Subgroups — C₂×C₄×D₆₈₈ #25249859

Order 11008.

806 conjugacy classes of subgroups.

The subgroup lattice has 228 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	1	yes	maximal, characteristic
2	C₂×C₄×D₃₄₄	C₂	1	2	yes	maximal, complemented
3	C₄×D₆₈₈	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582725	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582701	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₄₄	C₂²	1	2	yes	—
8	C₂×C₄×D₁₇₂	—	2	2	no	—
9	C₂×D₆₈₈	C₄	1	8	yes	direct product factor
10	C₂×D₆₈₈	C₂²	1	4	yes	—
11	C₄×D₃₄₄	—	2	4	no	—
12	C₄×D₃₄₄	C₂²	1	8	yes	complemented
13	C₂×#13063259	C₂²	1	2	yes	—
14	C₂×#13063247	—	2	2	no	—
15	#14582725	C₂²	1	4	yes	complemented
16	#14582701	C₂²	1	8	yes	complemented
17	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₄₃	C₂²	1	4	yes	complemented
21	C₂²×D₁₇₂	—	2	2	no	—
22	C₂²×C₄×D₄₃	—	4	2	no	—
23	C₂×D₃₄₄	—	2	2	no	—
24	C₂×D₃₄₄	—	2	4	no	—
25	C₂×D₃₄₄	C₂×C₄	1	8	yes	complemented
26	C₂×D₃₄₄	C₂³	1	4	yes	—
27	C₄×D₁₇₂	—	2	8	no	—
28	C₄×D₁₇₂	—	4	4	no	—
29	C₂×C₄⋊Dic₄₃	—	2	2	no	—
30	C₂×(C₂×D₄₃)⋊C₄	—	4	2	no	—
31	D₆₈₈	C₂×C₄	1	16	yes	direct product factor
32	#13063259	—	2	2	no	—
33	#13063259	C₂³	1	4	yes	—
34	#13063247	—	2	8	no	—
35	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₄₃	—	2	2	no	—
39	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
40	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
41	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
42	C₄×C₈×C₄₃	D₄	1	2	yes	—
43	C₄×C₈×C₄₃	C₂³	1	2	yes	—
44	C₂³×D₄₃	—	4	2	no	—
45	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₄₃	—	4	2	no	—
49	C₂×D₁₇₂	—	2	8	no	—
50	C₂×D₁₇₂	—	2	4	no	—
51	C₂×D₁₇₂	—	4	2	no	—
52	C₂×D₁₇₂	—	4	4	no	—
53	C₂×C₄×D₄₃	—	4	8	no	—
54	C₂×C₄×D₄₃	—	8	8	no	—
55	C₂×C₈×C₄₃	—	2	2	no	—
56	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₄₃	#41	1	2	yes	—
60	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
61	C₄²×C₄₃	D₈	1	2	yes	—
62	C₄²×C₄₃	C₂×D₄	1	2	yes	—
63	(C₂×D₄₃)⋊C₄	—	4	8	no	—
64	C₄⋊Dic₄₃	—	2	4	no	—
65	C₄⋊Dic₄₃	—	4	2	no	—
66	D₃₄₄	—	2	8	no	—
67	D₃₄₄	—	2	4	no	—
68	D₃₄₄	C₂²×C₄	1	8	yes	complemented
69	C₁₆×C₄₃	—	2	2	no	—
70	C₁₆×C₄₃	C₂²×C₄	1	4	yes	complemented
71	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
72	C₂²×D₄₃	—	4	4	no	—
73	C₂²×D₄₃	—	4	8	no	—
74	C₂²×D₄₃	—	8	4	no	—
75	C₂²×D₄₃	—	8	4	no	—
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	D₁₆	1	4	yes	complemented
78	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₄₃	#135	1	2	yes	—
83	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₄₃	—	4	4	no	—
85	C₂×Dic₄₃	—	8	4	no	—
86	D₁₇₂	—	2	8	no	—
87	D₁₇₂	—	4	4	no	—
88	D₁₇₂	—	4	8	no	—
89	C₄×D₄₃	—	8	16	no	—
90	C₈×C₄₃	—	2	2	no	—
91	C₈×C₄₃	C₄×D₄	1	2	yes	—
92	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
93	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
