Subgroups — #44559303

Order 32128.

348 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632240	C₂	1	1	yes	maximal, characteristic
2	#30631817	C₂	1	4	yes	maximal, complemented
3	#30631815	C₂	1	4	yes	maximal, complemented
4	#30631715	C₂	1	1	yes	maximal, characteristic
5	#30631714	C₂	1	1	yes	maximal, characteristic
6	#30631710	C₂	1	1	yes	maximal, characteristic
7	#30631642	C₂	1	1	yes	maximal, characteristic
8	C₂×#22548625	C₂	1	1	yes	maximal, characteristic
9	#405×C₂₅₁	C₂	1	1	yes	maximal, characteristic
10	C₂×#16860794	C₂²	1	1	yes	characteristic
11	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
12	#22548714	C₄	1	2	yes	—
13	#22548706	C₄	1	2	yes	—
14	#22548692	C₄	1	4	yes	—
15	C₄×Dic₅₀₂	C₂²	1	4	yes	—
16	#22548674	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
18	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	4	yes	—
19	Dic₂₅₁⋊C₈	C₂²	1	4	yes	—
20	Dic₂₅₁⋊C₈	C₂²	1	4	yes	—
21	#22548625	C₂²	1	2	yes	—
22	#22548625	C₂²	1	2	yes	—
23	#22548625	C₂²	1	2	yes	—
24	C₈×Dic₂₅₁	C₂²	1	2	yes	—
25	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
27	#117×C₂₅₁	C₂²	1	1	yes	characteristic
28	#97×C₂₅₁	C₂²	1	2	yes	—
29	C₄×C₈×C₂₅₁	C₂²	1	2	yes	—
30	C₂×Dic₅₀₂	C₂×C₄	1	4	yes	—
31	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
32	#16860794	—	2	2	no	—
33	C₂×C₂₅₁⋊C₈	—	2	2	no	—
34	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
35	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
36	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₂₅₁	C₂×C₄	1	4	yes	—
38	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	—
39	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	—
40	C₂²⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
41	C₄×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₂₅₁	C₂³	1	2	yes	—
44	C₄×Dic₂₅₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
46	M₁₆×C₂₅₁	—	2	2	no	—
47	C₂×C₈×C₂₅₁	Q₈	1	4	yes	complemented
48	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
49	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	—
53	C₄²×C₂₅₁	C₂³	1	2	yes	—
54	Dic₅₀₂	—	4	4	no	—
55	C₂×Dic₂₅₁	—	4	2	no	—
56	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
57	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
58	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
59	C₂₅₁⋊C₈	—	2	2	no	—
60	C₂₅₁⋊C₈	—	2	2	no	—
61	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₂₅₁	#41	1	2	yes	—
66	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
67	C₈×C₂₅₁	—	2	2	no	—
68	C₈×C₂₅₁	C₂×Q₈	1	2	yes	—
69	C₈×C₂₅₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₂₅₁	—	2	2	no	—
71	C₂²×C₂₅₁	C₄×Q₈	1	2	yes	—
72	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₂₅₁	—	2	4	no	—
74	C₄×C₂₅₁	C₄×Q₈	1	2	yes	—
75	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
76	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
77	Dic₂₅₁	—	2	2	no	—
78	Dic₂₅₁	—	2	2	no	—
79	Dic₂₅₁	—	2	4	no	—
80	C₂×C₂₅₁	—	2	2	no	—
81	C₂×C₂₅₁	C₂×C₄×Q₈	1	1	yes	characteristic
