Subgroups — #35303057

Order 21248.

382 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24908226	C₂	1	2	yes	maximal
2	#24907623	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127454	C₂	1	1	yes	maximal, characteristic
4	#24905347	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581726	—	2	1	no	—
7	C₂×#14581726	—	2	1	no	—
8	C₂×#14581723	—	2	2	no	—
9	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
10	#19127571	C₂²	1	2	yes	—
11	#19127463	—	2	2	no	—
12	#19127454	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₃	—	2	1	no	—
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
17	#336×C₈₃	—	2	1	no	—
18	C₂²×C₈₃⋊C₈	—	2	2	no	—
19	C₂×C₄⋊Dic₈₃	—	2	2	no	—
20	C₂×C₄⋊Dic₈₃	—	2	2	no	—
21	C₂×C₄⋊Dic₈₃	—	2	2	no	—
22	#14581726	—	4	2	no	—
23	#14581726	—	4	2	no	—
24	#14581723	—	4	4	no	—
25	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
26	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₈₃	—	2	1	no	—
28	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	—	2	1	no	—
33	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₈₃	—	2	1	no	—
35	C₄⋊C₈×C₈₃	Q₈	1	2	yes	—
36	C₄×C₈×C₈₃	—	4	2	no	—
37	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₈₃	Q₈	1	2	yes	—
39	#95×C₈₃	—	2	1	no	—
40	C₂²×Dic₈₃	—	4	2	no	—
41	C₂×C₈₃⋊C₈	—	2	4	no	—
42	C₂×C₈₃⋊C₈	—	4	2	no	—
43	C₂×C₈₃⋊C₈	—	4	2	no	—
44	C₄⋊Dic₈₃	—	4	2	no	—
45	C₄⋊Dic₈₃	—	4	2	no	—
46	C₄⋊Dic₈₃	—	4	2	no	—
47	C₄⋊Dic₈₃	—	4	2	no	—
48	C₄⋊Dic₈₃	—	8	2	no	—
49	C₂²×C₄×C₈₃	—	2	1	no	—
50	C₂²×C₄×C₈₃	—	2	1	no	—
51	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
54	C₂×C₈×C₈₃	—	2	1	no	—
55	C₂×C₈×C₈₃	—	2	1	no	—
56	C₂×C₈×C₈₃	—	2	2	no	—
57	C₂×C₈×C₈₃	—	4	1	no	—
58	C₂×C₈×C₈₃	—	4	1	no	—
59	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
62	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
63	C₄⋊C₄×C₈₃	—	2	1	no	—
64	C₄⋊C₄×C₈₃	—	2	1	no	—
65	C₄⋊C₄×C₈₃	D₈	1	2	yes	—
66	C₄⋊C₄×C₈₃	Q₁₆	1	2	yes	—
67	C₄²×C₈₃	—	4	1	no	—
68	C₄²×C₈₃	—	4	1	no	—
69	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₈₃	#41	1	1	yes	characteristic
73	C₂³×C₈₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₈₃	—	2	1	no	—
75	C₂×C₄×C₈₃	—	2	1	no	—
76	C₂×C₄×C₈₃	—	2	2	no	—
77	C₂×C₄×C₈₃	—	2	2	no	—
78	C₂×C₄×C₈₃	—	4	1	no	—
79	C₂×C₄×C₈₃	—	4	1	no	—
80	C₂×C₄×C₈₃	—	4	1	no	—
81	C₂×C₄×C₈₃	—	4	1	no	—
82	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
87	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
93	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
94	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
96	C₂×Dic₈₃	—	8	2	no	—
97	C₂×Dic₈₃	—	8	2	no	—
98	C₂×Dic₈₃	—	8	2	no	—
99	C₈×C₈₃	—	2	2	no	—
100	C₈×C₈₃	—	4	2	no	—
101	C₈×C₈₃	#106	1	4	yes	complemented
102	C₈₃⋊C₈	—	4	4	no	—
103	C₂²×C₈₃	#389	1	1	yes	characteristic
104	C₂²×C₈₃	#460	1	1	yes	characteristic
105	C₂²×C₈₃	#461	1	1	yes	characteristic
106	C₂²×C₈₃	#466	1	1	yes	characteristic
107	C₂²×C₈₃	#467	1	1	yes	characteristic
108	C₂²×C₈₃	#472	1	1	yes	characteristic
109	C₂²×C₈₃	#473	1	1	yes	characteristic
110	C₄×C₈₃	—	2	1	no	—
111	C₄×C₈₃	—	2	1	no	—
112	C₄×C₈₃	—	4	2	no	—
113	C₄×C₈₃	—	4	2	no	—
114	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
115	C₄×C₈₃	#428	1	1	yes	characteristic
116	C₄×C₈₃	C₄×D₈	1	1	yes	characteristic
117	C₄×C₈₃	C₄×Q₁₆	1	1	yes	characteristic
118	C₄×C₈₃	#475	1	1	yes	characteristic
119	C₄×C₈₃	#476	1	1	yes	characteristic
120	C₄×C₈₃	#499	1	1	yes	characteristic
121	C₄×C₈₃	#500	1	1	yes	characteristic
122	Dic₈₃	—	16	2	no	—
123	#9615	—	83	1	no	maximal, Sylow
124	C₂×C₈₃	#1668	1	1	yes	characteristic
125	C₂×C₈₃	#1670	1	1	yes	characteristic
126	C₂×C₈₃	#1673	1	1	yes	characteristic
127	C₂×C₈₃	#1675	1	1	yes	characteristic
128	C₂×C₈₃	#1942	1	1	yes	characteristic
129	C₂×C₈₃	#1948	1	1	yes	characteristic
130	C₂×C₈₃	#1952	1	1	yes	characteristic
131	#1948	—	83	2	no	—
132	#1845	—	83	1	no	—
133	C₈×C₄⋊C₄	D₈₃	1	1	yes	characteristic
134	C₂×#335	—	83	1	no	—
135	#1296	—	83	2	no	—
136	C₈₃	#9615	1	1	yes	characteristic, complemented, Sylow
137	C₂×#107	—	166	1	no	—
138	C₂×#107	—	166	1	no	—
139	C₂×#106	—	166	2	no	—
140	C₂×C₄⋊C₈	—	83	2	no	—
141	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
142	C₂×C₄×C₈	—	2	1	no	—
143	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
