Subgroups — C₄²×Q₈×D₈₉ #36712622

Order 22784.

3966 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×D₈₉	C₂	1	3	yes	maximal
2	C₄×Q₈×Dic₈₉	C₂	1	3	yes	maximal, complemented
3	C₄²×Dic₁₇₈	C₂	1	3	yes	maximal, complemented
4	C₄×#15667977	C₂	1	9	yes	maximal, complemented
5	C₄×C₄⋊C₄×D₈₉	C₂	1	9	yes	maximal
6	C₄³×D₈₉	C₂	1	3	yes	maximal
7	C₄²×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×D₈₉	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₈₉	C₂²	1	3	yes	—
10	C₂×C₄×Dic₁₇₈	C₂²	1	9	yes	—
11	C₂×#15667977	C₂²	1	9	yes	—
12	C₄×Q₈×D₈₉	C₄	1	48	yes	direct product factor
13	C₂×C₄⋊C₄×D₈₉	C₂²	1	9	yes	—
14	C₂×C₄²×D₈₉	C₂²	1	9	yes	—
15	C₂×C₄²×D₈₉	C₂²	1	9	yes	—
16	C₂×C₄²×D₈₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×Dic₈₉	C₂²	1	18	yes	—
18	C₄×#13083553	C₂²	1	18	yes	—
19	C₄×C₄⋊Dic₈₉	C₂²	1	9	yes	—
20	#19151449	C₂²	1	18	yes	—
21	#95×D₈₉	C₂²	1	6	yes	—
22	C₄²×Dic₈₉	C₂²	1	9	yes	—
23	C₄²×Dic₈₉	C₂²	1	9	yes	—
24	C₄²×Dic₈₉	C₂²	1	3	yes	—
25	C₂×C₄×Q₈×C₈₉	C₂²	1	3	yes	—
26	C₄×C₄⋊C₄×C₈₉	C₂²	1	9	yes	—
27	C₄³×C₈₉	C₂²	1	3	yes	—
28	C₂²×Dic₁₇₈	C₂³	1	3	yes	—
29	C₂×Q₈×D₈₉	C₂×C₄	1	24	yes	—
30	C₂²×C₄×D₈₉	C₂³	1	9	yes	—
31	C₂²×C₄×D₈₉	C₂³	1	3	yes	—
32	C₂²×C₄×D₈₉	C₂³	1	3	yes	—
33	Q₈×Dic₈₉	C₂×C₄	1	24	yes	complemented
34	C₄×Dic₁₇₈	C₂×C₄	1	72	yes	complemented
35	#15667977	C₂×C₄	1	72	yes	complemented
36	C₂×C₄⋊Dic₈₉	C₂³	1	9	yes	—
37	C₂×#13083553	C₂³	1	18	yes	—
38	C₄⋊C₄×D₈₉	C₂×C₄	1	72	yes	—
39	C₂×C₄×Dic₈₉	C₂³	1	18	yes	—
40	C₂×C₄×Dic₈₉	C₂³	1	9	yes	—
41	C₂×C₄×Dic₈₉	C₂³	1	9	yes	—
42	C₂×C₄×Dic₈₉	C₂³	1	9	yes	—
43	C₂×C₄×Dic₈₉	C₂³	1	9	yes	—
44	C₂×C₄×Dic₈₉	C₂³	1	3	yes	—
45	C₄²×D₈₉	—	2	36	no	—
46	C₄²×D₈₉	C₂×C₄	1	72	yes	—
47	C₄²×D₈₉	Q₈	1	8	yes	direct product factor
48	#15667919	C₂³	1	18	yes	—
49	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₈₉	C₂×C₄	1	24	yes	complemented
51	C₂×C₄⋊C₄×C₈₉	C₂³	1	9	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	9	yes	—
53	C₂×C₄²×C₈₉	C₂³	1	9	yes	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	#95×C₈₉	C₂³	1	6	yes	—
56	C₂³×D₈₉	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₁₇₈	C₂²×C₄	1	36	yes	—
58	Q₈×D₈₉	C₄²	1	64	yes	direct product factor
59	C₂²×Dic₈₉	C₂⁴	1	9	yes	—
60	C₂²×Dic₈₉	C₂⁴	1	3	yes	—
61	C₂²×Dic₈₉	C₂⁴	1	3	yes	—
