Subgroups — #35313384

Order 21248.

330 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14581731	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₆×D₈₃	C₂	1	1	yes	maximal, characteristic
3	#24905745	C₂	1	1	yes	maximal, characteristic
4	#24905380	C₂	1	1	yes	maximal, characteristic, complemented
5	#24905379	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581731	C₄	1	2	yes	—
9	C₂×C₁₆×D₈₃	C₄	1	2	yes	—
10	#19127752	C₄	1	2	yes	—
11	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
12	#19127476	C₄	1	2	yes	—
13	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₁₆×Dic₈₃	C₄	1	2	yes	—
16	#19127423	C₄	1	2	yes	—
17	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
20	C₈²×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×D₈₃	C₂×C₄	1	1	yes	characteristic
24	#14581731	C₈	1	8	yes	complemented
25	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
27	C₁₆×D₈₃	C₈	1	8	yes	complemented
28	Dic₈₃⋊C₈	C₈	1	4	yes	—
29	(C₂×D₈₃)⋊C₈	C₈	1	4	yes	—
30	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
32	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×D₈₃	C₄²	1	1	yes	characteristic
39	C₂×C₄×D₈₃	C₂×C₈	1	2	yes	—
40	C₂×C₈₃⋊C₈	C₄²	1	1	yes	characteristic
41	C₂×C₈₃⋊C₈	C₄²	1	1	yes	characteristic
42	C₂×C₈₃⋊C₈	C₂×C₈	1	4	yes	—
43	C₈×D₈₃	C₂×C₈	1	4	yes	—
44	C₈×D₈₃	C₂×C₈	1	4	yes	—
45	C₄×Dic₈₃	C₄²	1	1	yes	characteristic
46	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
47	C₈₃⋊C₁₆	C₂×C₈	1	4	yes	—
48	C₈₃⋊C₁₆	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₈₃	C₂×C₈	1	4	yes	—
50	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₈₃	C₂×C₈	1	4	yes	—
54	C₁₆×C₈₃	C₂×C₈	1	4	yes	—
55	C₂²×D₈₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
58	C₂×Dic₈₃	C₄×C₈	1	2	yes	—
59	C₂×Dic₈₃	C₄×C₈	1	1	yes	characteristic
60	C₄×D₈₃	—	2	4	no	—
61	C₄×D₈₃	C₄×C₈	1	2	yes	—
62	C₄×D₈₃	C₄×C₈	1	2	yes	—
63	C₈×C₈₃	—	2	4	no	—
64	C₈×C₈₃	C₂²×C₈	1	2	yes	—
65	C₈×C₈₃	C₂²×C₈	1	2	yes	—
66	C₈₃⋊C₈	—	2	4	no	—
67	C₈₃⋊C₈	C₄×C₈	1	2	yes	—
68	C₈₃⋊C₈	C₄×C₈	1	2	yes	—
69	C₂²×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
70	C₂×D₈₃	—	2	2	no	—
71	C₂×D₈₃	C₈²	1	2	yes	—
72	C₄×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
73	C₄×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
74	C₄×C₈₃	#504	1	4	yes	—
75	Dic₈₃	—	2	2	no	—
76	Dic₈₃	C₈²	1	2	yes	—
77	#11399	—	83	1	no	maximal, Sylow
78	C₂×C₈₃	C₂×C₈²	1	1	yes	characteristic
79	C₂×C₈₃	#2108	1	2	yes	—
80	D₈₃	—	2	2	no	—
81	C₄×M₃₂	—	83	1	no	—
82	C₂×C₄×C₁₆	—	83	1	no	—
83	#1450	—	83	1	no	—
84	#1311	—	83	1	no	—
85	#1311	—	83	1	no	—
86	C₈×C₁₆	—	83	1	no	—
87	C₈×C₁₆	D₈₃	1	1	yes	characteristic, complemented
88	C₈₃	#11399	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	—	83	2	no	—
