Subgroups — #35324591

Order 21248.

214 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127535	C₂	1	1	yes	maximal, characteristic
2	#24909059	C₂	1	1	yes	maximal, characteristic
3	#24905598	C₂	1	4	yes	maximal, complemented
4	C₂×M₆₄×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581732	C₄	1	1	yes	characteristic
6	#19127656	C₄	1	1	yes	characteristic
7	#19127535	C₄	1	2	yes	complemented
8	#19127535	C₂²	1	2	yes	—
9	C₂×C₈₃⋊C₃₂	C₄	1	2	yes	complemented
10	#19127476	C₂²	1	2	yes	—
11	C₁₆×Dic₈₃	C₄	1	2	yes	—
12	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
13	M₆₄×C₈₃	C₄	1	2	yes	complemented
14	M₆₄×C₈₃	C₂²	1	2	yes	—
15	C₂×C₃₂×C₈₃	C₄	1	2	yes	complemented
16	C₂×#13050574	C₈	1	1	yes	characteristic
17	#14581751	C₈	1	1	yes	characteristic
18	#14581732	C₈	1	2	yes	—
19	#14581732	C₈	1	2	yes	—
20	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
21	#14581702	C₈	1	2	yes	—
22	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
23	C₈₃⋊C₃₂	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
31	C₃₂×C₈₃	C₂×C₄	1	4	yes	—
32	#13050574	—	2	1	no	—
33	#13050574	—	2	1	no	—
34	C₂×C₈₃⋊C₈	C₂×C₈	1	2	yes	—
35	C₄⋊Dic₈₃	—	2	1	no	—
36	C₂²⋊Dic₈₃	—	2	1	no	—
37	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
38	C₈₃⋊C₁₆	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₁₆×C₈₃	C₄²	1	1	yes	characteristic
47	C₁₆×C₈₃	C₄²	1	1	yes	characteristic
48	C₁₆×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	C₁₆×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂³×C₈₃	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₈₃	M₃₂	1	1	yes	characteristic
54	C₂×C₄×C₈₃	M₃₂	1	1	yes	characteristic
55	C₂×C₄×C₈₃	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
57	C₂×Dic₈₃	—	2	2	no	—
58	C₈×C₈₃	C₄×C₈	1	1	yes	characteristic
59	C₈×C₈₃	C₄×C₈	1	1	yes	characteristic
60	C₈×C₈₃	C₂×C₄²	1	1	yes	characteristic
61	C₈×C₈₃	C₂²×C₈	1	1	yes	characteristic
62	C₈₃⋊C₈	—	2	2	no	—
63	C₂²×C₈₃	—	4	1	no	—
64	C₂²×C₈₃	#346	1	1	yes	characteristic
65	C₂²×C₈₃	#346	1	1	yes	characteristic
66	C₂²×C₈₃	C₂×M₃₂	1	1	yes	characteristic
67	C₄×C₈₃	#346	1	1	yes	characteristic
68	C₄×C₈₃	#346	1	1	yes	characteristic
69	C₄×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
70	C₄×C₈₃	C₂×M₃₂	1	1	yes	characteristic
71	Dic₈₃	—	4	2	no	—
72	#13551	—	83	1	no	maximal, Sylow
73	C₂×C₈₃	—	2	1	no	—
74	C₂×C₈₃	—	2	1	no	—
75	C₂×C₈₃	—	2	1	no	—
76	C₂×C₈₃	C₂×#346	1	1	yes	characteristic
77	C₂×M₆₄	—	83	1	no	—
78	C₂×M₆₄	D₈₃	1	1	yes	characteristic
79	#2108	—	83	1	no	—
80	#1398	—	83	4	no	—
81	C₈₃	#13551	1	1	yes	characteristic, complemented, Sylow
82	C₂×M₃₂	—	83	1	no	—
83	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
84	#405	—	83	1	no	—
85	M₆₄	—	83	2	no	—
86	M₆₄	—	83	2	no	—
87	M₆₄	Dic₈₃	1	2	yes	complemented
88	M₆₄	C₂×D₈₃	1	2	yes	—
89	C₂×C₃₂	—	83	2	no	—
90	C₂×C₃₂	Dic₈₃	1	2	yes	complemented
