Subgroups — #44559449

Order 32128.

356 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic
2	#30631816	C₂	1	4	yes	maximal, complemented
3	#30631815	C₂	1	4	yes	maximal, complemented
4	#30631744	C₂	1	1	yes	maximal, characteristic
5	#30631741	C₂	1	2	yes	maximal
6	#30631737	C₂	1	2	yes	maximal
7	#432×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₅₀₂	C₄	1	1	yes	characteristic
9	C₄×Dic₅₀₂	C₄	1	2	yes	—
10	C₄×Dic₅₀₂	C₄	1	2	yes	—
11	C₄×Dic₅₀₂	C₂²	1	2	yes	—
12	C₄×Dic₅₀₂	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₂₅₁	C₄	1	1	yes	characteristic
14	C₂×Dic₂₅₁⋊C₄	C₄	1	1	yes	characteristic
15	C₂×Dic₂₅₁⋊C₄	C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
18	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
19	Dic₂₅₁⋊C₈	C₂²	1	4	yes	—
20	Dic₂₅₁⋊C₈	C₂²	1	4	yes	—
21	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
22	#22548626	C₂²	1	2	yes	—
23	#22548625	C₂²	1	4	yes	—
24	C₈×Dic₂₅₁	C₂²	1	4	yes	—
25	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	—
27	C₂²⋊C₈×C₂₅₁	C₂²	1	2	yes	—
28	#97×C₂₅₁	C₂²	1	2	yes	—
29	C₂×Dic₅₀₂	—	2	4	no	—
30	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
31	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
32	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₂₅₁⋊C₈	C₂³	1	4	yes	—
37	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
39	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
40	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
41	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
42	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
43	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
44	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
45	C₄×Dic₂₅₁	C₂³	1	2	yes	—
46	C₄×Dic₂₅₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₂₅₁	C₂³	1	4	yes	—
51	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
54	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
55	Dic₅₀₂	—	2	4	no	—
56	Dic₅₀₂	—	2	4	no	—
57	C₂×Dic₂₅₁	—	2	2	no	—
58	C₂×Dic₂₅₁	—	4	1	no	—
59	C₂×Dic₂₅₁	—	4	1	no	—
60	C₂×Dic₂₅₁	M₁₆	1	4	yes	—
61	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
62	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
63	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
64	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
65	C₂₅₁⋊C₈	—	2	4	no	—
66	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₅₁	—	2	2	no	—
68	C₂×C₄×C₂₅₁	—	2	2	no	—
69	C₂×C₄×C₂₅₁	—	4	1	no	—
70	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
76	C₈×C₂₅₁	—	2	4	no	—
77	C₂²×C₂₅₁	—	2	1	no	—
78	C₂²×C₂₅₁	—	2	1	no	—
79	C₂²×C₂₅₁	C₂×M₁₆	1	2	yes	—
80	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
81	C₄×C₂₅₁	—	2	1	no	—
82	C₄×C₂₅₁	—	2	1	no	—
83	C₄×C₂₅₁	—	2	1	no	—
84	C₄×C₂₅₁	—	2	1	no	—
85	C₄×C₂₅₁	—	2	1	no	—
86	C₄×C₂₅₁	2₋¹⁺⁴	1	2	yes	—
87	Dic₂₅₁	—	2	2	no	—
88	Dic₂₅₁	—	2	2	no	—
89	Dic₂₅₁	—	2	2	no	—
90	Dic₂₅₁	C₂×M₁₆	1	4	yes	—
91	C₂×C₂₅₁	—	2	2	no	—
92	C₂×C₂₅₁	#519	1	1	yes	characteristic
93	C₂×C₂₅₁	C₂²×M₁₆	1	1	yes	characteristic
