Subgroups — #103×Dic₂₅₇ #45664142

Order 32896.

332 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#22551516	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
5	#30656500	C₂	1	1	yes	maximal, characteristic
6	#30656499	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₂₅₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#14523339	C₂²	1	1	yes	characteristic
9	Q₈×Dic₂₅₇	—	2	2	no	—
10	Q₈×Dic₂₅₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₂₅₇	—	2	1	no	—
12	C₂×C₄⋊Dic₂₅₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₂₅₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₂₅₇	—	2	1	no	—
15	C₂×C₄×Dic₂₅₇	—	2	1	no	—
16	C₂×C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
17	#22551516	C₄	1	4	yes	complemented
18	C₈×Dic₂₅₇	—	2	2	no	—
19	#22551499	—	2	1	no	—
20	C₂²×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₅₇	C₂²	1	1	yes	characteristic
23	#103×C₂₅₇	C₄	1	4	yes	complemented
24	C₂²×Dic₂₅₇	—	2	1	no	—
25	C₂²×Dic₂₅₇	—	2	1	no	—
26	C₂²×Dic₂₅₇	D₄	1	1	yes	characteristic
27	C₂²×Dic₂₅₇	D₄	1	1	yes	characteristic
28	C₂×#14523339	—	2	1	no	—
29	C₂×#14523339	—	2	1	no	—
30	C₂×#14523339	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₂₅₇	—	2	4	no	—
32	C₄⋊Dic₂₅₇	—	2	1	no	—
33	C₄⋊Dic₂₅₇	—	2	1	no	—
34	C₄⋊Dic₂₅₇	C₂×C₄	1	2	yes	—
35	C₄⋊Dic₂₅₇	C₂×C₄	1	2	yes	—
36	C₄×Dic₂₅₇	—	2	4	no	—
37	C₄×Dic₂₅₇	—	4	2	no	—
38	C₄×Dic₂₅₇	C₂×C₄	1	2	yes	—
39	C₄×Dic₂₅₇	D₄	1	2	yes	—
40	C₂×Q₈×C₂₅₇	—	2	1	no	—
41	C₂×Q₈×C₂₅₇	—	2	1	no	—
42	C₂×Q₈×C₂₅₇	—	2	1	no	—
43	C₂×Q₈×C₂₅₇	C₂×C₄	1	2	yes	—
44	C₂×Q₈×C₂₅₇	C₂×C₄	1	2	yes	—
45	C₂×Q₈×C₂₅₇	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₂₅₇	—	2	1	no	—
47	C₂²×C₄×C₂₅₇	—	2	1	no	—
48	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₂₅₇	—	2	1	no	—
50	C₂×C₈×C₂₅₇	—	2	1	no	—
51	C₂×C₈×C₂₅₇	C₂×C₄	1	2	yes	—
52	C₄⋊C₄×C₂₅₇	—	2	1	no	—
53	C₄⋊C₄×C₂₅₇	C₂×C₄	1	2	yes	—
54	C₂×Dic₂₅₇	—	2	1	no	—
55	C₂×Dic₂₅₇	—	2	1	no	—
56	C₂×Dic₂₅₇	—	2	2	no	—
57	C₂×Dic₂₅₇	—	2	2	no	—
58	C₂×Dic₂₅₇	—	4	1	no	—
59	C₂×Dic₂₅₇	—	4	1	no	—
60	C₂×Dic₂₅₇	—	4	1	no	—
61	C₂×Dic₂₅₇	—	4	1	no	—
62	C₂×Dic₂₅₇	C₂²⋊C₄	1	2	yes	—
63	C₂×Dic₂₅₇	C₂²⋊C₄	1	2	yes	—
64	C₂×Dic₂₅₇	SD₁₆	1	2	yes	—
65	C₂×Dic₂₅₇	Q₁₆	1	2	yes	—
66	#14523339	—	2	2	no	—
67	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
68	Q₈×C₂₅₇	—	2	2	no	—
69	Q₈×C₂₅₇	—	2	2	no	—
70	Q₈×C₂₅₇	—	2	2	no	—
71	Q₈×C₂₅₇	C₄²	1	4	yes	complemented
72	C₂×C₄×C₂₅₇	—	2	2	no	—
73	C₂×C₄×C₂₅₇	—	2	2	no	—
74	C₂×C₄×C₂₅₇	—	2	2	no	—
75	C₂×C₄×C₂₅₇	—	2	2	no	—
76	C₂×C₄×C₂₅₇	—	4	1	no	—
77	C₂×C₄×C₂₅₇	—	4	1	no	—
78	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
