Subgroups — #36462697

Order 22784.

642 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151622	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151363	C₂	1	2	yes	maximal
3	C₂×#19151363	C₂	1	2	yes	maximal
4	#26019946	C₂	1	1	yes	maximal, characteristic
5	C₂×#336×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	2	yes	—
7	C₂²×#13083558	C₂²	1	1	yes	characteristic
8	C₂²×#13083558	C₂²	1	1	yes	characteristic
9	#19151622	—	2	4	no	—
10	C₂×C₄⋊#10623494	—	2	2	no	—
11	#19151363	—	2	4	no	—
12	#19151363	—	2	4	no	—
13	#19151353	—	2	2	no	—
14	C₂³×C₈×C₈₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	#336×C₈₉	C₄	1	2	yes	—
17	#336×C₈₉	C₄	1	2	yes	—
18	C₂×#13083558	—	2	2	no	—
19	C₂×#13083558	—	2	2	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	1	no	—
22	C₂×#13083558	—	2	1	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	—	2	2	no	—
26	C₂×#13083558	—	2	2	no	—
27	C₂×#13083558	—	2	2	no	—
28	C₂×#13083558	—	2	2	no	—
29	C₂×#13083558	D₄	1	2	yes	—
30	C₂×#13083558	D₄	1	4	yes	—
31	C₂×#13083558	Q₈	1	2	yes	—
32	C₂²×#10623494	—	2	2	no	—
33	C₂²×#10623494	—	2	1	no	—
34	C₂²×#10623494	—	2	1	no	—
35	C₄⋊#10623494	—	2	8	no	—
36	C₂³×C₄×C₈₉	Q₈	1	2	yes	—
37	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	—	2	2	no	—
39	C₂²×C₈×C₈₉	—	2	2	no	—
40	C₂²×C₈×C₈₉	—	2	2	no	—
41	C₂²×C₈×C₈₉	—	2	2	no	—
42	C₂²×C₈×C₈₉	—	2	2	no	—
43	C₂²×C₈×C₈₉	—	2	2	no	—
44	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₈₉	—	2	2	no	—
46	C₂×C₄²×C₈₉	—	2	2	no	—
47	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	D₄	1	2	yes	—
50	#13083558	—	2	4	no	—
51	#13083558	—	2	4	no	—
52	#13083558	—	2	4	no	—
53	#13083558	—	2	4	no	—
54	#13083558	—	4	2	no	—
55	#13083558	—	4	2	no	—
56	#13083558	—	4	2	no	—
57	#13083558	—	4	2	no	—
58	C₂×#10623494	—	2	4	no	—
59	C₂×#10623494	—	2	2	no	—
60	C₂×#10623494	—	2	2	no	—
61	C₂×#10623494	—	4	1	no	—
62	C₂×#10623494	—	4	2	no	—
63	C₂×#10623494	—	4	1	no	—
64	C₂×#10623494	—	4	1	no	—
65	C₂×#10623494	—	4	2	no	—
66	C₂×#10623494	—	4	1	no	—
67	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	—	2	4	no	—
69	C₂²×C₄×C₈₉	—	2	4	no	—
70	C₂²×C₄×C₈₉	—	4	2	no	—
71	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	#41	1	2	yes	—
79	C₂²×C₄×C₈₉	#41	1	2	yes	—
80	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	#41	1	2	yes	—
82	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₈₉	—	2	4	no	—
84	C₂×C₈×C₈₉	—	2	4	no	—
85	C₂×C₈×C₈₉	—	2	4	no	—
86	C₂×C₈×C₈₉	—	4	2	no	—
87	C₂×C₈×C₈₉	—	4	2	no	—
88	C₂×C₈×C₈₉	—	4	2	no	—
89	C₂×C₈×C₈₉	—	4	2	no	—
90	C₄²×C₈₉	—	2	2	no	—
91	C₄²×C₈₉	—	2	2	no	—
92	C₄²×C₈₉	—	2	2	no	—
93	C₄²×C₈₉	—	2	2	no	—
94	C₂³×C₈₉	—	2	2	no	—
95	C₂³×C₈₉	—	2	2	no	—
96	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₈₉	#125	1	1	yes	characteristic
99	C₂³×C₈₉	#125	1	2	yes	—
100	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
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	—	2	2	no	—
