Subgroups — D₄×C₄⋊Dic₂₅₇ #45667222

Order 32896.

776 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Dic₂₅₇	C₂	1	2	yes	maximal
2	C₂²×C₄⋊Dic₂₅₇	C₂	1	2	yes	maximal, complemented
3	#30656636	C₂	1	2	yes	maximal, complemented
4	#30656612	C₂	1	4	yes	maximal, complemented
5	#30656607	C₂	1	1	yes	maximal, characteristic, complemented
6	#30656597	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₂₅₇	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₂₅₇	C₂²	1	4	yes	—
10	D₄×Dic₂₅₇	—	2	8	no	—
11	C₂×C₄⋊Dic₂₅₇	—	2	8	no	—
12	C₂×C₄⋊Dic₂₅₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₂₅₇	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₂₅₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₂₅₇	C₂²	1	4	yes	—
19	C₂×C₄×Dic₂₅₇	C₂²	1	2	yes	—
20	#22551499	C₂²	1	2	yes	—
21	#22551499	C₂²	1	4	yes	—
22	C₂²×D₄×C₂₅₇	C₂²	1	1	yes	characteristic
23	C₂³×C₄×C₂₅₇	C₂²	1	2	yes	—
24	C₄×D₄×C₂₅₇	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₄×C₂₅₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₂₅₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
28	C₂²×Dic₂₅₇	—	2	8	no	—
29	C₂²×Dic₂₅₇	—	2	8	no	—
30	C₂²×Dic₂₅₇	—	4	4	no	—
31	C₂²×Dic₂₅₇	C₂³	1	4	yes	—
32	C₂²×Dic₂₅₇	C₂³	1	2	yes	—
33	C₂²×Dic₂₅₇	C₂³	1	2	yes	—
34	C₄⋊Dic₂₅₇	—	2	8	no	—
35	C₄⋊Dic₂₅₇	—	2	4	no	—
36	C₄⋊Dic₂₅₇	—	2	4	no	—
37	C₄⋊Dic₂₅₇	—	2	2	no	—
38	C₄⋊Dic₂₅₇	—	4	2	no	—
39	C₄⋊Dic₂₅₇	D₄	1	4	yes	direct product factor
40	C₂²⋊Dic₂₅₇	—	2	8	no	—
41	C₄×Dic₂₅₇	—	2	4	no	—
42	C₂⁴×C₂₅₇	C₂³	1	2	yes	—
43	C₂×D₄×C₂₅₇	C₂×C₄	1	4	yes	—
44	C₂×D₄×C₂₅₇	D₄	1	4	yes	—
45	C₂×D₄×C₂₅₇	Q₈	1	4	yes	—
46	C₂²×C₄×C₂₅₇	—	2	4	no	—
47	C₂²×C₄×C₂₅₇	—	2	4	no	—
48	C₂²×C₄×C₂₅₇	C₂×C₄	1	8	yes	complemented
49	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₅₇	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₂₅₇	C₂×C₄	1	4	yes	complemented
54	C₂²⋊C₄×C₂₅₇	C₂×C₄	1	8	yes	complemented
55	C₄²×C₂₅₇	C₂×C₄	1	4	yes	complemented
56	C₂×Dic₂₅₇	—	2	2	no	—
57	C₂×Dic₂₅₇	—	2	2	no	—
58	C₂×Dic₂₅₇	—	2	2	no	—
59	C₂×Dic₂₅₇	—	2	4	no	—
60	C₂×Dic₂₅₇	—	2	4	no	—
61	C₂×Dic₂₅₇	—	2	2	no	—
62	C₂×Dic₂₅₇	—	2	4	no	—
63	C₂×Dic₂₅₇	—	2	2	no	—
64	C₂×Dic₂₅₇	—	4	16	no	—
65	C₂×Dic₂₅₇	C₂×D₄	1	4	yes	—
