Subgroups — D₄×D₆₈₈ #25253252

Order 11008.

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×D₆₈₈	C₂	1	2	yes	maximal, complemented
2	D₄×D₃₄₄	C₂	1	2	yes	maximal, complemented
3	#19136845	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136681	C₂	1	2	yes	maximal, complemented
5	#19136631	C₂	1	2	yes	maximal, complemented
6	#19136509	C₂	1	4	yes	maximal, complemented
7	C₄×D₆₈₈	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₄₄	C₂²	1	4	yes	complemented
10	D₄×D₁₇₂	—	2	2	no	—
11	C₂×D₆₈₈	—	2	8	no	—
12	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
13	C₂×D₆₈₈	C₂²	1	2	yes	complemented
14	C₂×D₆₈₈	C₂²	1	4	yes	complemented
15	C₂×D₆₈₈	C₂²	1	2	yes	complemented
16	#14583276	C₂²	1	2	yes	complemented
17	#14583135	C₂²	1	4	yes	complemented
18	#14583095	—	2	2	no	—
19	#14583035	—	2	4	no	—
20	C₄×D₃₄₄	C₂²	1	2	yes	complemented
21	#14582725	C₂²	1	1	yes	characteristic, complemented
22	#14582701	C₂²	1	4	yes	complemented
23	#14582701	C₂²	1	2	yes	complemented
24	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	complemented
25	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
27	C₄₃×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
28	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
29	C₂²×D₁₇₂	—	2	4	no	—
30	C₂×D₄×D₄₃	—	4	2	no	—
31	C₂×D₃₄₄	—	2	8	no	—
32	C₂×D₃₄₄	—	2	4	no	—
33	C₂×D₃₄₄	—	2	2	no	—
34	C₂×D₃₄₄	—	4	4	no	—
35	C₂×D₃₄₄	C₂³	1	4	yes	—
36	C₂×D₃₄₄	C₂³	1	2	yes	—
37	#13063322	—	2	2	no	—
38	#13063300	—	4	2	no	—
39	#13063299	—	2	4	no	—
40	#13063295	—	4	4	no	—
41	C₄×D₁₇₂	—	2	2	no	—
42	D₆₈₈	—	2	8	no	—
43	D₆₈₈	—	2	4	no	—
44	D₆₈₈	—	4	2	no	—
45	D₆₈₈	D₄	1	4	yes	direct product factor
46	#13063259	C₂³	1	2	yes	—
47	#13063247	—	2	4	no	—
48	#13063247	—	2	2	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	C₂³×D₄₃	—	4	4	no	—
59	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
61	C₂×C₂²⋊D₄₃	—	4	4	no	—
62	D₄×D₄₃	—	8	8	no	—
63	C₂×D₁₇₂	—	2	2	no	—
64	C₂×D₁₇₂	—	2	4	no	—
65	C₂×D₁₇₂	—	4	4	no	—
66	C₂×D₁₇₂	—	4	2	no	—
67	C₂×D₁₇₂	—	4	2	no	—
68	C₂×D₁₇₂	—	4	8	no	—
69	C₂×D₁₇₂	—	8	4	no	—
70	C₂×C₄×D₄₃	—	4	2	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
77	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
78	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
79	(C₂×D₄₃)⋊C₄	—	4	2	no	—
80	(C₂×D₄₃)⋊C₄	—	4	4	no	—
81	C₄⋊Dic₄₃	—	2	2	no	—
82	D₃₄₄	—	2	4	no	—
83	D₃₄₄	—	2	2	no	—
84	D₃₄₄	—	2	4	no	—
85	D₃₄₄	—	4	8	no	—
86	D₃₄₄	—	8	2	no	—
87	D₃₄₄	C₂×D₄	1	4	yes	complemented
88	C₁₆×C₄₃	—	2	2	no	—
89	C₁₆×C₄₃	—	2	1	no	—
90	C₁₆×C₄₃	C₂×D₄	1	2	yes	complemented
91	C₂³×C₄₃	C₂×D₈	1	2	yes	—
92	C₂²×D₄₃	—	4	2	no	—
93	C₂²×D₄₃	—	4	4	no	—
94	C₂²×D₄₃	—	8	8	no	—
95	C₂²×D₄₃	—	8	8	no	—
96	C₂²×D₄₃	—	16	4	no	—
97	D₄×C₄₃	D₁₆	1	4	yes	complemented
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
104	C₂²⋊D₄₃	—	8	8	no	—
105	C₂×Dic₄₃	—	4	2	no	—
106	D₁₇₂	—	2	4	no	—
107	D₁₇₂	—	4	4	no	—
108	D₁₇₂	—	4	2	no	—
109	D₁₇₂	—	4	4	no	—
110	D₁₇₂	—	8	8	no	—
111	D₁₇₂	—	8	4	no	—
112	D₁₇₂	—	16	2	no	—
113	C₄×D₄₃	—	8	4	no	—
114	C₈×C₄₃	—	2	2	no	—
115	C₈×C₄₃	—	2	1	no	—
116	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
