Subgroups — C₂²×C₄×D₄×C₂₅₁ #44548009

Order 32128.

2528 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	C₂³×D₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×C₂₅₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₄×C₂₅₁	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊C₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₂²⋊C₄×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄²×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×C₂₅₁	C₂²	1	2	yes	—
8	C₂²×D₄×C₂₅₁	C₄	1	16	yes	direct product factor
9	C₂²×D₄×C₂₅₁	C₂²	1	12	yes	—
10	C₂³×C₄×C₂₅₁	—	2	16	no	—
11	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₅₁	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₂₅₁	C₂²	1	24	yes	complemented
16	C₄×D₄×C₂₅₁	C₂²	1	64	yes	direct product factor
17	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	12	yes	complemented
18	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	24	yes	complemented
19	C₂×C₄²×C₂₅₁	C₂²	1	12	yes	complemented
20	C₂⁴×C₂₅₁	—	2	8	no	—
21	C₂⁴×C₂₅₁	—	2	8	no	—
22	C₂⁴×C₂₅₁	C₂×C₄	1	16	yes	complemented
23	C₂⁴×C₂₅₁	C₂³	1	12	yes	—
24	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
25	C₂×D₄×C₂₅₁	C₂×C₄	1	96	yes	direct product factor
26	C₂×D₄×C₂₅₁	C₂³	1	16	yes	—
27	C₂²×C₄×C₂₅₁	—	2	8	no	—
28	C₂²×C₄×C₂₅₁	—	2	4	no	—
29	C₂²×C₄×C₂₅₁	—	2	96	no	—
30	C₂²×C₄×C₂₅₁	C₂×C₄	1	8	yes	complemented
31	C₂²×C₄×C₂₅₁	D₄	1	8	yes	direct product factor
32	C₂²×C₄×C₂₅₁	C₂³	1	6	yes	—
33	C₂²×C₄×C₂₅₁	C₂³	1	12	yes	—
34	C₂²×C₄×C₂₅₁	C₂³	1	6	yes	—
35	C₂²×C₄×C₂₅₁	C₂³	1	6	yes	—
36	C₂²×C₄×C₂₅₁	C₂³	1	32	yes	complemented
37	C₄⋊C₄×C₂₅₁	C₂³	1	16	yes	complemented
38	C₂²⋊C₄×C₂₅₁	C₂³	1	32	yes	complemented
39	C₄²×C₂₅₁	C₂³	1	16	yes	complemented
40	C₂³×C₂₅₁	—	2	64	no	—
41	C₂³×C₂₅₁	—	2	24	no	—
42	C₂³×C₂₅₁	—	2	24	no	—
43	C₂³×C₂₅₁	C₂²×C₄	1	4	yes	—
44	C₂³×C₂₅₁	C₂²×C₄	1	48	yes	complemented
45	C₂³×C₂₅₁	C₂×D₄	1	4	yes	—
46	C₂³×C₂₅₁	#41	1	4	yes	—
47	C₂³×C₂₅₁	C₂⁴	1	8	yes	—
48	C₂³×C₂₅₁	C₂⁴	1	3	yes	—
49	D₄×C₂₅₁	C₂²×C₄	1	64	yes	direct product factor
50	C₂×C₄×C₂₅₁	—	2	24	no	—
51	C₂×C₄×C₂₅₁	—	2	12	no	—
52	C₂×C₄×C₂₅₁	—	2	64	no	—
