Subgroups — #44564860

Order 32128.

890 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅₁⋊#564	C₂	1	4	yes	maximal, complemented
2	#30632416	C₂	1	4	yes	maximal, complemented
3	#30632415	C₂	1	4	yes	maximal, complemented
4	#30632412	C₂	1	4	yes	maximal, complemented
5	#30632299	C₂	1	4	yes	maximal
6	#30632297	C₂	1	8	yes	maximal
7	#30632244	C₂	1	2	yes	maximal
8	#533×C₂₅₁	C₂	1	1	yes	maximal, characteristic
9	#22548715	C₂²	1	2	yes	—
10	#22548715	C₂²	1	2	yes	—
11	#22548715	C₂²	1	4	yes	—
12	#22548713	C₂²	1	8	yes	—
13	#22548708	C₂²	1	16	yes	—
14	#22548707	C₂²	1	4	yes	—
15	#22548707	C₂²	1	4	yes	—
16	#22548706	C₂²	1	8	yes	—
17	#22548706	C₂²	1	16	yes	—
18	#22548705	C₂²	1	16	yes	—
19	#22548705	C₂²	1	16	yes	—
20	#22548694	C₂²	1	4	yes	—
21	#22548694	C₂²	1	4	yes	—
22	Q₈×Dic₂₅₁	C₂²	1	8	yes	—
23	C₄×Dic₅₀₂	C₂²	1	4	yes	—
24	C₄×Dic₅₀₂	C₂²	1	4	yes	—
25	C₄×Dic₅₀₂	C₂²	1	8	yes	—
26	#22548680	C₂²	1	8	yes	—
27	#22548674	C₂²	1	4	yes	—
28	C₄⋊Q₈×C₂₅₁	C₂²	1	2	yes	—
29	#125×C₂₅₁	C₂²	1	2	yes	—
30	C₂²⋊Q₈×C₂₅₁	C₂²	1	4	yes	—
31	C₄×Q₈×C₂₅₁	C₂²	1	4	yes	—
32	#117×C₂₅₁	C₂²	1	2	yes	—
33	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
34	C₂×Dic₅₀₂	C₂³	1	4	yes	—
35	C₂×Dic₅₀₂	C₂³	1	4	yes	—
36	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
37	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
38	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
39	C₄⋊Dic₂₅₁	C₂³	1	4	yes	—
40	C₄⋊Dic₂₅₁	C₂³	1	16	yes	—
41	Dic₂₅₁⋊C₄	C₂³	1	16	yes	—
42	Dic₂₅₁⋊C₄	C₂³	1	8	yes	—
43	Dic₂₅₁⋊C₄	C₂³	1	8	yes	—
44	Dic₂₅₁⋊C₄	C₂³	1	16	yes	—
45	C₂²⋊Dic₂₅₁	C₂³	1	8	yes	—
46	C₄×Dic₂₅₁	C₂³	1	8	yes	—
47	C₄×Dic₂₅₁	C₂³	1	16	yes	—
48	C₂×Q₈×C₂₅₁	C₂³	1	4	yes	—
49	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
51	C₄⋊C₄×C₂₅₁	C₂³	1	4	yes	—
52	C₄⋊C₄×C₂₅₁	C₂³	1	8	yes	—
53	C₄⋊C₄×C₂₅₁	C₂³	1	4	yes	—
54	C₂²⋊C₄×C₂₅₁	C₂³	1	4	yes	—
55	C₄²×C₂₅₁	C₂³	1	2	yes	—
56	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
57	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
58	C₄²×C₂₅₁	C₂³	1	4	yes	—
59	Dic₅₀₂	—	2	8	no	—
60	Dic₅₀₂	—	2	8	no	—
61	C₂×Dic₂₅₁	C₂⁴	1	8	yes	—
62	C₂×Dic₂₅₁	C₂⁴	1	8	yes	—
63	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
64	Q₈×C₂₅₁	—	2	8	no	—
65	C₂×C₄×C₂₅₁	—	2	4	no	—
66	C₂×C₄×C₂₅₁	—	2	2	no	—
67	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
68	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
69	C₂×C₄×C₂₅₁	C₂⁴	1	8	yes	—
70	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
71	C₂×C₄×C₂₅₁	C₂⁴	1	2	yes	—
72	C₂²×C₂₅₁	—	2	2	no	—
73	C₂²×C₂₅₁	—	2	1	no	—
74	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
75	C₄×C₂₅₁	—	2	2	no	—
76	C₄×C₂₅₁	—	2	1	no	—
77	C₄×C₂₅₁	—	2	8	no	—
78	C₄×C₂₅₁	2₋¹⁺⁴	1	4	yes	—
79	C₄×C₂₅₁	2₋¹⁺⁴	1	2	yes	—
80	Dic₂₅₁	—	2	8	no	—
81	Dic₂₅₁	—	2	8	no	—
82	C₂×C₂₅₁	—	4	1	no	—
83	C₂×C₂₅₁	C₂×2₋¹⁺⁴	1	2	yes	—
84	C₂×C₂₅₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
85	C₂₅₁	#3556	1	1	yes	characteristic, complemented, Sylow
86	#3556	—	251	1	no	maximal, Sylow
