Subgroups — #35392105

Order 21248.

922 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918042	C₂	1	4	yes	maximal, complemented
2	#24917891	C₂	1	2	yes	maximal, complemented
3	#24907081	C₂	1	1	yes	maximal, characteristic, complemented
4	#24907078	C₂	1	1	yes	maximal, characteristic, complemented
5	#24907069	C₂	1	2	yes	maximal, complemented
6	#24907055	C₂	1	4	yes	maximal, complemented
7	C₂×C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581814	—	2	4	no	—
9	#19128251	C₂²	1	4	yes	—
10	#19128106	C₂²	1	4	yes	—
11	#19128085	—	2	4	no	—
12	#19128081	C₂²	1	2	yes	—
13	#19128078	C₂²	1	4	yes	—
14	#19128078	C₂²	1	2	yes	—
15	#19128074	C₂²	1	4	yes	—
16	#19128048	C₂²	1	8	yes	—
17	#19127937	—	2	8	no	—
18	C₄×D₆₆₄	—	2	8	no	—
19	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₈²×C₈₃	C₂²	1	2	yes	—
22	C₈²×C₈₃	C₂²	1	1	yes	characteristic
23	C₈²×C₈₃	C₂²	1	1	yes	characteristic
24	C₂×#13050590	—	2	4	no	—
25	#14581814	—	4	16	no	—
26	C₂×Dic₃₃₂	—	2	4	no	—
27	C₂×#13050579	—	2	8	no	—
28	C₂×D₆₆₄	—	2	4	no	—
29	C₂×D₆₆₄	—	2	8	no	—
30	#14581792	C₂³	1	4	yes	—
31	#14581789	C₂³	1	4	yes	—
32	#14581780	—	2	4	no	—
33	#14581766	—	2	8	no	—
34	C₄×D₃₃₂	—	2	8	no	—
35	#14581726	—	2	8	no	—
36	#14581714	—	2	16	no	—
37	C₂²×C₈×C₈₃	D₄	1	4	yes	—
38	C₂²×C₈×C₈₃	D₄	1	2	yes	—
39	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₈₃	D₄	1	2	yes	—
41	C₄×C₈×C₈₃	D₄	1	4	yes	—
42	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₈₃	C₂³	1	2	yes	—
45	C₄×C₈×C₈₃	C₂³	1	2	yes	—
46	#13050590	—	4	16	no	—
47	C₂×Dic₁₆₆	—	2	4	no	—
48	C₂×D₃₃₂	—	2	8	no	—
49	C₂×D₃₃₂	—	2	4	no	—
50	C₂×C₂²⋊D₈₃	—	4	4	no	—
51	C₂×C₄×D₈₃	—	4	4	no	—
52	Dic₃₃₂	—	4	8	no	—
53	#13050579	—	4	16	no	—
54	D₆₆₄	—	4	8	no	—
55	D₆₆₄	—	4	16	no	—
56	C₄⋊Dic₈₃	—	2	8	no	—
57	(C₂×D₈₃)⋊C₄	—	4	8	no	—
58	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	—	2	8	no	—
61	C₂×C₈×C₈₃	D₈	1	8	yes	complemented
62	C₂×C₈×C₈₃	C₂×D₄	1	4	yes	—
63	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₈₃	C₂×D₄	1	4	yes	—
66	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₈₃	C₂×D₄	1	2	yes	—
68	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
69	C₂³×C₈₃	#127	1	1	yes	characteristic
70	C₂²×D₈₃	—	4	4	no	—
71	C₂×C₄×C₈₃	—	2	4	no	—
72	C₂×C₄×C₈₃	#127	1	2	yes	—
73	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
77	C₂²⋊D₈₃	—	8	8	no	—
78	C₂×Dic₈₃	—	4	4	no	—
79	D₃₃₂	—	4	8	no	—
80	D₃₃₂	—	4	16	no	—
81	C₄×D₈₃	—	8	8	no	—
82	Dic₁₆₆	—	4	8	no	—
83	C₈×C₈₃	—	2	4	no	—
84	C₈×C₈₃	C₂×D₈	1	4	yes	—
85	C₈×C₈₃	C₂×D₈	1	4	yes	—
86	C₈×C₈₃	#135	1	8	yes	complemented
87	C₂²×C₈₃	—	2	2	no	—
88	C₂²×C₈₃	#493	1	2	yes	—
89	C₂²×C₈₃	C₂×#127	1	1	yes	characteristic
90	C₂×D₈₃	—	8	4	no	—
91	C₂×D₈₃	—	8	8	no	—
92	C₄×C₈₃	—	2	2	no	—
93	C₄×C₈₃	#493	1	2	yes	—
94	C₄×C₈₃	C₂²×D₈	1	2	yes	—
95	C₄×C₈₃	C₂×#135	1	4	yes	—
96	Dic₈₃	—	8	4	no	—
97	#19603	—	83	1	no	maximal, Sylow
98	C₂×C₈₃	—	2	2	no	—
99	C₂×C₈₃	C₂×#493	1	1	yes	characteristic
100	C₂×C₈₃	#3147	1	2	yes	—
101	D₈₃	—	16	4	no	—
102	#3169	—	83	4	no	—
103	#3147	—	83	2	no	—
104	#1713	—	83	1	no	—
105	#1712	—	83	1	no	—
106	#1709	—	83	2	no	—
107	#1706	—	83	4	no	—
108	C₂×C₈²	D₈₃	1	1	yes	characteristic, complemented
109	C₈₃	#19603	1	1	yes	characteristic, complemented, Sylow
110	C₂×#135	—	166	4	no	—
111	#532	—	83	4	no	—
112	#500	—	83	4	no	—
113	#495	—	166	4	no	—
114	#494	—	83	2	no	—
115	#493	—	83	2	no	—
116	#493	—	83	4	no	—
117	#492	—	83	4	no	—
