Subgroups — #44559491

Order 32128.

398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#16860810	C₂	1	1	yes	maximal, characteristic
2	#30631814	C₂	1	2	yes	maximal, complemented
3	#30631756	C₂	1	4	yes	maximal, complemented
4	#30631752	C₂	1	2	yes	maximal, complemented
5	#30631743	C₂	1	2	yes	maximal, complemented
6	#30631740	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₈×D₂₅₁	C₂	1	2	yes	maximal, complemented
8	#423×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16860810	C₄	1	1	yes	characteristic
10	C₂×C₈×D₂₅₁	C₂²	1	4	yes	—
11	C₄×Dic₅₀₂	C₄	1	1	yes	characteristic
12	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
13	C₄×D₁₀₀₄	C₄	1	1	yes	characteristic
14	C₄×D₁₀₀₄	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₂₅₁	C₄	1	2	yes	—
16	C₄×C₂²⋊D₂₅₁	C₂²	1	2	yes	—
17	#22548674	C₄	1	1	yes	characteristic
18	#22548672	C₄	1	2	yes	—
19	C₄²×D₂₅₁	C₂²	1	2	yes	—
20	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
21	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
22	Dic₂₅₁⋊C₈	C₂²	1	4	yes	—
23	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
24	(C₂×D₂₅₁)⋊C₈	C₂²	1	4	yes	—
25	#22548625	C₂²	1	4	yes	—
26	C₄×C₂₅₁⋊C₈	C₂²	1	2	yes	—
27	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	—
30	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	—
31	#16860810	—	2	4	no	—
32	C₂×Dic₅₀₂	C₂×C₄	1	1	yes	characteristic
33	C₂×D₁₀₀₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₂₅₁	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₂₅₁	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₂₅₁	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
38	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
39	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
40	C₈×D₂₅₁	—	4	4	no	—
41	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
42	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
43	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
44	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
45	C₂²⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
46	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
47	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
48	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
49	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
50	C₄×Dic₂₅₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
54	M₁₆×C₂₅₁	—	4	2	no	—
55	C₂×C₈×C₂₅₁	C₂³	1	4	yes	—
56	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
59	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
60	C₂²×D₂₅₁	C₂²×C₄	1	2	yes	—
61	Dic₅₀₂	—	2	2	no	—
62	D₁₀₀₄	M₁₆	1	4	yes	complemented
63	C₂²⋊D₂₅₁	—	2	4	no	—
64	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
65	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
66	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
67	C₄×D₂₅₁	—	2	4	no	—
68	C₄×D₂₅₁	—	2	4	no	—
69	C₄×D₂₅₁	—	2	2	no	—
70	C₄×D₂₅₁	—	2	2	no	—
71	C₂₅₁⋊C₈	—	2	2	no	—
72	C₂₅₁⋊C₈	—	2	2	no	—
73	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₅₁	—	2	2	no	—
75	C₂×C₄×C₂₅₁	—	2	2	no	—
76	C₂×C₄×C₂₅₁	—	2	1	no	—
77	C₂×C₄×C₂₅₁	—	2	1	no	—
78	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
84	C₈×C₂₅₁	—	2	4	no	—
85	C₂²×C₂₅₁	—	2	1	no	—
86	C₂²×C₂₅₁	—	2	1	no	—
87	C₂²×C₂₅₁	—	2	1	no	—
88	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
89	C₂×D₂₅₁	—	2	2	no	—
90	C₂×D₂₅₁	—	2	2	no	—
91	C₂×D₂₅₁	C₂×M₁₆	1	4	yes	—
92	C₄×C₂₅₁	—	2	1	no	—
93	C₄×C₂₅₁	—	2	1	no	—
94	C₄×C₂₅₁	—	2	1	no	—
95	C₄×C₂₅₁	C₂×M₁₆	1	2	yes	—
96	C₄×C₂₅₁	#131	1	2	yes	—
97	C₄×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
