Subgroups — #36523598

Order 22784.

642 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#432×D₈₉	C₂	1	2	yes	maximal
2	C₂×#19151649	C₂	1	1	yes	maximal, characteristic
3	#26019036	C₂	1	4	yes	maximal, complemented
4	#26019034	C₂	1	4	yes	maximal, complemented
5	#26019015	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019013	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₈₉	C₂²	1	1	yes	characteristic
9	#19151655	C₂²	1	2	yes	—
10	#19151654	C₂²	1	4	yes	—
11	#19151653	—	2	2	no	—
12	#19151652	—	2	2	no	—
13	#19151649	C₄	1	4	yes	—
14	#19151649	C₂²	1	2	yes	—
15	#19151649	C₂²	1	2	yes	—
16	C₂×#15667941	C₄	1	2	yes	—
17	C₄⋊C₈×D₈₉	C₂²	1	4	yes	—
18	C₂×#15667924	C₄	1	2	yes	—
19	C₂²⋊C₈×D₈₉	—	2	2	no	—
20	#97×D₈₉	—	2	2	no	—
21	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
22	#19151351	C₂²	1	2	yes	complemented
23	#19151350	C₂²	1	4	yes	complemented
24	#19151349	C₂²	1	2	yes	complemented
25	#19151348	C₂²	1	4	yes	complemented
26	#432×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	#335×C₈₉	C₂²	1	2	yes	complemented
29	#334×C₈₉	C₂²	1	2	yes	complemented
30	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×D₈₉	Q₈	1	1	yes	characteristic
33	C₂²×#10623494	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₈₉	—	2	4	no	—
35	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₈₉	Q₈	1	1	yes	characteristic
38	C₄²×D₈₉	C₂×C₄	1	2	yes	—
39	C₄²×D₈₉	D₄	1	2	yes	—
40	C₄²×D₈₉	Q₈	1	2	yes	—
41	C₄²×D₈₉	C₂³	1	2	yes	—
42	C₄⋊#10623494	C₂³	1	4	yes	—
43	#15667941	—	2	4	no	—
44	#15667941	C₂×C₄	1	4	yes	—
45	#15667941	C₂×C₄	1	4	yes	—
46	C₂²⋊#10623494	—	2	2	no	—
47	#15667924	—	2	4	no	—
48	#15667924	C₂×C₄	1	4	yes	—
49	#15667924	C₂×C₄	1	4	yes	—
50	#15667923	—	2	2	no	—
51	#15667922	—	2	4	no	—
52	C₄×#10623494	C₂×C₄	1	2	yes	—
53	C₄×#10623494	C₂³	1	2	yes	—
54	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
57	C₂²⋊C₈×C₈₉	—	2	2	no	—
58	#97×C₈₉	—	2	2	no	—
59	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂³×D₈₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×D₈₉	—	2	4	no	—
67	C₂×C₄×D₈₉	—	4	2	no	—
68	C₂×C₄×D₈₉	—	4	2	no	—
69	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
73	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
74	C₂×C₄×D₈₉	C₂×Q₈	1	2	yes	—
75	C₂×#10623494	—	2	4	no	—
76	C₂×#10623494	—	4	2	no	—
77	C₂×#10623494	C₂²×C₄	1	4	yes	—
78	C₈×D₈₉	—	4	8	no	—
79	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
80	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
81	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
82	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
83	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
84	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₈₉	—	2	4	no	—
89	C₂×C₈×C₈₉	—	4	2	no	—
90	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
92	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
95	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
