Subgroups — C₂²×#497×C₈₉ #36438482

Order 22784.

3350 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₈₉	C₂	1	2	yes	maximal, complemented
2	C₂³×SD₁₆×C₈₉	C₂	1	2	yes	maximal
3	C₂²×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#124×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#497×C₈₉	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#97×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₈₉	C₂²	1	2	yes	—
8	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
10	C₂²×Q₁₆×C₈₉	—	2	16	no	—
11	C₂²×Q₁₆×C₈₉	C₂²	1	24	yes	complemented
12	C₂²×SD₁₆×C₈₉	—	2	16	no	—
13	C₂²×SD₁₆×C₈₉	C₂²	1	24	yes	—
14	C₂³×C₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₈₉	—	2	6	no	—
16	C₂×C₄⋊Q₈×C₈₉	C₂²	1	12	yes	complemented
17	C₂×#124×C₈₉	—	2	6	no	—
18	C₂×#124×C₈₉	C₂²	1	12	yes	—
19	C₂²×C₄⋊C₄×C₈₉	—	2	2	no	—
20	C₂²×C₂²⋊C₄×C₈₉	—	2	2	no	—
21	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
22	#497×C₈₉	C₂²	1	64	yes	direct product factor
23	C₂×#97×C₈₉	C₂²	1	12	yes	complemented
24	C₂⁵×C₈₉	—	2	1	no	—
25	C₂²×Q₈×C₈₉	—	2	3	no	—
26	C₂²×Q₈×C₈₉	—	2	6	no	—
27	C₂²×Q₈×C₈₉	—	2	8	no	—
28	C₂²×Q₈×C₈₉	—	2	8	no	—
29	C₂²×Q₈×C₈₉	—	2	8	no	—
30	C₂²×Q₈×C₈₉	C₂³	1	6	yes	—
31	C₂²×Q₈×C₈₉	C₂³	1	12	yes	—
32	C₂²×D₄×C₈₉	—	2	8	no	—
33	C₂²×D₄×C₈₉	—	2	3	no	—
34	C₂²×D₄×C₈₉	C₂³	1	6	yes	—
35	C₂³×C₄×C₈₉	—	2	1	no	—
36	C₂³×C₄×C₈₉	—	2	2	no	—
37	C₂³×C₄×C₈₉	D₄	1	2	yes	—
38	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×Q₁₆×C₈₉	—	2	96	no	—
40	C₂×Q₁₆×C₈₉	C₂³	1	32	yes	complemented
41	C₂×SD₁₆×C₈₉	—	2	96	no	—
42	C₂×SD₁₆×C₈₉	C₂³	1	32	yes	—
43	C₂²×C₈×C₈₉	D₄	1	16	yes	—
44	C₂²×C₈×C₈₉	C₂³	1	12	yes	—
45	C₄⋊Q₈×C₈₉	—	2	24	no	—
46	C₄⋊Q₈×C₈₉	C₂³	1	16	yes	complemented
47	#124×C₈₉	—	2	24	no	—
48	#124×C₈₉	C₂³	1	16	yes	—
49	C₂×C₄⋊C₄×C₈₉	—	2	24	no	—
50	C₂×C₂²⋊C₄×C₈₉	—	2	24	no	—
51	C₂×C₄²×C₈₉	D₄	1	6	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	6	yes	—
53	#97×C₈₉	C₂³	1	16	yes	complemented
54	C₂⁴×C₈₉	—	2	6	no	—
55	C₂⁴×C₈₉	—	4	4	no	—
56	C₂⁴×C₈₉	—	4	8	no	—
57	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₈₉	—	2	6	no	—
59	C₂×Q₈×C₈₉	—	2	12	no	—
60	C₂×Q₈×C₈₉	—	2	24	no	—
61	C₂×Q₈×C₈₉	—	2	24	no	—
62	C₂×Q₈×C₈₉	—	2	24	no	—
63	C₂×Q₈×C₈₉	—	4	12	no	—
64	C₂×Q₈×C₈₉	—	4	24	no	—
65	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
66	C₂×Q₈×C₈₉	C₂⁴	1	8	yes	—
67	C₂×D₄×C₈₉	—	2	24	no	—
68	C₂×D₄×C₈₉	—	2	6	no	—
69	C₂×D₄×C₈₉	—	4	12	no	—
70	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
