Subgroups — #35313408

Order 21248.

282 conjugacy classes of subgroups.

The subgroup lattice has 136 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	#24905746	C₂	1	1	yes	maximal, characteristic
3	#24905383	C₂	1	4	yes	maximal, complemented
4	C₂×#346×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581732	C₄	1	2	yes	—
6	#19127754	C₄	1	2	yes	—
7	#19127477	C₂²	1	2	yes	—
8	C₄×C₈₃⋊C₁₆	C₄	1	2	yes	—
9	#19127422	C₂²	1	2	yes	—
10	C₈×C₈₃⋊C₈	C₄	1	2	yes	—
11	C₂²×C₁₆×C₈₃	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	#346×C₈₃	C₄	1	2	yes	—
14	#346×C₈₃	C₂²	1	2	yes	—
15	#14581732	—	2	4	no	—
16	C₂×C₈₃⋊C₁₆	C₂×C₄	1	4	yes	—
17	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
19	C₄×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₈₃	C₈	1	8	yes	complemented
24	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
29	C₂×C₈₃⋊C₈	—	2	4	no	—
30	C₈₃⋊C₁₆	—	2	4	no	—
31	C₂²×C₄×C₈₃	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₈₃	M₁₆	1	2	yes	—
33	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₈₃	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₈₃	M₁₆	1	4	yes	—
37	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₈₃	C₄²	1	1	yes	characteristic
40	C₄²×C₈₃	M₁₆	1	2	yes	—
41	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₈₃	C₂×C₈	1	8	yes	—
43	C₂³×C₈₃	#97	1	1	yes	characteristic
44	C₂×C₄×C₈₃	—	4	2	no	—
45	C₂×C₄×C₈₃	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₈₃	#97	1	2	yes	—
47	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
48	C₂×C₄×C₈₃	#97	1	2	yes	—
49	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
51	C₈×C₈₃	C₄×C₈	1	2	yes	—
52	C₈×C₈₃	#97	1	2	yes	—
53	C₈×C₈₃	C₂²×C₈	1	2	yes	—
54	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
55	C₈₃⋊C₈	—	4	4	no	—
56	C₂²×C₈₃	—	2	2	no	—
57	C₂²×C₈₃	#322	1	2	yes	—
58	C₂²×C₈₃	C₂×#97	1	1	yes	characteristic
59	C₄×C₈₃	—	2	2	no	—
60	C₄×C₈₃	—	2	2	no	—
61	C₄×C₈₃	#322	1	2	yes	—
62	C₄×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₈₃	C₂×#97	1	1	yes	characteristic
64	#11405	—	83	1	no	maximal, Sylow
65	C₂×C₈₃	—	2	2	no	—
66	C₂×C₈₃	C₂×#322	1	1	yes	characteristic
67	C₂×C₈₃	#2110	1	2	yes	—
68	C₄×M₃₂	—	83	1	no	—
69	C₂×#346	D₈₃	1	1	yes	characteristic
70	#1450	—	83	1	no	—
71	#1313	—	83	4	no	—
72	C₈₃	#11405	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₃₂	—	83	2	no	—
74	C₂²×C₁₆	Dic₈₃	1	2	yes	—
75	#431	—	83	2	no	—
76	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
77	#346	—	83	2	no	—
78	#346	Dic₈₃	1	2	yes	—
79	#346	C₂×D₈₃	1	2	yes	—
80	C₄×C₁₆	—	83	2	no	—
81	#322	—	83	2	no	—
82	C₈²	—	83	2	no	—
83	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
84	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
85	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
86	M₃₂	—	166	4	no	—
87	C₂×C₁₆	—	83	4	no	—
88	C₂×C₁₆	C₈₃⋊C₈	1	8	yes	complemented
89	C₂×C₁₆	C₂×Dic₈₃	1	4	yes	—
90	C₄⋊C₈	—	166	2	no	—
91	C₂²⋊C₈	—	166	2	no	—
92	C₄×C₈	—	83	4	no	—
93	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
94	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
95	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
96	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
97	C₂²×C₄	C₄×Dic₈₃	1	1	yes	characteristic
98	C₂²×C₄	#13050573	1	2	yes	—
99	C₂×C₈	—	166	4	no	—
100	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
101	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
102	C₂×C₈	C₂×C₈₃⋊C₈	1	4	yes	—
103	C₂×C₈	#13050573	1	4	yes	—
104	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
105	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
106	C₄²	C₄×Dic₈₃	1	1	yes	characteristic
107	C₄²	#13050573	1	2	yes	—
108	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
109	C₁₆	—	166	4	no	—
110	C₁₆	C₂×C₈₃⋊C₈	1	8	yes	—
111	C₂³	#14581702	1	1	yes	characteristic
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄×C₈₃⋊C₈	1	2	yes	—
114	C₂×C₄	#14581702	1	2	yes	—
115	C₂×C₄	#14581702	1	2	yes	—
116	C₂×C₄	#14581702	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
118	C₂×C₄	C₂×#13050573	1	2	yes	—
119	C₈	—	332	4	no	—
120	C₈	C₄×C₈₃⋊C₈	1	2	yes	—
121	C₈	#14581702	1	2	yes	—
122	C₈	C₂²×C₈₃⋊C₈	1	2	yes	—
123	C₈	C₂×#13050573	1	2	yes	—
124	C₂²	—	2	2	no	—
125	C₂²	#19127422	1	2	yes	—
126	C₂²	C₂×#14581702	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	#19127422	1	2	yes	—
130	C₄	C₂×C₄×C₈₃⋊C₈	1	1	yes	characteristic
131	C₄	C₂×#14581702	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#19127422	1	1	yes	characteristic
134	C₂	#24909067	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₈₃
commutator	C₄×C₈₃	C₂×C₄×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₂×#346×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#97
prime radical	C₂³×C₈₃	#97
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	C₂×#346	D₈₃
hypocenter	C₈₃	#11405
LCS	G ▷ C₄×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #24905383	C₂	semidirect
23. C₂×C₁₆×C₈₃	C₈	semidirect
72. C₈₃	#11405	semidirect
88. C₂×C₁₆	C₈₃⋊C₈	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
#24905746	2
#24905383	2
C₂×#346×C₈₃	2
#11405	83