Subgroups — C₂×C₄×D₈×C₂₅₁ #44547523

Order 32128.

616 conjugacy classes of subgroups.

The subgroup lattice has 166 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	C₂×C₄×D₄×C₂₅₁	C₂	1	2	yes	maximal, complemented
3	C₄×D₈×C₂₅₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×#107×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#102×C₂₅₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄×C₂₅₁	C₂²	1	2	yes	—
8	C₂³×C₄×C₂₅₁	—	2	2	no	—
9	C₂×D₈×C₂₅₁	C₄	1	8	yes	direct product factor
10	C₂×D₈×C₂₅₁	C₂²	1	4	yes	—
11	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₄×D₄×C₂₅₁	—	2	4	no	—
14	C₄×D₄×C₂₅₁	C₂²	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₂₅₁	—	2	2	no	—
17	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
18	#107×C₂₅₁	C₂²	1	4	yes	complemented
19	#102×C₂₅₁	C₂²	1	8	yes	complemented
20	C₄×C₈×C₂₅₁	C₂²	1	4	yes	complemented
21	C₂⁴×C₂₅₁	—	2	2	no	—
22	C₂×D₄×C₂₅₁	—	2	2	no	—
23	C₂×D₄×C₂₅₁	—	2	4	no	—
24	C₂×D₄×C₂₅₁	C₂×C₄	1	8	yes	complemented
25	C₂×D₄×C₂₅₁	C₂³	1	4	yes	—
26	C₂²×C₄×C₂₅₁	—	2	2	no	—
27	C₂²×C₄×C₂₅₁	—	2	8	no	—
28	C₂²×C₄×C₂₅₁	—	4	8	no	—
29	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
32	D₈×C₂₅₁	C₂×C₄	1	16	yes	direct product factor
33	C₂×C₈×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	2	yes	—
37	C₄⋊C₄×C₂₅₁	—	2	2	no	—
38	C₄⋊C₄×C₂₅₁	C₂³	1	4	yes	—
39	C₂²⋊C₄×C₂₅₁	—	2	8	no	—
40	C₄²×C₂₅₁	D₄	1	2	yes	—
41	C₄²×C₂₅₁	C₂³	1	2	yes	—
42	C₂³×C₂₅₁	—	2	8	no	—
43	C₂³×C₂₅₁	—	2	4	no	—
44	C₂³×C₂₅₁	—	4	4	no	—
45	C₂³×C₂₅₁	—	4	4	no	—
46	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
47	D₄×C₂₅₁	—	2	8	no	—
48	D₄×C₂₅₁	—	2	4	no	—
49	D₄×C₂₅₁	C₂²×C₄	1	8	yes	complemented
50	C₂×C₄×C₂₅₁	—	2	2	no	—
51	C₂×C₄×C₂₅₁	—	2	4	no	—
52	C₂×C₄×C₂₅₁	—	4	4	no	—
53	C₂×C₄×C₂₅₁	—	4	16	no	—
54	C₂×C₄×C₂₅₁	D₈	1	4	yes	direct product factor
55	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₅₁	#41	1	2	yes	—
60	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
61	C₈×C₂₅₁	—	2	2	no	—
62	C₈×C₂₅₁	C₂²×C₄	1	4	yes	complemented
63	C₂²×C₂₅₁	—	2	8	no	—
64	C₂²×C₂₅₁	—	4	4	no	—
65	C₂²×C₂₅₁	—	4	4	no	—
66	C₂²×C₂₅₁	—	4	16	no	—
67	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
68	C₂²×C₂₅₁	C₂×D₈	1	2	yes	—
69	C₂²×C₂₅₁	#135	1	2	yes	—
70	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₂₅₁	—	2	2	no	—
72	C₄×C₂₅₁	—	4	4	no	—
73	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
74	C₄×C₂₅₁	C₂×D₈	1	4	yes	direct product factor
75	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