144	#384	—	83	2	no	—
145	C₄×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
146	#340	—	166	2	no	—
147	#336	—	2	1	no	—
148	#335	—	83	4	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	166	2	no	—
151	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	166	2	no	—
154	C₂×C₄⋊C₄	—	166	2	no	—
155	C₂×C₄⋊C₄	—	166	2	no	—
156	C₂×C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
157	C₂×C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
160	#107	—	332	2	no	—
161	#107	—	332	2	no	—
162	#106	—	332	4	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	83	4	no	—
165	C₄⋊C₈	—	166	2	no	—
166	C₄⋊C₈	Dic₁₆₆	1	2	yes	—
167	C₄×C₈	—	4	2	no	—
168	C₄×C₈	Dic₁₆₆	1	2	yes	—
169	C₄×C₈	C₄×D₈₃	1	2	yes	—
170	#95	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	332	2	no	—
174	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
175	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
176	C₂²×C₄	#13050590	1	1	yes	characteristic
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	332	2	no	—
183	C₂×C₈	—	332	2	no	—
184	C₂×C₈	—	166	4	no	—
185	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
186	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
187	C₂×C₈	#13050577	1	2	yes	—
188	C₂×C₈	#13050578	1	2	yes	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	332	2	no	—
192	C₄⋊C₄	—	332	2	no	—
193	C₄⋊C₄	—	332	2	no	—
194	C₄⋊C₄	—	332	2	no	—
195	C₄⋊C₄	—	664	2	no	—
196	C₄⋊C₄	D₆₆₄	1	2	yes	—
197	C₄⋊C₄	Dic₃₃₂	1	2	yes	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
201	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
202	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
203	C₄²	#13050590	1	1	yes	characteristic
204	C₂³	#14581766	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	664	2	no	—
214	C₂×C₄	—	664	2	no	—
215	C₂×C₄	—	664	2	no	—
216	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
217	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
218	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
219	C₂×C₄	#14581771	1	1	yes	characteristic
220	C₂×C₄	#14581784	1	1	yes	characteristic
221	C₂×C₄	#14581792	1	1	yes	characteristic
222	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
223	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
224	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
225	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
226	C₂×C₄	#14581814	1	1	yes	characteristic
227	C₂×C₄	#14581814	1	1	yes	characteristic
228	C₂×C₄	#14581817	1	1	yes	characteristic
229	C₂×C₄	#14581822	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	4	2	no	—
232	C₈	—	332	4	no	—
233	C₈	#14581723	1	4	yes	complemented
234	C₂²	#19127599	1	1	yes	characteristic
235	C₂²	#19127901	1	1	yes	characteristic
236	C₂²	#19127906	1	1	yes	characteristic
237	C₂²	#19127937	1	1	yes	characteristic
238	C₂²	#19127942	1	1	yes	characteristic
239	C₂²	#19127971	1	1	yes	characteristic
240	C₂²	#19127976	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	1328	2	no	—
246	C₄	C₂×#14581723	1	1	yes	characteristic
247	C₄	#19127744	1	1	yes	characteristic
248	C₄	C₄×D₆₆₄	1	1	yes	characteristic
249	C₄	C₄×Dic₃₃₂	1	1	yes	characteristic
250	C₄	#19127986	1	1	yes	characteristic
251	C₄	#19127991	1	1	yes	characteristic
252	C₄	#19128103	1	1	yes	characteristic
253	C₄	#19128106	1	1	yes	characteristic
254	C₂	#24906823	1	1	yes	characteristic
255	C₂	#24906833	1	1	yes	characteristic
256	C₂	#24906852	1	1	yes	characteristic
257	C₂	#24906862	1	1	yes	characteristic
258	C₂	#24908196	1	1	yes	characteristic
259	C₂	#24908226	1	1	yes	characteristic
260	C₂	#24908248	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581766
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈×C₄⋊C₄×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#121
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₈₃
hypocenter	C₈₃	#9615
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. #19127454	C₄	semidirect
101. C₈×C₈₃	#106	semidirect
136. C₈₃	#9615	semidirect
233. C₈	#14581723	semidirect

Maximal subgroups

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

subgroup	index
#24908226	2
#24907623	2
C₂×#19127454	2
#24905347	2
C₈×C₄⋊C₄×C₈₃	2
#9615	83