Subgroups — #1729×C₂₅₁ #44546316

Order 32128.

416 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×M₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#117×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#336×C₂₅₁	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₂₅₁	—	2	2	no	—
7	C₂×M₁₆×C₂₅₁	—	2	2	no	—
8	C₂×M₁₆×C₂₅₁	C₄	1	4	yes	complemented
9	C₂²×C₈×C₂₅₁	—	2	2	no	—
10	C₂²×C₈×C₂₅₁	—	2	2	no	—
11	C₂²×C₈×C₂₅₁	C₄	1	4	yes	complemented
12	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
13	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
14	#117×C₂₅₁	—	2	2	no	—
15	#117×C₂₅₁	—	2	2	no	—
16	C₂×C₄⋊C₄×C₂₅₁	C₄	1	2	yes	—
17	C₂×C₂²⋊C₄×C₂₅₁	C₄	1	2	yes	—
18	C₂×C₄²×C₂₅₁	C₂²	1	2	yes	—
19	C₂⁴×C₂₅₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₅₁	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
25	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
26	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
27	C₂²×C₄×C₂₅₁	Q₈	1	2	yes	—
28	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
29	M₁₆×C₂₅₁	—	2	8	no	—
30	C₂×C₈×C₂₅₁	—	2	8	no	—
31	C₂×C₈×C₂₅₁	—	2	2	no	—
32	C₂×C₈×C₂₅₁	—	2	2	no	—
33	C₂×C₈×C₂₅₁	—	2	2	no	—
34	C₂×C₈×C₂₅₁	—	2	2	no	—
35	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	—
36	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	—
37	C₄⋊C₄×C₂₅₁	—	2	4	no	—
38	C₂²⋊C₄×C₂₅₁	—	2	4	no	—
39	C₄²×C₂₅₁	—	2	2	no	—
40	C₄²×C₂₅₁	—	2	2	no	—
41	C₂³×C₂₅₁	—	2	4	no	—
42	C₂³×C₂₅₁	C₄²	1	2	yes	—
43	C₂³×C₂₅₁	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₂₅₁	C₄⋊C₄	1	2	yes	—
45	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₅₁	—	2	4	no	—
47	C₂×C₄×C₂₅₁	—	2	8	no	—
48	C₂×C₄×C₂₅₁	C₄²	1	2	yes	—
49	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	4	yes	—
51	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	4	yes	—
52	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₂₅₁	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₂₅₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
61	C₈×C₂₅₁	—	2	4	no	—
62	C₈×C₂₅₁	—	2	4	no	—
63	C₂²×C₂₅₁	—	2	4	no	—
64	C₂²×C₂₅₁	—	2	4	no	—
65	C₂²×C₂₅₁	—	2	4	no	—
66	C₂²×C₂₅₁	#95	1	4	yes	—
67	C₂²×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₂₅₁	#131	1	4	yes	—
71	C₄×C₂₅₁	—	4	4	no	—
72	C₄×C₂₅₁	#95	1	4	yes	—
73	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₂₅₁	—	2	4	no	—
78	C₂×C₂₅₁	C₂×#95	1	1	yes	characteristic
79	C₂×C₂₅₁	#405	1	2	yes	—
80	C₂×C₂₅₁	#408	1	2	yes	—
81	C₂×C₂₅₁	#424	1	2	yes	—
82	C₂₅₁	#1729	1	1	yes	characteristic, direct product factor, Sylow
83	#1729	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×M₁₆	C₂×C₂₅₁	1	1	yes	characteristic
85	C₂³×C₈	C₂×C₂₅₁	1	1	yes	characteristic
86	C₂×#117	C₂×C₂₅₁	1	1	yes	characteristic
87	#336	C₂×C₂₅₁	1	4	yes	complemented
88	C₂³×C₄	C₂²×C₂₅₁	1	1	yes	characteristic