94	C₂×C₄×D₁₆	—	43	1	no	maximal, Sylow
95	C₂²×C₄₃	C₄×D₈	1	2	yes	—
96	C₂²×C₄₃	C₂×D₁₆	1	2	yes	—
97	C₂²×C₄₃	#508	1	2	yes	—
98	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
99	C₂×D₄₃	—	4	8	no	—
100	C₂×D₄₃	—	8	4	no	—
101	C₂×D₄₃	—	8	4	no	—
102	C₂×D₄₃	—	8	16	no	—
103	C₄×C₄₃	—	2	2	no	—
104	C₄×C₄₃	C₄×D₈	1	2	yes	—
105	C₄×C₄₃	C₂×D₁₆	1	4	yes	complemented
106	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
107	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
108	Dic₄₃	—	8	4	no	—
109	C₂²×D₁₆	—	43	1	no	—
110	C₂×C₄×D₈	—	43	2	no	—
111	C₄×D₁₆	—	43	8	no	—
112	C₂×#366	—	43	1	no	—
113	C₂×#357	—	43	2	no	—
114	C₂×C₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
115	C₂×C₄₃	C₄×D₁₆	1	4	yes	complemented
116	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
117	C₂×C₄₃	C₂²×D₁₆	1	1	yes	characteristic
118	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
119	D₄₃	—	8	8	no	—
120	C₂²×D₈	—	43	2	no	—
121	C₂×C₄×D₄	—	86	2	no	—
122	C₂×D₁₆	—	43	4	no	—
123	C₂×D₁₆	—	43	8	no	—
124	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
125	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
126	C₄×D₈	—	43	8	no	—
127	C₄×D₈	—	86	4	no	—
128	C₂×#107	—	43	2	no	—
129	C₂×#102	—	86	2	no	—
130	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
131	#366	—	43	4	no	—
132	#357	—	43	8	no	—
133	C₄×C₁₆	C₂×D₄₃	1	4	yes	complemented
134	C₄₃	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
135	C₂²×D₄	—	86	2	no	—
136	C₂³×C₄	—	172	2	no	—
137	C₂×D₈	—	43	8	no	—
138	C₂×D₈	—	43	4	no	—
139	C₂×D₈	—	86	2	no	—
140	C₂×D₈	—	86	4	no	—
141	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
143	C₄×D₄	—	86	8	no	—
144	C₄×D₄	—	172	4	no	—
145	C₂×C₄⋊C₄	—	86	2	no	—
146	C₂×C₂²⋊C₄	—	172	2	no	—
147	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
148	D₁₆	—	43	16	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	C₄×D₄₃	1	4	yes	complemented
151	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
152	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
153	#107	—	43	4	no	—
154	#107	—	86	2	no	—
155	#102	—	86	8	no	—
156	C₄×C₈	D₁₇₂	1	2	yes	—
157	C₄×C₈	C₂²×D₄₃	1	2	yes	—
158	C₂⁴	—	172	2	no	—
159	C₂×D₄	—	86	8	no	—
160	C₂×D₄	—	86	4	no	—
161	C₂×D₄	—	172	2	no	—
162	C₂×D₄	—	172	4	no	—
163	C₂²×C₄	—	344	8	no	—
164	C₂²×C₄	—	172	2	no	—
165	C₂²×C₄	—	172	8	no	—
166	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
167	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
169	D₈	—	43	8	no	—
170	D₈	—	86	8	no	—
171	D₈	—	86	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
174	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
175	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
176	C₂×C₈	#10621898	1	2	yes	—
177	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
178	C₄⋊C₄	—	86	4	no	—
179	C₄⋊C₄	—	172	2	no	—
180	C₂²⋊C₄	—	172	8	no	—
