Subgroups — #3208×C₂₅₁ #44547795

Order 32128.

488 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#127×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#492×C₂₅₁	C₂	1	2	yes	maximal, complemented
4	#486×C₂₅₁	C₂	1	2	yes	maximal, complemented
5	#460×C₂₅₁	C₂	1	4	yes	maximal, complemented
6	#450×C₂₅₁	C₂	1	4	yes	maximal, complemented
7	#431×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₂₅₁	—	2	1	no	—
9	C₂²×D₄×C₂₅₁	—	2	1	no	—
10	C₂²×D₄×C₂₅₁	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₂₅₁	C₂²	1	8	yes	complemented
12	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
13	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
14	#127×C₂₅₁	—	2	1	no	—
15	#127×C₂₅₁	—	2	1	no	—
16	#127×C₂₅₁	C₂²	1	2	yes	—
17	#127×C₂₅₁	C₂²	1	2	yes	—
18	#125×C₂₅₁	—	2	1	no	—
19	C₂²⋊Q₈×C₂₅₁	—	2	1	no	—
20	C₂²⋊Q₈×C₂₅₁	C₂²	1	2	yes	—
21	C₄×Q₈×C₂₅₁	—	2	1	no	—
22	C₄×Q₈×C₂₅₁	C₂²	1	2	yes	complemented
23	#117×C₂₅₁	—	2	1	no	—
24	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	complemented
26	#102×C₂₅₁	C₂²	1	8	yes	complemented
27	C₂²⋊C₈×C₂₅₁	C₂²	1	2	yes	—
28	C₄×C₈×C₂₅₁	C₂²	1	2	yes	complemented
29	C₂⁴×C₂₅₁	—	2	2	no	—
30	C₂×Q₈×C₂₅₁	—	2	1	no	—
31	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
32	C₂×D₄×C₂₅₁	—	2	2	no	—
33	C₂×D₄×C₂₅₁	—	2	2	no	—
34	C₂×D₄×C₂₅₁	—	2	2	no	—
35	C₂×D₄×C₂₅₁	—	2	2	no	—
36	C₂×D₄×C₂₅₁	—	2	2	no	—
37	C₂×D₄×C₂₅₁	—	4	4	no	—
38	C₂×D₄×C₂₅₁	—	4	4	no	—
39	C₂×D₄×C₂₅₁	—	4	4	no	—
40	C₂×D₄×C₂₅₁	C₂³	1	4	yes	—
41	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
44	SD₁₆×C₂₅₁	—	4	8	no	—
45	C₂×C₈×C₂₅₁	C₂³	1	4	yes	—
46	C₄⋊C₄×C₂₅₁	—	2	1	no	—
47	C₄⋊C₄×C₂₅₁	—	2	1	no	—
48	C₄⋊C₄×C₂₅₁	—	2	1	no	—
49	C₄⋊C₄×C₂₅₁	—	2	2	no	—
50	C₄⋊C₄×C₂₅₁	—	2	2	no	—
51	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
52	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
53	C₄²×C₂₅₁	—	2	2	no	—
54	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
55	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
56	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
57	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
58	C₂³×C₂₅₁	—	2	4	no	—
59	C₂³×C₂₅₁	—	4	4	no	—
60	C₂³×C₂₅₁	—	4	4	no	—
61	C₂³×C₂₅₁	—	4	4	no	—
62	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
63	Q₈×C₂₅₁	—	2	4	no	—
64	Q₈×C₂₅₁	—	4	2	no	—
65	D₄×C₂₅₁	—	2	8	no	—
66	D₄×C₂₅₁	—	4	4	no	—
67	D₄×C₂₅₁	—	4	4	no	—
68	D₄×C₂₅₁	—	4	4	no	—
69	D₄×C₂₅₁	—	4	4	no	—
70	D₄×C₂₅₁	—	4	4	no	—
71	C₂×C₄×C₂₅₁	—	2	2	no	—
72	C₂×C₄×C₂₅₁	—	2	2	no	—
73	C₂×C₄×C₂₅₁	—	2	2	no	—
74	C₂×C₄×C₂₅₁	—	2	2	no	—
75	C₂×C₄×C₂₅₁	SD₁₆	1	4	yes	complemented
76	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
