Subgroups — #35302986

Order 21248.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 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	C₄×C₄⋊C₈×C₈₃	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	1	yes	characteristic
9	C₂×C₄⋊(C₈₃⋊C₈)	C₄	1	1	yes	characteristic
10	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
12	#19127456	C₄	1	2	yes	—
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×C₄×C₈×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	2	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)	C₈	1	4	yes	complemented
28	C₄×C₈₃⋊C₈	—	2	2	no	—
29	C₄×C₈₃⋊C₈	—	2	2	no	—
30	C₂²×C₈×C₈₃	—	2	1	no	—
31	C₂²×C₈×C₈₃	—	2	1	no	—
32	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₃	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₈₃	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₈₃	—	2	1	no	—
40	C₄⋊C₈×C₈₃	—	2	1	no	—
41	C₄⋊C₈×C₈₃	D₄	1	2	yes	—
42	C₄⋊C₈×C₈₃	Q₈	1	2	yes	—
43	C₄×C₈×C₈₃	—	4	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	—	2	2	no	—
49	C₂×C₈₃⋊C₈	—	4	2	no	—
50	C₂×C₈₃⋊C₈	—	4	2	no	—
51	C₂×C₈₃⋊C₈	—	4	2	no	—
52	C₂×C₈₃⋊C₈	C₂×C₈	1	4	yes	—
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	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₈₃	—	2	2	no	—
61	C₂×C₈×C₈₃	—	2	2	no	—
62	C₂×C₈×C₈₃	—	4	1	no	—
63	C₂×C₈×C₈₃	—	4	1	no	—
64	C₂×C₈×C₈₃	—	4	1	no	—
65	C₂×C₈×C₈₃	—	4	1	no	—
66	C₄²×C₈₃	—	2	2	no	—
67	C₄²×C₈₃	—	2	1	no	—
68	C₄²×C₈₃	—	2	1	no	—
69	C₄²×C₈₃	—	2	1	no	—
70	C₄²×C₈₃	—	2	1	no	—
71	C₄²×C₈₃	—	4	1	no	—
72	C₄²×C₈₃	C₂×C₈	1	2	yes	—
73	C₄²×C₈₃	M₁₆	1	2	yes	—
74	C₄²×C₈₃	SD₁₆	1	2	yes	—
75	C₄²×C₈₃	SD₁₆	1	2	yes	—
76	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₈₃	#41	1	1	yes	characteristic
79	C₄²×C₈₃	#41	1	1	yes	characteristic
80	C₂³×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
90	C₂×C₄×C₈₃	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₈₃	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₈₃	#106	1	2	yes	—
93	C₂×C₄×C₈₃	#106	1	2	yes	—
94	C₂×C₄×C₈₃	#108	1	2	yes	—
95	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
100	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
104	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
105	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
109	C₈×C₈₃	—	4	2	no	—
110	C₈×C₈₃	—	4	2	no	—
111	C₈₃⋊C₈	—	2	4	no	—
112	C₈₃⋊C₈	—	4	2	no	—
113	C₈₃⋊C₈	—	8	2	no	—
114	C₂²×C₈₃	#389	1	1	yes	characteristic
115	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂²×C₈₃	#424	1	1	yes	characteristic
117	C₂²×C₈₃	C₂×#106	1	1	yes	characteristic
118	C₂²×C₈₃	#428	1	1	yes	characteristic
119	C₂²×C₈₃	C₂×#108	1	1	yes	characteristic
120	C₂²×C₈₃	#430	1	1	yes	characteristic
121	C₄×C₈₃	—	2	2	no	—
122	C₄×C₈₃	—	2	2	no	—
123	C₄×C₈₃	—	2	1	no	—
124	C₄×C₈₃	—	2	1	no	—
125	C₄×C₈₃	—	4	1	no	—
126	C₄×C₈₃	#334	1	4	yes	complemented
127	C₄×C₈₃	C₈×D₄	1	1	yes	characteristic
128	C₄×C₈₃	#436	1	1	yes	characteristic
129	C₄×C₈₃	C₈×Q₈	1	1	yes	characteristic
130	C₄×C₈₃	#446	1	1	yes	characteristic
131	C₄×C₈₃	#460	1	1	yes	characteristic
132	C₄×C₈₃	#461	1	1	yes	characteristic
133	C₄×C₈₃	#475	1	1	yes	characteristic
134	C₄×C₈₃	#476	1	1	yes	characteristic
135	#9602	—	83	1	no	maximal, Sylow
136	C₂×C₈₃	C₂×#334	1	1	yes	characteristic
137	C₂×C₈₃	#1565	1	1	yes	characteristic
138	C₂×C₈₃	#1570	1	1	yes	characteristic
139	C₂×C₈₃	#1574	1	1	yes	characteristic
140	C₂×C₈₃	#1939	1	1	yes	characteristic
141	C₂×C₈₃	#1948	1	1	yes	characteristic
142	C₂×C₈₃	#1950	1	1	yes	characteristic
143	#1842	—	83	1	no	—
144	C₄×C₄⋊C₈	—	83	1	no	—
145	C₄×C₄⋊C₈	D₈₃	1	1	yes	characteristic
146	C₂×#334	—	83	2	no	—
147	#1277	—	83	2	no	—
148	C₈₃	#9602	1	1	yes	characteristic, complemented, Sylow
149	C₂×C₄⋊C₈	—	83	1	no	—
150	C₂×C₄⋊C₈	—	83	1	no	—