66	C₂³×C₂₅₇	—	2	2	no	—
67	C₂³×C₂₅₇	—	2	2	no	—
68	C₂³×C₂₅₇	—	2	2	no	—
69	C₂³×C₂₅₇	—	2	2	no	—
70	C₂³×C₂₅₇	C₂²×C₄	1	4	yes	—
71	C₂³×C₂₅₇	C₂×D₄	1	4	yes	—
72	C₂³×C₂₅₇	C₂×Q₈	1	4	yes	—
73	C₂³×C₂₅₇	C₂⁴	1	1	yes	characteristic
74	D₄×C₂₅₇	C₄⋊C₄	1	16	yes	complemented
75	C₂×C₄×C₂₅₇	—	2	2	no	—
76	C₂×C₄×C₂₅₇	—	2	2	no	—
77	C₂×C₄×C₂₅₇	—	2	1	no	—
78	C₂×C₄×C₂₅₇	—	2	1	no	—
79	C₂×C₄×C₂₅₇	—	2	16	no	—
80	C₂×C₄×C₂₅₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₂₅₇	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₂₅₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₅₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₂₅₇	C₂×Q₈	1	2	yes	—
87	C₂×C₄×C₂₅₇	#41	1	2	yes	—
88	C₂²×C₂₅₇	—	2	4	no	—
89	C₂²×C₂₅₇	—	2	4	no	—
90	C₂²×C₂₅₇	—	2	4	no	—
91	C₂²×C₂₅₇	—	2	4	no	—
92	C₂²×C₂₅₇	—	2	4	no	—
93	C₂²×C₂₅₇	—	2	4	no	—
94	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	8	yes	complemented
95	C₂²×C₂₅₇	C₂³×C₄	1	1	yes	characteristic
96	C₂²×C₂₅₇	C₂²×D₄	1	1	yes	characteristic
97	C₂²×C₂₅₇	C₂²×D₄	1	1	yes	characteristic
98	C₂²×C₂₅₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×C₂₅₇	C₂×#41	1	1	yes	characteristic
100	C₂²×C₂₅₇	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂²×C₂₅₇	2₋¹⁺⁴	1	1	yes	characteristic
102	C₄×C₂₅₇	—	2	4	no	—
103	C₄×C₂₅₇	—	2	2	no	—
104	C₄×C₂₅₇	C₂×C₄⋊C₄	1	4	yes	complemented
105	C₄×C₂₅₇	C₄×D₄	1	4	yes	complemented
106	Dic₂₅₇	—	2	4	no	—
107	Dic₂₅₇	—	4	4	no	—
108	Dic₂₅₇	—	4	2	no	—
109	C₂×C₂₅₇	—	2	8	no	—
110	C₂×C₂₅₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₂₅₇	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂×C₂₅₇	#520	1	1	yes	characteristic
113	C₂×C₂₅₇	D₄²	1	1	yes	characteristic
114	C₂×C₂₅₇	#547	1	1	yes	characteristic
115	C₂×C₂₅₇	D₄×Q₈	1	1	yes	characteristic
116	C₂×C₂₅₇	#554	1	1	yes	characteristic
117	C₂₅₇	D₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
118	D₄×C₄⋊C₄	—	257	1	no	maximal, Sylow
119	C₂×C₄×D₄	—	257	2	no	—
120	C₂×C₄×D₄	D₂₅₇	1	1	yes	characteristic
121	C₂²×C₄⋊C₄	—	257	2	no	—
122	#389	—	257	2	no	—
123	#385	—	257	4	no	—
124	#384	—	257	1	no	—
125	#380	—	257	2	no	—
126	C₄×C₄⋊C₄	—	257	1	no	—
127	C₂²×D₄	C₂×D₂₅₇	1	1	yes	characteristic
128	C₂³×C₄	—	257	4	no	—
