Subgroups — #35559810

Order 21248.

834 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128133	C₂	1	4	yes	maximal, complemented
2	#24917891	C₂	1	2	yes	maximal, complemented
3	#24909724	C₂	1	4	yes	maximal, complemented
4	#24909719	C₂	1	2	yes	maximal, complemented
5	#24909716	C₂	1	1	yes	maximal, characteristic, complemented
6	#24909713	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581814	—	2	2	no	—
9	C₂×#14581814	C₂²	1	4	yes	—
10	#19128251	—	2	2	no	—
11	#19128133	—	2	16	no	—
12	C₂×Dic₆₆₄	C₂²	1	4	yes	—
13	C₂×#14581737	C₂²	1	8	yes	—
14	C₂×D₁₃₂₈	C₂²	1	4	yes	—
15	#19128085	—	2	2	no	—
16	#19128085	C₂²	1	4	yes	—
17	#19128081	C₂²	1	2	yes	—
18	#19128078	C₂²	1	2	yes	—
19	#19128074	—	2	2	no	—
20	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
24	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
25	C₂×#13050590	—	2	4	no	—
26	#14581814	—	2	16	no	—
27	#14581814	—	4	8	no	—
28	C₂×Dic₃₃₂	—	2	2	no	—
29	C₂×Dic₃₃₂	C₂³	1	4	yes	—
30	C₂×#13050579	—	2	4	no	—
31	C₂×#13050579	—	2	4	no	—
32	C₂×D₆₆₄	—	2	2	no	—
33	C₂×D₆₆₄	C₂³	1	4	yes	—
34	#14581792	—	2	2	no	—
35	#14581789	—	2	2	no	—
36	#14581780	—	2	4	no	—
37	#14581766	—	4	4	no	—
38	C₄×D₃₃₂	—	4	4	no	—
39	Dic₆₆₄	—	2	8	no	—
40	#14581737	—	2	16	no	—
41	D₁₃₂₈	—	2	8	no	—
42	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₈₃	D₄	1	8	yes	complemented
46	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
47	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
48	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
52	#13050590	—	4	8	no	—
53	#13050590	—	4	8	no	—
54	C₂×Dic₁₆₆	—	2	4	no	—
55	C₂×D₃₃₂	—	2	4	no	—
56	C₂×D₃₃₂	—	4	4	no	—
57	C₂×C₂²⋊D₈₃	—	4	4	no	—
58	C₂×C₄×D₈₃	—	4	4	no	—
59	Dic₃₃₂	—	2	8	no	—
60	Dic₃₃₂	—	4	4	no	—
61	#13050579	—	4	8	no	—
62	#13050579	—	4	8	no	—
63	D₆₆₄	—	2	8	no	—
64	D₆₆₄	—	4	4	no	—
65	C₄⋊Dic₈₃	—	4	4	no	—
66	(C₂×D₈₃)⋊C₄	—	4	8	no	—
67	C₂²×C₄×C₈₃	D₈	1	2	yes	—
68	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₈₃	D₈	1	4	yes	—
70	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₈₃	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
75	C₄²×C₈₃	D₈	1	2	yes	—
76	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₈₃	C₂×D₄	1	4	yes	—
79	C₁₆×C₈₃	C₂×D₄	1	4	yes	—
80	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
81	C₂²×D₈₃	—	4	4	no	—
82	C₂×C₄×C₈₃	—	2	4	no	—
83	C₂×C₄×C₈₃	#127	1	2	yes	—
84	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
89	C₂²⋊D₈₃	—	8	8	no	—
90	C₂×Dic₈₃	—	4	4	no	—
91	D₃₃₂	—	4	4	no	—
92	D₃₃₂	—	4	4	no	—
93	D₃₃₂	—	8	8	no	—
94	C₄×D₈₃	—	8	8	no	—
95	Dic₁₆₆	—	4	4	no	—
96	Dic₁₆₆	—	4	4	no	—
97	C₈×C₈₃	#127	1	2	yes	—
98	C₈×C₈₃	C₂×D₈	1	2	yes	—
99	C₈×C₈₃	C₂×D₈	1	2	yes	—
100	C₈×C₈₃	C₂²×D₄	1	2	yes	—
101	C₂²×C₈₃	—	2	2	no	—
102	C₂²×C₈₃	#493	1	2	yes	—
103	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
104	C₂×D₈₃	—	8	4	no	—
105	C₂×D₈₃	—	8	8	no	—
106	C₄×C₈₃	—	2	2	no	—
107	C₄×C₈₃	#493	1	2	yes	—
108	C₄×C₈₃	#508	1	4	yes	complemented
109	C₄×C₈₃	C₂×#127	1	1	yes	characteristic
110	C₄×C₈₃	C₂²×D₈	1	1	yes	characteristic
111	Dic₈₃	—	8	4	no	—
112	#33713	—	83	1	no	maximal, Sylow
113	C₂×C₈₃	—	2	2	no	—
114	C₂×C₈₃	C₂×#493	1	1	yes	characteristic
115	C₂×C₈₃	C₂×#508	1	2	yes	—
116	D₈₃	—	16	4	no	—
117	C₂×#508	—	83	4	no	—
118	#3147	—	83	2	no	—
119	#2249	—	83	4	no	—
120	#2248	—	83	2	no	—
121	#2247	—	83	1	no	—
122	#2246	—	83	1	no	—
123	C₂×C₄×C₁₆	D₈₃	1	1	yes	characteristic, complemented
124	C₈₃	#33713	1	1	yes	characteristic, complemented, Sylow
125	C₂×#135	—	83	4	no	—
126	C₂×#135	—	166	2	no	—
127	#532	—	166	2	no	—
128	#508	—	166	16	no	—
129	C₂×Q₃₂	—	83	4	no	—
130	C₂×SD₃₂	—	83	8	no	—
131	C₂×D₁₆	—	83	4	no	—
132	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
133	#495	—	83	4	no	—
134	#495	—	166	2	no	—
135	#494	—	83	2	no	—
136	#493	—	83	2	no	—
137	#492	—	166	2	no	—
