Subgroups — #36562636

Order 22784.

730 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030600	C₂	1	1	yes	maximal, characteristic
2	#26028965	C₂	1	1	yes	maximal, characteristic
3	#26021234	C₂	1	1	yes	maximal, characteristic, complemented
4	#26021083	C₂	1	1	yes	maximal, characteristic
5	#26020955	C₂	1	4	yes	maximal, complemented
6	#26020392	C₂	1	1	yes	maximal, characteristic, complemented
7	#26020098	C₂	1	4	yes	maximal, complemented
8	#26019938	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668018	C₂²	1	2	yes	—
11	#19152148	C₂²	1	1	yes	characteristic
12	#19151968	C₄	1	2	yes	complemented
13	#19151964	C₄	1	2	yes	complemented
14	#19151960	C₄	1	4	yes	complemented
15	#19151677	—	2	2	no	—
16	C₈×D₃₅₆	—	2	2	no	—
17	#19151622	C₂²	1	1	yes	characteristic
18	C₂×#15667938	C₂²	1	4	yes	—
19	#19151592	C₂²	1	4	yes	—
20	#19151564	C₂²	1	2	yes	—
21	C₂×#15667923	C₂²	1	1	yes	characteristic
22	C₂×#15667922	C₂²	1	2	yes	—
23	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
24	#19151444	C₂²	1	1	yes	characteristic
25	#19151362	C₂²	1	4	yes	—
26	#19151357	C₂²	1	4	yes	—
27	#533×C₈₉	C₂²	1	1	yes	characteristic
28	#446×C₈₉	—	2	2	no	—
29	C₈×Q₈×C₈₉	—	2	2	no	—
30	#424×C₈₉	C₂²	1	2	yes	—
31	#423×C₈₉	C₂²	1	1	yes	characteristic
32	#405×C₈₉	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#13083574	C₂³	1	2	yes	—
35	#15668018	—	2	8	no	—
36	C₂×#13083558	C₂³	1	2	yes	—
37	C₂×#13083557	—	2	2	no	—
38	C₂²×#10623494	C₂³	1	2	yes	—
39	C₂×C₈×D₈₉	—	2	2	no	—
40	#15668012	C₂×C₄	1	1	yes	characteristic
41	#15668009	C₂×C₄	1	1	yes	characteristic
42	#15668000	C₂×C₄	1	2	yes	—
43	#15667986	—	2	2	no	—
44	C₄×D₃₅₆	—	2	2	no	—
45	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
46	C₄⋊#10623494	—	2	2	no	—
47	#15667938	—	2	8	no	—
48	#15667935	C₂×C₄	1	8	yes	—
49	#15667932	C₂×C₄	1	8	yes	—
50	#15667924	—	2	4	no	—
51	#15667923	C₂×C₄	1	2	yes	—
52	#15667923	C₂×C₄	1	2	yes	—
53	#15667922	—	2	4	no	—
54	C₈×Dic₈₉	—	2	4	no	—
55	#15667919	—	2	2	no	—
56	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
57	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
60	#125×C₈₉	C₂×C₄	1	2	yes	—
61	C₂²⋊Q₈×C₈₉	—	2	2	no	—
62	C₄×Q₈×C₈₉	—	2	2	no	—
63	#117×C₈₉	C₂³	1	2	yes	—
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₈₉	—	2	4	no	—
66	C₄⋊C₈×C₈₉	—	2	2	no	—
67	#97×C₈₉	—	2	2	no	—
68	C₄×C₈×C₈₉	—	2	2	no	—
69	C₄×C₈×C₈₉	D₄	1	2	yes	—
70	C₄×C₈×C₈₉	D₄	1	2	yes	—
71	#13083574	—	2	4	no	—
72	#13083574	—	2	4	no	—
73	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
74	C₂×D₃₅₆	—	2	2	no	—
75	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
76	C₂×C₂²⋊D₈₉	—	2	2	no	—
77	C₂²×Dic₈₉	—	2	2	no	—
78	C₂×C₄×D₈₉	—	2	2	no	—
79	#13083558	—	2	4	no	—
80	#13083557	—	4	4	no	—
81	C₂×#10623494	—	2	2	no	—
82	C₂×#10623494	—	2	2	no	—
83	C₂×#10623494	—	2	2	no	—
84	C₂×#10623494	C₂²×C₄	1	4	yes	—
85	C₈×D₈₉	—	4	4	no	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	#13083551	—	2	4	no	—
88	C₄×Dic₈₉	—	2	2	no	—
89	C₄×Dic₈₉	#41	1	4	yes	—
90	C₂×Q₈×C₈₉	—	2	2	no	—
91	C₂²×C₄×C₈₉	#41	1	2	yes	—
92	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
93	M₁₆×C₈₉	—	2	4	no	—
94	C₂×C₈×C₈₉	—	2	2	no	—
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	#41	1	4	yes	—