129	C₂³×C₄	C₂×D₂₅₇	1	2	yes	—
130	C₄×D₄	—	514	8	no	—
131	C₄×D₄	Dic₂₅₇	1	8	yes	complemented
132	C₂×C₄⋊C₄	—	257	1	no	—
133	C₂×C₄⋊C₄	—	257	2	no	—
134	C₂×C₄⋊C₄	—	257	4	no	—
135	C₂×C₄⋊C₄	—	257	2	no	—
136	C₂×C₄⋊C₄	—	257	1	no	—
137	C₂×C₄⋊C₄	—	257	2	no	—
138	C₂×C₄⋊C₄	—	514	8	no	—
139	C₂×C₄⋊C₄	C₂×D₂₅₇	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	—	257	4	no	—
141	C₂×C₂²⋊C₄	C₂×D₂₅₇	1	2	yes	—
142	C₂×C₄²	—	257	2	no	—
143	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
144	#95	—	257	2	no	—
145	#95	—	257	4	no	—
146	C₂⁴	C₂²×D₂₅₇	1	2	yes	—
147	C₂×D₄	C₂×Dic₂₅₇	1	4	yes	—
148	C₂×D₄	D₁₀₂₈	1	4	yes	—
149	C₂×D₄	Dic₅₁₄	1	4	yes	—
150	C₂²×C₄	—	257	2	no	—
151	C₂²×C₄	—	257	2	no	—
152	C₂²×C₄	—	257	4	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	514	8	no	—
156	C₂²×C₄	—	514	8	no	—
157	C₂²×C₄	—	1028	4	no	—
158	C₂²×C₄	C₂×Dic₂₅₇	1	8	yes	complemented
159	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₂₅₇	1	2	yes	—
161	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
163	C₄⋊C₄	—	257	4	no	—
164	C₄⋊C₄	—	514	8	no	—
165	C₄⋊C₄	—	514	4	no	—
166	C₄⋊C₄	—	514	4	no	—
167	C₄⋊C₄	—	514	2	no	—
168	C₄⋊C₄	—	1028	2	no	—
169	C₄⋊C₄	C₂×Dic₂₅₇	1	4	yes	complemented
170	C₂²⋊C₄	—	514	8	no	—
171	C₂²⋊C₄	C₂×Dic₂₅₇	1	8	yes	complemented
172	C₄²	—	514	4	no	—
173	C₄²	C₂×Dic₂₅₇	1	4	yes	complemented
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	C₂²×Dic₂₅₇	1	4	yes	—
179	C₂³	C₂×D₁₀₂₈	1	4	yes	—
180	C₂³	C₂×Dic₅₁₄	1	4	yes	—
181	C₂³	C₂³×D₂₅₇	1	1	yes	characteristic
182	D₄	C₄⋊Dic₂₅₇	1	16	yes	direct product factor
183	C₂×C₄	—	257	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	16	no	—
189	C₂×C₄	—	514	2	no	—
190	C₂×C₄	—	514	2	no	—
191	C₂×C₄	—	514	4	no	—
192	C₂×C₄	—	514	2	no	—
193	C₂×C₄	—	514	4	no	—
194	C₂×C₄	—	514	2	no	—
195	C₂×C₄	—	514	4	no	—
196	C₂×C₄	—	514	2	no	—
197	C₂×C₄	—	1028	16	no	—
198	C₂×C₄	C₂²×Dic₂₅₇	1	2	yes	—
199	C₂×C₄	C₂²×Dic₂₅₇	1	4	yes	—
200	C₂×C₄	C₂²×Dic₂₅₇	1	2	yes	—
201	C₂×C₄	C₂²×Dic₂₅₇	1	2	yes	—
202	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
203	C₂×C₄	C₂×D₁₀₂₈	1	2	yes	—
204	C₂×C₄	C₂×Dic₅₁₄	1	2	yes	—
205	C₂×C₄	#16861617	1	2	yes	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	C₂×C₄⋊Dic₂₅₇	1	8	yes	complemented
