Subgroups — #44559402

Order 32128.

464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₀₀₄	C₂	1	1	yes	maximal, characteristic
2	#30631764	C₂	1	2	yes	maximal, complemented
3	#30631760	C₂	1	4	yes	maximal, complemented
4	#30631752	C₂	1	2	yes	maximal, complemented
5	#30631740	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631649	C₂	1	2	yes	maximal, complemented
7	C₂²⋊C₈×D₂₅₁	C₂	1	2	yes	maximal, complemented
8	#432×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₀₀₄	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₂₅₁	C₂²	1	2	yes	—
11	C₂×#16860793	C₂²	1	4	yes	complemented
12	C₂×C₈×D₂₅₁	C₂²	1	4	yes	complemented
13	C₄×D₁₀₀₄	C₄	1	2	yes	—
14	C₄×D₁₀₀₄	C₄	1	1	yes	characteristic
15	C₄×D₁₀₀₄	C₄	1	1	yes	characteristic
16	C₄×D₁₀₀₄	C₂²	1	1	yes	characteristic
17	C₄×D₁₀₀₄	C₂²	1	2	yes	—
18	C₄×D₁₀₀₄	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₂₅₁	C₄	1	1	yes	characteristic
20	C₂×(C₂×D₂₅₁)⋊C₄	C₄	1	2	yes	—
21	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	complemented
22	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
24	(C₂×D₂₅₁)⋊C₈	C₂²	1	4	yes	complemented
25	(C₂×D₂₅₁)⋊C₈	C₂²	1	4	yes	complemented
26	C₄×C₂₅₁⋊C₈	C₂²	1	2	yes	complemented
27	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	complemented
29	C₂²⋊C₈×C₂₅₁	C₂²	1	2	yes	—
30	#97×C₂₅₁	C₂²	1	2	yes	complemented
31	C₂³×D₂₅₁	C₂×C₄	1	2	yes	—
32	C₂×D₁₀₀₄	—	2	4	no	—
33	C₂×D₁₀₀₄	C₂×C₄	1	2	yes	—
34	C₂×D₁₀₀₄	C₂×C₄	1	1	yes	characteristic
35	C₂×D₁₀₀₄	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₂₅₁	—	4	4	no	—
38	C₂×C₄×D₂₅₁	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₂₅₁	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
41	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
42	#16860793	—	4	4	no	—
43	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
44	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
45	C₈×D₂₅₁	—	4	4	no	—
46	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
48	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
49	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
50	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
51	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
52	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
53	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₂₅₁	C₂³	1	4	yes	—
57	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
60	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
61	C₂²×D₂₅₁	—	2	4	no	—
62	C₂²×D₂₅₁	—	2	4	no	—
63	C₂²×D₂₅₁	—	4	2	no	—
64	C₂²×D₂₅₁	C₂²×C₄	1	2	yes	—
65	C₂²×D₂₅₁	C₂²×C₄	1	2	yes	—
66	D₁₀₀₄	—	2	4	no	—
67	D₁₀₀₄	—	2	2	no	—
68	D₁₀₀₄	M₁₆	1	4	yes	complemented
69	C₂×Dic₂₅₁	—	4	2	no	—
70	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
71	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
72	C₄×D₂₅₁	—	2	4	no	—
73	C₄×D₂₅₁	—	2	4	no	—
74	C₄×D₂₅₁	—	4	2	no	—
75	C₄×D₂₅₁	—	4	2	no	—
76	C₂₅₁⋊C₈	—	2	2	no	—
77	C₂₅₁⋊C₈	—	2	2	no	—
78	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₅₁	—	2	2	no	—
80	C₂×C₄×C₂₅₁	—	2	2	no	—
81	C₂×C₄×C₂₅₁	—	2	2	no	—
82	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
88	C₈×C₂₅₁	—	2	4	no	—
89	C₂²×C₂₅₁	—	2	1	no	—
90	C₂²×C₂₅₁	—	2	1	no	—
91	C₂²×C₂₅₁	#131	1	2	yes	—
92	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₂₅₁	—	2	2	no	—
94	C₂×D₂₅₁	—	2	2	no	—
95	C₂×D₂₅₁	—	2	2	no	—
96	C₂×D₂₅₁	—	2	2	no	—
97	C₂×D₂₅₁	—	2	2	no	—
98	C₂×D₂₅₁	—	4	8	no	—
99	C₂×D₂₅₁	C₂×M₁₆	1	4	yes	—
100	C₄×C₂₅₁	—	2	1	no	—
101	C₄×C₂₅₁	—	2	1	no	—
102	C₄×C₂₅₁	—	2	1	no	—
103	C₄×C₂₅₁	—	2	1	no	—
104	C₄×C₂₅₁	C₂×M₁₆	1	2	yes	—
105	C₄×C₂₅₁	H(5,𝔽₂)	1	2	yes	—
106	Dic₂₅₁	—	2	2	no	—
107	Dic₂₅₁	—	2	2	no	—
108	C₂×C₂₅₁	—	2	2	no	—
109	C₂×C₂₅₁	#518	1	1	yes	characteristic
110	C₂×C₂₅₁	C₂²×M₁₆	1	1	yes	characteristic
111	C₂×C₂₅₁	C₂×#131	1	1	yes	characteristic
112	D₂₅₁	—	2	4	no	—
113	D₂₅₁	—	4	2	no	—
114	C₂₅₁	#2974	1	1	yes	characteristic, complemented, Sylow