76	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
77	C₂×C₂₅₁	—	4	8	no	—
78	C₂×C₂₅₁	C₄×D₈	1	4	yes	direct product factor
79	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
80	C₂×C₂₅₁	C₂²×D₈	1	1	yes	characteristic
81	C₂×C₂₅₁	C₂×#135	1	1	yes	characteristic
82	C₂₅₁	C₂×C₄×D₈	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×C₄×D₈	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×D₈	C₂×C₂₅₁	1	1	yes	characteristic
85	C₂×C₄×D₄	C₂×C₂₅₁	1	2	yes	complemented
86	C₄×D₈	C₂×C₂₅₁	1	8	yes	direct product factor
87	C₂×#107	C₂×C₂₅₁	1	1	yes	characteristic, complemented
88	C₂×#102	C₂×C₂₅₁	1	2	yes	complemented
89	C₂×C₄×C₈	C₂×C₂₅₁	1	1	yes	characteristic, complemented
90	C₂²×D₄	C₂²×C₂₅₁	1	2	yes	—
91	C₂³×C₄	—	2	2	no	—
92	C₂×D₈	C₄×C₂₅₁	1	8	yes	direct product factor
93	C₂×D₈	C₂²×C₂₅₁	1	4	yes	—
94	C₂²×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
95	C₂²×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
96	C₄×D₄	—	2	4	no	—
97	C₄×D₄	C₂²×C₂₅₁	1	8	yes	complemented
98	C₂×C₄⋊C₄	C₂²×C₂₅₁	1	2	yes	—
99	C₂×C₂²⋊C₄	—	2	2	no	—
100	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
101	#107	C₂²×C₂₅₁	1	4	yes	complemented
102	#102	C₂²×C₂₅₁	1	8	yes	complemented
103	C₄×C₈	C₂²×C₂₅₁	1	4	yes	complemented
104	C₂⁴	—	2	2	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	4	no	—
107	C₂×D₄	C₂×C₄×C₂₅₁	1	8	yes	complemented
108	C₂×D₄	C₂³×C₂₅₁	1	4	yes	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	—	2	8	no	—
111	C₂²×C₄	—	4	8	no	—
112	C₂²×C₄	D₄×C₂₅₁	1	1	yes	characteristic
113	C₂²×C₄	D₄×C₂₅₁	1	1	yes	characteristic
114	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
115	D₈	C₂×C₄×C₂₅₁	1	16	yes	direct product factor
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	C₂×C₄×C₂₅₁	1	4	yes	complemented
118	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
119	C₂×C₈	C₂³×C₂₅₁	1	2	yes	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	C₂³×C₂₅₁	1	4	yes	—
122	C₂²⋊C₄	—	2	8	no	—
123	C₄²	D₄×C₂₅₁	1	2	yes	—
124	C₄²	C₂³×C₂₅₁	1	2	yes	—
125	C₂³	—	2	8	no	—
126	C₂³	—	2	4	no	—
127	C₂³	—	4	4	no	—
128	C₂³	—	4	4	no	—
129	C₂³	C₂×D₄×C₂₅₁	1	1	yes	characteristic
130	D₄	—	2	8	no	—
131	D₄	—	2	4	no	—
132	D₄	C₂²×C₄×C₂₅₁	1	8	yes	complemented
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	16	no	—
137	C₂×C₄	D₈×C₂₅₁	1	4	yes	direct product factor
138	C₂×C₄	C₂²×C₄×C₂₅₁	1	2	yes	—
139	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
140	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
141	C₂×C₄	C₂×D₄×C₂₅₁	1	2	yes	—
