Subgroups — #35322779

Order 21248.

438 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127527	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127506	C₂	1	2	yes	maximal
4	#24908324	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₃₂	C₂²	1	2	yes	—
7	C₂×Dic₆₆₄	—	2	2	no	—
8	C₂×Dic₆₆₄	—	2	2	no	—
9	C₂×Dic₆₆₄	C₄	1	4	yes	complemented
10	C₂×#14581726	C₂²	1	2	yes	—
11	C₂×#14581717	—	2	2	no	—
12	#19127607	—	2	2	no	—
13	#19127527	—	2	2	no	—
14	#19127506	—	2	4	no	—
15	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₈₃	—	2	2	no	—
19	C₂²×Dic₁₆₆	—	2	2	no	—
20	C₂×Dic₃₃₂	—	2	2	no	—
21	C₂×Dic₃₃₂	—	2	2	no	—
22	C₂×Dic₃₃₂	—	2	2	no	—
23	C₂×Dic₃₃₂	—	2	2	no	—
24	C₂×Dic₃₃₂	—	2	2	no	—
25	C₂×Dic₃₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₈₃	—	2	2	no	—
27	Dic₆₆₄	—	2	8	no	—
28	#14581726	—	2	2	no	—
29	#14581726	—	2	2	no	—
30	#14581717	—	4	4	no	—
31	C₈₃⋊#95	—	4	4	no	—
32	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₃	—	2	1	no	—
36	C₂×C₁₆×C₈₃	—	2	1	no	—
37	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
39	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
40	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
41	C₄×C₈×C₈₃	—	2	1	no	—
42	C₄×C₈×C₈₃	—	2	1	no	—
43	C₂×Dic₁₆₆	—	2	4	no	—
44	C₂×Dic₁₆₆	—	4	2	no	—
45	C₂×Dic₁₆₆	—	4	2	no	—
46	C₂×Dic₁₆₆	—	4	2	no	—
47	C₂×Dic₁₆₆	—	4	2	no	—
48	C₂×Dic₁₆₆	—	4	2	no	—
49	C₂²×Dic₈₃	—	4	2	no	—
50	C₂²×Dic₈₃	—	4	2	no	—
51	Dic₃₃₂	—	2	4	no	—
52	Dic₃₃₂	—	2	4	no	—
53	Dic₃₃₂	—	4	2	no	—
54	Dic₃₃₂	—	4	2	no	—
55	C₄⋊Dic₈₃	—	4	2	no	—
56	C₄⋊Dic₈₃	—	4	2	no	—
57	C₂²×C₄×C₈₃	D₈	1	2	yes	—
58	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₃	—	2	1	no	—
60	C₂×C₈×C₈₃	—	2	1	no	—
61	C₂×C₈×C₈₃	D₈	1	2	yes	—
62	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
67	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
68	C₄²×C₈₃	—	2	1	no	—
69	C₄²×C₈₃	—	2	1	no	—
70	C₁₆×C₈₃	—	2	2	no	—
71	C₁₆×C₈₃	C₂²⋊C₄	1	4	yes	—
72	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₈₃	—	2	2	no	—
74	C₂×C₄×C₈₃	—	2	2	no	—
75	C₂×C₄×C₈₃	Q₃₂	1	4	yes	complemented
76	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
79	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
81	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
82	C₂×Dic₈₃	—	4	4	no	—
83	C₂×Dic₈₃	—	8	2	no	—
84	C₂×Dic₈₃	—	8	2	no	—
85	C₂×Dic₈₃	—	8	2	no	—
86	C₂×Dic₈₃	—	8	4	no	—
87	Dic₁₆₆	—	4	4	no	—
88	Dic₁₆₆	—	4	4	no	—
89	Dic₁₆₆	—	8	2	no	—
90	Dic₁₆₆	—	8	2	no	—
91	C₈×C₈₃	—	2	2	no	—
92	C₈×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
94	C₈×C₈₃	#121	1	1	yes	characteristic
95	C₈×C₈₃	#124	1	1	yes	characteristic
96	C₂²×C₈₃	C₄×D₈	1	1	yes	characteristic
97	C₂²×C₈₃	#466	1	1	yes	characteristic
98	C₂²×C₈₃	#493	1	1	yes	characteristic
99	C₂²×C₈₃	C₂×Q₃₂	1	2	yes	—
100	C₂²×C₈₃	#508	1	2	yes	—
101	C₄×C₈₃	—	2	4	no	—
102	C₄×C₈₃	#390	1	1	yes	characteristic
103	C₄×C₈₃	C₄×D₈	1	1	yes	characteristic
104	C₄×C₈₃	#466	1	1	yes	characteristic
105	C₄×C₈₃	#495	1	1	yes	characteristic
106	Dic₈₃	—	8	4	no	—
107	Dic₈₃	—	16	2	no	—
108	#13236	—	83	1	no	maximal, Sylow
109	C₂×C₈₃	#1970	1	1	yes	characteristic
110	C₂×C₈₃	C₄×Q₃₂	1	2	yes	—
111	C₂×C₈₃	#2216	1	2	yes	—
112	C₂×C₈₃	#2247	1	1	yes	characteristic
113	C₂×C₈₃	#2249	1	1	yes	characteristic
114	C₂²×Q₃₂	—	83	1	no	—
115	C₂×#366	—	83	1	no	—
116	C₂×#358	—	83	2	no	—
117	C₂×C₄×C₁₆	D₈₃	1	1	yes	characteristic
118	#1969	—	83	2	no	—
119	C₈₃	#13236	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₁₆	—	83	2	no	—
121	C₂×Q₃₂	—	83	4	no	—
122	C₂×Q₃₂	—	166	2	no	—
123	C₂×Q₃₂	—	166	2	no	—
124	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
