Subgroups — #35309322

Order 21248.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24908227	C₂	1	2	yes	maximal
2	#24908181	C₂	1	1	yes	maximal, characteristic
3	#24907623	C₂	1	1	yes	maximal, characteristic
4	#24905347	C₂	1	2	yes	maximal
5	C₈₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581726	—	2	1	no	—
7	C₂×#14581726	—	2	1	no	—
8	C₂×#14581724	—	2	2	no	—
9	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×Dic₈₃⋊C₈	—	2	1	no	—
11	#19127571	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
13	#19127464	—	2	2	no	—
14	#19127455	—	2	1	no	—
15	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈₃⋊C₈	—	2	1	no	—
18	C₂×C₄⋊Dic₈₃	—	2	2	no	—
19	C₂×C₄⋊Dic₈₃	—	2	2	no	—
20	C₂×C₄⋊Dic₈₃	—	2	2	no	—
21	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₈₃	—	2	1	no	—
24	#14581726	—	4	2	no	—
25	#14581726	—	4	2	no	—
26	#14581724	—	4	4	no	—
27	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
28	Dic₈₃⋊C₈	—	4	2	no	—
29	C₈₃⋊#95	—	2	1	no	—
30	C₂²×C₈×C₈₃	—	2	2	no	—
31	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₈₃	—	2	2	no	—
35	C₄⋊C₈×C₈₃	Q₈	1	4	yes	—
36	C₄×C₈×C₈₃	—	2	2	no	—
37	C₂²×Dic₈₃	—	2	1	no	—
38	C₂²×Dic₈₃	—	2	1	no	—
39	C₂²×Dic₈₃	—	4	2	no	—
40	C₂×C₈₃⋊C₈	—	4	1	no	—
41	C₂×C₈₃⋊C₈	—	4	1	no	—
42	C₂×C₈₃⋊C₈	—	4	1	no	—
43	C₄⋊Dic₈₃	—	2	1	no	—
44	C₄⋊Dic₈₃	—	2	2	no	—
45	C₄⋊Dic₈₃	—	4	2	no	—
46	C₄⋊Dic₈₃	—	4	2	no	—
47	C₄⋊Dic₈₃	—	4	2	no	—
48	C₄⋊Dic₈₃	—	4	2	no	—
49	C₄⋊Dic₈₃	—	8	2	no	—
50	C₄⋊Dic₈₃	D₈	1	2	yes	—
51	C₄×Dic₈₃	—	4	1	no	—
52	C₄×Dic₈₃	—	4	1	no	—
53	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₈₃	—	2	1	no	—
57	C₂×C₈×C₈₃	—	2	1	no	—
58	C₂×C₈×C₈₃	—	2	1	no	—
59	C₂×C₈×C₈₃	—	2	1	no	—
60	C₂×C₈×C₈₃	—	2	4	no	—
61	C₂×C₈×C₈₃	—	4	2	no	—
62	C₂×C₈×C₈₃	—	4	2	no	—
63	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
64	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
65	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
66	C₄²×C₈₃	#41	1	2	yes	—
67	C₂³×C₈₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₈₃	—	2	1	no	—
69	C₂×C₄×C₈₃	—	2	1	no	—
70	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
72	C₂×C₄×C₈₃	#125	1	2	yes	—
73	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
78	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₈₃	#137	1	2	yes	—
80	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
81	C₂×Dic₈₃	—	2	2	no	—
82	C₂×Dic₈₃	—	4	1	no	—
83	C₂×Dic₈₃	—	4	1	no	—
84	C₂×Dic₈₃	—	4	1	no	—
85	C₂×Dic₈₃	—	4	1	no	—
86	C₂×Dic₈₃	—	4	1	no	—
87	C₂×Dic₈₃	—	8	2	no	—
88	C₂×Dic₈₃	—	8	2	no	—
89	C₂×Dic₈₃	—	8	2	no	—
90	C₈×C₈₃	—	2	2	no	—
91	C₈×C₈₃	—	2	2	no	—
92	C₈×C₈₃	—	4	4	no	—
93	C₈₃⋊C₈	—	8	1	no	—
94	C₂²×C₈₃	#398	1	1	yes	characteristic
95	C₂²×C₈₃	#460	1	1	yes	characteristic
96	C₂²×C₈₃	#461	1	1	yes	characteristic
97	C₂²×C₈₃	#466	1	1	yes	characteristic
98	C₂²×C₈₃	#470	1	1	yes	characteristic
99	C₂²×C₈₃	#472	1	1	yes	characteristic
100	C₂²×C₈₃	#473	1	1	yes	characteristic
101	C₄×C₈₃	—	2	2	no	—
102	C₄×C₈₃	#474	1	2	yes	—
103	C₄×C₈₃	#479	1	2	yes	—
104	C₄×C₈₃	#481	1	1	yes	characteristic
105	C₄×C₈₃	#483	1	1	yes	characteristic
106	C₄×C₈₃	#498	1	1	yes	characteristic
107	C₄×C₈₃	#501	1	1	yes	characteristic
108	Dic₈₃	—	4	2	no	—
109	Dic₈₃	—	8	1	no	—
110	Dic₈₃	—	16	2	no	—
111	#10651	—	83	1	no	maximal, Sylow
112	C₂×C₈₃	#1668	1	1	yes	characteristic
113	C₂×C₈₃	#1673	1	1	yes	characteristic
114	C₂×C₈₃	#1690	1	1	yes	characteristic
115	C₂×C₈₃	#1695	1	1	yes	characteristic
116	C₂×C₈₃	#2084	1	1	yes	characteristic
117	C₂×C₈₃	#2086	1	1	yes	characteristic
118	C₂×C₈₃	#2089	1	1	yes	characteristic
119	#1948	—	83	2	no	—
120	#1939	—	83	1	no	—
121	#1845	—	83	1	no	—
122	#1296	—	83	2	no	—
123	#1277	D₈₃	1	1	yes	characteristic
124	C₈₃	#10651	1	1	yes	characteristic, complemented, Sylow
