Subgroups — #3123×C₂₅₁ #44547710

Order 32128.

394 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#541×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#518×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	#485×C₂₅₁	C₂	1	2	yes	maximal, complemented
4	#478×C₂₅₁	C₂	1	2	yes	maximal, complemented
5	#461×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	#450×C₂₅₁	C₂	1	2	yes	maximal
7	#440×C₂₅₁	C₂	1	2	yes	maximal, complemented
8	C₄×SD₁₆×C₂₅₁	C₂	1	2	yes	maximal, complemented
9	#433×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₂₅₁	C₂²	1	2	yes	—
12	C₂×SD₁₆×C₂₅₁	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
14	#126×C₂₅₁	—	2	2	no	—
15	#125×C₂₅₁	—	2	2	no	—
16	#125×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
17	C₂²⋊Q₈×C₂₅₁	C₂²	1	2	yes	—
18	C₄×Q₈×C₂₅₁	C₂²	1	2	yes	complemented
19	C₄×D₄×C₂₅₁	—	2	2	no	—
20	C₄×D₄×C₂₅₁	C₂²	1	2	yes	—
21	C₄×D₄×C₂₅₁	C₂²	1	2	yes	—
22	#117×C₂₅₁	C₂²	1	1	yes	characteristic
23	#117×C₂₅₁	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₂₅₁	—	2	2	no	—
25	#107×C₂₅₁	C₂²	1	2	yes	complemented
26	#106×C₂₅₁	C₂²	1	2	yes	complemented
27	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	complemented
28	#103×C₂₅₁	C₂²	1	2	yes	complemented
29	#103×C₂₅₁	C₂²	1	2	yes	complemented
30	#102×C₂₅₁	C₂²	1	2	yes	—
31	#102×C₂₅₁	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₂₅₁	C₂²	1	2	yes	—
33	#97×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
34	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
35	C₂⁴×C₂₅₁	—	2	1	no	—
36	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
37	C₂×D₄×C₂₅₁	—	2	4	no	—
38	C₂×D₄×C₂₅₁	—	2	1	no	—
39	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₅₁	—	2	2	no	—
41	C₂²×C₄×C₂₅₁	—	2	2	no	—
42	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
43	SD₁₆×C₂₅₁	—	2	4	no	—
44	SD₁₆×C₂₅₁	—	2	4	no	—
45	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
46	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
47	C₄⋊C₄×C₂₅₁	—	2	2	no	—
48	C₄⋊C₄×C₂₅₁	—	2	2	no	—
49	C₄⋊C₄×C₂₅₁	—	2	2	no	—
50	C₄⋊C₄×C₂₅₁	—	2	2	no	—
51	C₄⋊C₄×C₂₅₁	D₄	1	2	yes	—
52	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
53	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
54	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
55	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
56	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
57	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
58	C₂²⋊C₄×C₂₅₁	D₄	1	2	yes	—
59	C₄²×C₂₅₁	—	2	1	no	—
60	C₄²×C₂₅₁	—	2	2	no	—
61	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
62	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
63	C₂³×C₂₅₁	—	2	4	no	—
