Subgroups — #36472170

Order 22784.

326 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151419	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151381	C₂	1	1	yes	maximal, characteristic
3	#26021865	C₂	1	1	yes	maximal, characteristic
4	#26021848	C₂	1	1	yes	maximal, characteristic, complemented
5	#26020089	C₂	1	1	yes	maximal, characteristic
6	#26018340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₂²	1	1	yes	characteristic
9	C₂×#15667952	C₂²	1	1	yes	characteristic
10	C₂×#15667952	C₂²	1	1	yes	characteristic
11	C₂×#15667940	C₄	1	1	yes	characteristic
12	#19151595	C₄	1	1	yes	characteristic
13	#19151553	C₂²	1	1	yes	characteristic
14	#19151419	—	2	2	no	—
15	#19151381	—	2	2	no	—
16	#19151377	—	2	1	no	—
17	#19151377	—	2	1	no	—
18	#19151373	—	2	1	no	—
19	C₁₆×Dic₈₉	—	2	1	no	—
20	#19151363	C₄	1	1	yes	characteristic
21	#19151358	C₄	1	1	yes	characteristic
22	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
26	#15668018	—	2	1	no	—
27	#15668018	—	2	1	no	—
28	#15668018	—	2	1	no	—
29	#15668018	—	2	1	no	—
30	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
31	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
32	C₂×#13083557	—	2	1	no	—
33	C₂×C₈×D₈₉	—	2	1	no	—
34	#15667971	C₂×C₄	1	1	yes	characteristic
35	#15667952	—	2	1	no	—
36	#15667952	—	2	1	no	—
37	#15667952	—	2	1	no	—
38	#15667952	D₄	1	2	yes	—
39	C₂×#13083549	—	2	1	no	—
40	C₂×#13083549	—	2	1	no	—
41	#15667940	—	2	2	no	—
42	#15667937	C₈	1	1	yes	characteristic
43	#15667937	C₈	1	1	yes	characteristic
44	#15667934	C₈	1	1	yes	characteristic
45	#15667934	C₈	1	1	yes	characteristic
46	#15667922	—	2	1	no	—
47	C₈×Dic₈₉	—	2	1	no	—
48	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₁₆×C₈₉	—	2	1	no	—
52	C₂×C₁₆×C₈₉	—	2	1	no	—
53	C₂×C₁₆×C₈₉	—	2	1	no	—
54	C₂×C₁₆×C₈₉	—	2	1	no	—
55	C₂×C₁₆×C₈₉	—	2	1	no	—
56	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
57	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₈₉	—	2	1	no	—
59	C₄×C₈×C₈₉	—	2	1	no	—
60	#13083574	—	2	1	no	—
61	#13083574	—	2	1	no	—
62	#13083574	—	2	1	no	—
63	#13083574	—	2	1	no	—
64	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
65	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	#13083558	—	2	1	no	—
69	#13083558	—	2	1	no	—
70	#13083558	—	2	1	no	—
71	#13083558	M₁₆	1	2	yes	—
72	#13083557	—	4	1	no	—
73	#13083557	—	4	1	no	—
74	C₂×#10623494	—	2	1	no	—
75	C₂×#10623494	—	2	1	no	—
76	C₈×D₈₉	—	4	1	no	—
77	C₈×D₈₉	—	4	1	no	—
78	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
79	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
80	C₂²⋊Dic₈₉	—	2	1	no	—
81	C₄×Dic₈₉	—	2	1	no	—
82	#13083549	—	2	2	no	—
83	#13083549	—	4	1	no	—
84	C₂²×C₄×C₈₉	—	2	1	no	—
85	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	—	2	1	no	—
87	C₂×C₈×C₈₉	—	2	1	no	—
88	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
90	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
93	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
94	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
95	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
96	C₄²×C₈₉	—	2	1	no	—
97	C₄²×C₈₉	—	2	1	no	—
98	C₁₆×C₈₉	—	2	1	no	—
99	C₁₆×C₈₉	—	2	1	no	—
