Subgroups — #36712215

Order 22784.

476 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033120	C₂	1	1	yes	maximal, characteristic
2	#26022497	C₂	1	2	yes	maximal, complemented
3	#26022479	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022416	C₂	1	2	yes	maximal, complemented
5	#26022404	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022294	C₂	1	2	yes	maximal
7	#26022294	C₂	1	1	yes	maximal, characteristic
8	#26022161	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022114	C₂	1	2	yes	maximal
10	C₄×#15667955	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#3395	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152308	—	2	1	no	—
13	C₂×#15667955	C₂²	1	1	yes	characteristic
14	C₂×#15667955	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152000	C₂²	1	2	yes	—
17	#19151899	—	2	2	no	—
18	#19151889	—	2	1	no	—
19	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
20	C₄×D₇₁₂	C₂²	1	2	yes	—
21	#19151425	C₂²	1	1	yes	characteristic
22	#19151423	C₂²	1	2	yes	—
23	C₄⋊#13083549	C₂²	1	1	yes	characteristic
24	C₄⋊#13083549	C₂²	1	2	yes	—
25	#19151402	C₂²	1	2	yes	—
26	#19151402	C₂²	1	2	yes	—
27	#19151402	C₂²	1	2	yes	—
28	#19151402	C₂²	1	2	yes	—
29	#19151402	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	1	yes	characteristic
31	#19151374	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	#557×C₈₉	—	2	1	no	—
34	#494×C₈₉	C₂²	1	1	yes	characteristic
35	#494×C₈₉	C₂²	1	2	yes	—
36	#462×C₈₉	—	2	1	no	—
37	#462×C₈₉	—	2	2	no	—
38	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
41	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
42	#15668012	—	2	1	no	—
43	#15668004	—	2	2	no	—
44	#15667990	—	4	2	no	—
45	#15667988	—	4	1	no	—
46	C₄×D₃₅₆	—	2	1	no	—
47	C₄×D₃₅₆	—	2	2	no	—
48	C₄⋊C₄×D₈₉	—	4	1	no	—
49	#15667955	—	2	2	no	—
50	#15667955	—	2	4	no	—
51	C₂×#13083549	C₂³	1	1	yes	characteristic
52	C₂×#13083549	C₂³	1	1	yes	characteristic
53	C₂×#13083549	C₂³	1	2	yes	—
54	#15667946	—	2	1	no	—
55	#15667946	C₂³	1	2	yes	—
56	#15667946	C₂³	1	1	yes	characteristic
57	#15667944	—	2	2	no	—
58	#15667934	—	2	2	no	—
59	#15667934	—	2	1	no	—
60	C₂×Q₁₆×C₈₉	—	2	2	no	—
61	C₂×Q₁₆×C₈₉	—	2	1	no	—
62	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
63	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₈₉	—	2	2	no	—
65	C₄⋊Q₈×C₈₉	—	2	1	no	—
66	#125×C₈₉	—	4	2	no	—
67	#125×C₈₉	—	4	1	no	—
68	C₄×Q₈×C₈₉	—	2	2	no	—
69	C₄×Q₈×C₈₉	—	2	1	no	—
70	C₄×Q₈×C₈₉	—	4	1	no	—
71	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
72	#107×C₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
75	#103×C₈₉	—	2	2	no	—
76	#103×C₈₉	—	2	1	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	2	yes	—
79	C₂×Dic₁₇₈	—	4	1	no	—
80	C₂×D₃₅₆	—	2	1	no	—
81	C₂×C₄×D₈₉	—	4	1	no	—
82	C₂×C₄×D₈₉	—	4	2	no	—
83	D₇₁₂	—	2	1	no	—
84	D₇₁₂	#41	1	2	yes	—
85	C₄⋊Dic₈₉	—	2	1	no	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	4	1	no	—
88	#13083553	—	4	2	no	—
89	#13083553	—	4	2	no	—
90	#13083551	—	4	2	no	—
91	#13083551	—	4	1	no	—
92	#13083549	—	2	1	no	—
93	#13083549	—	2	2	no	—
94	#13083549	—	2	1	no	—
95	C₂×Q₈×C₈₉	—	2	2	no	—
96	C₂×Q₈×C₈₉	—	2	1	no	—
97	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
