Subgroups — #44553242

Order 32128.

548 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	#30631586	C₂	1	2	yes	maximal, complemented
2	#30631573	C₂	1	4	yes	maximal, complemented
3	#30631547	C₂	1	1	yes	maximal, characteristic
4	#30631547	C₂	1	2	yes	maximal
5	C₄⋊C₄×Dic₂₅₁	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
7	C₄×C₂²⋊Dic₂₅₁	C₂	1	2	yes	maximal
8	C₂²×C₄⋊C₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₅₁	C₂²	1	4	yes	—
11	C₂×C₂²⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Dic₂₅₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
15	C₂×C₄×Dic₂₅₁	C₂²	1	4	yes	—
16	C₂₅₁⋊#95	C₂²	1	2	yes	—
17	C₂₅₁⋊#95	C₂²	1	4	yes	—
18	C₂₅₁⋊#95	C₂²	1	4	yes	—
19	C₂₅₁⋊#95	C₂²	1	2	yes	—
20	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₂³×C₄×C₂₅₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₂₅₁	C₄	1	8	yes	complemented
23	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
26	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
27	C₂²×Dic₂₅₁	C₂³	1	4	yes	—
28	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
29	C₄⋊Dic₂₅₁	—	2	4	no	—
30	C₄⋊Dic₂₅₁	—	4	4	no	—
31	C₂²⋊Dic₂₅₁	—	2	2	no	—
32	C₂²⋊Dic₂₅₁	—	2	4	no	—
33	C₂²⋊Dic₂₅₁	Q₈	1	4	yes	—
34	C₄×Dic₂₅₁	—	2	4	no	—
35	C₄×Dic₂₅₁	—	4	4	no	—
36	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₅₁	—	2	4	no	—
38	C₂²×C₄×C₂₅₁	—	4	4	no	—
39	C₂²×C₄×C₂₅₁	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₂₅₁	C₂×C₄	1	8	yes	—
41	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₂₅₁	C₂×C₄	1	8	yes	—
46	C₄⋊C₄×C₂₅₁	C₂×C₄	1	4	yes	—
47	C₄⋊C₄×C₂₅₁	C₂×C₄	1	4	yes	—
48	C₂×Dic₂₅₁	—	2	2	no	—
49	C₂×Dic₂₅₁	—	2	2	no	—
50	C₂×Dic₂₅₁	—	2	2	no	—
51	C₂×Dic₂₅₁	—	2	2	no	—
52	C₂×Dic₂₅₁	—	2	4	no	—
53	C₂×Dic₂₅₁	—	2	4	no	—
54	C₂×Dic₂₅₁	—	2	4	no	—
55	C₂×Dic₂₅₁	—	2	2	no	—
56	C₂×Dic₂₅₁	C₂×Q₈	1	4	yes	—
57	C₂³×C₂₅₁	—	2	2	no	—
58	C₂³×C₂₅₁	—	2	2	no	—
59	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
60	C₂³×C₂₅₁	C₂×Q₈	1	2	yes	—
61	C₂³×C₂₅₁	#41	1	2	yes	—
62	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
63	C₂×C₄×C₂₅₁	—	2	2	no	—
64	C₂×C₄×C₂₅₁	—	2	4	no	—
65	C₂×C₄×C₂₅₁	—	2	4	no	—
66	C₂×C₄×C₂₅₁	—	2	8	no	—
67	C₂×C₄×C₂₅₁	—	4	8	no	—
68	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
71	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₂₅₁	#41	1	4	yes	—
73	C₂²×C₂₅₁	—	2	2	no	—
74	C₂²×C₂₅₁	—	2	2	no	—
75	C₂²×C₂₅₁	—	2	4	no	—
76	C₂²×C₂₅₁	—	2	4	no	—
77	C₂²×C₂₅₁	C₄×Q₈	1	4	yes	—
78	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
79	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
81	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
82	C₂²×C₂₅₁	H(5,𝔽₂)	1	2	yes	—
83	C₄×C₂₅₁	—	2	2	no	—
84	C₄×C₂₅₁	—	2	4	no	—
85	C₄×C₂₅₁	—	2	4	no	—
86	C₄×C₂₅₁	#117	1	4	yes	—
87	Dic₂₅₁	—	2	4	no	—
88	Dic₂₅₁	—	4	2	no	—
89	Dic₂₅₁	—	4	4	no	—
90	C₂×C₂₅₁	—	2	4	no	—
91	C₂×C₂₅₁	C₂×#117	1	1	yes	characteristic
92	C₂×C₂₅₁	C₂×C₄×Q₈	1	1	yes	characteristic
93	C₂×C₂₅₁	#518	1	1	yes	characteristic
94	C₂×C₂₅₁	#553	1	2	yes	—
95	C₂×C₂₅₁	#554	1	2	yes	—
96	C₂₅₁	#2345	1	1	yes	characteristic, complemented, Sylow
97	#2345	—	251	1	no	maximal, Sylow
98	C₂²×C₄⋊C₄	D₂₅₁	1	1	yes	characteristic
99	#389	—	251	2	no	—
100	#387	—	251	4	no	—