151	C₂×C₄⋊C₈	—	83	2	no	—
152	C₂×C₄⋊C₈	—	166	1	no	—
153	C₂×C₄⋊C₈	—	166	1	no	—
154	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
156	C₂×C₄×C₈	—	2	1	no	—
157	C₂×C₄×C₈	—	83	2	no	—
158	C₄³	C₂×D₈₃	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	#336	—	83	2	no	—
161	#334	—	166	4	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	—	83	2	no	—
165	C₂²×C₈	—	83	2	no	—
166	C₂²×C₈	—	166	2	no	—
167	C₂×C₄²	Dic₁₆₆	1	1	yes	characteristic
168	C₂×C₄²	Dic₁₆₆	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	332	2	no	—
177	C₄⋊C₈	—	332	2	no	—
178	C₄⋊C₈	—	83	4	no	—
179	C₄⋊C₈	—	166	2	no	—
180	C₄⋊C₈	—	166	2	no	—
181	C₄⋊C₈	—	166	2	no	—
182	C₄⋊C₈	Dic₁₆₆	1	2	yes	—
183	C₄⋊C₈	D₃₃₂	1	2	yes	—
184	C₄×C₈	—	4	2	no	—
185	C₄×C₈	—	166	2	no	—
186	C₄×C₈	—	166	2	no	—
187	C₂²×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	332	2	no	—
201	C₂×C₈	—	332	2	no	—
202	C₂×C₈	—	332	2	no	—
203	C₂×C₈	—	83	4	no	—
204	C₂×C₈	—	166	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	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₈×D₈₃	1	2	yes	—
216	C₄²	#13050573	1	2	yes	—
217	C₄²	#13050577	1	2	yes	—
218	C₄²	#13050578	1	2	yes	—
219	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
220	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
221	C₄²	#13050590	1	1	yes	characteristic
222	C₄²	#13050590	1	1	yes	characteristic
223	C₂³	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	Dic₈₃⋊C₈	1	2	yes	—
234	C₂×C₄	Dic₈₃⋊C₈	1	2	yes	—
235	C₂×C₄	#14581723	1	2	yes	—
236	C₂×C₄	#14581723	1	2	yes	—
237	C₂×C₄	#14581727	1	2	yes	—
238	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
239	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
240	C₂×C₄	#14581766	1	1	yes	characteristic
241	C₂×C₄	#14581771	1	1	yes	characteristic
242	C₂×C₄	#14581784	1	1	yes	characteristic
243	C₂×C₄	#14581792	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
245	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
246	C₂×C₄	#14581798	1	1	yes	characteristic
247	C₂×C₄	#14581798	1	1	yes	characteristic
248	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
249	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
250	C₂×C₄	#14581817	1	1	yes	characteristic
251	C₂×C₄	#14581822	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₈	—	4	2	no	—
254	C₈	—	332	2	no	—
255	C₈	—	664	2	no	—
256	C₈	—	166	4	no	—
257	C₂²	#19127599	1	1	yes	characteristic
258	C₂²	C₂×Dic₈₃⋊C₈	1	1	yes	characteristic
259	C₂²	#19127727	1	1	yes	characteristic
260	C₂²	C₂×#14581723	1	1	yes	characteristic
261	C₂²	#19127744	1	1	yes	characteristic
262	C₂²	C₂×#14581727	1	1	yes	characteristic
263	C₂²	#19127749	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	1	no	—
269	C₄	#19127451	1	4	yes	complemented
270	C₄	C₈×D₃₃₂	1	1	yes	characteristic
271	C₄	#19127780	1	1	yes	characteristic
272	C₄	C₈×Dic₁₆₆	1	1	yes	characteristic
273	C₄	#19127831	1	1	yes	characteristic
274	C₄	#19127901	1	1	yes	characteristic
275	C₄	#19127906	1	1	yes	characteristic
276	C₄	#19127986	1	1	yes	characteristic
277	C₄	#19127991	1	1	yes	characteristic
278	C₂	C₂×#19127451	1	1	yes	characteristic
279	C₂	#24906216	1	1	yes	characteristic
280	C₂	#24906233	1	1	yes	characteristic
281	C₂	#24906251	1	1	yes	characteristic
282	C₂	#24908181	1	1	yes	characteristic
283	C₂	#24908226	1	1	yes	characteristic
284	C₂	#24908238	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₄×C₄⋊C₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₈₃
hypocenter	C₈₃	#9602
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
27. C₄⋊(C₈₃⋊C₈)	C₈	semidirect
126. C₄×C₈₃	#334	semidirect
148. C₈₃	#9602	semidirect
269. C₄	#19127451	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
C₄×C₄⋊C₈×C₈₃	2
#9602	83