Subgroups — #35557504

Order 21248.

582 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24909270	C₂	1	4	yes	maximal, complemented
2	#24909266	C₂	1	4	yes	maximal, complemented
3	#24909222	C₂	1	2	yes	maximal, complemented
4	#24909077	C₂	1	2	yes	maximal
5	C₄×#14581732	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19127477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#131×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14581732	C₂²	1	2	yes	—
9	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
10	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	8	yes	—
11	C₂²⋊(C₈₃⋊C₁₆)	C₂²	1	8	yes	—
12	#19127477	C₂²	1	2	yes	—
13	#19127477	C₂²	1	2	yes	—
14	#19127477	C₂²	1	2	yes	—
15	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	—
16	C₂×#131×C₈₃	C₄	1	1	yes	characteristic
17	C₂×#131×C₈₃	C₂²	1	1	yes	characteristic
18	C₄×#41×C₈₃	C₄	1	1	yes	characteristic
19	#405×C₈₃	C₄	1	2	yes	—
20	#405×C₈₃	C₄	1	1	yes	characteristic
21	#405×C₈₃	C₂²	1	1	yes	characteristic
22	#405×C₈₃	C₂²	1	2	yes	—
23	C₄×M₁₆×C₈₃	C₄	1	1	yes	characteristic
24	C₄×M₁₆×C₈₃	C₄	1	2	yes	—
25	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
27	#14581732	—	2	4	no	—
28	C₂×C₈₃⋊C₁₆	—	2	4	no	—
29	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
30	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
31	C₂×#41×C₈₃	C₂×C₄	1	1	yes	characteristic
32	#131×C₈₃	—	2	4	no	—
33	#131×C₈₃	—	2	4	no	—
34	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₈₃	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₈₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
42	C₄×Q₈×C₈₃	C₈	1	2	yes	—
43	C₄×D₄×C₈₃	C₈	1	4	yes	—
44	C₄×D₄×C₈₃	C₈	1	2	yes	—
45	#117×C₈₃	C₂×C₄	1	2	yes	—
46	#117×C₈₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
49	#97×C₈₃	C₂×C₄	1	2	yes	—
50	#97×C₈₃	C₂×C₄	1	2	yes	—
51	#97×C₈₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₈₃	C₂³	1	2	yes	—
59	C₈₃⋊C₁₆	—	2	4	no	—
60	C₈₃⋊C₁₆	—	2	4	no	—
61	#41×C₈₃	—	2	2	no	—
62	#41×C₈₃	—	2	2	no	—
63	C₂×Q₈×C₈₃	C₂×C₈	1	1	yes	characteristic
64	C₂×D₄×C₈₃	C₂×C₈	1	2	yes	—
65	C₂×D₄×C₈₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₈₃	C₂×C₈	1	4	yes	—
67	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
70	M₁₆×C₈₃	—	2	2	no	—
71	M₁₆×C₈₃	—	2	2	no	—
72	M₁₆×C₈₃	—	2	4	no	—
73	M₁₆×C₈₃	—	2	4	no	—
74	C₂×C₈×C₈₃	—	2	4	no	—
75	C₂×C₈×C₈₃	—	2	4	no	—
76	C₂×C₈×C₈₃	M₁₆	1	4	yes	—
77	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₈₃	#41	1	4	yes	—
83	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₈₃	C₂×C₈	1	4	yes	—
85	C₄⋊C₄×C₈₃	C₂×C₈	1	2	yes	—
86	C₂²⋊C₄×C₈₃	C₂×C₈	1	4	yes	—
87	C₂²⋊C₄×C₈₃	C₂×C₈	1	2	yes	—
88	C₄²×C₈₃	C₂×C₈	1	4	yes	—
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	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