213	C₂²	C₂³×Dic₂₅₇	1	1	yes	characteristic
214	C₂²	C₂×D₄×D₂₅₇	1	1	yes	characteristic
215	C₂²	C₂²×D₁₀₂₈	1	1	yes	characteristic
216	C₂²	C₂²×Dic₅₁₄	1	1	yes	characteristic
217	C₂²	C₂×#16861617	1	1	yes	characteristic
218	C₂²	#22551638	1	1	yes	characteristic
219	C₂²	#22551639	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	2	2	no	—
222	C₄	—	514	4	no	—
223	C₄	—	1028	4	no	—
224	C₄	—	1028	2	no	—
225	C₄	C₂×C₄⋊Dic₂₅₇	1	4	yes	complemented
226	C₄	D₄×Dic₂₅₇	1	4	yes	complemented
227	C₂	—	2	8	no	—
228	C₂	C₂²×C₄⋊Dic₂₅₇	1	1	yes	characteristic
229	C₂	C₂×D₄×Dic₂₅₇	1	1	yes	characteristic
230	C₂	#30657220	1	1	yes	characteristic
231	C₂	D₄×D₁₀₂₈	1	1	yes	characteristic
232	C₂	#30657391	1	1	yes	characteristic
233	C₂	D₄×Dic₅₁₄	1	1	yes	characteristic
234	C₂	#30657444	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₇
commutator	C₂²×C₂₅₇	C₂³×C₄
Fitting	C₂×C₄×D₄×C₂₅₇	C₂
socle, cosocle	C₂³×C₂₅₇	C₂⁴
prime radical	C₂²×D₄×C₂₅₇	C₂²
hypercenter	C₂×C₄×D₄	D₂₅₇
hypocenter	C₂₅₇	D₄×C₄⋊C₄
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊Dic₂₅₇	C₂	semidirect
3. #30656636	C₂	semidirect
4. #30656612	C₂	semidirect
5. #30656607	C₂	semidirect
6. #30656597	C₂	semidirect
7. C₄×C₄⋊Dic₂₅₇	C₂	semidirect
24. C₄×D₄×C₂₅₇	C₄	semidirect
39. C₄⋊Dic₂₅₇	D₄	direct (N × H)
48. C₂²×C₄×C₂₅₇	C₂×C₄	semidirect
53. C₄⋊C₄×C₂₅₇	C₂×C₄	semidirect
54. C₂²⋊C₄×C₂₅₇	C₂×C₄	semidirect
55. C₄²×C₂₅₇	C₂×C₄	semidirect
74. D₄×C₂₅₇	C₄⋊C₄	semidirect
94. C₂²×C₂₅₇	C₂×C₄⋊C₄	semidirect
104. C₄×C₂₅₇	C₂×C₄⋊C₄	semidirect
105. C₄×C₂₅₇	C₄×D₄	semidirect
117. C₂₅₇	D₄×C₄⋊C₄	semidirect
131. C₄×D₄	Dic₂₅₇	semidirect
158. C₂²×C₄	C₂×Dic₂₅₇	semidirect
169. C₄⋊C₄	C₂×Dic₂₅₇	semidirect
171. C₂²⋊C₄	C₂×Dic₂₅₇	semidirect
173. C₄²	C₂×Dic₂₅₇	semidirect
182. D₄	C₄⋊Dic₂₅₇	direct (N × H)
212. C₂²	C₂×C₄⋊Dic₂₅₇	semidirect
225. C₄	C₂×C₄⋊Dic₂₅₇	semidirect
226. C₄	D₄×Dic₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₂₅₇	2
C₂²×C₄⋊Dic₂₅₇	2
#30656636	2
#30656612	2
#30656607	2
#30656597	2
C₄×C₄⋊Dic₂₅₇	2
C₂×C₄×D₄×C₂₅₇	2
D₄×C₄⋊C₄	257

Subgroups — D4×C4⋊Dic257 #45667222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₄⋊Dic₂₅₇ #45667222