Subgroups — #35303760

Order 21248.

394 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917977	C₂	1	1	yes	maximal, characteristic
2	#24907330	C₂	1	1	yes	maximal, characteristic
3	#24905963	C₂	1	2	yes	maximal
4	#24905867	C₂	1	2	yes	maximal
5	C₈₃×#1564	C₂	1	1	yes	maximal, characteristic
6	#19128258	C₂²	1	2	yes	—
7	#19128106	C₄	1	1	yes	characteristic, complemented
8	#19128106	C₄	1	1	yes	characteristic, complemented
9	#19128103	—	2	1	no	—
10	#19128103	—	2	1	no	—
11	#19128099	C₄	1	2	yes	complemented
12	#19127739	—	2	1	no	—
13	#19127739	—	2	1	no	—
14	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×#14581702	—	2	1	no	—
16	C₄×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
17	#19127455	—	2	1	no	—
18	#19127439	—	2	4	no	—
19	#19127433	—	2	4	no	—
20	#431×C₈₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	#335×C₈₃	—	2	1	no	—
23	#334×C₈₃	—	2	1	no	—
24	C₂²×C₈₃⋊C₈	—	2	1	no	—
25	#14581792	C₂×C₄	1	2	yes	—
26	#14581792	C₂×C₄	1	2	yes	—
27	#14581784	—	2	2	no	—
28	#14581771	—	2	2	no	—
29	#14581771	—	2	2	no	—
30	C₄×Dic₁₆₆	—	2	2	no	—
31	C₄×Dic₁₆₆	—	2	2	no	—
32	#14581751	—	2	2	no	—
33	C₂×C₄⋊Dic₈₃	—	2	1	no	—
34	C₂×C₄×Dic₈₃	—	2	1	no	—
35	#14581726	—	2	1	no	—
36	#14581726	—	2	1	no	—
37	#14581726	—	2	1	no	—
38	#14581726	—	2	1	no	—
39	#14581724	—	2	2	no	—
40	#14581724	—	2	2	no	—
41	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
42	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
43	#14581702	—	4	2	no	—
44	C₈₃⋊#95	D₄	1	2	yes	—
45	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₈₃	—	2	2	no	—
49	C₂²⋊C₈×C₈₃	—	2	2	no	—
50	C₄×C₈×C₈₃	—	2	2	no	—
51	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
55	C₂×Dic₁₆₆	—	2	2	no	—
56	C₂×Dic₁₆₆	—	2	2	no	—
57	C₂²×Dic₈₃	—	2	2	no	—
58	C₂×C₈₃⋊C₈	—	2	2	no	—
59	C₂×C₈₃⋊C₈	—	4	1	no	—
60	C₂×C₈₃⋊C₈	—	4	1	no	—
61	C₄⋊Dic₈₃	—	2	2	no	—
62	C₄⋊Dic₈₃	—	2	2	no	—
63	C₄⋊Dic₈₃	—	2	2	no	—
64	C₄⋊Dic₈₃	—	2	2	no	—
65	C₄⋊Dic₈₃	—	4	2	no	—
66	Dic₈₃⋊C₄	—	4	2	no	—
67	Dic₈₃⋊C₄	—	4	2	no	—
68	C₂²⋊Dic₈₃	—	4	2	no	—
69	C₄×Dic₈₃	—	4	2	no	—
70	C₄×Dic₈₃	—	4	1	no	—
71	C₄×Dic₈₃	—	4	1	no	—
72	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
75	C₂×C₈×C₈₃	—	2	1	no	—
76	C₂×C₈×C₈₃	—	2	1	no	—
77	C₂×C₈×C₈₃	—	4	2	no	—
78	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
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	SD₁₆	1	2	yes	—
83	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
84	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₈₃	#41	1	1	yes	characteristic
88	C₂³×C₈₃	#121	1	1	yes	characteristic
89	C₂×C₄×C₈₃	—	2	2	no	—
90	C₂×C₄×C₈₃	—	4	1	no	—
91	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
96	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
100	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
101	C₂×Dic₈₃	—	4	2	no	—
102	C₂×Dic₈₃	—	4	2	no	—
103	C₂×Dic₈₃	—	4	2	no	—
104	C₂×Dic₈₃	—	4	2	no	—
105	C₂×Dic₈₃	—	4	2	no	—
106	Dic₁₆₆	—	4	4	no	—
107	Dic₁₆₆	—	8	2	no	—
108	C₈×C₈₃	—	2	1	no	—
109	C₈×C₈₃	—	2	1	no	—
110	C₈×C₈₃	—	8	2	no	—
111	C₈×C₈₃	#103	1	2	yes	—
112	C₈×C₈₃	#103	1	2	yes	—
113	C₈₃⋊C₈	—	4	2	no	—
114	C₂²×C₈₃	—	2	1	no	—
115	C₂²×C₈₃	—	2	1	no	—
116	C₂²×C₈₃	#390	1	1	yes	characteristic
117	C₂²×C₈₃	#461	1	1	yes	characteristic
118	C₂²×C₈₃	#462	1	1	yes	characteristic
119	C₄×C₈₃	—	2	1	no	—
120	C₄×C₈₃	—	2	1	no	—
121	C₄×C₈₃	—	2	1	no	—
122	C₄×C₈₃	C₂×#103	1	1	yes	characteristic
123	C₄×C₈₃	#419	1	1	yes	characteristic
124	C₄×C₈₃	#444	1	2	yes	—
125	C₄×C₈₃	#461	1	1	yes	characteristic
126	C₄×C₈₃	#462	1	1	yes	characteristic
127	Dic₈₃	—	8	2	no	—
128	Dic₈₃	—	8	2	no	—
129	Dic₈₃	—	8	2	no	—
130	#9752	—	83	1	no	maximal, Sylow
131	C₂×C₈₃	—	2	1	no	—
132	C₂×C₈₃	—	2	1	no	—
133	C₂×C₈₃	#1975	1	1	yes	characteristic
134	C₂×C₈₃	#1976	1	1	yes	characteristic
135	C₂×C₈₃	#1981	1	1	yes	characteristic
136	#3160	—	83	1	no	—
137	#1770	—	83	1	no	—
138	#1564	D₈₃	1	1	yes	characteristic
139	#1503	—	83	2	no	—
140	#1481	—	83	2	no	—
141	C₈₃	#9752	1	1	yes	characteristic, complemented, Sylow
142	#533	—	83	2	no	—
143	#500	—	83	1	no	—
144	#500	—	83	1	no	—
145	#499	—	166	1	no	—
