Subgroups — D₄×Dic₄₇₂ #28745419

Order 15104.

664 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₇₂	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₂₃₆	C₂	1	2	yes	maximal
3	#21426349	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426187	C₂	1	2	yes	maximal, complemented
5	#21426147	C₂	1	2	yes	maximal, complemented
6	#21426032	C₂	1	4	yes	maximal, complemented
7	C₄×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₃₆	C₂²	1	4	yes	—
10	D₄×Dic₁₁₈	—	2	2	no	—
11	C₂×Dic₄₇₂	—	2	8	no	—
12	C₂×Dic₄₇₂	C₂²	1	2	yes	—
13	C₂×Dic₄₇₂	C₂²	1	4	yes	—
14	C₂×Dic₄₇₂	C₂²	1	2	yes	—
15	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
16	#16837782	C₂²	1	2	yes	—
17	#16837643	C₂²	1	4	yes	—
18	#16837613	—	2	2	no	—
19	#16837560	—	2	4	no	—
20	C₄×Dic₂₃₆	C₂²	1	2	yes	—
21	#16837228	C₂²	1	1	yes	characteristic
22	#16837207	C₂²	1	4	yes	—
23	#16837207	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
25	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₅₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₁₈	—	2	4	no	—
30	C₂×Dic₂₃₆	—	2	8	no	—
31	C₂×Dic₂₃₆	—	2	4	no	—
32	C₂×Dic₂₃₆	—	2	2	no	—
33	C₂×Dic₂₃₆	—	4	4	no	—
34	C₂×Dic₂₃₆	C₂³	1	4	yes	—
35	C₂×Dic₂₃₆	C₂³	1	2	yes	—
36	#14273727	—	2	2	no	—
37	#14273726	—	4	2	no	—
38	#14273706	—	2	4	no	—
39	#14273704	—	4	4	no	—
40	C₄×Dic₁₁₈	—	2	2	no	—
41	D₄×Dic₅₉	—	4	2	no	—
42	Dic₄₇₂	—	2	8	no	—
43	Dic₄₇₂	—	2	4	no	—
44	Dic₄₇₂	—	4	2	no	—
45	Dic₄₇₂	D₄	1	4	yes	direct product factor
46	#14273661	C₂³	1	2	yes	—
47	#14273652	—	2	4	no	—
48	#14273652	—	2	2	no	—
49	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
50	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₅₉	—	2	4	no	—
52	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
54	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
56	C₂²⋊C₈×C₅₉	D₄	1	2	yes	—
57	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
60	C₂²×Dic₅₉	—	4	4	no	—
61	C₂×Dic₁₁₈	—	2	2	no	—
62	C₂×Dic₁₁₈	—	2	4	no	—
63	C₂×Dic₁₁₈	—	4	8	no	—
64	C₂×Dic₁₁₈	—	4	4	no	—
65	C₂×Dic₁₁₈	—	8	4	no	—
66	C₂×C₈×C₅₉	—	2	2	no	—
67	C₂×C₈×C₅₉	—	2	2	no	—
68	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
72	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
73	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂²⋊Dic₅₉	—	4	4	no	—
75	C₄⋊Dic₅₉	—	2	2	no	—
76	C₄⋊Dic₅₉	—	4	2	no	—
77	C₄⋊Dic₅₉	—	4	2	no	—
78	Dic₅₉⋊C₄	—	4	2	no	—
79	Dic₅₉⋊C₄	—	4	4	no	—
80	C₄×Dic₅₉	—	4	2	no	—
81	Dic₂₃₆	—	2	4	no	—
82	Dic₂₃₆	—	2	2	no	—
83	Dic₂₃₆	—	2	4	no	—
84	Dic₂₃₆	—	4	8	no	—
85	Dic₂₃₆	—	8	2	no	—
86	Dic₂₃₆	C₂×D₄	1	4	yes	—
87	C₁₆×C₅₉	—	2	2	no	—
88	C₁₆×C₅₉	—	2	1	no	—
89	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
90	C₂³×C₅₉	C₂×D₈	1	2	yes	—
91	D₄×C₅₉	Q₃₂	1	4	yes	complemented
