Subgroups — #36709119

Order 22784.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030792	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029104	C₂	1	1	yes	maximal, characteristic
3	#26022411	C₂	1	2	yes	maximal, complemented
4	#26022407	C₂	1	2	yes	maximal, complemented
5	#26022397	C₂	1	2	yes	maximal, complemented
6	#26022387	C₂	1	2	yes	maximal, complemented
7	#26022040	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022034	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022012	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151399	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2140	C₂	1	1	yes	maximal, characteristic
12	#19152226	—	2	1	no	—
13	#19152073	—	2	1	no	—
14	C₂²×#13083549	C₂²	1	1	yes	characteristic
15	#19152005	—	2	2	no	—
16	#19152002	C₂²	1	2	yes	complemented
17	#19151999	C₂²	1	2	yes	complemented
18	#19151710	—	2	2	no	—
19	D₈×Dic₈₉	C₂²	1	2	yes	—
20	D₈×Dic₈₉	C₂²	1	2	yes	—
21	#19151641	—	2	1	no	—
22	#19151636	C₂²	1	1	yes	characteristic
23	C₂×#15667946	C₂²	1	1	yes	characteristic
24	#19151603	—	2	1	no	—
25	#19151553	C₂²	1	1	yes	characteristic
26	#19151426	C₂²	1	2	yes	complemented
27	#19151425	C₂²	1	1	yes	characteristic, complemented
28	#19151425	C₂²	1	1	yes	characteristic, complemented
29	#19151424	C₂²	1	2	yes	complemented
30	#19151423	C₂²	1	1	yes	characteristic, complemented
31	#19151423	C₂²	1	1	yes	characteristic, complemented
32	#19151416	C₂²	1	2	yes	complemented
33	#19151416	C₂²	1	2	yes	complemented
34	#19151399	C₂²	1	2	yes	—
35	#19151399	C₂²	1	2	yes	—
36	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
38	#427×C₈₉	C₂²	1	1	yes	characteristic
39	#357×C₈₉	C₂²	1	2	yes	—
40	#357×C₈₉	C₂²	1	2	yes	—
41	C₂×#13083558	D₄	1	1	yes	characteristic
42	#15668012	—	4	1	no	—
43	#15668011	—	2	2	no	—
44	#15668004	—	4	1	no	—
45	#15668003	—	2	2	no	—
46	#15667989	—	4	2	no	—
47	D₄×Dic₈₉	—	2	2	no	—
48	D₄×Dic₈₉	—	2	2	no	—
49	D₄×Dic₈₉	—	4	1	no	—
50	D₄×Dic₈₉	—	4	1	no	—
51	#15667971	—	2	1	no	—
52	#15667971	D₄	1	1	yes	characteristic
53	C₂×C₄⋊Dic₈₉	—	2	1	no	—
54	C₂×C₂²⋊Dic₈₉	—	4	2	no	—
55	C₂×#13083549	—	2	2	no	—
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	#15667946	—	2	1	no	—
59	#15667946	C₂³	1	2	yes	—
60	#15667946	C₂³	1	1	yes	characteristic
61	#15667946	C₂³	1	1	yes	characteristic
62	#15667945	—	2	2	no	—
63	#15667944	—	2	1	no	—
64	#15667943	—	2	2	no	—
65	#15667933	—	2	2	no	—
66	#15667933	—	2	2	no	—
67	#15667922	D₄	1	2	yes	—
68	C₈×Dic₈₉	C₂³	1	2	yes	—
69	C₂²×D₄×C₈₉	—	2	1	no	—
70	C₂×D₈×C₈₉	—	2	1	no	—
71	C₂×D₈×C₈₉	—	2	2	no	—
72	C₂×D₈×C₈₉	Q₈	1	4	yes	complemented
73	C₂×D₈×C₈₉	C₂³	1	2	yes	—
74	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	#117×C₈₉	—	2	1	no	—
