Subgroups — D₄×Dic₃₄₄ #25253327

Order 11008.

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×Dic₃₄₄	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₁₇₂	C₂	1	2	yes	maximal
3	#19136848	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136686	C₂	1	2	yes	maximal, complemented
5	#19136646	C₂	1	2	yes	maximal, complemented
6	#19136531	C₂	1	4	yes	maximal, complemented
7	C₄×Dic₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₂²	1	4	yes	—
10	D₄×Dic₈₆	—	2	2	no	—
11	C₂×Dic₃₄₄	—	2	8	no	—
12	C₂×Dic₃₄₄	C₂²	1	4	yes	—
13	C₂×Dic₃₄₄	C₂²	1	2	yes	—
14	C₂×Dic₃₄₄	C₂²	1	2	yes	—
15	C₂×Dic₃₄₄	C₂²	1	1	yes	characteristic
16	#14583279	C₂²	1	2	yes	—
17	#14583140	C₂²	1	4	yes	—
18	#14583110	—	2	2	no	—
19	#14583057	—	2	4	no	—
20	C₄×Dic₁₇₂	C₂²	1	2	yes	—
21	#14582725	C₂²	1	1	yes	characteristic
22	#14582704	C₂²	1	2	yes	—
23	#14582704	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
25	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₄₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₈₆	—	2	4	no	—
30	C₂×Dic₁₇₂	—	2	8	no	—
31	C₂×Dic₁₇₂	—	2	4	no	—
32	C₂×Dic₁₇₂	—	2	2	no	—
33	C₂×Dic₁₇₂	—	4	4	no	—
34	C₂×Dic₁₇₂	C₂³	1	4	yes	—
35	C₂×Dic₁₇₂	C₂³	1	2	yes	—
36	#13063325	—	2	2	no	—
37	#13063324	—	4	2	no	—
38	#13063304	—	2	4	no	—
39	#13063302	—	4	4	no	—
40	C₄×Dic₈₆	—	2	2	no	—
41	D₄×Dic₄₃	—	4	2	no	—
42	Dic₃₄₄	—	2	8	no	—
43	Dic₃₄₄	—	2	4	no	—
44	Dic₃₄₄	—	4	2	no	—
45	Dic₃₄₄	D₄	1	4	yes	direct product factor
46	#13063259	C₂³	1	2	yes	—
47	#13063250	—	2	2	no	—
48	#13063250	—	2	4	no	—
49	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
50	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₄₃	—	2	4	no	—
52	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
53	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
56	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
57	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
60	C₂²×Dic₄₃	—	4	4	no	—
61	C₂×Dic₈₆	—	2	2	no	—
62	C₂×Dic₈₆	—	2	4	no	—
63	C₂×Dic₈₆	—	4	8	no	—
64	C₂×Dic₈₆	—	4	4	no	—
65	C₂×Dic₈₆	—	8	4	no	—
66	C₂×C₈×C₄₃	—	2	2	no	—
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
72	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
73	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²⋊Dic₄₃	—	4	4	no	—
75	C₄⋊Dic₄₃	—	2	2	no	—
76	C₄⋊Dic₄₃	—	4	2	no	—
77	C₄⋊Dic₄₃	—	4	2	no	—
78	Dic₄₃⋊C₄	—	4	2	no	—
79	Dic₄₃⋊C₄	—	4	4	no	—
80	C₄×Dic₄₃	—	4	2	no	—
81	Dic₁₇₂	—	2	4	no	—
82	Dic₁₇₂	—	2	2	no	—
83	Dic₁₇₂	—	2	4	no	—
84	Dic₁₇₂	—	4	8	no	—
85	Dic₁₇₂	—	8	2	no	—
86	Dic₁₇₂	C₂×D₄	1	4	yes	—
87	C₁₆×C₄₃	—	2	2	no	—
88	C₁₆×C₄₃	—	2	1	no	—
89	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
90	C₂³×C₄₃	C₂×D₈	1	2	yes	—
91	D₄×C₄₃	Q₃₂	1	4	yes	complemented
92	C₂×C₄×C₄₃	—	2	2	no	—