92	C₂×C₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₅₉	—	4	2	no	—
99	C₂×Dic₅₉	—	4	2	no	—
100	C₂×Dic₅₉	—	4	4	no	—
101	C₂×Dic₅₉	—	8	8	no	—
102	Dic₁₁₈	—	2	4	no	—
103	Dic₁₁₈	—	4	2	no	—
104	Dic₁₁₈	—	4	4	no	—
105	Dic₁₁₈	—	4	4	no	—
106	Dic₁₁₈	—	8	8	no	—
107	C₈×C₅₉	—	2	2	no	—
108	C₈×C₅₉	—	2	1	no	—
109	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
111	D₄×Q₃₂	—	59	1	no	maximal, Sylow
112	C₂²×C₅₉	—	2	2	no	—
113	C₂²×C₅₉	—	2	2	no	—
114	C₂²×C₅₉	C₂×Q₃₂	1	4	yes	complemented
115	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₅₉	—	2	1	no	—
117	C₄×C₅₉	—	2	2	no	—
118	C₄×C₅₉	C₂×Q₃₂	1	2	yes	complemented
119	C₄×C₅₉	D₄²	1	1	yes	characteristic
120	C₄×C₅₉	#576	1	1	yes	characteristic
121	Dic₅₉	—	4	4	no	—
122	Dic₅₉	—	8	2	no	—
123	Dic₅₉	—	8	4	no	—
124	C₂²×Q₃₂	—	59	2	no	—
125	D₄×Q₁₆	—	59	2	no	—
126	#2247	—	59	1	no	—
127	#2216	—	59	2	no	—
128	#2209	—	59	2	no	—
129	#2188	—	59	4	no	—
130	C₄×Q₃₂	—	59	1	no	—
131	D₄×C₁₆	D₅₉	1	1	yes	characteristic
132	C₂×C₅₉	—	2	4	no	—
133	C₂×C₅₉	D₄×D₈	1	1	yes	characteristic
134	C₂×C₅₉	C₂²×Q₃₂	1	1	yes	characteristic
135	C₂×C₅₉	#3417	1	1	yes	characteristic
136	C₂²×Q₁₆	—	59	4	no	—
137	D₄×Q₈	—	118	2	no	—
138	C₂×Q₃₂	—	59	1	no	—
139	C₂×Q₃₂	—	59	4	no	—
140	C₂×Q₃₂	—	59	2	no	—
141	C₂×Q₃₂	—	59	2	no	—
142	C₂×Q₃₂	—	118	8	no	—
143	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
144	#494	—	59	2	no	—
145	#467	—	59	4	no	—
146	#462	—	118	2	no	—
147	#451	—	118	4	no	—
148	C₄×Q₁₆	—	59	2	no	—
149	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
150	#366	—	59	1	no	—
151	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
152	#358	—	59	2	no	—
153	#358	—	59	4	no	—
154	C₂²⋊C₁₆	C₂×D₅₉	1	2	yes	—
155	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
156	C₅₉	D₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
157	C₂²×Q₈	—	118	4	no	—
158	C₂×Q₁₆	—	59	2	no	—
159	C₂×Q₁₆	—	59	4	no	—
160	C₂×Q₁₆	—	118	8	no	—
161	C₂×Q₁₆	—	118	2	no	—
162	C₂×Q₁₆	—	118	4	no	—
163	C₂×Q₁₆	—	236	4	no	—
164	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
165	C₄⋊Q₈	—	118	2	no	—
166	C₄⋊Q₈	—	236	2	no	—
167	C₂²⋊Q₈	—	118	4	no	—
168	C₂²⋊Q₈	—	236	4	no	—
169	C₄×Q₈	—	118	2	no	—
170	C₄×D₄	—	236	2	no	—
171	C₄×D₄	D₂₃₆	1	1	yes	characteristic
172	Q₃₂	—	59	4	no	—
173	Q₃₂	—	118	8	no	—
174	Q₃₂	—	118	4	no	—
175	Q₃₂	—	236	2	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
178	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
179	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
180	#107	—	59	2	no	—
181	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
182	#103	—	118	2	no	—
183	#103	—	118	4	no	—
184	C₂²⋊C₈	D₂₃₆	1	2	yes	—
185	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
186	C₂×Q₈	—	118	2	no	—
187	C₂×Q₈	—	118	4	no	—
188	C₂×Q₈	—	472	4	no	—
