Subgroups — #35557490

Order 21248.

590 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127655	C₂	1	1	yes	maximal, characteristic
2	#24909269	C₂	1	8	yes	maximal, complemented
3	#24909219	C₂	1	1	yes	maximal, characteristic, complemented
4	#24909073	C₂	1	2	yes	maximal
5	M₃₂×Dic₈₃	C₂	1	2	yes	maximal, complemented
6	C₈₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#13050573	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₈₃	C₂²	1	1	yes	characteristic
9	C₂×#14581749	C₄	1	1	yes	characteristic
10	C₂×#14581732	C₂²	1	2	yes	—
11	#19127655	C₄	1	4	yes	—
12	#19127655	C₂²	1	4	yes	—
13	C₂×#14581703	C₄	1	1	yes	characteristic
14	C₂×#14581702	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
16	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	4	yes	complemented
17	(C₂×D₈₃)⋊C₁₆	C₂²	1	8	yes	—
18	#19127476	C₂²	1	4	yes	complemented
19	C₁₆×Dic₈₃	C₂²	1	4	yes	complemented
20	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₈₃	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#13050573	—	2	4	no	—
25	C₂×#13050573	C₂×C₄	1	4	yes	—
26	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₈₃	—	2	2	no	—
29	C₂×C₈×D₈₃	—	2	2	no	—
30	C₂×C₈×D₈₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₈₃	C₂³	1	2	yes	—
32	#14581749	C₈	1	4	yes	—
33	#14581749	C₂×C₄	1	4	yes	—
34	C₂×Dic₈₃⋊C₄	C₈	1	2	yes	—
35	C₂×(C₂×D₈₃)⋊C₄	C₈	1	2	yes	—
36	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
37	#14581732	—	2	4	no	—
38	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
39	#14581703	C₂×C₄	1	4	yes	—
40	#14581702	C₂×C₄	1	4	yes	—
41	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
42	C₈×Dic₈₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
46	M₃₂×C₈₃	—	2	4	no	—
47	C₂×C₁₆×C₈₃	C₂³	1	4	yes	—
48	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₈₃	C₂³	1	2	yes	—
50	C₂³×D₈₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×Dic₈₃	C₂×C₈	1	2	yes	—
53	C₂×C₄×D₈₃	—	2	2	no	—
54	C₂×C₄×D₈₃	—	2	2	no	—
55	C₂×C₄×D₈₃	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₈₃	C₂²×C₄	1	2	yes	—
57	#13050573	—	2	8	no	—
58	C₂×C₈₃⋊C₈	—	2	2	no	—
59	C₂×C₈₃⋊C₈	M₁₆	1	4	yes	—
60	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
61	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
62	C₈×D₈₃	—	2	4	no	—
63	C₈×D₈₃	—	2	2	no	—
64	C₈×D₈₃	—	2	2	no	—
65	Dic₈₃⋊C₄	C₂×C₈	1	8	yes	—
66	(C₂×D₈₃)⋊C₄	C₂×C₈	1	8	yes	—
67	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
68	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
69	C₈₃⋊C₁₆	—	2	4	no	—
70	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₈₃	—	2	2	no	—
73	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₈₃	#41	1	2	yes	—
76	C₂×C₈×C₈₃	#41	1	2	yes	—
77	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
