Subgroups — #36537689

Order 22784.

570 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031597	C₂	1	2	yes	maximal
2	#26030766	C₂	1	1	yes	maximal, characteristic
3	#26029186	C₂	1	1	yes	maximal, characteristic
4	#26019781	C₂	1	2	yes	maximal, complemented
5	#26019771	C₂	1	2	yes	maximal, complemented
6	#26019635	C₂	1	4	yes	maximal, complemented
7	#26018940	C₂	1	2	yes	maximal
8	C₂×#334×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19152155	C₂²	1	2	yes	—
10	#19152155	C₂²	1	1	yes	characteristic
11	#19152003	C₂²	1	1	yes	characteristic
12	#19152003	C₂²	1	1	yes	characteristic
13	#19152000	—	2	1	no	—
14	#19152000	—	2	1	no	—
15	#19151996	C₂²	1	2	yes	—
16	#19151894	—	2	2	no	—
17	#19151894	C₂²	1	4	yes	—
18	#19151839	—	2	4	no	—
19	#19151809	—	2	2	no	—
20	#19151809	C₂²	1	4	yes	—
21	#19151728	—	2	1	no	—
22	#19151728	C₂²	1	2	yes	—
23	C₈×Dic₁₇₈	—	2	1	no	—
24	C₈×Dic₁₇₈	C₂²	1	2	yes	—
25	#19151636	—	2	1	no	—
26	#19151636	—	2	1	no	—
27	#19151624	—	2	1	no	—
28	#19151622	C₂²	1	1	yes	characteristic
29	#19151553	—	2	1	no	—
30	#19151346	C₂²	1	4	yes	—
31	#19151345	C₂²	1	4	yes	—
32	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
34	#334×C₈₉	C₂²	1	2	yes	—
35	#334×C₈₉	C₂²	1	2	yes	—
36	C₂×Dic₃₅₆	—	2	4	no	—
37	C₂×#13083558	—	2	1	no	—
38	#15668012	D₄	1	1	yes	characteristic
39	#15668012	D₄	1	1	yes	characteristic
40	#15668012	C₂³	1	2	yes	—
41	#15668012	C₂³	1	2	yes	—
42	#15668004	—	2	2	no	—
43	#15668004	—	2	1	no	—
44	#15667991	—	2	2	no	—
45	#15667991	—	2	1	no	—
46	#15667991	—	2	2	no	—
47	#15667991	D₄	1	2	yes	—
48	C₄×Dic₁₇₈	—	2	1	no	—
49	C₄×Dic₁₇₈	—	2	2	no	—
50	C₄×Dic₁₇₈	—	2	2	no	—
51	C₄×Dic₁₇₈	C₂³	1	2	yes	—
52	#15667971	—	2	1	no	—
53	#15667971	—	2	2	no	—
54	#15667946	—	2	4	no	—
55	#15667946	—	2	1	no	—
56	#15667946	—	2	1	no	—
57	#15667946	—	2	1	no	—
58	#15667946	—	2	1	no	—
59	#15667944	—	2	2	no	—
60	#15667944	—	2	2	no	—
61	C₄⋊#10623494	—	2	1	no	—
62	C₄⋊#10623494	C₂³	1	2	yes	—
63	#15667941	—	2	2	no	—
64	#15667941	—	2	2	no	—
65	#15667937	—	2	4	no	—
66	#15667937	—	2	4	no	—
67	#15667922	—	2	2	no	—
68	C₈×Dic₈₉	—	2	2	no	—
69	C₂²×C₈×C₈₉	—	2	2	no	—
70	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₈₉	—	2	2	no	—
74	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
75	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
76	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×Dic₁₇₈	—	2	2	no	—
81	C₂×Dic₁₇₈	—	2	2	no	—
82	C₂×Dic₁₇₈	C₂×D₄	1	2	yes	—
83	Dic₃₅₆	—	8	4	no	—
84	#13083558	—	4	2	no	—
85	C₂×#10623494	—	2	2	no	—
86	C₄⋊Dic₈₉	—	2	1	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	—	2	2	no	—
90	C₄⋊Dic₈₉	—	2	2	no	—
91	C₄⋊Dic₈₉	—	2	2	no	—
92	C₄⋊Dic₈₉	—	2	2	no	—
93	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
94	#13083553	—	2	2	no	—
95	#13083553	—	2	2	no	—
96	#13083553	—	4	2	no	—
97	#13083553	—	4	2	no	—
98	C₂²⋊Dic₈₉	—	2	2	no	—
99	C₂²⋊Dic₈₉	—	4	2	no	—
100	C₄×Dic₈₉	—	2	2	no	—
101	C₄×Dic₈₉	—	4	2	no	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
105	C₂×C₈×C₈₉	—	2	2	no	—
106	C₂×C₈×C₈₉	—	2	4	no	—
107	C₂×C₈×C₈₉	—	8	2	no	—
108	C₂×C₈×C₈₉	Q₁₆	1	4	yes	complemented
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₉	#41	1	1	yes	characteristic