101	#380	—	251	2	no	—
102	#380	—	251	1	no	—
103	C₄×C₄⋊C₄	—	251	2	no	—
104	C₄×C₂²⋊C₄	—	251	1	no	—
105	C₄×C₂²⋊C₄	—	251	2	no	—
106	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
107	C₂³×C₄	C₂×D₂₅₁	1	2	yes	—
108	C₂×C₄⋊C₄	—	251	4	no	—
109	C₂×C₄⋊C₄	—	251	2	no	—
110	C₂×C₄⋊C₄	Dic₂₅₁	1	8	yes	complemented
111	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
113	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
114	C₂×C₂²⋊C₄	—	251	1	no	—
115	C₂×C₂²⋊C₄	—	251	2	no	—
116	C₂×C₂²⋊C₄	—	251	1	no	—
117	C₂×C₄²	—	251	4	no	—
118	C₂×C₄²	—	251	2	no	—
119	#95	—	251	2	no	—
120	#95	—	251	4	no	—
121	#95	—	251	4	no	—
122	#95	—	251	2	no	—
123	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	4	4	no	—
126	C₂²×C₄	—	251	2	no	—
127	C₂²×C₄	—	251	4	no	—
128	C₂²×C₄	—	251	2	no	—
129	C₂²×C₄	C₂×Dic₂₅₁	1	4	yes	—
130	C₂²×C₄	C₂×Dic₂₅₁	1	8	yes	—
131	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
133	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
134	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
135	C₄⋊C₄	—	1004	4	no	—
136	C₄⋊C₄	—	502	4	no	—
137	C₄⋊C₄	C₂×Dic₂₅₁	1	8	yes	—
138	C₄⋊C₄	C₂×Dic₂₅₁	1	4	yes	—
139	C₄⋊C₄	C₂×Dic₂₅₁	1	4	yes	—
140	C₂²⋊C₄	—	502	2	no	—
141	C₂²⋊C₄	—	502	4	no	—
142	C₂²⋊C₄	—	251	4	no	—
143	C₄²	—	1004	4	no	—
144	C₄²	—	502	4	no	—
145	C₂³	—	2	2	no	—
146	C₂³	—	2	2	no	—
147	C₂³	C₂²×Dic₂₅₁	1	2	yes	—
148	C₂³	Q₈×D₂₅₁	1	2	yes	—
149	C₂³	#16860811	1	2	yes	—
150	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	8	no	—
155	C₂×C₄	—	4	8	no	—
156	C₂×C₄	—	502	2	no	—
157	C₂×C₄	—	502	4	no	—
158	C₂×C₄	—	502	2	no	—
159	C₂×C₄	—	502	4	no	—
160	C₂×C₄	—	502	4	no	—
161	C₂×C₄	—	502	2	no	—
162	C₂×C₄	—	502	2	no	—
163	C₂×C₄	—	502	2	no	—
164	C₂×C₄	—	251	4	no	—
165	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
166	C₂×C₄	C₂²×Dic₂₅₁	1	4	yes	—
167	C₂×C₄	C₂²×Dic₂₅₁	1	4	yes	—
168	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
169	C₂×C₄	#16860810	1	4	yes	—
170	C₂²	—	2	4	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	Q₈×Dic₂₅₁	1	4	yes	—
175	C₂²	C₂³×Dic₂₅₁	1	1	yes	characteristic
176	C₂²	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
177	C₂²	C₂×#16860810	1	2	yes	—
178	C₂²	C₂×#16860811	1	1	yes	characteristic
179	C₂²	#22548760	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	4	no	—
182	C₄	—	2	4	no	—
183	C₄	—	1004	2	no	—
184	C₄	—	1004	4	no	—
185	C₄	—	502	4	no	—
186	C₄	#22548674	1	4	yes	—
187	C₂	—	2	4	no	—
188	C₂	C₂×#22548674	1	1	yes	characteristic
189	C₂	C₂×Q₈×Dic₂₅₁	1	1	yes	characteristic
190	C₂	#30632162	1	1	yes	characteristic
191	C₂	#30632385	1	2	yes	—
192	C₂	#30632392	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator	C₂²×C₂₅₁	C₂³×C₄
Fitting	C₂²×C₄⋊C₄×C₂₅₁	C₂
socle, cosocle	C₂³×C₂₅₁	C₂⁴
prime radical	C₂⁴×C₂₅₁	C₂³
hypercenter	C₂²×C₄⋊C₄	D₂₅₁
hypocenter	C₂₅₁	#2345
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #30631586	C₂	semidirect
2. #30631573	C₂	semidirect
5. C₄⋊C₄×Dic₂₅₁	C₂	semidirect
22. C₂×C₄⋊C₄×C₂₅₁	C₄	semidirect
96. C₂₅₁	#2345	semidirect
110. C₂×C₄⋊C₄	Dic₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30631586	2
#30631573	2
#30631547	2
C₄⋊C₄×Dic₂₅₁	2
C₄×C₂²⋊Dic₂₅₁	2
C₂²×C₄⋊C₄×C₂₅₁	2
#2345	251