Subgroups — #24918789

Order 10624.

398 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128251	C₂	1	1	yes	maximal, characteristic, complemented
2	#19127998	C₂	1	2	yes	maximal, complemented
3	#19127996	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127991	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127935	C₂	1	2	yes	maximal, complemented
6	#19127932	C₂	1	2	yes	maximal, complemented
7	#19127913	C₂	1	2	yes	maximal, complemented
8	#19127901	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127896	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#529×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13050590	—	2	1	no	—
13	C₂×#13050577	C₂²	1	2	yes	—
14	C₂×#13050575	C₂²	1	2	yes	—
15	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
16	#14581792	C₂²	1	1	yes	characteristic
17	#14581789	C₂²	1	1	yes	characteristic
18	#14581780	—	2	1	no	—
19	#14581766	—	2	1	no	—
20	#14581766	C₂²	1	2	yes	—
21	C₄×D₃₃₂	—	2	1	no	—
22	C₄×D₃₃₂	C₂²	1	2	yes	—
23	#14581725	C₂²	1	2	yes	—
24	#14581723	C₂²	1	2	yes	—
25	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
28	#14581716	C₂²	1	2	yes	—
29	#14581713	C₂²	1	2	yes	—
30	#14581712	C₂²	1	2	yes	—
31	#14581712	C₂²	1	2	yes	—
32	#126×C₈₃	—	2	1	no	—
33	#125×C₈₃	C₂²	1	1	yes	characteristic
34	#124×C₈₃	C₂²	1	1	yes	characteristic
35	#123×C₈₃	—	2	1	no	—
36	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
37	#121×C₈₃	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₈₃	—	2	1	no	—
40	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
41	#117×C₈₃	—	2	1	no	—
42	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
43	#13050590	—	4	4	no	—
44	C₂×Dic₁₆₆	—	2	1	no	—
45	C₂×D₃₃₂	—	2	1	no	—
46	C₂×D₃₃₂	—	2	1	no	—
47	C₂×D₃₃₂	C₂³	1	2	yes	—
48	C₂×C₂²⋊D₈₃	—	2	2	no	—
49	C₂×C₄×D₈₃	—	2	2	no	—
50	#13050577	—	4	2	no	—
51	#13050575	—	4	2	no	—
52	C₂×C₈₃⋊C₈	—	4	2	no	—
53	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
54	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
55	C₄⋊Dic₈₃	—	2	1	no	—
56	C₄⋊Dic₈₃	C₂³	1	2	yes	—
57	(C₂×D₈₃)⋊C₄	—	2	2	no	—
58	(C₂×D₈₃)⋊C₄	—	2	2	no	—
59	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₈₃	—	2	1	no	—
61	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₈₃	—	2	1	no	—
63	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₈₃	—	2	1	no	—
67	C₄⋊C₄×C₈₃	—	2	1	no	—
68	C₄⋊C₄×C₈₃	—	2	1	no	—
69	C₄⋊C₄×C₈₃	—	2	1	no	—
70	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₈₃	—	2	1	no	—
73	C₂²⋊C₄×C₈₃	—	2	1	no	—
74	C₂²⋊C₄×C₈₃	—	2	1	no	—
75	C₂²⋊C₄×C₈₃	—	2	1	no	—
76	C₂²⋊C₄×C₈₃	—	2	1	no	—
77	C₄²×C₈₃	—	2	1	no	—
78	C₄²×C₈₃	D₄	1	1	yes	characteristic
79	C₄²×C₈₃	D₄	1	1	yes	characteristic
80	C₄²×C₈₃	C₂³	1	1	yes	characteristic
81	C₄²×C₈₃	C₂³	1	1	yes	characteristic
82	C₂³×C₈₃	—	2	1	no	—
83	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₈₃	—	2	2	no	—
85	Q₈×C₈₃	—	2	1	no	—
86	Q₈×C₈₃	—	2	1	no	—
87	D₄×C₈₃	—	2	1	no	—
88	D₄×C₈₃	—	2	1	no	—
89	D₄×C₈₃	—	4	2	no	—
90	C₂×C₄×C₈₃	—	2	2	no	—
91	C₂×C₄×C₈₃	—	2	2	no	—
92	C₂×C₄×C₈₃	—	2	1	no	—
93	C₂×C₄×C₈₃	—	2	1	no	—
94	C₂×C₄×C₈₃	—	2	1	no	—
95	C₂×C₄×C₈₃	—	2	1	no	—
96	C₂×C₄×C₈₃	—	2	1	no	—
97	C₂×C₄×C₈₃	—	4	2	no	—
98	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₈₃	—	4	4	no	—
105	C₂×Dic₈₃	—	2	2	no	—
106	D₃₃₂	—	2	2	no	—
107	D₃₃₂	—	2	2	no	—
108	D₃₃₂	—	4	2	no	—
109	D₃₃₂	—	4	2	no	—
110	C₄×D₈₃	—	4	4	no	—
111	Dic₁₆₆	—	4	2	no	—
112	C₈₃⋊C₈	—	2	2	no	—
113	C₈₃⋊C₈	—	2	2	no	—
114	C₂²×C₈₃	—	2	1	no	—
115	C₂²×C₈₃	—	2	1	no	—
116	C₂²×C₈₃	—	4	1	no	—
117	C₂²×C₈₃	—	4	1	no	—
118	C₂²×C₈₃	—	4	1	no	—
119	C₂²×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
120	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₈₃	—	4	2	no	—
122	C₂×D₈₃	—	4	2	no	—
123	C₂×D₈₃	—	4	2	no	—
124	C₄×C₈₃	—	2	1	no	—
125	C₄×C₈₃	—	2	1	no	—
126	C₄×C₈₃	—	2	1	no	—
127	C₄×C₈₃	—	2	1	no	—
128	C₄×C₈₃	—	4	1	no	—
129	C₄×C₈₃	—	4	1	no	—
130	C₄×C₈₃	—	4	1	no	—
131	C₄×C₈₃	#135	1	2	yes	—
132	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
133	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
134	Dic₈₃	—	4	2	no	—
135	C₂×C₈₃	—	2	2	no	—
136	C₂×C₈₃	—	8	1	no	—