93	C₂³×C₈₃	C₂²×C₈	1	2	yes	—
94	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
95	Q₈×C₈₃	—	2	2	no	—
96	D₄×C₈₃	—	2	4	no	—
97	D₄×C₈₃	—	2	2	no	—
98	C₂×C₄×C₈₃	—	2	2	no	—
99	C₂×C₄×C₈₃	—	2	2	no	—
100	C₂×C₄×C₈₃	—	4	2	no	—
101	C₂×C₄×C₈₃	—	4	4	no	—
102	C₂×C₄×C₈₃	#117	1	2	yes	—
103	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₈₃	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₈₃	—	2	2	no	—
111	C₈×C₈₃	—	2	2	no	—
112	C₈×C₈₃	#117	1	2	yes	—
113	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
114	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
115	C₈×C₈₃	C₂×#41	1	2	yes	—
116	C₂²×C₈₃	—	2	2	no	—
117	C₂²×C₈₃	—	2	4	no	—
118	C₂²×C₈₃	—	2	2	no	—
119	C₂²×C₈₃	#431	1	2	yes	—
120	C₂²×C₈₃	C₂³×C₈	1	1	yes	characteristic
121	C₄×C₈₃	—	2	2	no	—
122	C₄×C₈₃	—	2	2	no	—
123	C₄×C₈₃	—	2	4	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	#33421	—	83	1	no	maximal, Sylow
129	C₂×C₈₃	—	2	2	no	—
130	C₂×C₈₃	—	4	2	no	—
131	C₂×C₈₃	C₂×#431	1	1	yes	characteristic
132	C₂×C₈₃	#3407	1	2	yes	—
133	C₄×#131	D₈₃	1	1	yes	characteristic
134	#2164	—	83	4	no	—
135	#2163	—	83	4	no	—
136	#2151	—	83	2	no	—
137	#2113	—	83	2	no	—
138	C₄×M₃₂	—	83	1	no	—
139	C₂×#346	—	83	1	no	—
140	C₈₃	#33421	1	1	yes	characteristic, complemented, Sylow
141	C₂×#131	Dic₈₃	1	1	yes	characteristic
142	C₂×#131	C₂×D₈₃	1	1	yes	characteristic
143	C₄×#41	Dic₈₃	1	1	yes	characteristic
144	C₂×M₃₂	—	83	2	no	—
145	C₂²×C₁₆	—	83	2	no	—
146	#405	Dic₈₃	1	2	yes	—
147	#405	Dic₈₃	1	1	yes	characteristic
148	#405	C₂×D₈₃	1	1	yes	characteristic
149	#405	C₂×D₈₃	1	2	yes	—
150	C₄×M₁₆	Dic₈₃	1	1	yes	characteristic
151	C₄×M₁₆	Dic₈₃	1	2	yes	—
152	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₈₃	1	2	yes	—
154	C₄⋊C₁₆	—	83	8	no	—
155	C₂²⋊C₁₆	—	83	8	no	—
156	#346	—	83	2	no	—
157	#346	—	83	2	no	—
158	#346	—	83	2	no	—
159	C₄×C₁₆	—	83	2	no	—
160	C₂×#41	C₂×Dic₈₃	1	1	yes	characteristic
161	#131	—	2	4	no	—
162	#131	—	2	4	no	—
163	C₂×M₁₆	C₂×Dic₈₃	1	1	yes	characteristic
164	C₂×M₁₆	C₂×Dic₈₃	1	2	yes	—
165	C₂×M₁₆	C₂×Dic₈₃	1	1	yes	characteristic
166	C₂×M₁₆	C₂×Dic₈₃	1	2	yes	—
167	C₂²×C₈	C₂×Dic₈₃	1	2	yes	—
168	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₈₃	1	2	yes	—
170	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
171	C₄×Q₈	C₈₃⋊C₈	1	2	yes	—
172	C₄×D₄	C₈₃⋊C₈	1	4	yes	—
173	C₄×D₄	C₈₃⋊C₈	1	2	yes	—
174	#117	C₂×Dic₈₃	1	2	yes	—
175	#117	C₂×Dic₈₃	1	1	yes	characteristic
176	C₂×C₄²	C₂×Dic₈₃	1	2	yes	—
177	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
178	M₃₂	—	166	4	no	—
179	C₂×C₁₆	—	83	4	no	—
180	C₂×C₁₆	—	83	4	no	—
181	C₂×C₁₆	—	166	4	no	—
182	#97	C₂×Dic₈₃	1	2	yes	—
183	#97	C₂×Dic₈₃	1	2	yes	—
184	#97	C₂×Dic₈₃	1	2	yes	—
185	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