91	#346	—	83	2	no	—
92	C₄×C₁₆	—	83	2	no	—
93	C₂×M₁₆	—	83	1	no	—
94	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
95	#117	—	83	1	no	—
96	M₃₂	—	83	2	no	—
97	M₃₂	—	83	2	no	—
98	C₂×C₁₆	—	83	2	no	—
99	C₂×C₁₆	C₄×D₈₃	1	1	yes	characteristic
100	C₂×C₁₆	C₄×D₈₃	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₈₃	1	1	yes	characteristic
102	C₂×C₁₆	C₂×Dic₈₃	1	1	yes	characteristic
103	C₂×C₁₆	C₂×Dic₈₃	1	1	yes	characteristic
104	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
105	#97	—	83	2	no	—
106	C₄×C₈	—	83	2	no	—
107	C₃₂	—	83	4	no	—
108	C₃₂	C₂×Dic₈₃	1	4	yes	—
109	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
110	M₁₆	—	166	1	no	—
111	M₁₆	—	166	1	no	—
112	C₂×C₈	—	83	2	no	—
113	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
114	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
115	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
116	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
117	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
119	C₄⋊C₄	—	166	1	no	—
120	C₂²⋊C₄	—	166	1	no	—
121	C₄²	—	83	2	no	—
122	C₁₆	—	83	4	no	—
123	C₁₆	C₄×Dic₈₃	1	1	yes	characteristic
124	C₁₆	C₄×Dic₈₃	1	1	yes	characteristic
125	C₁₆	C₂×C₄×D₈₃	1	1	yes	characteristic
126	C₁₆	C₂²×Dic₈₃	1	1	yes	characteristic
127	C₂³	#14581731	1	1	yes	characteristic
128	C₂×C₄	—	166	2	no	—
129	C₂×C₄	C₈×Dic₈₃	1	1	yes	characteristic
130	C₂×C₄	C₈×Dic₈₃	1	1	yes	characteristic
131	C₂×C₄	#14581731	1	1	yes	characteristic
132	C₂×C₄	#14581731	1	1	yes	characteristic
133	C₂×C₄	#14581731	1	1	yes	characteristic
134	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
135	C₈	—	166	2	no	—
136	C₈	C₈×Dic₈₃	1	1	yes	characteristic
137	C₈	C₈×Dic₈₃	1	1	yes	characteristic
138	C₈	C₂×C₄×Dic₈₃	1	1	yes	characteristic
139	C₈	C₂×C₈×D₈₃	1	1	yes	characteristic
140	C₂²	—	4	1	no	—
141	C₂²	#19127476	1	1	yes	characteristic
142	C₂²	#19127476	1	1	yes	characteristic
143	C₂²	C₂×#14581731	1	1	yes	characteristic
144	C₄	—	332	2	no	—
145	C₄	#19127476	1	1	yes	characteristic
146	C₄	#19127476	1	1	yes	characteristic
147	C₄	C₂×C₈×Dic₈₃	1	1	yes	characteristic
148	C₄	C₂×#14581731	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	—	2	1	no	—
152	C₂	C₂×#19127476	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₈₃
commutator	C₄×C₈₃	C₂×C₄×C₈
Frattini	C₂×C₁₆	C₂²×D₈₃
Fitting	C₂×M₆₄×C₈₃	C₂
socle	C₂×C₈₃	C₂×#346
prime radical	C₂³×C₈₃	M₃₂
cosocle	C₂×C₁₆×C₈₃	C₂³
hypercenter	C₂×M₆₄	D₈₃
hypocenter	C₈₃	#13551
LCS	G ▷ C₄×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #24905598	C₂	semidirect
7. #19127535	C₄	semidirect
9. C₂×C₈₃⋊C₃₂	C₄	semidirect
13. M₆₄×C₈₃	C₄	semidirect
15. C₂×C₃₂×C₈₃	C₄	semidirect
81. C₈₃	#13551	semidirect
87. M₆₄	Dic₈₃	semidirect
90. C₂×C₃₂	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127535	2
#24909059	2
#24905598	2
C₂×M₆₄×C₈₃	2
#13551	83