Subgroups — C₂²×C₄×C₄⋊Dic₈₉ #36712606

Order 22784.

5758 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	C₂³×C₄⋊Dic₈₉	C₂	1	2	yes	maximal
2	C₂³×C₄×Dic₈₉	C₂	1	2	yes	maximal
3	C₂×C₄×C₄⋊Dic₈₉	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#15667919	C₂	1	2	yes	maximal
5	C₂³×C₄²×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₈₉	C₂²	1	4	yes	—
7	C₂²×C₄⋊Dic₈₉	C₄	1	32	yes	direct product factor
8	C₂²×C₄⋊Dic₈₉	C₂²	1	24	yes	—
9	C₂²×C₄×Dic₈₉	—	2	16	no	—
10	C₂²×C₄×Dic₈₉	C₂²	1	24	yes	—
11	C₄×C₄⋊Dic₈₉	C₂²	1	64	yes	direct product factor
12	C₂×#15667919	C₂²	1	24	yes	—
13	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₈₉	C₄	1	16	yes	complemented
17	C₂²×C₄²×C₈₉	C₂²	1	12	yes	—
18	C₂³×Dic₈₉	—	2	16	no	—
19	C₂³×Dic₈₉	—	2	16	no	—
20	C₂³×Dic₈₉	C₂×C₄	1	32	yes	—
21	C₂³×Dic₈₉	C₂³	1	24	yes	—
22	C₂×C₄⋊Dic₈₉	C₂×C₄	1	192	yes	direct product factor
23	C₂×C₄⋊Dic₈₉	C₂³	1	32	yes	—
24	C₂×C₄×Dic₈₉	—	2	96	no	—
25	C₂×C₄×Dic₈₉	C₂³	1	32	yes	—
26	#15667919	C₂³	1	32	yes	—
27	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₈₉	—	2	8	no	—
29	C₂³×C₄×C₈₉	C₂×C₄	1	16	yes	—
30	C₂³×C₄×C₈₉	C₂×C₄	1	8	yes	—
31	C₂³×C₄×C₈₉	C₂×C₄	1	8	yes	—
32	C₂³×C₄×C₈₉	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₈₉	D₄	1	8	yes	—
34	C₂³×C₄×C₈₉	Q₈	1	8	yes	—
35	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
36	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
37	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
38	C₂×C₄²×C₈₉	C₂×C₄	1	96	yes	complemented
39	C₂×C₄²×C₈₉	C₂³	1	16	yes	—
40	C₂²×Dic₈₉	—	2	128	no	—
41	C₂²×Dic₈₉	—	2	48	no	—
42	C₂²×Dic₈₉	—	2	48	no	—
43	C₂²×Dic₈₉	C₂²×C₄	1	96	yes	—
44	C₂²×Dic₈₉	C₂⁴	1	16	yes	—
45	C₄⋊Dic₈₉	C₂²×C₄	1	128	yes	direct product factor
46	C₄×Dic₈₉	—	2	64	no	—
47	C₂⁴×C₈₉	C₂²×C₄	1	4	yes	—
48	C₂⁴×C₈₉	C₂²×C₄	1	8	yes	—
49	C₂⁴×C₈₉	C₂×D₄	1	4	yes	—
50	C₂⁴×C₈₉	C₂×Q₈	1	4	yes	—
51	C₂⁴×C₈₉	#41	1	8	yes	—
52	C₂⁴×C₈₉	C₂⁴	1	3	yes	—
53	C₂²×C₄×C₈₉	—	2	32	no	—
54	C₂²×C₄×C₈₉	—	2	24	no	—
55	C₂²×C₄×C₈₉	C₄²	1	64	yes	complemented
56	C₂²×C₄×C₈₉	C₄⋊C₄	1	64	yes	complemented
57	C₂²×C₄×C₈₉	C₂²×C₄	1	24	yes	—
58	C₂²×C₄×C₈₉	C₂²×C₄	1	48	yes	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	24	yes	—
60	C₂²×C₄×C₈₉	C₂²×C₄	1	24	yes	—
61	C₂²×C₄×C₈₉	C₂×D₄	1	24	yes	—
62	C₂²×C₄×C₈₉	C₂×Q₈	1	24	yes	—
63	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
64	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