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	4	8	no	—
107	C₂×C₄×C₈₉	#108	1	4	yes	—
108	C₂×C₄×C₈₉	#117	1	2	yes	—
109	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#121	1	2	yes	—
114	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#123	1	2	yes	—
118	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
119	C₂×C₄×C₈₉	#123	1	2	yes	—
120	C₂×C₄×C₈₉	#124	1	2	yes	—
121	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#125	1	2	yes	—
124	C₂×C₄×C₈₉	#126	1	2	yes	—
125	C₂×C₄×C₈₉	#126	1	2	yes	—
126	C₈×C₈₉	—	4	4	no	—
127	#10623494	—	4	2	no	—
128	#10623494	—	4	4	no	—
129	#10623494	—	4	2	no	—
130	C₂²×C₈₉	—	2	2	no	—
131	C₂²×C₈₉	—	2	2	no	—
132	C₂²×C₈₉	—	2	4	no	—
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	#382	1	1	yes	characteristic
136	C₂²×C₈₉	#385	1	1	yes	characteristic
137	C₂²×C₈₉	#389	1	1	yes	characteristic
138	C₂²×C₈₉	#398	1	2	yes	—
139	C₂²×C₈₉	#400	1	2	yes	—
140	C₂²×C₈₉	C₂×#108	1	2	yes	—
141	C₂²×C₈₉	#430	1	2	yes	—
142	C₄×C₈₉	—	2	4	no	—
143	C₄×C₈₉	—	4	2	no	—
144	C₄×C₈₉	#387	1	2	yes	—
145	C₄×C₈₉	#388	1	2	yes	—
146	C₄×C₈₉	#393	1	1	yes	characteristic
147	C₄×C₈₉	#394	1	1	yes	characteristic
148	C₄×C₈₉	#397	1	1	yes	characteristic
149	C₄×C₈₉	#400	1	1	yes	characteristic
150	#11269	—	89	1	no	maximal, Sylow
151	C₂×C₈₉	—	2	2	no	—
152	C₂×C₈₉	—	2	2	no	—
153	C₂×C₈₉	#1442	1	1	yes	characteristic
154	C₂×C₈₉	#1837	1	1	yes	characteristic
155	C₂×C₈₉	#1838	1	1	yes	characteristic
156	C₂×C₈₉	#1855	1	2	yes	—
157	C₂×C₈₉	#1950	1	2	yes	—
158	C₂×#423	—	89	1	no	—
159	C₂×#341	—	89	2	no	—
160	C₂×#341	—	89	2	no	—
161	#1728	—	89	1	no	—
162	C₂×#336	D₈₉	1	1	yes	characteristic
163	C₈₉	#11269	1	1	yes	characteristic, complemented, Sylow
164	C₂²×M₁₆	—	89	2	no	—
165	C₂²×M₁₆	—	89	1	no	—
166	C₂²×M₁₆	—	89	1	no	—
167	C₂³×C₈	C₂×D₈₉	1	2	yes	—
168	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
169	#423	—	178	4	no	—
170	C₂×C₄⋊C₈	—	178	2	no	—
171	#341	—	178	4	no	—
172	#341	—	178	4	no	—
173	#336	—	178	2	no	—
174	#336	Dic₈₉	1	2	yes	—
175	#336	Dic₈₉	1	2	yes	—
176	C₂³×C₄	Dic₁₇₈	1	2	yes	—
177	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
178	C₂×M₁₆	—	89	2	no	—
179	C₂×M₁₆	—	89	4	no	—
180	C₂×M₁₆	—	89	2	no	—
181	C₂×M₁₆	—	178	2	no	—
182	C₂×M₁₆	—	178	2	no	—
183	C₂×M₁₆	—	178	2	no	—
184	C₂×M₁₆	—	178	2	no	—
185	C₂×M₁₆	—	178	2	no	—
186	C₂×M₁₆	—	178	2	no	—
187	C₂×M₁₆	—	178	2	no	—
188	C₂×M₁₆	—	178	2	no	—
189	C₂×M₁₆	—	178	2	no	—
190	C₂×M₁₆	—	178	1	no	—
191	C₂×M₁₆	—	178	1	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	178	1	no	—
199	C₂²×C₈	—	178	1	no	—
200	C₂²×C₈	—	178	2	no	—
201	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	D₃₅₆	1	2	yes	—
205	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
206	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
207	C₄⋊C₈	—	178	8	no	—
208	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
