Subgroups — C₂×C₄×D₁₄₂₄ #36707690

Order 22784.

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×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₇₁₂	C₂	1	2	yes	maximal, complemented
3	C₄×D₁₄₂₄	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151400	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₇₁₂	C₂²	1	2	yes	—
8	C₂×C₄×D₃₅₆	—	2	2	no	—
9	C₂×D₁₄₂₄	C₄	1	8	yes	direct product factor
10	C₂×D₁₄₂₄	C₂²	1	4	yes	—
11	C₄×D₇₁₂	—	2	4	no	—
12	C₄×D₇₁₂	C₂²	1	8	yes	complemented
13	C₂×#15667946	C₂²	1	2	yes	—
14	C₂×#15667934	—	2	2	no	—
15	#19151424	C₂²	1	4	yes	complemented
16	#19151400	C₂²	1	8	yes	complemented
17	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₈₉	C₂²	1	4	yes	complemented
21	C₂²×D₃₅₆	—	2	2	no	—
22	C₂²×C₄×D₈₉	—	4	2	no	—
23	C₂×D₇₁₂	—	2	2	no	—
24	C₂×D₇₁₂	—	2	4	no	—
25	C₂×D₇₁₂	C₂×C₄	1	8	yes	complemented
26	C₂×D₇₁₂	C₂³	1	4	yes	—
27	C₄×D₃₅₆	—	2	8	no	—
28	C₄×D₃₅₆	—	4	4	no	—
29	C₂×C₄⋊Dic₈₉	—	2	2	no	—
30	C₂×#13083551	—	4	2	no	—
31	D₁₄₂₄	C₂×C₄	1	16	yes	direct product factor
32	#15667946	—	2	2	no	—
33	#15667946	C₂³	1	4	yes	—
34	#15667934	—	2	8	no	—
35	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₈₉	—	2	2	no	—
39	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
40	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
41	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
42	C₄×C₈×C₈₉	D₄	1	2	yes	—
43	C₄×C₈×C₈₉	C₂³	1	2	yes	—
44	C₂³×D₈₉	—	4	2	no	—
45	C₂×D₃₅₆	—	2	8	no	—
46	C₂×D₃₅₆	—	2	4	no	—
47	C₂×D₃₅₆	—	4	2	no	—
48	C₂×D₃₅₆	—	4	4	no	—
49	C₂²×Dic₈₉	—	4	2	no	—
50	C₂×C₄×D₈₉	—	4	8	no	—
51	C₂×C₄×D₈₉	—	8	8	no	—
52	D₇₁₂	—	2	8	no	—
53	D₇₁₂	—	2	4	no	—
54	D₇₁₂	C₂²×C₄	1	8	yes	complemented
55	C₄⋊Dic₈₉	—	2	4	no	—
56	C₄⋊Dic₈₉	—	4	2	no	—
57	#13083551	—	4	8	no	—
58	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	—	2	2	no	—
62	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	#41	1	2	yes	—
66	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
67	C₄²×C₈₉	D₈	1	2	yes	—
68	C₄²×C₈₉	C₂×D₄	1	2	yes	—
69	C₁₆×C₈₉	—	2	2	no	—
70	C₁₆×C₈₉	C₂²×C₄	1	4	yes	complemented
71	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
72	C₂²×D₈₉	—	4	4	no	—
73	C₂²×D₈₉	—	4	8	no	—
74	C₂²×D₈₉	—	8	4	no	—
75	C₂²×D₈₉	—	8	4	no	—
76	C₂×C₄×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	D₁₆	1	4	yes	complemented
78	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₈₉	#135	1	2	yes	—
83	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₈₉	—	4	4	no	—
85	C₂×Dic₈₉	—	8	4	no	—
86	D₃₅₆	—	2	8	no	—
87	D₃₅₆	—	4	4	no	—
88	D₃₅₆	—	4	8	no	—
89	C₄×D₈₉	—	8	16	no	—
90	C₈×C₈₉	—	2	2	no	—
91	C₈×C₈₉	C₄×D₄	1	2	yes	—
92	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
93	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
94	C₂²×C₈₉	C₄×D₈	1	2	yes	—
95	C₂²×C₈₉	C₂×D₁₆	1	2	yes	—
96	C₂²×C₈₉	#508	1	2	yes	—
97	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
98	C₂×D₈₉	—	4	8	no	—
99	C₂×D₈₉	—	8	4	no	—
100	C₂×D₈₉	—	8	4	no	—
101	C₂×D₈₉	—	8	16	no	—
102	C₄×C₈₉	—	2	2	no	—
103	C₄×C₈₉	C₄×D₈	1	2	yes	—
104	C₄×C₈₉	C₂×D₁₆	1	4	yes	complemented
105	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
106	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
107	Dic₈₉	—	8	4	no	—
108	C₂×C₄×D₁₆	—	89	1	no	maximal, Sylow
109	C₂×C₈₉	C₄×D₁₆	1	4	yes	complemented
110	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
111	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
112	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
113	D₈₉	—	8	8	no	—
114	C₂²×D₁₆	—	89	1	no	—
115	C₂×C₄×D₈	—	89	2	no	—
116	C₄×D₁₆	—	89	8	no	—
117	C₂×#366	—	89	1	no	—
118	C₂×#357	—	89	2	no	—
119	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
120	C₈₉	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
121	C₂²×D₈	—	89	2	no	—
122	C₂×C₄×D₄	—	178	2	no	—
123	C₂×D₁₆	—	89	8	no	—
124	C₂×D₁₆	—	89	4	no	—
125	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
126	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
127	C₄×D₈	—	89	8	no	—
128	C₄×D₈	—	178	4	no	—
129	C₂×#107	—	89	2	no	—
130	C₂×#102	—	178	2	no	—
131	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
132	#366	—	89	4	no	—
133	#357	—	89	8	no	—
134	C₄×C₁₆	C₂×D₈₉	1	4	yes	complemented
135	C₂²×D₄	—	178	2	no	—