65	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
66	C₄²×C₈₉	C₂²×C₄	1	64	yes	complemented
67	C₂³×C₈₉	C₂×C₄²	1	16	yes	—
68	C₂³×C₈₉	C₂×C₄⋊C₄	1	16	yes	—
69	C₂³×C₈₉	#117	1	16	yes	—
70	C₂³×C₈₉	C₄×D₄	1	32	yes	—
71	C₂³×C₈₉	C₄×Q₈	1	32	yes	—
72	C₂³×C₈₉	C₂³×C₄	1	12	yes	—
73	C₂³×C₈₉	C₂³×C₄	1	6	yes	—
74	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
75	C₂³×C₈₉	C₂²×Q₈	1	6	yes	—
76	C₂³×C₈₉	C₂×#41	1	12	yes	—
77	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
78	C₂×C₄×C₈₉	—	2	48	no	—
79	C₂×C₄×C₈₉	—	2	8	no	—
80	C₂×C₄×C₈₉	C₂×C₄²	1	96	yes	complemented
81	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	96	yes	complemented
82	C₂×C₄×C₈₉	C₂³×C₄	1	16	yes	—
83	C₂×C₄×C₈₉	C₂³×C₄	1	8	yes	—
84	C₂×C₄×C₈₉	C₂³×C₄	1	8	yes	—
85	C₂×C₄×C₈₉	C₂³×C₄	1	8	yes	—
86	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
87	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
88	C₂×Dic₈₉	—	2	192	no	—
89	C₂×Dic₈₉	—	2	16	no	—
90	C₂×Dic₈₉	—	2	16	no	—
91	C₂×Dic₈₉	C₂³×C₄	1	32	yes	—
92	C₂²×C₈₉	C₄×C₄⋊C₄	1	64	yes	complemented
93	C₂²×C₈₉	C₂²×C₄²	1	12	yes	—
94	C₂²×C₈₉	C₂²×C₄⋊C₄	1	12	yes	—
95	C₂²×C₈₉	C₂×#117	1	12	yes	—
96	C₂²×C₈₉	C₂×C₄×D₄	1	24	yes	—
97	C₂²×C₈₉	C₂×C₄×Q₈	1	24	yes	—
98	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
99	C₂²×C₈₉	C₂⁴×C₄	1	2	yes	—
100	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
101	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
102	C₂²×C₈₉	C₂²×#41	1	2	yes	—
103	C₄×C₈₉	—	2	8	no	—
104	C₄×C₈₉	C₂²×C₄²	1	16	yes	complemented
105	C₄×C₈₉	C₂²×C₄⋊C₄	1	16	yes	complemented
106	Dic₈₉	—	2	32	no	—
107	C₂²×C₄×C₄⋊C₄	—	89	1	no	maximal, Sylow
108	C₂×C₈₉	C₂×C₄×C₄⋊C₄	1	24	yes	complemented
109	C₂×C₈₉	C₂³×C₄²	1	1	yes	characteristic
110	C₂×C₈₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₈₉	C₂²×#117	1	1	yes	characteristic
112	C₂×C₈₉	C₂²×C₄×D₄	1	2	yes	—
113	C₂×C₈₉	C₂²×C₄×Q₈	1	2	yes	—
114	C₂³×C₄⋊C₄	—	89	2	no	—
115	C₂³×C₄²	—	89	2	no	—
116	C₂³×C₄²	D₈₉	1	1	yes	characteristic
117	C₂×C₄×C₄⋊C₄	—	89	24	no	—
118	C₂²×#95	—	89	2	no	—
119	C₈₉	C₂²×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
120	C₂⁴×C₄	—	89	4	no	—
121	C₂⁴×C₄	C₂×D₈₉	1	1	yes	characteristic
122	C₂⁴×C₄	C₂×D₈₉	1	1	yes	characteristic
123	C₂⁴×C₄	C₂×D₈₉	1	1	yes	characteristic
124	C₂²×C₄⋊C₄	—	89	32	no	—
125	C₂²×C₄⋊C₄	—	89	24	no	—
126	C₂²×C₄²	—	89	24	no	—
127	C₂²×C₄²	—	178	16	no	—