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	C₄×C₂₅₁	1	4	yes	complemented
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	C₄×C₂₅₁	1	4	yes	complemented
95	C₂²×C₈	C₂²×C₂₅₁	1	2	yes	—
96	C₂²×C₈	C₂²×C₂₅₁	1	2	yes	—
97	#117	—	2	2	no	—
98	#117	—	2	2	no	—
99	C₂×C₄⋊C₄	C₄×C₂₅₁	1	2	yes	—
100	C₂×C₂²⋊C₄	C₄×C₂₅₁	1	2	yes	—
101	C₂×C₄²	C₂²×C₂₅₁	1	2	yes	—
102	C₂⁴	C₂×C₄×C₂₅₁	1	1	yes	characteristic
103	C₂²×C₄	C₂×C₄×C₂₅₁	1	4	yes	—
104	C₂²×C₄	C₂×C₄×C₂₅₁	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄×C₂₅₁	1	2	yes	—
106	C₂²×C₄	C₂×C₄×C₂₅₁	1	2	yes	—
107	C₂²×C₄	D₄×C₂₅₁	1	2	yes	—
108	C₂²×C₄	D₄×C₂₅₁	1	2	yes	—
109	C₂²×C₄	D₄×C₂₅₁	1	2	yes	—
110	C₂²×C₄	Q₈×C₂₅₁	1	2	yes	—
111	C₂²×C₄	C₂³×C₂₅₁	1	1	yes	characteristic
112	M₁₆	—	2	8	no	—
113	C₂×C₈	—	2	8	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	C₂×C₄×C₂₅₁	1	4	yes	—
119	C₂×C₈	C₂×C₄×C₂₅₁	1	4	yes	—
120	C₄⋊C₄	—	2	4	no	—
121	C₂²⋊C₄	—	2	4	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₂³	—	2	4	no	—
125	C₂³	C₄²×C₂₅₁	1	2	yes	—
126	C₂³	C₂²⋊C₄×C₂₅₁	1	2	yes	—
127	C₂³	C₄⋊C₄×C₂₅₁	1	2	yes	—
128	C₂³	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	C₄²×C₂₅₁	1	2	yes	—
132	C₂×C₄	C₂²⋊C₄×C₂₅₁	1	2	yes	—
133	C₂×C₄	C₂²⋊C₄×C₂₅₁	1	4	yes	—
134	C₂×C₄	C₂²⋊C₄×C₂₅₁	1	4	yes	—
135	C₂×C₄	C₄⋊C₄×C₂₅₁	1	2	yes	—
136	C₂×C₄	C₄⋊C₄×C₂₅₁	1	4	yes	—
137	C₂×C₄	C₄⋊C₄×C₂₅₁	1	4	yes	—
138	C₂×C₄	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
139	C₂×C₄	C₂²×C₄×C₂₅₁	1	1	yes	characteristic
140	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
141	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
142	C₂×C₄	C₂×D₄×C₂₅₁	1	1	yes	characteristic
143	C₂×C₄	C₂×Q₈×C₂₅₁	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	2	4	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	2	4	no	—
149	C₂²	#95×C₂₅₁	1	4	yes	—
150	C₂²	C₂×C₄²×C₂₅₁	1	1	yes	characteristic
151	C₂²	C₂×C₂²⋊C₄×C₂₅₁	1	1	yes	characteristic
152	C₂²	C₂×C₄⋊C₄×C₂₅₁	1	1	yes	characteristic
153	C₂²	#131×C₂₅₁	1	4	yes	—
154	C₄	—	4	4	no	—
155	C₄	#95×C₂₅₁	1	4	yes	—
156	C₄	C₂×C₂²⋊C₄×C₂₅₁	1	1	yes	characteristic
157	C₄	C₂×C₂²⋊C₄×C₂₅₁	1	1	yes	characteristic
158	C₄	C₂×C₄⋊C₄×C₂₅₁	1	1	yes	characteristic
159	C₄	C₂×C₄⋊C₄×C₂₅₁	1	1	yes	characteristic
160	C₂	—	2	4	no	—
161	C₂	C₂×#95×C₂₅₁	1	1	yes	characteristic
162	C₂	#405×C₂₅₁	1	2	yes	—
163	C₂	#408×C₂₅₁	1	2	yes	—
164	C₂	#424×C₂₅₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #336×C₂₅₁	C₂	semidirect
8. C₂×M₁₆×C₂₅₁	C₄	semidirect
11. C₂²×C₈×C₂₅₁	C₄	semidirect
82. C₂₅₁	#1729	direct (N × H)
83. #1729	C₂₅₁	direct (N × H)
87. #336	C₂×C₂₅₁	semidirect
91. C₂×M₁₆	C₄×C₂₅₁	semidirect
94. C₂²×C₈	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×M₁₆×C₂₅₁	2
C₂³×C₈×C₂₅₁	2
C₂×#117×C₂₅₁	2
#336×C₂₅₁	2
#1729	251

Subgroups — #1729×C251 #44546316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1729×C₂₅₁ #44546316