125	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
126	C₂×#107	—	83	2	no	—
127	C₂×#103	—	166	2	no	—
128	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
129	#391	—	166	2	no	—
130	#366	—	166	2	no	—
131	#358	—	166	4	no	—
132	C₄×C₁₆	—	2	2	no	—
133	C₂²×Q₈	—	166	2	no	—
134	C₂×Q₁₆	—	83	4	no	—
135	C₂×Q₁₆	—	166	2	no	—
136	C₂×Q₁₆	—	166	2	no	—
137	C₂×Q₁₆	—	166	2	no	—
138	C₂×Q₁₆	—	166	2	no	—
139	C₂×Q₁₆	—	166	2	no	—
140	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	166	2	no	—
143	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
144	Q₃₂	—	166	8	no	—
145	C₂×C₁₆	—	2	1	no	—
146	C₂×C₁₆	—	2	1	no	—
147	C₂×C₁₆	C₄×D₈₃	1	2	yes	—
148	C₂×C₁₆	D₃₃₂	1	2	yes	—
149	C₂×C₁₆	D₃₃₂	1	2	yes	—
150	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
151	#107	—	166	2	no	—
152	#107	—	166	2	no	—
153	#103	—	332	4	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	#95	—	332	4	no	—
157	C₂×Q₈	—	332	2	no	—
158	C₂×Q₈	—	332	2	no	—
159	C₂×Q₈	—	332	2	no	—
160	C₂×Q₈	—	332	2	no	—
161	C₂×Q₈	—	332	2	no	—
162	C₂×Q₈	—	166	4	no	—
163	C₂²×C₄	—	332	2	no	—
164	C₂²×C₄	—	332	2	no	—
165	C₂²×C₄	D₆₆₄	1	2	yes	—
166	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
167	Q₁₆	—	332	2	no	—
168	Q₁₆	—	332	2	no	—
169	Q₁₆	—	166	4	no	—
170	Q₁₆	—	166	4	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	D₆₆₄	1	2	yes	—
174	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
176	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
177	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
178	C₂×C₈	#13050590	1	1	yes	characteristic
179	C₂×C₈	#13050590	1	1	yes	characteristic
180	C₄⋊C₄	—	332	2	no	—
181	C₄⋊C₄	—	332	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₁₆	—	2	2	no	—
185	C₁₆	(C₂×D₈₃)⋊C₄	1	4	yes	—
186	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
187	Q₈	—	332	4	no	—
188	Q₈	—	332	4	no	—
189	Q₈	—	664	2	no	—
190	Q₈	—	664	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	332	4	no	—
194	C₂×C₄	—	664	2	no	—
195	C₂×C₄	—	664	4	no	—
196	C₂×C₄	—	664	2	no	—
197	C₂×C₄	—	664	2	no	—
198	C₂×C₄	Dic₆₆₄	1	4	yes	complemented
199	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
200	C₂×C₄	#14581766	1	1	yes	characteristic
201	C₂×C₄	#14581789	1	1	yes	characteristic
202	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
203	C₂×C₄	#14581814	1	1	yes	characteristic
204	C₂×C₄	#14581814	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
207	C₈	C₄×D₃₃₂	1	1	yes	characteristic
208	C₈	#14581766	1	1	yes	characteristic
209	C₈	#14581780	1	1	yes	characteristic
210	C₂²	C₄×D₆₆₄	1	1	yes	characteristic
211	C₂²	#19127937	1	1	yes	characteristic
212	C₂²	#19128078	1	1	yes	characteristic
213	C₂²	C₂×Dic₆₆₄	1	2	yes	—
214	C₂²	#19128133	1	2	yes	—
215	C₄	—	2	4	no	—
216	C₄	—	1328	2	no	—
217	C₄	—	664	4	no	—
218	C₄	#19127603	1	1	yes	characteristic
219	C₄	C₄×D₆₆₄	1	1	yes	characteristic
220	C₄	#19127937	1	1	yes	characteristic
221	C₄	#19128085	1	1	yes	characteristic
222	C₂	#24908329	1	1	yes	characteristic
223	C₂	C₄×Dic₆₆₄	1	2	yes	—
224	C₂	#24909554	1	2	yes	—
225	C₂	#24909716	1	1	yes	characteristic
226	C₂	#24909724	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₆₄
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×C₄×C₁₆×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₈₃
hypocenter	C₈₃	#13236
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. C₂×Dic₆₆₄	C₄	semidirect
75. C₂×C₄×C₈₃	Q₃₂	semidirect
119. C₈₃	#13236	semidirect
198. C₂×C₄	Dic₆₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₆₄	2
C₂×#19127527	2
C₂×#19127506	2
#24908324	2
C₂×C₄×C₁₆×C₈₃	2
#13236	83