Subgroups — #35320784

Order 21248.

342 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24907623	C₂	1	2	yes	maximal
2	#24905555	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581726	—	2	2	no	—
5	C₂×#14581726	C₂²	1	4	yes	—
6	#19127571	—	2	2	no	—
7	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₈₃	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₈₃	—	2	4	no	—
11	C₂×C₄⋊Dic₈₃	—	4	2	no	—
12	C₂×C₄⋊Dic₈₃	—	4	2	no	—
13	#14581726	—	2	8	no	—
14	#14581726	—	4	4	no	—
15	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₈₃	—	2	4	no	—
19	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₈₃	Q₈	1	2	yes	—
22	C₂²×Dic₈₃	—	4	4	no	—
23	C₄⋊Dic₈₃	—	4	4	no	—
24	C₄⋊Dic₈₃	—	4	4	no	—
25	C₄⋊Dic₈₃	—	8	4	no	—
26	C₄⋊Dic₈₃	—	8	4	no	—
27	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
28	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
29	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
32	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₈₃	#41	1	4	yes	—
35	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
36	C₄²×C₈₃	Q₁₆	1	2	yes	—
37	C₁₆×C₈₃	—	2	4	no	—
38	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
39	C₂×C₄×C₈₃	#107	1	2	yes	—
40	C₂×C₄×C₈₃	#107	1	2	yes	—
41	C₂×C₄×C₈₃	D₁₆	1	2	yes	—
42	C₂×C₄×C₈₃	SD₃₂	1	4	yes	—
43	C₂×C₄×C₈₃	Q₃₂	1	2	yes	—
44	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	#124	1	2	yes	—
46	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
47	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×Dic₈₃	—	8	4	no	—
49	C₂×Dic₈₃	—	8	4	no	—
50	C₂×Dic₈₃	—	8	4	no	—
51	C₈×C₈₃	#106	1	4	yes	—
52	C₈×C₈₃	#117	1	2	yes	—
53	C₈×C₈₃	#125	1	2	yes	—
54	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
55	C₂²×C₈₃	#486	1	2	yes	—
56	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
57	C₂²×C₈₃	C₂×SD₃₂	1	2	yes	—
58	C₂²×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
59	C₄×C₈₃	#366	1	4	yes	—
60	C₄×C₈₃	#367	1	4	yes	—
61	C₄×C₈₃	#382	1	1	yes	characteristic
62	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
63	C₄×C₈₃	#487	1	2	yes	—
64	Dic₈₃	—	16	4	no	—
65	#12888	—	83	1	no	maximal, Sylow
66	C₂×C₈₃	#1834	1	1	yes	characteristic
67	C₂×C₈₃	C₂×#366	1	1	yes	characteristic
68	C₂×C₈₃	C₂×#367	1	1	yes	characteristic
69	C₂×C₈₃	#2240	1	2	yes	—
70	C₂×C₈₃	#2241	1	2	yes	—
71	C₂×C₄×C₁₆	D₈₃	1	1	yes	characteristic
72	#1845	—	83	2	no	—
73	#1380	—	83	4	no	—
74	C₈₃	#12888	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
76	C₂×#107	—	83	4	no	—
77	C₂×#107	—	166	2	no	—
78	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
79	#384	—	166	2	no	—
80	C₄×C₁₆	Dic₈₃	1	4	yes	complemented
81	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
82	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
83	C₂×C₄⋊C₄	—	332	2	no	—
84	C₂×C₄⋊C₄	—	332	2	no	—
85	C₂×C₄⋊C₄	—	166	4	no	—
86	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	C₂×Dic₈₃	1	4	yes	—
89	#107	—	332	4	no	—
90	#107	—	166	8	no	—
91	C₄×C₈	Dic₁₆₆	1	2	yes	—
92	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
93	C₂²×C₄	—	332	4	no	—
94	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
95	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
96	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
97	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
98	C₂×C₈	#13050579	1	4	yes	—
99	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
100	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
101	C₂×C₈	#13050590	1	4	yes	—
102	C₄⋊C₄	—	332	4	no	—
103	C₄⋊C₄	—	332	4	no	—
104	C₄⋊C₄	—	664	4	no	—
105	C₄⋊C₄	—	664	4	no	—
106	C₄²	C₄⋊Dic₈₃	1	2	yes	—
107	C₄²	Dic₃₃₂	1	2	yes	—
108	C₁₆	—	2	4	no	—
109	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
110	C₂×C₄	—	664	4	no	—
111	C₂×C₄	—	664	4	no	—
112	C₂×C₄	—	664	4	no	—
113	C₂×C₄	#14581726	1	2	yes	—
114	C₂×C₄	#14581726	1	2	yes	—
115	C₂×C₄	D₁₃₂₈	1	2	yes	—
116	C₂×C₄	#14581737	1	4	yes	—
117	C₂×C₄	Dic₆₆₄	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
119	C₂×C₄	#14581780	1	2	yes	—
120	C₂×C₄	C₂×#13050579	1	2	yes	—
121	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
122	C₈	#14581724	1	4	yes	—
123	C₈	#14581751	1	2	yes	—
124	C₈	#14581784	1	2	yes	—
125	C₂²	C₂×#14581726	1	1	yes	characteristic
126	C₂²	#19128048	1	2	yes	—
127	C₂²	C₂×D₁₃₂₈	1	1	yes	characteristic
128	C₂²	C₂×#14581737	1	2	yes	—
129	C₂²	C₂×Dic₆₆₄	1	1	yes	characteristic
130	C₄	—	1328	4	no	—
131	C₄	#19127527	1	4	yes	—
132	C₄	#19127529	1	4	yes	—
133	C₄	#19127567	1	1	yes	characteristic
134	C₄	C₂×#14581724	1	1	yes	characteristic
135	C₄	#19128052	1	2	yes	—
136	C₂	#24907591	1	1	yes	characteristic
137	C₂	C₂×#19127527	1	1	yes	characteristic
138	C₂	C₂×#19127529	1	1	yes	characteristic
139	C₂	#24909687	1	2	yes	—
140	C₂	#24909691	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₆₆₄
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×C₄×C₁₆×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₈₃
hypocenter	C₈₃	#12888
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×C₁₆×C₈₃	C₄	semidirect
74. C₈₃	#12888	semidirect
80. 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
#24907623	2
#24905555	2
C₂×C₄×C₁₆×C₈₃	2
#12888	83