96	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×D₈₉	—	4	1	no	—
98	C₂²×D₈₉	—	4	1	no	—
99	C₂²×D₈₉	#108	1	4	yes	—
100	C₂²×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₈₉	—	2	2	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
110	C₂×Dic₈₉	—	2	2	no	—
111	C₂×Dic₈₉	—	4	1	no	—
112	C₂×Dic₈₉	—	4	1	no	—
113	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
116	C₄×D₈₉	—	2	4	no	—
117	C₄×D₈₉	—	2	4	no	—
118	C₄×D₈₉	—	2	4	no	—
119	C₄×D₈₉	—	2	4	no	—
120	C₄×D₈₉	—	4	2	no	—
121	C₄×D₈₉	—	4	2	no	—
122	C₈×C₈₉	—	2	2	no	—
123	C₈×C₈₉	—	2	2	no	—
124	C₈×C₈₉	—	4	4	no	—
125	#10623494	—	2	4	no	—
126	#10623494	—	4	4	no	—
127	C₂²×C₈₉	—	2	1	no	—
128	C₂²×C₈₉	—	2	1	no	—
129	C₂²×C₈₉	C₂×#108	1	2	yes	—
130	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₂×D₈₉	—	2	2	no	—
132	C₂×D₈₉	—	2	2	no	—
133	C₂×D₈₉	—	4	4	no	—
134	C₂×D₈₉	C₂×#108	1	4	yes	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	—	2	1	no	—
139	C₄×C₈₉	#568	1	2	yes	—
140	C₄×C₈₉	#575	1	2	yes	—
141	Dic₈₉	—	2	2	no	—
142	Dic₈₉	—	2	2	no	—
143	Dic₈₉	—	2	2	no	—
144	#17818	—	89	1	no	maximal, Sylow
145	C₂×C₈₉	—	2	2	no	—
146	C₂×C₈₉	#2905	1	1	yes	characteristic
147	C₂×C₈₉	#2911	1	1	yes	characteristic
148	C₂×C₈₉	C₂²×#108	1	1	yes	characteristic
149	D₈₉	—	2	4	no	—
150	C₂×#432	—	89	2	no	—
151	C₂×#431	—	89	1	no	—
152	#1573	—	89	4	no	—
153	#1573	—	89	4	no	—
154	#1568	—	89	2	no	—
155	#1568	—	89	1	no	—
156	#1568	D₈₉	1	1	yes	characteristic, complemented
157	C₈₉	#17818	1	1	yes	characteristic, complemented, Sylow
158	C₂²×C₄²	—	89	1	no	—
159	#432	—	89	2	no	—
160	#432	—	89	4	no	—
161	#432	—	178	2	no	—
162	#432	—	178	2	no	—
163	#432	C₂×D₈₉	1	2	yes	—
164	#431	—	89	4	no	—
165	#431	—	89	2	no	—
166	#431	—	89	2	no	—
167	C₂×C₄⋊C₈	—	89	2	no	—
168	C₂×C₄⋊C₈	—	89	4	no	—
169	C₂×C₂²⋊C₈	—	89	2	no	—
170	C₂×C₂²⋊C₈	—	178	2	no	—
171	C₂×#97	—	178	2	no	—
172	C₂×C₄×C₈	—	89	1	no	—
173	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
174	#335	—	89	4	no	—
175	#335	—	89	2	no	—
176	#335	C₂×D₈₉	1	2	yes	complemented
177	#334	—	89	4	no	—
178	#334	—	89	2	no	—
179	#334	C₂×D₈₉	1	2	yes	complemented
180	C₂³×C₄	—	89	1	no	—
181	C₂³×C₄	—	89	1	no	—
182	C₂³×C₄	—	89	1	no	—
183	C₂²×C₈	—	89	2	no	—
184	C₂²×C₈	—	178	4	no	—
185	C₂²×C₈	C₄×D₈₉	1	2	yes	—
186	C₂×C₄²	—	89	1	no	—
187	C₂×C₄²	—	89	1	no	—
188	C₂×C₄²	—	89	1	no	—
189	C₂×C₄²	—	89	2	no	—
190	C₂×C₄²	—	89	2	no	—
191	C₂×C₄²	—	89	2	no	—
192	C₂×C₄²	—	89	2	no	—
193	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
194	C₄⋊C₈	—	89	4	no	—
195	C₄⋊C₈	—	89	4	no	—
196	C₄⋊C₈	—	89	4	no	—
197	C₄⋊C₈	—	178	4	no	—
198	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	89	4	no	—
201	C₂²⋊C₈	—	89	4	no	—
202	C₂²⋊C₈	—	178	4	no	—
203	C₂²⋊C₈	—	178	2	no	—
204	#97	—	2	2	no	—
205	#97	—	178	4	no	—
206	#97	—	178	2	no	—
207	C₄×C₈	—	89	2	no	—
208	C₄×C₈	—	89	2	no	—
209	C₄×C₈	C₄×D₈₉	1	2	yes	—
210	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