71	C₂²×C₄×C₈₉	—	2	8	no	—
72	C₂²×C₄×C₈₉	—	2	6	no	—
73	C₂²×C₄×C₈₉	—	2	12	no	—
74	C₂²×C₄×C₈₉	—	4	4	no	—
75	C₂²×C₄×C₈₉	—	4	8	no	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
77	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
79	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
80	Q₁₆×C₈₉	—	2	64	no	—
81	SD₁₆×C₈₉	—	2	64	no	—
82	C₂×C₈×C₈₉	C₂×D₄	1	48	yes	—
83	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
84	C₄⋊C₄×C₈₉	—	2	32	no	—
85	C₂²⋊C₄×C₈₉	—	2	32	no	—
86	C₄²×C₈₉	C₂×D₄	1	12	yes	—
87	C₄²×C₈₉	C₂⁴	1	4	yes	—
88	C₂³×C₈₉	—	2	4	no	—
89	C₂³×C₈₉	—	4	12	no	—
90	C₂³×C₈₉	—	4	24	no	—
91	C₂³×C₈₉	—	8	16	no	—
92	C₂³×C₈₉	#127	1	4	yes	—
93	C₂³×C₈₉	#137	1	8	yes	—
94	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
95	Q₈×C₈₉	—	2	8	no	—
96	Q₈×C₈₉	—	2	8	no	—
97	Q₈×C₈₉	—	2	8	no	—
98	Q₈×C₈₉	—	4	12	no	—
99	Q₈×C₈₉	—	4	24	no	—
100	D₄×C₈₉	—	2	8	no	—
101	D₄×C₈₉	—	4	12	no	—
102	C₂×C₄×C₈₉	—	2	24	no	—
103	C₂×C₄×C₈₉	—	2	4	no	—
104	C₂×C₄×C₈₉	—	2	8	no	—
105	C₂×C₄×C₈₉	—	4	12	no	—
106	C₂×C₄×C₈₉	—	4	24	no	—
107	C₂×C₄×C₈₉	#127	1	12	yes	—
108	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
109	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
110	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
111	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
112	C₈×C₈₉	C₂²×D₄	1	16	yes	—
113	C₂²×C₈₉	—	4	4	no	—
114	C₂²×C₈₉	—	4	8	no	—
115	C₂²×C₈₉	—	8	24	no	—
116	C₂²×C₈₉	#497	1	16	yes	direct product factor
117	C₂²×C₈₉	C₂×#127	1	6	yes	—
118	C₂²×C₈₉	C₂×#137	1	12	yes	—
119	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
120	C₄×C₈₉	—	2	8	no	—
121	C₄×C₈₉	—	4	4	no	—
122	C₄×C₈₉	—	4	8	no	—
123	C₄×C₈₉	C₂×#127	1	6	yes	—
124	C₄×C₈₉	C₂³×D₄	1	2	yes	—
125	C₂²×#497	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₈₉	—	8	4	no	—
127	C₂×C₈₉	C₂×#497	1	12	yes	direct product factor
128	C₂×C₈₉	C₂²×#127	1	1	yes	characteristic
129	C₂×C₈₉	C₂²×#137	1	2	yes	—
130	C₂³×Q₁₆	C₂×C₈₉	1	2	yes	complemented
131	C₂³×SD₁₆	C₂×C₈₉	1	2	yes	—
132	C₂²×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
133	C₂²×#124	C₂×C₈₉	1	1	yes	characteristic
134	C₂×#497	C₂×C₈₉	1	24	yes	direct product factor
135	C₂²×#97	C₂×C₈₉	1	1	yes	characteristic, complemented
136	C₈₉	C₂²×#497	1	1	yes	characteristic, direct product factor, Sylow
137	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
138	C₂³×Q₈	C₂²×C₈₉	1	2	yes	—
139	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
140	C₂²×Q₁₆	—	2	16	no	—