82	C₈×C₂₅₁	—	2	4	no	—
83	C₂²×C₂₅₁	—	2	1	no	—
84	C₂²×C₂₅₁	—	2	1	no	—
85	C₂²×C₂₅₁	—	4	4	no	—
86	C₂²×C₂₅₁	—	4	4	no	—
87	C₂²×C₂₅₁	—	4	4	no	—
88	C₂²×C₂₅₁	—	8	8	no	—
89	C₂²×C₂₅₁	C₂×SD₁₆	1	2	yes	—
90	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
91	C₄×C₂₅₁	—	2	1	no	—
92	C₄×C₂₅₁	—	2	1	no	—
93	C₄×C₂₅₁	—	4	2	no	—
94	C₄×C₂₅₁	—	4	2	no	—
95	C₄×C₂₅₁	C₂×SD₁₆	1	2	yes	—
96	C₄×C₂₅₁	C₂×SD₁₆	1	2	yes	—
97	C₄×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
98	C₄×C₂₅₁	H(5,𝔽₂)	1	2	yes	—
99	C₂×C₂₅₁	—	2	2	no	—
100	C₂×C₂₅₁	—	8	4	no	—
101	C₂×C₂₅₁	#535	1	1	yes	characteristic
102	C₂×C₂₅₁	C₂²×SD₁₆	1	1	yes	characteristic
103	C₂×C₂₅₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂₅₁	#3208	1	1	yes	characteristic, direct product factor, Sylow
105	#3208	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	#533	C₂×C₂₅₁	1	1	yes	characteristic, complemented
107	C₂×#127	C₂×C₂₅₁	1	1	yes	characteristic
108	#492	C₂×C₂₅₁	1	2	yes	complemented
109	#486	C₂×C₂₅₁	1	2	yes	complemented
110	#460	C₂×C₂₅₁	1	4	yes	complemented
111	#450	C₂×C₂₅₁	1	4	yes	complemented
112	#431	C₂×C₂₅₁	1	1	yes	characteristic, complemented
113	C₂²×D₄	—	2	1	no	—
114	C₂²×D₄	—	2	1	no	—
115	C₂²×D₄	C₂²×C₂₅₁	1	2	yes	—
116	C₂×SD₁₆	C₂²×C₂₅₁	1	8	yes	complemented
117	C₄⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic, complemented
118	C₄⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic, complemented
119	#127	—	2	1	no	—
120	#127	—	2	1	no	—
121	#127	C₂²×C₂₅₁	1	2	yes	—
122	#127	C₂²×C₂₅₁	1	2	yes	—
123	#125	—	2	1	no	—
124	C₂²⋊Q₈	—	2	1	no	—
125	C₂²⋊Q₈	C₂²×C₂₅₁	1	2	yes	—
126	C₄×Q₈	—	2	1	no	—
127	C₄×Q₈	C₂²×C₂₅₁	1	2	yes	complemented
128	#117	—	2	1	no	—
129	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
130	C₄⋊C₈	C₂²×C₂₅₁	1	2	yes	complemented
131	#102	C₂²×C₂₅₁	1	8	yes	complemented
132	C₂²⋊C₈	C₂²×C₂₅₁	1	2	yes	—
133	C₄×C₈	C₂²×C₂₅₁	1	2	yes	complemented
134	C₂⁴	—	2	2	no	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	C₂³×C₂₅₁	1	2	yes	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	—	4	4	no	—
143	C₂×D₄	—	4	4	no	—
144	C₂×D₄	—	4	4	no	—
145	C₂×D₄	C₂³×C₂₅₁	1	4	yes	—
146	C₂²×C₄	D₄×C₂₅₁	1	1	yes	characteristic
147	C₂²×C₄	D₄×C₂₅₁	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
149	SD₁₆	—	4	8	no	—
150	C₂×C₈	C₂³×C₂₅₁	1	4	yes	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	C₂³×C₂₅₁	1	2	yes	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	D₄×C₂₅₁	1	1	yes	characteristic
160	C₄²	D₄×C₂₅₁	1	1	yes	characteristic
161	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
162	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
