Subgroups — #35319513

Order 21248.

418 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128119	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127484	C₂	1	1	yes	maximal, characteristic
3	#24909060	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19127476	C₂	1	1	yes	maximal, characteristic, complemented
5	#24907302	C₂	1	1	yes	maximal, characteristic
6	#24905544	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581798	C₂²	1	1	yes	characteristic
9	#19128119	—	2	2	no	—
10	#19128119	C₄	1	4	yes	complemented
11	C₂×#14581732	C₂²	1	1	yes	characteristic
12	C₂×#14581731	—	2	1	no	—
13	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₈₃	—	2	1	no	—
15	C₂×#14581718	C₄	1	1	yes	characteristic
16	#19127695	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
18	#19127484	C₄	1	4	yes	complemented
19	#19127477	—	2	1	no	—
20	#19127476	—	2	2	no	—
21	C₄×C₈₃⋊C₁₆	—	2	1	no	—
22	#19127465	C₄	1	1	yes	characteristic
23	#19127460	C₄	1	1	yes	characteristic
24	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
26	#405×C₈₃	C₂²	1	1	yes	characteristic
27	C₂×#13050590	C₂×C₄	1	1	yes	characteristic
28	#14581798	—	2	1	no	—
29	#14581798	—	2	1	no	—
30	#14581798	C₂×C₄	1	2	yes	—
31	#14581798	C₂×C₄	1	2	yes	—
32	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
33	C₂×#13050573	—	2	1	no	—
34	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₈₃	—	2	1	no	—
36	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
37	#14581732	—	2	1	no	—
38	#14581732	C₂×C₄	1	2	yes	—
39	#14581731	—	2	4	no	—
40	C₂×C₈₃⋊C₁₆	—	2	1	no	—
41	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
42	C₂×C₈₃⋊C₁₆	D₄	1	1	yes	characteristic
43	C₂×C₈₃⋊C₁₆	D₄	1	1	yes	characteristic
44	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
45	C₁₆×D₈₃	—	2	4	no	—
46	#14581718	C₈	1	4	yes	—
47	#14581715	C₈	1	2	yes	—
48	#14581712	C₈	1	2	yes	—
49	C₈×Dic₈₃	—	2	1	no	—
50	C₈×Dic₈₃	—	2	1	no	—
51	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
53	#117×C₈₃	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₈₃	—	2	1	no	—
55	M₃₂×C₈₃	C₂×C₄	1	2	yes	—
56	C₂×C₁₆×C₈₃	—	2	1	no	—
57	C₂×C₁₆×C₈₃	—	2	1	no	—
58	C₂×C₁₆×C₈₃	—	2	1	no	—
59	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
60	#97×C₈₃	—	2	1	no	—
61	C₄×C₈×C₈₃	—	2	1	no	—
62	#13050590	—	2	1	no	—
63	#13050590	—	2	1	no	—
64	#13050590	C₄²	1	2	yes	—
65	#13050590	C₂×C₈	1	2	yes	—
66	C₂×Dic₁₆₆	C₂×C₈	1	1	yes	characteristic
67	C₂×D₃₃₂	C₂×C₈	1	1	yes	characteristic
68	C₂×C₂²⋊D₈₃	—	2	1	no	—
69	C₂²×Dic₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	#13050574	—	2	1	no	—
72	#13050574	C₄²	1	2	yes	—
73	#13050573	—	2	2	no	—
74	#13050573	—	2	2	no	—
75	C₂×C₈₃⋊C₈	—	2	1	no	—
76	C₂×C₈₃⋊C₈	—	2	1	no	—
77	C₂×C₈₃⋊C₈	—	2	1	no	—
78	C₂×C₈₃⋊C₈	C₂×C₈	1	2	yes	—
79	C₈×D₈₃	—	2	2	no	—
80	C₈×D₈₃	—	2	2	no	—
81	C₄×Dic₈₃	—	2	1	no	—
82	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
83	C₈₃⋊C₁₆	—	2	2	no	—
84	C₈₃⋊C₁₆	C₂²⋊C₄	1	2	yes	—
85	C₈₃⋊C₁₆	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
87	M₁₆×C₈₃	—	2	1	no	—
88	M₁₆×C₈₃	C₂×C₈	1	2	yes	—
89	C₂×C₈×C₈₃	—	2	1	no	—