100	C₁₆×C₈₉	—	4	1	no	—
101	C₂³×C₈₉	#131	1	1	yes	characteristic
102	C₂²×D₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	—	4	1	no	—
105	C₂×C₄×C₈₉	—	4	1	no	—
106	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
112	C₂²⋊D₈₉	—	4	1	no	—
113	C₂²⋊D₈₉	—	4	1	no	—
114	C₂×Dic₈₉	—	2	1	no	—
115	C₂×Dic₈₉	—	2	1	no	—
116	C₂×Dic₈₉	—	2	1	no	—
117	D₃₅₆	—	2	1	no	—
118	D₃₅₆	—	2	1	no	—
119	C₄×D₈₉	—	4	1	no	—
120	C₄×D₈₉	—	4	1	no	—
121	Dic₁₇₈	—	2	1	no	—
122	Dic₁₇₈	—	2	1	no	—
123	C₈×C₈₉	—	2	1	no	—
124	C₈×C₈₉	—	2	1	no	—
125	C₈×C₈₉	#121	1	1	yes	characteristic
126	C₈×C₈₉	#123	1	1	yes	characteristic
127	C₈×C₈₉	#124	1	1	yes	characteristic
128	C₈×C₈₉	#126	1	1	yes	characteristic
129	#10623494	—	2	2	no	—
130	#10623494	—	4	1	no	—
131	C₂²×C₈₉	—	2	1	no	—
132	C₂²×C₈₉	—	2	1	no	—
133	C₂²×C₈₉	#433	1	1	yes	characteristic
134	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
135	C₂²×C₈₉	#436	1	1	yes	characteristic
136	C₂×D₈₉	—	4	1	no	—
137	C₂×D₈₉	—	4	1	no	—
138	C₂×D₈₉	—	4	1	no	—
139	C₄×C₈₉	—	4	2	no	—
140	C₄×C₈₉	#388	1	1	yes	characteristic
141	C₄×C₈₉	#431	1	1	yes	characteristic
142	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
143	C₄×C₈₉	#435	1	1	yes	characteristic
144	Dic₈₉	—	4	1	no	—
145	Dic₈₉	—	4	1	no	—
146	Dic₈₉	—	4	1	no	—
147	#13122	—	89	1	no	maximal, Sylow
148	C₂×C₈₉	—	2	1	no	—
149	C₂×C₈₉	—	2	1	no	—
150	C₂×C₈₉	#1923	1	1	yes	characteristic
151	C₂×C₈₉	#2170	1	1	yes	characteristic
152	C₂×C₈₉	#2171	1	1	yes	characteristic
153	D₈₉	—	8	1	no	—
154	C₂×#364	—	89	1	no	—
155	C₂×#350	—	89	1	no	—
156	#2113	—	89	1	no	—
157	#2108	—	89	1	no	—
158	#1764	—	89	1	no	—
159	#1379	—	89	1	no	—
160	#1374	D₈₉	1	1	yes	characteristic, complemented
161	C₈₉	#13122	1	1	yes	characteristic, complemented, Sylow
162	C₂×#131	—	89	1	no	—
163	C₂×M₃₂	—	89	1	no	—
164	C₂×M₃₂	—	89	1	no	—
165	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
166	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
167	C₂×#104	—	89	1	no	—
168	#416	—	89	1	no	—
169	#405	—	89	1	no	—
170	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
171	#364	—	178	2	no	—
172	#350	—	178	2	no	—
173	C₂²⋊C₁₆	—	178	1	no	—
174	C₂²⋊C₁₆	—	178	1	no	—
175	#346	—	178	1	no	—
176	C₄×C₁₆	—	178	1	no	—
177	#341	—	89	1	no	—
178	#339	—	89	1	no	—
179	C₂×#41	—	89	1	no	—
180	#131	—	178	1	no	—
181	#131	—	178	1	no	—
182	#131	—	178	1	no	—
183	#131	—	178	1	no	—
184	C₂×M₁₆	—	89	1	no	—
185	C₂×M₁₆	—	89	1	no	—
186	C₂×M₁₆	—	178	1	no	—
187	C₂²×C₈	—	178	1	no	—
188	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
190	#117	—	89	1	no	—
191	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
192	M₃₂	—	89	2	no	—
193	M₃₂	—	178	1	no	—
194	M₃₂	—	178	1	no	—
195	M₃₂	—	178	1	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	178	1	no	—
202	C₂×C₁₆	—	178	1	no	—
203	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
204	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
205	#104	—	178	2	no	—
206	#103	—	89	1	no	—
207	#103	—	89	1	no	—
208	#102	—	89	1	no	—
209	#102	—	89	1	no	—
210	#97	—	178	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	178	1	no	—
214	#41	—	178	1	no	—
215	#41	—	178	1	no	—
216	#41	—	178	1	no	—
217	#41	—	178	1	no	—
