Subgroups — #35320783

Order 21248.

342 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	C₂×C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#24907623	C₂	1	1	yes	maximal, characteristic
3	#24905554	C₂	1	4	yes	maximal
4	C₈₃×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14581726	—	2	1	no	—
7	C₂×#14581726	C₂²	1	2	yes	—
8	#19127571	—	2	1	no	—
9	C₄×C₈₃⋊C₁₆	C₄	1	4	yes	complemented
10	C₂×#107×C₈₃	—	2	1	no	—
11	C₂×#107×C₈₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	#384×C₈₃	—	2	1	no	—
14	C₂×C₄⋊Dic₈₃	—	2	2	no	—
15	C₂×C₄⋊Dic₈₃	—	4	1	no	—
16	C₂×C₄⋊Dic₈₃	—	4	1	no	—
17	C₂×C₈₃⋊C₁₆	—	2	2	no	—
18	C₂×C₈₃⋊C₁₆	—	2	2	no	—
19	C₂×C₈₃⋊C₁₆	C₂×C₄	1	4	yes	—
20	#14581726	—	2	4	no	—
21	#14581726	—	4	2	no	—
22	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
26	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
27	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
28	#107×C₈₃	—	2	4	no	—
29	#107×C₈₃	—	4	2	no	—
30	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₈₃	Q₈	1	2	yes	—
32	C₂²×Dic₈₃	—	4	2	no	—
33	C₄⋊Dic₈₃	—	4	2	no	—
34	C₄⋊Dic₈₃	—	4	2	no	—
35	C₄⋊Dic₈₃	—	8	2	no	—
36	C₄⋊Dic₈₃	—	8	2	no	—
37	C₈₃⋊C₁₆	—	2	4	no	—
38	C₂²×C₄×C₈₃	—	4	2	no	—
39	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
44	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
45	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₈₃	#41	1	2	yes	—
48	C₂×C₈×C₈₃	#41	1	2	yes	—
49	C₄⋊C₄×C₈₃	—	4	2	no	—
50	C₄⋊C₄×C₈₃	—	4	2	no	—
51	C₄⋊C₄×C₈₃	—	8	2	no	—
52	C₄⋊C₄×C₈₃	—	8	2	no	—
53	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
54	C₄²×C₈₃	Q₁₆	1	2	yes	—
55	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₈₃	—	8	2	no	—
57	C₂×C₄×C₈₃	—	8	2	no	—
58	C₂×C₄×C₈₃	—	8	2	no	—
59	C₂×C₄×C₈₃	#107	1	2	yes	—
60	C₂×C₄×C₈₃	#107	1	2	yes	—
61	C₂×C₄×C₈₃	D₁₆	1	2	yes	—
62	C₂×C₄×C₈₃	SD₃₂	1	2	yes	—
63	C₂×C₄×C₈₃	SD₃₂	1	2	yes	—
64	C₂×C₄×C₈₃	Q₃₂	1	2	yes	—
65	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
67	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₈₃	—	8	2	no	—
72	C₂×Dic₈₃	—	8	2	no	—
73	C₂×Dic₈₃	—	8	2	no	—
74	C₈×C₈₃	#106	1	4	yes	—
75	C₈×C₈₃	#117	1	2	yes	—
76	C₈×C₈₃	#125	1	2	yes	—
77	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
78	C₂²×C₈₃	#486	1	1	yes	characteristic
79	C₂²×C₈₃	#486	1	1	yes	characteristic
80	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
81	C₂²×C₈₃	C₂×SD₃₂	1	1	yes	characteristic
82	C₂²×C₈₃	C₂×SD₃₂	1	1	yes	characteristic
83	C₂²×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
84	C₄×C₈₃	—	16	2	no	—
85	C₄×C₈₃	#366	1	4	yes	—
86	C₄×C₈₃	#367	1	4	yes	—
87	C₄×C₈₃	#382	1	1	yes	characteristic
88	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
89	C₄×C₈₃	#487	1	1	yes	characteristic
90	C₄×C₈₃	#487	1	1	yes	characteristic
91	Dic₈₃	—	16	2	no	—
92	#12888	—	83	1	no	maximal, Sylow
93	C₂×C₈₃	#1834	1	1	yes	characteristic
94	C₂×C₈₃	C₂×#366	1	1	yes	characteristic
95	C₂×C₈₃	C₂×#367	1	1	yes	characteristic
96	C₂×C₈₃	#2240	1	1	yes	characteristic
97	C₂×C₈₃	#2240	1	1	yes	characteristic
98	C₂×C₈₃	#2241	1	1	yes	characteristic
99	C₂×C₈₃	#2241	1	1	yes	characteristic
100	C₂×C₄×C₁₆	—	83	1	no	—
101	#1845	—	83	1	no	—
102	#1845	D₈₃	1	1	yes	characteristic
103	#1380	—	83	4	no	—
104	C₈₃	#12888	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₁₆	—	83	2	no	—
106	C₂×#107	—	2	1	no	—