90	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
96	C₄⋊C₄×C₈₃	C₂×C₈	1	2	yes	—
97	C₂²⋊C₄×C₈₃	—	2	1	no	—
98	C₄²×C₈₃	—	2	1	no	—
99	C₁₆×C₈₃	—	2	2	no	—
100	C₁₆×C₈₃	—	2	2	no	—
101	C₂³×C₈₃	#131	1	1	yes	characteristic
102	C₂²×D₈₃	—	2	1	no	—
103	C₂×C₄×C₈₃	—	2	2	no	—
104	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
110	C₂²⋊D₈₃	—	2	2	no	—
111	C₂²⋊D₈₃	—	2	2	no	—
112	C₂×Dic₈₃	—	2	1	no	—
113	C₂×Dic₈₃	—	2	2	no	—
114	C₂×Dic₈₃	—	4	2	no	—
115	D₃₃₂	—	2	1	no	—
116	D₃₃₂	C₄×C₈	1	2	yes	—
117	C₄×D₈₃	—	2	2	no	—
118	C₄×D₈₃	—	2	2	no	—
119	Dic₁₆₆	—	2	1	no	—
120	Dic₁₆₆	C₄×C₈	1	2	yes	—
121	C₈×C₈₃	—	2	2	no	—
122	C₈×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₈×C₈₃	#117	1	1	yes	characteristic
124	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
125	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
126	C₈₃⋊C₈	—	2	2	no	—
127	C₈₃⋊C₈	—	2	2	no	—
128	C₂²×C₈₃	—	2	2	no	—
129	C₂²×C₈₃	#405	1	1	yes	characteristic
130	C₂²×C₈₃	C₈×D₄	1	1	yes	characteristic
131	C₂²×C₈₃	C₈×D₄	1	1	yes	characteristic
132	C₂×D₈₃	—	2	2	no	—
133	C₂×D₈₃	—	4	2	no	—
134	C₄×C₈₃	—	4	2	no	—
135	C₄×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
137	C₄×C₈₃	C₈×D₄	1	1	yes	characteristic
138	C₄×C₈₃	C₈×D₄	1	1	yes	characteristic
139	Dic₈₃	—	2	2	no	—
140	Dic₈₃	—	4	2	no	—
141	#12615	—	83	1	no	maximal, Sylow
142	C₂×C₈₃	—	2	1	no	—
143	C₂×C₈₃	—	2	1	no	—
144	C₂×C₈₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₈₃	#2171	1	2	yes	—
146	D₈₃	—	4	2	no	—
147	C₂×#504	—	83	1	no	—
148	C₂×#350	—	83	1	no	—
149	#2108	—	83	1	no	—
150	C₂×#346	—	83	1	no	—
151	#1764	—	83	1	no	—
152	#1379	D₈₃	1	1	yes	characteristic, complemented
153	#1376	—	83	1	no	—
154	C₈₃	#12615	1	1	yes	characteristic, complemented, Sylow
155	C₂×#131	—	83	1	no	—
156	#504	—	83	4	no	—
157	#504	—	166	2	no	—
158	C₂×M₃₂	—	83	1	no	—
159	C₂×M₃₂	—	166	1	no	—
160	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
161	C₂²×C₁₆	—	83	1	no	—
162	C₂²×C₁₆	—	166	1	no	—
163	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
164	C₂×#104	—	83	1	no	—
165	#416	—	83	1	no	—
166	#405	C₂×D₈₃	1	1	yes	characteristic
167	C₂×C₄×C₈	—	83	1	no	—
168	#350	—	83	4	no	—
169	#346	—	166	1	no	—
170	#346	—	166	2	no	—
171	C₄×C₁₆	—	166	1	no	—
172	#341	—	83	1	no	—
173	#339	—	83	1	no	—
174	C₂×#41	—	83	1	no	—
175	#131	—	83	2	no	—
176	#131	—	83	2	no	—
177	#131	—	166	1	no	—
178	#131	—	166	1	no	—
179	C₂×M₁₆	—	83	1	no	—
180	C₂×M₁₆	—	166	1	no	—
181	C₂×M₁₆	C₄×D₈₃	1	1	yes	characteristic
182	C₂²×C₈	—	83	1	no	—
183	C₂²×C₈	—	166	1	no	—
184	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
185	#117	C₄×D₈₃	1	1	yes	characteristic
186	C₂×C₄²	—	83	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	83	2	no	—
189	M₃₂	—	166	4	no	—
190	M₃₂	—	166	1	no	—
191	M₃₂	C₄×D₈₃	1	2	yes	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	83	2	no	—
196	C₂×C₁₆	—	83	2	no	—
197	C₂×C₁₆	—	83	1	no	—
198	C₂×C₁₆	—	83	1	no	—
199	C₂×C₁₆	—	166	4	no	—
200	C₂×C₁₆	—	166	1	no	—
201	C₂×C₁₆	C₄×D₈₃	1	2	yes	—
202	#104	—	83	4	no	—
203	#103	—	83	2	no	—
204	#102	—	83	2	no	—
205	#97	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	166	1	no	—
