Subgroups — #35557489

Order 21248.

590 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#24909271	C₂	1	4	yes	maximal, complemented
3	#24909268	C₂	1	4	yes	maximal, complemented
4	#24909218	C₂	1	2	yes	maximal, complemented
5	#24909073	C₂	1	2	yes	maximal
6	C₄×#14581732	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13050573	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₈₃	C₂²	1	1	yes	characteristic
10	C₂×#14581749	C₄	1	1	yes	characteristic
11	C₂×#14581732	C₂²	1	2	yes	—
12	#19127655	C₄	1	2	yes	—
13	#19127655	C₄	1	2	yes	—
14	#19127655	C₂²	1	2	yes	—
15	#19127655	C₂²	1	2	yes	—
16	C₂×#14581703	C₄	1	1	yes	characteristic
17	C₂×#14581702	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
19	Dic₈₃⋊C₁₆	C₂²	1	8	yes	complemented
20	(C₂×D₈₃)⋊C₁₆	C₂²	1	8	yes	—
21	#19127477	C₂²	1	2	yes	complemented
22	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	complemented
23	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
25	#346×C₈₃	C₂²	1	2	yes	complemented
26	C₄×C₁₆×C₈₃	C₂²	1	2	yes	complemented
27	C₂²×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#13050573	—	2	4	no	—
29	C₂×#13050573	C₂×C₄	1	2	yes	—
30	C₂×#13050573	C₂×C₄	1	2	yes	—
31	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₈₃	—	2	4	no	—
34	C₂×C₈×D₈₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₈₃	C₂³	1	2	yes	—
36	#14581749	C₈	1	4	yes	—
37	#14581749	C₂×C₄	1	2	yes	—
38	#14581749	C₂×C₄	1	2	yes	—
39	C₂×Dic₈₃⋊C₄	C₈	1	2	yes	—
40	C₂×(C₂×D₈₃)⋊C₄	C₈	1	2	yes	—
41	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
42	#14581732	—	2	4	no	—
43	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
44	#14581703	C₂×C₄	1	2	yes	—
45	#14581703	C₂×C₄	1	2	yes	—
46	#14581702	C₂×C₄	1	2	yes	—
47	#14581702	C₂×C₄	1	2	yes	—
48	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₈₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₈₃	—	2	4	no	—
54	C₂×C₁₆×C₈₃	C₂³	1	4	yes	—
55	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
59	C₂³×D₈₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₈₃	C₂×C₈	1	2	yes	—
62	C₂×C₄×D₈₃	—	2	2	no	—
63	C₂×C₄×D₈₃	—	2	2	no	—
64	C₂×C₄×D₈₃	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₈₃	C₂²×C₄	1	2	yes	—
66	#13050573	—	2	4	no	—
67	#13050573	—	2	4	no	—
68	C₂×C₈₃⋊C₈	—	2	2	no	—
69	C₂×C₈₃⋊C₈	—	2	2	no	—
70	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
72	C₈×D₈₃	—	2	4	no	—
73	C₈×D₈₃	—	2	4	no	—
74	Dic₈₃⋊C₄	C₂×C₈	1	4	yes	—
75	Dic₈₃⋊C₄	C₂×C₈	1	4	yes	—
76	(C₂×D₈₃)⋊C₄	C₂×C₈	1	4	yes	—
77	(C₂×D₈₃)⋊C₄	C₂×C₈	1	4	yes	—
78	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
79	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
80	C₈₃⋊C₁₆	—	2	4	no	—
81	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
82	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₈₃	M₁₆	1	4	yes	—
84	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₃	#41	1	4	yes	—
88	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
89	C₄²×C₈₃	C₂×C₈	1	2	yes	—