141	C₂²×Q₁₆	C₂²×C₈₉	1	24	yes	complemented
142	C₂²×SD₁₆	—	2	16	no	—
143	C₂²×SD₁₆	C₂²×C₈₉	1	24	yes	—
144	C₂³×C₈	C₂²×C₈₉	1	2	yes	—
145	C₂×C₄⋊Q₈	—	2	6	no	—
146	C₂×C₄⋊Q₈	C₂²×C₈₉	1	12	yes	complemented
147	C₂×#124	—	2	6	no	—
148	C₂×#124	C₂²×C₈₉	1	12	yes	—
149	C₂²×C₄⋊C₄	—	2	2	no	—
150	C₂²×C₂²⋊C₄	—	2	2	no	—
151	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
152	#497	C₂²×C₈₉	1	64	yes	direct product factor
153	C₂×#97	C₂²×C₈₉	1	12	yes	complemented
154	C₂⁵	—	2	1	no	—
155	C₂²×Q₈	—	2	3	no	—
156	C₂²×Q₈	—	2	6	no	—
157	C₂²×Q₈	—	2	8	no	—
158	C₂²×Q₈	—	2	8	no	—
159	C₂²×Q₈	—	2	8	no	—
160	C₂²×Q₈	C₂³×C₈₉	1	6	yes	—
161	C₂²×Q₈	C₂³×C₈₉	1	12	yes	—
162	C₂²×D₄	—	2	8	no	—
163	C₂²×D₄	—	2	3	no	—
164	C₂²×D₄	C₂³×C₈₉	1	6	yes	—
165	C₂³×C₄	—	2	1	no	—
166	C₂³×C₄	—	2	2	no	—
167	C₂³×C₄	D₄×C₈₉	1	2	yes	—
168	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
169	C₂×Q₁₆	—	2	96	no	—
170	C₂×Q₁₆	C₂³×C₈₉	1	32	yes	complemented
171	C₂×SD₁₆	—	2	96	no	—
172	C₂×SD₁₆	C₂³×C₈₉	1	32	yes	—
173	C₂²×C₈	D₄×C₈₉	1	16	yes	—
174	C₂²×C₈	C₂³×C₈₉	1	12	yes	—
175	C₄⋊Q₈	—	2	24	no	—
176	C₄⋊Q₈	C₂³×C₈₉	1	16	yes	complemented
177	#124	—	2	24	no	—
178	#124	C₂³×C₈₉	1	16	yes	—
179	C₂×C₄⋊C₄	—	2	24	no	—
180	C₂×C₂²⋊C₄	—	2	24	no	—
181	C₂×C₄²	D₄×C₈₉	1	6	yes	—
182	C₂×C₄²	C₂³×C₈₉	1	6	yes	—
183	#97	C₂³×C₈₉	1	16	yes	complemented
184	C₂⁴	—	2	6	no	—
185	C₂⁴	—	4	4	no	—
186	C₂⁴	—	4	8	no	—
187	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
188	C₂×Q₈	—	2	6	no	—
189	C₂×Q₈	—	2	12	no	—
190	C₂×Q₈	—	2	24	no	—
191	C₂×Q₈	—	2	24	no	—
192	C₂×Q₈	—	2	24	no	—
193	C₂×Q₈	—	4	12	no	—
194	C₂×Q₈	—	4	24	no	—
195	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
196	C₂×Q₈	C₂⁴×C₈₉	1	8	yes	—
197	C₂×D₄	—	2	24	no	—
198	C₂×D₄	—	2	6	no	—
199	C₂×D₄	—	4	12	no	—
200	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	6	no	—
203	C₂²×C₄	—	2	12	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	8	no	—
206	C₂²×C₄	C₂×D₄×C₈₉	1	12	yes	—
207	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
208	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
209	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
210	Q₁₆	—	2	64	no	—
211	SD₁₆	—	2	64	no	—
212	C₂×C₈	C₂×D₄×C₈₉	1	48	yes	—
213	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
214	C₄⋊C₄	—	2	32	no	—
215	C₂²⋊C₄	—	2	32	no	—
216	C₄²	C₂×D₄×C₈₉	1	12	yes	—
217	C₄²	C₂⁴×C₈₉	1	4	yes	—
218	C₂³	—	2	4	no	—
219	C₂³	—	4	12	no	—
220	C₂³	—	4	24	no	—
221	C₂³	—	8	16	no	—
222	C₂³	#127×C₈₉	1	4	yes	—
