Subgroups — D₄×#14268655 #28407277

Order 14464.

616 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14268655	C₂	1	2	yes	maximal, complemented
2	#20299626	C₂	1	2	yes	maximal, complemented
3	#20299625	C₂	1	4	yes	maximal, complemented
4	D₄×#11744753	C₂	1	2	yes	maximal, complemented
5	#20299599	C₂	1	1	yes	maximal, characteristic, complemented
6	#20299566	C₂	1	2	yes	maximal, complemented
7	C₄×#14268655	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₄×D₁₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#14268655	—	2	8	no	—
10	C₂×#14268655	C₂²	1	1	yes	characteristic
11	C₂×#14268655	C₂²	1	2	yes	—
12	C₂×#14268655	C₂²	1	1	yes	characteristic
13	C₂×#14268655	C₂²	1	4	yes	complemented
14	C₂²×#11744753	C₂²	1	4	yes	complemented
15	#16813689	C₂²	1	2	yes	complemented
16	C₄⋊#11744753	C₂²	1	2	yes	complemented
17	#16813679	C₂²	1	4	yes	complemented
18	C₂²⋊#11744753	C₂²	1	4	yes	complemented
19	C₄×#11744753	C₂²	1	2	yes	complemented
20	C₄×#11744753	C₂²	1	2	yes	complemented
21	C₂×D₄×D₁₁₃	C₄	1	1	yes	characteristic
22	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
23	C₄×D₄₅₂	C₄	1	1	yes	characteristic
24	D₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
25	#16813584	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊D₁₁₃	C₄	1	2	yes	—
27	#16813582	C₂²	1	2	yes	—
28	C₄⋊C₄×D₁₁₃	C₄	1	1	yes	characteristic
29	C₂²⋊C₄×D₁₁₃	C₄	1	2	yes	—
30	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₁₁₃	C₄	1	1	yes	characteristic
32	#14268655	—	2	4	no	—
33	#14268655	—	2	2	no	—
34	#14268655	—	2	8	no	—
35	#14268655	D₄	1	4	yes	direct product factor
36	C₂×#11744753	—	2	8	no	—
37	C₂×#11744753	C₂³	1	2	yes	—
38	C₂×#11744753	C₂³	1	4	yes	—
39	C₂×#11744753	C₂³	1	2	yes	—
40	C₂³×D₁₁₃	C₂×C₄	1	2	yes	—
41	C₂×D₄₅₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	2	yes	—
43	D₄×D₁₁₃	C₈	1	8	yes	complemented
44	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
45	C₂×C₄×D₁₁₃	—	2	4	no	—
46	C₂×C₄×D₁₁₃	—	2	4	no	—
47	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
52	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
53	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
54	#14268626	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₁₃	C₂×C₄	1	2	yes	—
56	#14268624	C₂×C₄	1	2	yes	—
57	#14268624	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
60	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
63	C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
65	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
66	C₂³×C₁₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×D₁₁₃	—	2	4	no	—
68	C₂²×D₁₁₃	—	2	4	no	—
69	C₂²×D₁₁₃	C₂×C₈	1	8	yes	complemented
70	C₂²×D₁₁₃	C₂²×C₄	1	2	yes	—
71	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
72	D₄×C₁₁₃	C₂×C₈	1	4	yes	complemented
73	C₂×C₄×C₁₁₃	—	2	4	no	—
74	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²⋊D₁₁₃	C₂×C₈	1	8	yes	complemented
79	C₂×Dic₁₁₃	—	2	2	no	—
80	C₂×Dic₁₁₃	—	2	2	no	—
81	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
