Subgroups — #44559532

Order 32128.

346 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	#30632221	C₂	1	1	yes	maximal, characteristic
2	#30631816	C₂	1	2	yes	maximal, complemented
3	#30631763	C₂	1	2	yes	maximal, complemented
4	#30631757	C₂	1	2	yes	maximal, complemented
5	#30631756	C₂	1	2	yes	maximal, complemented
6	#30631747	C₂	1	2	yes	maximal, complemented
7	#30631744	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631712	C₂	1	2	yes	maximal, complemented
9	#423×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16860793	C₂²	1	2	yes	—
11	C₂×C₈×D₂₅₁	C₂²	1	2	yes	—
12	#22548709	C₄	1	2	yes	—
13	#22548707	C₄	1	1	yes	characteristic
14	#22548703	C₄	1	1	yes	characteristic
15	#22548698	C₄	1	2	yes	—
16	#22548694	C₄	1	1	yes	characteristic
17	#22548689	C₄	1	1	yes	characteristic
18	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
19	C₄×D₁₀₀₄	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₂₅₁	C₂²	1	2	yes	—
21	#22548672	C₂²	1	2	yes	—
22	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
24	Dic₂₅₁⋊C₈	C₂²	1	2	yes	—
25	Dic₂₅₁⋊C₈	C₂²	1	2	yes	—
26	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
27	(C₂×D₂₅₁)⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₂₅₁)⋊C₈	C₂²	1	2	yes	—
29	#22548626	C₂²	1	2	yes	—
30	#22548625	C₂²	1	2	yes	—
31	C₈×Dic₂₅₁	C₂²	1	2	yes	—
32	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
33	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	—
36	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	—
37	C₂×Dic₅₀₂	C₂×C₄	1	1	yes	characteristic
38	C₂×D₁₀₀₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₂₅₁	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
41	#16860793	—	4	2	no	—
42	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
43	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
44	C₈×D₂₅₁	—	4	2	no	—
45	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
47	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
48	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
49	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
50	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
51	C₂²⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
52	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
53	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄×Dic₂₅₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
60	M₁₆×C₂₅₁	—	4	1	no	—
61	M₁₆×C₂₅₁	—	4	1	no	—
62	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
63	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
64	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
67	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
68	C₂²×D₂₅₁	C₂²×C₄	1	2	yes	—
69	Dic₅₀₂	—	4	1	no	—
70	D₁₀₀₄	—	4	1	no	—
71	C₂²⋊D₂₅₁	—	4	2	no	—
72	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
73	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
74	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
75	C₄×D₂₅₁	—	2	2	no	—
76	C₄×D₂₅₁	—	2	2	no	—
77	C₂₅₁⋊C₈	—	2	2	no	—
78	C₂₅₁⋊C₈	—	2	2	no	—
79	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₅₁	—	2	1	no	—
81	C₂×C₄×C₂₅₁	—	2	1	no	—
82	C₂×C₄×C₂₅₁	—	4	1	no	—
83	C₂×C₄×C₂₅₁	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
90	C₈×C₂₅₁	—	2	2	no	—
91	C₈×C₂₅₁	—	2	2	no	—
92	C₂²×C₂₅₁	—	2	1	no	—
93	C₂²×C₂₅₁	—	2	1	no	—
94	C₂²×C₂₅₁	—	2	1	no	—
95	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₂₅₁	—	2	2	no	—
97	C₂×D₂₅₁	—	2	2	no	—
98	C₂×D₂₅₁	—	2	2	no	—
99	C₄×C₂₅₁	—	2	1	no	—
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	H(5,𝔽₂)	1	1	yes	characteristic
105	C₄×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
106	Dic₂₅₁	—	2	2	no	—
107	Dic₂₅₁	—	2	2	no	—
108	Dic₂₅₁	—	2	2	no	—
109	C₂×C₂₅₁	—	4	1	no	—
110	C₂×C₂₅₁	#520	1	1	yes	characteristic
