Subgroups — #35319415

Order 21248.

354 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×#14581732	C₂	1	1	yes	maximal, characteristic
2	C₄×#14581702	C₂	1	1	yes	maximal, characteristic
3	#24905542	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581732	C₂²	1	2	yes	—
6	C₂×#14581702	C₄	1	2	yes	—
7	C₂×#14581702	C₂²	1	2	yes	—
8	C₂×C₄×C₈₃⋊C₈	C₄	1	1	yes	characteristic
9	C₄²×Dic₈₃	C₄	1	1	yes	characteristic
10	#19127477	C₄	1	2	yes	complemented
11	C₄×C₈₃⋊C₁₆	C₄	1	2	yes	complemented
12	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	#346×C₈₃	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₈₃	C₄	1	2	yes	complemented
16	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₈₃	—	2	2	no	—
18	C₂×C₄×Dic₈₃	C₂×C₄	1	2	yes	—
19	#14581732	—	2	4	no	—
20	C₂×C₈₃⋊C₁₆	C₂×C₄	1	4	yes	—
21	#14581702	—	2	4	no	—
22	#14581702	C₈	1	8	yes	—
23	C₄×C₈₃⋊C₈	—	2	2	no	—
24	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₈₃	—	2	4	no	—
28	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₈₃	—	2	2	no	—
34	C₂×C₈₃⋊C₈	—	2	2	no	—
35	C₂×C₈₃⋊C₈	—	2	2	no	—
36	C₂×C₈₃⋊C₈	C₂×C₈	1	4	yes	—
37	C₄×Dic₈₃	—	2	2	no	—
38	C₄×Dic₈₃	—	4	4	no	—
39	C₄×Dic₈₃	C₂×C₈	1	4	yes	—
40	C₈₃⋊C₁₆	—	2	4	no	—
41	C₂²×C₄×C₈₃	M₁₆	1	2	yes	—
42	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₈₃	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₃	#41	1	4	yes	—
49	C₄²×C₈₃	C₄²	1	1	yes	characteristic
50	C₄²×C₈₃	C₄²	1	1	yes	characteristic
51	C₄²×C₈₃	M₁₆	1	2	yes	—
52	C₁₆×C₈₃	—	2	4	no	—
53	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₄×C₈₃	—	2	4	no	—
55	C₂×C₄×C₈₃	#97	1	2	yes	—
56	C₂×C₄×C₈₃	#97	1	2	yes	—
57	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₈₃	#117	1	2	yes	—
59	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
61	C₂×Dic₈₃	—	2	4	no	—
62	C₂×Dic₈₃	—	4	4	no	—
63	C₈×C₈₃	C₄×C₈	1	2	yes	—
64	C₈×C₈₃	C₄×C₈	1	2	yes	—
65	C₈×C₈₃	#117	1	2	yes	—
66	C₈×C₈₃	#117	1	2	yes	—
67	C₈₃⋊C₈	—	2	4	no	—
68	C₂²×C₈₃	—	2	2	no	—
69	C₂²×C₈₃	C₂×#97	1	1	yes	characteristic
70	C₂²×C₈₃	#431	1	2	yes	—
71	C₄×C₈₃	—	2	2	no	—
72	C₄×C₈₃	#347	1	4	yes	—
73	C₄×C₈₃	#376	1	1	yes	characteristic
74	C₄×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
75	C₄×C₈₃	#431	1	2	yes	—
76	Dic₈₃	—	4	4	no	—
77	#12594	—	83	1	no	maximal, Sylow
78	C₂×C₈₃	—	2	2	no	—
79	C₂×C₈₃	#1744	1	1	yes	characteristic
80	C₂×C₈₃	C₂×#347	1	2	yes	—
81	C₄×M₃₂	—	83	1	no	—
82	C₄×M₃₂	D₈₃	1	1	yes	characteristic
83	C₄×#97	—	83	1	no	—
84	#1375	—	83	4	no	—
85	C₈₃	#12594	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₃₂	—	83	2	no	—
87	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
88	C₂×#97	—	83	2	no	—
89	C₂×#97	—	83	2	no	—
90	C₂×C₄×C₈	—	83	1	no	—
91	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
92	C₄³	—	83	1	no	—
93	#346	—	83	2	no	—
94	#346	Dic₈₃	1	2	yes	complemented
95	C₄×C₁₆	—	83	2	no	—
