃	—	2	2	no	—
58	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
59	M₁₆×C₈₃	—	2	2	no	—
60	C₂×C₈×C₈₃	—	2	2	no	—
61	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
68	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
69	C₄²×C₈₃	C₂³	1	1	yes	characteristic
70	C₄²×C₈₃	C₂³	1	1	yes	characteristic
71	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×D₈₃	—	2	2	no	—
73	C₂×C₄×C₈₃	—	2	1	no	—
74	C₂×C₄×C₈₃	—	2	1	no	—
75	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₈₃	—	8	2	no	—
82	C₂×Dic₈₃	—	2	2	no	—
83	C₂×Dic₈₃	—	2	2	no	—
84	C₂×Dic₈₃	—	2	2	no	—
85	D₃₃₂	—	2	2	no	—
86	D₃₃₂	—	2	2	no	—
87	C₄×D₈₃	—	8	2	no	—
88	Dic₁₆₆	—	2	2	no	—
89	Dic₁₆₆	—	2	2	no	—
90	Dic₁₆₆	—	8	2	no	—
91	C₈×C₈₃	—	2	1	no	—
92	C₈×C₈₃	—	2	1	no	—
93	C₈×C₈₃	#41	1	2	yes	—
94	C₈×C₈₃	#41	1	2	yes	—
95	C₂²×C₈₃	—	2	1	no	—
96	C₂²×C₈₃	—	2	1	no	—
97	C₂²×C₈₃	—	2	1	no	—
98	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×D₈₃	—	4	2	no	—
100	C₂×D₈₃	—	4	2	no	—
101	C₂×D₈₃	—	4	2	no	—
102	C₄×C₈₃	—	2	1	no	—
103	C₄×C₈₃	—	2	1	no	—
104	C₄×C₈₃	—	2	1	no	—
105	C₄×C₈₃	—	2	1	no	—
106	C₄×C₈₃	—	2	1	no	—
107	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
108	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
109	Dic₈₃	—	4	2	no	—
110	Dic₈₃	—	4	2	no	—
111	Dic₈₃	—	4	2	no	—
112	C₂×C₈₃	—	4	1	no	—
113	C₂×C₈₃	#532	1	1	yes	characteristic
114	C₂×C₈₃	#576	1	1	yes	characteristic
115	C₂×C₈₃	#578	1	1	yes	characteristic
116	D₈₃	—	8	2	no	—
117	#3175	—	83	1	no	maximal, Sylow
118	C₈₃	#3175	1	1	yes	characteristic, complemented, Sylow
119	#542	—	83	2	no	—
120	#500	—	83	1	no	—
121	#495	—	83	1	no	—
122	#489	—	83	2	no	—
123	#468	—	83	2	no	—
124	#467	—	83	1	no	—
125	#466	—	83	1	no	—
126	#440	—	83	2	no	—
127	C₄×Q₁₆	—	83	1	no	—
128	C₄×D₈	—	83	1	no	—
129	#405	D₈₃	1	1	yes	characteristic, complemented
130	C₂×Q₁₆	—	83	1	no	—
131	C₂×SD₁₆	—	83	2	no	—
132	C₂×D₈	—	83	1	no	—
133	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
134	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	83	2	no	—
136	#126	—	166	2	no	—
137	#124	—	83	2	no	—
138	#124	—	166	2	no	—
139	#123	—	166	2	no	—
140	C₂²⋊Q₈	—	83	2	no	—
141	C₂²⋊Q₈	—	166	2	no	—
142	#121	—	83	2	no	—
143	C₄×Q₈	—	83	2	no	—
144	C₄×D₄	—	83	2	no	—
145	#117	C₂×D₈₃	1	1	yes	characteristic
146	#107	—	83	1	no	—
147	#107	—	83	1	no	—
148	#107	—	83	2	no	—
149	#103	—	83	2	no	—
150	#103	—	83	2	no	—
151	#102	—	83	2	no	—
152	#102	—	83	2	no	—
153	#97	C₂×D₈₃	1	1	yes	characteristic
154	#97	C₂×D₈₃	1	1	yes	characteristic
155	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
156	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
157	C₂×Q₈	—	83	2	no	—
158	C₂×Q₈	—	166	2	no	—
159	C₂×D₄	—	83	2	no	—
160	C₂×D₄	—	166	2	no	—
161	C₂²×C₄	—	166	2	no	—
162	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
163	Q₁₆	—	166	2	no	—
164	SD₁₆	—	166	4	no	—
165	D₈	—	166	2	no	—
166	M₁₆	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
169	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
170	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
172	C₄⋊C₄	—	83	2	no	—
173	C₄⋊C₄	—	83	2	no	—
174	C₄⋊C₄	—	166	2	no	—
175	C₄⋊C₄	—	166	2	no	—
176	C₄⋊C₄	—	166	2	no	—
177	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
178	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
179	C₂²⋊C₄	—	166	2	no	—
180	C₂²⋊C₄	—	166	2	no	—
181	C₂²⋊C₄	—	166	2	no	—
182	C₂²⋊C₄	—	166	2	no	—
183	C₂²⋊C₄	—	166	2	no	—
184	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
185	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
186	C₄²	—	166	2	no	—
187	C₄²	C₂²×D₈₃	1	1	yes	characteristic
188	C₄²	C₂²×D₈₃	1	1	yes	characteristic
189	C₂³	—	166	2	no	—
190	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
191	Q₈	—	664	2	no	—
192	Q₈	—	166	2	no	—
193	Q₈	—	166	2	no	—
194	D₄	—	664	2	no	—
195	D₄	—	166	2	no	—
196	D₄	—	166	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	664	2	no	—
200	C₂×C₄	—	166	2	no	—
201	C₂×C₄	—	166	2	no	—
202	C₂×C₄	—	166	2	no	—
203	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
204	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
205	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
206	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
207	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	#13050590	1	2	yes	—
212	C₈	#13050590	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	332	2	no	—
217	C₂²	—	332	2	no	—
218	C₂²	—	332	2	no	—
219	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	332	2	no	—
226	C₄	—	332	2	no	—
227	C₄	—	332	2	no	—
228	C₄	C₂×#13050590	1	1	yes	characteristic
229	C₄	C₂×#13050590	1	1	yes	characteristic
230	C₂	—	4	1	no	—
231	C₂	—	664	2	no	—
232	C₂	#19128251	1	1	yes	characteristic
233	C₂	#19128506	1	1	yes	characteristic
234	C₂	#19128518	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#405×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19127937	C₂
hypercenter	#405	D₈₃
hypocenter	C₈₃	#3175
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
1. #19128322	C₂	semidirect
2. #19128106	C₂	semidirect
3. #19128085	C₂	semidirect
4. #19128061	C₂	semidirect
5. #19127947	C₂	semidirect
6. #19127942	C₂	semidirect
8. #19127803	C₂	semidirect
9. C₄×Dic₃₃₂	C₂	semidirect
10. C₄×D₆₆₄	C₂	semidirect
11. #405×C₈₃	C₂	semidirect
118. C₈₃	#3175	semidirect
129. #405	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128322	2
#19128106	2
#19128085	2
#19128061	2
#19127947	2
#19127942	2
#19127937	2
#19127803	2
C₄×Dic₃₃₂	2
C₄×D₆₆₄	2
#405×C₈₃	2
#3175	83