Subgroups — #35324022

Order 21248.

382 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128106	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127529	C₂	1	2	yes	maximal
3	#24908196	C₂	1	1	yes	maximal, characteristic
4	#24905555	C₂	1	2	yes	maximal
5	C₈₃×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581792	—	2	1	no	—
7	#19128106	—	2	4	no	—
8	C₂×#14581726	—	2	1	no	—
9	C₂×#14581726	C₂²	1	1	yes	characteristic
10	C₂×#14581726	C₂²	1	2	yes	—
11	C₂×#14581726	C₂²	1	1	yes	characteristic
12	#19127599	—	2	1	no	—
13	#19127529	—	2	4	no	—
14	#19127464	—	2	1	no	—
15	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₆₆	—	4	1	no	—
18	#14581792	—	2	4	no	—
19	#14581792	—	4	2	no	—
20	C₂×C₄⋊Dic₈₃	—	2	1	no	—
21	C₂×C₄⋊Dic₈₃	—	2	2	no	—
22	C₂×C₄⋊Dic₈₃	—	2	1	no	—
23	C₂×C₄⋊Dic₈₃	—	4	1	no	—
24	C₂×Dic₈₃⋊C₄	—	4	1	no	—
25	#14581726	—	2	4	no	—
26	#14581726	—	2	2	no	—
27	#14581726	—	2	2	no	—
28	#14581726	—	2	1	no	—
29	#14581726	D₄	1	4	yes	—
30	#14581726	Q₈	1	2	yes	complemented
31	C₈₃⋊#95	—	4	1	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	2	no	—
36	C₂×C₁₆×C₈₃	—	2	2	no	—
37	C₂×C₁₆×C₈₃	Q₈	1	4	yes	complemented
38	C₄×C₈×C₈₃	—	2	2	no	—
39	C₂×Dic₁₆₆	—	4	4	no	—
40	C₂×Dic₁₆₆	—	8	2	no	—
41	C₂×Dic₁₆₆	—	8	2	no	—
42	C₂²×Dic₈₃	—	4	2	no	—
43	C₂²×Dic₈₃	—	4	1	no	—
44	C₂²×Dic₈₃	—	4	1	no	—
45	C₄⋊Dic₈₃	—	2	4	no	—
46	C₄⋊Dic₈₃	—	2	2	no	—
47	C₄⋊Dic₈₃	—	4	4	no	—
48	C₄⋊Dic₈₃	—	4	2	no	—
49	C₄⋊Dic₈₃	—	4	1	no	—
50	C₄⋊Dic₈₃	—	4	1	no	—
51	C₄⋊Dic₈₃	—	4	1	no	—
52	Dic₈₃⋊C₄	—	8	2	no	—
53	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	—	2	1	no	—
57	C₂×C₈×C₈₃	—	2	1	no	—
58	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
59	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
63	C₄²×C₈₃	—	2	1	no	—
64	C₄²×C₈₃	—	2	1	no	—
65	C₁₆×C₈₃	—	2	4	no	—
66	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	—	2	1	no	—
68	C₂×C₄×C₈₃	—	2	1	no	—
69	C₂×C₄×C₈₃	—	2	1	no	—
70	C₂×C₄×C₈₃	—	2	1	no	—
71	C₂×C₄×C₈₃	SD₃₂	1	2	yes	—
72	C₂×C₄×C₈₃	SD₃₂	1	2	yes	—
73	C₂×C₄×C₈₃	#120	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
79	C₂×Dic₈₃	—	4	4	no	—
80	C₂×Dic₈₃	—	4	2	no	—
81	C₂×Dic₈₃	—	8	1	no	—
82	C₂×Dic₈₃	—	8	2	no	—
83	C₂×Dic₈₃	—	8	1	no	—
84	C₂×Dic₈₃	—	8	2	no	—
85	C₂×Dic₈₃	—	8	1	no	—
86	C₂×Dic₈₃	—	8	1	no	—
87	C₂×Dic₈₃	—	8	1	no	—
88	Dic₁₆₆	—	8	8	no	—
89	C₈×C₈₃	—	2	2	no	—
90	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
91	C₈×C₈₃	C₄⋊Q₈	1	2	yes	—
92	C₂²×C₈₃	#447	1	1	yes	characteristic
93	C₂²×C₈₃	#474	1	1	yes	characteristic
94	C₂²×C₈₃	#500	1	1	yes	characteristic
95	C₂²×C₈₃	C₂×SD₃₂	1	1	yes	characteristic
96	C₂²×C₈₃	C₂×SD₃₂	1	1	yes	characteristic
97	C₂²×C₈₃	#509	1	1	yes	characteristic
98	C₂²×C₈₃	#510	1	1	yes	characteristic
99	C₄×C₈₃	—	2	2	no	—
100	C₄×C₈₃	—	2	2	no	—
101	C₄×C₈₃	#395	1	1	yes	characteristic
102	C₄×C₈₃	#451	1	1	yes	characteristic
103	C₄×C₈₃	#477	1	1	yes	characteristic
104	C₄×C₈₃	#501	1	1	yes	characteristic
105	Dic₈₃	—	8	4	no	—
106	Dic₈₃	—	8	2	no	—
107	Dic₈₃	—	16	1	no	—
108	#13452	—	83	1	no	maximal, Sylow
109	C₂×C₈₃	#2057	1	1	yes	characteristic
110	C₂×C₈₃	#2185	1	1	yes	characteristic
111	C₂×C₈₃	#2187	1	1	yes	characteristic
112	C₂×C₈₃	#2225	1	1	yes	characteristic
113	C₂×C₈₃	#2226	1	1	yes	characteristic
114	C₂×C₈₃	#2254	1	1	yes	characteristic
115	C₂×C₈₃	#2255	1	1	yes	characteristic
116	C₂×#500	—	83	1	no	—
117	C₂×#367	—	83	2	no	—
118	#1942	—	83	1	no	—
119	#1380	—	83	2	no	—
120	#1374	D₈₃	1	1	yes	characteristic
121	C₈₃	#13452	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄⋊Q₈	—	166	1	no	—
123	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