90	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
92	C₁₆×C₈₃	—	2	4	no	—
93	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
94	C₂²×D₈₃	—	2	2	no	—
95	C₂²×D₈₃	—	4	2	no	—
96	C₂²×D₈₃	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₈₃	—	4	2	no	—
98	C₂×C₄×C₈₃	#117	1	2	yes	—
99	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×Dic₈₃	—	2	2	no	—
105	C₂×Dic₈₃	—	4	2	no	—
106	C₂×Dic₈₃	C₂²×C₈	1	4	yes	—
107	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
108	C₄×D₈₃	—	2	2	no	—
109	C₄×D₈₃	—	2	2	no	—
110	C₄×D₈₃	—	2	4	no	—
111	C₈×C₈₃	#117	1	2	yes	—
112	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
113	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
114	C₈×C₈₃	C₂×#41	1	2	yes	—
115	C₈₃⋊C₈	—	2	2	no	—
116	C₈₃⋊C₈	—	2	2	no	—
117	C₂²×C₈₃	—	2	2	no	—
118	C₂²×C₈₃	#431	1	2	yes	—
119	C₂²×C₈₃	C₂³×C₈	1	1	yes	characteristic
120	C₂×D₈₃	—	2	4	no	—
121	C₂×D₈₃	—	2	2	no	—
122	C₂×D₈₃	—	4	4	no	—
123	C₄×C₈₃	—	2	2	no	—
124	C₄×C₈₃	—	2	2	no	—
125	C₄×C₈₃	#431	1	2	yes	—
126	C₄×C₈₃	C₂×#117	1	1	yes	characteristic
127	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
128	Dic₈₃	—	2	4	no	—
129	Dic₈₃	—	2	2	no	—
130	#33419	—	83	1	no	maximal, Sylow
131	C₂×C₈₃	—	2	2	no	—
132	C₂×C₈₃	C₂×#431	1	1	yes	characteristic
133	C₂×C₈₃	#3407	1	2	yes	—
134	D₈₃	—	4	2	no	—
135	C₂×#405	—	83	1	no	—
136	#2164	—	83	4	no	—
137	#2164	—	83	4	no	—
138	#2150	—	83	2	no	—
139	#2112	—	83	2	no	—
140	C₄×M₃₂	—	83	1	no	—
141	C₄×M₃₂	D₈₃	1	1	yes	characteristic, complemented
142	C₈₃	#33419	1	1	yes	characteristic, complemented, Sylow
143	C₂²×M₁₆	—	83	1	no	—
144	C₂³×C₈	—	83	1	no	—
145	C₂×#117	—	83	1	no	—
146	C₂×M₃₂	—	83	2	no	—
147	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
148	#405	—	83	2	no	—
149	#405	—	83	2	no	—
150	#405	—	83	2	no	—
151	#405	—	83	2	no	—
152	C₂×#97	—	83	1	no	—
153	C₂×#97	—	83	1	no	—
154	C₂×C₄×C₈	—	83	1	no	—
155	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
156	C₄⋊C₁₆	—	83	8	no	—
157	C₂²⋊C₁₆	—	83	8	no	—
158	#346	—	83	2	no	—
159	#346	C₂×D₈₃	1	2	yes	complemented
160	C₄×C₁₆	—	83	2	no	—
161	C₄×C₁₆	C₂×D₈₃	1	2	yes	complemented
162	C₂³×C₄	—	83	1	no	—
163	C₂×M₁₆	—	83	2	no	—
164	C₂×M₁₆	—	83	2	no	—
165	C₂×M₁₆	—	166	4	no	—
166	C₂²×C₈	—	83	1	no	—
167	C₂²×C₈	—	83	1	no	—
168	C₂²×C₈	—	83	2	no	—
169	C₂²×C₈	—	83	2	no	—
170	C₂²×C₈	—	166	4	no	—
171	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
173	#117	—	83	4	no	—
174	#117	—	83	2	no	—
175	#117	—	83	2	no	—
176	C₂×C₄⋊C₄	—	83	2	no	—
177	C₂×C₂²⋊C₄	—	83	2	no	—
178	C₂×C₄²	—	83	1	no	—
179	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
180	M₃₂	—	2	4	no	—
181	M₃₂	—	166	4	no	—
182	C₂×C₁₆	—	83	4	no	—
183	C₂×C₁₆	C₂²×D₈₃	1	4	yes	—
184	#97	—	83	2	no	—
185	#97	—	83	2	no	—
186	#97	—	83	2	no	—
187	#97	—	83	2	no	—
188	C₄×C₈	—	83	2	no	—
189	C₄×C₈	—	83	2	no	—
190	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
191	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
192	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
