Subgroups — #35324339

Order 21248.

310 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127526	C₂	1	2	yes	maximal
2	#24907623	C₂	1	1	yes	maximal, characteristic
3	#24905554	C₂	1	2	yes	maximal
4	#24905541	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1942	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14581726	—	2	1	no	—
8	C₂×#14581726	C₂²	1	2	yes	—
9	#19127571	—	2	1	no	—
10	#19127526	—	2	4	no	—
11	C₂×#107×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×#107×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	#389×C₈₃	—	2	1	no	—
15	#340×C₈₃	—	2	1	no	—
16	C₂×C₄⋊Dic₈₃	—	2	2	no	—
17	C₂×C₄⋊Dic₈₃	—	4	1	no	—
18	C₂×C₄⋊Dic₈₃	—	4	1	no	—
19	C₂×C₈₃⋊C₁₆	—	2	2	no	—
20	C₂×C₈₃⋊C₁₆	—	2	2	no	—
21	C₂×C₈₃⋊C₁₆	Q₈	1	4	yes	—
22	#14581726	—	2	2	no	—
23	#14581726	—	2	2	no	—
24	#14581726	—	4	2	no	—
25	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
30	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
31	#107×C₈₃	—	2	1	no	—
32	#107×C₈₃	—	2	2	no	—
33	#107×C₈₃	D₄	1	2	yes	—
34	C₄×C₈×C₈₃	—	2	2	no	—
35	#95×C₈₃	—	4	1	no	—
36	C₂²×Dic₈₃	—	4	2	no	—
37	C₄⋊Dic₈₃	—	4	2	no	—
38	C₄⋊Dic₈₃	—	4	2	no	—
39	C₄⋊Dic₈₃	—	8	2	no	—
40	C₄⋊Dic₈₃	—	8	2	no	—
41	C₈₃⋊C₁₆	—	2	4	no	—
42	C₂²×C₄×C₈₃	—	4	1	no	—
43	C₂²×C₄×C₈₃	—	4	1	no	—
44	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₈₃	—	2	1	no	—
48	C₂×C₈×C₈₃	—	2	1	no	—
49	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
50	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₈₃	#41	1	2	yes	—
53	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₈₃	—	2	2	no	—
55	C₄⋊C₄×C₈₃	—	4	1	no	—
56	C₄⋊C₄×C₈₃	—	4	1	no	—
57	C₄⋊C₄×C₈₃	—	4	1	no	—
58	C₄⋊C₄×C₈₃	—	8	2	no	—
59	C₄²×C₈₃	—	2	1	no	—
60	C₄²×C₈₃	—	2	1	no	—
61	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₈₃	—	2	1	no	—
63	C₂×C₄×C₈₃	—	2	1	no	—
64	C₂×C₄×C₈₃	—	2	1	no	—
65	C₂×C₄×C₈₃	—	2	1	no	—
66	C₂×C₄×C₈₃	—	4	2	no	—
67	C₂×C₄×C₈₃	—	8	1	no	—
68	C₂×C₄×C₈₃	—	8	1	no	—
69	C₂×C₄×C₈₃	—	8	1	no	—
70	C₂×C₄×C₈₃	—	8	1	no	—
71	C₂×C₄×C₈₃	—	8	1	no	—
72	C₂×C₄×C₈₃	D₁₆	1	2	yes	—
73	C₂×C₄×C₈₃	Q₃₂	1	2	yes	—
74	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
80	C₂×Dic₈₃	—	8	2	no	—
81	C₂×Dic₈₃	—	8	2	no	—
82	C₂×Dic₈₃	—	8	2	no	—
83	C₈×C₈₃	—	2	2	no	—
84	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
85	C₈×C₈₃	#125	1	2	yes	—
86	C₂²×C₈₃	#459	1	1	yes	characteristic
87	C₂²×C₈₃	#480	1	1	yes	characteristic
88	C₂²×C₈₃	#500	1	1	yes	characteristic
89	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
90	C₂²×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
91	C₂²×C₈₃	#509	1	1	yes	characteristic
92	C₂²×C₈₃	#510	1	1	yes	characteristic
93	C₄×C₈₃	—	2	2	no	—
94	C₄×C₈₃	—	2	2	no	—
95	C₄×C₈₃	—	8	2	no	—
96	C₄×C₈₃	—	16	1	no	—
97	C₄×C₈₃	#398	1	1	yes	characteristic
98	C₄×C₈₃	#462	1	1	yes	characteristic
99	C₄×C₈₃	#484	1	1	yes	characteristic
100	C₄×C₈₃	#501	1	1	yes	characteristic
101	Dic₈₃	—	16	2	no	—
102	#13505	—	83	1	no	maximal, Sylow
103	C₂×C₈₃	#2083	1	1	yes	characteristic
104	C₂×C₈₃	#2206	1	1	yes	characteristic
105	C₂×C₈₃	#2209	1	1	yes	characteristic
106	C₂×C₈₃	#2232	1	1	yes	characteristic
107	C₂×C₈₃	#2236	1	1	yes	characteristic
108	C₂×C₈₃	#2252	1	1	yes	characteristic
109	C₂×C₈₃	#2255	1	1	yes	characteristic
110	C₂×#366	—	83	2	no	—
111	#1942	D₈₃	1	1	yes	characteristic
112	#1845	—	83	1	no	—
113	#1380	—	83	2	no	—
114	#1374	—	83	1	no	—
115	C₈₃	#13505	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₁₆	—	83	2	no	—