90	C₂²×C₁₆	—	83	2	no	—
91	#431	—	83	2	no	—
92	C₂×C₄×C₈	—	83	1	no	—
93	#346	—	83	2	no	—
94	C₄×C₁₆	—	83	2	no	—
95	C₄×C₁₆	—	83	1	no	—
96	C₄×C₁₆	—	83	1	no	—
97	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
98	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
99	#322	—	83	2	no	—
100	C₈²	—	83	1	no	—
101	C₈²	C₂×D₈₃	1	1	yes	characteristic
102	C₂²×C₈	—	83	1	no	—
103	C₂²×C₈	—	83	1	no	—
104	C₂×C₄²	—	83	1	no	—
105	M₃₂	—	83	8	no	—
106	C₂×C₁₆	—	83	2	no	—
107	C₂×C₁₆	—	83	2	no	—
108	C₂×C₁₆	—	83	8	no	—
109	C₂×C₁₆	C₄×D₈₃	1	2	yes	—
110	C₂×C₁₆	C₄×D₈₃	1	2	yes	—
111	C₄⋊C₈	—	83	4	no	—
112	C₂²⋊C₈	—	83	4	no	—
113	C₄×C₈	—	83	1	no	—
114	C₄×C₈	—	83	1	no	—
115	C₄×C₈	—	83	2	no	—
116	C₄×C₈	—	83	1	no	—
117	C₄×C₈	C₄×D₈₃	1	2	yes	—
118	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
119	C₂²×C₄	—	83	1	no	—
120	C₂²×C₄	—	83	2	no	—
121	C₂×C₈	—	83	1	no	—
122	C₂×C₈	—	83	1	no	—
123	C₂×C₈	—	83	4	no	—
124	C₂×C₈	—	83	4	no	—
125	C₂×C₈	—	83	4	no	—
126	C₂×C₈	C₈×D₈₃	1	4	yes	—
127	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
129	C₄²	—	83	2	no	—
130	C₄²	—	83	1	no	—
131	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
132	C₁₆	—	83	4	no	—
133	C₁₆	—	83	4	no	—
134	C₁₆	C₈×D₈₃	1	4	yes	—
135	C₁₆	C₈×D₈₃	1	4	yes	—
136	C₂³	—	83	1	no	—
137	C₂×C₄	—	83	1	no	—
138	C₂×C₄	—	83	2	no	—
139	C₂×C₄	—	83	2	no	—
140	C₂×C₄	—	83	2	no	—
141	C₂×C₄	—	166	4	no	—
142	C₂×C₄	C₄²×D₈₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	83	2	no	—
146	C₈	—	83	2	no	—
147	C₈	—	166	4	no	—
148	C₈	C₂×C₈×D₈₃	1	2	yes	—
149	C₈	C₂×C₈×D₈₃	1	2	yes	—
150	C₂²	—	83	2	no	—
151	C₂²	—	166	2	no	—
152	C₂²	C₄×C₈×D₈₃	1	1	yes	characteristic
153	C₄	—	83	2	no	—
154	C₄	—	166	2	no	—
155	C₄	C₄×C₈×D₈₃	1	1	yes	characteristic
156	C₄	C₄×C₈×D₈₃	1	1	yes	characteristic
157	C₄	#19128117	1	4	yes	—
158	C₂	—	166	2	no	—
159	C₂	C₈²×D₈₃	1	1	yes	characteristic
160	C₂	#24909058	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₂×D₈₃
commutator	C₂×C₈₃	C₂×C₈²
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈×C₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄×C₈
prime radical	C₂²×D₈₃	C₄×C₈
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	C₈×C₁₆	D₈₃
hypocenter	C₈₃	#11399
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄×C₁₆ ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
4. #24905380	C₂	semidirect
5. #24905379	C₂	semidirect
6. C₁₆×C₈₃⋊C₈	C₂	semidirect
7. C₈×C₁₆×C₈₃	C₂	semidirect
24. #14581731	C₈	semidirect
27. C₁₆×D₈₃	C₈	semidirect
87. C₈×C₁₆	D₈₃	semidirect
88. C₈₃	#11399	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14581731	2
C₄×C₁₆×D₈₃	2
#24905745	2
#24905380	2
#24905379	2
C₁₆×C₈₃⋊C₈	2
C₈×C₁₆×C₈₃	2
#11399	83