Subgroups — #35320777

Order 21248.

302 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14581732	C₂	1	1	yes	maximal, characteristic
2	#24907620	C₂	1	1	yes	maximal, characteristic
3	#24905563	C₂	1	4	yes	maximal
4	C₈₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581732	C₂²	1	2	yes	—
6	C₂×#14581724	—	2	1	no	—
7	C₂×#14581724	C₂²	1	2	yes	—
8	#19127571	—	2	1	no	—
9	#19127477	C₄	1	2	yes	complemented
10	C₄×C₈₃⋊C₁₆	C₄	1	2	yes	complemented
11	C₂×#108×C₈₃	—	2	1	no	—
12	C₂×#108×C₈₃	C₂²	1	2	yes	—
13	#423×C₈₃	—	2	1	no	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₈₃	—	2	2	no	—
16	C₂×C₄⋊Dic₈₃	—	4	2	no	—
17	#14581732	—	2	2	no	—
18	#14581732	—	2	2	no	—
19	C₂×C₈₃⋊C₁₆	C₂×C₄	1	4	yes	—
20	#14581724	—	2	4	no	—
21	#14581724	—	4	2	no	—
22	C₂×M₁₆×C₈₃	—	2	2	no	—
23	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
26	#108×C₈₃	—	2	4	no	—
27	#108×C₈₃	—	4	2	no	—
28	C₄⋊C₈×C₈₃	—	4	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	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
33	C₂²×Dic₈₃	—	4	2	no	—
34	C₄⋊Dic₈₃	—	4	2	no	—
35	C₄⋊Dic₈₃	—	4	2	no	—
36	C₄⋊Dic₈₃	—	8	4	no	—
37	C₈₃⋊C₁₆	—	2	4	no	—
38	C₂²×C₄×C₈₃	SD₁₆	1	2	yes	—
39	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
40	M₁₆×C₈₃	—	4	2	no	—
41	M₁₆×C₈₃	—	4	2	no	—
42	C₂×C₈×C₈₃	—	4	2	no	—
43	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₃	D₈	1	2	yes	—
46	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
47	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₈₃	#41	1	2	yes	—
50	C₂×C₈×C₈₃	#41	1	2	yes	—
51	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₈₃	SD₁₆	1	2	yes	—
54	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₈₃	—	2	4	no	—
56	C₂×C₄×C₈₃	#106	1	2	yes	—
57	C₂×C₄×C₈₃	#106	1	2	yes	—
58	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×Dic₈₃	—	8	4	no	—
65	C₂×Dic₈₃	—	8	2	no	—
66	C₈×C₈₃	—	8	2	no	—
67	C₈×C₈₃	#107	1	2	yes	—
68	C₈×C₈₃	#107	1	2	yes	—
69	C₈×C₈₃	#117	1	2	yes	—
70	C₈×C₈₃	#125	1	2	yes	—
71	C₂²×C₈₃	—	2	2	no	—
72	C₂²×C₈₃	C₂×#106	1	1	yes	characteristic
73	C₂²×C₈₃	#486	1	1	yes	characteristic
74	C₂²×C₈₃	#487	1	1	yes	characteristic
75	C₄×C₈₃	—	2	2	no	—
76	C₄×C₈₃	#365	1	4	yes	—
77	C₄×C₈₃	#382	1	1	yes	characteristic
78	C₄×C₈₃	C₂×#107	1	1	yes	characteristic
79	C₄×C₈₃	#486	1	1	yes	characteristic
80	C₄×C₈₃	#487	1	1	yes	characteristic
81	Dic₈₃	—	16	2	no	—
82	#12886	—	83	1	no	maximal, Sylow
83	C₂×C₈₃	—	2	1	no	—
84	C₂×C₈₃	—	2	1	no	—
85	C₂×C₈₃	#1834	1	1	yes	characteristic
86	C₂×C₈₃	C₂×#365	1	2	yes	—
87	C₄×M₃₂	—	83	1	no	—
88	#1848	D₈₃	1	1	yes	characteristic
89	#1844	—	83	1	no	—
90	#1383	—	83	4	no	—
91	C₈₃	#12886	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₃₂	—	83	2	no	—
93	C₂×#108	—	2	1	no	—
94	C₂×#108	C₂×D₈₃	1	2	yes	—
95	C₂×#106	—	83	2	no	—
96	C₂×#106	—	166	1	no	—
97	#423	—	2	1	no	—
98	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
99	#384	—	166	1	no	—
100	#346	—	83	2	no	—
101	C₄×C₁₆	—	83	2	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	332	2	no	—
106	C₂×C₄⋊C₄	—	166	2	no	—
107	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
108	M₃₂	—	166	2	no	—
109	M₃₂	—	166	2	no	—
110	C₂×C₁₆	—	83	4	no	—
111	#108	—	2	4	no	—
112	#108	—	4	2	no	—
113	#106	—	332	2	no	—
114	#106	—	166	4	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
117	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
118	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
119	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
120	C₂²×C₄	—	332	2	no	—
121	C₂²×C₄	#13050577	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
127	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
128	C₂×C₈	#13050575	1	2	yes	—
129	C₂×C₈	#13050580	1	2	yes	—
130	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
131	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
132	C₂×C₈	#13050589	1	2	yes	—
133	C₂×C₈	#13050591	1	2	yes	—
134	C₄⋊C₄	—	332	2	no	—
135	C₄⋊C₄	—	332	2	no	—
136	C₄⋊C₄	—	664	4	no	—
137	C₄²	Dic₈₃⋊C₄	1	1	yes	characteristic
138	C₄²	Dic₈₃⋊C₄	1	1	yes	characteristic
139	C₄²	#13050578	1	2	yes	—
140	C₁₆	—	166	4	no	—
141	C₂³	C₂×#13050577	1	1	yes	characteristic
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	664	4	no	—
144	C₂×C₄	—	664	2	no	—
145	C₂×C₄	#14581723	1	2	yes	—
146	C₂×C₄	#14581723	1	2	yes	—
147	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
148	C₂×C₄	#14581779	1	1	yes	characteristic
149	C₂×C₄	#14581781	1	1	yes	characteristic
150	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
151	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
152	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
153	C₈	—	8	2	no	—
154	C₈	#14581725	1	2	yes	—
155	C₈	#14581725	1	2	yes	—
156	C₈	#14581750	1	2	yes	—
157	C₈	#14581783	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂×#14581723	1	1	yes	characteristic
160	C₂²	#19128049	1	1	yes	characteristic
161	C₂²	#19128051	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	1328	2	no	—
164	C₄	#19127525	1	4	yes	—
165	C₄	#19127566	1	1	yes	characteristic
166	C₄	C₂×#14581725	1	1	yes	characteristic
167	C₄	#19128047	1	1	yes	characteristic
168	C₄	#19128053	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#24907590	1	1	yes	characteristic
172	C₂	C₂×#19127525	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14581723
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1848	C₂
socle	C₂²×C₈₃	C₂×#106
prime radical	C₂³×C₈₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1848	D₈₃
hypocenter	C₈₃	#12886
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
9. #19127477	C₄	semidirect
10. C₄×C₈₃⋊C₁₆	C₄	semidirect
91. C₈₃	#12886	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14581732	2
#24907620	2
#24905563	2
C₈₃×#1848	2
#12886	83