Subgroups — #44553324

Order 32128.

480 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic
2	#30631592	C₂	1	2	yes	maximal
3	#30631568	C₂	1	4	yes	maximal
4	#30631552	C₂	1	2	yes	maximal
5	C₄⋊C₄×Dic₂₅₁	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₂₅₁	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
8	#384×C₂₅₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
10	C₄×Dic₅₀₂	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₅₁⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
16	C₂×C₄×Dic₂₅₁	C₂²	1	4	yes	—
17	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
18	C₂₅₁⋊#95	C₂²	1	2	yes	—
19	C₂₅₁⋊#95	C₂²	1	4	yes	—
20	C₂₅₁⋊#95	C₂²	1	4	yes	—
21	C₂₅₁⋊#95	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
26	C₂×Dic₅₀₂	—	2	2	no	—
27	C₂×Dic₅₀₂	—	2	2	no	—
28	C₂×Dic₅₀₂	C₂×C₄	1	4	yes	—
29	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
30	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
31	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
32	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
33	C₄⋊Dic₂₅₁	—	2	2	no	—
34	C₄⋊Dic₂₅₁	C₂×C₄	1	4	yes	—
35	C₄⋊Dic₂₅₁	Q₈	1	4	yes	complemented
36	Dic₂₅₁⋊C₄	C₂×C₄	1	8	yes	—
37	C₄×Dic₂₅₁	—	2	4	no	—
38	C₄×Dic₂₅₁	—	4	4	no	—
39	C₄×Dic₂₅₁	C₂×C₄	1	8	yes	—
40	C₂²×C₄×C₂₅₁	C₂³	1	4	yes	—
41	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₂₅₁	—	2	4	no	—
45	C₄⋊C₄×C₂₅₁	—	2	4	no	—
46	C₄⋊C₄×C₂₅₁	—	4	2	no	—
47	C₄²×C₂₅₁	C₂×C₄	1	4	yes	—
48	Dic₅₀₂	—	2	8	no	—
49	C₂×Dic₂₅₁	—	2	2	no	—
50	C₂×Dic₂₅₁	—	2	2	no	—
51	C₂×Dic₂₅₁	—	2	2	no	—
52	C₂×Dic₂₅₁	—	2	2	no	—
53	C₂×Dic₂₅₁	—	2	2	no	—
54	C₂×Dic₂₅₁	—	2	2	no	—
55	C₂×Dic₂₅₁	—	2	2	no	—
56	C₂×Dic₂₅₁	—	2	2	no	—
57	C₂×Dic₂₅₁	—	2	2	no	—
58	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
59	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
60	C₂×Dic₂₅₁	C₂×Q₈	1	4	yes	—
61	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
62	C₂×C₄×C₂₅₁	—	2	1	no	—
63	C₂×C₄×C₂₅₁	—	2	1	no	—
64	C₂×C₄×C₂₅₁	—	2	4	no	—
65	C₂×C₄×C₂₅₁	—	2	4	no	—
66	C₂×C₄×C₂₅₁	—	2	4	no	—
67	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
71	C₂×C₄×C₂₅₁	#41	1	2	yes	—
72	C₂×C₄×C₂₅₁	#41	1	2	yes	—
73	C₂×C₄×C₂₅₁	#41	1	2	yes	—
74	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
77	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
78	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
79	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
81	C₄×C₂₅₁	—	2	2	no	—
82	C₄×C₂₅₁	—	4	4	no	—
83	C₄×C₂₅₁	#117	1	4	yes	—
84	C₄×C₂₅₁	C₄×Q₈	1	4	yes	—
85	Dic₂₅₁	—	2	4	no	—
86	Dic₂₅₁	—	2	4	no	—
87	Dic₂₅₁	—	2	4	no	—
88	Dic₂₅₁	—	4	2	no	—
89	C₂×C₂₅₁	C₂×#117	1	1	yes	characteristic
90	C₂×C₂₅₁	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂×C₂₅₁	#520	1	1	yes	characteristic
92	C₂×C₂₅₁	#554	1	1	yes	characteristic
93	C₂×C₂₅₁	#556	1	1	yes	characteristic
94	C₂×C₂₅₁	#557	1	1	yes	characteristic
95	C₂×C₂₅₁	#559	1	1	yes	characteristic
96	C₂₅₁	#2355	1	1	yes	characteristic, complemented, Sylow
97	#2355	—	251	1	no	maximal, Sylow
98	C₂×C₄×Q₈	—	251	1	no	—