218	C₂×Q₈	—	89	1	no	—
219	C₂×D₄	—	89	1	no	—
220	C₂×D₄	—	178	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	178	1	no	—
223	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
224	M₁₆	—	89	2	no	—
225	M₁₆	—	356	1	no	—
226	M₁₆	—	356	1	no	—
227	M₁₆	—	178	1	no	—
228	M₁₆	—	178	1	no	—
229	M₁₆	—	178	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	356	1	no	—
233	C₂×C₈	—	356	1	no	—
234	C₂×C₈	—	178	1	no	—
235	C₂×C₈	—	178	1	no	—
236	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
237	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
238	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
239	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
240	C₂×C₈	#13083573	1	1	yes	characteristic
241	C₂×C₈	#13083573	1	1	yes	characteristic
242	C₂×C₈	#13083574	1	1	yes	characteristic
243	C₂×C₈	#13083575	1	1	yes	characteristic
244	C₄⋊C₄	—	89	1	no	—
245	C₄⋊C₄	—	89	1	no	—
246	C₂²⋊C₄	—	178	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	178	1	no	—
250	C₁₆	—	2	1	no	—
251	C₁₆	—	2	1	no	—
252	C₁₆	—	4	1	no	—
253	C₁₆	—	356	1	no	—
254	C₁₆	—	178	2	no	—
255	C₂³	—	178	1	no	—
256	C₂³	#15668018	1	1	yes	characteristic
257	Q₈	—	178	1	no	—
258	Q₈	—	178	1	no	—
259	D₄	—	356	1	no	—
260	D₄	—	356	1	no	—
261	D₄	—	178	1	no	—
262	D₄	—	178	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	356	1	no	—
267	C₂×C₄	—	356	1	no	—
268	C₂×C₄	—	178	1	no	—
269	C₂×C₄	—	178	1	no	—
270	C₂×C₄	—	178	1	no	—
271	C₂×C₄	#15667970	1	1	yes	characteristic
272	C₂×C₄	#15667972	1	1	yes	characteristic
273	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
275	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
276	C₂×C₄	#15668020	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	356	1	no	—
280	C₈	—	178	2	no	—
281	C₈	#15667987	1	1	yes	characteristic
282	C₈	#15667996	1	1	yes	characteristic
283	C₈	#15667998	1	1	yes	characteristic
284	C₈	#15668007	1	1	yes	characteristic
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	356	1	no	—
288	C₂²	—	356	1	no	—
289	C₂²	—	356	1	no	—
290	C₂²	#19151658	1	1	yes	characteristic
291	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
292	C₂²	#19151673	1	1	yes	characteristic
293	C₄	—	4	2	no	—
294	C₄	—	356	1	no	—
295	C₄	—	356	1	no	—
296	C₄	—	356	1	no	—
297	C₄	#19151490	1	1	yes	characteristic
298	C₄	#19151650	1	1	yes	characteristic
299	C₄	#19151661	1	1	yes	characteristic
300	C₄	#19151671	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	712	1	no	—
304	C₂	#26020859	1	1	yes	characteristic
305	C₂	#26022090	1	1	yes	characteristic
306	C₂	#26022095	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1374	C₂
socle	C₂²×C₈₉	C₈×D₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1374	D₈₉
hypocenter	C₈₉	#13122
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ #1374

Complements

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

N	H ≅ G/N	type
1. C₂×#19151419	C₂	semidirect
4. #26021848	C₂	semidirect
6. #26018340	C₂	semidirect
7. C₈₉×#1374	C₂	semidirect
160. #1374	D₈₉	semidirect
161. C₈₉	#13122	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151419	2
C₂×#19151381	2
#26021865	2
#26021848	2
#26020089	2
#26018340	2
C₈₉×#1374	2
#13122	89