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₄₃	—	4	2	no	—
99	C₂×Dic₄₃	—	4	2	no	—
100	C₂×Dic₄₃	—	4	4	no	—
101	C₂×Dic₄₃	—	8	8	no	—
102	Dic₈₆	—	2	4	no	—
103	Dic₈₆	—	4	2	no	—
104	Dic₈₆	—	4	4	no	—
105	Dic₈₆	—	4	4	no	—
106	Dic₈₆	—	8	8	no	—
107	C₈×C₄₃	—	2	2	no	—
108	C₈×C₄₃	—	2	1	no	—
109	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
111	D₄×Q₃₂	—	43	1	no	maximal, Sylow
112	C₂²×C₄₃	—	2	2	no	—
113	C₂²×C₄₃	—	2	2	no	—
114	C₂²×C₄₃	C₂×Q₃₂	1	4	yes	complemented
115	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₄₃	—	2	1	no	—
117	C₄×C₄₃	—	2	2	no	—
118	C₄×C₄₃	C₂×Q₃₂	1	2	yes	complemented
119	C₄×C₄₃	D₄²	1	1	yes	characteristic
120	C₄×C₄₃	#576	1	1	yes	characteristic
121	Dic₄₃	—	4	4	no	—
122	Dic₄₃	—	8	2	no	—
123	Dic₄₃	—	8	4	no	—
124	C₂²×Q₃₂	—	43	2	no	—
125	D₄×Q₁₆	—	43	2	no	—
126	#2247	—	43	1	no	—
127	#2216	—	43	2	no	—
128	#2209	—	43	2	no	—
129	#2188	—	43	4	no	—
130	C₄×Q₃₂	—	43	1	no	—
131	D₄×C₁₆	D₄₃	1	1	yes	characteristic
132	C₂×C₄₃	—	2	4	no	—
133	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
134	C₂×C₄₃	C₂²×Q₃₂	1	1	yes	characteristic
135	C₂×C₄₃	#3417	1	1	yes	characteristic
136	C₂²×Q₁₆	—	43	4	no	—
137	D₄×Q₈	—	86	2	no	—
138	C₂×Q₃₂	—	43	1	no	—
139	C₂×Q₃₂	—	43	4	no	—
140	C₂×Q₃₂	—	43	2	no	—
141	C₂×Q₃₂	—	43	2	no	—
142	C₂×Q₃₂	—	86	8	no	—
143	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
144	#494	—	43	2	no	—
145	#467	—	43	4	no	—
146	#462	—	86	2	no	—
147	#451	—	86	4	no	—
148	C₄×Q₁₆	—	43	2	no	—
149	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
150	#366	—	43	1	no	—
151	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
152	#358	—	43	2	no	—
153	#358	—	43	4	no	—
154	C₂²⋊C₁₆	C₂×D₄₃	1	2	yes	—
155	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
156	C₄₃	D₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
157	C₂²×Q₈	—	86	4	no	—
158	C₂×Q₁₆	—	43	2	no	—
159	C₂×Q₁₆	—	43	4	no	—
160	C₂×Q₁₆	—	86	8	no	—
161	C₂×Q₁₆	—	86	2	no	—
162	C₂×Q₁₆	—	86	4	no	—
163	C₂×Q₁₆	—	172	4	no	—
164	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
165	C₄⋊Q₈	—	86	2	no	—
166	C₄⋊Q₈	—	172	2	no	—
167	C₂²⋊Q₈	—	86	4	no	—
168	C₂²⋊Q₈	—	172	4	no	—
169	C₄×Q₈	—	86	2	no	—
170	C₄×D₄	—	172	2	no	—
171	C₄×D₄	D₁₇₂	1	1	yes	characteristic
172	Q₃₂	—	43	4	no	—
173	Q₃₂	—	86	8	no	—
174	Q₃₂	—	86	4	no	—
175	Q₃₂	—	172	2	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
178	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
179	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
180	#107	—	43	2	no	—
181	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
182	#103	—	86	2	no	—
183	#103	—	86	4	no	—
184	C₂²⋊C₈	D₁₇₂	1	2	yes	—
185	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
186	C₂×Q₈	—	86	2	no	—
187	C₂×Q₈	—	86	4	no	—
188	C₂×Q₈	—	344	4	no	—
189	C₂×Q₈	—	172	8	no	—
190	C₂×Q₈	—	172	4	no	—