213	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₈₉	1	2	yes	—
216	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
218	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
219	C₂²×C₄	#13083573	1	1	yes	characteristic
220	C₂²×C₄	#13083573	1	2	yes	—
221	C₂²×C₄	#13083574	1	2	yes	—
222	C₂²×C₄	#13083574	1	2	yes	—
223	C₂²×C₄	#13083575	1	1	yes	characteristic
224	M₁₆	—	356	2	no	—
225	M₁₆	—	356	2	no	—
226	M₁₆	—	356	2	no	—
227	M₁₆	—	356	2	no	—
228	M₁₆	—	178	4	no	—
229	M₁₆	—	178	4	no	—
230	M₁₆	—	178	4	no	—
231	M₁₆	—	178	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	356	1	no	—
240	C₂×C₈	—	356	2	no	—
241	C₂×C₈	—	356	1	no	—
242	C₂×C₈	—	356	1	no	—
243	C₂×C₈	—	356	2	no	—
244	C₂×C₈	—	356	1	no	—
245	C₂×C₈	—	178	4	no	—
246	C₂×C₈	—	178	2	no	—
247	C₂×C₈	—	178	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
255	C₂³	#15667989	1	2	yes	—
256	C₂³	#15668003	1	1	yes	characteristic
257	C₂³	#15668004	1	2	yes	—
258	C₂³	#15668011	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	#15667948	1	4	yes	—
266	C₂×C₄	#15667971	1	2	yes	—
267	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
268	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
269	C₂×C₄	Q₈×Dic₈₉	1	1	yes	characteristic
270	C₂×C₄	#15667982	1	1	yes	characteristic
271	C₂×C₄	#15667985	1	2	yes	—
272	C₂×C₄	#15667987	1	1	yes	characteristic
273	C₂×C₄	#15667990	1	2	yes	—
274	C₂×C₄	#15667992	1	1	yes	characteristic
275	C₂×C₄	#15667994	1	2	yes	—
276	C₂×C₄	#15667996	1	1	yes	characteristic
277	C₂×C₄	#15667997	1	2	yes	—
278	C₂×C₄	#15667998	1	2	yes	—
279	C₂×C₄	#15668000	1	1	yes	characteristic
280	C₂×C₄	#15668000	1	1	yes	characteristic
281	C₂×C₄	#15668002	1	2	yes	—
282	C₂×C₄	#15668005	1	2	yes	—
283	C₂×C₄	#15668007	1	2	yes	—
284	C₈	—	4	4	no	—
285	C₈	—	356	2	no	—
286	C₈	—	356	4	no	—
287	C₈	—	356	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	#19151464	1	1	yes	characteristic
294	C₂²	#19151473	1	1	yes	characteristic
295	C₂²	#19151497	1	1	yes	characteristic
296	C₂²	#19151532	1	2	yes	—
297	C₂²	#19151540	1	2	yes	—
298	C₂²	C₂×#15667948	1	2	yes	—
299	C₂²	#19151647	1	2	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	4	2	no	—
302	C₄	#19151484	1	2	yes	—
303	C₄	#19151491	1	2	yes	—
304	C₄	#19151511	1	1	yes	characteristic
305	C₄	#19151516	1	1	yes	characteristic
306	C₄	#19151525	1	1	yes	characteristic
307	C₄	#19151538	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	#26018509	1	1	yes	characteristic
311	C₂	#26020390	1	1	yes	characteristic
312	C₂	#26020393	1	1	yes	characteristic
313	C₂	#26020459	1	2	yes	—
314	C₂	#26021028	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂×#336×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×#336	D₈₉
hypocenter	C₈₉	#11269
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
163. C₈₉	#11269	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151622	2
C₂×#19151363	2
#26019946	2
C₂×#336×C₈₉	2
#11269	89