Subgroups — #35291602

Order 21248.

246 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24907620	C₂	1	1	yes	maximal, characteristic
2	#24906211	C₂	1	1	yes	maximal, characteristic
3	C₈₃×#2150	C₂	1	1	yes	maximal, characteristic
4	#19127754	—	2	1	no	—
5	C₂×#14581724	—	2	1	no	—
6	C₂×#14581724	C₄	1	2	yes	—
7	#19127571	—	2	1	no	—
8	#19127454	C₄	1	2	yes	complemented
9	#19127452	—	2	1	no	—
10	C₂×M₃₂×C₈₃	—	2	1	no	—
11	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₈₃	—	2	1	no	—
13	C₄⋊C₁₆×C₈₃	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₈₃	—	2	2	no	—
15	C₂×C₄⋊Dic₈₃	—	4	1	no	—
16	C₂×C₄⋊Dic₈₃	—	4	1	no	—
17	#14581724	—	4	2	no	—
18	#14581724	—	4	1	no	—
19	#14581724	—	4	1	no	—
20	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
21	C₂²⋊(C₈₃⋊C₈)	—	4	1	no	—
22	C₄×C₈₃⋊C₈	—	4	1	no	—
23	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
26	M₃₂×C₈₃	—	4	1	no	—
27	M₃₂×C₈₃	—	4	1	no	—
28	C₂×C₁₆×C₈₃	—	2	2	no	—
29	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
33	C₂²×Dic₈₃	—	4	2	no	—
34	C₂×C₈₃⋊C₈	—	4	2	no	—
35	C₄⋊Dic₈₃	—	4	2	no	—
36	C₄⋊Dic₈₃	—	4	2	no	—
37	C₄⋊Dic₈₃	—	8	2	no	—
38	C₄⋊Dic₈₃	—	8	2	no	—
39	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
42	C₂×C₈×C₈₃	—	2	1	no	—
43	C₂×C₈×C₈₃	—	2	1	no	—
44	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
49	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₈₃	Q₁₆	1	1	yes	characteristic
52	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₈₃	SD₁₆	1	1	yes	characteristic
55	C₄²×C₈₃	Q₁₆	1	1	yes	characteristic
56	C₁₆×C₈₃	—	4	2	no	—
57	C₂³×C₈₃	#102	1	1	yes	characteristic
58	C₂×C₄×C₈₃	—	2	2	no	—
59	C₂×C₄×C₈₃	—	2	2	no	—
60	C₂×C₄×C₈₃	#95	1	1	yes	characteristic
61	C₂×C₄×C₈₃	#100	1	1	yes	characteristic
62	C₂×C₄×C₈₃	#101	1	1	yes	characteristic
63	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
64	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
65	C₂×C₄×C₈₃	#103	1	1	yes	characteristic
66	C₂×C₄×C₈₃	#103	1	1	yes	characteristic
67	C₂×C₄×C₈₃	#103	1	1	yes	characteristic
68	C₂×C₄×C₈₃	#106	1	1	yes	characteristic
69	C₂×C₄×C₈₃	#107	1	1	yes	characteristic
70	C₂×Dic₈₃	—	8	2	no	—
71	C₂×Dic₈₃	—	8	2	no	—
72	C₂×Dic₈₃	—	8	2	no	—
73	C₈×C₈₃	—	2	1	no	—
74	C₈×C₈₃	—	2	1	no	—
75	C₈×C₈₃	#102	1	1	yes	characteristic
76	C₈×C₈₃	#103	1	1	yes	characteristic
77	C₈×C₈₃	#106	1	1	yes	characteristic
78	C₈×C₈₃	#107	1	1	yes	characteristic
79	C₈₃⋊C₈	—	8	2	no	—
80	C₂²×C₈₃	—	2	1	no	—
81	C₂²×C₈₃	—	2	1	no	—
82	C₂²×C₈₃	#331	1	1	yes	characteristic
83	C₂²×C₈₃	#332	1	1	yes	characteristic
84	C₂²×C₈₃	#340	1	1	yes	characteristic
85	C₄×C₈₃	—	2	1	no	—
86	C₄×C₈₃	—	2	1	no	—
87	C₄×C₈₃	#332	1	1	yes	characteristic
88	C₄×C₈₃	#333	1	1	yes	characteristic
89	C₄×C₈₃	#340	1	1	yes	characteristic
90	C₄×C₈₃	#343	1	1	yes	characteristic
91	C₄×C₈₃	#360	1	2	yes	—
92	C₄×C₈₃	#365	1	2	yes	—
93	Dic₈₃	—	16	2	no	—
94	#7278	—	83	1	no	maximal, Sylow
95	C₂×C₈₃	—	2	1	no	—
96	C₂×C₈₃	—	2	1	no	—
97	C₂×C₈₃	#1296	1	1	yes	characteristic
98	C₂×C₈₃	#1383	1	1	yes	characteristic
99	C₂×C₈₃	#1386	1	1	yes	characteristic
100	#2150	D₈₃	1	1	yes	characteristic
101	#1844	—	83	1	no	—
102	#1564	—	83	1	no	—
103	C₈₃	#7278	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₃₂	—	2	1	no	—
105	#431	—	166	1	no	—
106	C₂×#106	—	83	2	no	—
107	C₂×#106	—	166	1	no	—
108	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
109	#384	—	166	1	no	—
110	C₄⋊C₁₆	—	2	1	no	—
111	C₄⋊C₁₆	Dic₈₃	1	2	yes	complemented
112	#335	—	83	2	no	—
