Subgroups — #36553022

Order 22784.

998 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	C₂×#19151720	C₂	1	2	yes	maximal
2	#26028680	C₂	1	2	yes	maximal, complemented
3	C₄×#15667948	C₂	1	1	yes	maximal, characteristic, complemented
4	#26020138	C₂	1	2	yes	maximal, complemented
5	#26020092	C₂	1	4	yes	maximal, complemented
6	#26020062	C₂	1	2	yes	maximal, complemented
7	#26019938	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#15668019	C₂²	1	4	yes	—
10	#19151720	C₄	1	16	yes	complemented
11	C₂×#15667948	C₂²	1	2	yes	—
12	C₂×#15667939	C₂²	1	4	yes	—
13	#19151554	—	2	4	no	—
14	#19151554	C₂²	1	4	yes	—
15	C₄×#13083558	—	2	2	no	—
16	C₄×#13083558	C₂²	1	2	yes	—
17	C₂×#15667923	—	2	2	no	—
18	C₂×#15667923	C₂²	1	1	yes	characteristic
19	C₂×#15667922	C₂²	1	2	yes	—
20	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
21	#19151444	C₂²	1	1	yes	characteristic
22	#19151363	C₂²	1	2	yes	—
23	#19151357	C₂²	1	4	yes	—
24	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
25	C₄×#41×C₈₉	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₈₉	C₄	1	2	yes	complemented
27	C₄×SD₁₆×C₈₉	C₄	1	4	yes	complemented
28	C₄×D₈×C₈₉	C₄	1	2	yes	complemented
29	#427×C₈₉	C₂²	1	1	yes	characteristic
30	#416×C₈₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	#15668019	—	2	8	no	—
33	#15668019	C₂×C₄	1	16	yes	complemented
34	C₂×#13083558	—	2	4	no	—
35	C₂×#13083558	C₂³	1	4	yes	—
36	C₂²×#10623494	—	2	4	no	—
37	C₂²×#10623494	C₂³	1	2	yes	—
38	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
39	#15667948	C₂×C₄	1	8	yes	complemented
40	#15667939	C₂×C₄	1	16	yes	complemented
41	#15667923	—	2	4	no	—
42	#15667923	—	2	2	no	—
43	#15667923	D₄	1	2	yes	—
44	#15667923	D₄	1	2	yes	—
45	#15667922	C₂×C₄	1	8	yes	complemented
46	C₄×#10623494	—	2	4	no	—
47	C₄×#10623494	—	2	2	no	—
48	C₈×Dic₈₉	—	2	4	no	—
49	#15667919	—	2	2	no	—
50	C₂×#41×C₈₉	C₂³	1	2	yes	—
51	#135×C₈₉	C₂×C₄	1	8	yes	—
52	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₉	C₂×C₄	1	2	yes	—
54	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
58	C₄×D₄×C₈₉	—	2	4	no	—
59	C₄×D₄×C₈₉	C₂×C₄	1	4	yes	—
60	#117×C₈₉	—	2	2	no	—
61	#117×C₈₉	C₂³	1	2	yes	—
62	C₂×C₄²×C₈₉	—	2	2	no	—
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	#107×C₈₉	C₂×C₄	1	2	yes	—
65	#106×C₈₉	C₂×C₄	1	2	yes	—
66	#103×C₈₉	C₂×C₄	1	4	yes	—
67	#102×C₈₉	C₂×C₄	1	4	yes	—
68	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
70	C₂²×Dic₈₉	—	2	2	no	—
71	#13083558	—	2	4	no	—
72	#13083558	—	2	16	no	—
73	#13083558	C₂²×C₄	1	8	yes	—
74	C₂×#10623494	—	2	4	no	—
75	C₂×#10623494	—	2	16	no	—
76	C₂×#10623494	C₂²×C₄	1	4	yes	—
77	C₂×#10623494	C₂×D₄	1	4	yes	—
78	C₂×#10623494	#41	1	4	yes	—
79	C₄×Dic₈₉	—	2	2	no	—
80	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
81	#41×C₈₉	—	2	4	no	—
82	#41×C₈₉	C₂²×C₄	1	8	yes	—
83	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
84	C₂×D₄×C₈₉	—	2	2	no	—
85	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₈₉	—	2	2	no	—
87	C₂²×C₄×C₈₉	—	2	4	no	—
88	C₂²×C₄×C₈₉	#41	1	2	yes	—
89	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
90	Q₁₆×C₈₉	C₄²	1	4	yes	—
91	SD₁₆×C₈₉	C₄²	1	8	yes	—
92	D₈×C₈₉	C₄²	1	4	yes	—
93	C₂×C₈×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
96	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	—	2	4	no	—