85	D₄₅₂	C₂×C₈	1	4	yes	complemented
86	C₄×D₁₁₃	—	2	4	no	—
87	C₄×D₁₁₃	—	2	2	no	—
88	C₄×D₁₁₃	—	2	4	no	—
89	C₄×D₁₁₃	—	2	2	no	—
90	C₄×D₁₁₃	—	2	2	no	—
91	C₄×D₁₁₃	C₂×C₈	1	4	yes	complemented
92	C₄×D₁₁₃	C₂×D₄	1	2	yes	—
93	C₄×D₁₁₃	#41	1	2	yes	—
94	#11744753	—	2	4	no	—
95	#11744753	—	2	2	no	—
96	#11744753	C₂×D₄	1	4	yes	complemented
97	C₂²×C₁₁₃	—	2	2	no	—
98	C₂²×C₁₁₃	—	2	2	no	—
99	C₂²×C₁₁₃	C₂²×C₈	1	4	yes	complemented
100	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₁₁₃	—	2	2	no	—
102	C₂×D₁₁₃	—	2	2	no	—
103	C₂×D₁₁₃	—	2	16	no	—
104	C₂×D₁₁₃	C₄×D₄	1	2	yes	—
105	C₂×D₁₁₃	C₂²×C₈	1	2	yes	—
106	C₂×D₁₁₃	C₂²×C₈	1	4	yes	complemented
107	C₂×D₁₁₃	#131	1	2	yes	—
108	C₄×C₁₁₃	—	2	2	no	—
109	C₄×C₁₁₃	—	2	1	no	—
110	C₄×C₁₁₃	C₄×D₄	1	2	yes	—
111	C₄×C₁₁₃	C₂²×C₈	1	2	yes	complemented
112	Dic₁₁₃	—	2	2	no	—
113	Dic₁₁₃	—	2	1	no	—
114	Dic₁₁₃	C₂²×C₈	1	2	yes	complemented
115	Dic₁₁₃	C₂²×D₄	1	1	yes	characteristic
116	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₁₁₃	—	2	4	no	—
118	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₁₁₃	C₂³×C₈	1	1	yes	characteristic
120	C₂×C₁₁₃	C₂×#131	1	1	yes	characteristic
121	D₁₁₃	—	2	4	no	—
122	D₁₁₃	C₈×D₄	1	4	yes	complemented
123	C₂×C₈×D₄	—	113	1	no	maximal, Sylow
124	C₁₁₃	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₈	—	113	2	no	—
126	C₂×C₄×D₄	—	113	1	no	—
127	C₈×D₄	—	113	2	no	—
128	C₈×D₄	—	113	4	no	—
129	C₈×D₄	—	113	2	no	—
130	C₂×C₄⋊C₈	—	113	1	no	—
131	C₂×C₂²⋊C₈	—	113	2	no	—
132	C₂×C₄×C₈	—	113	1	no	—
133	C₂²×D₄	—	113	1	no	—
134	C₂³×C₄	—	113	2	no	—
135	C₂²×C₈	—	113	1	no	—
136	C₂²×C₈	—	113	4	no	—
137	C₂²×C₈	—	113	2	no	—
138	C₂²×C₈	—	113	4	no	—
139	C₂²×C₈	—	113	1	no	—
140	C₂²×C₈	—	226	8	no	—
141	C₄×D₄	—	113	1	no	—
142	C₄×D₄	—	113	2	no	—
143	C₄×D₄	—	113	1	no	—
144	C₄×D₄	—	113	2	no	—
145	C₄×D₄	—	113	1	no	—
146	C₄×D₄	#90673	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	113	1	no	—
148	C₂×C₂²⋊C₄	—	113	2	no	—
149	C₂×C₄²	—	113	1	no	—
150	C₄⋊C₈	—	113	2	no	—
151	C₄⋊C₈	—	113	2	no	—
152	C₂²⋊C₈	—	113	4	no	—
153	C₂²⋊C₈	—	113	4	no	—
154	C₄×C₈	—	113	2	no	—
155	C₄×C₈	—	113	2	no	—
156	C₂⁴	—	113	2	no	—
157	C₂×D₄	—	113	2	no	—
158	C₂×D₄	—	113	8	no	—
159	C₂×D₄	—	113	1	no	—
160	C₂×D₄	C₂×#90673	1	1	yes	characteristic
161	C₂²×C₄	—	113	1	no	—
162	C₂²×C₄	—	113	1	no	—
163	C₂²×C₄	—	113	2	no	—
164	C₂²×C₄	—	113	2	no	—
165	C₂²×C₄	—	113	1	no	—
166	C₂²×C₄	—	113	2	no	—
167	C₂²×C₄	—	113	2	no	—
168	C₂²×C₄	—	226	4	no	—
169	C₂²×C₄	—	226	4	no	—
170	C₂²×C₄	C₂×#90673	1	2	yes	—
171	C₂×C₈	—	113	2	no	—
172	C₂×C₈	—	113	4	no	—
173	C₂×C₈	—	113	2	no	—
174	C₂×C₈	—	113	4	no	—
175	C₂×C₈	—	226	8	no	—
176	C₂×C₈	—	226	8	no	—
177	C₂×C₈	—	226	4	no	—
178	C₂×C₈	—	226	2	no	—
179	C₄⋊C₄	—	113	2	no	—
180	C₄⋊C₄	—	113	1	no	—
181	C₄⋊C₄	C₂×#90673	1	1	yes	characteristic
182	C₂²⋊C₄	—	113	2	no	—
183	C₂²⋊C₄	—	113	2	no	—
184	C₂²⋊C₄	—	113	2	no	—
185	C₂²⋊C₄	C₂×#90673	1	2	yes	—
186	C₄²	—	113	1	no	—
187	C₄²	—	113	1	no	—
188	C₄²	—	113	1	no	—
189	C₄²	C₂×#90673	1	1	yes	characteristic
190	C₂³	—	113	1	no	—
191	C₂³	—	113	8	no	—
192	C₂³	—	113	2	no	—