163	C₂³	—	2	4	no	—
164	C₂³	—	4	4	no	—
165	C₂³	—	4	4	no	—
166	C₂³	—	4	4	no	—
167	C₂³	C₂×D₄×C₂₅₁	1	1	yes	characteristic
168	Q₈	—	2	4	no	—
169	Q₈	—	4	2	no	—
170	D₄	—	2	8	no	—
171	D₄	—	4	4	no	—
172	D₄	—	4	4	no	—
173	D₄	—	4	4	no	—
174	D₄	—	4	4	no	—
175	D₄	—	4	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	SD₁₆×C₂₅₁	1	4	yes	complemented
181	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
183	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
184	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
185	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
186	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
187	C₈	—	2	4	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	4	4	no	—
191	C₂²	—	4	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	—	8	8	no	—
194	C₂²	C₂×SD₁₆×C₂₅₁	1	2	yes	—
195	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	C₂×SD₁₆×C₂₅₁	1	2	yes	—
201	C₄	C₂×SD₁₆×C₂₅₁	1	2	yes	—
202	C₄	AGL(1,ℤ/8)×C₂₅₁	1	2	yes	—
203	C₄	H(5,𝔽₂)×C₂₅₁	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	8	4	no	—
206	C₂	#535×C₂₅₁	1	1	yes	characteristic
207	C₂	C₂²×SD₁₆×C₂₅₁	1	1	yes	characteristic
208	C₂	C₂×AGL(1,ℤ/8)×C₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₅₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₅₁
prime radical	C₂×#127×C₂₅₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₁ ◁ C₂×C₄²×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #533×C₂₅₁	C₂	semidirect
3. #492×C₂₅₁	C₂	semidirect
4. #486×C₂₅₁	C₂	semidirect
5. #460×C₂₅₁	C₂	semidirect
6. #450×C₂₅₁	C₂	semidirect
7. #431×C₂₅₁	C₂	semidirect
11. C₂×SD₁₆×C₂₅₁	C₂²	semidirect
12. C₄⋊Q₈×C₂₅₁	C₂²	semidirect
13. C₄⋊Q₈×C₂₅₁	C₂²	semidirect
22. C₄×Q₈×C₂₅₁	C₂²	semidirect
25. C₄⋊C₈×C₂₅₁	C₂²	semidirect
26. #102×C₂₅₁	C₂²	semidirect
28. C₄×C₈×C₂₅₁	C₂²	semidirect
75. C₂×C₄×C₂₅₁	SD₁₆	semidirect
104. C₂₅₁	#3208	direct (N × H)
105. #3208	C₂₅₁	direct (N × H)
106. #533	C₂×C₂₅₁	semidirect
108. #492	C₂×C₂₅₁	semidirect
109. #486	C₂×C₂₅₁	semidirect
110. #460	C₂×C₂₅₁	semidirect
111. #450	C₂×C₂₅₁	semidirect
112. #431	C₂×C₂₅₁	semidirect
116. C₂×SD₁₆	C₂²×C₂₅₁	semidirect
117. C₄⋊Q₈	C₂²×C₂₅₁	semidirect
118. C₄⋊Q₈	C₂²×C₂₅₁	semidirect
127. C₄×Q₈	C₂²×C₂₅₁	semidirect
130. C₄⋊C₈	C₂²×C₂₅₁	semidirect
131. #102	C₂²×C₂₅₁	semidirect
133. C₄×C₈	C₂²×C₂₅₁	semidirect
180. C₂×C₄	SD₁₆×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#533×C₂₅₁	2
C₂×#127×C₂₅₁	2
#492×C₂₅₁	2
#486×C₂₅₁	2
#460×C₂₅₁	2
#450×C₂₅₁	2
#431×C₂₅₁	2
#3208	251

Subgroups — #3208×C251 #44547795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3208×C₂₅₁ #44547795