Subgroups — #35295607

Order 21248.

414 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	#24907316	C₂	1	1	yes	maximal, characteristic
2	C₄×#14581703	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127453	C₂	1	1	yes	maximal, characteristic
4	C₂×#19127451	C₂	1	1	yes	maximal, characteristic
5	#24905297	C₂	1	1	yes	maximal, characteristic
6	#24905297	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#1767	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#14581703	—	2	1	no	—
11	C₂×#14581703	C₄	1	2	yes	—
12	C₂×#14581703	C₄	1	2	yes	—
13	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
15	#19127456	—	2	1	no	—
16	#19127453	C₄	1	4	yes	complemented
17	#19127451	C₄	1	4	yes	complemented
18	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×#97×C₈₃	—	2	1	no	—
21	C₄³×C₈₃	C₂²	1	1	yes	characteristic
22	#336×C₈₃	—	2	1	no	—
23	C₂²×C₈₃⋊C₈	—	2	1	no	—
24	C₂²×C₈₃⋊C₈	—	2	1	no	—
25	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
29	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
30	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
31	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
32	#14581703	—	2	4	no	—
33	#14581703	—	2	4	no	—
34	#14581703	—	4	2	no	—
35	C₄×C₈₃⋊C₈	—	2	1	no	—
36	C₄×C₈₃⋊C₈	—	2	1	no	—
37	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
38	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₈₃	—	2	1	no	—
40	C₂²×C₈×C₈₃	—	2	1	no	—
41	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₃	Q₈	1	2	yes	—
45	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₈₃	—	2	1	no	—
47	C₄⋊C₈×C₈₃	—	2	1	no	—
48	C₄⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
49	C₄⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
50	#97×C₈₃	—	4	2	no	—
51	C₂×C₈₃⋊C₈	—	2	2	no	—
52	C₂×C₈₃⋊C₈	—	2	2	no	—
53	C₂×C₈₃⋊C₈	—	2	2	no	—
54	C₂×C₈₃⋊C₈	—	2	2	no	—
55	C₂×C₈₃⋊C₈	—	2	2	no	—
56	C₂×C₈₃⋊C₈	—	4	1	no	—
57	C₂×C₈₃⋊C₈	—	4	1	no	—
58	C₂×C₈₃⋊C₈	—	4	1	no	—
59	C₂×C₈₃⋊C₈	—	4	1	no	—
60	C₂×C₈₃⋊C₈	C₄²	1	2	yes	—
61	C₂×C₈₃⋊C₈	C₄²	1	2	yes	—
62	C₂×C₈₃⋊C₈	M₁₆	1	4	yes	—
63	C₂×C₈₃⋊C₈	M₁₆	1	4	yes	—
64	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₈₃	—	2	2	no	—
70	C₂×C₈×C₈₃	—	2	2	no	—
71	C₂×C₈×C₈₃	—	4	1	no	—
72	C₂×C₈×C₈₃	—	4	1	no	—
73	C₂×C₈×C₈₃	—	4	1	no	—
74	C₂×C₈×C₈₃	—	4	1	no	—
75	C₄²×C₈₃	—	2	2	no	—
76	C₄²×C₈₃	—	2	2	no	—
77	C₄²×C₈₃	—	2	1	no	—
78	C₄²×C₈₃	—	2	1	no	—
79	C₄²×C₈₃	—	2	1	no	—
80	C₄²×C₈₃	—	2	1	no	—
81	C₄²×C₈₃	—	4	2	no	—
82	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₈₃	#41	1	1	yes	characteristic
85	C₄²×C₈₃	#41	1	1	yes	characteristic
86	C₂³×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₈₃	—	2	2	no	—
88	C₂×C₄×C₈₃	—	2	2	no	—
89	C₂×C₄×C₈₃	—	2	2	no	—
90	C₂×C₄×C₈₃	—	2	2	no	—
91	C₂×C₄×C₈₃	—	2	2	no	—
92	C₂×C₄×C₈₃	—	2	2	no	—
93	C₂×C₄×C₈₃	—	2	2	no	—
94	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
105	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
106	C₈×C₈₃	—	4	2	no	—
107	C₈×C₈₃	—	4	2	no	—
108	C₈₃⋊C₈	—	2	2	no	—
109	C₈₃⋊C₈	—	2	4	no	—
110	C₈₃⋊C₈	—	4	2	no	—
111	C₈₃⋊C₈	—	4	2	no	—
112	C₈₃⋊C₈	#97	1	4	yes	complemented
113	C₂²×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₈₃	#423	1	2	yes	—
115	C₂²×C₈₃	#428	1	1	yes	characteristic
116	C₂²×C₈₃	#428	1	1	yes	characteristic
117	C₂²×C₈₃	#430	1	2	yes	—
118	C₄×C₈₃	—	2	1	no	—
119	C₄×C₈₃	—	2	1	no	—
120	C₄×C₈₃	—	4	2	no	—
121	C₄×C₈₃	—	4	2	no	—
122	C₄×C₈₃	C₂×#97	1	1	yes	characteristic
123	C₄×C₈₃	C₄×M₁₆	1	1	yes	characteristic
124	C₄×C₈₃	#432	1	2	yes	—
125	C₄×C₈₃	#440	1	1	yes	characteristic
126	C₄×C₈₃	#440	1	1	yes	characteristic
127	C₄×C₈₃	#441	1	1	yes	characteristic
128	C₄×C₈₃	#442	1	1	yes	characteristic
129	#8276	—	83	1	no	maximal, Sylow
130	C₂×C₈₃	#1569	1	2	yes	—
131	C₂×C₈₃	#1569	1	2	yes	—
132	C₂×C₈₃	#1767	1	1	yes	characteristic
133	C₂×C₈₃	#1776	1	1	yes	characteristic
134	C₂×C₈₃	#1778	1	1	yes	characteristic
135	#1767	—	83	1	no	—
