Subgroups — #44548592

Order 32128.

568 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×C₂₅₁⋊#95	C₂	1	2	yes	maximal
2	C₂×C₂₅₁⋊#95	C₂	1	2	yes	maximal
3	C₂×C₂₅₁⋊#95	C₂	1	2	yes	maximal
4	C₂²×C₄²×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₅₁	C₂²	1	4	yes	—
6	C₂₅₁⋊#95	—	2	4	no	—
7	C₂₅₁⋊#95	—	2	4	no	—
8	C₂₅₁⋊#95	C₄	1	8	yes	complemented
9	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₂₅₁	—	2	4	no	—
13	C₂×C₄²×C₂₅₁	C₄	1	4	yes	complemented
14	C₂²×Dic₂₅₁	—	2	4	no	—
15	C₂²×Dic₂₅₁	—	2	4	no	—
16	C₂²×Dic₂₅₁	—	2	4	no	—
17	C₂²×Dic₂₅₁	—	2	4	no	—
18	C₂²×Dic₂₅₁	—	2	4	no	—
19	C₂²×Dic₂₅₁	—	2	4	no	—
20	C₂²×Dic₂₅₁	C₂×C₄	1	8	yes	—
21	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₂₅₁	—	2	2	no	—
23	C₂²×C₄×C₂₅₁	—	2	2	no	—
24	C₂²×C₄×C₂₅₁	—	2	2	no	—
25	C₂²×C₄×C₂₅₁	—	2	2	no	—
26	C₂²×C₄×C₂₅₁	—	2	2	no	—
27	C₂²×C₄×C₂₅₁	—	2	2	no	—
28	C₂²×C₄×C₂₅₁	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₂₅₁	D₄	1	4	yes	—
33	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
34	C₂²×C₄×C₂₅₁	Q₈	1	2	yes	—
35	C₄²×C₂₅₁	—	2	8	no	—
36	C₂×Dic₂₅₁	—	2	8	no	—
37	C₂×Dic₂₅₁	—	2	8	no	—
38	C₂×Dic₂₅₁	—	2	8	no	—
39	C₂×Dic₂₅₁	—	4	4	no	—
40	C₂×Dic₂₅₁	—	4	4	no	—
41	C₂×Dic₂₅₁	—	4	4	no	—
42	C₂×Dic₂₅₁	—	4	4	no	—
43	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
44	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
45	C₂³×C₂₅₁	C₂×D₄	1	2	yes	—
46	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
47	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
48	C₂³×C₂₅₁	#41	1	2	yes	—
49	C₂³×C₂₅₁	#41	1	2	yes	—
50	C₂³×C₂₅₁	#41	1	2	yes	—
51	C₂³×C₂₅₁	#41	1	2	yes	—
52	C₂×C₄×C₂₅₁	—	2	2	no	—
53	C₂×C₄×C₂₅₁	—	2	2	no	—
54	C₂×C₄×C₂₅₁	—	2	2	no	—
55	C₂×C₄×C₂₅₁	—	2	2	no	—
56	C₂×C₄×C₂₅₁	—	2	2	no	—
57	C₂×C₄×C₂₅₁	—	2	2	no	—
58	C₂×C₄×C₂₅₁	—	2	8	no	—
59	C₂×C₄×C₂₅₁	—	2	4	no	—
60	C₂×C₄×C₂₅₁	—	2	4	no	—
61	C₂×C₄×C₂₅₁	—	2	8	no	—
62	C₂×C₄×C₂₅₁	C₄²	1	4	yes	—
63	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	4	yes	—
64	C₂×C₄×C₂₅₁	C₄⋊C₄	1	4	yes	—
65	C₂²×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₂₅₁	#117	1	4	yes	—
69	C₂²×C₂₅₁	#117	1	1	yes	characteristic
70	C₂²×C₂₅₁	#117	1	1	yes	characteristic
71	C₂²×C₂₅₁	#117	1	1	yes	characteristic
72	C₂²×C₂₅₁	#117	1	1	yes	characteristic
73	C₂²×C₂₅₁	C₄×D₄	1	4	yes	—
74	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
75	C₂²×C₂₅₁	C₄×Q₈	1	2	yes	—
76	C₂²×C₂₅₁	#121	1	2	yes	—
77	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
78	C₂²×C₂₅₁	#123	1	4	yes	—
79	C₂²×C₂₅₁	#124	1	2	yes	—
80	C₂²×C₂₅₁	#125	1	2	yes	—
81	C₂²×C₂₅₁	#126	1	4	yes	—
82	C₄×C₂₅₁	—	2	4	no	—
83	C₄×C₂₅₁	—	2	4	no	—
84	C₄×C₂₅₁	—	2	4	no	—
85	Dic₂₅₁	—	4	8	no	—
86	C₂×C₂₅₁	#376	1	1	yes	characteristic
87	C₂×C₂₅₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₂₅₁	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₂₅₁	#380	1	1	yes	characteristic
90	C₂×C₂₅₁	#381	1	1	yes	characteristic
91	C₂×C₂₅₁	#382	1	1	yes	characteristic
92	C₂×C₂₅₁	#383	1	1	yes	characteristic
93	C₂×C₂₅₁	#387	1	4	yes	—
94	C₂×C₂₅₁	#388	1	4	yes	—
95	C₂₅₁	#1434	1	1	yes	characteristic, complemented, Sylow