116	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₂³×C₈₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	4	1	no	—
120	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₈₉	#137	1	2	yes	—
125	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₈₉	—	2	2	no	—
129	C₂×Dic₈₉	—	2	2	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	C₂×Dic₈₉	—	4	2	no	—
132	Dic₁₇₈	—	2	4	no	—
133	Dic₁₇₈	—	4	2	no	—
134	Dic₁₇₈	—	4	2	no	—
135	Dic₁₇₈	—	4	2	no	—
136	Dic₁₇₈	—	8	2	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	—	4	4	no	—
140	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
141	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
142	#10623494	—	4	2	no	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	#462	1	2	yes	—
146	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	#462	1	2	yes	—
151	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
152	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
153	C₄×C₈₉	#577	1	2	yes	—
154	Dic₈₉	—	4	2	no	—
155	Dic₈₉	—	4	2	no	—
156	Dic₈₉	—	8	2	no	—
157	Dic₈₉	—	8	2	no	—
158	#19210	—	89	1	no	maximal, Sylow
159	C₂×C₈₉	—	2	2	no	—
160	C₂×C₈₉	C₂×#462	1	1	yes	characteristic
161	C₂×C₈₉	#3060	1	1	yes	characteristic
162	C₂×C₈₉	#3067	1	1	yes	characteristic
163	#3249	—	89	2	no	—
164	#3160	—	89	1	no	—
165	#2960	—	89	1	no	—
166	#1696	—	89	2	no	—
167	#1694	—	89	2	no	—
168	#1672	—	89	4	no	—
169	C₂×#334	D₈₉	1	1	yes	characteristic
170	#1550	—	89	2	no	—
171	C₈₉	#19210	1	1	yes	characteristic, complemented, Sylow
172	#533	—	89	1	no	—
173	#533	—	89	2	no	—
174	#500	—	89	1	no	—
175	#500	—	89	1	no	—
176	#499	—	178	1	no	—
177	#499	—	178	1	no	—
178	#498	—	89	2	no	—
179	#477	—	89	4	no	—
180	#477	—	178	2	no	—
181	#467	—	178	4	no	—
182	#462	—	89	4	no	—
183	#462	—	178	2	no	—
184	#446	—	89	2	no	—
185	#446	—	178	1	no	—
186	C₈×Q₈	—	89	2	no	—
187	C₈×Q₈	—	178	1	no	—
188	#427	—	178	1	no	—
189	#427	—	178	1	no	—
190	#424	—	178	1	no	—
191	#423	—	89	1	no	—
192	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
193	#405	—	178	1	no	—
194	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
195	#334	C₂×D₈₉	1	2	yes	—
196	#334	C₂×D₈₉	1	2	yes	—
197	#333	—	89	4	no	—
198	#332	—	89	4	no	—
199	C₂×Q₁₆	—	178	4	no	—
200	C₂×M₁₆	—	178	1	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
203	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
204	C₄⋊Q₈	—	89	1	no	—
205	C₄⋊Q₈	—	89	1	no	—
206	C₄⋊Q₈	—	89	2	no	—
207	C₄⋊Q₈	—	89	2	no	—
208	#125	—	178	2	no	—
209	#125	—	178	1	no	—
210	C₂²⋊Q₈	—	89	2	no	—
211	C₂²⋊Q₈	—	178	2	no	—
212	C₂²⋊Q₈	—	178	2	no	—
213	C₂²⋊Q₈	—	178	1	no	—
214	C₄×Q₈	—	89	2	no	—
215	C₄×Q₈	—	178	2	no	—
216	C₄×Q₈	—	178	2	no	—
217	C₄×Q₈	—	178	1	no	—
218	#117	—	178	2	no	—
219	#117	—	178	1	no	—
220	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
221	#107	—	178	4	no	—
222	#107	—	178	1	no	—
223	#107	—	178	1	no	—
224	#107	—	178	1	no	—
225	#107	—	178	1	no	—
226	#106	—	178	2	no	—
227	#106	—	178	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	89	2	no	—
230	C₄⋊C₈	—	178	2	no	—
231	C₄⋊C₈	—	178	2	no	—
232	C₄⋊C₈	—	178	1	no	—
233	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
234	#103	—	178	4	no	—