208	C₄×C₈	—	166	1	no	—
209	#41	—	83	2	no	—
210	#41	—	83	2	no	—
211	#41	—	166	1	no	—
212	#41	—	166	1	no	—
213	C₂×Q₈	—	83	1	no	—
214	C₂×D₄	—	83	1	no	—
215	C₂×D₄	—	166	1	no	—
216	C₂²×C₄	—	166	1	no	—
217	C₂²×C₄	—	166	1	no	—
218	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	83	2	no	—
221	M₁₆	—	166	1	no	—
222	M₁₆	—	166	2	no	—
223	M₁₆	—	166	2	no	—
224	M₁₆	C₈×D₈₃	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	83	2	no	—
227	C₂×C₈	—	166	1	no	—
228	C₂×C₈	—	166	1	no	—
229	C₂×C₈	—	166	1	no	—
230	C₂×C₈	—	166	2	no	—
231	C₂×C₈	—	166	2	no	—
232	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
234	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
235	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
236	C₂×C₈	#13050589	1	1	yes	characteristic
237	C₂×C₈	#13050591	1	1	yes	characteristic
238	C₄⋊C₄	C₈×D₈₃	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	83	2	no	—
242	C₄²	—	166	1	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	83	2	no	—
246	C₁₆	—	83	2	no	—
247	C₁₆	—	166	2	no	—
248	C₂³	—	166	1	no	—
249	C₂³	#14581800	1	1	yes	characteristic
250	Q₈	—	83	2	no	—
251	Q₈	—	166	1	no	—
252	D₄	—	83	2	no	—
253	D₄	—	166	2	no	—
254	D₄	—	166	2	no	—
255	D₄	—	166	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	332	2	no	—
258	C₂×C₄	—	166	2	no	—
259	C₂×C₄	—	166	1	no	—
260	C₂×C₄	—	166	2	no	—
261	C₂×C₄	—	166	2	no	—
262	C₂×C₄	C₄²×D₈₃	1	1	yes	characteristic
263	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
264	C₂×C₄	#14581754	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
267	C₂×C₄	#14581800	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	166	2	no	—
270	C₈	—	166	2	no	—
271	C₈	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
272	C₈	#14581750	1	1	yes	characteristic
273	C₈	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
274	C₈	#14581754	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	332	2	no	—
277	C₂²	—	166	2	no	—
278	C₂²	#19127658	1	1	yes	characteristic
279	C₂²	(C₂²⋊D₈₃)⋊C₈	1	1	yes	characteristic
280	C₂²	#19127766	1	1	yes	characteristic
281	C₄	—	4	2	no	—
282	C₄	—	332	2	no	—
283	C₄	—	166	2	no	—
284	C₄	#19127548	1	1	yes	characteristic
285	C₄	C₄×C₈×D₈₃	1	1	yes	characteristic
286	C₄	(C₂²⋊D₈₃)⋊C₈	1	1	yes	characteristic
287	C₄	#19127766	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	332	2	no	—
291	C₂	#24907233	1	1	yes	characteristic
292	C₂	#24909302	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₈₃
commutator	C₄×C₈₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1379	C₂
socle	C₂²×C₈₃	C₈×D₄
prime radical	C₂×#13050590	C₂×C₄
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1379	D₈₃
hypocenter	C₈₃	#12615
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
3. #24909060	C₂	semidirect
4. C₂×#19127476	C₂	semidirect
6. #24905544	C₂	semidirect
7. C₈₃×#1379	C₂	semidirect
10. #19128119	C₄	semidirect
18. #19127484	C₄	semidirect
152. #1379	D₈₃	semidirect
154. C₈₃	#12615	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128119	2
C₂×#19127484	2
#24909060	2
C₂×#19127476	2
#24907302	2
#24905544	2
C₈₃×#1379	2
#12615	83