136	C₂³×C₄	—	356	2	no	—
137	C₂×D₈	—	89	8	no	—
138	C₂×D₈	—	89	4	no	—
139	C₂×D₈	—	178	2	no	—
140	C₂×D₈	—	178	4	no	—
141	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
143	C₄×D₄	—	356	4	no	—
144	C₄×D₄	—	178	8	no	—
145	C₂×C₄⋊C₄	—	178	2	no	—
146	C₂×C₂²⋊C₄	—	356	2	no	—
147	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
148	D₁₆	—	89	16	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	C₄×D₈₉	1	4	yes	complemented
151	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
152	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
153	#107	—	89	4	no	—
154	#107	—	178	2	no	—
155	#102	—	178	8	no	—
156	C₄×C₈	D₃₅₆	1	2	yes	—
157	C₄×C₈	C₂²×D₈₉	1	2	yes	—
158	C₂⁴	—	356	2	no	—
159	C₂×D₄	—	356	2	no	—
160	C₂×D₄	—	356	4	no	—
161	C₂×D₄	—	178	8	no	—
162	C₂×D₄	—	178	4	no	—
163	C₂²×C₄	—	356	2	no	—
164	C₂²×C₄	—	356	8	no	—
165	C₂²×C₄	—	712	8	no	—
166	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
167	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
169	D₈	—	89	8	no	—
170	D₈	—	178	8	no	—
171	D₈	—	178	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
174	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
175	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
176	C₂×C₈	#13083574	1	2	yes	—
177	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
178	C₄⋊C₄	—	356	2	no	—
179	C₄⋊C₄	—	178	4	no	—
180	C₂²⋊C₄	—	356	8	no	—
181	C₄²	D₇₁₂	1	2	yes	—
182	C₄²	C₂×D₃₅₆	1	2	yes	—
183	C₁₆	—	2	2	no	—
184	C₁₆	C₂×C₄×D₈₉	1	4	yes	complemented
185	C₂³	—	356	8	no	—
186	C₂³	—	356	4	no	—
187	C₂³	—	712	4	no	—
188	C₂³	—	712	4	no	—
189	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
190	D₄	—	356	8	no	—
191	D₄	—	356	4	no	—
192	D₄	—	178	8	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	356	4	no	—
195	C₂×C₄	—	712	4	no	—
196	C₂×C₄	—	712	16	no	—
197	C₂×C₄	D₁₄₂₄	1	4	yes	direct product factor
198	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
199	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
200	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
201	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
202	C₂×C₄	#15668034	1	2	yes	—
203	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	C₄×D₃₅₆	1	2	yes	—
206	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
207	C₈	C₂×#13083574	1	1	yes	characteristic
208	C₂²	—	356	8	no	—
209	C₂²	—	712	4	no	—
210	C₂²	—	712	16	no	—
211	C₂²	—	712	4	no	—
212	C₂²	C₄×D₇₁₂	1	2	yes	—
213	C₂²	C₂×D₁₄₂₄	1	2	yes	—
214	C₂²	#19152030	1	2	yes	—
215	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	712	4	no	—
218	C₄	C₄×D₇₁₂	1	2	yes	—
219	C₄	C₂×D₁₄₂₄	1	4	yes	direct product factor
220	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
221	C₄	C₂×#15668034	1	1	yes	characteristic
222	C₂	—	712	8	no	—
223	C₂	C₄×D₁₄₂₄	1	4	yes	direct product factor
224	C₂	C₂×C₄×D₇₁₂	1	1	yes	characteristic
225	C₂	C₂²×D₁₄₂₄	1	1	yes	characteristic
226	C₂	C₂×#19152030	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₇₁₂
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂²×D₁₄₂₄	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
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×D₇₁₂	C₂	semidirect
3. C₄×D₁₄₂₄	C₂	direct (N × H)
4. C₂×#19151424	C₂	semidirect
5. C₂×#19151400	C₂	semidirect
6. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
9. C₂×D₁₄₂₄	C₄	direct (N × H)
12. C₄×D₇₁₂	C₂²	semidirect
15. #19151424	C₂²	semidirect
16. #19151400	C₂²	semidirect
20. C₄×C₁₆×C₈₉	C₂²	semidirect
25. C₂×D₇₁₂	C₂×C₄	semidirect
31. D₁₄₂₄	C₂×C₄	direct (N × H)
39. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
54. D₇₁₂	C₂²×C₄	semidirect
70. C₁₆×C₈₉	C₂²×C₄	semidirect
77. C₂×C₄×C₈₉	D₁₆	semidirect
104. C₄×C₈₉	C₂×D₁₆	semidirect
109. C₂×C₈₉	C₄×D₁₆	semidirect
119. C₂×C₄×C₁₆	D₈₉	semidirect
120. C₈₉	C₂×C₄×D₁₆	semidirect
134. C₄×C₁₆	C₂×D₈₉	semidirect
150. C₂×C₁₆	C₄×D₈₉	semidirect
184. C₁₆	C₂×C₄×D₈₉	semidirect
197. C₂×C₄	D₁₄₂₄	direct (N × H)
219. C₄	C₂×D₁₄₂₄	direct (N × H)
223. C₂	C₄×D₁₄₂₄	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×D₁₄₂₄	2
C₂×C₄×D₇₁₂	2
C₄×D₁₄₂₄	2
C₂×#19151424	2
C₂×#19151400	2
C₂×C₄×C₁₆×C₈₉	2
C₂×C₄×D₁₆	89

Subgroups — C2×C4×D1424 #36707690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₁₄₂₄ #36707690