Subgroups — #2948×C₂₅₁ #44547535

Order 32128.

528 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	#518×C₂₅₁	C₂	1	2	yes	maximal, complemented
3	#442×C₂₅₁	C₂	1	4	yes	maximal, complemented
4	C₄×D₈×C₂₅₁	C₂	1	4	yes	maximal, complemented
5	#427×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#418×C₂₅₁	C₂	1	2	yes	maximal, complemented
7	#405×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₂₅₁	C₂²	1	2	yes	—
9	C₂×D₈×C₂₅₁	C₄	1	8	yes	complemented
10	C₂×D₈×C₂₅₁	C₂²	1	2	yes	—
11	C₂×D₈×C₂₅₁	C₂²	1	2	yes	—
12	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₂₅₁	—	2	4	no	—
15	C₄×D₄×C₂₅₁	C₂²	1	8	yes	complemented
16	#117×C₂₅₁	C₂²	1	2	yes	—
17	#117×C₂₅₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₅₁	—	2	4	no	—
19	#107×C₂₅₁	C₂²	1	2	yes	complemented
20	#106×C₂₅₁	C₂²	1	2	yes	complemented
21	#102×C₂₅₁	C₂²	1	8	yes	complemented
22	#97×C₂₅₁	C₂²	1	2	yes	complemented
23	C₄×C₈×C₂₅₁	C₂²	1	2	yes	complemented
24	C₂⁴×C₂₅₁	—	2	2	no	—
25	C₂×D₄×C₂₅₁	—	2	2	no	—
26	C₂×D₄×C₂₅₁	—	2	4	no	—
27	C₂×D₄×C₂₅₁	C₂×C₄	1	8	yes	complemented
28	C₂×D₄×C₂₅₁	C₂³	1	4	yes	—
29	C₂²×C₄×C₂₅₁	—	2	8	no	—
30	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
31	D₈×C₂₅₁	C₂×C₄	1	8	yes	complemented
32	D₈×C₂₅₁	C₂×C₄	1	8	yes	complemented
33	M₁₆×C₂₅₁	—	2	2	no	—
34	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	complemented
35	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
36	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₅₁	D₄	1	2	yes	—
39	C₄⋊C₄×C₂₅₁	C₂³	1	4	yes	—
40	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
41	C₂²⋊C₄×C₂₅₁	—	2	8	no	—
42	C₂²⋊C₄×C₂₅₁	D₄	1	2	yes	—
43	C₄²×C₂₅₁	—	2	2	no	—
44	C₄²×C₂₅₁	C₂³	1	2	yes	—
45	C₂³×C₂₅₁	—	2	4	no	—
46	C₂³×C₂₅₁	—	2	8	no	—
47	C₂³×C₂₅₁	—	8	4	no	—
48	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
49	D₄×C₂₅₁	—	2	8	no	—
50	D₄×C₂₅₁	—	2	4	no	—
51	D₄×C₂₅₁	C₂²×C₄	1	8	yes	—
52	C₂×C₄×C₂₅₁	—	2	4	no	—
53	C₂×C₄×C₂₅₁	—	8	8	no	—
54	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
56	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₅₁	#41	1	2	yes	—
58	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
59	C₈×C₂₅₁	—	2	2	no	—
60	C₈×C₂₅₁	C₂²×C₄	1	2	yes	—
61	C₈×C₂₅₁	C₂²×C₄	1	2	yes	—
62	C₂²×C₂₅₁	—	2	2	no	—
63	C₂²×C₂₅₁	—	2	8	no	—
64	C₂²×C₂₅₁	—	4	8	no	—
65	C₂²×C₂₅₁	—	8	8	no	—
66	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
67	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₂₅₁	—	2	4	no	—
69	C₄×C₂₅₁	—	4	4	no	—
70	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
71	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
72	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
73	C₂×C₂₅₁	—	2	2	no	—
74	C₂×C₂₅₁	—	4	8	no	—
75	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
76	C₂×C₂₅₁	#576	1	2	yes	—
77	C₂₅₁	#2948	1	1	yes	characteristic, direct product factor, Sylow
78	#2948	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×D₈	C₂×C₂₅₁	1	1	yes	characteristic
80	#518	C₂×C₂₅₁	1	2	yes	complemented
81	#442	C₂×C₂₅₁	1	4	yes	complemented
82	C₄×D₈	C₂×C₂₅₁	1	4	yes	complemented
83	#427	C₂×C₂₅₁	1	1	yes	characteristic, complemented
84	#418	C₂×C₂₅₁	1	2	yes	complemented
85	#405	C₂×C₂₅₁	1	1	yes	characteristic, complemented
86	C₂²×D₄	C₂²×C₂₅₁	1	2	yes	—
87	C₂×D₈	C₄×C₂₅₁	1	8	yes	complemented
88	C₂×D₈	C₂²×C₂₅₁	1	2	yes	—
89	C₂×D₈	C₂²×C₂₅₁	1	2	yes	—
90	C₂×M₁₆	C₂²×C₂₅₁	1	1	yes	characteristic
91	C₂²×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
92	C₄×D₄	—	2	4	no	—
93	C₄×D₄	C₂²×C₂₅₁	1	8	yes	complemented