62	C₂×C₄×D₈₉	—	2	36	no	—
63	C₂×C₄×D₈₉	C₂²×C₄	1	36	yes	—
64	C₂×C₄×D₈₉	C₂²×C₄	1	36	yes	—
65	C₂×C₄×D₈₉	C₂²×C₄	1	12	yes	—
66	C₂×C₄×D₈₉	C₂×Q₈	1	12	yes	—
67	C₂×C₄×D₈₉	#41	1	12	yes	—
68	C₄⋊Dic₈₉	C₂²×C₄	1	36	yes	—
69	#13083553	C₂²×C₄	1	72	yes	—
70	C₄×Dic₈₉	—	2	18	no	—
71	C₄×Dic₈₉	—	2	36	no	—
72	C₄×Dic₈₉	C₂²×C₄	1	36	yes	—
73	C₄×Dic₈₉	C₂²×C₄	1	36	yes	—
74	C₄×Dic₈₉	C₂²×C₄	1	36	yes	—
75	C₄×Dic₈₉	C₂×Q₈	1	12	yes	complemented
76	C₂×Q₈×C₈₉	C₂²×C₄	1	12	yes	—
77	C₂²×C₄×C₈₉	C₂⁴	1	9	yes	—
78	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
79	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
80	C₄⋊C₄×C₈₉	C₂²×C₄	1	36	yes	—
81	C₄²×C₈₉	—	2	18	no	—
82	C₄²×C₈₉	C₂²×C₄	1	36	yes	—
83	C₄²×C₈₉	C₂×Q₈	1	4	yes	complemented
84	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
85	C₂²×D₈₉	C₂³×C₄	1	6	yes	—
86	C₂²×D₈₉	C₂²×Q₈	1	2	yes	—
87	C₂²×D₈₉	C₂×#41	1	6	yes	—
88	Q₈×C₈₉	C₂×C₄²	1	16	yes	complemented
89	C₂×C₄×C₈₉	—	2	18	no	—
90	C₂×C₄×C₈₉	C₂³×C₄	1	18	yes	—
91	C₂×C₄×C₈₉	C₂³×C₄	1	6	yes	—
92	C₂×C₄×C₈₉	C₂³×C₄	1	18	yes	—
93	C₂×C₄×C₈₉	C₂²×Q₈	1	6	yes	—
94	C₂×C₄×C₈₉	C₂×#41	1	6	yes	—
95	C₂×Dic₈₉	—	2	18	no	—
96	C₂×Dic₈₉	C₂³×C₄	1	18	yes	—
97	C₂×Dic₈₉	C₂³×C₄	1	6	yes	—
98	C₂×Dic₈₉	C₂³×C₄	1	18	yes	—
99	C₂×Dic₈₉	C₂²×Q₈	1	6	yes	—
100	C₂×Dic₈₉	C₂×#41	1	6	yes	—
101	C₄×D₈₉	—	2	72	no	—
102	C₄×D₈₉	C₂×C₄²	1	48	yes	—
103	C₄×D₈₉	C₄×Q₈	1	48	yes	direct product factor
104	Dic₁₇₈	C₂×C₄²	1	48	yes	complemented
105	C₂²×C₈₉	C₂⁴×C₄	1	3	yes	—
106	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
107	C₂²×C₈₉	C₂²×#41	1	3	yes	—
108	C₂×D₈₉	C₂²×C₄²	1	4	yes	—
109	C₂×D₈₉	C₂×C₄×Q₈	1	12	yes	—
110	C₂×D₈₉	C₄×#41	1	12	yes	—
111	C₄×C₈₉	—	2	18	no	—
112	C₄×C₈₉	C₂²×C₄²	1	12	yes	—
113	C₄×C₈₉	C₂×C₄×Q₈	1	12	yes	complemented
114	Dic₈₉	—	2	18	no	—
115	Dic₈₉	C₂²×C₄²	1	12	yes	—
116	Dic₈₉	C₂×C₄×Q₈	1	12	yes	complemented
117	C₂×C₄²×Q₈	—	89	1	no	maximal, Sylow
118	C₂×C₈₉	C₂³×C₄²	1	1	yes	characteristic
119	C₂×C₈₉	C₂²×C₄×Q₈	1	3	yes	—
120	C₂×C₈₉	C₂×C₄×#41	1	3	yes	—
121	D₈₉	C₄²×Q₈	1	8	yes	direct product factor
122	C₂²×C₄×Q₈	—	89	3	no	—
123	C₄²×Q₈	—	89	9	no	—
124	C₄²×Q₈	—	89	3	no	—
125	C₄²×Q₈	—	89	3	no	—
126	C₄²×Q₈	D₈₉	1	1	yes	characteristic, direct product factor
127	C₂×C₄×C₄⋊C₄	—	89	9	no	—