136	#1767	D₈₃	1	1	yes	characteristic
137	C₄×#97	—	83	1	no	—
138	C₂×#335	—	83	1	no	—
139	C₂×#334	—	83	1	no	—
140	#1277	—	83	1	no	—
141	#1277	—	83	1	no	—
142	C₈₃	#8276	1	1	yes	characteristic, complemented, Sylow
143	C₂×C₄⋊C₈	—	83	1	no	—
144	C₂×C₄⋊C₈	—	83	1	no	—
145	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
146	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
147	C₂×#97	—	2	1	no	—
148	C₂×#97	—	83	2	no	—
149	C₂×#97	—	83	2	no	—
150	C₂×#97	—	166	1	no	—
151	C₂×C₄×C₈	—	83	1	no	—
152	C₂×C₄×C₈	—	83	1	no	—
153	C₄³	C₂×D₈₃	1	1	yes	characteristic
154	#336	—	2	1	no	—
155	#336	—	166	1	no	—
156	#335	—	83	4	no	—
157	#334	—	83	4	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	—	83	1	no	—
161	C₂²×C₈	—	83	2	no	—
162	C₂²×C₈	—	83	1	no	—
163	C₂²×C₈	—	166	1	no	—
164	C₂²×C₈	—	166	1	no	—
165	C₂×C₄²	Dic₁₆₆	1	2	yes	—
166	C₂×C₄²	C₄×D₈₃	1	2	yes	—
167	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	83	2	no	—
173	C₄⋊C₈	—	83	2	no	—
174	C₄⋊C₈	—	166	1	no	—
175	C₄⋊C₈	—	166	1	no	—
176	C₄⋊C₈	C₄×D₈₃	1	2	yes	—
177	C₄⋊C₈	C₄×D₈₃	1	2	yes	—
178	#97	—	4	2	no	—
179	#97	—	332	2	no	—
180	#97	—	166	4	no	—
181	#97	—	166	4	no	—
182	C₄×C₈	—	83	2	no	—
183	C₄×C₈	—	83	2	no	—
184	C₄×C₈	—	166	1	no	—
185	C₄×C₈	—	166	1	no	—
186	C₂²×C₄	Dic₈₃⋊C₄	1	2	yes	—
187	C₂²×C₄	Dic₈₃⋊C₄	1	2	yes	—
188	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	332	1	no	—
198	C₂×C₈	—	332	1	no	—
199	C₂×C₈	—	332	1	no	—
200	C₂×C₈	—	332	1	no	—
201	C₂×C₈	—	83	4	no	—
202	C₂×C₈	—	83	2	no	—
203	C₂×C₈	—	83	4	no	—
204	C₂×C₈	—	83	2	no	—
205	C₂×C₈	—	166	2	no	—
206	C₂×C₈	—	166	2	no	—
207	C₂×C₈	—	166	2	no	—
208	C₂×C₈	—	166	2	no	—
209	C₂×C₈	—	166	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	4	2	no	—
217	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
218	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
219	C₄²	#13050590	1	1	yes	characteristic
220	C₄²	#13050590	1	1	yes	characteristic
221	C₂³	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	C₄²×D₈₃	1	1	yes	characteristic
230	C₂×C₄	#14581749	1	1	yes	characteristic
231	C₂×C₄	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
232	C₂×C₄	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
233	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
234	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
235	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
236	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
237	C₂×C₄	#14581817	1	1	yes	characteristic
238	C₂×C₄	#14581818	1	1	yes	characteristic
239	C₂×C₄	#14581822	1	1	yes	characteristic
240	C₂×C₄	#14581823	1	1	yes	characteristic
241	C₈	—	4	2	no	—
242	C₈	—	4	2	no	—
243	C₈	—	332	2	no	—
244	C₈	—	332	2	no	—
245	C₈	—	83	4	no	—
246	C₈	—	166	2	no	—
247	C₈	—	166	4	no	—
248	C₂²	C₄×Dic₈₃⋊C₄	1	1	yes	characteristic
249	C₂²	#19127724	1	2	yes	—
250	C₂²	#19127741	1	1	yes	characteristic
251	C₂²	#19127744	1	1	yes	characteristic
252	C₂²	#19127749	1	2	yes	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	#97×D₈₃	1	1	yes	characteristic
258	C₄	#19127653	1	1	yes	characteristic
259	C₄	#19127756	1	2	yes	—
260	C₄	#19127799	1	1	yes	characteristic
261	C₄	#19127801	1	1	yes	characteristic
262	C₄	#19127806	1	1	yes	characteristic
263	C₄	#19127811	1	1	yes	characteristic
264	C₂	#24906228	1	2	yes	—
265	C₂	#24906231	1	2	yes	—
266	C₂	#24907317	1	1	yes	characteristic
267	C₂	#24907361	1	1	yes	characteristic
268	C₂	#24907370	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1767	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1767	D₈₃
hypocenter	C₈₃	#8276
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
16. #19127453	C₄	semidirect
17. #19127451	C₄	semidirect
112. C₈₃⋊C₈	#97	semidirect
142. C₈₃	#8276	semidirect

Maximal subgroups

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

subgroup	index
#24907316	2
C₄×#14581703	2
C₂×#19127453	2
C₂×#19127451	2
#24905297	2
C₈₃×#1767	2
#8276	83