115	#2974	—	251	1	no	maximal, Sylow
116	C₂×C₄×D₄	—	251	1	no	—
117	#436	—	251	2	no	—
118	#435	—	251	4	no	—
119	C₈×D₄	—	251	2	no	—
120	#432	D₂₅₁	1	1	yes	characteristic, complemented
121	#431	—	251	1	no	—
122	#407	—	251	2	no	—
123	C₂×C₂²⋊C₈	—	251	2	no	—
124	C₂²×D₄	—	251	1	no	—
125	C₂³×C₄	—	251	2	no	—
126	C₂×M₁₆	—	251	4	no	—
127	C₂²×C₈	—	251	4	no	—
128	C₄×D₄	—	251	2	no	—
129	C₄×D₄	—	251	1	no	—
130	C₄×D₄	—	251	1	no	—
131	C₄×D₄	—	251	1	no	—
132	C₄×D₄	—	251	2	no	—
133	C₄×D₄	—	251	1	no	—
134	C₂×C₄⋊C₄	—	251	1	no	—
135	C₂×C₂²⋊C₄	—	251	2	no	—
136	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
137	C₄⋊C₈	—	251	2	no	—
138	C₄⋊C₈	C₂×D₂₅₁	1	2	yes	complemented
139	C₂²⋊C₈	—	251	1	no	—
140	C₂²⋊C₈	—	251	1	no	—
141	C₂²⋊C₈	—	251	4	no	—
142	C₂²⋊C₈	—	251	4	no	—
143	C₂²⋊C₈	C₂×D₂₅₁	1	2	yes	—
144	#97	C₂×D₂₅₁	1	2	yes	complemented
145	C₄×C₈	—	251	2	no	—
146	C₂⁴	—	251	2	no	—
147	C₂×D₄	—	502	4	no	—
148	C₂×D₄	—	251	2	no	—
149	C₂×D₄	—	251	1	no	—
150	C₂×D₄	—	251	1	no	—
151	C₂²×C₄	—	1004	4	no	—
152	C₂²×C₄	—	251	2	no	—
153	C₂²×C₄	—	251	2	no	—
154	C₂²×C₄	—	251	2	no	—
155	C₂²×C₄	—	251	2	no	—
156	C₂²×C₄	—	251	2	no	—
157	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
158	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
160	M₁₆	—	1004	4	no	—
161	C₂×C₈	—	1004	4	no	—
162	C₂×C₈	—	251	2	no	—
163	C₂×C₈	—	251	2	no	—
164	C₂×C₈	C₂²×D₂₅₁	1	4	yes	—
165	C₄⋊C₄	—	251	2	no	—
166	C₄⋊C₄	—	251	1	no	—
167	C₄⋊C₄	—	251	1	no	—
168	C₂²⋊C₄	—	251	2	no	—
169	C₂²⋊C₄	—	251	2	no	—
170	C₂²⋊C₄	—	251	2	no	—
171	C₂²⋊C₄	—	251	2	no	—
172	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
173	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
174	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
175	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
176	C₂³	—	1004	2	no	—
177	C₂³	—	502	4	no	—
178	C₂³	—	502	4	no	—
179	C₂³	—	251	2	no	—
180	C₂³	—	251	2	no	—
181	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
182	D₄	—	502	4	no	—
183	D₄	—	502	2	no	—
184	D₄	—	251	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	1004	2	no	—
189	C₂×C₄	—	1004	2	no	—
190	C₂×C₄	—	1004	2	no	—
191	C₂×C₄	—	502	4	no	—
192	C₂×C₄	—	502	4	no	—
193	C₂×C₄	—	251	2	no	—
194	C₂×C₄	—	251	2	no	—
195	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
201	C₈	—	2	4	no	—
202	C₈	—	502	2	no	—
203	C₈	—	502	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	1004	8	no	—
207	C₂²	—	502	2	no	—
208	C₂²	—	502	2	no	—
209	C₂²	—	502	2	no	—
210	C₂²	—	502	2	no	—
211	C₂²	—	502	2	no	—
212	C₂²	—	251	4	no	—
213	C₂²	#22548723	1	2	yes	—
214	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	502	2	no	—
220	C₄	—	502	2	no	—
221	C₄	M₁₆×D₂₅₁	1	2	yes	—
222	C₄	#22548761	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	1004	2	no	—
225	C₂	—	502	4	no	—
226	C₂	#30632161	1	1	yes	characteristic
227	C₂	C₂×M₁₆×D₂₅₁	1	1	yes	characteristic
228	C₂	C₂×#22548723	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	C₂²×D₁₀₀₄	C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#432	D₂₅₁
hypocenter	C₂₅₁	#2974
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. #30631764	C₂	semidirect
3. #30631760	C₂	semidirect
4. #30631752	C₂	semidirect
5. #30631740	C₂	semidirect
6. #30631649	C₂	semidirect
7. C₂²⋊C₈×D₂₅₁	C₂	semidirect
8. #432×C₂₅₁	C₂	semidirect
11. C₂×#16860793	C₂²	semidirect
12. C₂×C₈×D₂₅₁	C₂²	semidirect
21. C₄⋊(C₂₅₁⋊C₈)	C₂²	semidirect
24. (C₂×D₂₅₁)⋊C₈	C₂²	semidirect
25. (C₂×D₂₅₁)⋊C₈	C₂²	semidirect
26. C₄×C₂₅₁⋊C₈	C₂²	semidirect
28. C₄⋊C₈×C₂₅₁	C₂²	semidirect
30. #97×C₂₅₁	C₂²	semidirect
68. D₁₀₀₄	M₁₆	semidirect
114. C₂₅₁	#2974	semidirect
120. #432	D₂₅₁	semidirect
138. C₄⋊C₈	C₂×D₂₅₁	semidirect
144. #97	C₂×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₀₀₄	2
#30631764	2
#30631760	2
#30631752	2
#30631740	2
#30631649	2
C₂²⋊C₈×D₂₅₁	2
#432×C₂₅₁	2
#2974	251