124	#500	—	166	4	no	—
125	C₂×#107	—	83	2	no	—
126	C₂×#107	—	83	1	no	—
127	C₂×#107	—	83	1	no	—
128	C₂×#107	—	166	1	no	—
129	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
130	#389	—	166	1	no	—
131	#367	—	166	4	no	—
132	#340	—	166	1	no	—
133	C₂²×Q₈	—	332	1	no	—
134	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	332	2	no	—
137	C₄⋊Q₈	—	166	4	no	—
138	C₂×C₄⋊C₄	—	332	1	no	—
139	C₂×C₄⋊C₄	—	332	1	no	—
140	C₂×C₄⋊C₄	—	166	1	no	—
141	C₂×C₄⋊C₄	—	166	2	no	—
142	C₂×C₄⋊C₄	—	166	1	no	—
143	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	Dic₁₆₆	1	4	yes	complemented
147	#107	—	83	4	no	—
148	#107	—	83	2	no	—
149	#107	—	166	2	no	—
150	#107	—	166	4	no	—
151	#107	—	166	2	no	—
152	#107	—	166	1	no	—
153	C₄×C₈	—	2	2	no	—
154	#95	—	332	1	no	—
155	C₂×Q₈	—	332	4	no	—
156	C₂×Q₈	—	664	2	no	—
157	C₂×Q₈	—	664	2	no	—
158	C₂²×C₄	—	332	1	no	—
159	C₂²×C₄	—	332	2	no	—
160	C₂²×C₄	—	332	1	no	—
161	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
162	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	Dic₃₃₂	1	2	yes	—
167	C₂×C₈	D₄×D₈₃	1	2	yes	—
168	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
169	C₂×C₈	C₂×Dic₁₆₆	1	2	yes	—
170	C₂×C₈	#13050589	1	1	yes	characteristic
171	C₄⋊C₄	—	332	4	no	—
172	C₄⋊C₄	—	332	2	no	—
173	C₄⋊C₄	—	332	1	no	—
174	C₄⋊C₄	—	332	1	no	—
175	C₄⋊C₄	—	332	1	no	—
176	C₄⋊C₄	—	664	2	no	—
177	C₄⋊C₄	—	166	4	no	—
178	C₄⋊C₄	—	166	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₁₆	—	2	4	no	—
182	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
183	Q₈	—	664	8	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	332	4	no	—
189	C₂×C₄	—	332	2	no	—
190	C₂×C₄	—	664	1	no	—
191	C₂×C₄	—	664	1	no	—
192	C₂×C₄	—	664	1	no	—
193	C₂×C₄	—	664	1	no	—
194	C₂×C₄	—	664	2	no	—
195	C₂×C₄	—	664	2	no	—
196	C₂×C₄	—	664	1	no	—
197	C₂×C₄	#14581737	1	2	yes	—
198	C₂×C₄	#14581737	1	2	yes	—
199	C₂×C₄	#14581762	1	1	yes	characteristic
200	C₂×C₄	#14581772	1	1	yes	characteristic
201	C₂×C₄	#14581792	1	1	yes	characteristic
202	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
203	C₂×C₄	#14581820	1	1	yes	characteristic
204	C₂×C₄	#14581825	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	#14581769	1	2	yes	—
207	C₈	#14581791	1	2	yes	—
208	C₂²	#19127837	1	1	yes	characteristic
209	C₂²	#19127982	1	1	yes	characteristic
210	C₂²	#19128106	1	1	yes	characteristic
211	C₂²	C₂×#14581737	1	1	yes	characteristic
212	C₂²	C₂×#14581737	1	1	yes	characteristic
213	C₂²	#19128139	1	1	yes	characteristic
214	C₂²	#19128144	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	1328	1	no	—
218	C₄	—	664	2	no	—
219	C₄	—	664	4	no	—
220	C₄	#19127622	1	1	yes	characteristic
221	C₄	#19127859	1	1	yes	characteristic
222	C₄	#19127997	1	1	yes	characteristic
223	C₄	#19128110	1	1	yes	characteristic
224	C₂	#24908792	1	1	yes	characteristic
225	C₂	#24909382	1	1	yes	characteristic
226	C₂	#24909394	1	1	yes	characteristic
227	C₂	#24909606	1	1	yes	characteristic
228	C₂	#24909611	1	1	yes	characteristic
229	C₂	#24909745	1	1	yes	characteristic
230	C₂	#24909749	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₆₄
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1374	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
hypercenter	#1374	D₈₃
hypocenter	C₈₃	#13452
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
30. #14581726	Q₈	semidirect
37. C₂×C₁₆×C₈₃	Q₈	semidirect
121. C₈₃	#13452	semidirect
146. C₂×C₁₆	Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128106	2
C₂×#19127529	2
#24908196	2
#24905555	2
C₈₃×#1374	2
#13452	83