78	C₄²×C₈₃	C₂×C₈	1	2	yes	—
79	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
80	C₁₆×C₈₃	—	2	4	no	—
81	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₈₃	—	2	2	no	—
83	C₂²×D₈₃	—	4	2	no	—
84	C₂²×D₈₃	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₈₃	—	4	2	no	—
86	C₂×C₄×C₈₃	#117	1	2	yes	—
87	C₂×C₄×C₈₃	C₂²×C₈	1	4	yes	—
88	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
90	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
91	C₂×Dic₈₃	—	2	2	no	—
92	C₂×Dic₈₃	—	4	2	no	—
93	C₂×Dic₈₃	C₂²×C₈	1	4	yes	—
94	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
95	C₄×D₈₃	—	2	2	no	—
96	C₄×D₈₃	—	2	2	no	—
97	C₄×D₈₃	—	2	4	no	—
98	C₈×C₈₃	—	2	2	no	—
99	C₈×C₈₃	#117	1	2	yes	—
100	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
101	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
102	C₈₃⋊C₈	—	2	2	no	—
103	C₈₃⋊C₈	C₂×M₁₆	1	4	yes	—
104	C₂²×C₈₃	—	2	2	no	—
105	C₂²×C₈₃	#431	1	2	yes	—
106	C₂²×C₈₃	C₂³×C₈	1	1	yes	characteristic
107	C₂×D₈₃	—	2	4	no	—
108	C₂×D₈₃	—	2	2	no	—
109	C₂×D₈₃	—	4	4	no	—
110	C₄×C₈₃	—	2	4	no	—
111	C₄×C₈₃	#431	1	2	yes	—
112	C₄×C₈₃	C₂×#117	1	1	yes	characteristic
113	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
114	Dic₈₃	—	2	4	no	—
115	Dic₈₃	—	2	2	no	—
116	#33419	—	83	1	no	maximal, Sylow
117	C₂×C₈₃	—	2	2	no	—
118	C₂×C₈₃	C₂×#431	1	1	yes	characteristic
119	C₂×C₈₃	#3407	1	2	yes	—
120	D₈₃	—	4	2	no	—
121	C₂×#405	—	83	1	no	—
122	#2164	—	83	8	no	—
123	#2150	—	83	1	no	—
124	#2150	D₈₃	1	1	yes	characteristic, complemented
125	#2112	—	83	2	no	—
126	C₄×M₃₂	—	83	2	no	—
127	C₈₃	#33419	1	1	yes	characteristic, complemented, Sylow
128	C₂²×M₁₆	—	83	1	no	—
129	C₂³×C₈	—	83	1	no	—
130	C₂×#117	—	83	1	no	—
131	C₂×M₃₂	—	83	2	no	—
132	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
133	#405	—	83	4	no	—
134	#405	—	83	4	no	—
135	C₂×#97	—	83	1	no	—
136	C₂×#97	—	83	1	no	—
137	C₂×C₄×C₈	—	83	1	no	—
138	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
139	C₄⋊C₁₆	—	83	4	no	—
140	C₄⋊C₁₆	C₂×D₈₃	1	4	yes	complemented
141	C₂²⋊C₁₆	—	83	8	no	—
142	#346	—	83	4	no	—
143	C₄×C₁₆	—	83	4	no	—
144	C₂³×C₄	—	83	1	no	—
145	C₂×M₁₆	—	83	4	no	—
146	C₂×M₁₆	—	166	4	no	—
147	C₂²×C₈	—	83	1	no	—
148	C₂²×C₈	—	83	1	no	—
149	C₂²×C₈	—	83	2	no	—
150	C₂²×C₈	—	83	2	no	—
151	C₂²×C₈	—	166	2	no	—
152	C₂²×C₈	—	166	2	no	—
153	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
155	#117	—	83	4	no	—
156	#117	—	83	4	no	—
157	C₂×C₄⋊C₄	—	83	2	no	—
158	C₂×C₂²⋊C₄	—	83	2	no	—
159	C₂×C₄²	—	83	1	no	—
160	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
161	M₃₂	—	2	4	no	—
162	M₃₂	—	166	4	no	—
163	C₂×C₁₆	—	83	4	no	—
164	C₂×C₁₆	C₂²×D₈₃	1	4	yes	—
165	#97	—	83	4	no	—
166	#97	—	83	4	no	—
167	C₄×C₈	—	83	2	no	—
168	C₄×C₈	—	83	2	no	—