99	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
100	C₂²⋊C₄×C₈₉	—	2	2	no	—
101	C₂²⋊C₄×C₈₉	—	2	4	no	—
102	C₄²×C₈₉	—	2	4	no	—
103	C₄²×C₈₉	—	2	2	no	—
104	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
105	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₈₉	—	2	2	no	—
108	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
109	Q₈×C₈₉	—	2	2	no	—
110	Q₈×C₈₉	C₂×C₄²	1	4	yes	—
111	D₄×C₈₉	—	2	8	no	—
112	D₄×C₈₉	—	2	2	no	—
113	D₄×C₈₉	C₂×C₄²	1	4	yes	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	8	no	—
116	C₂×C₄×C₈₉	—	2	4	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	—	8	4	no	—
119	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
125	C₂×Dic₈₉	—	2	4	no	—
126	C₂×Dic₈₉	—	4	4	no	—
127	C₈×C₈₉	—	2	2	no	—
128	C₈×C₈₉	C₂×C₄²	1	2	yes	—
129	C₈×C₈₉	C₂×C₄²	1	2	yes	—
130	#10623494	—	2	8	no	—
131	#10623494	C₄×D₄	1	8	yes	—
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	2	4	no	—
134	C₂²×C₈₉	—	4	4	no	—
135	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₈₉	C₄×#41	1	2	yes	—
137	C₄×C₈₉	—	2	2	no	—
138	C₄×C₈₉	—	2	2	no	—
139	C₄×C₈₉	—	2	4	no	—
140	C₄×C₈₉	—	4	4	no	—
141	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
142	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
143	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
144	Dic₈₉	—	4	4	no	—
145	#20513	—	89	1	no	maximal, Sylow
146	C₂×C₈₉	—	2	2	no	—
147	C₂×C₈₉	—	4	4	no	—
148	C₂×C₈₉	C₄²×D₄	1	1	yes	characteristic
149	C₂×C₈₉	#2957	1	2	yes	—
150	C₂×#444	—	89	2	no	—
151	C₄×#135	D₈₉	1	1	yes	characteristic
152	#2875	—	89	2	no	—
153	C₄×#108	—	89	1	no	—
154	#1773	—	89	2	no	—
155	#1764	—	89	4	no	—
156	#1759	—	89	2	no	—
157	#1726	—	89	1	no	—
158	C₈₉	#20513	1	1	yes	characteristic, complemented, Sylow
159	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
160	C₂×#131	—	89	4	no	—
161	C₄×#41	C₂×D₈₉	1	2	yes	—
162	#444	—	89	16	no	—
163	C₄×Q₁₆	Dic₈₉	1	2	yes	complemented
164	C₄×SD₁₆	Dic₈₉	1	4	yes	complemented
165	C₄×D₈	Dic₈₉	1	2	yes	complemented
166	C₂×#108	—	89	2	no	—
167	#427	C₂×D₈₉	1	1	yes	characteristic
168	C₂×#104	—	89	4	no	—
169	#416	C₂×D₈₉	1	2	yes	—
170	#405	—	89	4	no	—
171	#405	—	178	4	no	—
172	C₄×M₁₆	—	89	2	no	—
173	C₄×M₁₆	—	178	2	no	—
174	C₂×#97	—	89	2	no	—
175	C₂×#97	—	89	1	no	—
176	C₂×#97	—	178	2	no	—
177	C₂×C₄×C₈	—	89	2	no	—
178	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
179	#376	—	89	1	no	—
180	#341	—	89	2	no	—
181	#339	—	89	4	no	—
182	C₂×#41	C₂²×D₈₉	1	2	yes	—
183	#135	C₄×D₈₉	1	8	yes	—
184	C₂×Q₁₆	C₂×Dic₈₉	1	1	yes	characteristic
185	C₂×SD₁₆	C₂×Dic₈₉	1	2	yes	—
186	C₂×D₈	C₂×Dic₈₉	1	1	yes	characteristic
187	#131	—	89	16	no	—
188	#131	—	178	8	no	—
189	C₂×M₁₆	—	89	4	no	—
190	C₂×M₁₆	—	178	4	no	—
191	C₂²×C₈	—	89	2	no	—
192	C₂²×C₈	—	178	4	no	—
193	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
195	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
196	C₄×D₄	—	2	4	no	—
197	C₄×D₄	C₂×Dic₈₉	1	4	yes	—
198	#117	—	2	2	no	—
199	#117	C₂²×D₈₉	1	2	yes	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	89	2	no	—
202	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
203	#108	—	89	8	no	—
204	#107	C₂×Dic₈₉	1	2	yes	—
205	#106	C₂×Dic₈₉	1	2	yes	—
206	#104	—	89	16	no	—
207	#103	C₂×Dic₈₉	1	4	yes	—
208	#102	C₂×Dic₈₉	1	4	yes	—