137	C₂×C₈₃	#537	1	1	yes	characteristic
138	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
139	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
140	D₈₃	—	8	2	no	—
141	#3247	—	83	1	no	maximal, Sylow
142	C₈₃	#3247	1	1	yes	characteristic, complemented, Sylow
143	#532	—	83	1	no	—
144	#529	D₈₃	1	1	yes	characteristic, complemented
145	#478	—	83	2	no	—
146	#477	—	83	1	no	—
147	#476	—	83	1	no	—
148	#466	—	83	2	no	—
149	#465	—	83	2	no	—
150	#463	—	83	2	no	—
151	#460	—	83	1	no	—
152	#459	—	83	1	no	—
153	C₂×C₄⋊C₈	—	83	1	no	—
154	C₂×#41	—	166	1	no	—
155	C₂×SD₁₆	—	83	2	no	—
156	C₂×D₈	—	83	2	no	—
157	C₂²×C₈	—	83	2	no	—
158	C₄⋊Q₈	—	83	1	no	—
159	#127	—	83	1	no	—
160	#126	—	2	1	no	—
161	#125	C₂×D₈₃	1	1	yes	characteristic
162	#124	—	166	1	no	—
163	#124	C₂×D₈₃	1	1	yes	characteristic
164	#123	—	2	1	no	—
165	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
166	#121	—	83	2	no	—
167	#121	—	166	1	no	—
168	#121	C₂×D₈₃	1	1	yes	characteristic
169	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	—	83	2	no	—
172	C₄×D₄	—	166	1	no	—
173	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
174	#117	—	2	1	no	—
175	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
176	#107	—	83	2	no	—
177	#106	—	83	2	no	—
178	C₄⋊C₈	—	83	1	no	—
179	C₄⋊C₈	—	83	1	no	—
180	C₄⋊C₈	—	83	2	no	—
181	#103	—	83	2	no	—
182	#102	—	83	2	no	—
183	#102	—	83	2	no	—
184	#102	—	83	2	no	—
185	#41	—	332	4	no	—
186	C₂×Q₈	—	166	1	no	—
187	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	83	2	no	—
190	C₂×D₄	—	166	2	no	—
191	C₂×D₄	—	166	1	no	—
192	C₂×D₄	—	166	1	no	—
193	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	166	2	no	—
196	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
199	SD₁₆	—	332	2	no	—
200	D₈	—	332	2	no	—
201	C₂×C₈	—	332	2	no	—
202	C₂×C₈	—	83	2	no	—
203	C₂×C₈	—	83	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	83	2	no	—
209	C₄⋊C₄	—	166	1	no	—
210	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
211	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	166	2	no	—
218	C₂²⋊C₄	—	166	2	no	—
219	C₄²	—	2	1	no	—
220	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
221	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
222	C₄²	C₂²×D₈₃	1	1	yes	characteristic
223	C₄²	C₂²×D₈₃	1	1	yes	characteristic
224	C₂³	—	2	1	no	—
225	C₂³	—	166	2	no	—
226	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
227	Q₈	—	2	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	332	2	no	—
230	D₄	—	2	1	no	—
231	D₄	—	2	1	no	—
232	D₄	—	4	2	no	—
233	D₄	—	332	2	no	—
234	D₄	—	332	4	no	—
235	D₄	—	332	2	no	—
236	D₄	—	166	2	no	—
237	D₄	—	166	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	332	4	no	—
247	C₂×C₄	—	166	2	no	—
248	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
253	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
254	C₈	—	166	2	no	—
255	C₈	—	166	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	332	2	no	—
262	C₂²	—	332	2	no	—
263	C₂²	—	332	2	no	—
264	C₂²	#14581817	1	2	yes	—
265	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	332	2	no	—
274	C₄	#14581812	1	2	yes	—
275	C₄	#14581838	1	1	yes	characteristic
276	C₄	#14581839	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	8	1	no	—
279	C₂	—	664	2	no	—
280	C₂	#19128279	1	1	yes	characteristic
281	C₂	C₂×#14581812	1	1	yes	characteristic
282	C₂	C₂×#14581817	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#529×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#529	D₈₃
hypocenter	C₈₃	#3247
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #19128251	C₂	semidirect
2. #19127998	C₂	semidirect
3. #19127996	C₂	semidirect
4. #19127991	C₂	semidirect
5. #19127935	C₂	semidirect
6. #19127932	C₂	semidirect
7. #19127913	C₂	semidirect
8. #19127901	C₂	semidirect
9. #19127896	C₂	semidirect
10. C₂×C₄⋊(C₈₃⋊C₈)	C₂	semidirect
11. #529×C₈₃	C₂	semidirect
142. C₈₃	#3247	semidirect
144. #529	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128251	2
#19127998	2
#19127996	2
#19127991	2
#19127935	2
#19127932	2
#19127913	2
#19127901	2
#19127896	2
C₂×C₄⋊(C₈₃⋊C₈)	2
#529×C₈₃	2
#3247	83