169	C₄×C₈	C₄×D₈₃	1	2	yes	—
170	C₄×C₈	C₂²×D₈₃	1	2	yes	—
171	C₂⁴	—	83	1	no	—
172	C₂²×C₄	—	83	2	no	—
173	C₂²×C₄	—	83	1	no	—
174	C₂²×C₄	—	83	2	no	—
175	C₂²×C₄	—	83	2	no	—
176	C₂²×C₄	—	166	2	no	—
177	C₂²×C₄	—	166	2	no	—
178	C₂²×C₄	C₈×D₈₃	1	2	yes	—
179	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
180	M₁₆	—	166	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	83	4	no	—
183	C₂×C₈	—	83	2	no	—
184	C₂×C₈	—	83	2	no	—
185	C₂×C₈	—	166	2	no	—
186	C₂×C₈	—	166	2	no	—
187	C₂×C₈	—	166	2	no	—
188	C₂×C₈	—	166	4	no	—
189	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
190	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
191	C₂×C₈	#13050589	1	2	yes	—
192	C₂×C₈	#13050591	1	2	yes	—
193	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
194	C₄⋊C₄	—	83	8	no	—
195	C₂²⋊C₄	—	83	8	no	—
196	C₄²	—	83	2	no	—
197	C₄²	—	83	2	no	—
198	C₄²	C₈×D₈₃	1	2	yes	—
199	C₄²	C₂×C₄×D₈₃	1	2	yes	—
200	C₁₆	—	2	4	no	—
201	C₁₆	—	166	4	no	—
202	C₂³	—	332	2	no	—
203	C₂³	—	83	2	no	—
204	C₂³	—	166	2	no	—
205	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	332	2	no	—
208	C₂×C₄	—	83	4	no	—
209	C₂×C₄	—	83	2	no	—
210	C₂×C₄	—	166	2	no	—
211	C₂×C₄	—	166	2	no	—
212	C₂×C₄	—	166	2	no	—
213	C₂×C₄	—	166	4	no	—
214	C₂×C₄	#14581750	1	2	yes	—
215	C₂×C₄	C₂×C₈×D₈₃	1	4	yes	—
216	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
217	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
218	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	83	4	no	—
221	C₈	—	166	2	no	—
222	C₈	#14581750	1	2	yes	—
223	C₈	C₂×#13050589	1	1	yes	characteristic
224	C₈	C₂×#13050591	1	1	yes	characteristic
225	C₂²	—	2	2	no	—
226	C₂²	—	332	4	no	—
227	C₂²	—	166	2	no	—
228	C₂²	—	166	4	no	—
229	C₂²	#19127753	1	2	yes	—
230	C₂²	C₂²×C₈×D₈₃	1	1	yes	characteristic
231	C₄	—	2	4	no	—
232	C₄	—	166	2	no	—
233	C₄	—	166	4	no	—
234	C₄	#19127753	1	2	yes	—
235	C₄	C₂×#14581750	1	1	yes	characteristic
236	C₄	C₂×M₁₆×D₈₃	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	—	332	2	no	—
239	C₂	C₂×#19127753	1	1	yes	characteristic
240	C₂	#24920565	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₈
Fitting	C₈₃×#2150	C₂
prime radical	C₂³×D₈₃	C₂×C₈
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	#2150	D₈₃
hypocenter	C₈₃	#33419
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #24909269	C₂	semidirect
3. #24909219	C₂	semidirect
5. M₃₂×Dic₈₃	C₂	semidirect
6. C₈₃×#2150	C₂	semidirect
16. C₄⋊(C₈₃⋊C₁₆)	C₂²	semidirect
18. #19127476	C₂²	semidirect
19. C₁₆×Dic₈₃	C₂²	semidirect
22. C₄⋊C₁₆×C₈₃	C₂²	semidirect
124. #2150	D₈₃	semidirect
127. C₈₃	#33419	semidirect
140. C₄⋊C₁₆	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127655	2
#24909269	2
#24909219	2
#24909073	2
M₃₂×Dic₈₃	2
C₈₃×#2150	2
#33419	83