Subgroups — #35313398

Order 21248.

322 conjugacy classes of subgroups.

The subgroup lattice has 146 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	C₂×C₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#24905382	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581732	C₄	1	2	yes	—
6	C₂×C₄×C₈₃⋊C₈	C₄	1	2	yes	—
7	#19127477	C₄	1	2	yes	—
8	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	—
9	C₈×C₈₃⋊C₈	C₄	1	1	yes	characteristic
10	C₈×C₈₃⋊C₈	C₄	1	1	yes	characteristic
11	C₈×C₈₃⋊C₈	C₂²	1	2	yes	—
12	C₂×M₃₂×C₈₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	#346×C₈₃	C₄	1	2	yes	—
15	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
16	C₂²×C₈₃⋊C₈	C₈	1	4	yes	—
17	#14581732	—	2	4	no	—
18	C₂×C₈₃⋊C₁₆	C₂×C₄	1	4	yes	—
19	C₄×C₈₃⋊C₈	C₈	1	4	yes	—
20	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
21	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
25	M₃₂×C₈₃	—	2	4	no	—
26	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
31	C₂×C₈₃⋊C₈	—	2	8	no	—
32	C₂×C₈₃⋊C₈	C₂×C₈	1	4	yes	—
33	C₂×C₈₃⋊C₈	C₂×C₈	1	4	yes	—
34	C₈₃⋊C₁₆	—	2	4	no	—
35	C₂²×C₄×C₈₃	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₈₃	M₁₆	1	4	yes	—
40	C₂×C₈×C₈₃	M₁₆	1	4	yes	—
41	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₈₃	C₄²	1	1	yes	characteristic
44	C₄²×C₈₃	C₂×C₈	1	2	yes	—
45	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₁₆×C₈₃	—	2	4	no	—
47	C₂³×C₈₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₈₃	—	2	4	no	—
49	C₂×C₄×C₈₃	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₈₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₈₃	#97	1	2	yes	—
52	C₂×C₄×C₈₃	#97	1	2	yes	—
53	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
55	C₈×C₈₃	#97	1	2	yes	—
56	C₈×C₈₃	#97	1	2	yes	—
57	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
58	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
59	C₈₃⋊C₈	—	2	4	no	—
60	C₈₃⋊C₈	M₃₂	1	8	yes	complemented
61	C₂²×C₈₃	—	2	2	no	—
62	C₂²×C₈₃	#322	1	2	yes	—
63	C₂²×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₈₃	—	2	2	no	—
65	C₄×C₈₃	#322	1	2	yes	—
66	C₄×C₈₃	C₂×#97	1	1	yes	characteristic
67	C₄×C₈₃	C₂×#97	1	1	yes	characteristic
68	C₄×C₈₃	C₂×M₃₂	1	4	yes	—
69	#11402	—	83	1	no	maximal, Sylow
70	C₂×C₈₃	—	2	2	no	—
71	C₂×C₈₃	C₂×#322	1	1	yes	characteristic
72	C₂×C₈₃	C₄×M₃₂	1	2	yes	—
73	C₄×M₃₂	—	83	1	no	—
74	C₄×M₃₂	D₈₃	1	1	yes	characteristic
75	C₂×C₈²	—	83	1	no	—
76	#1312	—	83	4	no	—
77	C₈₃	#11402	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₃₂	—	83	2	no	—
79	C₂×M₃₂	Dic₈₃	1	2	yes	—
80	C₂×C₄×C₈	—	83	2	no	—
81	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
82	#346	—	83	2	no	—
83	#346	Dic₈₃	1	2	yes	—
84	C₄×C₁₆	—	83	2	no	—
85	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
86	C₈²	—	83	1	no	—
87	C₈²	—	83	1	no	—
88	C₈²	—	83	2	no	—
89	C₂²×C₈	—	83	4	no	—
90	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
92	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
93	M₃₂	—	2	4	no	—
94	M₃₂	—	166	4	no	—
95	C₂×C₁₆	—	83	4	no	—
96	C₂×C₁₆	C₂×Dic₈₃	1	4	yes	—
97	C₄×C₈	—	83	4	no	—
98	C₄×C₈	—	83	2	no	—
99	C₄×C₈	—	83	2	no	—
100	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
101	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
104	C₂²×C₄	C₄×Dic₈₃	1	1	yes	characteristic
105	C₂²×C₄	C₈×D₈₃	1	2	yes	—
106	C₂×C₈	—	83	4	no	—
107	C₂×C₈	—	83	4	no	—
108	C₂×C₈	—	166	8	no	—
109	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
110	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
111	C₂×C₈	#13050573	1	4	yes	—
112	C₂×C₈	#13050574	1	4	yes	—
113	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
114	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
115	C₄²	C₄×Dic₈₃	1	1	yes	characteristic
116	C₄²	C₈×D₈₃	1	2	yes	—
117	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
118	C₁₆	—	2	4	no	—
119	C₁₆	—	166	4	no	—
120	C₂³	C₈×Dic₈₃	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	C₈×Dic₈₃	1	2	yes	—
123	C₂×C₄	C₈×Dic₈₃	1	1	yes	characteristic
124	C₂×C₄	#14581702	1	2	yes	—
125	C₂×C₄	#14581703	1	2	yes	—
126	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
127	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
128	C₈	—	83	8	no	—
129	C₈	—	166	4	no	—
130	C₈	#14581702	1	2	yes	—
131	C₈	#14581703	1	2	yes	—
132	C₈	C₂×#13050573	1	2	yes	—
133	C₈	C₂×#13050574	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#19127423	1	2	yes	—
136	C₂²	C₂×C₈×Dic₈₃	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	#19127423	1	2	yes	—
139	C₄	C₂×#14581702	1	1	yes	characteristic
140	C₄	C₂×#14581703	1	1	yes	characteristic
141	C₄	M₃₂×D₈₃	1	4	yes	—
142	C₂	—	2	2	no	—
143	C₂	C₂×#19127423	1	1	yes	characteristic
144	C₂	M₃₂×Dic₈₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₈₃
commutator, socle	C₂²×C₈₃	C₂×C₄×C₈
Fitting	C₄×M₃₂×C₈₃	C₂
prime radical	C₂³×C₈₃	C₄×C₈
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	C₄×M₃₂	D₈₃
hypocenter	C₈₃	#11402
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #24905382	C₂	semidirect
60. C₈₃⋊C₈	M₃₂	semidirect
77. C₈₃	#11402	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
C₂×C₈×C₈₃⋊C₈	2
#24905382	2
C₄×M₃₂×C₈₃	2
#11402	83