118	D₄×D₁₆	—	43	1	no	maximal, Sylow
119	C₂²×C₄₃	—	2	2	no	—
120	C₂²×C₄₃	—	2	2	no	—
121	C₂²×C₄₃	C₂×D₁₆	1	4	yes	complemented
122	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
123	C₂×D₄₃	—	4	4	no	—
124	C₂×D₄₃	—	8	2	no	—
125	C₂×D₄₃	—	8	2	no	—
126	C₂×D₄₃	—	8	4	no	—
127	C₂×D₄₃	—	8	4	no	—
128	C₂×D₄₃	—	8	4	no	—
129	C₂×D₄₃	—	16	16	no	—
130	C₄×C₄₃	—	2	1	no	—
131	C₄×C₄₃	—	2	2	no	—
132	C₄×C₄₃	C₂×D₁₆	1	2	yes	complemented
133	C₄×C₄₃	D₄²	1	1	yes	characteristic
134	C₄×C₄₃	#576	1	1	yes	characteristic
135	Dic₄₃	—	8	2	no	—
136	C₂²×D₁₆	—	43	2	no	—
137	D₄×D₈	—	43	2	no	—
138	#2246	—	43	1	no	—
139	#2215	—	43	2	no	—
140	#2206	—	43	2	no	—
141	#2184	—	43	4	no	—
142	C₄×D₁₆	—	43	1	no	—
143	D₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
144	C₂×C₄₃	—	2	4	no	—
145	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
146	C₂×C₄₃	C₂²×D₁₆	1	1	yes	characteristic
147	C₂×C₄₃	#3415	1	1	yes	characteristic
148	D₄₃	—	8	4	no	—
149	D₄₃	—	16	4	no	—
150	C₂²×D₈	—	43	4	no	—
151	D₄²	—	86	2	no	—
152	C₂×D₁₆	—	43	1	no	—
153	C₂×D₁₆	—	43	4	no	—
154	C₂×D₁₆	—	43	2	no	—
155	C₂×D₁₆	—	43	2	no	—
156	C₂×D₁₆	—	86	8	no	—
157	C₂²×C₁₆	C₂×D₄₃	1	2	yes	complemented
158	#493	—	43	2	no	—
159	#466	—	43	4	no	—
160	#459	—	86	2	no	—
161	#447	—	86	4	no	—
162	C₄×D₈	—	43	2	no	—
163	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
164	#366	—	43	1	no	—
165	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
166	#357	—	43	2	no	—
167	#357	—	43	4	no	—
168	C₂²⋊C₁₆	C₂×D₄₃	1	2	yes	complemented
169	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
170	C₄₃	D₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
171	C₂²×D₄	—	86	4	no	—
172	C₂²×D₄	—	172	2	no	—
173	C₂×D₈	—	43	2	no	—
174	C₂×D₈	—	43	4	no	—
175	C₂×D₈	—	86	8	no	—
176	C₂×D₈	—	86	2	no	—
177	C₂×D₈	—	86	4	no	—
178	C₂×D₈	—	172	4	no	—
179	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
180	#127	—	86	2	no	—
181	#121	—	86	4	no	—
182	#121	—	172	2	no	—
183	#120	—	172	4	no	—
184	C₄×D₄	—	86	2	no	—
185	C₄×D₄	D₁₇₂	1	1	yes	characteristic
186	D₁₆	—	43	4	no	—
187	D₁₆	—	86	8	no	—
188	D₁₆	—	86	4	no	—
189	D₁₆	—	172	2	no	—
190	C₂×C₁₆	—	2	4	no	—
191	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
192	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
193	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
194	#107	—	43	2	no	—
195	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
196	#102	—	86	2	no	—
197	#102	—	86	4	no	—
198	C₂²⋊C₈	D₁₇₂	1	2	yes	—
199	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
200	C₂⁴	—	172	4	no	—
201	C₂×D₄	—	86	2	no	—
202	C₂×D₄	—	86	4	no	—
203	C₂×D₄	—	344	4	no	—
204	C₂×D₄	—	344	8	no	—
205	C₂×D₄	—	172	8	no	—
206	C₂×D₄	—	172	4	no	—
207	C₂×D₄	—	172	4	no	—
208	C₂×D₄	—	172	2	no	—
209	C₂×D₄	—	172	2	no	—
210	C₂×D₄	D₃₄₄	1	1	yes	characteristic
211	C₂²×C₄	—	172	2	no	—
212	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
213	D₈	—	43	4	no	—
214	D₈	—	86	4	no	—
215	D₈	—	86	2	no	—
216	D₈	—	86	4	no	—
217	D₈	—	344	2	no	—
218	D₈	—	172	8	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
222	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
