Subgroups — #35319639

Order 21248.

438 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127817	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127509	C₂	1	1	yes	maximal, characteristic
3	#24909168	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#24907344	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905556	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581814	C₂²	1	1	yes	characteristic
9	C₂×#14581800	—	2	1	no	—
10	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	#19127817	—	2	2	no	—
12	#19127817	C₄	1	4	yes	complemented
13	C₂×#14581728	C₂²	1	1	yes	characteristic
14	C₂×#14581718	—	2	1	no	—
15	#19127657	—	2	1	no	—
16	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
17	#19127509	C₄	1	4	yes	complemented
18	#19127504	C₄	1	2	yes	complemented
19	#19127501	C₄	1	2	yes	complemented
20	C₁₆×Dic₈₃	—	2	2	no	—
21	#19127460	—	2	1	no	—
22	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
24	#427×C₈₃	C₂²	1	1	yes	characteristic
25	C₂×#13050590	—	2	1	no	—
26	#14581814	—	2	1	no	—
27	#14581814	—	2	1	no	—
28	#14581814	C₂×C₄	1	2	yes	—
29	#14581814	C₂×C₄	1	2	yes	—
30	C₂×Dic₃₃₂	C₂×C₄	1	1	yes	characteristic
31	C₂×#13050579	—	2	1	no	—
32	C₂×D₆₆₄	C₂×C₄	1	1	yes	characteristic
33	#14581800	—	2	4	no	—
34	#14581800	—	4	2	no	—
35	C₂×#13050574	—	2	1	no	—
36	C₂×#13050573	—	4	1	no	—
37	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
38	C₂×C₈×D₈₃	—	4	1	no	—
39	C₂×C₄×Dic₈₃	D₄	1	1	yes	characteristic
40	C₂×C₈₃⋊C₁₆	—	2	1	no	—
41	C₂×C₈₃⋊C₁₆	—	2	1	no	—
42	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
43	#14581728	—	2	1	no	—
44	#14581728	C₂×C₄	1	2	yes	—
45	#14581718	—	2	4	no	—
46	#14581703	—	4	1	no	—
47	C₄×C₈₃⋊C₈	—	4	1	no	—
48	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₈₃	D₄	1	2	yes	—
50	C₂×M₁₆×C₈₃	—	2	1	no	—
51	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
52	#117×C₈₃	—	2	1	no	—
53	C₂×C₁₆×C₈₃	—	2	1	no	—
54	C₂×C₁₆×C₈₃	—	2	1	no	—
55	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
56	#108×C₈₃	—	2	1	no	—
57	#108×C₈₃	C₂×C₄	1	2	yes	—
58	#107×C₈₃	C₂×C₄	1	2	yes	—
59	#106×C₈₃	—	2	1	no	—
60	#13050590	—	2	2	no	—
61	#13050590	—	2	2	no	—
62	#13050590	—	4	1	no	—
63	#13050590	—	4	1	no	—
64	C₂×Dic₁₆₆	—	2	1	no	—
65	C₂×D₃₃₂	—	2	1	no	—
66	C₂×C₂²⋊D₈₃	—	4	1	no	—
67	C₂²×Dic₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	4	1	no	—
69	Dic₃₃₂	—	2	1	no	—
70	Dic₃₃₂	C₄²	1	2	yes	—
71	#13050579	—	2	2	no	—
72	#13050579	—	2	2	no	—
73	D₆₆₄	—	2	1	no	—
74	D₆₆₄	C₄²	1	2	yes	—
75	#13050574	—	2	2	no	—
76	#13050574	—	4	1	no	—
77	#13050573	—	4	4	no	—
78	C₂×C₈₃⋊C₈	—	4	1	no	—
79	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈₃⋊C₈	D₈	1	1	yes	characteristic
81	C₂×C₈₃⋊C₈	D₈	1	1	yes	characteristic
82	C₂×C₈₃⋊C₈	SD₁₆	1	2	yes	—
83	C₈×D₈₃	—	4	4	no	—
84	C₄×Dic₈₃	—	2	1	no	—
85	C₄×Dic₈₃	C₂²⋊C₄	1	2	yes	—
86	C₈₃⋊C₁₆	—	2	2	no	—
87	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
88	M₁₆×C₈₃	—	2	2	no	—
89	M₁₆×C₈₃	—	4	1	no	—
90	C₂×C₈×C₈₃	—	4	1	no	—
91	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
97	C₄⋊C₄×C₈₃	—	2	2	no	—
98	C₂²⋊C₄×C₈₃	—	4	1	no	—
99	C₄²×C₈₃	—	4	1	no	—