111	C₂×C₂₅₁	#568	1	1	yes	characteristic
112	C₂×C₂₅₁	#568	1	1	yes	characteristic
113	D₂₅₁	—	4	2	no	—
114	C₂₅₁	#2995	1	1	yes	characteristic, complemented, Sylow
115	#2995	—	251	1	no	maximal, Sylow
116	#529	—	251	1	no	—
117	#446	—	251	2	no	—
118	#436	—	251	2	no	—
119	#435	—	251	2	no	—
120	#435	—	251	2	no	—
121	#433	—	251	2	no	—
122	#432	—	251	1	no	—
123	#424	—	251	2	no	—
124	#423	D₂₅₁	1	1	yes	characteristic, complemented
125	C₂×M₁₆	—	251	2	no	—
126	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
127	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
128	C₂²×C₈	—	251	2	no	—
129	#126	—	251	2	no	—
130	#125	—	251	1	no	—
131	#124	—	251	1	no	—
132	#123	—	251	2	no	—
133	C₂²⋊Q₈	—	251	1	no	—
134	#121	—	251	1	no	—
135	C₄×Q₈	—	251	1	no	—
136	C₄×D₄	—	251	2	no	—
137	C₄×D₄	—	251	1	no	—
138	#117	—	251	2	no	—
139	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
140	C₄⋊C₈	—	251	2	no	—
141	C₄⋊C₈	—	251	2	no	—
142	C₄⋊C₈	—	251	1	no	—
143	C₄⋊C₈	—	251	1	no	—
144	C₄⋊C₈	C₂×D₂₅₁	1	2	yes	—
145	C₄⋊C₈	C₂×D₂₅₁	1	2	yes	—
146	C₂²⋊C₈	—	251	2	no	—
147	C₂²⋊C₈	—	251	2	no	—
148	C₂²⋊C₈	—	251	2	no	—
149	#97	—	251	2	no	—
150	#97	—	251	2	no	—
151	C₄×C₈	—	251	2	no	—
152	C₂×Q₈	—	251	1	no	—
153	C₂×D₄	—	251	2	no	—
154	C₂×D₄	—	251	1	no	—
155	C₂²×C₄	—	251	2	no	—
156	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
157	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
159	M₁₆	—	4	1	no	—
160	M₁₆	—	4	1	no	—
161	M₁₆	—	1004	2	no	—
162	C₂×C₈	—	1004	2	no	—
163	C₂×C₈	—	251	2	no	—
164	C₂×C₈	—	251	2	no	—
165	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
166	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
167	C₄⋊C₄	—	251	2	no	—
168	C₄⋊C₄	—	251	2	no	—
169	C₄⋊C₄	—	251	2	no	—
170	C₄⋊C₄	—	251	2	no	—
171	C₄⋊C₄	—	251	1	no	—
172	C₄⋊C₄	—	251	1	no	—
173	C₂²⋊C₄	—	251	2	no	—
174	C₂²⋊C₄	—	251	2	no	—
175	C₂²⋊C₄	—	251	2	no	—
176	C₂²⋊C₄	—	251	2	no	—
177	C₂²⋊C₄	—	251	2	no	—
178	C₄²	—	251	2	no	—
179	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
180	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
181	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
182	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
183	C₂³	—	251	2	no	—
184	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
185	Q₈	—	1004	1	no	—
186	D₄	—	1004	1	no	—
187	D₄	—	1004	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	502	2	no	—
193	C₂×C₄	—	502	2	no	—
194	C₂×C₄	—	251	2	no	—
195	C₂×C₄	—	251	2	no	—
196	C₂×C₄	—	251	2	no	—
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	C₂×C₄×D₂₅₁	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	502	2	no	—
206	C₈	—	502	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	502	2	no	—
211	C₂²	—	502	2	no	—
212	C₂²	—	502	2	no	—
213	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
214	C₄	—	2	1	no	—
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	—	502	2	no	—
222	C₄	#22548760	1	1	yes	characteristic
223	C₄	#22548763	1	1	yes	characteristic
224	C₂	—	4	1	no	—
225	C₂	—	1004	2	no	—
226	C₂	#30632168	1	1	yes	characteristic
227	C₂	#30632453	1	1	yes	characteristic
228	C₂	#30632453	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	#22548689	C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#423	D₂₅₁
hypocenter	C₂₅₁	#2995
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. #30631816	C₂	semidirect
3. #30631763	C₂	semidirect
4. #30631757	C₂	semidirect
5. #30631756	C₂	semidirect
6. #30631747	C₂	semidirect
7. #30631744	C₂	semidirect
8. #30631712	C₂	semidirect
9. #423×C₂₅₁	C₂	semidirect
114. C₂₅₁	#2995	semidirect
124. #423	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632221	2
#30631816	2
#30631763	2
#30631757	2
#30631756	2
#30631747	2
#30631744	2
#30631712	2
#423×C₂₅₁	2
#2995	251