83	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
84	C₈×C₂₅₇	—	2	2	no	—
85	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₂₅₇	C₄×D₄	1	1	yes	characteristic
87	C₂²×C₂₅₇	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₂₅₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₂²×C₂₅₇	C₂×Q₁₆	1	1	yes	characteristic
90	C₂²×C₂₅₇	#135	1	1	yes	characteristic
91	C₂²×C₂₅₇	#135	1	1	yes	characteristic
92	C₄×C₂₅₇	—	2	4	no	—
93	C₄×C₂₅₇	—	4	2	no	—
94	C₄×C₂₅₇	C₂×C₄²	1	1	yes	characteristic
95	C₄×C₂₅₇	#117	1	1	yes	characteristic
96	C₄×C₂₅₇	C₄×D₄	1	1	yes	characteristic
97	C₄×C₂₅₇	C₄×D₄	1	1	yes	characteristic
98	Dic₂₅₇	—	2	2	no	—
99	Dic₂₅₇	—	4	2	no	—
100	Dic₂₅₇	—	4	2	no	—
101	Dic₂₅₇	#103	1	4	yes	direct product factor
102	C₂×C₂₅₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₂₅₇	C₂×#103	1	1	yes	characteristic
104	C₂×C₂₅₇	#416	1	1	yes	characteristic
105	C₂×C₂₅₇	C₄×SD₁₆	1	1	yes	characteristic
106	C₂×C₂₅₇	C₄×SD₁₆	1	1	yes	characteristic
107	C₂×C₂₅₇	C₄×Q₁₆	1	1	yes	characteristic
108	C₂×C₂₅₇	C₄×Q₁₆	1	1	yes	characteristic
109	C₂₅₇	C₄×#103	1	1	yes	characteristic, complemented, Sylow
110	C₄×#103	—	257	1	no	maximal, Sylow
111	C₂×C₄×Q₈	—	257	1	no	—
112	C₂×#103	—	257	1	no	—
113	C₂×#103	D₂₅₇	1	1	yes	characteristic
114	C₂×C₄×C₈	—	257	1	no	—
115	C₄×C₄⋊C₄	—	257	1	no	—
116	#340	—	257	1	no	—
117	#340	—	257	1	no	—
118	C₂²×Q₈	C₂×D₂₅₇	1	1	yes	characteristic
119	C₂²×C₈	—	257	1	no	—
120	C₂²×C₈	C₂×D₂₅₇	1	1	yes	characteristic
121	C₄×Q₈	—	257	4	no	—
122	C₄×Q₈	—	514	2	no	—
123	C₂×C₄⋊C₄	—	257	1	no	—
124	C₂×C₄⋊C₄	—	257	1	no	—
125	C₂×C₄⋊C₄	—	514	1	no	—
126	C₂×C₄⋊C₄	C₂×D₂₅₇	1	1	yes	characteristic
127	C₂×C₄²	—	257	1	no	—
128	C₂×C₄²	—	514	1	no	—
129	C₂×C₄²	—	514	1	no	—
130	#103	—	257	4	no	—
131	#103	Dic₂₅₇	1	4	yes	direct product factor
132	C₄×C₈	—	514	2	no	—
133	#95	—	514	1	no	—
134	C₂×Q₈	—	2	1	no	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	—	2	1	no	—
137	C₂×Q₈	C₄×D₂₅₇	1	2	yes	—
138	C₂×Q₈	C₄×D₂₅₇	1	2	yes	—
139	C₂×Q₈	C₂×Dic₂₅₇	1	2	yes	—
140	C₂²×C₄	—	257	1	no	—
141	C₂²×C₄	—	257	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	514	1	no	—
145	C₂²×C₄	—	514	1	no	—
146	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
147	C₂×C₈	—	257	2	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	514	1	no	—
151	C₂×C₈	—	514	1	no	—
152	C₂×C₈	C₂×Dic₂₅₇	1	2	yes	—
153	C₄⋊C₄	—	257	2	no	—
154	C₄⋊C₄	—	257	2	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	514	4	no	—
157	C₄⋊C₄	—	514	1	no	—
158	C₄⋊C₄	—	514	1	no	—
159	C₄⋊C₄	C₂×Dic₂₅₇	1	2	yes	—
