Subgroups — #35303003

Order 21248.

388 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	#24907615	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127451	C₂	1	2	yes	maximal
4	#24905297	C₂	1	2	yes	maximal
5	C₈₃×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
9	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
11	#19127456	C₄	1	1	yes	characteristic
12	#19127456	C₄	1	1	yes	characteristic
13	#19127451	—	2	4	no	—
14	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×#97×C₈₃	—	2	1	no	—
17	C₄³×C₈₃	C₂²	1	1	yes	characteristic
18	#336×C₈₃	—	2	1	no	—
19	C₂²×C₈₃⋊C₈	—	2	2	no	—
20	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
25	C₄⋊(C₈₃⋊C₈)	—	4	2	no	—
26	C₄⋊(C₈₃⋊C₈)	—	4	2	no	—
27	C₄×C₈₃⋊C₈	—	2	2	no	—
28	C₄×C₈₃⋊C₈	—	2	2	no	—
29	C₂²×C₈×C₈₃	—	2	1	no	—
30	C₂²×C₈×C₈₃	—	2	1	no	—
31	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₃	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₈₃	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₈₃	—	2	1	no	—
39	C₄⋊C₈×C₈₃	—	2	1	no	—
40	C₄⋊C₈×C₈₃	D₄	1	2	yes	—
41	C₄⋊C₈×C₈₃	Q₈	1	2	yes	—
42	#97×C₈₃	—	4	2	no	—
43	C₂×C₈₃⋊C₈	—	2	2	no	—
44	C₂×C₈₃⋊C₈	—	2	2	no	—
45	C₂×C₈₃⋊C₈	—	2	2	no	—
46	C₂×C₈₃⋊C₈	—	2	2	no	—
47	C₂×C₈₃⋊C₈	—	2	2	no	—
48	C₂×C₈₃⋊C₈	—	4	2	no	—
49	C₂×C₈₃⋊C₈	—	4	2	no	—
50	C₂×C₈₃⋊C₈	—	4	2	no	—
51	C₂×C₈₃⋊C₈	M₁₆	1	4	yes	—
52	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₈₃	—	2	2	no	—
60	C₂×C₈×C₈₃	—	2	2	no	—
61	C₂×C₈×C₈₃	—	4	1	no	—
62	C₂×C₈×C₈₃	—	4	1	no	—
63	C₂×C₈×C₈₃	—	4	1	no	—
64	C₂×C₈×C₈₃	—	4	1	no	—
65	C₄²×C₈₃	—	2	2	no	—
66	C₄²×C₈₃	—	2	2	no	—
67	C₄²×C₈₃	—	2	2	no	—
68	C₄²×C₈₃	—	2	1	no	—
69	C₄²×C₈₃	—	2	1	no	—
70	C₄²×C₈₃	—	4	1	no	—
71	C₄²×C₈₃	SD₁₆	1	2	yes	—
72	C₄²×C₈₃	SD₁₆	1	2	yes	—
73	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₈₃	#41	1	1	yes	characteristic
76	C₄²×C₈₃	#41	1	1	yes	characteristic
77	C₂³×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₈₃	—	2	1	no	—
79	C₂×C₄×C₈₃	—	2	1	no	—
80	C₂×C₄×C₈₃	—	2	1	no	—
81	C₂×C₄×C₈₃	—	2	1	no	—
82	C₂×C₄×C₈₃	—	2	1	no	—
83	C₂×C₄×C₈₃	—	2	1	no	—
84	C₂×C₄×C₈₃	—	2	1	no	—
85	C₂×C₄×C₈₃	—	2	1	no	—
86	C₂×C₄×C₈₃	—	2	1	no	—
87	C₂×C₄×C₈₃	—	2	1	no	—
88	C₂×C₄×C₈₃	—	2	1	no	—
89	C₂×C₄×C₈₃	#106	1	2	yes	—
90	C₂×C₄×C₈₃	#106	1	2	yes	—
91	C₂×C₄×C₈₃	#108	1	2	yes	—
92	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₈₃	#131	1	2	yes	—
100	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
104	C₈×C₈₃	—	4	2	no	—
105	C₈×C₈₃	—	4	2	no	—
106	C₈₃⋊C₈	—	2	4	no	—
107	C₈₃⋊C₈	—	4	2	no	—
108	C₈₃⋊C₈	—	8	2	no	—
109	C₂²×C₈₃	#389	1	1	yes	characteristic
110	C₂²×C₈₃	#423	1	1	yes	characteristic
111	C₂²×C₈₃	#424	1	1	yes	characteristic
112	C₂²×C₈₃	C₂×#106	1	1	yes	characteristic
113	C₂²×C₈₃	#428	1	1	yes	characteristic
114	C₂²×C₈₃	C₂×#108	1	1	yes	characteristic
115	C₂²×C₈₃	#430	1	1	yes	characteristic
116	C₄×C₈₃	—	2	2	no	—
117	C₄×C₈₃	—	2	2	no	—
118	C₄×C₈₃	—	2	2	no	—
119	C₄×C₈₃	—	2	1	no	—
120	C₄×C₈₃	—	2	1	no	—
121	C₄×C₈₃	—	4	1	no	—
122	C₄×C₈₃	#435	1	2	yes	—
123	C₄×C₈₃	#446	1	2	yes	—
124	C₄×C₈₃	#460	1	1	yes	characteristic
125	C₄×C₈₃	#461	1	1	yes	characteristic
126	C₄×C₈₃	#475	1	1	yes	characteristic
127	C₄×C₈₃	#476	1	1	yes	characteristic
128	#9605	—	83	1	no	maximal, Sylow
129	C₂×C₈₃	#1566	1	1	yes	characteristic
130	C₂×C₈₃	#1569	1	1	yes	characteristic
131	C₂×C₈₃	#1570	1	1	yes	characteristic
132	C₂×C₈₃	#1573	1	1	yes	characteristic
133	C₂×C₈₃	#1940	1	1	yes	characteristic
134	C₂×C₈₃	#1948	1	1	yes	characteristic
135	C₂×C₈₃	#1950	1	1	yes	characteristic
136	#1842	—	83	1	no	—
137	#1767	D₈₃	1	1	yes	characteristic
138	C₄×C₄⋊C₈	—	83	1	no	—
139	C₂×#334	—	83	2	no	—
140	#1277	—	83	2	no	—
141	C₈₃	#9605	1	1	yes	characteristic, complemented, Sylow