189	C₂×Q₈	—	236	8	no	—
190	C₂×Q₈	—	236	4	no	—
191	C₂×D₄	D₄₇₂	1	1	yes	characteristic
192	C₂²×C₄	—	236	4	no	—
193	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
194	Q₁₆	—	59	4	no	—
195	Q₁₆	—	118	4	no	—
196	Q₁₆	—	118	2	no	—
197	Q₁₆	—	118	4	no	—
198	Q₁₆	—	472	2	no	—
199	Q₁₆	—	236	8	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
203	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
204	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
205	C₄⋊C₄	—	118	2	no	—
206	C₄⋊C₄	—	236	2	no	—
207	C₄⋊C₄	—	236	2	no	—
208	C₄⋊C₄	—	236	4	no	—
209	C₄⋊C₄	—	236	2	no	—
210	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
211	C₂²⋊C₄	—	236	4	no	—
212	C₂²⋊C₄	D₄₇₂	1	2	yes	—
213	C₄²	—	236	2	no	—
214	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
215	C₁₆	—	2	2	no	—
216	C₁₆	—	2	1	no	—
217	C₁₆	D₄×D₅₉	1	2	yes	—
218	C₂³	C₂×D₄₇₂	1	2	yes	—
219	Q₈	—	118	4	no	—
220	Q₈	—	472	8	no	—
221	Q₈	—	236	4	no	—
222	Q₈	—	236	4	no	—
223	Q₈	—	236	2	no	—
224	D₄	Dic₄₇₂	1	4	yes	direct product factor
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	472	8	no	—
228	C₂×C₄	—	236	2	no	—
229	C₂×C₄	—	236	4	no	—
230	C₂×C₄	—	236	2	no	—
231	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
232	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
233	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	2	no	—
237	C₈	C₂×D₄×D₅₉	1	1	yes	characteristic
238	C₈	#14273773	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	C₂×Dic₄₇₂	1	4	yes	complemented
242	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	2	no	—
245	C₄	—	472	2	no	—
246	C₄	—	472	4	no	—
247	C₄	—	236	4	no	—
248	C₄	C₂×Dic₄₇₂	1	2	yes	complemented
249	C₄	D₄×D₂₃₆	1	1	yes	characteristic
250	C₄	#16838207	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	D₄×D₄₇₂	1	1	yes	characteristic
253	C₂	C₂²×Dic₄₇₂	1	1	yes	characteristic
254	C₂	#21437249	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	D₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂×D₄×C₅₉	D₈
hypercenter	D₄×C₁₆	D₅₉
hypocenter	C₅₉	D₄×Q₃₂
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₄₇₂	C₂	semidirect
3. #21426349	C₂	semidirect
4. #21426187	C₂	semidirect
5. #21426147	C₂	semidirect
6. #21426032	C₂	semidirect
7. C₄×Dic₄₇₂	C₂	semidirect
45. Dic₄₇₂	D₄	direct (N × H)
91. D₄×C₅₉	Q₃₂	semidirect
114. C₂²×C₅₉	C₂×Q₃₂	semidirect
118. C₄×C₅₉	C₂×Q₃₂	semidirect
156. C₅₉	D₄×Q₃₂	semidirect
224. D₄	Dic₄₇₂	direct (N × H)
241. C₂²	C₂×Dic₄₇₂	semidirect
248. C₄	C₂×Dic₄₇₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₇₂	2
D₄×Dic₂₃₆	2
#21426349	2
#21426187	2
#21426147	2
#21426032	2
C₄×Dic₄₇₂	2
D₄×C₁₆×C₅₉	2
D₄×Q₃₂	59

Subgroups — D4×Dic472 #28745419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Dic₄₇₂ #28745419