98	Q₁₆×C₈₉	—	2	1	no	—
99	Q₁₆×C₈₉	—	2	2	no	—
100	Q₁₆×C₈₉	—	2	1	no	—
101	Q₁₆×C₈₉	—	2	2	no	—
102	Q₁₆×C₈₉	—	8	2	no	—
103	Q₁₆×C₈₉	—	8	1	no	—
104	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	—	2	1	no	—
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₄⋊C₄×C₈₉	—	4	2	no	—
110	C₄⋊C₄×C₈₉	—	4	2	no	—
111	C₄⋊C₄×C₈₉	—	4	1	no	—
112	C₄⋊C₄×C₈₉	—	4	1	no	—
113	C₄⋊C₄×C₈₉	—	4	2	no	—
114	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
115	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
116	C₄²×C₈₉	—	4	2	no	—
117	C₄²×C₈₉	—	4	1	no	—
118	C₄²×C₈₉	C₂×D₄	1	2	yes	—
119	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂²×D₈₉	—	4	1	no	—
121	Q₈×C₈₉	—	2	2	no	—
122	Q₈×C₈₉	—	4	1	no	—
123	Q₈×C₈₉	—	4	2	no	—
124	Q₈×C₈₉	—	4	1	no	—
125	Q₈×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₈₉	—	4	1	no	—
135	C₂×Dic₈₉	—	4	2	no	—
136	D₃₅₆	—	2	2	no	—
137	D₃₅₆	—	4	1	no	—
138	C₄×D₈₉	—	8	2	no	—
139	C₄×D₈₉	—	16	2	no	—
140	Dic₁₇₈	—	16	1	no	—
141	C₈×C₈₉	—	2	2	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
144	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₈₉	—	4	2	no	—
147	C₂×D₈₉	—	8	1	no	—
148	C₂×D₈₉	—	8	1	no	—
149	C₄×C₈₉	—	2	2	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	8	2	no	—
153	C₄×C₈₉	—	8	1	no	—
154	C₄×C₈₉	—	8	1	no	—
155	C₄×C₈₉	#510	1	2	yes	—
156	C₄×C₈₉	#548	1	1	yes	characteristic
157	C₄×C₈₉	#577	1	1	yes	characteristic
158	Dic₈₉	—	8	1	no	—
159	Dic₈₉	—	8	2	no	—
160	#33939	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	#3358	1	1	yes	characteristic
162	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
163	C₂×C₈₉	#3416	1	1	yes	characteristic
164	D₈₉	—	8	2	no	—
165	#3395	D₈₉	1	1	yes	characteristic, complemented
166	#3380	—	89	1	no	—
167	#2245	—	89	2	no	—
168	#2241	—	89	1	no	—
169	#2229	—	89	2	no	—
170	#2227	—	89	1	no	—
171	#2208	—	89	2	no	—
172	#2208	—	89	1	no	—
173	#2183	—	89	1	no	—
174	#2175	—	89	2	no	—
175	C₄×SD₃₂	—	89	1	no	—
176	C₈₉	#33939	1	1	yes	characteristic, complemented, Sylow
177	#557	—	2	1	no	—
178	#554	—	178	1	no	—
179	C₂×SD₃₂	—	89	1	no	—
180	C₂×SD₃₂	—	89	2	no	—
181	#500	—	89	1	no	—
182	#499	—	89	2	no	—
183	#494	C₂×D₈₉	1	1	yes	characteristic
184	#494	C₂×D₈₉	1	2	yes	—
185	#478	—	178	2	no	—
186	#476	—	178	1	no	—
187	#462	—	2	1	no	—
188	#462	—	2	2	no	—
189	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
190	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
191	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
192	C₄×D₈	—	89	1	no	—
193	C₄×D₈	—	89	2	no	—
194	#367	—	89	1	no	—
195	#366	—	89	2	no	—
196	C₄⋊C₁₆	—	89	1	no	—
197	C₄⋊C₁₆	—	89	2	no	—
198	#358	—	89	1	no	—
199	#358	—	89	2	no	—
200	#358	—	89	2	no	—
201	#358	—	89	2	no	—
202	#358	—	89	2	no	—
203	#357	—	89	1	no	—
204	#346	—	89	2	no	—
205	C₄×C₁₆	—	89	1	no	—
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
209	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
210	C₂×D₈	—	89	1	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	178	1	no	—
214	#125	—	4	1	no	—
215	#125	—	4	2	no	—
216	#125	—	178	2	no	—
217	C₂²⋊Q₈	—	356	1	no	—
218	C₂²⋊Q₈	—	356	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	4	1	no	—
222	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
