Subgroups — #45664052

Order 32896.

420 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22551551	C₂	1	1	yes	maximal, characteristic
2	C₂×#22551551	C₂	1	1	yes	maximal, characteristic
3	#30656503	C₂	1	4	yes	maximal
4	C₂×#117×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	#22551551	—	2	1	no	—
6	#22551551	—	2	1	no	—
7	#22551551	—	2	2	no	—
8	#22551551	—	2	1	no	—
9	#22551551	—	2	1	no	—
10	#22551551	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₂₅₇	—	2	1	no	—
12	C₂×C₄⋊Dic₂₅₇	C₄	1	2	yes	complemented
13	C₂×C₂²⋊Dic₂₅₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₂₅₇	C₂²	1	2	yes	—
15	C₂×C₄×Dic₂₅₇	—	2	1	no	—
16	C₂×C₄×Dic₂₅₇	C₄	1	2	yes	complemented
17	C₂³×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
18	#117×C₂₅₇	—	2	2	no	—
19	#117×C₂₅₇	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₄×C₂₅₇	—	2	1	no	—
21	C₂×C₂²⋊C₄×C₂₅₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₅₇	—	2	1	no	—
23	C₂²×Dic₂₅₇	—	2	2	no	—
24	C₂²×Dic₂₅₇	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₂₅₇	—	2	4	no	—
26	C₄⋊Dic₂₅₇	—	2	2	no	—
27	C₄⋊Dic₂₅₇	—	2	2	no	—
28	C₂²⋊Dic₂₅₇	—	2	2	no	—
29	C₂²⋊Dic₂₅₇	—	2	2	no	—
30	C₂²⋊Dic₂₅₇	—	2	2	no	—
31	C₂²⋊Dic₂₅₇	C₂×C₄	1	4	yes	—
32	C₄×Dic₂₅₇	—	2	4	no	—
33	C₄×Dic₂₅₇	—	2	2	no	—
34	C₄×Dic₂₅₇	—	2	2	no	—
35	C₂⁴×C₂₅₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₅₇	—	2	2	no	—
37	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₂₅₇	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₅₇	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
44	C₂²×C₄×C₂₅₇	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₅₇	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₅₇	Q₈	1	2	yes	—
47	C₄⋊C₄×C₂₅₇	—	2	4	no	—
48	C₂²⋊C₄×C₂₅₇	—	2	2	no	—
49	C₂²⋊C₄×C₂₅₇	C₂×C₄	1	4	yes	—
50	C₄²×C₂₅₇	—	2	4	no	—
51	C₂×Dic₂₅₇	—	2	4	no	—
52	C₂×Dic₂₅₇	—	2	4	no	—
53	C₂×Dic₂₅₇	—	4	4	no	—
54	C₂×Dic₂₅₇	—	4	4	no	—
55	C₂³×C₂₅₇	—	4	2	no	—
56	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
57	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
58	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
59	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
60	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
61	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
62	C₂³×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₂₅₇	—	2	4	no	—
64	C₂×C₄×C₂₅₇	—	4	4	no	—
65	C₂×C₄×C₂₅₇	C₄²	1	2	yes	—
66	C₂×C₄×C₂₅₇	C₄²	1	2	yes	—
67	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
68	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
70	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
75	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
81	C₂²×C₂₅₇	—	2	2	no	—
82	C₂²×C₂₅₇	—	2	2	no	—
83	C₂²×C₂₅₇	—	2	2	no	—
84	C₂²×C₂₅₇	—	4	1	no	—
85	C₂²×C₂₅₇	—	4	1	no	—
86	C₂²×C₂₅₇	—	4	1	no	—
87	C₂²×C₂₅₇	—	4	1	no	—
88	C₂²×C₂₅₇	C₂×C₄²	1	1	yes	characteristic
89	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₂₅₇	—	4	4	no	—
96	C₄×C₂₅₇	#95	1	2	yes	—
97	C₄×C₂₅₇	#95	1	2	yes	—
98	C₄×C₂₅₇	#95	1	2	yes	—
99	C₄×C₂₅₇	#95	1	2	yes	—
100	Dic₂₅₇	—	4	4	no	—
101	Dic₂₅₇	—	4	4	no	—
102	C₂×C₂₅₇	—	2	1	no	—
103	C₂×C₂₅₇	—	2	1	no	—
104	C₂×C₂₅₇	—	2	1	no	—
105	C₂×C₂₅₇	—	2	1	no	—
106	C₂×C₂₅₇	—	2	1	no	—
107	C₂×C₂₅₇	—	2	1	no	—
108	C₂×C₂₅₇	C₂×#95	1	1	yes	characteristic
109	C₂×C₂₅₇	#410	1	2	yes	—
110	C₂₅₇	#1740	1	1	yes	characteristic, complemented, Sylow
111	#1740	—	257	1	no	maximal, Sylow
112	C₂×#117	—	257	1	no	—
113	C₂×#117	—	257	1	no	—
114	C₂×#117	D₂₅₇	1	1	yes	characteristic
115	#342	—	257	4	no	—
116	C₂³×C₄	C₂×D₂₅₇	1	1	yes	characteristic
117	#117	—	257	4	no	—
118	#117	—	2	2	no	—
119	#117	—	514	1	no	—
120	#117	—	514	2	no	—
121	#117	—	514	1	no	—
