Subgroups — C₂×#30631643 #44558778

Order 32128.

592 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16860794	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#30631643	C₂	1	8	yes	maximal, direct product factor
4	C₂×#22548626	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₂₅₁⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×#117×C₂₅₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#16860794	C₄	1	8	yes	complemented
8	C₂×#16860794	C₂²	1	4	yes	—
9	C₂²×C₂₅₁⋊C₈	C₄	1	8	yes	complemented
10	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₂₅₁⋊C₈	C₂²	1	4	yes	—
14	#22548626	C₂²	1	8	yes	complemented
15	C₄×C₂₅₁⋊C₈	C₂²	1	8	yes	complemented
16	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
17	#117×C₂₅₁	C₄	1	4	yes	—
18	#117×C₂₅₁	C₂²	1	4	yes	—
19	C₂×C₄⋊C₄×C₂₅₁	C₄	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₅₁	C₄	1	2	yes	—
21	C₂×C₄²×C₂₅₁	C₂²	1	2	yes	—
22	#16860794	C₂×C₄	1	16	yes	complemented
23	C₂×C₂₅₁⋊C₈	C₂×C₄	1	4	yes	—
24	C₂×C₂₅₁⋊C₈	C₂×C₄	1	4	yes	—
25	C₂×C₂₅₁⋊C₈	C₂×C₄	1	8	yes	—
26	C₂×C₂₅₁⋊C₈	C₂×C₄	1	16	yes	complemented
27	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
28	C₂×C₂₅₁⋊C₈	C₂³	1	4	yes	—
29	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
30	C₂⁴×C₂₅₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₅₁	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₂₅₁	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₂₅₁	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₅₁	C₂×C₄	1	8	yes	—
39	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	8	yes	—
40	C₄²×C₂₅₁	C₂×C₄	1	4	yes	—
41	C₄²×C₂₅₁	C₂³	1	4	yes	—
42	C₂₅₁⋊C₈	C₂²×C₄	1	4	yes	—
43	C₂₅₁⋊C₈	C₂²×C₄	1	4	yes	—
44	C₂₅₁⋊C₈	C₂²×C₄	1	8	yes	—
45	C₂³×C₂₅₁	—	2	4	no	—
46	C₂³×C₂₅₁	C₄²	1	4	yes	—
47	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
48	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₅₁	—	2	8	no	—
50	C₂×C₄×C₂₅₁	C₄²	1	8	yes	—
51	C₂×C₄×C₂₅₁	C₄²	1	4	yes	—
52	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
55	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₂₅₁	C₂²×C₄	1	8	yes	—
57	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
60	C₂²×C₂₅₁	—	2	4	no	—
61	C₂²×C₂₅₁	—	2	8	no	—
62	C₂²×C₂₅₁	C₂×C₄²	1	4	yes	—
63	C₂²×C₂₅₁	C₂×C₄²	1	2	yes	—
64	C₂²×C₂₅₁	#131	1	4	yes	—
65	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₂₅₁	—	2	8	no	—
67	C₄×C₂₅₁	C₂×C₄²	1	2	yes	—
68	C₄×C₂₅₁	C₂×C₄²	1	4	yes	—
69	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
70	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
71	C₂×C₂₅₁	—	2	4	no	—
72	C₂×C₂₅₁	#405	1	4	yes	complemented
73	C₂×C₂₅₁	C₂²×C₄²	1	1	yes	characteristic
74	C₂×C₂₅₁	C₂×#131	1	2	yes	—
75	C₂₅₁	C₂×#405	1	1	yes	characteristic, complemented, Sylow
76	C₂×#405	—	251	1	no	maximal, Sylow
77	C₂²×M₁₆	—	251	1	no	—
78	C₂³×C₈	—	251	1	no	—
79	C₂×#117	D₂₅₁	1	1	yes	characteristic
80	#405	—	251	8	no	—
81	C₂×#97	—	251	2	no	—
82	C₂×C₄×C₈	—	251	2	no	—
83	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
84	C₂×M₁₆	—	251	8	no	—
85	C₂×M₁₆	—	251	4	no	—
86	C₂²×C₈	—	251	8	no	—