98	C₄×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
99	Dic₂₅₁	—	2	2	no	—
100	Dic₂₅₁	—	2	2	no	—
101	Dic₂₅₁	—	2	2	no	—
102	C₂×C₂₅₁	—	4	1	no	—
103	C₂×C₂₅₁	#520	1	1	yes	characteristic
104	C₂×C₂₅₁	C₂²×M₁₆	1	1	yes	characteristic
105	C₂×C₂₅₁	C₂×#131	1	1	yes	characteristic
106	D₂₅₁	—	2	4	no	—
107	C₂₅₁	#2990	1	1	yes	characteristic, complemented, Sylow
108	#2990	—	251	1	no	maximal, Sylow
109	C₄×#41	—	251	1	no	—
110	#446	—	251	2	no	—
111	#435	—	251	4	no	—
112	C₈×D₄	—	251	2	no	—
113	#432	—	251	2	no	—
114	#431	—	251	1	no	—
115	#423	D₂₅₁	1	1	yes	characteristic, complemented
116	C₂×C₄⋊C₈	—	251	2	no	—
117	C₂×#41	—	251	1	no	—
118	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
119	C₂²×C₈	—	251	4	no	—
120	C₄×Q₈	—	251	1	no	—
121	C₄×Q₈	—	251	1	no	—
122	C₄×D₄	—	251	2	no	—
123	C₄×D₄	—	251	2	no	—
124	C₄×D₄	—	251	1	no	—
125	C₄×D₄	—	251	1	no	—
126	#117	—	251	1	no	—
127	#117	—	251	2	no	—
128	C₂×C₄²	—	251	2	no	—
129	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
130	C₄⋊C₈	—	251	4	no	—
131	C₄⋊C₈	—	251	1	no	—
132	C₄⋊C₈	—	251	1	no	—
133	C₄⋊C₈	C₂×D₂₅₁	1	2	yes	—
134	C₄⋊C₈	C₂×D₂₅₁	1	2	yes	—
135	C₂²⋊C₈	—	251	2	no	—
136	C₂²⋊C₈	—	251	4	no	—
137	#97	—	251	4	no	—
138	C₄×C₈	—	251	2	no	—
139	#41	—	502	4	no	—
140	C₂×Q₈	—	251	1	no	—
141	C₂×D₄	—	251	2	no	—
142	C₂×D₄	—	251	1	no	—
143	C₂²×C₄	—	251	2	no	—
144	C₂²×C₄	—	251	2	no	—
145	C₂²×C₄	—	251	2	no	—
146	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
149	M₁₆	—	4	2	no	—
150	C₂×C₈	—	1004	4	no	—
151	C₂×C₈	—	251	2	no	—
152	C₂×C₈	—	251	2	no	—
153	C₂×C₈	C₂²×D₂₅₁	1	4	yes	—
154	C₄⋊C₄	—	251	2	no	—
155	C₄⋊C₄	—	251	2	no	—
156	C₄⋊C₄	—	251	1	no	—
157	C₄⋊C₄	—	251	1	no	—
158	C₂²⋊C₄	—	251	2	no	—
159	C₂²⋊C₄	—	251	2	no	—
160	C₂²⋊C₄	—	251	2	no	—
161	C₄²	—	251	2	no	—
162	C₄²	—	251	2	no	—
163	C₄²	—	251	2	no	—
164	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
165	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
166	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
167	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
168	C₂³	—	251	2	no	—
169	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
170	Q₈	—	502	2	no	—
171	D₄	—	502	4	no	—
172	D₄	—	251	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	502	4	no	—
178	C₂×C₄	—	502	2	no	—
179	C₂×C₄	—	502	2	no	—
180	C₂×C₄	—	502	4	no	—
181	C₂×C₄	—	251	2	no	—
182	C₂×C₄	—	251	2	no	—
183	C₂×C₄	—	251	2	no	—
184	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	502	2	no	—
192	C₈	—	502	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	502	2	no	—
197	C₂²	—	502	2	no	—
198	C₂²	—	251	4	no	—
199	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	502	2	no	—
204	C₄	—	502	2	no	—
205	C₄	—	502	2	no	—
206	C₄	M₁₆×D₂₅₁	1	2	yes	—
207	C₄	#22548723	1	2	yes	—
208	C₄	#22548760	1	1	yes	characteristic
209	C₄	#22548763	1	1	yes	characteristic
210	C₂	—	4	1	no	—
211	C₂	—	502	4	no	—
212	C₂	#30632168	1	1	yes	characteristic
213	C₂	C₂×M₁₆×D₂₅₁	1	1	yes	characteristic
214	C₂	C₂×#22548723	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Fitting	#423×C₂₅₁	C₂
prime radical	C₂×#16860810	C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#423	D₂₅₁
hypocenter	C₂₅₁	#2990
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #30631814	C₂	semidirect
3. #30631756	C₂	semidirect
4. #30631752	C₂	semidirect
5. #30631743	C₂	semidirect
6. #30631740	C₂	semidirect
7. C₄⋊C₈×D₂₅₁	C₂	semidirect
8. #423×C₂₅₁	C₂	semidirect
62. D₁₀₀₄	M₁₆	semidirect
107. C₂₅₁	#2990	semidirect
115. #423	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#16860810	2
#30631814	2
#30631756	2
#30631752	2
#30631743	2
#30631740	2
C₄⋊C₈×D₂₅₁	2
#423×C₂₅₁	2
#2990	251