223	C₂³	#137×C₈₉	1	8	yes	—
224	C₂³	C₂²×D₄×C₈₉	1	3	yes	—
225	Q₈	—	2	8	no	—
226	Q₈	—	2	8	no	—
227	Q₈	—	2	8	no	—
228	Q₈	—	4	12	no	—
229	Q₈	—	4	24	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	12	no	—
232	C₂×C₄	—	2	24	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	4	12	no	—
236	C₂×C₄	—	4	24	no	—
237	C₂×C₄	#127×C₈₉	1	12	yes	—
238	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
239	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
240	C₂×C₄	C₂²×D₄×C₈₉	1	12	yes	—
241	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
242	C₈	C₂²×D₄×C₈₉	1	16	yes	—
243	C₂²	—	4	8	no	—
244	C₂²	—	4	4	no	—
245	C₂²	—	8	24	no	—
246	C₂²	#497×C₈₉	1	16	yes	direct product factor
247	C₂²	C₂×#127×C₈₉	1	6	yes	—
248	C₂²	C₂×#137×C₈₉	1	12	yes	—
249	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
250	C₄	—	2	8	no	—
251	C₄	—	4	4	no	—
252	C₄	—	4	8	no	—
253	C₄	C₂×#127×C₈₉	1	6	yes	—
254	C₄	C₂³×D₄×C₈₉	1	2	yes	—
255	C₂	—	8	4	no	—
256	C₂	C₂×#497×C₈₉	1	12	yes	direct product factor
257	C₂	C₂²×#127×C₈₉	1	1	yes	characteristic
258	C₂	C₂²×#137×C₈₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₈₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂³×D₄×C₈₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₈₉	C₂	semidirect
3. C₂²×C₄⋊Q₈×C₈₉	C₂	semidirect
5. C₂×#497×C₈₉	C₂	direct (N × H)
6. C₂²×#97×C₈₉	C₂	semidirect
11. C₂²×Q₁₆×C₈₉	C₂²	semidirect
16. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
22. #497×C₈₉	C₂²	direct (N × H)
23. C₂×#97×C₈₉	C₂²	semidirect
40. C₂×Q₁₆×C₈₉	C₂³	semidirect
46. C₄⋊Q₈×C₈₉	C₂³	semidirect
53. #97×C₈₉	C₂³	semidirect
116. C₂²×C₈₉	#497	direct (N × H)
125. C₂²×#497	C₈₉	direct (N × H)
127. C₂×C₈₉	C₂×#497	direct (N × H)
130. C₂³×Q₁₆	C₂×C₈₉	semidirect
132. C₂²×C₄⋊Q₈	C₂×C₈₉	semidirect
134. C₂×#497	C₂×C₈₉	direct (N × H)
135. C₂²×#97	C₂×C₈₉	semidirect
136. C₈₉	C₂²×#497	direct (N × H)
141. C₂²×Q₁₆	C₂²×C₈₉	semidirect
146. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
152. #497	C₂²×C₈₉	direct (N × H)
153. C₂×#97	C₂²×C₈₉	semidirect
170. C₂×Q₁₆	C₂³×C₈₉	semidirect
176. C₄⋊Q₈	C₂³×C₈₉	semidirect
183. #97	C₂³×C₈₉	semidirect
246. C₂²	#497×C₈₉	direct (N × H)
256. C₂	C₂×#497×C₈₉	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×Q₁₆×C₈₉	2
C₂³×SD₁₆×C₈₉	2
C₂²×C₄⋊Q₈×C₈₉	2
C₂²×#124×C₈₉	2
C₂×#497×C₈₉	2
C₂²×#97×C₈₉	2
C₂²×#497	89

Subgroups — C22×#497×C89 #36438482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#497×C₈₉ #36438482