138	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
139	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
140	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
141	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
142	C₂×#41	—	166	4	no	—
143	#135	—	332	8	no	—
144	#135	—	166	16	no	—
145	C₂×Q₁₆	—	83	4	no	—
146	C₂×Q₁₆	—	166	2	no	—
147	C₂×SD₁₆	—	166	4	no	—
148	C₂×SD₁₆	—	166	4	no	—
149	C₂×D₈	—	83	4	no	—
150	C₂×D₈	—	166	2	no	—
151	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
153	C₄⋊Q₈	—	166	2	no	—
154	#127	—	166	2	no	—
155	#124	—	166	4	no	—
156	#121	—	332	4	no	—
157	C₄×D₄	—	332	4	no	—
158	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
159	Q₃₂	—	166	8	no	—
160	SD₃₂	—	166	16	no	—
161	D₁₆	—	166	8	no	—
162	C₂×C₁₆	D₃₃₂	1	8	yes	complemented
163	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
164	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
165	C₄×C₈	D₃₃₂	1	1	yes	characteristic
166	C₄×C₈	D₃₃₂	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
169	#41	—	332	8	no	—
170	#41	—	332	8	no	—
171	C₂×Q₈	—	166	4	no	—
172	C₂×D₄	—	332	4	no	—
173	C₂×D₄	—	332	4	no	—
174	C₂×D₄	—	166	4	no	—
175	C₂²×C₄	—	332	4	no	—
176	C₂²×C₄	D₆₆₄	1	2	yes	—
177	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
178	Q₁₆	—	332	4	no	—
179	Q₁₆	—	166	8	no	—
180	SD₁₆	—	332	8	no	—
181	SD₁₆	—	332	8	no	—
182	D₈	—	332	4	no	—
183	D₈	—	166	8	no	—
184	C₂×C₈	D₆₆₄	1	4	yes	—
185	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
186	C₂×C₈	C₂×D₃₃₂	1	4	yes	—
187	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
188	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
189	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
190	C₄⋊C₄	—	332	4	no	—
191	C₂²⋊C₄	—	332	8	no	—
192	C₄²	D₆₆₄	1	2	yes	—
193	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
194	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
195	C₁₆	C₂×D₃₃₂	1	4	yes	—
196	C₁₆	C₂×D₃₃₂	1	4	yes	—
197	C₂³	—	332	4	no	—
198	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
199	Q₈	—	332	4	no	—
200	Q₈	—	332	4	no	—
201	D₄	—	332	4	no	—
202	D₄	—	332	4	no	—
203	D₄	—	664	8	no	—
204	D₄	—	664	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	332	4	no	—
207	C₂×C₄	—	664	8	no	—
208	C₂×C₄	#14581789	1	2	yes	—
209	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
210	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
211	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
212	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
214	C₈	#14581789	1	2	yes	—
215	C₈	C₂×D₆₆₄	1	2	yes	—
216	C₈	C₂×D₆₆₄	1	2	yes	—
217	C₈	C₂²×D₃₃₂	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	664	8	no	—
220	C₂²	—	664	4	no	—
221	C₂²	#19128078	1	2	yes	—
222	C₂²	C₂²×D₆₆₄	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	664	4	no	—
225	C₄	#19128078	1	2	yes	—
226	C₄	#19128133	1	4	yes	complemented
227	C₄	C₂×#14581789	1	1	yes	characteristic
228	C₄	C₂²×D₆₆₄	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	1328	4	no	—
231	C₂	C₂×#19128078	1	1	yes	characteristic
232	C₂	C₂×#19128133	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₆₄
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×C₄×C₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₈
hypercenter	C₂×C₄×C₁₆	D₈₃
hypocenter	C₈₃	#33713
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19128133	C₂	semidirect
2. #24917891	C₂	semidirect
3. #24909724	C₂	semidirect
4. #24909719	C₂	semidirect
5. #24909716	C₂	semidirect
6. #24909713	C₂	semidirect
7. C₂×C₄×C₁₆×C₈₃	C₂	semidirect
45. C₂×C₁₆×C₈₃	D₄	semidirect
108. C₄×C₈₃	#508	semidirect
123. C₂×C₄×C₁₆	D₈₃	semidirect
124. C₈₃	#33713	semidirect
162. C₂×C₁₆	D₃₃₂	semidirect
226. C₄	#19128133	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128133	2
#24917891	2
#24909724	2
#24909719	2
#24909716	2
#24909713	2
C₂×C₄×C₁₆×C₈₃	2
#33713	83