76	M₃₂×C₈₉	—	2	2	no	—
77	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
78	#107×C₈₉	C₂³	1	1	yes	characteristic
79	#107×C₈₉	C₂³	1	1	yes	characteristic
80	#106×C₈₉	—	2	1	no	—
81	C₂×Dic₁₇₈	—	4	2	no	—
82	C₂²×Dic₈₉	—	4	1	no	—
83	C₂²×Dic₈₉	—	4	2	no	—
84	C₂²×Dic₈₉	—	4	2	no	—
85	#13083558	—	2	2	no	—
86	C₂×#10623494	C₂×D₄	1	2	yes	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	—	2	2	no	—
90	C₄⋊Dic₈₉	—	4	1	no	—
91	C₄⋊Dic₈₉	—	4	2	no	—
92	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
93	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
94	#13083553	—	4	2	no	—
95	#13083553	—	4	2	no	—
96	C₂²⋊Dic₈₉	—	4	1	no	—
97	C₂²⋊Dic₈₉	—	4	2	no	—
98	C₂²⋊Dic₈₉	—	4	2	no	—
99	C₂²⋊Dic₈₉	D₈	1	2	yes	—
100	C₄×Dic₈₉	—	4	1	no	—
101	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
102	#13083549	—	2	1	no	—
103	#13083549	—	2	1	no	—
104	C₂⁴×C₈₉	—	4	1	no	—
105	C₂×D₄×C₈₉	—	2	4	no	—
106	C₂×D₄×C₈₉	—	2	2	no	—
107	C₂×D₄×C₈₉	—	4	1	no	—
108	C₂×D₄×C₈₉	—	4	2	no	—
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	D₈×C₈₉	—	2	4	no	—
111	D₈×C₈₉	—	2	2	no	—
112	D₈×C₈₉	C₂×Q₈	1	4	yes	—
113	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
115	C₂×C₈×C₈₉	#41	1	2	yes	—
116	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	—	2	1	no	—
118	C₄⋊C₄×C₈₉	—	2	1	no	—
119	C₂²⋊C₄×C₈₉	—	4	1	no	—
120	C₄²×C₈₉	—	4	1	no	—
121	C₁₆×C₈₉	—	2	2	no	—
122	C₂³×C₈₉	—	4	2	no	—
123	C₂³×C₈₉	—	4	4	no	—
124	C₂³×C₈₉	—	16	1	no	—
125	C₂³×C₈₉	—	16	1	no	—
126	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
127	D₄×C₈₉	—	2	4	no	—
128	D₄×C₈₉	—	4	2	no	—
129	D₄×C₈₉	—	4	4	no	—
130	C₂×C₄×C₈₉	—	4	1	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	#137	1	2	yes	—
135	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	—	4	1	no	—
137	C₂×Dic₈₉	—	4	1	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	16	2	no	—
140	C₂×Dic₈₉	—	16	2	no	—
141	C₂×Dic₈₉	—	16	1	no	—
142	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
143	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
144	Dic₁₇₈	—	16	2	no	—
145	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
146	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
148	#10623494	—	2	2	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	—	4	4	no	—
152	C₂²×C₈₉	—	8	2	no	—
153	C₂²×C₈₉	—	8	2	no	—
154	C₂²×C₈₉	—	16	4	no	—
155	C₂²×C₈₉	#474	1	2	yes	—
156	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
157	C₄×C₈₉	—	8	1	no	—
158	C₄×C₈₉	—	8	1	no	—
159	C₄×C₈₉	#474	1	2	yes	—
160	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
162	Dic₈₉	—	2	2	no	—
163	Dic₈₉	—	2	2	no	—
164	Dic₈₉	—	8	2	no	—
165	Dic₈₉	—	8	1	no	—
166	Dic₈₉	—	8	1	no	—