107	C₂×#107	—	83	2	no	—
108	C₂×#107	—	166	1	no	—
109	C₂×#107	C₂×D₈₃	1	2	yes	—
110	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
111	#384	—	2	1	no	—
112	#384	—	166	1	no	—
113	C₄×C₁₆	—	83	4	no	—
114	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	4	1	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄⋊C₄	—	332	1	no	—
120	C₂×C₄⋊C₄	—	332	1	no	—
121	C₂×C₄⋊C₄	—	166	2	no	—
122	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
123	C₂×C₁₆	—	83	4	no	—
124	C₂×C₁₆	—	166	2	no	—
125	C₂×C₁₆	—	166	2	no	—
126	#107	—	2	4	no	—
127	#107	—	4	2	no	—
128	#107	—	332	2	no	—
129	#107	—	166	4	no	—
130	C₄×C₈	Dic₁₆₆	1	2	yes	—
131	C₄×C₈	C₄×D₈₃	1	2	yes	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	332	2	no	—
134	C₂²×C₄	#13050575	1	1	yes	characteristic
135	C₂²×C₄	#13050575	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
137	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
138	C₂×C₈	#13050577	1	2	yes	—
139	C₂×C₈	#13050578	1	2	yes	—
140	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
142	C₂×C₈	#13050589	1	2	yes	—
143	C₂×C₈	#13050591	1	2	yes	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	8	2	no	—
147	C₄⋊C₄	—	8	2	no	—
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	Dic₈₃⋊C₄	1	2	yes	—
153	C₄²	#13050580	1	2	yes	—
154	C₁₆	—	166	4	no	—
155	C₂³	C₂×#13050575	1	1	yes	characteristic
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	664	2	no	—
160	C₂×C₄	—	664	2	no	—
161	C₂×C₄	—	664	2	no	—
162	C₂×C₄	#14581725	1	2	yes	—
163	C₂×C₄	#14581725	1	2	yes	—
164	C₂×C₄	#14581733	1	2	yes	—
165	C₂×C₄	#14581735	1	2	yes	—
166	C₂×C₄	#14581736	1	2	yes	—
167	C₂×C₄	#14581738	1	2	yes	—
168	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
169	C₂×C₄	#14581779	1	1	yes	characteristic
170	C₂×C₄	#14581781	1	1	yes	characteristic
171	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
172	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
173	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
174	C₈	#14581723	1	4	yes	—
175	C₈	#14581750	1	2	yes	—
176	C₈	#14581783	1	2	yes	—
177	C₂²	C₂×#14581725	1	1	yes	characteristic
178	C₂²	#19128047	1	1	yes	characteristic
179	C₂²	#19128049	1	1	yes	characteristic
180	C₂²	C₂×#14581733	1	1	yes	characteristic
181	C₂²	C₂×#14581735	1	1	yes	characteristic
182	C₂²	C₂×#14581736	1	1	yes	characteristic
183	C₂²	C₂×#14581738	1	1	yes	characteristic
184	C₄	—	16	2	no	—
185	C₄	—	1328	2	no	—
186	C₄	#19127526	1	4	yes	—
187	C₄	#19127528	1	4	yes	—
188	C₄	#19127566	1	1	yes	characteristic
189	C₄	C₂×#14581723	1	1	yes	characteristic
190	C₄	#19128051	1	1	yes	characteristic
191	C₄	#19128053	1	1	yes	characteristic
192	C₂	#24907590	1	1	yes	characteristic
193	C₂	C₂×#19127526	1	1	yes	characteristic
194	C₂	C₂×#19127528	1	1	yes	characteristic
195	C₂	#24909686	1	1	yes	characteristic
196	C₂	#24909688	1	1	yes	characteristic
197	C₂	#24909690	1	1	yes	characteristic
198	C₂	#24909692	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13050575
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1845	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1845	D₈₃
hypocenter	C₈₃	#12888
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
9. C₄×C₈₃⋊C₁₆	C₄	semidirect
104. C₈₃	#12888	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈₃⋊C₁₆	2
#24907623	2
#24905554	2
C₈₃×#1845	2
#12888	83