64	C₂³×C₂₅₁	—	2	2	no	—
65	C₂³×C₂₅₁	—	8	2	no	—
66	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₂₅₁	—	2	2	no	—
68	Q₈×C₂₅₁	—	2	2	no	—
69	D₄×C₂₅₁	—	2	4	no	—
70	D₄×C₂₅₁	—	2	2	no	—
71	D₄×C₂₅₁	#41	1	4	yes	—
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	—	2	2	no	—
76	C₂×C₄×C₂₅₁	—	8	2	no	—
77	C₂×C₄×C₂₅₁	—	8	2	no	—
78	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
82	C₈×C₂₅₁	—	2	2	no	—
83	C₈×C₂₅₁	—	2	2	no	—
84	C₂²×C₂₅₁	—	2	1	no	—
85	C₂²×C₂₅₁	—	2	4	no	—
86	C₂²×C₂₅₁	—	2	2	no	—
87	C₂²×C₂₅₁	—	4	2	no	—
88	C₂²×C₂₅₁	—	4	2	no	—
89	C₂²×C₂₅₁	—	8	4	no	—
90	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
91	C₄×C₂₅₁	—	2	1	no	—
92	C₄×C₂₅₁	—	2	2	no	—
93	C₄×C₂₅₁	—	2	2	no	—
94	C₄×C₂₅₁	—	4	2	no	—
95	C₄×C₂₅₁	—	4	2	no	—
96	C₄×C₂₅₁	—	4	2	no	—
97	C₄×C₂₅₁	C₂×#41	1	2	yes	—
98	C₂×C₂₅₁	—	4	1	no	—
99	C₂×C₂₅₁	—	4	4	no	—
100	C₂×C₂₅₁	#525	1	1	yes	characteristic
101	C₂×C₂₅₁	#577	1	2	yes	—
102	C₂₅₁	#3123	1	1	yes	characteristic, direct product factor, Sylow
103	#3123	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	#541	C₂×C₂₅₁	1	1	yes	characteristic, complemented
105	#518	C₂×C₂₅₁	1	1	yes	characteristic
106	#485	C₂×C₂₅₁	1	2	yes	complemented
107	#478	C₂×C₂₅₁	1	2	yes	complemented
108	#461	C₂×C₂₅₁	1	2	yes	complemented
109	#450	C₂×C₂₅₁	1	2	yes	—
110	#440	C₂×C₂₅₁	1	2	yes	complemented
111	C₄×SD₁₆	C₂×C₂₅₁	1	2	yes	complemented
112	#433	C₂×C₂₅₁	1	1	yes	characteristic, complemented
113	C₂²×D₄	C₂²×C₂₅₁	1	1	yes	characteristic
114	C₂×SD₁₆	C₂²×C₂₅₁	1	2	yes	—
115	C₂×SD₁₆	C₂²×C₂₅₁	1	2	yes	—
116	C₄⋊Q₈	C₂²×C₂₅₁	1	1	yes	characteristic, complemented
117	#126	—	2	2	no	—
118	#125	—	2	2	no	—
119	#125	C₂²×C₂₅₁	1	1	yes	characteristic, complemented
120	C₂²⋊Q₈	C₂²×C₂₅₁	1	2	yes	—
121	C₄×Q₈	C₂²×C₂₅₁	1	2	yes	complemented
122	C₄×D₄	—	2	2	no	—
123	C₄×D₄	C₂²×C₂₅₁	1	2	yes	—
124	C₄×D₄	C₂²×C₂₅₁	1	2	yes	—
125	#117	C₂²×C₂₅₁	1	1	yes	characteristic
126	#117	C₂²×C₂₅₁	1	1	yes	characteristic
127	C₂×C₂²⋊C₄	—	2	2	no	—
128	#107	C₂²×C₂₅₁	1	2	yes	complemented
129	#106	C₂²×C₂₅₁	1	2	yes	complemented
130	C₄⋊C₈	C₂²×C₂₅₁	1	2	yes	complemented
131	#103	C₂²×C₂₅₁	1	2	yes	complemented
132	#103	C₂²×C₂₅₁	1	2	yes	complemented
133	#102	C₂²×C₂₅₁	1	2	yes	—
134	#102	C₂²×C₂₅₁	1	2	yes	—
135	C₂²⋊C₈	C₂²×C₂₅₁	1	2	yes	—
136	#97	C₂²×C₂₅₁	1	1	yes	characteristic, complemented
137	C₄×C₈	C₂²×C₂₅₁	1	1	yes	characteristic, complemented
138	C₂⁴	—	2	1	no	—
139	C₂×Q₈	C₂³×C₂₅₁	1	2	yes	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	2	1	no	—
142	C₂×D₄	C₂³×C₂₅₁	1	2	yes	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
146	SD₁₆	—	2	4	no	—
147	SD₁₆	—	2	4	no	—
148	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