186	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
187	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
188	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₈₃	1	2	yes	—
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	C₂×Q₈	C₂×C₈₃⋊C₈	1	1	yes	characteristic
195	C₂×D₄	C₂×C₈₃⋊C₈	1	2	yes	—
196	C₂×D₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₈₃⋊C₈	1	4	yes	—
198	C₂²×C₄	C₂×C₈₃⋊C₈	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₈₃	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	#13050574	1	4	yes	—
208	C₂×C₈	C₂²×Dic₈₃	1	2	yes	—
209	C₂×C₈	C₂²×Dic₈₃	1	2	yes	—
210	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
211	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
212	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
213	C₂×C₈	#13050590	1	4	yes	—
214	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
215	C₄⋊C₄	C₂×C₈₃⋊C₈	1	4	yes	—
216	C₄⋊C₄	C₂×C₈₃⋊C₈	1	2	yes	—
217	C₂²⋊C₄	C₂×C₈₃⋊C₈	1	4	yes	—
218	C₂²⋊C₄	C₂×C₈₃⋊C₈	1	2	yes	—
219	C₄²	C₂×C₈₃⋊C₈	1	4	yes	—
220	C₄²	C₂×C₈₃⋊C₈	1	2	yes	—
221	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
222	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
223	C₄²	C₂²×Dic₈₃	1	2	yes	—
224	C₁₆	—	166	4	no	—
225	C₁₆	—	166	4	no	—
226	C₂³	C₂²×C₈₃⋊C₈	1	2	yes	—
227	C₂³	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	#14581751	1	2	yes	—
236	C₂×C₄	C₂²×C₈₃⋊C₈	1	2	yes	—
237	C₂×C₄	C₂²×C₈₃⋊C₈	1	2	yes	—
238	C₂×C₄	C₂²×C₈₃⋊C₈	1	4	yes	—
239	C₂×C₄	C₂²×C₈₃⋊C₈	1	2	yes	—
240	C₂×C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
241	C₂×C₄	C₂×#13050574	1	2	yes	—
242	C₂×C₄	C₂³×Dic₈₃	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	#14581751	1	2	yes	—
246	C₈	C₂×#13050574	1	2	yes	—
247	C₈	C₂×#13050574	1	2	yes	—
248	C₈	C₂×#13050590	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	#19127754	1	2	yes	—
253	C₂²	C₂³×C₈₃⋊C₈	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	4	no	—
258	C₄	#19127754	1	2	yes	—
259	C₄	C₂×#14581751	1	1	yes	characteristic
260	C₄	C₂²×#13050574	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	C₂×#19127754	1	1	yes	characteristic
264	C₂	#24920566	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₈
Fitting	C₄×#131×C₈₃	C₂
prime radical	C₂×#41×C₈₃	C₂×C₄
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₄×#131	D₈₃
hypocenter	C₈₃	#33421
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #24909270	C₂	semidirect
2. #24909266	C₂	semidirect
3. #24909222	C₂	semidirect
5. C₄×#14581732	C₂	semidirect
6. C₂×#19127477	C₂	semidirect
140. C₈₃	#33421	semidirect

Maximal subgroups

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

subgroup	index
#24909270	2
#24909266	2
#24909222	2
#24909077	2
C₄×#14581732	2
C₂×#19127477	2
C₄×#131×C₈₃	2
#33421	83