53	C₂×C₄×C₂₅₁	C₂²×C₄	1	24	yes	complemented
54	C₂×C₄×C₂₅₁	C₂×D₄	1	24	yes	direct product factor
55	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
56	C₂×C₄×C₂₅₁	C₂⁴	1	8	yes	—
57	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
58	C₂×C₄×C₂₅₁	C₂⁴	1	4	yes	—
59	C₂²×C₂₅₁	—	2	96	no	—
60	C₂²×C₂₅₁	—	2	8	no	—
61	C₂²×C₂₅₁	—	2	8	no	—
62	C₂²×C₂₅₁	C₄×D₄	1	16	yes	direct product factor
63	C₂²×C₂₅₁	C₂³×C₄	1	6	yes	—
64	C₂²×C₂₅₁	C₂³×C₄	1	16	yes	complemented
65	C₂²×C₂₅₁	C₂²×D₄	1	6	yes	—
66	C₂²×C₂₅₁	C₂×#41	1	6	yes	—
67	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
68	C₄×C₂₅₁	—	2	8	no	—
69	C₄×C₂₅₁	—	2	4	no	—
70	C₄×C₂₅₁	C₂³×C₄	1	8	yes	complemented
71	C₄×C₂₅₁	C₂²×D₄	1	8	yes	direct product factor
72	C₂×C₂₅₁	—	2	16	no	—
73	C₂×C₂₅₁	C₂×C₄×D₄	1	12	yes	direct product factor
74	C₂×C₂₅₁	C₂⁴×C₄	1	1	yes	characteristic
75	C₂×C₂₅₁	C₂³×D₄	1	1	yes	characteristic
76	C₂×C₂₅₁	C₂²×#41	1	1	yes	characteristic
77	C₂₅₁	C₂²×C₄×D₄	1	1	yes	characteristic, direct product factor, Sylow
78	C₂²×C₄×D₄	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×D₄	C₂×C₂₅₁	1	1	yes	characteristic
80	C₂⁴×C₄	C₂×C₂₅₁	1	2	yes	complemented
81	C₂×C₄×D₄	C₂×C₂₅₁	1	24	yes	direct product factor
82	C₂²×C₄⋊C₄	C₂×C₂₅₁	1	1	yes	characteristic, complemented
83	C₂²×C₂²⋊C₄	C₂×C₂₅₁	1	2	yes	complemented
84	C₂²×C₄²	C₂×C₂₅₁	1	1	yes	characteristic, complemented
85	C₂⁵	C₂²×C₂₅₁	1	2	yes	—
86	C₂²×D₄	C₄×C₂₅₁	1	16	yes	direct product factor
87	C₂²×D₄	C₂²×C₂₅₁	1	12	yes	—
88	C₂³×C₄	—	2	16	no	—
89	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
90	C₂³×C₄	C₂²×C₂₅₁	1	2	yes	—
91	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
92	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
93	C₂³×C₄	C₂²×C₂₅₁	1	24	yes	complemented
94	C₄×D₄	C₂²×C₂₅₁	1	64	yes	direct product factor
95	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	12	yes	complemented
96	C₂×C₂²⋊C₄	C₂²×C₂₅₁	1	24	yes	complemented
97	C₂×C₄²	C₂²×C₂₅₁	1	12	yes	complemented
98	C₂⁴	—	2	8	no	—
99	C₂⁴	—	2	8	no	—
100	C₂⁴	C₂×C₄×C₂₅₁	1	16	yes	complemented
101	C₂⁴	C₂³×C₂₅₁	1	12	yes	—
102	C₂⁴	C₂³×C₂₅₁	1	1	yes	characteristic
