Subgroups — D₄×Dic₆₃₂ #34444999

Order 20224.

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×Dic₆₃₂	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₃₁₆	C₂	1	2	yes	maximal
3	#23765653	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765491	C₂	1	2	yes	maximal, complemented
5	#23765451	C₂	1	2	yes	maximal, complemented
6	#23765336	C₂	1	4	yes	maximal, complemented
7	C₄×Dic₆₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₁₆	C₂²	1	4	yes	—
10	D₄×Dic₁₅₈	—	2	2	no	—
11	C₂×Dic₆₃₂	—	2	8	no	—
12	C₂×Dic₆₃₂	C₂²	1	4	yes	—
13	C₂×Dic₆₃₂	C₂²	1	2	yes	—
14	C₂×Dic₆₃₂	C₂²	1	2	yes	—
15	C₂×Dic₆₃₂	C₂²	1	1	yes	characteristic
16	#19098943	C₂²	1	2	yes	—
17	#19098804	C₂²	1	4	yes	—
18	#19098774	—	2	2	no	—
19	#19098721	—	2	4	no	—
20	C₄×Dic₃₁₆	C₂²	1	2	yes	—
21	#19098389	C₂²	1	1	yes	characteristic
22	#19098368	C₂²	1	2	yes	—
23	#19098368	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
25	C₈×D₄×C₇₉	C₂²	1	1	yes	characteristic
26	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
27	C₂²⋊C₁₆×C₇₉	C₂²	1	2	yes	—
28	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₅₈	—	2	4	no	—
30	C₂×Dic₃₁₆	—	2	8	no	—
31	C₂×Dic₃₁₆	—	2	4	no	—
32	C₂×Dic₃₁₆	—	2	2	no	—
33	C₂×Dic₃₁₆	—	4	4	no	—
34	C₂×Dic₃₁₆	C₂³	1	4	yes	—
35	C₂×Dic₃₁₆	C₂³	1	2	yes	—
36	#14579161	—	2	2	no	—
37	#14579160	—	4	2	no	—
38	#14579140	—	2	4	no	—
39	#14579138	—	4	4	no	—
40	C₄×Dic₁₅₈	—	2	2	no	—
41	D₄×Dic₇₉	—	4	2	no	—
42	Dic₆₃₂	—	2	4	no	—
43	Dic₆₃₂	—	2	8	no	—
44	Dic₆₃₂	—	4	2	no	—
45	Dic₆₃₂	D₄	1	4	yes	direct product factor
46	#14579095	C₂³	1	2	yes	—
47	#14579086	—	2	2	no	—
48	#14579086	—	2	4	no	—
49	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
50	C₄×D₄×C₇₉	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₇₉	—	2	4	no	—
52	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
53	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
56	C₂²⋊C₈×C₇₉	D₄	1	2	yes	—
57	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₅₈	—	2	2	no	—
59	C₂×Dic₁₅₈	—	2	4	no	—
60	C₂×Dic₁₅₈	—	4	8	no	—
61	C₂×Dic₁₅₈	—	4	4	no	—
62	C₂×Dic₁₅₈	—	8	4	no	—
63	C₂²×Dic₇₉	—	4	4	no	—
64	Dic₃₁₆	—	2	4	no	—
65	Dic₃₁₆	—	2	2	no	—
66	Dic₃₁₆	—	2	4	no	—
67	Dic₃₁₆	—	4	8	no	—
68	Dic₃₁₆	—	8	2	no	—
69	Dic₃₁₆	C₂×D₄	1	4	yes	—
70	C₄⋊Dic₇₉	—	2	2	no	—
71	C₄⋊Dic₇₉	—	4	2	no	—
72	C₄⋊Dic₇₉	—	4	2	no	—
73	Dic₇₉⋊C₄	—	4	2	no	—
74	Dic₇₉⋊C₄	—	4	4	no	—
75	C₂²⋊Dic₇₉	—	4	4	no	—
76	C₄×Dic₇₉	—	4	2	no	—
77	C₂×D₄×C₇₉	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₇₉	—	2	2	no	—
80	C₂×C₈×C₇₉	—	2	2	no	—
81	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
85	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
86	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
87	C₁₆×C₇₉	—	2	2	no	—
88	C₁₆×C₇₉	—	2	1	no	—
89	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
90	C₂³×C₇₉	C₂×D₈	1	2	yes	—