146	#499	—	166	1	no	—
147	#498	—	83	2	no	—
148	#431	C₂×D₈₃	1	2	yes	—
149	#427	—	166	1	no	—
150	#427	—	166	1	no	—
151	C₂×C₄⋊C₈	—	83	1	no	—
152	C₂×#97	—	166	1	no	—
153	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
154	C₄×C₄⋊C₄	—	83	1	no	—
155	#336	—	166	1	no	—
156	#335	—	2	1	no	—
157	#334	—	2	1	no	—
158	#328	—	166	4	no	—
159	Q₈⋊C₈	—	166	4	no	—
160	C₂²×C₈	—	166	1	no	—
161	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
162	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
163	C₄⋊Q₈	—	83	2	no	—
164	C₄⋊Q₈	—	83	2	no	—
165	#125	—	166	2	no	—
166	C₂²⋊Q₈	—	166	2	no	—
167	C₂²⋊Q₈	—	166	2	no	—
168	C₄×Q₈	—	166	2	no	—
169	C₄×Q₈	—	166	2	no	—
170	#117	—	166	2	no	—
171	C₂×C₄⋊C₄	—	166	1	no	—
172	C₂×C₄²	—	166	1	no	—
173	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
174	#107	—	166	1	no	—
175	#107	—	166	1	no	—
176	#107	—	166	1	no	—
177	#107	—	166	1	no	—
178	#106	—	166	2	no	—
179	#106	—	166	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	83	2	no	—
182	C₄⋊C₈	—	166	1	no	—
183	C₂²⋊C₈	—	2	2	no	—
184	#97	—	332	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
187	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
190	#95	—	83	2	no	—
191	C₂×Q₈	—	166	2	no	—
192	C₂×Q₈	—	166	2	no	—
193	C₂²×C₄	—	166	2	no	—
194	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
196	C₂²×C₄	#13050591	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	332	1	no	—
201	C₂×C₈	—	332	1	no	—
202	C₂×C₈	—	166	2	no	—
203	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
207	C₂×C₈	#13050579	1	2	yes	—
208	C₂×C₈	Dic₃₃₂	1	2	yes	—
209	C₄⋊C₄	—	332	2	no	—
210	C₄⋊C₄	—	332	2	no	—
211	C₄⋊C₄	—	332	2	no	—
212	C₄⋊C₄	—	166	2	no	—
213	C₄⋊C₄	—	166	2	no	—
214	C₄⋊C₄	—	166	2	no	—
215	C₄⋊C₄	—	166	2	no	—
216	C₂²⋊C₄	—	332	2	no	—
217	C₄²	—	332	2	no	—
218	C₄²	—	332	1	no	—
219	C₄²	—	332	1	no	—
220	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
221	C₄²	D₄×D₈₃	1	1	yes	characteristic
222	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
223	C₄²	#13050591	1	1	yes	characteristic
224	C₂³	#14581767	1	1	yes	characteristic
225	Q₈	—	332	4	no	—
226	Q₈	—	664	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	332	2	no	—
230	C₂×C₄	—	332	2	no	—
231	C₂×C₄	—	332	2	no	—
232	C₂×C₄	—	332	2	no	—
233	C₂×C₄	—	332	2	no	—
234	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
235	C₂×C₄	#14581754	1	1	yes	characteristic
236	C₂×C₄	#14581754	1	1	yes	characteristic
237	C₂×C₄	#14581767	1	1	yes	characteristic
238	C₂×C₄	#14581781	1	1	yes	characteristic
239	C₂×C₄	#14581788	1	1	yes	characteristic
240	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
241	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
242	C₂×C₄	#14581825	1	1	yes	characteristic
243	C₂×C₄	#14581825	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	8	2	no	—
247	C₈	—	332	2	no	—
248	C₈	#14581717	1	2	yes	—
249	C₈	#14581717	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#19127602	1	1	yes	characteristic
253	C₂²	#19127907	1	1	yes	characteristic
254	C₂²	#19127912	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	664	2	no	—
259	C₄	—	664	2	no	—
260	C₄	—	664	2	no	—
261	C₄	C₂×#14581717	1	1	yes	characteristic
262	C₄	#19127711	1	1	yes	characteristic
263	C₄	#19127822	1	2	yes	—
264	C₄	#19127907	1	1	yes	characteristic
265	C₄	#19127912	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	#24908354	1	1	yes	characteristic
269	C₂	#24908359	1	1	yes	characteristic
270	C₂	#24908385	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581767
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1564	C₂
socle	C₂²×C₈₃	#390
prime radical	C₂³×C₈₃	#121
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1564	D₈₃
hypocenter	C₈₃	#9752
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
7. #19128106	C₄	semidirect
8. #19128106	C₄	semidirect
11. #19128099	C₄	semidirect
141. C₈₃	#9752	semidirect

Maximal subgroups

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

subgroup	index
#24917977	2
#24907330	2
#24905963	2
#24905867	2
C₈₃×#1564	2
#9752	83