223	C₄×D₄	—	178	1	no	—
224	C₄×D₄	—	178	2	no	—
225	C₂×C₄⋊C₄	—	356	1	no	—
226	SD₃₂	—	178	2	no	—
227	SD₃₂	—	178	4	no	—
228	C₂×C₁₆	—	89	1	no	—
229	C₂×C₁₆	—	89	1	no	—
230	C₂×C₁₆	—	89	2	no	—
231	#107	—	89	2	no	—
232	#107	—	89	1	no	—
233	#107	—	178	1	no	—
234	#107	C₂²×D₈₉	1	1	yes	characteristic
235	#106	—	178	2	no	—
236	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
237	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
238	#103	—	2	2	no	—
239	#103	—	2	1	no	—
240	#102	—	178	2	no	—
241	#102	—	178	1	no	—
242	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₈₉	1	2	yes	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	356	1	no	—
247	C₂×Q₈	#13083559	1	1	yes	characteristic
248	C₂×D₄	—	178	1	no	—
249	C₂²×C₄	—	356	2	no	—
250	C₂²×C₄	—	356	1	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	2	2	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	8	2	no	—
256	Q₁₆	—	8	1	no	—
257	D₈	—	89	2	no	—
258	D₈	—	178	1	no	—
259	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
260	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	356	2	no	—
270	C₄⋊C₄	—	356	1	no	—
271	C₄⋊C₄	—	356	2	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	—	178	2	no	—
274	C₄⋊C₄	#13083559	1	1	yes	characteristic
275	C₄⋊C₄	#13083559	1	2	yes	—
276	C₂²⋊C₄	—	356	1	no	—
277	C₂²⋊C₄	—	356	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
281	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
282	C₁₆	—	178	1	no	—
283	C₁₆	—	178	2	no	—
284	C₁₆	—	178	1	no	—
285	C₂³	—	356	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	1424	1	no	—
292	D₄	—	356	1	no	—
293	D₄	—	178	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	356	1	no	—
298	C₂×C₄	—	356	2	no	—
299	C₂×C₄	—	1424	2	no	—
300	C₂×C₄	—	712	2	no	—
301	C₂×C₄	C₂×#13083559	1	2	yes	—
302	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
303	C₂×C₄	C₂×#13083559	1	2	yes	—
304	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	#41×D₈₉	1	1	yes	characteristic
309	C₈	#15668059	1	1	yes	characteristic
310	C₂²	—	356	2	no	—
311	C₂²	—	712	1	no	—
312	C₂²	—	712	1	no	—
313	C₂²	C₂²×#13083559	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	712	1	no	—
321	C₄	—	712	2	no	—
322	C₄	#19152039	1	2	yes	—
323	C₄	#19152255	1	1	yes	characteristic
324	C₄	#19152409	1	1	yes	characteristic
325	C₂	—	712	2	no	—
326	C₂	#26032883	1	1	yes	characteristic
327	C₂	C₂×#19152039	1	1	yes	characteristic
328	C₂	#26033396	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3395	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₇₁₂	C₂³
hypercenter	#3395	D₈₉
hypocenter	C₈₉	#33939
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

Complements

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

N	H ≅ G/N	type
2. #26022497	C₂	semidirect
3. #26022479	C₂	semidirect
4. #26022416	C₂	semidirect
5. #26022404	C₂	semidirect
8. #26022161	C₂	semidirect
11. C₈₉×#3395	C₂	semidirect
165. #3395	D₈₉	semidirect
176. C₈₉	#33939	semidirect

Maximal subgroups

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

subgroup	index
#26033120	2
#26022497	2
#26022479	2
#26022416	2
#26022404	2
#26022294	2
#26022161	2
#26022114	2
C₄×#15667955	2
C₈₉×#3395	2
#33939	89