Subgroups — #35319773

Order 21248.

318 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	#24909219	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127478	C₂	1	2	yes	maximal
3	#24907330	C₂	1	1	yes	maximal, characteristic
4	#24905542	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581732	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₈₃⋊C₈)	C₄	1	1	yes	characteristic
8	C₂×#14581702	C₂²	1	2	yes	—
9	C₄×C₄⋊Dic₈₃	C₄	1	1	yes	characteristic
10	C₄⋊(C₈₃⋊C₁₆)	C₄	1	2	yes	complemented
11	C₄⋊(C₈₃⋊C₁₆)	C₄	1	2	yes	complemented
12	#19127478	—	2	4	no	—
13	#19127455	C₄	1	2	yes	—
14	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#346×C₈₃	C₄	1	2	yes	complemented
18	C₄×C₁₆×C₈₃	C₄	1	2	yes	complemented
19	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₈₃	—	2	1	no	—
21	C₂×C₄×Dic₈₃	C₂×C₄	1	2	yes	—
22	#14581732	—	2	2	no	—
23	#14581732	—	2	2	no	—
24	C₂×C₈₃⋊C₁₆	C₂×C₄	1	4	yes	—
25	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
26	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
27	#14581702	—	2	2	no	—
28	#14581702	—	2	2	no	—
29	C₈₃⋊#95	—	2	1	no	—
30	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₈₃	—	2	1	no	—
34	M₃₂×C₈₃	—	2	1	no	—
35	M₃₂×C₈₃	D₄	1	2	yes	—
36	M₃₂×C₈₃	Q₈	1	2	yes	—
37	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₈₃	—	2	2	no	—
44	C₂×C₈₃⋊C₈	—	2	2	no	—
45	C₂×C₈₃⋊C₈	—	2	2	no	—
46	C₂×C₈₃⋊C₈	—	2	2	no	—
47	C₄⋊Dic₈₃	—	4	2	no	—
48	C₄×Dic₈₃	—	2	2	no	—
49	C₄×Dic₈₃	—	2	2	no	—
50	C₈₃⋊C₁₆	—	2	4	no	—
51	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₃	—	2	1	no	—
55	C₂×C₈×C₈₃	—	2	1	no	—
56	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
58	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
63	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
64	C₄²×C₈₃	C₄²	1	1	yes	characteristic
65	C₄²×C₈₃	C₄²	1	1	yes	characteristic
66	C₄²×C₈₃	M₁₆	1	1	yes	characteristic
67	C₄²×C₈₃	M₁₆	1	1	yes	characteristic
68	C₁₆×C₈₃	—	2	2	no	—
69	C₁₆×C₈₃	C₄⋊C₄	1	4	yes	complemented
70	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₈₃	—	2	2	no	—
72	C₂×C₄×C₈₃	—	4	1	no	—
73	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
74	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
75	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
76	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
82	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
83	C₂×Dic₈₃	—	4	2	no	—
84	C₂×Dic₈₃	—	4	2	no	—
85	C₂×Dic₈₃	—	4	2	no	—
86	C₈×C₈₃	—	2	1	no	—
87	C₈×C₈₃	—	2	1	no	—
88	C₈×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₈×C₈₃	#117	1	1	yes	characteristic
90	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
91	C₈×C₈₃	C₄×Q₈	1	1	yes	characteristic
92	C₈₃⋊C₈	—	4	2	no	—
93	C₂²×C₈₃	—	2	1	no	—
94	C₂²×C₈₃	—	2	1	no	—
95	C₂²×C₈₃	C₂×#97	1	1	yes	characteristic
96	C₂²×C₈₃	#436	1	1	yes	characteristic
97	C₂²×C₈₃	#446	1	1	yes	characteristic
98	C₄×C₈₃	—	2	1	no	—
99	C₄×C₈₃	—	2	1	no	—
100	C₄×C₈₃	—	2	1	no	—
101	C₄×C₈₃	#347	1	2	yes	complemented
102	C₄×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₄×C₈₃	#405	1	1	yes	characteristic
104	C₄×C₈₃	#436	1	1	yes	characteristic
105	C₄×C₈₃	#446	1	1	yes	characteristic
106	Dic₈₃	—	8	2	no	—
107	#12666	—	83	1	no	maximal, Sylow
108	C₂×C₈₃	—	2	1	no	—
109	C₂×C₈₃	—	2	1	no	—
110	C₂×C₈₃	#1770	1	1	yes	characteristic
111	C₂×C₈₃	C₂×#347	1	1	yes	characteristic
112	C₂×C₈₃	#2110	1	1	yes	characteristic
113	#2150	—	83	1	no	—
114	C₂×#347	—	83	2	no	—
115	C₄×M₃₂	D₈₃	1	1	yes	characteristic
116	#1770	—	83	1	no	—
117	#1375	—	83	2	no	—
118	C₈₃	#12666	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₃₂	—	83	2	no	—
120	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
121	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
122	C₂×C₄⋊C₈	—	83	1	no	—
123	C₂×#97	—	83	2	no	—
124	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
125	C₄×C₄⋊C₄	—	83	1	no	—
126	C₄⋊C₁₆	—	83	2	no	—
127	C₄⋊C₁₆	—	83	2	no	—
128	#347	—	166	4	no	—
129	#346	Dic₈₃	1	2	yes	complemented
130	C₄×C₁₆	Dic₈₃	1	2	yes	complemented
