Subgroups — #36711781

Order 22784.

548 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032600	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022525	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022432	C₂	1	2	yes	maximal, complemented
4	#26022430	C₂	1	2	yes	maximal, complemented
5	#26022389	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022386	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022336	C₂	1	2	yes	maximal, complemented
8	C₄×#15667953	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151423	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3334	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152252	—	2	1	no	—
13	C₂×#15667953	C₂²	1	1	yes	characteristic
14	C₂²×#13083549	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151912	—	2	2	no	—
17	#19151879	—	2	1	no	—
18	#19151834	C₂²	1	2	yes	—
19	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
20	C₂×#15667946	C₂²	1	2	yes	—
21	#19151423	C₂²	1	2	yes	—
22	#19151423	C₂²	1	2	yes	—
23	#19151423	C₂²	1	2	yes	—
24	#19151423	C₂²	1	2	yes	—
25	#19151423	C₂²	1	1	yes	characteristic
26	C₄⋊#13083549	C₂²	1	1	yes	characteristic
27	#19151399	C₂²	1	1	yes	characteristic
28	#19151399	C₂²	1	2	yes	—
29	#19151398	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	2	yes	—
31	C₂²⋊#13083549	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	#547×C₈₉	—	2	1	no	—
34	#500×C₈₉	C₂²	1	1	yes	characteristic
35	#480×C₈₉	—	2	2	no	—
36	#474×C₈₉	—	2	1	no	—
37	#466×C₈₉	C₂²	1	2	yes	—
38	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×#107×C₈₉	C₂²	1	2	yes	—
41	C₂×#13083573	—	4	1	no	—
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	#15668012	—	2	1	no	—
44	#15667996	—	4	2	no	—
45	#15667992	—	4	1	no	—
46	#15667986	—	2	2	no	—
47	C₄×D₃₅₆	—	2	1	no	—
48	C₂×C₄⋊Dic₈₉	—	2	2	no	—
49	#15667953	—	2	2	no	—
50	C₂×#13083549	—	2	4	no	—
51	C₂×#13083549	C₂³	1	1	yes	characteristic
52	C₂×#13083549	C₂³	1	2	yes	—
53	C₂×#13083549	C₂³	1	1	yes	characteristic
54	#15667946	—	2	2	no	—
55	#15667946	—	2	1	no	—
56	#15667946	C₂³	1	1	yes	characteristic
57	#15667946	C₂³	1	2	yes	—
58	#15667934	—	2	2	no	—
59	#15667934	—	2	1	no	—
60	C₂×#41×C₈₉	—	4	1	no	—
61	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₈₉	—	2	1	no	—
64	#123×C₈₉	—	4	2	no	—
65	C₂²⋊Q₈×C₈₉	—	4	1	no	—
66	#121×C₈₉	—	2	2	no	—
67	C₄×D₄×C₈₉	—	2	1	no	—
68	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
70	#107×C₈₉	—	2	2	no	—
71	#107×C₈₉	—	2	1	no	—
72	#107×C₈₉	C₂³	1	1	yes	characteristic
73	#107×C₈₉	C₂³	1	2	yes	—
74	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
75	#102×C₈₉	—	2	1	no	—
76	#102×C₈₉	—	2	2	no	—
77	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	#13083573	—	8	4	no	—
80	C₂×Dic₁₇₈	—	4	1	no	—
81	C₂×D₃₅₆	—	2	1	no	—
82	C₂×C₂²⋊D₈₉	—	4	2	no	—
83	C₂²×Dic₈₉	—	4	2	no	—
84	C₂×C₄×D₈₉	—	4	1	no	—
85	D₇₁₂	—	2	1	no	—
86	D₇₁₂	—	2	1	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	—	4	2	no	—
90	C₄⋊Dic₈₉	—	4	1	no	—
91	#13083551	—	4	1	no	—
92	#13083551	—	4	2	no	—
93	#13083549	—	2	2	no	—
94	#13083549	—	2	1	no	—
95	#13083549	#41	1	2	yes	complemented
96	#41×C₈₉	—	8	4	no	—
97	C₂×Q₈×C₈₉	—	4	1	no	—
98	C₂×D₄×C₈₉	—	2	1	no	—
99	C₂×D₄×C₈₉	—	4	2	no	—
100	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	—	4	2	no	—
102	C₂²×C₄×C₈₉	—	4	1	no	—
103	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
104	D₈×C₈₉	—	2	1	no	—
105	D₈×C₈₉	—	2	1	no	—
106	C₂×C₈×C₈₉	—	2	2	no	—
107	C₂×C₈×C₈₉	—	2	2	no	—
108	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	1	no	—
112	C₄⋊C₄×C₈₉	—	2	2	no	—
113	C₄⋊C₄×C₈₉	—	4	2	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
116	C₂²⋊C₄×C₈₉	—	4	1	no	—
