Subgroups — D₄×D₁₂₆₄ #34444924

Order 20224.

812 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₆₄	C₂	1	2	yes	maximal, complemented
2	D₄×D₆₃₂	C₂	1	2	yes	maximal, complemented
3	#23765650	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765486	C₂	1	2	yes	maximal, complemented
5	#23765436	C₂	1	2	yes	maximal, complemented
6	#23765314	C₂	1	4	yes	maximal, complemented
7	C₄×D₁₂₆₄	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₆₃₂	C₂²	1	4	yes	complemented
10	D₄×D₃₁₆	—	2	2	no	—
11	C₂×D₁₂₆₄	—	2	8	no	—
12	C₂×D₁₂₆₄	C₂²	1	1	yes	characteristic
13	C₂×D₁₂₆₄	C₂²	1	4	yes	complemented
14	C₂×D₁₂₆₄	C₂²	1	2	yes	complemented
15	C₂×D₁₂₆₄	C₂²	1	2	yes	complemented
16	#19098940	C₂²	1	2	yes	complemented
17	#19098799	C₂²	1	4	yes	complemented
18	#19098759	—	2	2	no	—
19	#19098699	—	2	4	no	—
20	C₄×D₆₃₂	C₂²	1	2	yes	complemented
21	#19098389	C₂²	1	1	yes	characteristic, complemented
22	#19098365	C₂²	1	2	yes	complemented
23	#19098365	C₂²	1	4	yes	complemented
24	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	complemented
25	C₈×D₄×C₇₉	C₂²	1	1	yes	characteristic
26	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
27	C₂²⋊C₁₆×C₇₉	C₂²	1	2	yes	complemented
28	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
29	C₂²×D₃₁₆	—	2	4	no	—
30	C₂×D₄×D₇₉	—	4	2	no	—
31	C₂×D₆₃₂	—	2	8	no	—
32	C₂×D₆₃₂	—	2	4	no	—
33	C₂×D₆₃₂	—	2	2	no	—
34	C₂×D₆₃₂	—	4	4	no	—
35	C₂×D₆₃₂	C₂³	1	4	yes	—
36	C₂×D₆₃₂	C₂³	1	2	yes	—
37	#14579158	—	2	2	no	—
38	#14579136	—	4	2	no	—
39	#14579135	—	2	4	no	—
40	#14579131	—	4	4	no	—
41	C₄×D₃₁₆	—	2	2	no	—
42	D₁₂₆₄	—	2	4	no	—
43	D₁₂₆₄	—	2	8	no	—
44	D₁₂₆₄	—	4	2	no	—
45	D₁₂₆₄	D₄	1	4	yes	direct product factor
46	#14579095	C₂³	1	2	yes	—
47	#14579083	—	2	2	no	—
48	#14579083	—	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	C₂³×D₇₉	—	4	4	no	—
59	C₂×D₃₁₆	—	2	2	no	—
60	C₂×D₃₁₆	—	2	4	no	—
61	C₂×D₃₁₆	—	4	8	no	—
62	C₂×D₃₁₆	—	4	4	no	—
63	C₂×D₃₁₆	—	4	2	no	—
64	C₂×D₃₁₆	—	4	2	no	—
65	C₂×D₃₁₆	—	8	4	no	—
66	C₂×C₂²⋊D₇₉	—	4	4	no	—
67	D₄×D₇₉	—	8	8	no	—
68	C₂×C₄×D₇₉	—	4	2	no	—
69	D₆₃₂	—	2	4	no	—
70	D₆₃₂	—	2	2	no	—
71	D₆₃₂	—	2	4	no	—
72	D₆₃₂	—	4	8	no	—
73	D₆₃₂	—	8	2	no	—
74	D₆₃₂	C₂×D₄	1	4	yes	complemented
75	C₄⋊Dic₇₉	—	2	2	no	—
76	(C₂×D₇₉)⋊C₄	—	4	2	no	—
77	(C₂×D₇₉)⋊C₄	—	4	4	no	—
78	C₂×D₄×C₇₉	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₇₉	—	2	2	no	—
81	C₂×C₈×C₇₉	—	2	2	no	—
