Subgroups — #35322402

Order 21248.

350 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₃⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	#24905540	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₈₃⋊C₈)	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₈₃	C₂²	1	2	yes	—
10	C₄×C₄⋊Dic₈₃	C₄	1	1	yes	characteristic
11	Dic₈₃⋊C₁₆	—	2	4	no	—
12	C₄×C₈₃⋊C₁₆	—	2	2	no	—
13	#19127455	C₄	1	2	yes	—
14	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₈₃	—	2	2	no	—
17	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄⋊Dic₈₃	C₈	1	1	yes	characteristic
19	C₂×C₄⋊Dic₈₃	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₈₃	C₂×C₄	1	2	yes	—
21	C₂×C₈₃⋊C₁₆	—	2	1	no	—
22	C₂×C₈₃⋊C₁₆	—	2	1	no	—
23	C₂×C₈₃⋊C₁₆	—	2	1	no	—
24	C₂×C₈₃⋊C₁₆	—	2	1	no	—
25	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
26	C₂×C₈₃⋊C₁₆	Q₈	1	2	yes	—
27	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
28	C₈×Dic₈₃	—	2	2	no	—
29	C₈×Dic₈₃	—	2	2	no	—
30	C₈₃⋊#95	C₈	1	2	yes	—
31	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₈₃	—	2	2	no	—
35	C₂×C₁₆×C₈₃	—	2	2	no	—
36	C₂×C₁₆×C₈₃	—	2	2	no	—
37	C₄×C₈×C₈₃	D₄	1	2	yes	—
38	C₄×C₈×C₈₃	Q₈	1	2	yes	—
39	C₂²×Dic₈₃	C₂×C₈	1	2	yes	—
40	C₂²×Dic₈₃	C₂×C₈	1	2	yes	—
41	C₂×C₈₃⋊C₈	—	2	2	no	—
42	C₂×C₈₃⋊C₈	—	2	2	no	—
43	C₂×C₈₃⋊C₈	—	2	2	no	—
44	C₄⋊Dic₈₃	—	2	2	no	—
45	C₄⋊Dic₈₃	C₁₆	1	4	yes	complemented
46	C₄×Dic₈₃	—	2	2	no	—
47	C₄×Dic₈₃	—	2	2	no	—
48	C₈₃⋊C₁₆	—	2	2	no	—
49	C₈₃⋊C₁₆	—	2	2	no	—
50	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₃	—	2	1	no	—
54	C₂×C₈×C₈₃	—	2	1	no	—
55	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
61	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
62	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
63	C₄²×C₈₃	M₁₆	1	2	yes	—
64	C₁₆×C₈₃	—	4	2	no	—
65	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₈₃	—	2	1	no	—
67	C₂×C₄×C₈₃	—	2	1	no	—
68	C₂×C₄×C₈₃	C₄⋊C₈	1	2	yes	—
69	C₂×C₄×C₈₃	C₄⋊C₈	1	2	yes	—
70	C₂×C₄×C₈₃	C₂×C₁₆	1	2	yes	—
71	C₂×C₄×C₈₃	M₃₂	1	2	yes	—
72	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
77	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
78	C₂×Dic₈₃	—	2	2	no	—
79	C₂×Dic₈₃	—	2	2	no	—
80	C₂×Dic₈₃	—	2	2	no	—
81	C₂×Dic₈₃	—	2	2	no	—
82	C₂×Dic₈₃	—	2	2	no	—
83	C₂×Dic₈₃	C₂×C₁₆	1	4	yes	—
84	C₈×C₈₃	—	2	2	no	—
85	C₈×C₈₃	#121	1	1	yes	characteristic
86	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₈×C₈₃	#125	1	1	yes	characteristic
88	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
89	C₈₃⋊C₈	—	4	2	no	—
90	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₂²×C₈₃	C₈×D₄	1	1	yes	characteristic
92	C₂²×C₈₃	C₈×Q₈	1	1	yes	characteristic
93	C₂²×C₈₃	C₂²×C₁₆	1	1	yes	characteristic
94	C₂²×C₈₃	C₂×M₃₂	1	1	yes	characteristic
95	C₂²×C₈₃	#504	1	1	yes	characteristic
96	C₂²×C₈₃	#504	1	1	yes	characteristic
97	C₄×C₈₃	—	2	2	no	—
98	C₄×C₈₃	C₄⋊C₁₆	1	4	yes	complemented
99	C₄×C₈₃	#389	1	1	yes	characteristic
100	C₄×C₈₃	#424	1	1	yes	characteristic
101	C₄×C₈₃	#436	1	1	yes	characteristic
102	C₄×C₈₃	#446	1	1	yes	characteristic
103	Dic₈₃	—	2	4	no	—
104	Dic₈₃	—	4	2	no	—
105	#13159	—	83	1	no	maximal, Sylow
106	C₂×C₈₃	#1939	1	1	yes	characteristic
107	C₂×C₈₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
108	C₂×C₈₃	#2151	1	1	yes	characteristic
109	C₂×C₈₃	D₄×C₁₆	1	1	yes	characteristic
110	C₂×C₈₃	#2169	1	1	yes	characteristic
111	C₂×C₈₃	Q₈×C₁₆	1	1	yes	characteristic
112	C₂×C₈₃	#2183	1	1	yes	characteristic
113	C₂×C₄⋊C₁₆	—	83	2	no	—
114	C₂×C₄⋊C₁₆	D₈₃	1	1	yes	characteristic
115	C₂×C₄×C₁₆	—	83	1	no	—
116	C₈×C₄⋊C₄	—	83	1	no	—
117	#1374	—	83	2	no	—
118	C₈₃	#13159	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₁₆	—	83	1	no	—
120	C₂²×C₁₆	—	83	1	no	—
121	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