100	C₄⋊C₄×C₈₉	—	2	4	no	—
101	C₄⋊C₄×C₈₉	—	2	2	no	—
102	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
103	C₂²⋊C₄×C₈₉	—	2	2	no	—
104	C₄²×C₈₉	—	2	2	no	—
105	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₈₉	C₂×D₄	1	2	yes	—
108	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
109	C₂²×D₈₉	—	2	2	no	—
110	Q₈×C₈₉	—	4	4	no	—
111	C₂×C₄×C₈₉	—	2	4	no	—
112	C₂×C₄×C₈₉	—	4	2	no	—
113	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
118	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
119	C₂²⋊D₈₉	—	4	4	no	—
120	C₂×Dic₈₉	—	2	2	no	—
121	C₂×Dic₈₉	—	2	4	no	—
122	C₂×Dic₈₉	—	4	4	no	—
123	D₃₅₆	—	2	4	no	—
124	D₃₅₆	—	4	4	no	—
125	C₄×D₈₉	—	4	2	no	—
126	C₄×D₈₉	—	4	2	no	—
127	Dic₁₇₈	—	2	4	no	—
128	C₈×C₈₉	—	2	2	no	—
129	C₈×C₈₉	—	4	2	no	—
130	C₈×C₈₉	#124	1	2	yes	—
131	C₈×C₈₉	#124	1	2	yes	—
132	#10623494	—	2	4	no	—
133	#10623494	—	4	2	no	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₈₉	#529	1	2	yes	—
137	C₂×D₈₉	—	4	4	no	—
138	C₂×D₈₉	—	4	2	no	—
139	C₄×C₈₉	—	2	2	no	—
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	—	4	4	no	—
142	C₄×C₈₉	C₄×#41	1	2	yes	—
143	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
144	C₄×C₈₉	#532	1	1	yes	characteristic
145	Dic₈₉	—	4	4	no	—
146	Dic₈₉	—	4	2	no	—
147	#21663	—	89	1	no	maximal, Sylow
148	C₂×C₈₉	—	2	2	no	—
149	C₂×C₈₉	C₄×#124	1	1	yes	characteristic
150	C₂×C₈₉	#2955	1	2	yes	—
151	D₈₉	—	8	2	no	—
152	#3135	—	89	1	no	—
153	#2960	D₈₉	1	1	yes	characteristic, complemented
154	#2931	—	89	1	no	—
155	#1995	—	89	1	no	—
156	#1963	—	89	1	no	—
157	#1937	—	89	4	no	—
158	#1838	—	89	1	no	—
159	#1765	—	89	4	no	—
160	#1726	—	89	1	no	—
161	C₈₉	#21663	1	1	yes	characteristic, complemented, Sylow
162	C₂×#131	—	89	2	no	—
163	#533	C₂×D₈₉	1	1	yes	characteristic
164	#532	—	89	1	no	—
165	#491	—	89	2	no	—
166	#490	—	89	2	no	—
167	#489	—	89	4	no	—
168	#446	—	2	2	no	—
169	C₈×Q₈	—	2	2	no	—
170	#436	—	178	2	no	—
171	C₈×D₄	—	178	2	no	—
172	#424	C₂×D₈₉	1	2	yes	—
173	#423	—	89	1	no	—
174	#423	C₂×D₈₉	1	1	yes	characteristic
175	C₂×#104	—	89	4	no	—
176	#416	—	89	4	no	—
177	#408	—	89	2	no	—
178	#405	C₂×D₈₉	1	2	yes	—
179	C₂×#97	—	89	1	no	—
180	C₂×#97	—	89	2	no	—
181	C₂×C₄×C₈	—	89	2	no	—
182	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
183	#376	—	89	1	no	—
184	#341	—	89	4	no	—
185	#339	—	89	4	no	—
186	C₂×#41	—	89	2	no	—
187	#131	—	178	8	no	—
188	C₂×M₁₆	—	89	2	no	—
189	C₂×M₁₆	—	178	2	no	—
190	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
191	C₂²×C₈	—	89	2	no	—
192	C₂²×C₈	—	178	2	no	—
193	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
195	C₄⋊Q₈	—	89	1	no	—
196	C₄⋊Q₈	C₄×D₈₉	1	2	yes	—
197	#127	—	89	1	no	—
198	#125	C₄×D₈₉	1	2	yes	—
199	#124	—	89	2	no	—
200	C₂²⋊Q₈	—	2	2	no	—
201	#121	—	178	2	no	—
202	C₄×Q₈	—	2	2	no	—
203	C₄×D₄	—	178	2	no	—
204	#117	C₂²×D₈₉	1	2	yes	—
205	C₂×C₄²	—	89	2	no	—
206	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
207	C₄⋊C₈	—	2	4	no	—
208	C₄⋊C₈	—	2	2	no	—
209	C₄⋊C₈	—	178	2	no	—
210	#104	—	178	8	no	—
211	#103	—	89	8	no	—
212	#102	—	89	8	no	—
213	C₂²⋊C₈	—	178	4	no	—
214	#97	—	2	2	no	—
215	#97	—	89	2	no	—
216	#97	—	89	2	no	—
217	#97	—	178	4	no	—
218	C₄×C₈	—	2	2	no	—