193	C₂³	—	226	4	no	—
194	C₂³	—	226	4	no	—
195	C₂³	C₂²×#90673	1	2	yes	—
196	D₄	—	113	8	no	—
197	D₄	—	113	4	no	—
198	D₄	#14268655	1	4	yes	direct product factor
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	113	1	no	—
201	C₂×C₄	—	113	1	no	—
202	C₂×C₄	—	113	2	no	—
203	C₂×C₄	—	113	2	no	—
204	C₂×C₄	—	113	4	no	—
205	C₂×C₄	—	113	1	no	—
206	C₂×C₄	—	113	2	no	—
207	C₂×C₄	—	226	2	no	—
208	C₂×C₄	—	226	2	no	—
209	C₂×C₄	—	226	4	no	—
210	C₂×C₄	—	226	2	no	—
211	C₂×C₄	—	226	2	no	—
212	C₂×C₄	—	226	2	no	—
213	C₂×C₄	—	226	4	no	—
214	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
215	C₂×C₄	C₂²×#90673	1	2	yes	—
216	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
217	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
218	C₈	—	113	4	no	—
219	C₈	—	226	4	no	—
220	C₈	—	226	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	113	2	no	—
224	C₂²	—	113	2	no	—
225	C₂²	—	113	2	no	—
226	C₂²	—	113	4	no	—
227	C₂²	—	226	16	no	—
228	C₂²	—	226	2	no	—
229	C₂²	—	226	2	no	—
230	C₂²	C₂×#14268655	1	4	yes	complemented
231	C₂²	C₂³×#90673	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	—	113	1	no	—
235	C₄	—	113	1	no	—
236	C₄	—	113	2	no	—
237	C₄	—	226	2	no	—
238	C₄	—	226	1	no	—
239	C₄	D₄×#90673	1	2	yes	—
240	C₄	C₂×#14268655	1	2	yes	complemented
241	C₂	—	2	4	no	—
242	C₂	—	113	4	no	—
243	C₂	—	226	4	no	—
244	C₂	C₂×D₄×#90673	1	1	yes	characteristic
245	C₂	C₂²×#14268655	1	1	yes	characteristic
246	C₂	D₁₁₃⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#90673
commutator	C₂×C₁₁₃	C₂³×C₈
Frattini	C₂²	C₂³×#90673
Fitting	C₄×D₄×C₁₁₃	C₄
socle	C₂²×C₁₁₃	C₂³×C₄
prime radical	C₂×D₄×D₁₁₃	C₄
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	C₄×D₄	#90673
hypocenter	C₁₁₃	C₂×C₈×D₄
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14268655	C₂	semidirect
2. #20299626	C₂	semidirect
3. #20299625	C₂	semidirect
4. D₄×#11744753	C₂	semidirect
5. #20299599	C₂	semidirect
6. #20299566	C₂	semidirect
7. C₄×#14268655	C₂	semidirect
13. C₂×#14268655	C₂²	semidirect
14. C₂²×#11744753	C₂²	semidirect
15. #16813689	C₂²	semidirect
16. C₄⋊#11744753	C₂²	semidirect
17. #16813679	C₂²	semidirect
18. C₂²⋊#11744753	C₂²	semidirect
19. C₄×#11744753	C₂²	semidirect
20. C₄×#11744753	C₂²	semidirect
35. #14268655	D₄	direct (N × H)
43. D₄×D₁₁₃	C₈	semidirect
69. C₂²×D₁₁₃	C₂×C₈	semidirect
72. D₄×C₁₁₃	C₂×C₈	semidirect
78. C₂²⋊D₁₁₃	C₂×C₈	semidirect
85. D₄₅₂	C₂×C₈	semidirect
91. C₄×D₁₁₃	C₂×C₈	semidirect
96. #11744753	C₂×D₄	semidirect
99. C₂²×C₁₁₃	C₂²×C₈	semidirect
106. C₂×D₁₁₃	C₂²×C₈	semidirect
111. C₄×C₁₁₃	C₂²×C₈	semidirect
114. Dic₁₁₃	C₂²×C₈	semidirect
122. D₁₁₃	C₈×D₄	semidirect
124. C₁₁₃	C₂×C₈×D₄	semidirect
198. D₄	#14268655	direct (N × H)
230. C₂²	C₂×#14268655	semidirect
240. C₄	C₂×#14268655	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14268655	2
#20299626	2
#20299625	2
D₄×#11744753	2
#20299599	2
#20299566	2
C₄×#14268655	2
C₄×D₄×D₁₁₃	2
C₂×C₈×D₄	113

Subgroups — D4×#14268655 #28407277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#14268655 #28407277