82	C₂×C₂₅₁	#568	1	2	yes	—
83	C₂₅₁	#2961	1	1	yes	characteristic, complemented, Sylow
84	#2961	—	251	1	no	maximal, Sylow
85	#533	—	251	1	no	—
86	#446	—	251	4	no	—
87	#446	—	251	4	no	—
88	#424	—	251	1	no	—
89	#424	—	251	1	no	—
90	#423	—	251	1	no	—
91	#405	—	251	1	no	—
92	#405	D₂₅₁	1	1	yes	characteristic
93	C₂×#97	—	251	1	no	—
94	C₂×M₁₆	—	251	1	no	—
95	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
96	C₂²×C₈	—	251	1	no	—
97	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
98	C₄⋊Q₈	—	251	2	no	—
99	#125	—	251	2	no	—
100	C₂²⋊Q₈	—	251	4	no	—
101	C₄×Q₈	—	251	4	no	—
102	#117	—	251	1	no	—
103	#117	C₂×D₂₅₁	1	1	yes	characteristic
104	C₂×C₄²	—	251	1	no	—
105	C₄⋊C₈	—	251	4	no	—
106	C₄⋊C₈	—	251	4	no	—
107	C₄⋊C₈	—	251	4	no	—
108	#97	—	251	2	no	—
109	#97	—	251	2	no	—
110	#97	—	251	2	no	—
111	#97	C₂×D₂₅₁	1	2	yes	—
112	C₄×C₈	—	251	2	no	—
113	C₄×C₈	C₂×D₂₅₁	1	2	yes	—
114	C₂×Q₈	—	251	4	no	—
115	C₂²×C₄	—	251	2	no	—
116	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	502	2	no	—
119	C₂×C₈	—	502	2	no	—
120	C₂×C₈	—	251	2	no	—
121	C₂×C₈	—	251	2	no	—
122	C₂×C₈	Dic₅₀₂	1	4	yes	complemented
123	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
124	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
126	C₄⋊C₄	—	251	2	no	—
127	C₄⋊C₄	—	251	4	no	—
128	C₄⋊C₄	—	251	4	no	—
129	C₄⋊C₄	—	251	4	no	—
130	C₄⋊C₄	C₄×D₂₅₁	1	2	yes	—
131	C₂²⋊C₄	—	251	2	no	—
132	C₂²⋊C₄	C₄×D₂₅₁	1	2	yes	—
133	C₄²	—	251	1	no	—
134	C₄²	—	251	1	no	—
135	C₄²	—	251	2	no	—
136	C₄²	—	251	2	no	—
137	C₄²	C₂²×D₂₅₁	1	2	yes	—
138	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
139	Q₈	—	1004	4	no	—
140	C₂×C₄	—	1004	2	no	—
141	C₂×C₄	—	251	2	no	—
142	C₂×C₄	—	251	2	no	—
143	C₂×C₄	—	251	4	no	—
144	C₂×C₄	C₂×C₄×D₂₅₁	1	4	yes	—
145	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
146	C₂×C₄	C₂×Dic₅₀₂	1	2	yes	—
147	C₂×C₄	#16860810	1	2	yes	—
148	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	502	2	no	—
151	C₈	—	502	2	no	—
152	C₈	C₂×Dic₅₀₂	1	2	yes	—
153	C₈	C₂×Dic₅₀₂	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₅₀₂	1	2	yes	—
156	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	502	2	no	—
159	C₄	—	502	2	no	—
160	C₄	—	502	4	no	—
161	C₄	C₄×Dic₅₀₂	1	2	yes	—
162	C₄	C₂²×Dic₅₀₂	1	1	yes	characteristic
163	C₄	C₂×#16860810	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₅₀₂	1	1	yes	characteristic
166	C₂	#30632452	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Fitting	#405×C₂₅₁	C₂
prime radical	C₂³×C₂₅₁	C₂²×C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#405	D₂₅₁
hypocenter	C₂₅₁	#2961
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #30631817	C₂	semidirect
3. #30631815	C₂	semidirect
47. C₂×C₈×C₂₅₁	Q₈	semidirect
83. C₂₅₁	#2961	semidirect
122. C₂×C₈	Dic₅₀₂	semidirect

Maximal subgroups

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

subgroup	index
#30632240	2
#30631817	2
#30631815	2
#30631715	2
#30631714	2
#30631710	2
#30631642	2
C₂×#22548625	2
#405×C₂₅₁	2
#2961	251