191	C₂×D₄	D₃₄₄	1	1	yes	characteristic
192	C₂²×C₄	—	172	4	no	—
193	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
194	Q₁₆	—	43	4	no	—
195	Q₁₆	—	86	4	no	—
196	Q₁₆	—	86	2	no	—
197	Q₁₆	—	86	4	no	—
198	Q₁₆	—	344	2	no	—
199	Q₁₆	—	172	8	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
203	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
204	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
205	C₄⋊C₄	—	86	2	no	—
206	C₄⋊C₄	—	172	2	no	—
207	C₄⋊C₄	—	172	2	no	—
208	C₄⋊C₄	—	172	4	no	—
209	C₄⋊C₄	—	172	2	no	—
210	C₄⋊C₄	D₃₄₄	1	1	yes	characteristic
211	C₂²⋊C₄	—	172	4	no	—
212	C₂²⋊C₄	D₃₄₄	1	2	yes	—
213	C₄²	—	172	2	no	—
214	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
215	C₁₆	—	2	2	no	—
216	C₁₆	—	2	1	no	—
217	C₁₆	D₄×D₄₃	1	2	yes	—
218	C₂³	C₂×D₃₄₄	1	2	yes	—
219	Q₈	—	86	4	no	—
220	Q₈	—	344	8	no	—
221	Q₈	—	172	4	no	—
222	Q₈	—	172	4	no	—
223	Q₈	—	172	2	no	—
224	D₄	Dic₃₄₄	1	4	yes	direct product factor
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	344	8	no	—
228	C₂×C₄	—	172	2	no	—
229	C₂×C₄	—	172	4	no	—
230	C₂×C₄	—	172	2	no	—
231	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
232	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
233	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	2	no	—
237	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
238	C₈	#13063371	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	C₂×Dic₃₄₄	1	4	yes	complemented
242	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	1	no	—
245	C₄	—	344	4	no	—
246	C₄	—	344	2	no	—
247	C₄	—	172	4	no	—
248	C₄	C₂×Dic₃₄₄	1	2	yes	complemented
249	C₄	D₄×D₁₇₂	1	1	yes	characteristic
250	C₄	#14583704	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	D₄×D₃₄₄	1	1	yes	characteristic
253	C₂	C₂²×Dic₃₄₄	1	1	yes	characteristic
254	C₂	#19147748	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	D₄×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×D₄×C₄₃	D₈
hypercenter	D₄×C₁₆	D₄₃
hypocenter	C₄₃	D₄×Q₃₂
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
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×Dic₃₄₄	C₂	semidirect
3. #19136848	C₂	semidirect
4. #19136686	C₂	semidirect
5. #19136646	C₂	semidirect
6. #19136531	C₂	semidirect
7. C₄×Dic₃₄₄	C₂	semidirect
45. Dic₃₄₄	D₄	direct (N × H)
91. D₄×C₄₃	Q₃₂	semidirect
114. C₂²×C₄₃	C₂×Q₃₂	semidirect
118. C₄×C₄₃	C₂×Q₃₂	semidirect
156. C₄₃	D₄×Q₃₂	semidirect
224. D₄	Dic₃₄₄	direct (N × H)
241. C₂²	C₂×Dic₃₄₄	semidirect
248. C₄	C₂×Dic₃₄₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₃₄₄	2
D₄×Dic₁₇₂	2
#19136848	2
#19136686	2
#19136646	2
#19136531	2
C₄×Dic₃₄₄	2
D₄×C₁₆×C₄₃	2
D₄×Q₃₂	43

Subgroups — D4×Dic344 #25253327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Dic₃₄₄ #25253327