91	D₄×C₇₉	Q₃₂	1	4	yes	complemented
92	C₂×C₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	—	2	2	no	—
94	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₇₉	—	4	2	no	—
99	C₂×Dic₇₉	—	4	2	no	—
100	C₂×Dic₇₉	—	4	4	no	—
101	C₂×Dic₇₉	—	8	8	no	—
102	Dic₁₅₈	—	2	4	no	—
103	Dic₁₅₈	—	4	2	no	—
104	Dic₁₅₈	—	4	4	no	—
105	Dic₁₅₈	—	4	4	no	—
106	Dic₁₅₈	—	8	8	no	—
107	C₈×C₇₉	—	2	2	no	—
108	C₈×C₇₉	—	2	1	no	—
109	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂²×C₇₉	—	2	2	no	—
112	C₂²×C₇₉	—	2	2	no	—
113	C₂²×C₇₉	C₂×Q₃₂	1	4	yes	complemented
114	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
115	C₄×C₇₉	—	2	1	no	—
116	C₄×C₇₉	—	2	2	no	—
117	C₄×C₇₉	C₂×Q₃₂	1	2	yes	complemented
118	C₄×C₇₉	D₄²	1	1	yes	characteristic
119	C₄×C₇₉	#576	1	1	yes	characteristic
120	Dic₇₉	—	4	4	no	—
121	Dic₇₉	—	8	2	no	—
122	Dic₇₉	—	8	4	no	—
123	D₄×Q₃₂	—	79	1	no	maximal, Sylow
124	C₂×C₇₉	—	2	4	no	—
125	C₂×C₇₉	D₄×D₈	1	1	yes	characteristic
126	C₂×C₇₉	C₂²×Q₃₂	1	1	yes	characteristic
127	C₂×C₇₉	#3417	1	1	yes	characteristic
128	C₂²×Q₃₂	—	79	2	no	—
129	D₄×Q₁₆	—	79	2	no	—
130	#2247	—	79	1	no	—
131	#2216	—	79	2	no	—
132	#2209	—	79	2	no	—
133	#2188	—	79	4	no	—
134	C₄×Q₃₂	—	79	1	no	—
135	D₄×C₁₆	D₇₉	1	1	yes	characteristic
136	C₇₉	D₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₁₆	—	79	4	no	—
138	D₄×Q₈	—	158	2	no	—
139	C₂×Q₃₂	—	79	2	no	—
140	C₂×Q₃₂	—	79	1	no	—
141	C₂×Q₃₂	—	79	4	no	—
142	C₂×Q₃₂	—	79	2	no	—
143	C₂×Q₃₂	—	158	8	no	—
144	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
145	#494	—	79	2	no	—
146	#467	—	79	4	no	—
147	#462	—	158	2	no	—
148	#451	—	158	4	no	—
149	C₄×Q₁₆	—	79	2	no	—
150	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
151	#366	—	79	1	no	—
152	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
153	#358	—	79	2	no	—
154	#358	—	79	4	no	—
155	C₂²⋊C₁₆	C₂×D₇₉	1	2	yes	—
156	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
157	C₂²×Q₈	—	158	4	no	—
158	C₂×Q₁₆	—	316	4	no	—
159	C₂×Q₁₆	—	79	2	no	—
160	C₂×Q₁₆	—	79	4	no	—
161	C₂×Q₁₆	—	158	8	no	—
162	C₂×Q₁₆	—	158	4	no	—
163	C₂×Q₁₆	—	158	2	no	—
164	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
165	C₄⋊Q₈	—	316	2	no	—
166	C₄⋊Q₈	—	158	2	no	—
167	C₂²⋊Q₈	—	316	4	no	—
168	C₂²⋊Q₈	—	158	4	no	—
169	C₄×Q₈	—	158	2	no	—
170	C₄×D₄	—	316	2	no	—
171	C₄×D₄	D₃₁₆	1	1	yes	characteristic
172	Q₃₂	—	316	2	no	—
173	Q₃₂	—	79	4	no	—
174	Q₃₂	—	158	8	no	—
175	Q₃₂	—	158	4	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
178	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
179	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
180	#107	—	79	2	no	—
181	C₄⋊C₈	D₃₁₆	1	1	yes	characteristic
182	#103	—	158	2	no	—
183	#103	—	158	4	no	—
184	C₂²⋊C₈	D₃₁₆	1	2	yes	—
185	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
186	C₂×Q₈	—	316	8	no	—
187	C₂×Q₈	—	316	4	no	—