113	#334	—	166	1	no	—
114	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	332	1	no	—
117	C₂×C₄⋊C₄	—	332	1	no	—
118	C₂×C₄⋊C₄	—	166	2	no	—
119	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
120	M₃₂	—	4	1	no	—
121	M₃₂	—	4	1	no	—
122	C₂×C₁₆	—	2	2	no	—
123	#106	—	332	1	no	—
124	#106	—	332	1	no	—
125	#106	—	332	2	no	—
126	C₄⋊C₈	—	166	2	no	—
127	C₂²⋊C₈	—	332	1	no	—
128	C₄×C₈	—	332	1	no	—
129	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
130	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
131	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
132	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
133	C₂²×C₄	—	332	2	no	—
134	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
135	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
136	C₂²×C₄	#13050579	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	332	2	no	—
140	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
141	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
142	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
143	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
144	C₂×C₈	#13050575	1	1	yes	characteristic
145	C₂×C₈	#13050577	1	1	yes	characteristic
146	C₂×C₈	#13050578	1	1	yes	characteristic
147	C₂×C₈	#13050580	1	1	yes	characteristic
148	C₄⋊C₄	—	332	2	no	—
149	C₄⋊C₄	—	332	2	no	—
150	C₄⋊C₄	—	664	2	no	—
151	C₄⋊C₄	—	664	2	no	—
152	C₄²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
153	C₄²	C₄⋊Dic₈₃	1	1	yes	characteristic
154	C₄²	#13050579	1	1	yes	characteristic
155	C₄²	Dic₃₃₂	1	1	yes	characteristic
156	C₁₆	—	4	2	no	—
157	C₂³	#14581714	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	664	2	no	—
161	C₂×C₄	—	664	2	no	—
162	C₂×C₄	—	664	2	no	—
163	C₂×C₄	C₈₃⋊#95	1	1	yes	characteristic
164	C₂×C₄	#14581708	1	1	yes	characteristic
165	C₂×C₄	#14581710	1	1	yes	characteristic
166	C₂×C₄	#14581712	1	1	yes	characteristic
167	C₂×C₄	#14581714	1	1	yes	characteristic
168	C₂×C₄	#14581715	1	1	yes	characteristic
169	C₂×C₄	#14581717	1	1	yes	characteristic
170	C₂×C₄	#14581717	1	1	yes	characteristic
171	C₂×C₄	#14581724	1	1	yes	characteristic
172	C₂×C₄	#14581726	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	664	2	no	—
176	C₈	#14581713	1	1	yes	characteristic
177	C₈	#14581716	1	1	yes	characteristic
178	C₈	#14581723	1	1	yes	characteristic
179	C₈	#14581725	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	#19127444	1	1	yes	characteristic
183	C₂²	#19127446	1	1	yes	characteristic
184	C₂²	#19127464	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	1328	2	no	—
188	C₄	#19127448	1	1	yes	characteristic
189	C₄	#19127449	1	1	yes	characteristic
190	C₄	#19127463	1	1	yes	characteristic
191	C₄	#19127469	1	1	yes	characteristic
192	C₄	#19127512	1	2	yes	—
193	C₄	#19127524	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#24905347	1	1	yes	characteristic
197	C₂	#24905562	1	1	yes	characteristic
198	C₂	#24905569	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19127464
commutator	C₂×C₈×C₈₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₈₃
Fitting	C₈₃×#2150	C₂
socle	C₂²×C₈₃	#340
prime radical	C₂³×C₈₃	#102
cosocle	C₂×C₄×C₈×C₈₃	C₂²
hypercenter	#2150	D₈₃
hypocenter	C₈₃	#7278
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
8. #19127454	C₄	semidirect
13. C₄⋊C₁₆×C₈₃	C₄	semidirect
103. C₈₃	#7278	semidirect
111. C₄⋊C₁₆	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24907620	2
#24906211	2
C₈₃×#2150	2
#7278	83