235	#103	—	178	4	no	—
236	#97	—	178	2	no	—
237	C₄×C₈	—	178	2	no	—
238	C₄×C₈	D₃₅₆	1	1	yes	characteristic
239	C₄×C₈	D₃₅₆	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×Q₈	—	89	2	no	—
243	C₂×Q₈	—	178	2	no	—
244	C₂×Q₈	—	178	2	no	—
245	C₂×Q₈	—	178	1	no	—
246	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
248	C₂²×C₄	#13083575	1	1	yes	characteristic
249	Q₁₆	—	712	4	no	—
250	M₁₆	—	356	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	8	2	no	—
254	C₂×C₈	—	178	2	no	—
255	C₂×C₈	Dic₃₅₆	1	4	yes	complemented
256	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
257	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
258	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
259	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
260	C₄⋊C₄	—	89	2	no	—
261	C₄⋊C₄	—	356	2	no	—
262	C₄⋊C₄	—	356	2	no	—
263	C₄⋊C₄	—	178	2	no	—
264	C₄⋊C₄	—	178	2	no	—
265	C₄⋊C₄	—	178	2	no	—
266	C₄⋊C₄	—	178	2	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₄⋊C₄	—	178	1	no	—
269	C₄⋊C₄	—	178	2	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	—	178	2	no	—
272	C₂²⋊C₄	—	356	2	no	—
273	C₂²⋊C₄	—	178	2	no	—
274	C₄²	—	356	2	no	—
275	C₄²	—	178	2	no	—
276	C₄²	D₄×D₈₉	1	1	yes	characteristic
277	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
278	C₄²	#13083575	1	1	yes	characteristic
279	C₄²	C₂³×D₈₉	1	1	yes	characteristic
280	C₂³	#15667987	1	1	yes	characteristic
281	Q₈	—	356	2	no	—
282	Q₈	—	356	2	no	—
283	Q₈	—	356	2	no	—
284	Q₈	—	178	4	no	—
285	Q₈	—	712	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	356	2	no	—
289	C₂×C₄	—	356	2	no	—
290	C₂×C₄	—	178	2	no	—
291	C₂×C₄	—	178	2	no	—
292	C₂×C₄	#15667987	1	1	yes	characteristic
293	C₂×C₄	#15667987	1	1	yes	characteristic
294	C₂×C₄	#15667987	1	1	yes	characteristic
295	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
296	C₂×C₄	#15668045	1	2	yes	—
297	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
299	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	4	4	no	—
303	C₈	—	356	2	no	—
304	C₈	C₂×Dic₃₅₆	1	2	yes	—
305	C₈	C₂×Dic₃₅₆	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#19151809	1	2	yes	—
309	C₂²	C₂×#15667987	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	356	2	no	—
314	C₄	—	356	2	no	—
315	C₄	—	712	2	no	—
316	C₄	—	712	2	no	—
317	C₄	#19151809	1	2	yes	—
318	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
319	C₄	C₂×#15668045	1	1	yes	characteristic
320	C₄	#19152404	1	2	yes	—
321	C₂	—	2	2	no	—
322	C₂	C₂×#19151809	1	1	yes	characteristic
323	C₂	#26029915	1	1	yes	characteristic
324	C₂	#26029973	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667987
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#334×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂³×C₈₉	#121
hypercenter	C₂×#334	D₈₉
hypocenter	C₈₉	#19210
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
4. #26019781	C₂	semidirect
5. #26019771	C₂	semidirect
6. #26019635	C₂	semidirect
108. C₂×C₈×C₈₉	Q₁₆	semidirect
171. C₈₉	#19210	semidirect
255. C₂×C₈	Dic₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
#26031597	2
#26030766	2
#26029186	2
#26019781	2
#26019771	2
#26019635	2
#26018940	2
C₂×#334×C₈₉	2
#19210	89