122	#117	—	514	1	no	—
123	#117	—	514	1	no	—
124	#117	Dic₂₅₇	1	4	yes	complemented
125	C₂×C₄⋊C₄	—	257	2	no	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	—	514	1	no	—
128	C₂×C₂²⋊C₄	—	257	2	no	—
129	C₂×C₂²⋊C₄	—	257	2	no	—
130	C₂×C₂²⋊C₄	C₂×D₂₅₇	1	2	yes	—
131	C₂×C₄²	—	257	2	no	—
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	—	514	1	no	—
134	C₂⁴	C₂²×D₂₅₇	1	1	yes	characteristic
135	C₂²×C₄	—	257	4	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	514	2	no	—
138	C₂²×C₄	C₄×D₂₅₇	1	1	yes	characteristic
139	C₂²×C₄	C₄×D₂₅₇	1	2	yes	—
140	C₂²×C₄	C₄×D₂₅₇	1	1	yes	characteristic
141	C₂²×C₄	C₂×Dic₂₅₇	1	2	yes	—
142	C₂²×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²⋊D₂₅₇	1	2	yes	—
144	C₂²×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
145	C₂²×C₄	D₁₀₂₈	1	1	yes	characteristic
146	C₂²×C₄	D₁₀₂₈	1	1	yes	characteristic
147	C₂²×C₄	Dic₅₁₄	1	2	yes	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	514	2	no	—
150	C₄⋊C₄	—	514	2	no	—
151	C₄⋊C₄	—	514	4	no	—
152	C₂²⋊C₄	—	257	4	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	—	514	2	no	—
155	C₂²⋊C₄	—	514	2	no	—
156	C₂²⋊C₄	—	514	2	no	—
157	C₂²⋊C₄	C₂×Dic₂₅₇	1	4	yes	—
158	C₄²	—	2	4	no	—
159	C₄²	—	514	2	no	—
160	C₄²	—	514	4	no	—
161	C₄²	—	514	2	no	—
162	C₂³	—	4	2	no	—
163	C₂³	C₂×C₄×D₂₅₇	1	1	yes	characteristic
164	C₂³	C₂×C₄×D₂₅₇	1	1	yes	characteristic
165	C₂³	C₂²×Dic₂₅₇	1	1	yes	characteristic
166	C₂³	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
167	C₂³	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
168	C₂³	C₂×D₁₀₂₈	1	1	yes	characteristic
169	C₂³	C₂×Dic₅₁₄	1	1	yes	characteristic
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	514	4	no	—
172	C₂×C₄	—	514	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	1028	4	no	—
175	C₂×C₄	—	1028	4	no	—
176	C₂×C₄	C₄×Dic₂₅₇	1	2	yes	—
177	C₂×C₄	C₄×Dic₂₅₇	1	2	yes	—
178	C₂×C₄	#16861595	1	1	yes	characteristic
179	C₂×C₄	#16861595	1	2	yes	—
180	C₂×C₄	#16861595	1	2	yes	—
181	C₂×C₄	#16861595	1	1	yes	characteristic
182	C₂×C₄	#16861595	1	1	yes	characteristic
183	C₂×C₄	#16861595	1	1	yes	characteristic
184	C₂×C₄	C₂²⋊Dic₂₅₇	1	2	yes	—
185	C₂×C₄	C₂²⋊Dic₂₅₇	1	2	yes	—
186	C₂×C₄	#16861597	1	2	yes	—
187	C₂×C₄	#16861597	1	2	yes	—
188	C₂×C₄	#16861597	1	2	yes	—
189	C₂×C₄	#16861597	1	2	yes	—
190	C₂×C₄	C₄⋊Dic₂₅₇	1	2	yes	—
191	C₂×C₄	C₄⋊Dic₂₅₇	1	2	yes	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	1	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	4	1	no	—
198	C₂²	—	4	1	no	—
199	C₂²	C₂×C₄×Dic₂₅₇	1	1	yes	characteristic
200	C₂²	C₂×#16861595	1	1	yes	characteristic
201	C₂²	C₂×#16861595	1	1	yes	characteristic
202	C₂²	C₂×C₂²⋊Dic₂₅₇	1	1	yes	characteristic
203	C₂²	C₂×#16861597	1	1	yes	characteristic
204	C₂²	C₂×#16861597	1	1	yes	characteristic
205	C₂²	C₂×C₄⋊Dic₂₅₇	1	1	yes	characteristic
206	C₄	—	4	4	no	—
207	C₄	—	1028	4	no	—
208	C₄	—	1028	4	no	—
209	C₄	#22551499	1	2	yes	—
210	C₄	#22551499	1	2	yes	—
211	C₄	#22551499	1	2	yes	—
212	C₄	#22551499	1	2	yes	—
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	—	2	1	no	—
219	C₂	C₂×#22551499	1	1	yes	characteristic
220	C₂	#30656710	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#16861595
commutator	C₂²×C₂₅₇	C₂×C₄²
Frattini	C₂⁴	C₂²×D₂₅₇
Fitting	C₂×#117×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₂²⋊C₄
prime radical, cosocle	C₂⁴×C₂₅₇	C₂³
hypercenter	C₂×#117	D₂₅₇
hypocenter	C₂₅₇	#1740
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
10. #22551551	C₄	semidirect
12. C₂×C₄⋊Dic₂₅₇	C₄	semidirect
16. C₂×C₄×Dic₂₅₇	C₄	semidirect
19. #117×C₂₅₇	C₄	semidirect
110. C₂₅₇	#1740	semidirect
124. #117	Dic₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22551551	2
#30656503	2
C₂×#117×C₂₅₇	2
#1740	257