100	C₁₆×C₈₃	—	2	2	no	—
101	C₂³×C₈₃	#135	1	1	yes	characteristic
102	C₂²×D₈₃	—	4	1	no	—
103	C₂×C₄×C₈₃	—	4	2	no	—
104	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	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	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
110	C₂²⋊D₈₃	—	4	2	no	—
111	C₂²⋊D₈₃	—	4	2	no	—
112	C₂×Dic₈₃	—	2	4	no	—
113	C₂×Dic₈₃	—	2	2	no	—
114	C₂×Dic₈₃	—	4	1	no	—
115	D₃₃₂	—	2	2	no	—
116	D₃₃₂	—	4	1	no	—
117	C₄×D₈₃	—	4	2	no	—
118	C₄×D₈₃	—	4	2	no	—
119	Dic₁₆₆	—	2	2	no	—
120	Dic₁₆₆	—	4	1	no	—
121	C₈×C₈₃	—	4	2	no	—
122	C₈×C₈₃	C₂×C₄²	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	—	4	2	no	—
127	C₈₃⋊C₈	#102	1	2	yes	—
128	C₈₃⋊C₈	#102	1	2	yes	—
129	C₂²×C₈₃	—	2	2	no	—
130	C₂²×C₈₃	#416	1	1	yes	characteristic
131	C₂²×C₈₃	C₄×D₈	1	1	yes	characteristic
132	C₂²×C₈₃	C₄×SD₁₆	1	1	yes	characteristic
133	C₂×D₈₃	—	4	2	no	—
134	C₂×D₈₃	—	8	2	no	—
135	C₄×C₈₃	—	8	2	no	—
136	C₄×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₈₃	C₂×#102	1	1	yes	characteristic
138	C₄×C₈₃	C₄×D₈	1	1	yes	characteristic
139	C₄×C₈₃	C₄×SD₁₆	1	1	yes	characteristic
140	Dic₈₃	—	2	4	no	—
141	Dic₈₃	—	4	2	no	—
142	#12641	—	83	1	no	maximal, Sylow
143	C₂×C₈₃	—	2	1	no	—
144	C₂×C₈₃	—	2	1	no	—
145	C₂×C₈₃	C₄×#102	1	1	yes	characteristic
146	C₂×C₈₃	#2178	1	2	yes	—
147	D₈₃	—	8	2	no	—
148	C₂×#443	—	83	1	no	—
149	C₂×#359	—	83	1	no	—
150	#2138	—	83	1	no	—
151	C₂×C₄×C₁₆	—	83	1	no	—
152	#1772	—	83	1	no	—
153	#1382	D₈₃	1	1	yes	characteristic, complemented
154	#1381	—	83	1	no	—
155	C₈₃	#12641	1	1	yes	characteristic, complemented, Sylow
156	C₂×#135	—	83	1	no	—
157	C₂×#131	—	166	1	no	—
158	C₂²×C₁₆	—	83	1	no	—
159	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
160	#443	—	83	4	no	—
161	#443	—	166	2	no	—
162	C₂×#108	—	83	1	no	—
163	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
164	#427	C₂×D₈₃	1	1	yes	characteristic
165	C₂×#104	—	166	1	no	—
166	#405	—	166	1	no	—
167	C₂×C₄×C₈	—	83	1	no	—
168	#359	—	83	4	no	—
169	#358	—	83	2	no	—
170	#357	—	83	2	no	—
171	C₄×C₁₆	—	166	2	no	—
172	#339	—	166	1	no	—
173	C₂×#41	—	166	1	no	—
174	#135	—	83	2	no	—
175	#135	—	83	2	no	—
176	#135	—	166	1	no	—
177	#135	—	166	1	no	—
178	C₂×Q₁₆	—	83	1	no	—
179	C₂×SD₁₆	—	166	1	no	—
180	C₂×D₈	—	83	1	no	—
181	#131	—	332	2	no	—
182	#131	—	166	4	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	332	1	no	—
185	C₂×M₁₆	—	166	1	no	—
186	C₂²×C₈	—	332	1	no	—
187	C₂²×C₈	—	83	1	no	—
188	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
189	#117	—	2	1	no	—
190	C₂×C₄²	—	83	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	83	2	no	—
194	C₂×C₁₆	—	166	1	no	—
195	C₂×C₁₆	—	166	1	no	—
196	C₂×C₁₆	C₄×D₈₃	1	2	yes	—
197	#108	—	2	1	no	—
198	#108	—	83	2	no	—
199	#108	—	166	1	no	—
200	#108	C₄×D₈₃	1	2	yes	—
201	#107	C₄×D₈₃	1	2	yes	—
202	#106	—	2	1	no	—
203	#104	—	166	4	no	—
204	#97	—	332	1	no	—
205	C₄×C₈	—	332	1	no	—
206	C₄×C₈	—	83	2	no	—
207	C₄×C₈	—	83	2	no	—
208	#41	—	332	1	no	—
209	#41	—	332	1	no	—
210	#41	—	166	2	no	—
211	#41	—	166	2	no	—
212	C₂×Q₈	—	166	1	no	—
213	C₂×D₄	—	332	1	no	—
214	C₂×D₄	—	166	1	no	—
215	C₂²×C₄	—	332	1	no	—
