Subgroups — #35291237

Order 21248.

262 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	C₄×#14581732	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127422	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581732	C₄	1	2	yes	—
5	C₂×C₄×C₈₃⋊C₈	C₄	1	2	yes	—
6	#19127477	—	2	1	no	—
7	C₄×C₈₃⋊C₁₆	—	2	1	no	—
8	#19127422	—	2	1	no	—
9	#19127422	—	2	1	no	—
10	C₂×M₃₂×C₈₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
12	#346×C₈₃	—	2	1	no	—
13	C₄×C₁₆×C₈₃	—	2	1	no	—
14	C₂²×C₈₃⋊C₈	C₈	1	4	yes	—
15	#14581732	—	2	2	no	—
16	#14581732	C₈	1	4	yes	complemented
17	C₂×C₈₃⋊C₁₆	—	2	2	no	—
18	C₄×C₈₃⋊C₈	—	2	2	no	—
19	C₄×C₈₃⋊C₈	—	2	2	no	—
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	M₃₂×C₈₃	—	2	2	no	—
24	M₃₂×C₈₃	C₈	1	4	yes	complemented
25	C₂×C₁₆×C₈₃	—	2	2	no	—
26	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
30	C₂×C₈₃⋊C₈	—	2	4	no	—
31	C₂×C₈₃⋊C₈	—	4	4	no	—
32	C₈₃⋊C₁₆	—	2	4	no	—
33	C₂²×C₄×C₈₃	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₈₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
42	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
43	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₈₃	M₁₆	1	2	yes	—
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	#95	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
53	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
54	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₈₃	C₄⋊C₈	1	2	yes	—
57	C₈×C₈₃	C₂²⋊C₈	1	2	yes	—
58	C₈×C₈₃	C₂²⋊C₈	1	2	yes	—
59	C₈×C₈₃	C₄⋊C₈	1	2	yes	—
60	C₈×C₈₃	C₄⋊C₈	1	2	yes	—
61	C₈₃⋊C₈	—	4	4	no	—
62	C₂²×C₈₃	—	2	2	no	—
63	C₂²×C₈₃	C₈²	1	1	yes	characteristic
64	C₂²×C₈₃	#322	1	1	yes	characteristic
65	C₂²×C₈₃	#336	1	1	yes	characteristic
66	C₄×C₈₃	—	2	2	no	—
67	C₄×C₈₃	#322	1	1	yes	characteristic
68	C₄×C₈₃	#322	1	1	yes	characteristic
69	C₄×C₈₃	#336	1	1	yes	characteristic
70	C₄×C₈₃	#336	1	1	yes	characteristic
71	C₄×C₈₃	#349	1	4	yes	—
72	#7139	—	83	1	no	maximal, Sylow
73	C₂×C₈₃	—	2	1	no	—
74	C₂×C₈₃	—	2	1	no	—
75	C₂×C₈₃	#1273	1	1	yes	characteristic
76	C₂×C₈₃	#1377	1	2	yes	—
77	C₄×M₃₂	—	83	1	no	—
78	C₄×M₃₂	D₈₃	1	1	yes	characteristic
79	C₂×#322	—	83	1	no	—
80	C₈₃	#7139	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	83	2	no	—
82	C₂×M₃₂	Dic₈₃	1	2	yes	—
83	C₂×C₄×C₈	—	83	2	no	—
84	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
85	#346	—	2	1	no	—
86	#346	—	166	1	no	—
87	C₄×C₁₆	—	2	1	no	—
88	C₄×C₁₆	—	166	1	no	—
89	#322	—	166	1	no	—
90	#322	—	166	1	no	—
91	C₂²×C₈	—	83	4	no	—
92	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
93	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
94	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
95	M₃₂	—	2	2	no	—
96	M₃₂	—	83	4	no	—
97	M₃₂	—	166	2	no	—
98	M₃₂	C₈₃⋊C₈	1	4	yes	complemented
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	166	2	no	—
101	C₄×C₈	—	166	2	no	—
102	C₄×C₈	—	166	2	no	—
103	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
104	C₄×C₈	D₃₃₂	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
106	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
107	C₂²×C₄	C₄×Dic₈₃	1	1	yes	characteristic
108	C₂²×C₄	C₈×D₈₃	1	2	yes	—
109	C₂×C₈	—	332	4	no	—
110	C₂×C₈	—	166	4	no	—
111	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
113	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
114	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
115	C₂×C₈	C₈×D₈₃	1	2	yes	—
116	C₂×C₈	C₂×C₈₃⋊C₈	1	2	yes	—
117	C₂×C₈	#13050573	1	2	yes	—
118	C₂×C₈	#13050574	1	2	yes	—
119	C₄²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
120	C₄²	Dic₈₃⋊C₄	1	1	yes	characteristic
121	C₄²	#13050573	1	2	yes	—
122	C₁₆	—	2	4	no	—
123	C₁₆	—	166	4	no	—
124	C₂³	C₈×Dic₈₃	1	1	yes	characteristic
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	C₈₃⋊#95	1	1	yes	characteristic
127	C₂×C₄	C₈×Dic₈₃	1	1	yes	characteristic
128	C₂×C₄	C₄×C₈₃⋊C₈	1	1	yes	characteristic
129	C₂×C₄	#14581702	1	1	yes	characteristic
130	C₂×C₄	#14581702	1	1	yes	characteristic
131	C₂×C₄	#14581703	1	1	yes	characteristic
132	C₂×C₄	(C₂×D₈₃)⋊C₈	1	2	yes	—
133	C₂×C₄	Dic₈₃⋊C₈	1	2	yes	—
134	C₈	—	332	4	no	—
135	C₈	(C₂×D₈₃)⋊C₈	1	2	yes	—
136	C₈	C₂²⋊(C₈₃⋊C₈)	1	2	yes	—
137	C₈	Dic₈₃⋊C₈	1	2	yes	—
138	C₈	C₄⋊(C₈₃⋊C₈)	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	C₈×C₈₃⋊C₈	1	1	yes	characteristic
141	C₂²	#19127423	1	1	yes	characteristic
142	C₂²	#19127455	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	#19127422	1	1	yes	characteristic
145	C₄	#19127423	1	1	yes	characteristic
146	C₄	#19127455	1	1	yes	characteristic
147	C₄	#19127456	1	1	yes	characteristic
148	C₄	#19127482	1	4	yes	—
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	C₈₃⋊#1273	1	1	yes	characteristic
152	C₂	#24905546	1	2	yes	—

	subgroup	G/H
center	C₄²	(C₂×D₈₃)⋊C₄
commutator	C₂²×C₈₃	C₈²
Frattini	C₂×C₄×C₈	C₂×D₈₃
Fitting	C₄×M₃₂×C₈₃	C₂
socle	C₂²×C₈₃	#336
prime radical	C₂³×C₈₃	C₄×C₈
cosocle	C₂×C₄×C₈×C₈₃	C₂²
hypercenter	C₄×M₃₂	D₈₃
hypocenter	C₈₃	#7139
LCS	G ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
16. #14581732	C₈	semidirect
24. M₃₂×C₈₃	C₈	semidirect
80. C₈₃	#7139	semidirect
98. M₃₂	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
C₂×#19127422	2
C₄×M₃₂×C₈₃	2
#7139	83