131	#336	—	83	2	no	—
132	C₂²×C₈	—	83	2	no	—
133	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	166	1	no	—
136	C₂×C₄²	—	83	2	no	—
137	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
138	M₃₂	—	2	1	no	—
139	M₃₂	—	2	1	no	—
140	M₃₂	—	166	2	no	—
141	M₃₂	—	166	2	no	—
142	M₃₂	Dic₁₆₆	1	2	yes	—
143	M₃₂	D₃₃₂	1	2	yes	—
144	C₂×C₁₆	—	83	4	no	—
145	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
146	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
147	C₄⋊C₈	—	166	1	no	—
148	C₄⋊C₈	—	166	1	no	—
149	#97	—	166	2	no	—
150	#97	—	166	2	no	—
151	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
152	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
153	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
154	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
155	#95	—	166	1	no	—
156	C₂²×C₄	—	166	2	no	—
157	C₂²×C₄	#13050573	1	1	yes	characteristic
158	C₂²×C₄	#13050573	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	166	2	no	—
163	C₂×C₈	—	166	2	no	—
164	C₂×C₈	—	166	2	no	—
165	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
166	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
167	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
168	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
169	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
170	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
171	C₂×C₈	#13050590	1	1	yes	characteristic
172	C₂×C₈	#13050590	1	1	yes	characteristic
173	C₄⋊C₄	—	332	2	no	—
174	C₄²	—	166	2	no	—
175	C₄²	—	166	2	no	—
176	C₄²	C₄×Dic₈₃	1	1	yes	characteristic
177	C₄²	C₄×Dic₈₃	1	1	yes	characteristic
178	C₄²	#13050573	1	1	yes	characteristic
179	C₄²	#13050573	1	1	yes	characteristic
180	C₁₆	—	2	2	no	—
181	C₁₆	—	166	4	no	—
182	C₁₆	C₄⋊Dic₈₃	1	4	yes	complemented
183	C₂³	C₂×#13050573	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	332	2	no	—
187	C₂×C₄	—	332	2	no	—
188	C₂×C₄	—	332	2	no	—
189	C₂×C₄	#14581702	1	1	yes	characteristic
190	C₂×C₄	#14581702	1	1	yes	characteristic
191	C₂×C₄	#14581702	1	1	yes	characteristic
192	C₂×C₄	#14581702	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
194	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
195	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
196	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
197	C₂×C₄	#14581798	1	1	yes	characteristic
198	C₂×C₄	#14581798	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	332	2	no	—
202	C₈	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
203	C₈	#14581751	1	1	yes	characteristic
204	C₈	C₄×D₃₃₂	1	1	yes	characteristic
205	C₈	C₄×Dic₁₆₆	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×#14581702	1	1	yes	characteristic
209	C₂²	#19127780	1	1	yes	characteristic
210	C₂²	#19127831	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	664	2	no	—
215	C₄	#19127478	1	2	yes	complemented
216	C₄	C₄×C₄⋊Dic₈₃	1	1	yes	characteristic
217	C₄	#19127656	1	1	yes	characteristic
218	C₄	#19127780	1	1	yes	characteristic
219	C₄	#19127831	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#24907330	1	1	yes	characteristic
223	C₂	C₂×#19127478	1	1	yes	characteristic
224	C₂	#24909067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₈₃
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₄×M₃₂×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#97
prime radical	C₂³×C₈₃	C₂×M₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₄×M₃₂	D₈₃
hypocenter	C₈₃	#12666
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. C₄⋊(C₈₃⋊C₁₆)	C₄	semidirect
11. C₄⋊(C₈₃⋊C₁₆)	C₄	semidirect
17. #346×C₈₃	C₄	semidirect
18. C₄×C₁₆×C₈₃	C₄	semidirect
69. C₁₆×C₈₃	C₄⋊C₄	semidirect
101. C₄×C₈₃	#347	semidirect
118. C₈₃	#12666	semidirect
129. #346	Dic₈₃	semidirect
130. C₄×C₁₆	Dic₈₃	semidirect
182. C₁₆	C₄⋊Dic₈₃	semidirect
215. C₄	#19127478	semidirect

Maximal subgroups

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

subgroup	index
#24909219	2
C₂×#19127478	2
#24907330	2
#24905542	2
C₄×M₃₂×C₈₃	2
#12666	83