188	C₂×Q₈	—	632	4	no	—
189	C₂×Q₈	—	158	4	no	—
190	C₂×Q₈	—	158	2	no	—
191	C₂×D₄	D₆₃₂	1	1	yes	characteristic
192	C₂²×C₄	—	316	4	no	—
193	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
194	Q₁₆	—	316	8	no	—
195	Q₁₆	—	79	4	no	—
196	Q₁₆	—	632	2	no	—
197	Q₁₆	—	158	4	no	—
198	Q₁₆	—	158	2	no	—
199	Q₁₆	—	158	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
203	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
204	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
205	C₄⋊C₄	—	316	2	no	—
206	C₄⋊C₄	—	316	4	no	—
207	C₄⋊C₄	—	316	2	no	—
208	C₄⋊C₄	—	316	2	no	—
209	C₄⋊C₄	—	158	2	no	—
210	C₄⋊C₄	D₆₃₂	1	1	yes	characteristic
211	C₂²⋊C₄	—	316	4	no	—
212	C₂²⋊C₄	D₆₃₂	1	2	yes	—
213	C₄²	—	316	2	no	—
214	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
215	C₁₆	—	2	1	no	—
216	C₁₆	—	2	2	no	—
217	C₁₆	D₄×D₇₉	1	2	yes	—
218	C₂³	C₂×D₆₃₂	1	2	yes	—
219	Q₈	—	316	2	no	—
220	Q₈	—	316	4	no	—
221	Q₈	—	316	4	no	—
222	Q₈	—	632	8	no	—
223	Q₈	—	158	4	no	—
224	D₄	Dic₆₃₂	1	4	yes	direct product factor
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	316	2	no	—
228	C₂×C₄	—	316	2	no	—
229	C₂×C₄	—	316	4	no	—
230	C₂×C₄	—	632	8	no	—
231	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
232	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
233	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	—	2	1	no	—
237	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
238	C₈	#14579207	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	C₂×Dic₆₃₂	1	4	yes	complemented
242	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	2	no	—
245	C₄	—	316	4	no	—
246	C₄	—	632	2	no	—
247	C₄	—	632	4	no	—
248	C₄	C₂×Dic₆₃₂	1	2	yes	complemented
249	C₄	D₄×D₃₁₆	1	1	yes	characteristic
250	C₄	#19099368	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	D₄×D₆₃₂	1	1	yes	characteristic
253	C₂	C₂²×Dic₆₃₂	1	1	yes	characteristic
254	C₂	#23776553	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	D₄×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×D₄×C₇₉	D₈
hypercenter	D₄×C₁₆	D₇₉
hypocenter	C₇₉	D₄×Q₃₂
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
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×Dic₆₃₂	C₂	semidirect
3. #23765653	C₂	semidirect
4. #23765491	C₂	semidirect
5. #23765451	C₂	semidirect
6. #23765336	C₂	semidirect
7. C₄×Dic₆₃₂	C₂	semidirect
45. Dic₆₃₂	D₄	direct (N × H)
91. D₄×C₇₉	Q₃₂	semidirect
113. C₂²×C₇₉	C₂×Q₃₂	semidirect
117. C₄×C₇₉	C₂×Q₃₂	semidirect
136. C₇₉	D₄×Q₃₂	semidirect
224. D₄	Dic₆₃₂	direct (N × H)
241. C₂²	C₂×Dic₆₃₂	semidirect
248. C₄	C₂×Dic₆₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₃₂	2
D₄×Dic₃₁₆	2
#23765653	2
#23765491	2
#23765451	2
#23765336	2
C₄×Dic₆₃₂	2
D₄×C₁₆×C₇₉	2
D₄×Q₃₂	79

Subgroups — D4×Dic632 #34444999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Dic₆₃₂ #34444999