160	C₄²	—	257	2	no	—
161	C₄²	—	257	2	no	—
162	C₄²	—	514	4	no	—
163	C₄²	—	1028	2	no	—
164	C₂³	D₄×D₂₅₇	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	Q₈	—	2	2	no	—
167	Q₈	—	2	2	no	—
168	Q₈	C₄×Dic₂₅₇	1	4	yes	complemented
169	C₂×C₄	—	257	2	no	—
170	C₂×C₄	—	257	2	no	—
171	C₂×C₄	—	257	2	no	—
172	C₂×C₄	—	257	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	514	1	no	—
178	C₂×C₄	—	514	1	no	—
179	C₂×C₄	—	514	2	no	—
180	C₂×C₄	—	514	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	1028	1	no	—
184	C₂×C₄	—	1028	1	no	—
185	C₂×C₄	—	1028	1	no	—
186	C₂×C₄	—	1028	1	no	—
187	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₂₅₇	1	1	yes	characteristic
190	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
191	C₂×C₄	#16861617	1	1	yes	characteristic
192	C₂×C₄	#16861617	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	—	514	2	no	—
195	C₂²	C₂²⋊C₄×D₂₅₇	1	1	yes	characteristic
196	C₂²	#22551552	1	1	yes	characteristic
197	C₂²	D₄×Dic₂₅₇	1	1	yes	characteristic
198	C₂²	SD₁₆×D₂₅₇	1	1	yes	characteristic
199	C₂²	Q₁₆×D₂₅₇	1	1	yes	characteristic
200	C₂²	#22551613	1	1	yes	characteristic
201	C₂²	#22551615	1	1	yes	characteristic
202	C₄	—	257	4	no	—
203	C₄	—	2	4	no	—
204	C₄	—	514	2	no	—
205	C₄	—	4	2	no	—
206	C₄	—	1028	2	no	—
207	C₄	—	1028	2	no	—
208	C₄	C₂×C₄×Dic₂₅₇	1	1	yes	characteristic
209	C₄	#22551548	1	1	yes	characteristic
210	C₄	#22551552	1	1	yes	characteristic
211	C₄	D₄×Dic₂₅₇	1	1	yes	characteristic
212	C₂	C₂²⋊C₄×Dic₂₅₇	1	1	yes	characteristic
213	C₂	#103×D₂₅₇	1	1	yes	characteristic
214	C₂	#30656732	1	1	yes	characteristic
215	C₂	#30656822	1	1	yes	characteristic
216	C₂	SD₁₆×Dic₂₅₇	1	1	yes	characteristic
217	C₂	#30656829	1	1	yes	characteristic
218	C₂	Q₁₆×Dic₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₂₅₇
commutator	C₄×C₂₅₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	C₂×#103×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂×D₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	C₂×#103	D₂₅₇
hypocenter	C₂₅₇	C₄×#103
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
10. Q₈×Dic₂₅₇	C₄	semidirect
17. #22551516	C₄	semidirect
23. #103×C₂₅₇	C₄	semidirect
71. Q₈×C₂₅₇	C₄²	semidirect
101. Dic₂₅₇	#103	direct (N × H)
109. C₂₅₇	C₄×#103	semidirect
131. #103	Dic₂₅₇	direct (N × H)
168. Q₈	C₄×Dic₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₂₅₇	2
C₂×#22551516	2
C₂×C₈×Dic₂₅₇	2
C₄⋊C₄×Dic₂₅₇	2
#30656500	2
#30656499	2
C₂×#103×C₂₅₇	2
C₄×#103	257

Subgroups — #103×Dic257 #45664142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #103×Dic₂₅₇ #45664142