Subgroups — #36553030

Order 22784.

998 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151715	C₂	1	2	yes	maximal
2	#26028680	C₂	1	2	yes	maximal, complemented
3	C₄×#15667948	C₂	1	1	yes	maximal, characteristic, complemented
4	#26020133	C₂	1	2	yes	maximal, complemented
5	#26020099	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	#19151715	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	1	yes	characteristic
44	#15667923	D₄	1	1	yes	characteristic
45	#15667923	D₄	1	2	yes	—
46	#15667922	—	2	4	no	—
47	C₄×#10623494	—	2	4	no	—
48	C₄×#10623494	—	2	2	no	—
49	C₈×Dic₈₉	C₂×C₄	1	8	yes	complemented
50	#15667919	—	2	2	no	—
51	C₂×#41×C₈₉	C₂³	1	2	yes	—
52	#135×C₈₉	C₂×C₄	1	8	yes	—
53	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₈₉	C₂×C₄	1	2	yes	—
55	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₈₉	—	2	4	no	—
60	C₄×D₄×C₈₉	C₂×C₄	1	4	yes	—
61	#117×C₈₉	—	2	2	no	—
62	#117×C₈₉	C₂³	1	2	yes	—
63	C₂×C₄²×C₈₉	—	2	2	no	—
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	#107×C₈₉	C₂×C₄	1	2	yes	—
66	#106×C₈₉	C₂×C₄	1	2	yes	—
67	#103×C₈₉	C₂×C₄	1	4	yes	—
68	#102×C₈₉	C₂×C₄	1	4	yes	—
69	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
71	C₂²×Dic₈₉	—	2	2	no	—
72	#13083558	—	2	4	no	—
73	#13083558	—	2	16	no	—
74	#13083558	C₂²×C₄	1	8	yes	—
75	C₂×#10623494	—	2	4	no	—
76	C₂×#10623494	—	2	16	no	—
77	C₂×#10623494	C₂²×C₄	1	4	yes	—
78	C₂×#10623494	C₂×D₄	1	2	yes	—
79	C₂×#10623494	C₂×D₄	1	2	yes	—
80	C₂×#10623494	#41	1	4	yes	—
81	C₄×Dic₈₉	—	2	2	no	—
82	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
83	#41×C₈₉	—	2	4	no	—
84	#41×C₈₉	C₂²×C₄	1	8	yes	—
85	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
86	C₂×D₄×C₈₉	—	2	2	no	—
87	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₈₉	—	2	2	no	—
89	C₂²×C₄×C₈₉	—	2	4	no	—
90	C₂²×C₄×C₈₉	#41	1	2	yes	—
91	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
92	Q₁₆×C₈₉	C₄²	1	4	yes	—
93	SD₁₆×C₈₉	C₄²	1	8	yes	—
94	D₈×C₈₉	C₄²	1	4	yes	—
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	—	2	4	no	—
101	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
102	C₂²⋊C₄×C₈₉	—	2	2	no	—
103	C₂²⋊C₄×C₈₉	—	2	4	no	—
104	C₄²×C₈₉	—	2	4	no	—
105	C₄²×C₈₉	—	2	2	no	—
106	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
107	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₈₉	—	2	2	no	—
110	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
111	Q₈×C₈₉	—	2	2	no	—
112	Q₈×C₈₉	C₂×C₄²	1	4	yes	—
113	D₄×C₈₉	—	2	8	no	—
114	D₄×C₈₉	—	2	2	no	—
115	D₄×C₈₉	C₂×C₄²	1	4	yes	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	8	no	—
118	C₂×C₄×C₈₉	—	2	4	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	8	4	no	—
121	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
124	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
127	C₂×Dic₈₉	—	2	4	no	—
128	C₂×Dic₈₉	—	4	4	no	—
129	C₈×C₈₉	—	2	2	no	—
130	C₈×C₈₉	C₂×C₄²	1	2	yes	—
131	C₈×C₈₉	C₂×C₄²	1	2	yes	—
132	#10623494	—	2	8	no	—
133	#10623494	C₄×D₄	1	4	yes	—
134	#10623494	C₄×D₄	1	4	yes	—
135	C₂²×C₈₉	—	2	2	no	—
136	C₂²×C₈₉	—	2	4	no	—
137	C₂²×C₈₉	—	4	4	no	—
138	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₈₉	C₄×#41	1	2	yes	—
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	—	2	2	no	—
142	C₄×C₈₉	—	2	4	no	—
143	C₄×C₈₉	—	4	4	no	—
144	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
145	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
146	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
147	Dic₈₉	—	4	4	no	—
148	#20514	—	89	1	no	maximal, Sylow
149	C₂×C₈₉	—	2	2	no	—
150	C₂×C₈₉	—	4	4	no	—
151	C₂×C₈₉	C₄²×D₄	1	1	yes	characteristic
152	C₂×C₈₉	#2957	1	2	yes	—
153	C₂×#443	—	89	2	no	—
154	C₄×#135	D₈₉	1	1	yes	characteristic
155	#2875	—	89	2	no	—
156	C₄×#108	—	89	1	no	—
157	#1772	—	89	2	no	—
158	#1765	—	89	4	no	—
159	#1759	—	89	2	no	—
160	#1726	—	89	1	no	—
161	C₈₉	#20514	1	1	yes	characteristic, complemented, Sylow
162	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
163	C₂×#131	—	89	4	no	—
164	C₄×#41	C₂×D₈₉	1	2	yes	—
165	#443	—	89	16	no	—
166	C₄×Q₁₆	Dic₈₉	1	2	yes	complemented
167	C₄×SD₁₆	Dic₈₉	1	4	yes	complemented
168	C₄×D₈	Dic₈₉	1	2	yes	complemented
169	C₂×#108	—	89	2	no	—
170	#427	C₂×D₈₉	1	1	yes	characteristic