122	C₂×C₄⋊C₈	—	83	1	no	—
123	C₂×C₄×C₈	—	83	2	no	—
124	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
125	C₄×C₄⋊C₄	—	83	1	no	—
126	C₄⋊C₁₆	—	2	2	no	—
127	C₄⋊C₁₆	—	166	4	no	—
128	C₄×C₁₆	—	166	2	no	—
129	#336	—	83	2	no	—
130	C₂²×C₈	—	83	2	no	—
131	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
133	C₂×C₄⋊C₄	—	83	1	no	—
134	C₂×C₄⋊C₄	—	83	1	no	—
135	C₂×C₄²	—	83	2	no	—
136	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	83	2	no	—
141	C₂×C₁₆	—	83	2	no	—
142	C₂×C₁₆	—	166	1	no	—
143	C₂×C₁₆	—	166	1	no	—
144	C₂×C₁₆	—	166	1	no	—
145	C₂×C₁₆	—	166	1	no	—
146	C₄⋊C₈	—	166	2	no	—
147	C₄×C₈	—	166	2	no	—
148	C₄×C₈	—	166	2	no	—
149	C₄×C₈	Dic₁₆₆	1	2	yes	—
150	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
151	#95	—	83	2	no	—
152	C₂²×C₄	—	83	2	no	—
153	C₂²×C₄	—	83	2	no	—
154	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
155	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	166	2	no	—
160	C₂×C₈	—	166	2	no	—
161	C₂×C₈	—	166	2	no	—
162	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
163	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
165	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
166	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
167	C₂×C₈	#13050589	1	1	yes	characteristic
168	C₂×C₈	#13050591	1	1	yes	characteristic
169	C₄⋊C₄	—	83	4	no	—
170	C₄⋊C₄	—	166	2	no	—
171	C₄²	—	166	2	no	—
172	C₄²	—	166	2	no	—
173	C₄²	Dic₈₃⋊C₄	1	2	yes	—
174	C₄²	#13050573	1	2	yes	—
175	C₁₆	—	4	2	no	—
176	C₁₆	—	166	2	no	—
177	C₁₆	—	166	2	no	—
178	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	83	4	no	—
182	C₂×C₄	—	166	2	no	—
183	C₂×C₄	—	166	2	no	—
184	C₂×C₄	—	166	2	no	—
185	C₂×C₄	—	166	2	no	—
186	C₂×C₄	—	166	2	no	—
187	C₂×C₄	Dic₈₃⋊C₈	1	2	yes	—
188	C₂×C₄	Dic₈₃⋊C₈	1	2	yes	—
189	C₂×C₄	C₁₆×D₈₃	1	2	yes	—
190	C₂×C₄	#14581731	1	2	yes	—
191	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
192	C₂×C₄	#14581754	1	1	yes	characteristic
193	C₂×C₄	#14581757	1	1	yes	characteristic
194	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
195	C₂×C₄	#14581800	1	1	yes	characteristic
196	C₂×C₄	#14581800	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	—	332	2	no	—
199	C₈	#14581763	1	1	yes	characteristic
200	C₈	#14581768	1	1	yes	characteristic
201	C₈	#14581783	1	1	yes	characteristic
202	C₈	#14581791	1	1	yes	characteristic
203	C₂²	C₂×Dic₈₃⋊C₈	1	1	yes	characteristic
204	C₂²	#19127766	1	1	yes	characteristic
205	C₂²	#19127825	1	1	yes	characteristic
206	C₂²	C₂×C₁₆×D₈₃	1	1	yes	characteristic
207	C₂²	C₂×#14581731	1	1	yes	characteristic
208	C₂²	#19128119	1	1	yes	characteristic
209	C₂²	#19128119	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	332	2	no	—
212	C₄	—	166	4	no	—
213	C₄	Dic₈₃⋊C₁₆	1	4	yes	complemented
214	C₄	#19127597	1	1	yes	characteristic
215	C₄	#19127729	1	1	yes	characteristic
216	C₄	#19127778	1	1	yes	characteristic
217	C₄	#19127828	1	1	yes	characteristic
218	C₂	#24908179	1	1	yes	characteristic
219	C₂	C₂×Dic₈₃⋊C₁₆	1	1	yes	characteristic
220	C₂	#24909223	1	1	yes	characteristic
221	C₂	#24909282	1	1	yes	characteristic
222	C₂	#24909294	1	1	yes	characteristic
223	C₂	#24909365	1	1	yes	characteristic
224	C₂	#24909368	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂²×C₁₆
Fitting	C₂×C₄⋊C₁₆×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×C₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₃
hypocenter	C₈₃	#13159
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
45. C₄⋊Dic₈₃	C₁₆	semidirect
98. C₄×C₈₃	C₄⋊C₁₆	semidirect
118. C₈₃	#13159	semidirect
213. C₄	Dic₈₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₃⋊C₁₆	2
C₂×C₄×C₈₃⋊C₁₆	2
C₈×C₄⋊Dic₈₃	2
#24905540	2
C₂×C₄⋊C₁₆×C₈₃	2
#13159	83