87	C₂²×C₈	—	251	4	no	—
88	C₂²×C₈	—	251	1	no	—
89	C₂²×C₈	—	251	1	no	—
90	C₂²×C₈	—	251	2	no	—
91	#117	Dic₂₅₁	1	4	yes	—
92	#117	C₂×D₂₅₁	1	4	yes	—
93	C₂×C₄⋊C₄	Dic₂₅₁	1	2	yes	—
94	C₂×C₂²⋊C₄	Dic₂₅₁	1	2	yes	—
95	C₂×C₄²	C₂×D₂₅₁	1	2	yes	—
96	#97	—	251	8	no	—
97	C₄×C₈	—	251	8	no	—
98	C₂⁴	C₂×Dic₂₅₁	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₂₅₁	1	4	yes	—
100	C₂²×C₄	C₄×D₂₅₁	1	4	yes	—
101	C₂²×C₄	C₂×Dic₂₅₁	1	4	yes	—
102	C₂²×C₄	C₂×Dic₂₅₁	1	1	yes	characteristic
103	C₂²×C₄	C₂×Dic₂₅₁	1	2	yes	—
104	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
105	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
106	M₁₆	—	251	16	no	—
107	C₂×C₈	—	251	4	no	—
108	C₂×C₈	—	251	4	no	—
109	C₂×C₈	—	251	8	no	—
110	C₂×C₈	—	251	16	no	—
111	C₂×C₈	—	251	2	no	—
112	C₂×C₈	—	251	4	no	—
113	C₂×C₈	—	251	2	no	—
114	C₄⋊C₄	C₂×Dic₂₅₁	1	8	yes	—
115	C₂²⋊C₄	C₂×Dic₂₅₁	1	8	yes	—
116	C₄²	C₂×Dic₂₅₁	1	4	yes	—
117	C₄²	C₂²×D₂₅₁	1	4	yes	—
118	C₂³	—	2	4	no	—
119	C₂³	C₄×Dic₂₅₁	1	4	yes	—
120	C₂³	C₂²×Dic₂₅₁	1	2	yes	—
121	C₂³	C₂²×Dic₂₅₁	1	1	yes	characteristic
122	C₂×C₄	—	2	8	no	—
123	C₂×C₄	C₄×Dic₂₅₁	1	8	yes	—
124	C₂×C₄	C₄×Dic₂₅₁	1	4	yes	—
125	C₂×C₄	C₂×C₄×D₂₅₁	1	4	yes	—
126	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
127	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
128	C₂×C₄	C₂×C₄×D₂₅₁	1	4	yes	—
129	C₂×C₄	C₂²×Dic₂₅₁	1	8	yes	—
130	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
131	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
132	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
133	C₈	—	251	4	no	—
134	C₈	—	251	4	no	—
135	C₈	—	251	8	no	—
136	C₂²	—	2	4	no	—
137	C₂²	—	2	8	no	—
138	C₂²	C₂×C₄×Dic₂₅₁	1	4	yes	—
139	C₂²	C₂×C₄×Dic₂₅₁	1	2	yes	—
140	C₂²	#22548722	1	4	yes	—
141	C₂²	C₂³×Dic₂₅₁	1	1	yes	characteristic
142	C₄	—	2	8	no	—
143	C₄	C₂×C₄×Dic₂₅₁	1	4	yes	—
144	C₄	C₂×C₄×Dic₂₅₁	1	2	yes	—
145	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
146	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
147	C₂	—	2	4	no	—
148	C₂	#30631643	1	4	yes	direct product factor
149	C₂	C₂²×C₄×Dic₂₅₁	1	1	yes	characteristic
150	C₂	C₂×#22548722	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₅₁
commutator	C₂×C₂₅₁	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×#117×C₂₅₁	C₂
socle	C₂³×C₂₅₁	C₂²×C₄
prime radical	C₂⁴×C₂₅₁	C₂×C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×#117	D₂₅₁
hypocenter	C₂₅₁	C₂×#405
LCS	G ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #30631643	C₂	direct (N × H)
4. C₂×#22548626	C₂	semidirect
5. C₂×C₄×C₂₅₁⋊C₈	C₂	semidirect
7. C₂×#16860794	C₄	semidirect
9. C₂²×C₂₅₁⋊C₈	C₄	semidirect
14. #22548626	C₂²	semidirect
15. C₄×C₂₅₁⋊C₈	C₂²	semidirect
22. #16860794	C₂×C₄	semidirect
26. C₂×C₂₅₁⋊C₈	C₂×C₄	semidirect
72. C₂×C₂₅₁	#405	semidirect
75. C₂₅₁	C₂×#405	semidirect
148. C₂	#30631643	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×#16860794	2
C₂³×C₂₅₁⋊C₈	2
#30631643	2
C₂×#22548626	2
C₂×C₄×C₂₅₁⋊C₈	2
C₂×#117×C₂₅₁	2
C₂×#405	251

Subgroups — C2×#30631643 #44558778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30631643 #44558778