167	#33631	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	—	2	2	no	—
169	C₂×C₈₉	—	8	4	no	—
170	C₂×C₈₉	C₂×#474	1	1	yes	characteristic
171	C₂×C₈₉	#3415	1	1	yes	characteristic
172	C₂×C₈₉	#3416	1	1	yes	characteristic
173	#3164	—	89	1	no	—
174	#2948	—	89	1	no	—
175	#2228	—	89	2	no	—
176	#2228	—	89	2	no	—
177	#2226	—	89	2	no	—
178	#2224	—	89	2	no	—
179	#2159	—	89	1	no	—
180	#2158	—	89	1	no	—
181	#2151	—	89	1	no	—
182	#2140	D₈₉	1	1	yes	characteristic
183	C₂×#357	—	89	1	no	—
184	C₈₉	#33631	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₈	C₂×D₈₉	1	1	yes	characteristic
186	#544	—	178	1	no	—
187	#518	—	178	1	no	—
188	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
189	C₂²×C₁₆	—	89	1	no	—
190	#501	—	178	2	no	—
191	#500	—	89	2	no	—
192	#499	—	89	2	no	—
193	#442	—	178	2	no	—
194	C₄×D₈	—	89	2	no	—
195	C₄×D₈	—	89	2	no	—
196	#428	—	178	1	no	—
197	#427	—	89	1	no	—
198	#427	C₂×D₈₉	1	1	yes	characteristic
199	C₂×#107	—	89	1	no	—
200	#418	—	178	1	no	—
201	#405	—	89	1	no	—
202	#367	—	89	2	no	—
203	#367	—	89	1	no	—
204	#367	—	89	1	no	—
205	#366	—	89	2	no	—
206	#366	—	89	1	no	—
207	#366	—	89	1	no	—
208	C₄⋊C₁₆	—	89	2	no	—
209	C₄⋊C₁₆	—	89	2	no	—
210	#357	—	89	2	no	—
211	#357	—	89	2	no	—
212	#357	C₂×D₈₉	1	2	yes	—
213	#357	C₂×D₈₉	1	2	yes	—
214	C₂²×D₄	—	2	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	Dic₁₇₈	1	4	yes	complemented
218	C₂×D₈	C₂²×D₈₉	1	2	yes	—
219	C₂×M₁₆	—	89	1	no	—
220	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
221	C₄⋊Q₈	—	356	1	no	—
222	C₄⋊Q₈	—	178	2	no	—
223	#125	—	356	1	no	—
224	#125	—	178	2	no	—
225	C₂²⋊Q₈	—	356	2	no	—
226	C₄×D₄	—	356	1	no	—
227	C₄×D₄	—	356	1	no	—
228	C₄×D₄	—	178	2	no	—
229	C₄×D₄	—	178	2	no	—
230	#117	—	2	1	no	—
231	#117	—	89	1	no	—
232	#117	—	178	1	no	—
233	C₂×C₄⋊C₄	—	178	1	no	—
234	C₂×C₂²⋊C₄	—	356	2	no	—
235	M₃₂	—	2	2	no	—
236	C₂×C₁₆	—	89	1	no	—
237	C₂×C₁₆	—	89	1	no	—
238	C₂×C₁₆	—	178	2	no	—
239	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
240	#107	—	89	2	no	—
241	#107	—	89	1	no	—
242	#107	—	89	1	no	—
243	#107	—	178	2	no	—
244	#107	—	178	1	no	—
245	#107	C₂²×D₈₉	1	1	yes	characteristic
246	#107	C₂²×D₈₉	1	1	yes	characteristic
247	#106	—	2	1	no	—
248	#106	—	178	2	no	—
249	#106	—	178	1	no	—
250	#102	—	178	2	no	—
251	#102	—	178	2	no	—
252	#97	—	89	2	no	—
253	C₄×C₈	—	89	2	no	—
254	C₂⁴	—	4	1	no	—
255	C₂×Q₈	—	356	2	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂²×C₄	—	356	2	no	—
261	C₂²×C₄	—	356	2	no	—
262	C₂²×C₄	—	356	1	no	—
263	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
264	D₈	—	2	4	no	—
265	D₈	—	2	2	no	—
266	D₈	C₂×Dic₁₇₈	1	4	yes	—
267	M₁₆	—	178	2	no	—
268	C₂×C₈	—	89	2	no	—
269	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