171	C₂×#104	—	89	4	no	—
172	#416	C₂×D₈₉	1	2	yes	—
173	#405	—	89	4	no	—
174	#405	—	178	4	no	—
175	C₄×M₁₆	—	89	2	no	—
176	C₄×M₁₆	—	178	2	no	—
177	C₂×#97	—	89	1	no	—
178	C₂×#97	—	89	2	no	—
179	C₂×#97	—	178	2	no	—
180	C₂×C₄×C₈	—	89	2	no	—
181	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
182	#376	—	89	1	no	—
183	#341	—	89	2	no	—
184	#339	—	89	4	no	—
185	C₂×#41	C₂²×D₈₉	1	2	yes	—
186	#135	C₄×D₈₉	1	8	yes	—
187	C₂×Q₁₆	C₂×Dic₈₉	1	1	yes	characteristic
188	C₂×SD₁₆	C₂×Dic₈₉	1	2	yes	—
189	C₂×D₈	C₂×Dic₈₉	1	1	yes	characteristic
190	#131	—	89	16	no	—
191	#131	—	178	8	no	—
192	C₂×M₁₆	—	89	4	no	—
193	C₂×M₁₆	—	178	4	no	—
194	C₂²×C₈	—	89	2	no	—
195	C₂²×C₈	—	178	4	no	—
196	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
198	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
199	C₄×D₄	—	2	4	no	—
200	C₄×D₄	C₂×Dic₈₉	1	4	yes	—
201	#117	—	2	2	no	—
202	#117	C₂²×D₈₉	1	2	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	89	2	no	—
205	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
206	#108	—	89	8	no	—
207	#107	C₂×Dic₈₉	1	2	yes	—
208	#106	C₂×Dic₈₉	1	2	yes	—
209	#104	—	89	16	no	—
210	#103	C₂×Dic₈₉	1	4	yes	—
211	#102	C₂×Dic₈₉	1	4	yes	—
212	#97	—	89	1	no	—
213	#97	—	89	1	no	—
214	#97	—	89	2	no	—
215	#97	—	178	4	no	—
216	#97	—	178	4	no	—
217	#97	—	178	2	no	—
218	C₄×C₈	—	89	8	no	—
219	C₄×C₈	—	178	4	no	—
220	C₄×C₈	—	178	2	no	—
221	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
222	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
223	#95	—	178	2	no	—
224	#41	—	2	4	no	—
225	#41	C₂×C₄×D₈₉	1	8	yes	—
226	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₂²×Dic₈₉	1	2	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	178	2	no	—
232	C₂²×C₄	#13083573	1	2	yes	—
233	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
234	Q₁₆	C₄×Dic₈₉	1	4	yes	—
235	SD₁₆	C₄×Dic₈₉	1	8	yes	—
236	D₈	C₄×Dic₈₉	1	4	yes	—
237	M₁₆	—	89	8	no	—
238	M₁₆	—	178	16	no	—
239	M₁₆	—	178	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	89	4	no	—
242	C₂×C₈	—	89	2	no	—
243	C₂×C₈	—	89	2	no	—
244	C₂×C₈	—	89	4	no	—
245	C₂×C₈	—	178	16	no	—
246	C₂×C₈	—	178	4	no	—
247	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
248	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
249	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
250	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	C₂²×Dic₈₉	1	4	yes	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	89	4	no	—
258	C₄²	—	178	2	no	—
259	C₄²	C₂²×Dic₈₉	1	2	yes	—
260	C₄²	D₄×D₈₉	1	1	yes	characteristic
261	C₄²	D₄×D₈₉	1	1	yes	characteristic
262	C₂³	—	2	2	no	—
263	C₂³	C₂×#13083573	1	1	yes	characteristic
264	Q₈	—	2	2	no	—
265	Q₈	C₂×C₄×Dic₈₉	1	4	yes	—
266	D₄	—	2	8	no	—
267	D₄	—	2	2	no	—
268	D₄	C₂×C₄×Dic₈₉	1	4	yes	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	4	no	—
274	C₂×C₄	—	356	4	no	—
275	C₂×C₄	—	178	4	no	—
276	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
277	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
279	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
281	C₂×C₄	#41×D₈₉	1	2	yes	—
282	C₈	—	2	2	no	—
283	C₈	—	89	4	no	—
284	C₈	—	89	4	no	—
285	C₈	—	178	8	no	—
286	C₈	C₂×C₄×Dic₈₉	1	2	yes	—
287	C₈	C₂×C₄×Dic₈₉	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
292	C₂²	C₄×#13083573	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	4	no	—
296	C₄	—	4	4	no	—
297	C₄	—	356	4	no	—
298	C₄	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
299	C₄	C₄×D₄×D₈₉	1	2	yes	—
300	C₄	#41×Dic₈₉	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	4	4	no	—
303	C₂	C₄×D₄×Dic₈₉	1	1	yes	characteristic
304	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	#20514
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. #26020133	C₂	semidirect
5. #26020099	C₂	semidirect
6. #26020062	C₂	semidirect
7. #26019938	C₂	semidirect
10. #19151715	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
49. C₈×Dic₈₉	C₂×C₄	semidirect
161. C₈₉	#20514	semidirect
166. C₄×Q₁₆	Dic₈₉	semidirect
167. C₄×SD₁₆	Dic₈₉	semidirect
168. 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×#19151715	2
#26028680	2
C₄×#15667948	2
#26020133	2
#26020099	2
#26020062	2
#26019938	2
C₄×#135×C₈₉	2
#20514	89