117	C₂²⋊C₄×C₈₉	—	4	2	no	—
118	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
119	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂³×C₈₉	—	4	1	no	—
121	C₂³×C₈₉	C₂×D₈	1	2	yes	—
122	C₂²×D₈₉	—	4	1	no	—
123	Q₈×C₈₉	—	8	2	no	—
124	D₄×C₈₉	—	4	1	no	—
125	D₄×C₈₉	—	4	1	no	—
126	D₄×C₈₉	—	8	4	no	—
127	D₄×C₈₉	D₁₆	1	4	yes	complemented
128	C₂×C₄×C₈₉	—	2	2	no	—
129	C₂×C₄×C₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	—	4	1	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	—	8	4	no	—
133	C₂×C₄×C₈₉	—	8	2	no	—
134	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
138	C₂²⋊D₈₉	—	8	4	no	—
139	C₂×Dic₈₉	—	4	1	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	C₂×Dic₈₉	—	8	4	no	—
142	D₃₅₆	—	4	1	no	—
143	D₃₅₆	—	4	1	no	—
144	C₄×D₈₉	—	8	2	no	—
145	Dic₁₇₈	—	8	2	no	—
146	C₈×C₈₉	—	2	2	no	—
147	C₈×C₈₉	—	2	1	no	—
148	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
149	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
150	C₂²×C₈₉	—	2	2	no	—
151	C₂²×C₈₉	—	2	2	no	—
152	C₂²×C₈₉	—	8	1	no	—
153	C₂²×C₈₉	—	8	1	no	—
154	C₂²×C₈₉	—	8	1	no	—
155	C₂²×C₈₉	C₂×D₁₆	1	4	yes	—
156	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₈₉	—	8	1	no	—
158	C₂×D₈₉	—	8	1	no	—
159	C₂×D₈₉	—	8	1	no	—
160	C₄×C₈₉	—	2	2	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	8	1	no	—
163	C₄×C₈₉	—	8	2	no	—
164	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
165	C₄×C₈₉	#547	1	1	yes	characteristic
166	C₄×C₈₉	#578	1	1	yes	characteristic
167	Dic₈₉	—	8	2	no	—
168	Dic₈₉	—	8	1	no	—
169	#33901	—	89	1	no	maximal, Sylow
170	C₂×C₈₉	—	2	4	no	—
171	C₂×C₈₉	—	16	1	no	—
172	C₂×C₈₉	#3334	1	1	yes	characteristic
173	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
174	C₂×C₈₉	#3417	1	1	yes	characteristic
175	D₈₉	—	16	1	no	—
176	#3334	—	89	1	no	—
177	#3334	D₈₉	1	1	yes	characteristic, complemented
178	#2252	—	89	1	no	—
179	#2232	—	89	2	no	—
180	#2232	—	89	2	no	—
181	#2224	—	89	1	no	—
182	#2224	—	89	1	no	—
183	#2215	—	89	2	no	—
184	C₄×D₁₆	—	89	1	no	—
185	D₄×C₁₆	—	89	1	no	—
186	C₂×#366	—	89	2	no	—
187	C₈₉	#33901	1	1	yes	characteristic, complemented, Sylow
188	#547	—	2	1	no	—
189	#547	—	178	1	no	—
190	C₂×D₁₆	—	89	1	no	—
191	C₂²×C₁₆	—	89	2	no	—
192	#500	—	89	1	no	—
193	#500	C₂×D₈₉	1	1	yes	characteristic
194	#480	—	2	2	no	—
195	#480	—	178	2	no	—
196	#474	—	2	1	no	—
197	#474	—	178	1	no	—
198	#466	—	89	2	no	—
199	#466	C₂×D₈₉	1	2	yes	—
200	C₄×D₈	—	89	1	no	—
201	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
202	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
203	C₂×#107	—	89	2	no	—
204	C₂×#107	C₂×D₈₉	1	2	yes	—
205	#366	—	89	1	no	—
206	#366	—	89	2	no	—
207	#366	—	89	2	no	—
208	#366	—	89	2	no	—
209	#366	—	89	2	no	—
210	C₄⋊C₁₆	—	89	1	no	—
211	#357	—	89	1	no	—
212	#357	—	89	2	no	—
213	#357	—	89	1	no	—
214	#357	—	89	2	no	—
215	C₂²⋊C₁₆	—	89	2	no	—
216	C₄×C₁₆	—	89	1	no	—
217	C₂×#41	—	4	1	no	—
218	C₂×#41	—	356	1	no	—
219	C₂×D₈	—	89	1	no	—
220	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
221	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	178	1	no	—
224	#123	—	4	2	no	—
225	#123	—	356	2	no	—
226	C₂²⋊Q₈	—	4	1	no	—
227	C₂²⋊Q₈	—	356	1	no	—
228	#121	—	2	2	no	—
229	#121	—	178	2	no	—
230	C₄×D₄	—	2	1	no	—
231	C₄×D₄	—	178	1	no	—
232	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	178	2	no	—
235	D₁₆	—	178	2	no	—
236	C₂×C₁₆	—	89	1	no	—
237	C₂×C₁₆	—	89	1	no	—
238	C₂×C₁₆	—	89	2	no	—
239	C₂×C₁₆	—	178	4	no	—
240	#107	—	2	2	no	—
241	#107	—	2	1	no	—
242	#107	—	89	1	no	—
243	#107	—	89	2	no	—
244	#107	—	178	2	no	—
245	#107	—	178	1	no	—
246	#107	C₂²×D₈₉	1	1	yes	characteristic
247	#107	C₂²×D₈₉	1	2	yes	—