82	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
86	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
87	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
88	C₁₆×C₇₉	—	2	2	no	—
89	C₁₆×C₇₉	—	2	1	no	—
90	C₁₆×C₇₉	C₂×D₄	1	2	yes	complemented
91	C₂³×C₇₉	C₂×D₈	1	2	yes	—
92	C₂²×D₇₉	—	4	2	no	—
93	C₂²×D₇₉	—	4	4	no	—
94	C₂²×D₇₉	—	8	8	no	—
95	C₂²×D₇₉	—	8	8	no	—
96	C₂²×D₇₉	—	16	4	no	—
97	D₄×C₇₉	D₁₆	1	4	yes	complemented
98	C₂×C₄×C₇₉	—	2	2	no	—
99	C₂×C₄×C₇₉	—	2	2	no	—
100	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
104	C₂²⋊D₇₉	—	8	8	no	—
105	C₂×Dic₇₉	—	4	2	no	—
106	D₃₁₆	—	2	4	no	—
107	D₃₁₆	—	4	4	no	—
108	D₃₁₆	—	4	2	no	—
109	D₃₁₆	—	4	4	no	—
110	D₃₁₆	—	8	8	no	—
111	D₃₁₆	—	8	4	no	—
112	D₃₁₆	—	16	2	no	—
113	C₄×D₇₉	—	8	4	no	—
114	C₈×C₇₉	—	2	2	no	—
115	C₈×C₇₉	—	2	1	no	—
116	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
117	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₇₉	—	2	2	no	—
119	C₂²×C₇₉	—	2	2	no	—
120	C₂²×C₇₉	C₂×D₁₆	1	4	yes	complemented
121	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
122	C₂×D₇₉	—	4	4	no	—
123	C₂×D₇₉	—	8	2	no	—
124	C₂×D₇₉	—	8	2	no	—
125	C₂×D₇₉	—	8	4	no	—
126	C₂×D₇₉	—	8	4	no	—
127	C₂×D₇₉	—	8	4	no	—
128	C₂×D₇₉	—	16	16	no	—
129	C₄×C₇₉	—	2	1	no	—
130	C₄×C₇₉	—	2	2	no	—
131	C₄×C₇₉	C₂×D₁₆	1	2	yes	complemented
132	C₄×C₇₉	D₄²	1	1	yes	characteristic
133	C₄×C₇₉	#576	1	1	yes	characteristic
134	Dic₇₉	—	8	2	no	—
135	D₄×D₁₆	—	79	1	no	maximal, Sylow
136	C₂×C₇₉	—	2	4	no	—
137	C₂×C₇₉	D₄×D₈	1	1	yes	characteristic
138	C₂×C₇₉	C₂²×D₁₆	1	1	yes	characteristic
139	C₂×C₇₉	#3415	1	1	yes	characteristic
140	D₇₉	—	8	4	no	—
141	D₇₉	—	16	4	no	—
142	C₂²×D₁₆	—	79	2	no	—
143	D₄×D₈	—	79	2	no	—
144	#2246	—	79	1	no	—
145	#2215	—	79	2	no	—
146	#2206	—	79	2	no	—
147	#2184	—	79	4	no	—
148	C₄×D₁₆	—	79	1	no	—
149	D₄×C₁₆	D₇₉	1	1	yes	characteristic, complemented
150	C₇₉	D₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
151	C₂²×D₈	—	79	4	no	—
152	D₄²	—	158	2	no	—
153	C₂×D₁₆	—	79	2	no	—
154	C₂×D₁₆	—	79	1	no	—
155	C₂×D₁₆	—	79	4	no	—
156	C₂×D₁₆	—	79	2	no	—
157	C₂×D₁₆	—	158	8	no	—
158	C₂²×C₁₆	C₂×D₇₉	1	2	yes	complemented
159	#493	—	79	2	no	—
160	#466	—	79	4	no	—
161	#459	—	158	2	no	—
162	#447	—	158	4	no	—
163	C₄×D₈	—	79	2	no	—
164	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
165	#366	—	79	1	no	—
166	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