194	C₂⁴	—	83	1	no	—
195	C₂²×C₄	—	83	2	no	—
196	C₂²×C₄	—	83	1	no	—
197	C₂²×C₄	—	83	2	no	—
198	C₂²×C₄	—	83	2	no	—
199	C₂²×C₄	—	166	2	no	—
200	C₂²×C₄	—	166	2	no	—
201	C₂²×C₄	C₈×D₈₃	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
203	M₁₆	—	166	4	no	—
204	M₁₆	—	166	4	no	—
205	C₂×C₈	—	83	2	no	—
206	C₂×C₈	—	83	2	no	—
207	C₂×C₈	—	166	2	no	—
208	C₂×C₈	—	166	2	no	—
209	C₂×C₈	—	166	4	no	—
210	C₂×C₈	—	166	4	no	—
211	C₂×C₈	#13050573	1	4	yes	—
212	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
214	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
215	C₂×C₈	#13050590	1	4	yes	—
216	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
217	C₄⋊C₄	—	83	4	no	—
218	C₄⋊C₄	—	83	4	no	—
219	C₂²⋊C₄	—	83	4	no	—
220	C₂²⋊C₄	—	83	4	no	—
221	C₄²	—	83	2	no	—
222	C₄²	—	83	2	no	—
223	C₄²	C₈×D₈₃	1	2	yes	—
224	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
225	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
226	C₁₆	—	2	4	no	—
227	C₁₆	—	166	4	no	—
228	C₂³	—	332	2	no	—
229	C₂³	—	83	2	no	—
230	C₂³	—	166	2	no	—
231	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	332	2	no	—
234	C₂×C₄	—	83	4	no	—
235	C₂×C₄	—	83	2	no	—
236	C₂×C₄	—	166	2	no	—
237	C₂×C₄	—	166	2	no	—
238	C₂×C₄	—	166	2	no	—
239	C₂×C₄	—	166	4	no	—
240	C₂×C₄	#14581749	1	2	yes	—
241	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
242	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
243	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#13050573	1	2	yes	—
245	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
246	C₈	—	166	2	no	—
247	C₈	—	166	2	no	—
248	C₈	#14581749	1	2	yes	—
249	C₈	C₂×#13050573	1	2	yes	—
250	C₈	C₂×#13050573	1	2	yes	—
251	C₈	C₂×#13050590	1	2	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	332	4	no	—
254	C₂²	—	166	2	no	—
255	C₂²	—	166	4	no	—
256	C₂²	#19127752	1	2	yes	—
257	C₂²	C₂²×C₈×D₈₃	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	166	2	no	—
261	C₄	—	166	4	no	—
262	C₄	#19127752	1	2	yes	—
263	C₄	C₂×#14581749	1	1	yes	characteristic
264	C₄	C₂²×#13050573	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	332	2	no	—
267	C₂	C₂×#19127752	1	1	yes	characteristic
268	C₂	#24920565	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₈
Fitting	C₄×M₃₂×C₈₃	C₂
prime radical	C₂³×D₈₃	C₂×C₈
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₄×M₃₂	D₈₃
hypocenter	C₈₃	#33419
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #24909271	C₂	semidirect
3. #24909268	C₂	semidirect
4. #24909218	C₂	semidirect
6. C₄×#14581732	C₂	semidirect
7. C₄×M₃₂×C₈₃	C₂	semidirect
19. Dic₈₃⋊C₁₆	C₂²	semidirect
21. #19127477	C₂²	semidirect
22. C₄×C₈₃⋊C₁₆	C₂²	semidirect
25. #346×C₈₃	C₂²	semidirect
26. C₄×C₁₆×C₈₃	C₂²	semidirect
141. C₄×M₃₂	D₈₃	semidirect
142. C₈₃	#33419	semidirect
159. #346	C₂×D₈₃	semidirect
161. 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
#24909271	2
#24909268	2
#24909218	2
#24909073	2
C₄×#14581732	2
C₄×M₃₂×C₈₃	2
#33419	83