99	#391	—	251	2	no	—
100	#387	—	251	4	no	—
101	#384	D₂₅₁	1	1	yes	characteristic
102	#382	—	251	2	no	—
103	C₄×C₄⋊C₄	—	251	2	no	—
104	C₄×C₄⋊C₄	—	251	2	no	—
105	C₄×C₄⋊C₄	—	251	1	no	—
106	C₂²×Q₈	—	251	1	no	—
107	C₄×Q₈	—	251	8	no	—
108	C₂×C₄⋊C₄	—	251	1	no	—
109	C₂×C₄⋊C₄	—	251	1	no	—
110	C₂×C₄⋊C₄	—	251	2	no	—
111	C₂×C₄⋊C₄	—	251	1	no	—
112	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
113	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
115	C₂×C₄²	—	251	4	no	—
116	C₂×C₄²	—	251	2	no	—
117	C₂×C₄²	—	251	2	no	—
118	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
119	#95	—	251	4	no	—
120	#95	—	251	2	no	—
121	#95	—	251	4	no	—
122	#95	—	251	4	no	—
123	C₂×Q₈	—	502	2	no	—
124	C₂×Q₈	—	502	2	no	—
125	C₂×Q₈	—	251	4	no	—
126	C₂²×C₄	—	251	2	no	—
127	C₂²×C₄	—	251	2	no	—
128	C₂²×C₄	—	251	2	no	—
129	C₂²×C₄	—	251	2	no	—
130	C₂²×C₄	C₂²×D₂₅₁	1	4	yes	—
131	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	502	2	no	—
138	C₄⋊C₄	—	251	8	no	—
139	C₄⋊C₄	—	251	4	no	—
140	C₄⋊C₄	—	251	4	no	—
141	C₄²	—	1004	4	no	—
142	C₄²	—	502	4	no	—
143	C₄²	—	251	8	no	—
144	C₄²	C₄×D₂₅₁	1	4	yes	—
145	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
146	Q₈	—	502	8	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	502	2	no	—
153	C₂×C₄	—	502	2	no	—
154	C₂×C₄	—	502	2	no	—
155	C₂×C₄	—	502	2	no	—
156	C₂×C₄	—	502	2	no	—
157	C₂×C₄	—	502	2	no	—
158	C₂×C₄	—	502	2	no	—
159	C₂×C₄	—	502	2	no	—
160	C₂×C₄	—	502	2	no	—
161	C₂×C₄	—	251	4	no	—
162	C₂×C₄	—	251	4	no	—
163	C₂×C₄	—	251	4	no	—
164	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
166	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
167	C₂×C₄	Q₈×D₂₅₁	1	2	yes	—
168	C₂×C₄	#16860809	1	2	yes	—
169	C₂×C₄	#16860809	1	2	yes	—
170	C₂×C₄	#16860811	1	2	yes	—
171	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
172	C₂²	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
173	C₂²	C₂×#16860809	1	1	yes	characteristic
174	C₂²	C₂×#16860809	1	1	yes	characteristic
175	C₂²	C₂×#16860811	1	1	yes	characteristic
176	C₂²	#22548760	1	1	yes	characteristic
177	C₂²	#22548763	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	4	4	no	—
180	C₄	—	1004	2	no	—
181	C₄	—	502	4	no	—
182	C₄	—	502	4	no	—
183	C₄	—	502	4	no	—
184	C₄	#22548673	1	4	yes	—
185	C₄	#22548680	1	4	yes	—
186	C₂	C₂×#22548673	1	1	yes	characteristic
187	C₂	C₂×#22548680	1	1	yes	characteristic
188	C₂	#30632168	1	1	yes	characteristic
189	C₂	#30632398	1	1	yes	characteristic
190	C₂	#30632411	1	1	yes	characteristic
191	C₂	#30632412	1	1	yes	characteristic
192	C₂	#30632423	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator	C₂²×C₂₅₁	C₂³×C₄
Fitting	#384×C₂₅₁	C₂
socle, prime radical, cosocle	C₂³×C₂₅₁	C₂⁴
hypercenter	#384	D₂₅₁
hypocenter	C₂₅₁	#2355
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₅₀₂	C₄	semidirect
35. C₄⋊Dic₂₅₁	Q₈	semidirect
96. C₂₅₁	#2355	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₀₂	2
#30631592	2
#30631568	2
#30631552	2
C₄⋊C₄×Dic₂₅₁	2
C₄×C₄⋊Dic₂₅₁	2
#384×C₂₅₁	2
#2355	251