94	#117	C₂²×C₂₅₁	1	2	yes	—
95	#117	C₂²×C₂₅₁	1	1	yes	characteristic
96	C₂×C₂²⋊C₄	—	2	4	no	—
97	#107	C₂²×C₂₅₁	1	2	yes	complemented
98	#106	C₂²×C₂₅₁	1	2	yes	complemented
99	#102	C₂²×C₂₅₁	1	8	yes	complemented
100	#97	C₂²×C₂₅₁	1	2	yes	complemented
101	C₄×C₈	C₂²×C₂₅₁	1	2	yes	complemented
102	C₂⁴	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	4	no	—
105	C₂×D₄	C₂×C₄×C₂₅₁	1	8	yes	complemented
106	C₂×D₄	C₂³×C₂₅₁	1	4	yes	—
107	C₂²×C₄	—	2	8	no	—
108	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
109	D₈	C₂×C₄×C₂₅₁	1	8	yes	complemented
110	D₈	C₂×C₄×C₂₅₁	1	8	yes	complemented
111	M₁₆	—	2	2	no	—
112	C₂×C₈	C₂×C₄×C₂₅₁	1	4	yes	complemented
113	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
114	C₂×C₈	C₂³×C₂₅₁	1	1	yes	characteristic
115	C₂×C₈	C₂³×C₂₅₁	1	1	yes	characteristic
116	C₄⋊C₄	D₄×C₂₅₁	1	2	yes	—
117	C₄⋊C₄	C₂³×C₂₅₁	1	4	yes	—
118	C₂²⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	2	8	no	—
120	C₂²⋊C₄	D₄×C₂₅₁	1	2	yes	—
121	C₄²	—	2	2	no	—
122	C₄²	C₂³×C₂₅₁	1	2	yes	—
123	C₂³	—	2	4	no	—
124	C₂³	—	2	8	no	—
125	C₂³	—	8	4	no	—
126	C₂³	C₂×D₄×C₂₅₁	1	1	yes	characteristic
127	D₄	—	2	8	no	—
128	D₄	—	2	4	no	—
129	D₄	C₂²×C₄×C₂₅₁	1	8	yes	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	8	8	no	—
132	C₂×C₄	C₂²×C₄×C₂₅₁	1	2	yes	—
133	C₂×C₄	C₂×D₄×C₂₅₁	1	4	yes	—
134	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
135	C₂×C₄	#41×C₂₅₁	1	2	yes	—
136	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	C₂²×C₄×C₂₅₁	1	2	yes	—
139	C₈	C₂²×C₄×C₂₅₁	1	2	yes	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	8	no	—
142	C₂²	—	4	8	no	—
143	C₂²	—	8	8	no	—
144	C₂²	C₄×D₄×C₂₅₁	1	2	yes	—
145	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	—	4	4	no	—
148	C₄	C₄×D₄×C₂₅₁	1	2	yes	—
149	C₄	C₂³×C₄×C₂₅₁	1	1	yes	characteristic
150	C₄	C₂×#41×C₂₅₁	1	1	yes	characteristic
151	C₂	—	2	2	no	—
152	C₂	—	4	8	no	—
153	C₂	C₂×C₄×D₄×C₂₅₁	1	1	yes	characteristic
154	C₂	#576×C₂₅₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₅₁	C₂²×D₄
commutator	C₄	C₂³×C₄×C₂₅₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₅₁
prime radical	C₂²×D₈×C₂₅₁	C₂
LCS	G ▷ 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
2. #518×C₂₅₁	C₂	semidirect
3. #442×C₂₅₁	C₂	semidirect
4. C₄×D₈×C₂₅₁	C₂	semidirect
5. #427×C₂₅₁	C₂	semidirect
6. #418×C₂₅₁	C₂	semidirect
7. #405×C₂₅₁	C₂	semidirect
9. C₂×D₈×C₂₅₁	C₄	semidirect
15. C₄×D₄×C₂₅₁	C₂²	semidirect
19. #107×C₂₅₁	C₂²	semidirect
20. #106×C₂₅₁	C₂²	semidirect
21. #102×C₂₅₁	C₂²	semidirect
22. #97×C₂₅₁	C₂²	semidirect
23. C₄×C₈×C₂₅₁	C₂²	semidirect
27. C₂×D₄×C₂₅₁	C₂×C₄	semidirect
31. D₈×C₂₅₁	C₂×C₄	semidirect
32. D₈×C₂₅₁	C₂×C₄	semidirect
34. C₂×C₈×C₂₅₁	C₂×C₄	semidirect
77. C₂₅₁	#2948	direct (N × H)
78. #2948	C₂₅₁	direct (N × H)
80. #518	C₂×C₂₅₁	semidirect
81. #442	C₂×C₂₅₁	semidirect
82. C₄×D₈	C₂×C₂₅₁	semidirect
83. #427	C₂×C₂₅₁	semidirect
84. #418	C₂×C₂₅₁	semidirect
85. #405	C₂×C₂₅₁	semidirect
87. C₂×D₈	C₄×C₂₅₁	semidirect
93. C₄×D₄	C₂²×C₂₅₁	semidirect
97. #107	C₂²×C₂₅₁	semidirect
98. #106	C₂²×C₂₅₁	semidirect
99. #102	C₂²×C₂₅₁	semidirect
100. #97	C₂²×C₂₅₁	semidirect
101. C₄×C₈	C₂²×C₂₅₁	semidirect
105. C₂×D₄	C₂×C₄×C₂₅₁	semidirect
109. D₈	C₂×C₄×C₂₅₁	semidirect
110. D₈	C₂×C₄×C₂₅₁	semidirect
112. C₂×C₈	C₂×C₄×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₈×C₂₅₁	2
#518×C₂₅₁	2
#442×C₂₅₁	2
C₄×D₈×C₂₅₁	2
#427×C₂₅₁	2
#418×C₂₅₁	2
#405×C₂₅₁	2
#2948	251

Subgroups — #2948×C251 #44547535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2948×C₂₅₁ #44547535