181	C₄²	D₃₄₄	1	2	yes	—
182	C₄²	C₂×D₁₇₂	1	2	yes	—
183	C₁₆	—	2	2	no	—
184	C₁₆	C₂×C₄×D₄₃	1	4	yes	complemented
185	C₂³	—	344	4	no	—
186	C₂³	—	344	4	no	—
187	C₂³	—	172	4	no	—
188	C₂³	—	172	8	no	—
189	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
190	D₄	—	86	8	no	—
191	D₄	—	172	4	no	—
192	D₄	—	172	8	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	344	4	no	—
195	C₂×C₄	—	344	16	no	—
196	C₂×C₄	—	172	4	no	—
197	C₂×C₄	D₆₈₈	1	4	yes	direct product factor
198	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
199	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
200	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
201	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
202	C₂×C₄	#13063347	1	2	yes	—
203	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	C₄×D₁₇₂	1	2	yes	—
206	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
207	C₈	C₂×#10621898	1	1	yes	characteristic
208	C₂²	—	344	4	no	—
209	C₂²	—	344	16	no	—
210	C₂²	—	344	4	no	—
211	C₂²	—	172	8	no	—
212	C₂²	C₄×D₃₄₄	1	2	yes	—
213	C₂²	C₂×D₆₈₈	1	2	yes	—
214	C₂²	#14583331	1	2	yes	—
215	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	344	4	no	—
218	C₄	C₄×D₃₄₄	1	2	yes	—
219	C₄	C₂×D₆₈₈	1	4	yes	direct product factor
220	C₄	C₂×C₄×D₁₇₂	1	1	yes	characteristic
221	C₄	C₂×#13063347	1	1	yes	characteristic
222	C₂	—	344	8	no	—
223	C₂	C₄×D₆₈₈	1	4	yes	direct product factor
224	C₂	C₂×C₄×D₃₄₄	1	1	yes	characteristic
225	C₂	C₂²×D₆₈₈	1	1	yes	characteristic
226	C₂	C₂×#14583331	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₃₄₄
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄×C₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₆₈₈	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₄₃
hypocenter	C₄₃	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₃₄₄	C₂	semidirect
3. C₄×D₆₈₈	C₂	direct (N × H)
4. C₂×#14582725	C₂	semidirect
5. C₂×#14582701	C₂	semidirect
6. C₂×C₄×C₁₆×C₄₃	C₂	semidirect
9. C₂×D₆₈₈	C₄	direct (N × H)
12. C₄×D₃₄₄	C₂²	semidirect
15. #14582725	C₂²	semidirect
16. #14582701	C₂²	semidirect
20. C₄×C₁₆×C₄₃	C₂²	semidirect
25. C₂×D₃₄₄	C₂×C₄	semidirect
31. D₆₈₈	C₂×C₄	direct (N × H)
39. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
68. D₃₄₄	C₂²×C₄	semidirect
70. C₁₆×C₄₃	C₂²×C₄	semidirect
77. C₂×C₄×C₄₃	D₁₆	semidirect
105. C₄×C₄₃	C₂×D₁₆	semidirect
114. C₂×C₄×C₁₆	D₄₃	semidirect
115. C₂×C₄₃	C₄×D₁₆	semidirect
133. C₄×C₁₆	C₂×D₄₃	semidirect
134. C₄₃	C₂×C₄×D₁₆	semidirect
150. C₂×C₁₆	C₄×D₄₃	semidirect
184. C₁₆	C₂×C₄×D₄₃	semidirect
197. C₂×C₄	D₆₈₈	direct (N × H)
219. C₄	C₂×D₆₈₈	direct (N × H)
223. C₂	C₄×D₆₈₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₆₈₈	2
C₂×C₄×D₃₄₄	2
C₄×D₆₈₈	2
C₂×#14582725	2
C₂×#14582701	2
C₂×C₄×C₁₆×C₄₃	2
C₂×C₄×D₁₆	43

Subgroups — C2×C4×D688 #25249859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₆₈₈ #25249859