142	C₂×C₄	#41×C₂₅₁	1	2	yes	—
143	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	C₂²×C₄×C₂₅₁	1	4	yes	complemented
146	C₂²	—	2	8	no	—
147	C₂²	—	4	16	no	—
148	C₂²	—	4	4	no	—
149	C₂²	—	4	4	no	—
150	C₂²	C₄×D₄×C₂₅₁	1	2	yes	—
151	C₂²	C₂×D₈×C₂₅₁	1	2	yes	—
152	C₂²	#135×C₂₅₁	1	2	yes	—
153	C₂²	C₂²×D₄×C₂₅₁	1	1	yes	characteristic
154	C₄	—	2	2	no	—
155	C₄	—	4	4	no	—
156	C₄	C₄×D₄×C₂₅₁	1	2	yes	—
157	C₄	C₂×D₈×C₂₅₁	1	4	yes	direct product factor
158	C₄	C₂³×C₄×C₂₅₁	1	1	yes	characteristic
159	C₄	C₂×#41×C₂₅₁	1	1	yes	characteristic
160	C₂	—	4	8	no	—
161	C₂	C₄×D₈×C₂₅₁	1	4	yes	direct product factor
162	C₂	C₂×C₄×D₄×C₂₅₁	1	1	yes	characteristic
163	C₂	C₂²×D₈×C₂₅₁	1	1	yes	characteristic
164	C₂	C₂×#135×C₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₅₁	D₄
commutator	C₄	C₂³×C₄×C₂₅₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₅₁
socle	C₂³×C₂₅₁	C₂×D₄
prime radical	C₂²×D₈×C₂₅₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₅₁ ◁ C₂×C₄²×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄×C₂₅₁	C₂	semidirect
3. C₄×D₈×C₂₅₁	C₂	direct (N × H)
4. C₂×#107×C₂₅₁	C₂	semidirect
5. C₂×#102×C₂₅₁	C₂	semidirect
6. C₂×C₄×C₈×C₂₅₁	C₂	semidirect
9. C₂×D₈×C₂₅₁	C₄	direct (N × H)
14. C₄×D₄×C₂₅₁	C₂²	semidirect
18. #107×C₂₅₁	C₂²	semidirect
19. #102×C₂₅₁	C₂²	semidirect
20. C₄×C₈×C₂₅₁	C₂²	semidirect
24. C₂×D₄×C₂₅₁	C₂×C₄	semidirect
32. D₈×C₂₅₁	C₂×C₄	direct (N × H)
34. C₂×C₈×C₂₅₁	C₂×C₄	semidirect
49. D₄×C₂₅₁	C₂²×C₄	semidirect
54. C₂×C₄×C₂₅₁	D₈	direct (N × H)
62. C₈×C₂₅₁	C₂²×C₄	semidirect
74. C₄×C₂₅₁	C₂×D₈	direct (N × H)
78. C₂×C₂₅₁	C₄×D₈	direct (N × H)
82. C₂₅₁	C₂×C₄×D₈	direct (N × H)
83. C₂×C₄×D₈	C₂₅₁	direct (N × H)
85. C₂×C₄×D₄	C₂×C₂₅₁	semidirect
86. C₄×D₈	C₂×C₂₅₁	direct (N × H)
87. C₂×#107	C₂×C₂₅₁	semidirect
88. C₂×#102	C₂×C₂₅₁	semidirect
89. C₂×C₄×C₈	C₂×C₂₅₁	semidirect
92. C₂×D₈	C₄×C₂₅₁	direct (N × H)
97. C₄×D₄	C₂²×C₂₅₁	semidirect
101. #107	C₂²×C₂₅₁	semidirect
102. #102	C₂²×C₂₅₁	semidirect
103. C₄×C₈	C₂²×C₂₅₁	semidirect
107. C₂×D₄	C₂×C₄×C₂₅₁	semidirect
115. D₈	C₂×C₄×C₂₅₁	direct (N × H)
117. C₂×C₈	C₂×C₄×C₂₅₁	semidirect
132. D₄	C₂²×C₄×C₂₅₁	semidirect
137. C₂×C₄	D₈×C₂₅₁	direct (N × H)
145. C₈	C₂²×C₄×C₂₅₁	semidirect
157. C₄	C₂×D₈×C₂₅₁	direct (N × H)
161. C₂	C₄×D₈×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
C₂²×D₈×C₂₅₁	2
C₂×C₄×D₄×C₂₅₁	2
C₄×D₈×C₂₅₁	2
C₂×#107×C₂₅₁	2
C₂×#102×C₂₅₁	2
C₂×C₄×C₈×C₂₅₁	2
C₂×C₄×D₈	251

Subgroups — C2×C4×D8×C251 #44547523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₈×C₂₅₁ #44547523