118	#486	—	83	8	no	—
119	#466	—	166	8	no	—
120	C₄×D₈	—	166	8	no	—
121	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂×D₈₃	1	2	yes	—
123	C₈²	C₂×D₈₃	1	2	yes	—
124	C₈²	C₂×D₈₃	1	1	yes	characteristic
125	C₈²	C₂×D₈₃	1	1	yes	characteristic
126	C₂×#41	—	166	4	no	—
127	#135	—	332	16	no	—
128	C₂×Q₁₆	—	166	4	no	—
129	C₂×SD₁₆	—	166	8	no	—
130	C₂×D₈	—	166	4	no	—
131	C₂×D₈	—	166	8	no	—
132	C₂²×C₈	D₃₃₂	1	4	yes	—
133	C₂²×C₈	D₃₃₂	1	2	yes	—
134	C₄⋊Q₈	—	83	4	no	—
135	#127	—	83	4	no	—
136	#124	—	166	4	no	—
137	#121	—	166	8	no	—
138	C₄×D₄	—	166	8	no	—
139	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
140	#107	—	166	8	no	—
141	#102	—	166	16	no	—
142	C₄×C₈	D₃₃₂	1	4	yes	—
143	C₄×C₈	D₃₃₂	1	2	yes	—
144	C₄×C₈	C₂²×D₈₃	1	2	yes	—
145	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
146	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
147	C₄×C₈	C₂²×D₈₃	1	2	yes	—
148	#41	—	332	16	no	—
149	C₂×Q₈	—	166	4	no	—
150	C₂×D₄	—	332	4	no	—
151	C₂×D₄	—	166	8	no	—
152	C₂×D₄	—	166	4	no	—
153	C₂²×C₄	—	332	4	no	—
154	C₂²×C₄	C₂×D₃₃₂	1	2	yes	—
155	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
156	Q₁₆	—	332	8	no	—
157	SD₁₆	—	332	16	no	—
158	D₈	—	332	8	no	—
159	D₈	—	332	16	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	D₆₆₄	1	8	yes	complemented
162	C₂×C₈	C₂×D₃₃₂	1	4	yes	—
163	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
164	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
165	C₂×C₈	C₂×D₃₃₂	1	4	yes	—
166	C₄⋊C₄	—	166	8	no	—
167	C₂²⋊C₄	—	332	8	no	—
168	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
169	C₄²	C₂×D₃₃₂	1	2	yes	—
170	C₄²	C₂³×D₈₃	1	1	yes	characteristic
171	C₂³	—	332	4	no	—
172	C₂³	#14581789	1	1	yes	characteristic
173	Q₈	—	332	8	no	—
174	D₄	—	332	8	no	—
175	D₄	—	332	16	no	—
176	D₄	—	664	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	332	4	no	—
179	C₂×C₄	—	664	8	no	—
180	C₂×C₄	#14581789	1	2	yes	—
181	C₂×C₄	#14581789	1	1	yes	characteristic
182	C₂×C₄	C₂×D₆₆₄	1	4	yes	—
183	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
184	C₂×C₄	C₂²×D₃₃₂	1	2	yes	—
185	C₈	—	2	4	no	—
186	C₈	C₂×D₆₆₄	1	4	yes	—
187	C₈	C₂×D₆₆₄	1	4	yes	—
188	C₈	#14581814	1	8	yes	complemented
189	C₂²	—	2	2	no	—
190	C₂²	—	664	4	no	—
191	C₂²	—	664	8	no	—
192	C₂²	#19128078	1	2	yes	—
193	C₂²	C₂×#14581789	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	664	4	no	—
196	C₄	#19128078	1	2	yes	—
197	C₄	C₂²×D₆₆₄	1	2	yes	—
198	C₄	C₂×#14581814	1	4	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	1328	4	no	—
201	C₂	C₂×#19128078	1	1	yes	characteristic
202	C₂	#24917891	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14581789
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×C₈²×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#127
hypercenter	C₂×C₈²	D₈₃
hypocenter	C₈₃	#19603
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. #24918042	C₂	semidirect
2. #24917891	C₂	semidirect
3. #24907081	C₂	semidirect
4. #24907078	C₂	semidirect
5. #24907069	C₂	semidirect
6. #24907055	C₂	semidirect
7. C₂×C₈²×C₈₃	C₂	semidirect
61. C₂×C₈×C₈₃	D₈	semidirect
86. C₈×C₈₃	#135	semidirect
108. C₂×C₈²	D₈₃	semidirect
109. C₈₃	#19603	semidirect
161. C₂×C₈	D₆₆₄	semidirect
188. C₈	#14581814	semidirect

Maximal subgroups

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

subgroup	index
#24918042	2
#24917891	2
#24907081	2
#24907078	2
#24907069	2
#24907055	2
C₂×C₈²×C₈₃	2
#19603	83