96	#1434	—	251	1	no	maximal, Sylow
97	C₂²×C₄²	D₂₅₁	1	1	yes	characteristic
98	C₂×#95	—	251	2	no	—
99	C₂×#95	—	251	2	no	—
100	C₂×#95	—	251	2	no	—
101	C₂³×C₄	—	251	4	no	—
102	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
103	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
104	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
105	C₂×C₄²	—	2	4	no	—
106	C₂×C₄²	Dic₂₅₁	1	4	yes	complemented
107	#95	—	502	4	no	—
108	#95	—	502	4	no	—
109	#95	—	251	8	no	—
110	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	502	4	no	—
118	C₂²×C₄	—	502	4	no	—
119	C₂²×C₄	—	502	4	no	—
120	C₂²×C₄	—	502	4	no	—
121	C₂²×C₄	—	502	4	no	—
122	C₂²×C₄	—	502	4	no	—
123	C₂²×C₄	—	251	8	no	—
124	C₂²×C₄	C₄×D₂₅₁	1	4	yes	—
125	C₂²×C₄	C₂×Dic₂₅₁	1	2	yes	—
126	C₂²×C₄	C₂×Dic₂₅₁	1	2	yes	—
127	C₂²×C₄	C₂×Dic₂₅₁	1	2	yes	—
128	C₂²×C₄	C₂²⋊D₂₅₁	1	4	yes	—
129	C₂²×C₄	D₁₀₀₄	1	2	yes	—
130	C₂²×C₄	Dic₅₀₂	1	2	yes	—
131	C₄²	—	2	8	no	—
132	C₂³	C₂×C₄×D₂₅₁	1	2	yes	—
133	C₂³	C₂²×Dic₂₅₁	1	1	yes	characteristic
134	C₂³	C₂×C₂²⋊D₂₅₁	1	2	yes	—
135	C₂³	C₂×D₁₀₀₄	1	1	yes	characteristic
136	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
137	C₂³	#16860810	1	2	yes	—
138	C₂³	#16860810	1	2	yes	—
139	C₂³	#16860810	1	2	yes	—
140	C₂³	#16860810	1	2	yes	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	1004	4	no	—
152	C₂×C₄	—	1004	4	no	—
153	C₂×C₄	—	1004	4	no	—
154	C₂×C₄	—	1004	4	no	—
155	C₂×C₄	—	502	8	no	—
156	C₂×C₄	—	502	8	no	—
157	C₂×C₄	—	502	8	no	—
158	C₂×C₄	C₄×Dic₂₅₁	1	4	yes	—
159	C₂×C₄	C₂²⋊Dic₂₅₁	1	4	yes	—
160	C₂×C₄	C₄⋊Dic₂₅₁	1	4	yes	—
161	C₂²	C₂×C₄×Dic₂₅₁	1	1	yes	characteristic
162	C₂²	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
163	C₂²	C₂×C₄⋊Dic₂₅₁	1	1	yes	characteristic
164	C₂²	#22548672	1	4	yes	—
165	C₂²	#22548674	1	1	yes	characteristic
166	C₂²	#22548674	1	1	yes	characteristic
167	C₂²	#22548674	1	1	yes	characteristic
168	C₂²	#22548674	1	1	yes	characteristic
169	C₂²	C₄×C₂²⋊D₂₅₁	1	4	yes	—
170	C₂²	C₄×D₁₀₀₄	1	2	yes	—
171	C₂²	C₄×Dic₅₀₂	1	2	yes	—
172	C₂²	#22548689	1	2	yes	—
173	C₂²	#22548694	1	2	yes	—
174	C₂²	#22548698	1	4	yes	—
175	C₂²	#22548703	1	2	yes	—
176	C₂²	#22548707	1	2	yes	—
177	C₂²	#22548709	1	4	yes	—
178	C₄	—	2	4	no	—
179	C₄	—	2	4	no	—
180	C₄	—	2	4	no	—
181	C₄	—	1004	8	no	—
182	C₂	#30631533	1	1	yes	characteristic
183	C₂	C₄×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
184	C₂	C₄×C₄⋊Dic₂₅₁	1	1	yes	characteristic
185	C₂	#30631547	1	1	yes	characteristic
186	C₂	#30631550	1	1	yes	characteristic
187	C₂	#30631553	1	1	yes	characteristic
188	C₂	#30631555	1	1	yes	characteristic
189	C₂	#30631571	1	4	yes	—
190	C₂	#30631578	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₂₅₁
commutator	C₂²×C₂₅₁	C₂×C₄²
Fitting	C₂²×C₄²×C₂₅₁	C₂
socle, prime radical, cosocle	C₂⁴×C₂₅₁	C₂³
hypercenter	C₂²×C₄²	D₂₅₁
hypocenter	C₂₅₁	#1434
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
8. C₂₅₁⋊#95	C₄	semidirect
13. C₂×C₄²×C₂₅₁	C₄	semidirect
95. C₂₅₁	#1434	semidirect
106. C₂×C₄²	Dic₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₅₁⋊#95	2
C₂²×C₄²×C₂₅₁	2
#1434	251