216	C₂²×C₄	—	166	1	no	—
217	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
218	Q₁₆	—	83	2	no	—
219	Q₁₆	—	166	1	no	—
220	SD₁₆	—	166	2	no	—
221	SD₁₆	—	166	2	no	—
222	D₈	—	83	2	no	—
223	D₈	—	166	1	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	332	4	no	—
227	M₁₆	—	332	1	no	—
228	M₁₆	—	166	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	332	4	no	—
231	C₂×C₈	—	332	1	no	—
232	C₂×C₈	—	83	1	no	—
233	C₂×C₈	—	83	1	no	—
234	C₂×C₈	—	83	2	no	—
235	C₂×C₈	—	83	2	no	—
236	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
239	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
240	C₂×C₈	#13050589	1	1	yes	characteristic
241	C₂×C₈	#13050591	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	83	2	no	—
246	C₄²	—	166	1	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	—	166	2	no	—
249	C₂³	—	332	1	no	—
250	C₂³	#14581813	1	1	yes	characteristic
251	Q₈	—	332	1	no	—
252	Q₈	—	166	2	no	—
253	D₄	—	332	2	no	—
254	D₄	—	332	2	no	—
255	D₄	—	332	1	no	—
256	D₄	—	166	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	332	1	no	—
259	C₂×C₄	—	332	2	no	—
260	C₂×C₄	—	332	2	no	—
261	C₂×C₄	—	166	4	no	—
262	C₂×C₄	—	166	2	no	—
263	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
264	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
265	C₂×C₄	#14581754	1	1	yes	characteristic
266	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
267	C₂×C₄	SD₁₆×D₈₃	1	1	yes	characteristic
268	C₂×C₄	#14581815	1	1	yes	characteristic
269	C₈	—	4	2	no	—
270	C₈	—	332	2	no	—
271	C₈	—	83	2	no	—
272	C₈	—	83	2	no	—
273	C₈	C₄²×D₈₃	1	1	yes	characteristic
274	C₈	#14581750	1	1	yes	characteristic
275	C₈	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
276	C₈	#14581754	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	—	332	2	no	—
279	C₂²	—	664	2	no	—
280	C₂²	#19127696	1	1	yes	characteristic
281	C₂²	#19127781	1	1	yes	characteristic
282	C₂²	#19127788	1	1	yes	characteristic
283	C₄	—	8	2	no	—
284	C₄	—	332	2	no	—
285	C₄	—	166	4	no	—
286	C₄	#19127548	1	1	yes	characteristic
287	C₄	#102×D₈₃	1	1	yes	characteristic
288	C₄	#19127783	1	1	yes	characteristic
289	C₄	#19127786	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	664	2	no	—
293	C₂	#24907277	1	1	yes	characteristic
294	C₂	#24909341	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₈₃
commutator	C₈×C₈₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1382	C₂
socle	C₂²×C₈₃	C₄×D₈
prime radical	C₂×#14581814	C₂²
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1382	D₈₃
hypocenter	C₈₃	#12641
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1382

Complements

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

N	H ≅ G/N	type
4. C₂×C₁₆×Dic₈₃	C₂	semidirect
5. #24907344	C₂	semidirect
6. #24905556	C₂	semidirect
7. C₈₃×#1382	C₂	semidirect
12. #19127817	C₄	semidirect
17. #19127509	C₄	semidirect
18. #19127504	C₄	semidirect
19. #19127501	C₄	semidirect
153. #1382	D₈₃	semidirect
155. C₈₃	#12641	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127817	2
C₂×#19127509	2
#24909168	2
C₂×C₁₆×Dic₈₃	2
#24907344	2
#24905556	2
C₈₃×#1382	2
#12641	83