167	#357	—	79	2	no	—
168	#357	—	79	4	no	—
169	C₂²⋊C₁₆	C₂×D₇₉	1	2	yes	complemented
170	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
171	C₂²×D₄	—	316	2	no	—
172	C₂²×D₄	—	158	4	no	—
173	C₂×D₈	—	316	4	no	—
174	C₂×D₈	—	79	2	no	—
175	C₂×D₈	—	79	4	no	—
176	C₂×D₈	—	158	8	no	—
177	C₂×D₈	—	158	4	no	—
178	C₂×D₈	—	158	2	no	—
179	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
180	#127	—	158	2	no	—
181	#121	—	316	2	no	—
182	#121	—	158	4	no	—
183	#120	—	316	4	no	—
184	C₄×D₄	—	158	2	no	—
185	C₄×D₄	D₃₁₆	1	1	yes	characteristic
186	D₁₆	—	316	2	no	—
187	D₁₆	—	79	4	no	—
188	D₁₆	—	158	8	no	—
189	D₁₆	—	158	4	no	—
190	C₂×C₁₆	—	2	4	no	—
191	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
192	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
193	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
194	#107	—	79	2	no	—
195	C₄⋊C₈	D₃₁₆	1	1	yes	characteristic
196	#102	—	158	2	no	—
197	#102	—	158	4	no	—
198	C₂²⋊C₈	D₃₁₆	1	2	yes	—
199	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
200	C₂⁴	—	316	4	no	—
201	C₂×D₄	—	316	8	no	—
202	C₂×D₄	—	316	4	no	—
203	C₂×D₄	—	316	4	no	—
204	C₂×D₄	—	316	2	no	—
205	C₂×D₄	—	316	2	no	—
206	C₂×D₄	—	632	4	no	—
207	C₂×D₄	—	632	8	no	—
208	C₂×D₄	—	158	4	no	—
209	C₂×D₄	—	158	2	no	—
210	C₂×D₄	D₆₃₂	1	1	yes	characteristic
211	C₂²×C₄	—	316	2	no	—
212	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
213	D₈	—	316	8	no	—
214	D₈	—	79	4	no	—
215	D₈	—	632	2	no	—
216	D₈	—	158	4	no	—
217	D₈	—	158	2	no	—
218	D₈	—	158	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
222	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
223	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
224	C₄⋊C₄	—	158	2	no	—
225	C₄⋊C₄	D₆₃₂	1	1	yes	characteristic
226	C₂²⋊C₄	—	316	2	no	—
227	C₂²⋊C₄	—	316	4	no	—
228	C₂²⋊C₄	D₆₃₂	1	2	yes	—
229	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
230	C₁₆	—	2	1	no	—
231	C₁₆	—	2	2	no	—
232	C₁₆	D₄×D₇₉	1	2	yes	complemented
233	C₂³	—	316	2	no	—
234	C₂³	—	316	4	no	—
235	C₂³	—	632	8	no	—
236	C₂³	—	632	8	no	—
237	C₂³	—	1264	4	no	—
238	C₂³	C₂×D₆₃₂	1	2	yes	—
239	D₄	—	316	4	no	—
240	D₄	—	316	2	no	—
241	D₄	—	316	4	no	—
242	D₄	—	632	8	no	—
243	D₄	—	632	8	no	—
244	D₄	—	632	4	no	—
245	D₄	—	158	4	no	—
246	D₄	—	1264	2	no	—
247	D₄	D₁₂₆₄	1	4	yes	direct product factor
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	316	2	no	—
251	C₂×C₄	—	632	4	no	—