128	C₂²×C₄²	Dic₈₉	1	16	yes	complemented
129	C₂²×C₄²	C₂×D₈₉	1	12	yes	—
130	C₄×C₄⋊C₄	—	89	64	no	—
131	C₂×#95	—	89	24	no	—
132	C₂⁵	C₂²×D₈₉	1	1	yes	characteristic
133	C₂³×C₄	—	2	8	no	—
134	C₂³×C₄	—	89	32	no	—
135	C₂³×C₄	—	89	24	no	—
136	C₂³×C₄	—	178	16	no	—
137	C₂³×C₄	—	178	16	no	—
138	C₂³×C₄	Dic₁₇₈	1	8	yes	—
139	C₂³×C₄	C₄×D₈₉	1	16	yes	—
140	C₂³×C₄	D₃₅₆	1	8	yes	—
141	C₂³×C₄	C₂×Dic₈₉	1	8	yes	—
142	C₂³×C₄	C₂×Dic₈₉	1	8	yes	—
143	C₂³×C₄	C₂×Dic₈₉	1	8	yes	—
144	C₂³×C₄	C₂²×D₈₉	1	6	yes	—
145	C₂³×C₄	C₂²×D₈₉	1	6	yes	—
146	C₂³×C₄	C₂²×D₈₉	1	6	yes	—
147	C₂×C₄⋊C₄	—	89	192	no	—
148	C₂×C₄⋊C₄	—	89	32	no	—
149	C₂×C₄²	—	89	32	no	—
150	C₂×C₄²	—	178	96	no	—
151	C₂×C₄²	C₂×Dic₈₉	1	96	yes	complemented
152	C₂×C₄²	C₂²×D₈₉	1	16	yes	—
153	#95	—	89	32	no	—
154	C₂⁴	C₂×C₄×D₈₉	1	8	yes	—
155	C₂⁴	C₂²×Dic₈₉	1	4	yes	—
156	C₂⁴	C₂×D₃₅₆	1	4	yes	—
157	C₂⁴	C₂×Dic₁₇₈	1	4	yes	—
158	C₂⁴	#13083574	1	8	yes	—
159	C₂⁴	C₂³×D₈₉	1	3	yes	—
160	C₂²×C₄	—	2	32	no	—
161	C₂²×C₄	—	2	24	no	—
162	C₂²×C₄	—	89	96	no	—
163	C₂²×C₄	—	89	16	no	—
164	C₂²×C₄	—	178	128	no	—
165	C₂²×C₄	—	178	48	no	—
166	C₂²×C₄	—	178	48	no	—
167	C₂²×C₄	C₄×Dic₈₉	1	64	yes	complemented
168	C₂²×C₄	C₄⋊Dic₈₉	1	64	yes	direct product factor
169	C₂²×C₄	C₂×C₄×D₈₉	1	48	yes	—
170	C₂²×C₄	C₂²×Dic₈₉	1	24	yes	—
171	C₂²×C₄	C₂²×Dic₈₉	1	24	yes	—
172	C₂²×C₄	C₂²×Dic₈₉	1	24	yes	—
173	C₂²×C₄	C₂×D₃₅₆	1	24	yes	—
174	C₂²×C₄	C₂×Dic₁₇₈	1	24	yes	—
175	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
176	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
177	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
178	C₄⋊C₄	—	89	128	no	—
179	C₄²	—	178	64	no	—
180	C₄²	C₂²×Dic₈₉	1	64	yes	complemented
181	C₂³	C₂×C₄×Dic₈₉	1	16	yes	—
182	C₂³	C₂×C₄⋊Dic₈₉	1	16	yes	—
183	C₂³	#15667971	1	16	yes	—
184	C₂³	C₄×D₃₅₆	1	32	yes	—
185	C₂³	C₄×Dic₁₇₈	1	32	yes	—
186	C₂³	C₂²×C₄×D₈₉	1	12	yes	—
187	C₂³	C₂³×Dic₈₉	1	6	yes	—
188	C₂³	C₂²×D₃₅₆	1	6	yes	—
189	C₂³	C₂²×Dic₁₇₈	1	6	yes	—
190	C₂³	C₂×#13083574	1	12	yes	—
191	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	—	2	48	no	—
194	C₂×C₄	—	89	32	no	—
195	C₂×C₄	—	178	16	no	—
196	C₂×C₄	—	178	16	no	—
197	C₂×C₄	—	178	192	no	—
198	C₂×C₄	C₂×C₄×Dic₈₉	1	96	yes	complemented
199	C₂×C₄	C₂×C₄⋊Dic₈₉	1	96	yes	direct product factor
200	C₂×C₄	C₂²×C₄×D₈₉	1	16	yes	—
201	C₂×C₄	C₂³×Dic₈₉	1	8	yes	—
202	C₂×C₄	C₂³×Dic₈₉	1	8	yes	—
203	C₂×C₄	C₂³×Dic₈₉	1	8	yes	—