209	#97	—	89	8	no	—
210	#97	—	89	2	no	—
211	#97	—	89	2	no	—
212	#97	—	178	4	no	—
213	#97	—	178	2	no	—
214	C₄×C₈	—	178	4	no	—
215	C₄×C₈	—	178	4	no	—
216	C₄×C₈	—	178	2	no	—
217	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
218	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
219	#95	—	178	2	no	—
220	#41	—	2	4	no	—
221	#41	C₂×C₄×D₈₉	1	8	yes	—
222	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	C₂²×Dic₈₉	1	2	yes	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	178	2	no	—
228	C₂²×C₄	#13083573	1	2	yes	—
229	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
230	Q₁₆	C₄×Dic₈₉	1	4	yes	—
231	SD₁₆	C₄×Dic₈₉	1	8	yes	—
232	D₈	C₄×Dic₈₉	1	4	yes	—
233	M₁₆	—	89	8	no	—
234	M₁₆	—	178	16	no	—
235	M₁₆	—	178	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	89	4	no	—
238	C₂×C₈	—	89	4	no	—
239	C₂×C₈	—	89	4	no	—
240	C₂×C₈	—	178	16	no	—
241	C₂×C₈	—	178	4	no	—
242	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
243	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
244	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
245	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	C₂²×Dic₈₉	1	4	yes	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	89	4	no	—
253	C₄²	—	178	2	no	—
254	C₄²	C₂²×Dic₈₉	1	2	yes	—
255	C₄²	D₄×D₈₉	1	1	yes	characteristic
256	C₄²	D₄×D₈₉	1	1	yes	characteristic
257	C₂³	—	2	2	no	—
258	C₂³	C₂×#13083573	1	1	yes	characteristic
259	Q₈	—	2	2	no	—
260	Q₈	C₂×C₄×Dic₈₉	1	4	yes	—
261	D₄	—	2	8	no	—
262	D₄	—	2	2	no	—
263	D₄	C₂×C₄×Dic₈₉	1	4	yes	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	356	4	no	—
270	C₂×C₄	—	178	4	no	—
271	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
272	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
274	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
276	C₂×C₄	#41×D₈₉	1	2	yes	—
277	C₈	—	2	2	no	—
278	C₈	—	89	8	no	—
279	C₈	—	178	8	no	—
280	C₈	C₂×C₄×Dic₈₉	1	2	yes	—
281	C₈	C₂×C₄×Dic₈₉	1	2	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
286	C₂²	C₄×#13083573	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	356	4	no	—
292	C₄	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
293	C₄	C₄×D₄×D₈₉	1	2	yes	—
294	C₄	#41×Dic₈₉	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	C₄×D₄×Dic₈₉	1	1	yes	characteristic
298	C₂	#26029169	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄×#135×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂×#135×C₈₉	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×#135	D₈₉
hypocenter	C₈₉	#20513
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. #26028680	C₂	semidirect
3. C₄×#15667948	C₂	semidirect
4. #26020138	C₂	semidirect
5. #26020092	C₂	semidirect
6. #26020062	C₂	semidirect
7. #26019938	C₂	semidirect
10. #19151720	C₄	semidirect
26. C₄×Q₁₆×C₈₉	C₄	semidirect
27. C₄×SD₁₆×C₈₉	C₄	semidirect
28. C₄×D₈×C₈₉	C₄	semidirect
33. #15668019	C₂×C₄	semidirect
39. #15667948	C₂×C₄	semidirect
40. #15667939	C₂×C₄	semidirect
45. #15667922	C₂×C₄	semidirect
158. C₈₉	#20513	semidirect
163. C₄×Q₁₆	Dic₈₉	semidirect
164. C₄×SD₁₆	Dic₈₉	semidirect
165. C₄×D₈	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151720	2
#26028680	2
C₄×#15667948	2
#26020138	2
#26020092	2
#26020062	2
#26019938	2
C₄×#135×C₈₉	2
#20513	89