248	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
249	#102	—	2	1	no	—
250	#102	—	2	2	no	—
251	#102	—	178	1	no	—
252	#102	—	178	2	no	—
253	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
254	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
255	#41	—	8	4	no	—
256	#41	—	712	4	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	356	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	356	2	no	—
262	C₂×D₄	—	178	1	no	—
263	C₂×D₄	#13083559	1	1	yes	characteristic
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	—	4	1	no	—
266	C₂²×C₄	—	356	2	no	—
267	C₂²×C₄	—	356	1	no	—
268	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
269	D₈	—	2	1	no	—
270	D₈	—	2	1	no	—
271	D₈	—	178	1	no	—
272	D₈	—	178	1	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
276	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
277	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	356	2	no	—
283	C₄⋊C₄	—	356	1	no	—
284	C₄⋊C₄	—	178	1	no	—
285	C₄⋊C₄	—	178	2	no	—
286	C₄⋊C₄	#13083559	1	1	yes	characteristic
287	C₂²⋊C₄	—	4	1	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	356	1	no	—
290	C₂²⋊C₄	—	356	2	no	—
291	C₂²⋊C₄	#13083559	1	2	yes	—
292	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
293	C₁₆	—	89	2	no	—
294	C₁₆	—	178	2	no	—
295	C₁₆	—	178	1	no	—
296	C₂³	—	4	1	no	—
297	C₂³	—	356	1	no	—
298	C₂³	C₂×#13083559	1	2	yes	—
299	Q₈	—	8	2	no	—
300	Q₈	—	712	2	no	—
301	D₄	—	4	1	no	—
302	D₄	—	4	1	no	—
303	D₄	—	8	4	no	—
304	D₄	—	356	1	no	—
305	D₄	—	356	1	no	—
306	D₄	—	712	4	no	—
307	D₄	#15667953	1	4	yes	complemented
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	4	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	356	1	no	—
315	C₂×C₄	—	356	2	no	—
316	C₂×C₄	—	712	4	no	—
317	C₂×C₄	—	712	2	no	—
318	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
319	C₂×C₄	C₂×#13083559	1	2	yes	—
320	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
321	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	2	no	—
324	C₈	#41×D₈₉	1	1	yes	characteristic
325	C₈	#15668059	1	1	yes	characteristic
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	—	712	1	no	—
332	C₂²	—	712	1	no	—
333	C₂²	—	712	1	no	—
334	C₂²	C₂×#15667953	1	4	yes	—
335	C₂²	C₂²×#13083559	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	1	no	—
338	C₄	—	8	1	no	—
339	C₄	—	8	2	no	—
340	C₄	—	712	1	no	—
341	C₄	—	712	2	no	—
342	C₄	C₂×#15667953	1	2	yes	—
343	C₄	#19152247	1	1	yes	characteristic
344	C₄	#19152413	1	1	yes	characteristic
345	C₂	—	2	4	no	—
346	C₂	—	16	1	no	—
347	C₂	—	1424	1	no	—
348	C₂	#26032595	1	1	yes	characteristic
349	C₂	C₂²×#15667953	1	1	yes	characteristic
350	C₂	#26033403	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3334	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#3334	D₈₉
hypocenter	C₈₉	#33901
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3334

Complements

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

N	H ≅ G/N	type
1. #26032600	C₂	semidirect
2. #26022525	C₂	semidirect
3. #26022432	C₂	semidirect
4. #26022430	C₂	semidirect
5. #26022389	C₂	semidirect
6. #26022386	C₂	semidirect
7. #26022336	C₂	semidirect
8. C₄×#15667953	C₂	semidirect
9. D₄×#13083549	C₂	semidirect
10. C₂×#19151423	C₂	semidirect
11. C₈₉×#3334	C₂	semidirect
95. #13083549	#41	semidirect
127. D₄×C₈₉	D₁₆	semidirect
177. #3334	D₈₉	semidirect
187. C₈₉	#33901	semidirect
307. D₄	#15667953	semidirect

Maximal subgroups

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

subgroup	index
#26032600	2
#26022525	2
#26022432	2
#26022430	2
#26022389	2
#26022386	2
#26022336	2
C₄×#15667953	2
D₄×#13083549	2
C₂×#19151423	2
C₈₉×#3334	2
#33901	89