223	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
224	C₄⋊C₄	—	86	2	no	—
225	C₄⋊C₄	D₃₄₄	1	1	yes	characteristic
226	C₂²⋊C₄	—	172	2	no	—
227	C₂²⋊C₄	—	172	4	no	—
228	C₂²⋊C₄	D₃₄₄	1	2	yes	—
229	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
230	C₁₆	—	2	2	no	—
231	C₁₆	—	2	1	no	—
232	C₁₆	D₄×D₄₃	1	2	yes	complemented
233	C₂³	—	344	8	no	—
234	C₂³	—	344	8	no	—
235	C₂³	—	172	4	no	—
236	C₂³	—	172	2	no	—
237	C₂³	—	688	4	no	—
238	C₂³	C₂×D₃₄₄	1	2	yes	—
239	D₄	—	86	4	no	—
240	D₄	—	344	8	no	—
241	D₄	—	344	8	no	—
242	D₄	—	344	4	no	—
243	D₄	—	172	4	no	—
244	D₄	—	172	2	no	—
245	D₄	—	172	4	no	—
246	D₄	—	688	2	no	—
247	D₄	D₆₈₈	1	4	yes	direct product factor
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	344	4	no	—
251	C₂×C₄	—	172	2	no	—
252	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
253	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
254	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	2	no	—
258	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
259	C₈	#13063371	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	344	2	no	—
263	C₂²	—	344	2	no	—
264	C₂²	—	344	4	no	—
265	C₂²	—	344	4	no	—
266	C₂²	—	344	4	no	—
267	C₂²	—	172	4	no	—
268	C₂²	—	688	16	no	—
269	C₂²	C₂×D₆₈₈	1	4	yes	complemented
270	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	1	no	—
273	C₄	—	344	2	no	—
274	C₄	C₂×D₆₈₈	1	2	yes	complemented
275	C₄	D₄×D₁₇₂	1	1	yes	characteristic
276	C₄	#14583704	1	1	yes	characteristic
277	C₂	—	2	4	no	—
278	C₂	—	344	4	no	—
279	C₂	—	688	4	no	—
280	C₂	D₄×D₃₄₄	1	1	yes	characteristic
281	C₂	C₂²×D₆₈₈	1	1	yes	characteristic
282	C₂	#19147736	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
hypercenter	D₄×C₁₆	D₄₃
hypocenter	C₄₃	D₄×D₁₆
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×D₆₈₈	C₂	semidirect
2. D₄×D₃₄₄	C₂	semidirect
3. #19136845	C₂	semidirect
4. #19136681	C₂	semidirect
5. #19136631	C₂	semidirect
6. #19136509	C₂	semidirect
7. C₄×D₆₈₈	C₂	semidirect
8. D₄×C₁₆×C₄₃	C₂	semidirect
9. C₂²×D₃₄₄	C₂²	semidirect
13. C₂×D₆₈₈	C₂²	semidirect
14. C₂×D₆₈₈	C₂²	semidirect
15. C₂×D₆₈₈	C₂²	semidirect
16. #14583276	C₂²	semidirect
17. #14583135	C₂²	semidirect
20. C₄×D₃₄₄	C₂²	semidirect
21. #14582725	C₂²	semidirect
22. #14582701	C₂²	semidirect
23. #14582701	C₂²	semidirect
24. C₂²×C₁₆×C₄₃	C₂²	semidirect
26. C₄₃×C₄⋊C₁₆	C₂²	semidirect
27. C₄₃×C₂²⋊C₁₆	C₂²	semidirect
28. C₄×C₁₆×C₄₃	C₂²	semidirect
45. D₆₈₈	D₄	direct (N × H)
87. D₃₄₄	C₂×D₄	semidirect
90. C₁₆×C₄₃	C₂×D₄	semidirect
97. D₄×C₄₃	D₁₆	semidirect
121. C₂²×C₄₃	C₂×D₁₆	semidirect
132. C₄×C₄₃	C₂×D₁₆	semidirect
143. D₄×C₁₆	D₄₃	semidirect
157. C₂²×C₁₆	C₂×D₄₃	semidirect
165. C₄⋊C₁₆	C₂×D₄₃	semidirect
168. C₂²⋊C₁₆	C₂×D₄₃	semidirect
169. C₄×C₁₆	C₂×D₄₃	semidirect
170. C₄₃	D₄×D₁₆	semidirect
232. C₁₆	D₄×D₄₃	semidirect
247. D₄	D₆₈₈	direct (N × H)
269. C₂²	C₂×D₆₈₈	semidirect
274. C₄	C₂×D₆₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₆₈₈	2
D₄×D₃₄₄	2
#19136845	2
#19136681	2
#19136631	2
#19136509	2
C₄×D₆₈₈	2
D₄×C₁₆×C₄₃	2
D₄×D₁₆	43

Subgroups — D4×D688 #25253252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×D₆₈₈ #25253252