128	C₂×C₄³	—	89	3	no	—
129	C₈₉	C₂×C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
130	C₂³×Q₈	—	89	1	no	—
131	C₂×C₄×Q₈	—	89	3	no	—
132	C₂×C₄×Q₈	—	89	48	no	—
133	C₂×C₄×Q₈	—	89	9	no	—
134	C₂×C₄×Q₈	—	89	9	no	—
135	C₂×C₄×Q₈	C₂×D₈₉	1	3	yes	—
136	C₂²×C₄⋊C₄	—	89	9	no	—
137	C₂²×C₄²	—	89	9	no	—
138	C₂²×C₄²	—	89	1	no	—
139	C₂²×C₄²	—	89	9	no	—
140	C₄×C₄⋊C₄	—	89	18	no	—
141	C₄×C₄⋊C₄	—	89	9	no	—
142	C₄×C₄⋊C₄	—	89	18	no	—
143	C₄×C₄⋊C₄	—	89	18	no	—
144	C₄×C₄⋊C₄	C₂×D₈₉	1	9	yes	—
145	C₂×#95	—	89	6	no	—
146	C₄³	—	89	9	no	—
147	C₄³	—	89	9	no	—
148	C₄³	—	89	3	no	—
149	C₄³	C₂×D₈₉	1	3	yes	—
150	C₂²×Q₈	—	89	24	no	—
151	C₂²×Q₈	—	89	3	no	—
152	C₂²×Q₈	C₂²×D₈₉	1	1	yes	characteristic
153	C₂³×C₄	—	89	3	no	—
154	C₂³×C₄	—	89	3	no	—
155	C₂³×C₄	—	89	9	no	—
156	C₄×Q₈	—	89	24	no	—
157	C₄×Q₈	—	89	72	no	—
158	C₄×Q₈	—	89	72	no	—
159	C₄×Q₈	C₄×D₈₉	1	24	yes	direct product factor
160	C₂×C₄⋊C₄	—	89	18	no	—
161	C₂×C₄⋊C₄	—	89	9	no	—
162	C₂×C₄⋊C₄	—	89	72	no	—
163	C₂×C₄⋊C₄	C₂²×D₈₉	1	9	yes	—
164	C₂×C₄²	—	89	9	no	—
165	C₂×C₄²	—	89	9	no	—
166	C₂×C₄²	—	89	9	no	—
167	C₂×C₄²	—	89	3	no	—
168	C₂×C₄²	—	89	72	no	—
169	C₂×C₄²	—	89	8	no	—
170	C₂×C₄²	—	89	18	no	—
171	C₂×C₄²	—	89	9	no	—
172	C₂×C₄²	—	178	36	no	—
173	C₂×C₄²	C₂²×D₈₉	1	9	yes	—
174	C₂×C₄²	C₂²×D₈₉	1	9	yes	—
175	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
176	#95	—	89	18	no	—
177	#95	C₂²×D₈₉	1	6	yes	—
178	C₂⁴	—	89	1	no	—
179	C₂×Q₈	—	89	64	no	—
180	C₂×Q₈	—	89	36	no	—
181	C₂×Q₈	C₂×C₄×D₈₉	1	12	yes	—
182	C₂²×C₄	—	89	3	no	—
183	C₂²×C₄	—	89	3	no	—
184	C₂²×C₄	—	89	12	no	—
185	C₂²×C₄	—	89	36	no	—
186	C₂²×C₄	—	89	12	no	—
187	C₂²×C₄	—	89	12	no	—
188	C₂²×C₄	—	89	9	no	—
189	C₂²×C₄	—	89	36	no	—
190	C₂²×C₄	—	178	36	no	—
191	C₂²×C₄	C₂³×D₈₉	1	9	yes	—
192	C₂²×C₄	C₂³×D₈₉	1	3	yes	—
193	C₂²×C₄	C₂³×D₈₉	1	3	yes	—
194	C₄⋊C₄	—	89	72	no	—
195	C₄⋊C₄	—	89	36	no	—
196	C₄⋊C₄	C₂×C₄×D₈₉	1	36	yes	—
197	C₄²	—	2	18	no	—
198	C₄²	—	89	36	no	—
199	C₄²	—	89	36	no	—
200	C₄²	—	89	36	no	—
201	C₄²	—	89	12	no	—
202	C₄²	—	178	36	no	—
203	C₄²	—	178	18	no	—
204	C₄²	C₂×C₄×D₈₉	1	36	yes	—
205	C₄²	Q₈×D₈₉	1	4	yes	direct product factor
206	C₂³	—	89	6	no	—
207	C₂³	—	89	2	no	—
208	C₂³	—	89	6	no	—
209	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