96	C₄×C₁₆	Dic₈₃	1	2	yes	complemented
97	C₂²×C₈	—	83	2	no	—
98	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
100	C₂×C₄²	—	83	2	no	—
101	C₂×C₄²	—	166	2	no	—
102	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
103	M₃₂	—	2	4	no	—
104	M₃₂	—	166	4	no	—
105	C₂×C₁₆	—	83	4	no	—
106	C₂×C₁₆	C₂×Dic₈₃	1	4	yes	—
107	#97	—	83	8	no	—
108	#97	—	166	4	no	—
109	C₄×C₈	—	166	2	no	—
110	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
111	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
114	C₂²×C₄	—	166	2	no	—
115	C₂²×C₄	#13050573	1	2	yes	—
116	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
117	C₂×C₈	—	83	4	no	—
118	C₂×C₈	—	166	2	no	—
119	C₂×C₈	—	166	2	no	—
120	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
121	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
122	C₂×C₈	C₈×D₈₃	1	4	yes	—
123	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
124	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
125	C₂×C₈	#13050590	1	4	yes	—
126	C₄²	—	332	4	no	—
127	C₄²	—	83	4	no	—
128	C₄²	—	166	2	no	—
129	C₄²	C₄×Dic₈₃	1	1	yes	characteristic
130	C₄²	C₄×Dic₈₃	1	1	yes	characteristic
131	C₄²	#13050573	1	2	yes	—
132	C₁₆	—	2	4	no	—
133	C₁₆	—	166	4	no	—
134	C₂³	C₂×#13050573	1	1	yes	characteristic
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	332	4	no	—
137	C₂×C₄	—	166	4	no	—
138	C₂×C₄	#14581702	1	2	yes	—
139	C₂×C₄	#14581702	1	2	yes	—
140	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
141	C₂×C₄	#14581749	1	2	yes	—
142	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
143	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
144	C₈	—	166	4	no	—
145	C₈	C₈×Dic₈₃	1	2	yes	—
146	C₈	C₈×Dic₈₃	1	2	yes	—
147	C₈	#14581749	1	2	yes	—
148	C₈	#14581751	1	2	yes	—
149	C₂²	—	2	2	no	—
150	C₂²	C₂×#14581702	1	1	yes	characteristic
151	C₂²	#19127752	1	2	yes	—
152	C₄	—	2	2	no	—
153	C₄	—	332	4	no	—
154	C₄	#19127478	1	4	yes	—
155	C₄	#19127547	1	1	yes	characteristic
156	C₄	C₂×C₈×Dic₈₃	1	1	yes	characteristic
157	C₄	#19127752	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	#24907208	1	1	yes	characteristic
160	C₂	C₂×#19127478	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₈₃
commutator	C₄×C₈₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₄×M₃₂×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#97
prime radical	C₂³×C₈₃	C₂×M₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₄×M₃₂	D₈₃
hypocenter	C₈₃	#12594
LCS	G ▷ C₄×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄² ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. #19127477	C₄	semidirect
11. C₄×C₈₃⋊C₁₆	C₄	semidirect
14. #346×C₈₃	C₄	semidirect
15. C₄×C₁₆×C₈₃	C₄	semidirect
85. C₈₃	#12594	semidirect
94. #346	Dic₈₃	semidirect
96. 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
C₄×#14581732	2
C₄×#14581702	2
#24905542	2
C₄×M₃₂×C₈₃	2
#12594	83