Subgroups — #35316049

Order 21248.

242 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24906222	C₂	1	2	yes	maximal
2	#24905537	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
4	#19127725	—	2	2	no	—
5	#19127454	—	2	2	no	—
6	#19127454	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₈₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₈₃	C₄	1	1	yes	characteristic
10	C₄×C₁₆×C₈₃	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
12	C₂×#13050574	—	4	2	no	—
13	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
14	C₄⋊(C₈₃⋊C₈)	—	4	2	no	—
15	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₈₃	Q₈	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	2	yes	—
20	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
25	#13050574	—	8	4	no	—
26	C₂×C₈₃⋊C₈	—	4	4	no	—
27	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
29	C₂²×C₄×C₈₃	Q₁₆	1	1	yes	characteristic
30	C₂×C₈×C₈₃	—	2	2	no	—
31	C₂×C₈×C₈₃	—	2	2	no	—
32	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₈₃	D₈	1	1	yes	characteristic
38	C₄²×C₈₃	Q₁₆	1	1	yes	characteristic
39	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₈₃	—	2	2	no	—
41	C₁₆×C₈₃	M₁₆	1	4	yes	complemented
42	C₂³×C₈₃	#107	1	1	yes	characteristic
43	C₂×C₄×C₈₃	—	2	2	no	—
44	C₂×C₄×C₈₃	—	2	1	no	—
45	C₂×C₄×C₈₃	—	2	1	no	—
46	C₂×C₄×C₈₃	#107	1	1	yes	characteristic
47	C₂×C₄×C₈₃	#107	1	1	yes	characteristic
48	C₂×C₄×C₈₃	#107	1	1	yes	characteristic
49	C₂×C₄×C₈₃	SD₃₂	1	4	yes	—
50	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
53	C₈×C₈₃	—	2	1	no	—
54	C₈×C₈₃	—	2	1	no	—
55	C₈×C₈₃	#108	1	2	yes	—
56	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
57	C₈₃⋊C₈	—	8	4	no	—
58	C₂²×C₈₃	—	2	1	no	—
59	C₂²×C₈₃	—	2	1	no	—
60	C₂²×C₈₃	#367	1	2	yes	—
61	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
62	C₄×C₈₃	—	2	1	no	—
63	C₄×C₈₃	—	2	1	no	—
64	C₄×C₈₃	#367	1	2	yes	—
65	C₄×C₈₃	#368	1	2	yes	—
66	C₄×C₈₃	#423	1	1	yes	characteristic
67	C₄×C₈₃	C₂×#108	1	1	yes	characteristic
68	C₄×C₈₃	C₂×SD₃₂	1	2	yes	—
69	#12023	—	83	1	no	maximal, Sylow
70	C₂×C₈₃	—	2	2	no	—
71	C₂×C₈₃	#1567	1	1	yes	characteristic
72	C₂×C₈₃	C₂×#367	1	1	yes	characteristic
73	C₂×C₈₃	C₂×#368	1	1	yes	characteristic
74	C₂×C₄×C₁₆	D₈₃	1	1	yes	characteristic
75	#1567	—	83	2	no	—
76	#1372	—	83	4	no	—
77	C₈₃	#12023	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₁₆	Dic₈₃	1	2	yes	—
79	#423	—	166	2	no	—
80	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
81	C₄×C₁₆	Dic₈₃	1	1	yes	characteristic
82	C₄×C₁₆	Dic₈₃	1	1	yes	characteristic
83	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
84	#335	—	83	4	no	—
85	#335	—	166	2	no	—
86	C₂×M₁₆	—	332	2	no	—
87	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
88	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
89	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
90	C₂×C₁₆	—	2	4	no	—
91	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
92	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
93	C₄⋊C₈	—	332	2	no	—
94	C₄⋊C₈	—	166	4	no	—
95	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
96	C₄×C₈	D₃₃₂	1	1	yes	characteristic
97	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
99	C₂²×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
100	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
101	C₂²×C₄	Dic₃₃₂	1	1	yes	characteristic
102	M₁₆	—	664	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	332	4	no	—
106	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
107	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
108	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
109	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
110	C₄²	C₄⋊Dic₈₃	1	1	yes	characteristic
111	C₄²	D₆₆₄	1	1	yes	characteristic
112	C₄²	Dic₃₃₂	1	1	yes	characteristic
113	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
114	C₁₆	—	2	2	no	—
115	C₁₆	#13050574	1	4	yes	complemented
116	C₂³	#14581726	1	1	yes	characteristic
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	#14581726	1	1	yes	characteristic
121	C₂×C₄	#14581726	1	1	yes	characteristic
122	C₂×C₄	#14581726	1	1	yes	characteristic
123	C₂×C₄	#14581737	1	4	yes	—
124	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
125	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
126	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
127	C₈	—	2	1	no	—
128	C₈	—	2	1	no	—
129	C₈	—	664	4	no	—
130	C₈	#14581728	1	2	yes	—
131	C₈	C₂×#13050574	1	2	yes	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#19127529	1	2	yes	—
135	C₂²	C₂×#14581726	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	#19127529	1	2	yes	—
139	C₄	#19127531	1	2	yes	—
140	C₄	#19127725	1	1	yes	characteristic
141	C₄	C₂×#14581728	1	1	yes	characteristic
142	C₄	C₂×#14581737	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	#24906222	1	1	yes	characteristic
145	C₂	C₂×#19127529	1	1	yes	characteristic
146	C₂	C₂×#19127531	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₆₄
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₂×C₄×C₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#107
prime radical	C₂³×C₈₃	#107
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₈₃
hypocenter	C₈₃	#12023
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #24905537	C₂	semidirect
41. C₁₆×C₈₃	M₁₆	semidirect
77. C₈₃	#12023	semidirect
115. C₁₆	#13050574	semidirect

Maximal subgroups

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

subgroup	index
#24906222	2
#24905537	2
C₂×C₄×C₁₆×C₈₃	2
#12023	83