125	C₂×#107	—	166	1	no	—
126	C₂×#107	—	166	1	no	—
127	C₂×#106	—	166	2	no	—
128	C₂×C₄⋊C₈	—	83	1	no	—
129	C₂×C₄⋊C₈	—	166	1	no	—
130	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
131	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
132	#384	—	83	2	no	—
133	C₄×C₄⋊C₄	—	83	1	no	—
134	#340	—	166	2	no	—
135	#336	—	166	1	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	166	1	no	—
138	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
139	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	83	1	no	—
141	C₂×C₄⋊C₄	—	83	1	no	—
142	C₂×C₄⋊C₄	—	166	2	no	—
143	C₂×C₄⋊C₄	—	166	2	no	—
144	C₂×C₄⋊C₄	—	166	2	no	—
145	C₂×C₄²	—	166	1	no	—
146	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
147	#107	—	332	2	no	—
148	#107	—	332	2	no	—
149	#106	—	332	4	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	332	2	no	—
152	C₄⋊C₈	—	166	2	no	—
153	C₄⋊C₈	Dic₁₆₆	1	4	yes	—
154	C₄×C₈	—	2	2	no	—
155	#95	—	166	1	no	—
156	C₂²×C₄	—	332	2	no	—
157	C₂²×C₄	—	166	1	no	—
158	C₂²×C₄	—	166	1	no	—
159	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
161	C₂²×C₄	#13050591	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	332	1	no	—
170	C₂×C₈	—	332	1	no	—
171	C₂×C₈	—	332	1	no	—
172	C₂×C₈	#13050579	1	2	yes	—
173	C₂×C₈	#13050579	1	2	yes	—
174	C₄⋊C₄	—	332	2	no	—
175	C₄⋊C₄	—	332	2	no	—
176	C₄⋊C₄	—	332	2	no	—
177	C₄⋊C₄	—	332	2	no	—
178	C₄⋊C₄	—	83	2	no	—
179	C₄⋊C₄	—	664	2	no	—
180	C₄⋊C₄	—	166	2	no	—
181	C₄⋊C₄	—	166	1	no	—
182	C₄²	—	332	1	no	—
183	C₄²	—	332	1	no	—
184	C₄²	C₂×Dic₁₆₆	1	2	yes	—
185	C₄²	#13050589	1	2	yes	—
186	C₂³	#14581767	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	332	1	no	—
190	C₂×C₄	—	332	1	no	—
191	C₂×C₄	—	332	1	no	—
192	C₂×C₄	—	332	1	no	—
193	C₂×C₄	—	332	1	no	—
194	C₂×C₄	—	664	2	no	—
195	C₂×C₄	—	664	2	no	—
196	C₂×C₄	—	664	2	no	—
197	C₂×C₄	—	166	2	no	—
198	C₂×C₄	#14581769	1	2	yes	—
199	C₂×C₄	#14581776	1	1	yes	characteristic
200	C₂×C₄	#14581783	1	2	yes	—
201	C₂×C₄	#14581792	1	1	yes	characteristic
202	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
203	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
204	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
205	C₂×C₄	#14581815	1	1	yes	characteristic
206	C₂×C₄	#14581820	1	1	yes	characteristic
207	C₂×C₄	#14581821	1	2	yes	—
208	C₂×C₄	#14581825	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	4	no	—
212	C₈	—	664	1	no	—
213	C₂²	#19127633	1	1	yes	characteristic
214	C₂²	#19127902	1	1	yes	characteristic
215	C₂²	#19127907	1	1	yes	characteristic
216	C₂²	#19127938	1	1	yes	characteristic
217	C₂²	#19127960	1	1	yes	characteristic
218	C₂²	#19127972	1	1	yes	characteristic
219	C₂²	#19127977	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	1328	2	no	—
222	C₄	—	332	2	no	—
223	C₄	—	664	1	no	—
224	C₄	#19127979	1	2	yes	—
225	C₄	#19128010	1	2	yes	—
226	C₄	#19128020	1	1	yes	characteristic
227	C₄	#19128032	1	1	yes	characteristic
228	C₄	#19128099	1	1	yes	characteristic
229	C₄	#19128110	1	1	yes	characteristic
230	C₂	#24906824	1	1	yes	characteristic
231	C₂	#24906853	1	1	yes	characteristic
232	C₂	#24906958	1	1	yes	characteristic
233	C₂	#24906987	1	1	yes	characteristic
234	C₂	#24908941	1	1	yes	characteristic
235	C₂	#24908953	1	1	yes	characteristic
236	C₂	#24908970	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581767
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1277	C₂
socle, prime radical	C₂³×C₈₃	#121
hypercenter	#1277	D₈₃
hypocenter	C₈₃	#10651
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
124. C₈₃	#10651	semidirect

Maximal subgroups

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

subgroup	index
#24908227	2
#24908181	2
#24907623	2
#24905347	2
C₈₃×#1277	2
#10651	83