210	Q₈	—	89	48	no	—
211	Q₈	C₄²×D₈₉	1	16	yes	direct product factor
212	C₂×C₄	—	2	18	no	—
213	C₂×C₄	—	89	6	no	—
214	C₂×C₄	—	89	18	no	—
215	C₂×C₄	—	89	6	no	—
216	C₂×C₄	—	89	6	no	—
217	C₂×C₄	—	89	48	no	—
218	C₂×C₄	—	89	48	no	—
219	C₂×C₄	—	89	18	no	—
220	C₂×C₄	—	178	72	no	—
221	C₂×C₄	—	178	18	no	—
222	C₂×C₄	C₂²×C₄×D₈₉	1	18	yes	—
223	C₂×C₄	C₂²×C₄×D₈₉	1	18	yes	—
224	C₂×C₄	C₂²×C₄×D₈₉	1	6	yes	—
225	C₂×C₄	C₂×Q₈×D₈₉	1	6	yes	—
226	C₂×C₄	#41×D₈₉	1	6	yes	—
227	C₂²	—	89	12	no	—
228	C₂²	—	89	12	no	—
229	C₂²	—	89	4	no	—
230	C₂²	C₂³×C₄×D₈₉	1	3	yes	—
231	C₂²	C₂²×Q₈×D₈₉	1	1	yes	characteristic
232	C₂²	C₂×#41×D₈₉	1	3	yes	—
233	C₄	—	2	18	no	—
234	C₄	—	89	12	no	—
235	C₄	—	89	12	no	—
236	C₄	—	178	18	no	—
237	C₄	C₂×C₄²×D₈₉	1	12	yes	—
238	C₄	C₄×Q₈×D₈₉	1	12	yes	direct product factor
239	C₂	—	89	8	no	—
240	C₂	C₂²×C₄²×D₈₉	1	1	yes	characteristic
241	C₂	C₂×C₄×Q₈×D₈₉	1	3	yes	—
242	C₂	C₄×#41×D₈₉	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₈₉
commutator	C₂×C₈₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₄²×Q₈×C₈₉	C₂
socle, cosocle	C₂³×C₈₉	C₂⁵
prime radical	C₂³×D₈₉	C₂⁴
hypercenter	C₄²×Q₈	D₈₉
hypocenter	C₈₉	C₂×C₄²×Q₈
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×Dic₈₉	C₂	semidirect
3. C₄²×Dic₁₇₈	C₂	semidirect
4. C₄×#15667977	C₂	semidirect
7. C₄²×Q₈×C₈₉	C₂	semidirect
12. C₄×Q₈×D₈₉	C₄	direct (N × H)
33. Q₈×Dic₈₉	C₂×C₄	semidirect
34. C₄×Dic₁₇₈	C₂×C₄	semidirect
35. #15667977	C₂×C₄	semidirect
47. C₄²×D₈₉	Q₈	direct (N × H)
50. C₄×Q₈×C₈₉	C₂×C₄	semidirect
58. Q₈×D₈₉	C₄²	direct (N × H)
75. C₄×Dic₈₉	C₂×Q₈	semidirect
83. C₄²×C₈₉	C₂×Q₈	semidirect
88. Q₈×C₈₉	C₂×C₄²	semidirect
103. C₄×D₈₉	C₄×Q₈	direct (N × H)
104. Dic₁₇₈	C₂×C₄²	semidirect
113. C₄×C₈₉	C₂×C₄×Q₈	semidirect
116. Dic₈₉	C₂×C₄×Q₈	semidirect
121. D₈₉	C₄²×Q₈	direct (N × H)
126. C₄²×Q₈	D₈₉	direct (N × H)
129. C₈₉	C₂×C₄²×Q₈	semidirect
159. C₄×Q₈	C₄×D₈₉	direct (N × H)
205. C₄²	Q₈×D₈₉	direct (N × H)
211. Q₈	C₄²×D₈₉	direct (N × H)
238. C₄	C₄×Q₈×D₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×D₈₉	2
C₄×Q₈×Dic₈₉	2
C₄²×Dic₁₇₈	2
C₄×#15667977	2
C₄×C₄⋊C₄×D₈₉	2
C₄³×D₈₉	2
C₄²×Q₈×C₈₉	2
C₂×C₄²×Q₈	89

Subgroups — C42×Q8×D89 #36712622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×Q₈×D₈₉ #36712622