142	C₂×C₄⋊C₈	—	83	2	no	—
143	C₂×C₄⋊C₈	—	83	2	no	—
144	C₂×C₄⋊C₈	—	166	1	no	—
145	C₂×C₄⋊C₈	—	166	1	no	—
146	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
147	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
148	C₂×#97	—	2	1	no	—
149	C₂×C₄×C₈	—	83	2	no	—
150	C₄³	C₂×D₈₃	1	1	yes	characteristic
151	#336	—	2	1	no	—
152	#336	—	83	1	no	—
153	#336	—	83	1	no	—
154	#334	—	166	4	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	83	2	no	—
158	C₂²×C₈	—	83	2	no	—
159	C₂²×C₈	—	166	2	no	—
160	C₂×C₄²	Dic₁₆₆	1	1	yes	characteristic
161	C₂×C₄²	Dic₁₆₆	1	1	yes	characteristic
162	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
163	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
167	C₄⋊C₈	—	2	1	no	—
168	C₄⋊C₈	—	2	1	no	—
169	C₄⋊C₈	—	332	2	no	—
170	C₄⋊C₈	—	332	2	no	—
171	C₄⋊C₈	—	166	4	no	—
172	C₄⋊C₈	—	166	2	no	—
173	C₄⋊C₈	—	166	2	no	—
174	C₄⋊C₈	Dic₁₆₆	1	2	yes	—
175	C₄⋊C₈	D₃₃₂	1	2	yes	—
176	#97	—	4	2	no	—
177	C₄×C₈	—	166	2	no	—
178	C₄×C₈	—	166	2	no	—
179	C₂²×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	332	2	no	—
193	C₂×C₈	—	332	2	no	—
194	C₂×C₈	—	332	2	no	—
195	C₂×C₈	—	83	4	no	—
196	C₂×C₈	—	166	2	no	—
197	C₂×C₈	—	166	2	no	—
198	C₂×C₈	—	166	2	no	—
199	C₂×C₈	—	166	2	no	—
200	C₂×C₈	—	166	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	4	1	no	—
207	C₄²	#13050577	1	2	yes	—
208	C₄²	#13050578	1	2	yes	—
209	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
210	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
211	C₄²	#13050590	1	1	yes	characteristic
212	C₄²	#13050590	1	1	yes	characteristic
213	C₂³	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	#14581723	1	2	yes	—
226	C₂×C₄	#14581723	1	2	yes	—
227	C₂×C₄	#14581727	1	2	yes	—
228	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
229	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
230	C₂×C₄	#14581766	1	1	yes	characteristic
231	C₂×C₄	#14581771	1	1	yes	characteristic
232	C₂×C₄	#14581784	1	1	yes	characteristic
233	C₂×C₄	#14581792	1	1	yes	characteristic
234	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
235	C₂×C₄	#14581800	1	2	yes	—
236	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
237	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
238	C₂×C₄	#14581817	1	1	yes	characteristic
239	C₂×C₄	#14581822	1	1	yes	characteristic
240	C₈	—	4	2	no	—
241	C₈	—	4	2	no	—
242	C₈	—	332	2	no	—
243	C₈	—	664	2	no	—
244	C₈	—	166	4	no	—
245	C₂²	#19127599	1	1	yes	characteristic
246	C₂²	#19127724	1	1	yes	characteristic
247	C₂²	#19127729	1	1	yes	characteristic
248	C₂²	C₂×#14581723	1	1	yes	characteristic
249	C₂²	#19127744	1	1	yes	characteristic
250	C₂²	C₂×#14581727	1	1	yes	characteristic
251	C₂²	#19127749	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	#19127774	1	2	yes	—
259	C₄	#19127829	1	2	yes	—
260	C₄	#19127901	1	1	yes	characteristic
261	C₄	#19127906	1	1	yes	characteristic
262	C₄	#19127986	1	1	yes	characteristic
263	C₄	#19127991	1	1	yes	characteristic
264	C₂	#24906218	1	1	yes	characteristic
265	C₂	#24906231	1	1	yes	characteristic
266	C₂	#24906235	1	1	yes	characteristic
267	C₂	#24906249	1	1	yes	characteristic
268	C₂	#24908186	1	1	yes	characteristic
269	C₂	#24908226	1	1	yes	characteristic
270	C₂	#24908238	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₃
commutator	C₂²×C₄×C₈₃	C₂²×C₄
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	#9605
LCS	G ▷ C₂²×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
141. C₈₃	#9605	semidirect

Maximal subgroups

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

subgroup	index
#24907615	2
C₄×C₄⋊(C₈₃⋊C₈)	2
C₂×#19127451	2
#24905297	2
C₈₃×#1767	2
#9605	83