204	C₂×C₄	C₂²×D₃₅₆	1	8	yes	—
205	C₂×C₄	C₂²×Dic₁₇₈	1	8	yes	—
206	C₂²	C₄×C₄⋊Dic₈₉	1	64	yes	direct product factor
207	C₂²	C₂²×C₄×Dic₈₉	1	12	yes	—
208	C₂²	C₂²×C₄⋊Dic₈₉	1	12	yes	—
209	C₂²	C₂×#15667971	1	12	yes	—
210	C₂²	C₂×C₄×D₃₅₆	1	24	yes	—
211	C₂²	C₂×C₄×Dic₁₇₈	1	24	yes	—
212	C₂²	C₂³×C₄×D₈₉	1	2	yes	—
213	C₂²	C₂⁴×Dic₈₉	1	1	yes	characteristic
214	C₂²	C₂³×D₃₅₆	1	1	yes	characteristic
215	C₂²	C₂³×Dic₁₇₈	1	1	yes	characteristic
216	C₂²	C₂²×#13083574	1	2	yes	—
217	C₄	—	2	8	no	—
218	C₄	—	178	32	no	—
219	C₄	C₂²×C₄×Dic₈₉	1	16	yes	complemented
220	C₄	C₂²×C₄⋊Dic₈₉	1	16	yes	direct product factor
221	C₂	C₂×C₄×C₄⋊Dic₈₉	1	24	yes	direct product factor
222	C₂	C₂³×C₄×Dic₈₉	1	1	yes	characteristic
223	C₂	C₂³×C₄⋊Dic₈₉	1	1	yes	characteristic
224	C₂	C₂²×#15667971	1	1	yes	characteristic
225	C₂	C₂²×C₄×D₃₅₆	1	2	yes	—
226	C₂	C₂²×C₄×Dic₁₇₈	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₈₉
commutator	C₂×C₈₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂³×C₄²×C₈₉	C₂
socle, prime radical	C₂⁵×C₈₉	C₂³
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂³×C₄²	D₈₉
hypocenter	C₈₉	C₂²×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₄⋊Dic₈₉	C₂	direct (N × H)
7. C₂²×C₄⋊Dic₈₉	C₄	direct (N × H)
11. C₄×C₄⋊Dic₈₉	C₂²	direct (N × H)
16. C₂²×C₄²×C₈₉	C₄	semidirect
22. C₂×C₄⋊Dic₈₉	C₂×C₄	direct (N × H)
38. C₂×C₄²×C₈₉	C₂×C₄	semidirect
45. C₄⋊Dic₈₉	C₂²×C₄	direct (N × H)
55. C₂²×C₄×C₈₉	C₄²	semidirect
56. C₂²×C₄×C₈₉	C₄⋊C₄	semidirect
66. C₄²×C₈₉	C₂²×C₄	semidirect
80. C₂×C₄×C₈₉	C₂×C₄²	semidirect
81. C₂×C₄×C₈₉	C₂×C₄⋊C₄	semidirect
92. C₂²×C₈₉	C₄×C₄⋊C₄	semidirect
104. C₄×C₈₉	C₂²×C₄²	semidirect
105. C₄×C₈₉	C₂²×C₄⋊C₄	semidirect
108. C₂×C₈₉	C₂×C₄×C₄⋊C₄	semidirect
119. C₈₉	C₂²×C₄×C₄⋊C₄	semidirect
128. C₂²×C₄²	Dic₈₉	semidirect
151. C₂×C₄²	C₂×Dic₈₉	semidirect
167. C₂²×C₄	C₄×Dic₈₉	semidirect
168. C₂²×C₄	C₄⋊Dic₈₉	direct (N × H)
180. C₄²	C₂²×Dic₈₉	semidirect
198. C₂×C₄	C₂×C₄×Dic₈₉	semidirect
199. C₂×C₄	C₂×C₄⋊Dic₈₉	direct (N × H)
206. C₂²	C₄×C₄⋊Dic₈₉	direct (N × H)
219. C₄	C₂²×C₄×Dic₈₉	semidirect
220. C₄	C₂²×C₄⋊Dic₈₉	direct (N × H)
221. C₂	C₂×C₄×C₄⋊Dic₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊Dic₈₉	2
C₂³×C₄×Dic₈₉	2
C₂×C₄×C₄⋊Dic₈₉	2
C₂²×#15667919	2
C₂³×C₄²×C₈₉	2
C₂²×C₄×C₄⋊C₄	89

Subgroups — C22×C4×C4⋊Dic89 #36712606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₄⋊Dic₈₉ #36712606