149	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	D₄×C₂₅₁	1	2	yes	—
155	C₄⋊C₄	C₂³×C₂₅₁	1	2	yes	—
156	C₄⋊C₄	C₂³×C₂₅₁	1	2	yes	—
157	C₂²⋊C₄	—	2	1	no	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	D₄×C₂₅₁	1	2	yes	—
162	C₄²	—	2	1	no	—
163	C₄²	—	2	2	no	—
164	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
165	C₄²	C₂³×C₂₅₁	1	1	yes	characteristic
166	C₂³	—	2	4	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	8	2	no	—
169	C₂³	C₂×D₄×C₂₅₁	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	D₄	—	2	4	no	—
173	D₄	—	2	2	no	—
174	D₄	#41×C₂₅₁	1	4	yes	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
182	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
183	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
184	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	2	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	8	4	no	—
193	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	C₂×#41×C₂₅₁	1	2	yes	—
201	C₂	—	4	1	no	—
202	C₂	—	4	4	no	—
203	C₂	#525×C₂₅₁	1	1	yes	characteristic
204	C₂	#577×C₂₅₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₅₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₅₁
prime radical	C₂²×D₄×C₂₅₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₁ ◁ #117×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #541×C₂₅₁	C₂	semidirect
3. #485×C₂₅₁	C₂	semidirect
4. #478×C₂₅₁	C₂	semidirect
5. #461×C₂₅₁	C₂	semidirect
7. #440×C₂₅₁	C₂	semidirect
8. C₄×SD₁₆×C₂₅₁	C₂	semidirect
9. #433×C₂₅₁	C₂	semidirect
13. C₄⋊Q₈×C₂₅₁	C₂²	semidirect
16. #125×C₂₅₁	C₂²	semidirect
18. C₄×Q₈×C₂₅₁	C₂²	semidirect
25. #107×C₂₅₁	C₂²	semidirect
26. #106×C₂₅₁	C₂²	semidirect
27. C₄⋊C₈×C₂₅₁	C₂²	semidirect
28. #103×C₂₅₁	C₂²	semidirect
29. #103×C₂₅₁	C₂²	semidirect
33. #97×C₂₅₁	C₂²	semidirect
34. C₄×C₈×C₂₅₁	C₂²	semidirect
102. C₂₅₁	#3123	direct (N × H)
103. #3123	C₂₅₁	direct (N × H)
104. #541	C₂×C₂₅₁	semidirect
106. #485	C₂×C₂₅₁	semidirect
107. #478	C₂×C₂₅₁	semidirect
108. #461	C₂×C₂₅₁	semidirect
110. #440	C₂×C₂₅₁	semidirect
111. C₄×SD₁₆	C₂×C₂₅₁	semidirect
112. #433	C₂×C₂₅₁	semidirect
116. C₄⋊Q₈	C₂²×C₂₅₁	semidirect
119. #125	C₂²×C₂₅₁	semidirect
121. C₄×Q₈	C₂²×C₂₅₁	semidirect
128. #107	C₂²×C₂₅₁	semidirect
129. #106	C₂²×C₂₅₁	semidirect
130. C₄⋊C₈	C₂²×C₂₅₁	semidirect
131. #103	C₂²×C₂₅₁	semidirect
132. #103	C₂²×C₂₅₁	semidirect
136. #97	C₂²×C₂₅₁	semidirect
137. C₄×C₈	C₂²×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#541×C₂₅₁	2
#518×C₂₅₁	2
#485×C₂₅₁	2
#478×C₂₅₁	2
#461×C₂₅₁	2
#450×C₂₅₁	2
#440×C₂₅₁	2
C₄×SD₁₆×C₂₅₁	2
#433×C₂₅₁	2
#3123	251

Subgroups — #3123×C251 #44547710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3123×C₂₅₁ #44547710