219	C₄×C₈	—	178	4	no	—
220	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
221	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
222	#95	—	178	2	no	—
223	#41	—	178	4	no	—
224	#41	—	178	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	89	2	no	—
227	C₂×D₄	—	89	2	no	—
228	C₂×D₄	—	178	2	no	—
229	C₂×D₄	—	178	2	no	—
230	C₂²×C₄	—	178	2	no	—
231	C₂²×C₄	—	178	2	no	—
232	C₂²×C₄	#13083574	1	2	yes	—
233	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	356	4	no	—
236	M₁₆	—	178	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	89	4	no	—
240	C₂×C₈	—	356	4	no	—
241	C₂×C₈	—	178	2	no	—
242	C₂×C₈	—	178	2	no	—
243	C₂×C₈	—	178	2	no	—
244	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
245	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
246	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
247	C₂×C₈	#13083575	1	4	yes	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	178	2	no	—
251	C₄⋊C₄	C₂×C₄×D₈₉	1	4	yes	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	178	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	89	4	no	—
256	C₄²	—	178	2	no	—
257	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
258	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
260	C₂³	—	178	2	no	—
261	C₂³	C₂×#13083574	1	1	yes	characteristic
262	Q₈	—	4	4	no	—
263	Q₈	—	178	4	no	—
264	D₄	—	356	4	no	—
265	D₄	—	356	4	no	—
266	D₄	—	178	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	356	4	no	—
270	C₂×C₄	—	356	2	no	—
271	C₂×C₄	—	356	2	no	—
272	C₂×C₄	—	178	4	no	—
273	C₂×C₄	—	178	2	no	—
274	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
275	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
276	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
278	C₂×C₄	#41×D₈₉	1	2	yes	—
279	C₂×C₄	C₂×#13083575	1	2	yes	—
280	C₈	—	2	2	no	—
281	C₈	—	4	2	no	—
282	C₈	—	356	2	no	—
283	C₈	—	178	4	no	—
284	C₈	#15668001	1	2	yes	—
285	C₈	#15668001	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	356	4	no	—
288	C₂²	—	356	2	no	—
289	C₂²	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
290	C₂²	#19152135	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	—	356	4	no	—
295	C₄	—	356	2	no	—
296	C₄	C₄×#13083575	1	2	yes	—
297	C₄	C₂×#15668001	1	1	yes	characteristic
298	C₄	#19152144	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	712	2	no	—
301	C₂	C₄×#15668001	1	1	yes	characteristic
302	C₂	#26029153	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#2960	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#19152148	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#2960	D₈₉
hypocenter	C₈₉	#21663
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #26021234	C₂	semidirect
5. #26020955	C₂	semidirect
6. #26020392	C₂	semidirect
7. #26020098	C₂	semidirect
8. #26019938	C₂	semidirect
9. C₈₉×#2960	C₂	semidirect
12. #19151968	C₄	semidirect
13. #19151964	C₄	semidirect
14. #19151960	C₄	semidirect
153. #2960	D₈₉	semidirect
161. C₈₉	#21663	semidirect

Maximal subgroups

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

subgroup	index
#26030600	2
#26028965	2
#26021234	2
#26021083	2
#26020955	2
#26020392	2
#26020098	2
#26019938	2
C₈₉×#2960	2
#21663	89