103	C₂×D₄	C₂×C₄×C₂₅₁	1	96	yes	direct product factor
104	C₂×D₄	C₂³×C₂₅₁	1	16	yes	—
105	C₂²×C₄	—	2	8	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	2	96	no	—
108	C₂²×C₄	C₂×C₄×C₂₅₁	1	8	yes	complemented
109	C₂²×C₄	D₄×C₂₅₁	1	8	yes	direct product factor
110	C₂²×C₄	C₂³×C₂₅₁	1	6	yes	—
111	C₂²×C₄	C₂³×C₂₅₁	1	12	yes	—
112	C₂²×C₄	C₂³×C₂₅₁	1	6	yes	—
113	C₂²×C₄	C₂³×C₂₅₁	1	6	yes	—
114	C₂²×C₄	C₂³×C₂₅₁	1	32	yes	complemented
115	C₄⋊C₄	C₂³×C₂₅₁	1	16	yes	complemented
116	C₂²⋊C₄	C₂³×C₂₅₁	1	32	yes	complemented
117	C₄²	C₂³×C₂₅₁	1	16	yes	complemented
118	C₂³	—	2	64	no	—
119	C₂³	—	2	24	no	—
120	C₂³	—	2	24	no	—
121	C₂³	C₂²×C₄×C₂₅₁	1	4	yes	—
122	C₂³	C₂²×C₄×C₂₅₁	1	48	yes	complemented
123	C₂³	C₂×D₄×C₂₅₁	1	4	yes	—
124	C₂³	#41×C₂₅₁	1	4	yes	—
125	C₂³	C₂⁴×C₂₅₁	1	8	yes	—
126	C₂³	C₂⁴×C₂₅₁	1	3	yes	—
127	D₄	C₂²×C₄×C₂₅₁	1	64	yes	direct product factor
128	C₂×C₄	—	2	24	no	—
129	C₂×C₄	—	2	12	no	—
130	C₂×C₄	—	2	64	no	—
131	C₂×C₄	C₂²×C₄×C₂₅₁	1	24	yes	complemented
132	C₂×C₄	C₂×D₄×C₂₅₁	1	24	yes	direct product factor
133	C₂×C₄	C₂⁴×C₂₅₁	1	4	yes	—
134	C₂×C₄	C₂⁴×C₂₅₁	1	8	yes	—
135	C₂×C₄	C₂⁴×C₂₅₁	1	4	yes	—
136	C₂×C₄	C₂⁴×C₂₅₁	1	4	yes	—
137	C₂²	—	2	96	no	—
138	C₂²	—	2	8	no	—
139	C₂²	—	2	8	no	—
140	C₂²	C₄×D₄×C₂₅₁	1	16	yes	direct product factor
141	C₂²	C₂³×C₄×C₂₅₁	1	6	yes	—
142	C₂²	C₂³×C₄×C₂₅₁	1	16	yes	complemented
143	C₂²	C₂²×D₄×C₂₅₁	1	6	yes	—
144	C₂²	C₂×#41×C₂₅₁	1	6	yes	—
145	C₂²	C₂⁵×C₂₅₁	1	1	yes	characteristic
146	C₄	—	2	8	no	—
147	C₄	—	2	4	no	—
148	C₄	C₂³×C₄×C₂₅₁	1	8	yes	complemented
149	C₄	C₂²×D₄×C₂₅₁	1	8	yes	direct product factor
150	C₂	—	2	16	no	—
151	C₂	C₂×C₄×D₄×C₂₅₁	1	12	yes	direct product factor
152	C₂	C₂⁴×C₄×C₂₅₁	1	1	yes	characteristic
153	C₂	C₂³×D₄×C₂₅₁	1	1	yes	characteristic
154	C₂	C₂²×#41×C₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₂₅₁	C₂²
commutator	C₂	C₂⁴×C₄×C₂₅₁
Frattini, cosocle	C₂²	C₂⁵×C₂₅₁
socle	C₂⁴×C₂₅₁	C₂³
prime radical	C₂³×D₄×C₂₅₁	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×C₂₅₁	C₂	semidirect
3. C₂×C₄×D₄×C₂₅₁	C₂	direct (N × H)
4. C₂²×C₄⋊C₄×C₂₅₁	C₂	semidirect
5. C₂²×C₂²⋊C₄×C₂₅₁	C₂	semidirect