252	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
253	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
254	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	2	1	no	—
258	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
259	C₈	#14579207	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	316	4	no	—
263	C₂²	—	632	2	no	—
264	C₂²	—	632	2	no	—
265	C₂²	—	632	4	no	—
266	C₂²	—	632	4	no	—
267	C₂²	—	632	4	no	—
268	C₂²	—	1264	16	no	—
269	C₂²	C₂×D₁₂₆₄	1	4	yes	complemented
270	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	2	no	—
273	C₄	—	632	2	no	—
274	C₄	C₂×D₁₂₆₄	1	2	yes	complemented
275	C₄	D₄×D₃₁₆	1	1	yes	characteristic
276	C₄	#19099368	1	1	yes	characteristic
277	C₂	—	2	4	no	—
278	C₂	—	632	4	no	—
279	C₂	—	1264	4	no	—
280	C₂	D₄×D₆₃₂	1	1	yes	characteristic
281	C₂	C₂²×D₁₂₆₄	1	1	yes	characteristic
282	C₂	#23776541	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
hypercenter	D₄×C₁₆	D₇₉
hypocenter	C₇₉	D₄×D₁₆
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×D₁₂₆₄	C₂	semidirect
2. D₄×D₆₃₂	C₂	semidirect
3. #23765650	C₂	semidirect
4. #23765486	C₂	semidirect
5. #23765436	C₂	semidirect
6. #23765314	C₂	semidirect
7. C₄×D₁₂₆₄	C₂	semidirect
8. D₄×C₁₆×C₇₉	C₂	semidirect
9. C₂²×D₆₃₂	C₂²	semidirect
13. C₂×D₁₂₆₄	C₂²	semidirect
14. C₂×D₁₂₆₄	C₂²	semidirect
15. C₂×D₁₂₆₄	C₂²	semidirect
16. #19098940	C₂²	semidirect
17. #19098799	C₂²	semidirect
20. C₄×D₆₃₂	C₂²	semidirect
21. #19098389	C₂²	semidirect
22. #19098365	C₂²	semidirect
23. #19098365	C₂²	semidirect
24. C₂²×C₁₆×C₇₉	C₂²	semidirect
26. C₄⋊C₁₆×C₇₉	C₂²	semidirect
27. C₂²⋊C₁₆×C₇₉	C₂²	semidirect
28. C₄×C₁₆×C₇₉	C₂²	semidirect
45. D₁₂₆₄	D₄	direct (N × H)
74. D₆₃₂	C₂×D₄	semidirect
90. C₁₆×C₇₉	C₂×D₄	semidirect
97. D₄×C₇₉	D₁₆	semidirect
120. C₂²×C₇₉	C₂×D₁₆	semidirect
131. C₄×C₇₉	C₂×D₁₆	semidirect
149. D₄×C₁₆	D₇₉	semidirect
150. C₇₉	D₄×D₁₆	semidirect
158. C₂²×C₁₆	C₂×D₇₉	semidirect
166. C₄⋊C₁₆	C₂×D₇₉	semidirect
169. C₂²⋊C₁₆	C₂×D₇₉	semidirect
170. C₄×C₁₆	C₂×D₇₉	semidirect
232. C₁₆	D₄×D₇₉	semidirect
247. D₄	D₁₂₆₄	direct (N × H)
269. C₂²	C₂×D₁₂₆₄	semidirect
274. C₄	C₂×D₁₂₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₂₆₄	2
D₄×D₆₃₂	2
#23765650	2
#23765486	2
#23765436	2
#23765314	2
C₄×D₁₂₆₄	2
D₄×C₁₆×C₇₉	2
D₄×D₁₆	79

Subgroups — D4×D1264 #34444924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×D₁₂₆₄ #34444924