87	#564	—	251	4	no	—
88	#557	—	251	4	no	—
89	#557	—	251	4	no	—
90	#557	—	251	4	no	—
91	#541	—	251	8	no	—
92	#541	—	251	4	no	—
93	#533	—	251	2	no	—
94	#533	D₂₅₁	1	1	yes	characteristic
95	C₄⋊Q₈	—	251	8	no	—
96	C₄⋊Q₈	—	251	2	no	—
97	C₄⋊Q₈	—	251	2	no	—
98	C₄⋊Q₈	—	251	4	no	—
99	C₄⋊Q₈	C₂×D₂₅₁	1	2	yes	—
100	#126	—	251	16	no	—
101	#125	—	251	8	no	—
102	#125	—	251	16	no	—
103	#125	—	251	16	no	—
104	#125	—	251	16	no	—
105	#125	—	251	4	no	—
106	#125	—	251	4	no	—
107	#125	C₂×D₂₅₁	1	2	yes	—
108	C₂²⋊Q₈	—	251	4	no	—
109	C₂²⋊Q₈	—	251	4	no	—
110	C₂²⋊Q₈	C₂×D₂₅₁	1	4	yes	—
111	C₄×Q₈	—	251	8	no	—
112	C₄×Q₈	—	251	8	no	—
113	C₄×Q₈	—	251	4	no	—
114	C₄×Q₈	—	251	4	no	—
115	C₄×Q₈	—	251	8	no	—
116	C₄×Q₈	C₂×D₂₅₁	1	4	yes	—
117	#117	—	251	4	no	—
118	#117	C₂×D₂₅₁	1	2	yes	—
119	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
120	C₂×Q₈	—	251	4	no	—
121	C₂×Q₈	—	251	4	no	—
122	C₂×Q₈	C₂²×D₂₅₁	1	4	yes	—
123	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
124	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
125	C₄⋊C₄	—	251	4	no	—
126	C₄⋊C₄	—	251	4	no	—
127	C₄⋊C₄	—	251	8	no	—
128	C₄⋊C₄	—	251	8	no	—
129	C₄⋊C₄	—	251	16	no	—
130	C₄⋊C₄	—	251	16	no	—
131	C₄⋊C₄	—	251	4	no	—
132	C₄⋊C₄	—	251	4	no	—
133	C₄⋊C₄	—	251	16	no	—
134	C₄⋊C₄	C₂²×D₂₅₁	1	4	yes	—
135	C₄⋊C₄	C₂²×D₂₅₁	1	4	yes	—
136	C₄⋊C₄	C₂²×D₂₅₁	1	8	yes	—
137	C₂²⋊C₄	—	251	8	no	—
138	C₂²⋊C₄	C₂²×D₂₅₁	1	4	yes	—
139	C₄²	—	251	8	no	—
140	C₄²	—	251	16	no	—
141	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
142	C₄²	C₂²×D₂₅₁	1	2	yes	—
143	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
144	C₄²	C₂²×D₂₅₁	1	4	yes	—
145	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
146	Q₈	—	2	8	no	—
147	Q₈	—	502	8	no	—
148	Q₈	—	502	8	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	251	8	no	—
152	C₂×C₄	—	251	8	no	—
153	C₂×C₄	C₂³×D₂₅₁	1	8	yes	—
154	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
155	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
156	C₂×C₄	C₂³×D₂₅₁	1	2	yes	—
157	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
158	C₂²	—	2	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₂⁴×D₂₅₁	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	8	no	—
163	C₄	—	2	1	no	—
164	C₄	—	502	8	no	—
165	C₄	—	502	8	no	—
166	C₄	#22548762	1	4	yes	—
167	C₄	#22548763	1	2	yes	—
168	C₂	—	4	1	no	—
169	C₂	C₂×#22548762	1	2	yes	—
170	C₂	C₂×#22548763	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₅₁
commutator, socle, cosocle	C₂²×C₂₅₁	C₂⁵
Fitting	#533×C₂₅₁	C₂
prime radical	C₂³×C₂₅₁	C₂⁴
hypercenter	#533	D₂₅₁
hypocenter	C₂₅₁	#3556
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₂₅₁⋊#564	C₂	semidirect
2. #30632416	C₂	semidirect
3. #30632415	C₂	semidirect
4. #30632412	C₂	semidirect
85. C₂₅₁	#3556	semidirect

Maximal subgroups

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

subgroup	index
C₂₅₁⋊#564	2
#30632416	2
#30632415	2
#30632412	2
#30632299	2
#30632297	2
#30632244	2
#533×C₂₅₁	2
#3556	251