6. C₂²×C₄²×C₂₅₁	C₂	semidirect
8. C₂²×D₄×C₂₅₁	C₄	direct (N × H)
15. C₂³×C₄×C₂₅₁	C₂²	semidirect
16. C₄×D₄×C₂₅₁	C₂²	direct (N × H)
17. C₂×C₄⋊C₄×C₂₅₁	C₂²	semidirect
18. C₂×C₂²⋊C₄×C₂₅₁	C₂²	semidirect
19. C₂×C₄²×C₂₅₁	C₂²	semidirect
22. C₂⁴×C₂₅₁	C₂×C₄	semidirect
25. C₂×D₄×C₂₅₁	C₂×C₄	direct (N × H)
30. C₂²×C₄×C₂₅₁	C₂×C₄	semidirect
31. C₂²×C₄×C₂₅₁	D₄	direct (N × H)
36. C₂²×C₄×C₂₅₁	C₂³	semidirect
37. C₄⋊C₄×C₂₅₁	C₂³	semidirect
38. C₂²⋊C₄×C₂₅₁	C₂³	semidirect
39. C₄²×C₂₅₁	C₂³	semidirect
44. C₂³×C₂₅₁	C₂²×C₄	semidirect
49. D₄×C₂₅₁	C₂²×C₄	direct (N × H)
53. C₂×C₄×C₂₅₁	C₂²×C₄	semidirect
54. C₂×C₄×C₂₅₁	C₂×D₄	direct (N × H)
62. C₂²×C₂₅₁	C₄×D₄	direct (N × H)
64. C₂²×C₂₅₁	C₂³×C₄	semidirect
70. C₄×C₂₅₁	C₂³×C₄	semidirect
71. C₄×C₂₅₁	C₂²×D₄	direct (N × H)
73. C₂×C₂₅₁	C₂×C₄×D₄	direct (N × H)
77. C₂₅₁	C₂²×C₄×D₄	direct (N × H)
78. C₂²×C₄×D₄	C₂₅₁	direct (N × H)
80. C₂⁴×C₄	C₂×C₂₅₁	semidirect
81. C₂×C₄×D₄	C₂×C₂₅₁	direct (N × H)
82. C₂²×C₄⋊C₄	C₂×C₂₅₁	semidirect
83. C₂²×C₂²⋊C₄	C₂×C₂₅₁	semidirect
84. C₂²×C₄²	C₂×C₂₅₁	semidirect
86. C₂²×D₄	C₄×C₂₅₁	direct (N × H)
93. C₂³×C₄	C₂²×C₂₅₁	semidirect
94. C₄×D₄	C₂²×C₂₅₁	direct (N × H)
95. C₂×C₄⋊C₄	C₂²×C₂₅₁	semidirect
96. C₂×C₂²⋊C₄	C₂²×C₂₅₁	semidirect
97. C₂×C₄²	C₂²×C₂₅₁	semidirect
100. C₂⁴	C₂×C₄×C₂₅₁	semidirect
103. C₂×D₄	C₂×C₄×C₂₅₁	direct (N × H)
108. C₂²×C₄	C₂×C₄×C₂₅₁	semidirect
109. C₂²×C₄	D₄×C₂₅₁	direct (N × H)
114. C₂²×C₄	C₂³×C₂₅₁	semidirect
115. C₄⋊C₄	C₂³×C₂₅₁	semidirect
116. C₂²⋊C₄	C₂³×C₂₅₁	semidirect
117. C₄²	C₂³×C₂₅₁	semidirect
122. C₂³	C₂²×C₄×C₂₅₁	semidirect
127. D₄	C₂²×C₄×C₂₅₁	direct (N × H)
131. C₂×C₄	C₂²×C₄×C₂₅₁	semidirect
132. C₂×C₄	C₂×D₄×C₂₅₁	direct (N × H)
140. C₂²	C₄×D₄×C₂₅₁	direct (N × H)
142. C₂²	C₂³×C₄×C₂₅₁	semidirect
148. C₄	C₂³×C₄×C₂₅₁	semidirect
149. C₄	C₂²×D₄×C₂₅₁	direct (N × H)
151. C₂	C₂×C₄×D₄×C₂₅₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₂₅₁	2
C₂⁴×C₄×C₂₅₁	2
C₂×C₄×D₄×C₂₅₁	2
C₂²×C₄⋊C₄×C₂₅₁	2
C₂²×C₂²⋊C₄×C₂₅₁	2
C₂²×C₄²×C₂₅₁	2
C₂²×C₄×D₄	251

Subgroups — C22×C4×D4×C251 #44548009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×D₄×C₂₅₁ #44548009