Subgroups — #44559466

Order 32128.

330 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632244	C₂	1	1	yes	maximal, characteristic
2	#30631817	C₂	1	2	yes	maximal, complemented
3	#30631814	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₅₀₂	C₂	1	2	yes	maximal, complemented
5	#30631811	C₂	1	2	yes	maximal, complemented
6	#30631745	C₂	1	4	yes	maximal
7	#30631740	C₂	1	1	yes	maximal, characteristic
8	#431×C₂₅₁	C₂	1	1	yes	maximal, characteristic
9	#22548715	C₄	1	1	yes	characteristic
10	#22548715	C₄	1	1	yes	characteristic
11	#22548707	C₄	1	2	yes	—
12	#22548694	C₄	1	2	yes	—
13	#22548694	C₄	1	2	yes	—
14	C₄×Dic₅₀₂	C₂²	1	2	yes	—
15	C₄×Dic₅₀₂	C₂²	1	2	yes	—
16	#22548674	C₂²	1	2	yes	—
17	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
19	Dic₂₅₁⋊C₈	C₂²	1	4	yes	—
20	Dic₂₅₁⋊C₈	C₂²	1	4	yes	—
21	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
22	#22548625	C₂²	1	4	yes	—
23	C₄×C₂₅₁⋊C₈	C₂²	1	2	yes	—
24	C₈×Dic₂₅₁	C₂²	1	4	yes	—
25	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
28	C₂²⋊C₈×C₂₅₁	C₂²	1	2	yes	—
29	C₄×C₈×C₂₅₁	C₂²	1	2	yes	—
30	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
31	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
32	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
33	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
34	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
35	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
36	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
38	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	—
39	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	—
40	C₂²⋊Dic₂₅₁	C₂×C₄	1	4	yes	—
41	C₄×Dic₂₅₁	C₂³	1	4	yes	—
42	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
46	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
47	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
50	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
51	Dic₅₀₂	—	2	4	no	—
52	Dic₅₀₂	—	2	4	no	—
53	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
54	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
55	C₂₅₁⋊C₈	—	2	2	no	—
56	C₂₅₁⋊C₈	—	2	2	no	—
57	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₅₁	—	2	2	no	—
59	C₂×C₄×C₂₅₁	—	2	2	no	—
60	C₂×C₄×C₂₅₁	—	2	1	no	—
61	C₂×C₄×C₂₅₁	—	2	1	no	—
62	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
68	C₈×C₂₅₁	—	2	2	no	—
69	C₈×C₂₅₁	—	2	2	no	—
70	C₂²×C₂₅₁	—	2	1	no	—
71	C₂²×C₂₅₁	—	2	1	no	—
72	C₂²×C₂₅₁	—	2	1	no	—
73	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₂₅₁	—	2	1	no	—
75	C₄×C₂₅₁	—	2	1	no	—
76	C₄×C₂₅₁	—	2	1	no	—
77	C₄×C₂₅₁	#131	1	2	yes	—
78	C₄×C₂₅₁	#131	1	2	yes	—
79	C₄×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
80	C₄×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
81	Dic₂₅₁	—	2	4	no	—
82	Dic₂₅₁	—	2	4	no	—
83	C₂×C₂₅₁	—	4	1	no	—
84	C₂×C₂₅₁	#519	1	1	yes	characteristic
85	C₂×C₂₅₁	C₂×#131	1	1	yes	characteristic
86	C₂×C₂₅₁	C₂×#131	1	1	yes	characteristic
87	C₂₅₁	#2986	1	1	yes	characteristic, complemented, Sylow
88	#2986	—	251	1	no	maximal, Sylow
89	#533	—	251	1	no	—
90	#446	—	251	2	no	—
91	#446	—	251	2	no	—
92	C₈×Q₈	—	251	2	no	—
93	C₈×Q₈	—	251	2	no	—
94	#433	—	251	4	no	—
95	#431	—	251	1	no	—
96	#431	D₂₅₁	1	1	yes	characteristic
97	C₄⋊Q₈	—	251	1	no	—
98	C₄⋊Q₈	—	251	1	no	—
99	#125	—	251	2	no	—
100	C₂²⋊Q₈	—	251	2	no	—
101	C₂²⋊Q₈	—	251	2	no	—
102	C₄×Q₈	—	251	2	no	—
103	C₄×Q₈	—	251	2	no	—
104	#117	—	251	2	no	—
105	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
106	C₄⋊C₈	—	251	4	no	—
107	C₄⋊C₈	—	251	4	no	—
108	C₄⋊C₈	—	251	1	no	—
109	C₄⋊C₈	—	251	1	no	—
110	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
112	C₂²⋊C₈	—	251	2	no	—
113	C₂²⋊C₈	C₂×D₂₅₁	1	2	yes	—
114	#97	—	251	4	no	—
115	C₄×C₈	—	251	4	no	—
116	C₄×C₈	—	251	2	no	—
117	C₄×C₈	C₂×D₂₅₁	1	2	yes	—
118	C₂×Q₈	—	251	2	no	—
119	C₂×Q₈	—	251	2	no	—
120	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
123	C₂×C₈	—	251	2	no	—
124	C₂×C₈	—	251	2	no	—
125	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
126	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
127	C₄⋊C₄	—	251	2	no	—
128	C₄⋊C₄	—	251	2	no	—
129	C₄⋊C₄	—	251	4	no	—
130	C₄⋊C₄	—	251	4	no	—
131	C₄⋊C₄	—	251	2	no	—
132	C₄⋊C₄	—	251	2	no	—
133	C₂²⋊C₄	—	251	4	no	—
134	C₄²	—	251	4	no	—
135	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
136	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
137	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
138	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
139	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
140	Q₈	—	502	4	no	—
141	Q₈	—	502	4	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	251	4	no	—
147	C₂×C₄	—	251	4	no	—
148	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₈	—	502	2	no	—
157	C₈	—	502	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	502	4	no	—
166	C₄	—	502	4	no	—
167	C₄	#22548721	1	2	yes	—
168	C₄	#22548723	1	2	yes	—
169	C₄	#22548762	1	1	yes	characteristic
170	C₄	#22548762	1	1	yes	characteristic
171	C₂	—	4	1	no	—
172	C₂	#30632164	1	1	yes	characteristic
173	C₂	C₂×#22548721	1	1	yes	characteristic
174	C₂	C₂×#22548723	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Fitting	#431×C₂₅₁	C₂
prime radical	C₂³×C₂₅₁	C₂²×C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#431	D₂₅₁
hypocenter	C₂₅₁	#2986
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #30631817	C₂	semidirect
3. #30631814	C₂	semidirect
4. C₈×Dic₅₀₂	C₂	semidirect
5. #30631811	C₂	semidirect
87. C₂₅₁	#2986	semidirect

Maximal subgroups

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

subgroup	index
#30632244	2
#30631817	2
#30631814	2
C₈×Dic₅₀₂	2
#30631811	2
#30631745	2
#30631740	2
#431×C₂₅₁	2
#2986	251