212	C₂⁴	—	89	1	no	—
213	C₂²×C₄	—	89	1	no	—
214	C₂²×C₄	—	89	1	no	—
215	C₂²×C₄	—	89	1	no	—
216	C₂²×C₄	—	89	2	no	—
217	C₂²×C₄	—	89	2	no	—
218	C₂²×C₄	—	89	2	no	—
219	C₂²×C₄	—	89	2	no	—
220	C₂²×C₄	—	89	2	no	—
221	C₂²×C₄	—	89	2	no	—
222	C₂²×C₄	—	356	2	no	—
223	C₂²×C₄	—	356	2	no	—
224	C₂²×C₄	—	178	4	no	—
225	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
226	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
227	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	89	4	no	—
231	C₂×C₈	—	356	8	no	—
232	C₂×C₈	—	356	2	no	—
233	C₂×C₈	—	178	4	no	—
234	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
236	C₄²	—	89	2	no	—
237	C₄²	—	89	2	no	—
238	C₄²	—	89	2	no	—
239	C₄²	—	89	2	no	—
240	C₄²	—	89	2	no	—
241	C₄²	—	89	2	no	—
242	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
243	C₄²	D₄×D₈₉	1	1	yes	characteristic
244	C₄²	Q₈×D₈₉	1	1	yes	characteristic
245	C₄²	C₂³×D₈₉	1	1	yes	characteristic
246	C₂³	—	89	4	no	—
247	C₂³	—	89	2	no	—
248	C₂³	—	356	1	no	—
249	C₂³	—	356	1	no	—
250	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	89	2	no	—
255	C₂×C₄	—	89	2	no	—
256	C₂×C₄	—	89	2	no	—
257	C₂×C₄	—	356	2	no	—
258	C₂×C₄	—	356	2	no	—
259	C₂×C₄	—	356	1	no	—
260	C₂×C₄	—	356	1	no	—
261	C₂×C₄	—	178	4	no	—
262	C₂×C₄	—	178	4	no	—
263	C₂×C₄	—	178	2	no	—
264	C₂×C₄	—	178	4	no	—
265	C₂×C₄	—	178	4	no	—
266	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
268	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
270	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
271	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₈	—	4	4	no	—
275	C₈	—	356	4	no	—
276	C₈	—	178	4	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	89	4	no	—
280	C₂²	—	356	4	no	—
281	C₂²	—	178	2	no	—
282	C₂²	—	178	2	no	—
283	C₂²	#108×D₈₉	1	2	yes	—
284	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	178	2	no	—
290	C₄	—	178	2	no	—
291	C₄	—	178	2	no	—
292	C₄	#19152364	1	2	yes	—
293	C₄	#19152395	1	2	yes	—
294	C₂	—	2	2	no	—
295	C₂	—	178	4	no	—
296	C₂	#26028814	1	1	yes	characteristic
297	C₂	#26028840	1	1	yes	characteristic
298	C₂	C₂×#108×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1568	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂³×D₈₉	C₄⋊C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1568	D₈₉
hypocenter	C₈₉	#17818
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #26019036	C₂	semidirect
4. #26019034	C₂	semidirect
5. #26019015	C₂	semidirect
6. #26019013	C₂	semidirect
7. C₈₉×#1568	C₂	semidirect
22. #19151351	C₂²	semidirect
23. #19151350	C₂²	semidirect
24. #19151349	C₂²	semidirect
25. #19151348	C₂²	semidirect
28. #335×C₈₉	C₂²	semidirect
29. #334×C₈₉	C₂²	semidirect
156. #1568	D₈₉	semidirect
157. C₈₉	#17818	semidirect
176. #335	C₂×D₈₉	semidirect
179. #334	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#432×D₈₉	2
C₂×#19151649	2
#26019036	2
#26019034	2
#26019015	2
#26019013	2
C₈₉×#1568	2
#17818	89