117	C₂×#107	—	83	2	no	—
118	C₂×#107	—	166	1	no	—
119	C₂×#107	C₂×D₈₃	1	1	yes	characteristic
120	C₂×#107	C₂×D₈₃	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
122	#389	—	2	1	no	—
123	#384	—	166	1	no	—
124	#366	—	166	4	no	—
125	#340	—	2	1	no	—
126	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	332	1	no	—
132	C₂×C₄⋊C₄	—	332	1	no	—
133	C₂×C₄⋊C₄	—	166	2	no	—
134	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
135	C₂×C₁₆	—	83	4	no	—
136	C₂×C₁₆	—	166	2	no	—
137	C₂×C₁₆	—	166	2	no	—
138	#107	—	2	1	no	—
139	#107	—	2	2	no	—
140	#107	—	332	2	no	—
141	#107	—	166	2	no	—
142	#107	—	166	2	no	—
143	#107	D₃₃₂	1	2	yes	—
144	C₄×C₈	—	2	2	no	—
145	#95	—	4	1	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	332	2	no	—
149	C₂²×C₄	#13050575	1	1	yes	characteristic
150	C₂²×C₄	#13050575	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	#13050580	1	2	yes	—
155	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
156	C₂×C₈	Q₈×D₈₃	1	2	yes	—
157	C₂×C₈	#13050589	1	2	yes	—
158	C₂×C₈	#13050590	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	4	1	no	—
161	C₄⋊C₄	—	4	1	no	—
162	C₄⋊C₄	—	4	1	no	—
163	C₄⋊C₄	—	8	2	no	—
164	C₄⋊C₄	—	332	2	no	—
165	C₄⋊C₄	—	332	2	no	—
166	C₄⋊C₄	—	664	2	no	—
167	C₄⋊C₄	—	664	2	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	2	1	no	—
170	C₁₆	—	166	4	no	—
171	C₂³	C₂×#13050575	1	1	yes	characteristic
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	8	1	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	—	8	1	no	—
181	C₂×C₄	—	8	1	no	—
182	C₂×C₄	—	664	2	no	—
183	C₂×C₄	—	664	2	no	—
184	C₂×C₄	—	664	2	no	—
185	C₂×C₄	#14581733	1	2	yes	—
186	C₂×C₄	#14581738	1	2	yes	—
187	C₂×C₄	#14581766	1	1	yes	characteristic
188	C₂×C₄	#14581773	1	1	yes	characteristic
189	C₂×C₄	#14581790	1	1	yes	characteristic
190	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
191	C₂×C₄	#14581817	1	1	yes	characteristic
192	C₂×C₄	#14581822	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	#14581772	1	2	yes	—
195	C₈	#14581782	1	2	yes	—
196	C₂²	#19127896	1	1	yes	characteristic
197	C₂²	#19128012	1	1	yes	characteristic
198	C₂²	#19128104	1	1	yes	characteristic
199	C₂²	C₂×#14581733	1	1	yes	characteristic
200	C₂²	C₂×#14581738	1	1	yes	characteristic
201	C₂²	#19128136	1	1	yes	characteristic
202	C₂²	#19128141	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	2	no	—
206	C₄	—	16	1	no	—
207	C₄	—	1328	2	no	—
208	C₄	#19127631	1	1	yes	characteristic
209	C₄	#19127911	1	1	yes	characteristic
210	C₄	#19128034	1	1	yes	characteristic
211	C₄	#19128111	1	1	yes	characteristic
212	C₂	#24908934	1	1	yes	characteristic
213	C₂	#24909498	1	1	yes	characteristic
214	C₂	#24909513	1	1	yes	characteristic
215	C₂	#24909641	1	1	yes	characteristic
216	C₂	#24909663	1	1	yes	characteristic
217	C₂	#24909736	1	1	yes	characteristic
218	C₂	#24909750	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13050575
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1942	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
hypercenter	#1942	D₈₃
hypocenter	C₈₃	#13505
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1942

Complements

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

N	H ≅ G/N	type
115. C₈₃	#13505	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127526	2
#24907623	2
#24905554	2
#24905541	2
C₈₃×#1942	2
#13505	83