94	C₂×C₂₅₁	#568	1	1	yes	characteristic
95	C₂₅₁	#2983	1	1	yes	characteristic, complemented, Sylow
96	#2983	—	251	1	no	maximal, Sylow
97	C₂×C₄×Q₈	—	251	1	no	—
98	#446	—	251	4	no	—
99	#446	—	251	4	no	—
100	#432	—	251	1	no	—
101	#432	—	251	2	no	—
102	#432	D₂₅₁	1	1	yes	characteristic
103	#431	—	251	2	no	—
104	C₂²×Q₈	—	251	1	no	—
105	C₄×Q₈	—	251	2	no	—
106	C₄×Q₈	—	251	2	no	—
107	C₄×Q₈	—	251	2	no	—
108	C₄×Q₈	—	251	2	no	—
109	C₂×C₄⋊C₄	—	251	1	no	—
110	C₂×C₄⋊C₄	—	251	1	no	—
111	C₂×C₄⋊C₄	—	251	1	no	—
112	C₂×C₄²	—	251	1	no	—
113	C₂×C₄²	—	251	1	no	—
114	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
115	C₄⋊C₈	—	251	4	no	—
116	C₄⋊C₈	—	251	4	no	—
117	C₄⋊C₈	—	251	2	no	—
118	C₄⋊C₈	C₂×D₂₅₁	1	2	yes	—
119	C₂²⋊C₈	—	251	2	no	—
120	C₂²⋊C₈	C₂×D₂₅₁	1	2	yes	—
121	#97	—	251	4	no	—
122	#97	—	251	2	no	—
123	#97	C₂×D₂₅₁	1	2	yes	—
124	C₄×C₈	—	251	4	no	—
125	C₂×Q₈	—	502	4	no	—
126	C₂×Q₈	—	251	2	no	—
127	C₂×Q₈	—	251	2	no	—
128	C₂²×C₄	—	251	1	no	—
129	C₂²×C₄	—	251	1	no	—
130	C₂²×C₄	—	251	1	no	—
131	C₂²×C₄	—	251	1	no	—
132	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
133	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
135	C₂×C₈	—	251	4	no	—
136	C₂×C₈	C₂²×D₂₅₁	1	4	yes	—
137	C₄⋊C₄	—	251	2	no	—
138	C₄⋊C₄	—	251	2	no	—
139	C₄⋊C₄	—	251	2	no	—
140	C₄⋊C₄	—	251	2	no	—
141	C₄⋊C₄	—	251	2	no	—
142	C₄⋊C₄	—	251	2	no	—
143	C₄²	—	251	2	no	—
144	C₄²	—	251	2	no	—
145	C₄²	—	251	2	no	—
146	C₄²	—	251	2	no	—
147	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
148	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
149	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
150	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
151	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
152	Q₈	—	502	4	no	—
153	Q₈	—	502	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	1004	1	no	—
158	C₂×C₄	—	1004	1	no	—
159	C₂×C₄	—	502	2	no	—
160	C₂×C₄	—	251	2	no	—
161	C₂×C₄	—	251	2	no	—
162	C₂×C₄	—	251	2	no	—
163	C₂×C₄	—	251	2	no	—
164	C₂×C₄	—	251	4	no	—
165	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
170	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
171	C₈	—	2	4	no	—
172	C₈	—	502	4	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	M₁₆×D₂₅₁	1	2	yes	—
176	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	502	2	no	—
183	C₄	—	502	2	no	—
184	C₄	—	502	2	no	—
185	C₄	—	251	4	no	—
186	C₄	#22548762	1	2	yes	—
187	C₂	—	2	2	no	—
188	C₂	#30632164	1	1	yes	characteristic
189	C₂	C₂×M₁₆×D₂₅₁	1	1	yes	characteristic
190	C₂	#30632453	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Fitting	#432×C₂₅₁	C₂
prime radical	C₂³×C₂₅₁	C₂²×C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#432	D₂₅₁
hypocenter	C₂₅₁	#2983
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #30631816	C₂	semidirect
3. #30631815	C₂	semidirect
95. C₂₅₁	#2983	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₀₂	2
#30631816	2
#30631815	2
#30631744	2
#30631741	2
#30631737	2
#432×C₂₅₁	2
#2983	251