271	C₂×C₈	#13083573	1	2	yes	—
272	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	89	1	no	—
276	C₄⋊C₄	—	89	1	no	—
277	C₄⋊C₄	—	356	2	no	—
278	C₄⋊C₄	—	356	2	no	—
279	C₄⋊C₄	—	356	2	no	—
280	C₄⋊C₄	—	356	1	no	—
281	C₄⋊C₄	—	178	1	no	—
282	C₄⋊C₄	—	178	1	no	—
283	C₄⋊C₄	—	178	2	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	89	2	no	—
286	C₂²⋊C₄	—	356	2	no	—
287	C₂²⋊C₄	—	356	2	no	—
288	C₂²⋊C₄	—	356	1	no	—
289	C₄²	—	4	1	no	—
290	C₄²	—	89	2	no	—
291	C₄²	—	356	1	no	—
292	C₁₆	—	2	2	no	—
293	C₁₆	—	178	1	no	—
294	C₁₆	—	178	1	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	16	1	no	—
298	C₂³	—	16	1	no	—
299	C₂³	D₈×D₈₉	1	1	yes	characteristic
300	Q₈	—	1424	2	no	—
301	D₄	—	2	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	89	2	no	—
307	C₂×C₄	—	89	2	no	—
308	C₂×C₄	—	356	1	no	—
309	C₂×C₄	—	356	1	no	—
310	C₂×C₄	—	356	2	no	—
311	C₂×C₄	—	1424	2	no	—
312	C₂×C₄	—	1424	2	no	—
313	C₂×C₄	—	1424	1	no	—
314	C₂×C₄	#15667989	1	2	yes	—
315	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
316	C₂×C₄	#15668041	1	2	yes	—
317	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
318	C₈	—	178	2	no	—
319	C₈	#15667989	1	2	yes	—
320	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
321	C₈	C₂×#13083573	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	4	no	—
328	C₂²	#19151876	1	2	yes	—
329	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
330	C₄	—	8	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	178	2	no	—
333	C₄	—	178	2	no	—
334	C₄	—	712	1	no	—
335	C₄	—	712	2	no	—
336	C₄	—	712	1	no	—
337	C₄	#19151876	1	2	yes	—
338	C₄	C₂×#15667989	1	1	yes	characteristic
339	C₄	C₂×#15668041	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	8	4	no	—
342	C₂	C₂×#19151876	1	1	yes	characteristic
343	C₂	#26033392	1	1	yes	characteristic
344	C₂	#26033398	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2140	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈×C₈₉	C₂²
hypercenter	#2140	D₈₉
hypocenter	C₈₉	#33631
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2140

Complements

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

N	H ≅ G/N	type
1. #26030792	C₂	semidirect
3. #26022411	C₂	semidirect
4. #26022407	C₂	semidirect
5. #26022397	C₂	semidirect
6. #26022387	C₂	semidirect
7. #26022040	C₂	semidirect
8. #26022034	C₂	semidirect
9. #26022012	C₂	semidirect
16. #19152002	C₂²	semidirect
17. #19151999	C₂²	semidirect
26. #19151426	C₂²	semidirect
27. #19151425	C₂²	semidirect
28. #19151425	C₂²	semidirect
29. #19151424	C₂²	semidirect
30. #19151423	C₂²	semidirect
31. #19151423	C₂²	semidirect
32. #19151416	C₂²	semidirect
33. #19151416	C₂²	semidirect
72. C₂×D₈×C₈₉	Q₈	semidirect
184. C₈₉	#33631	semidirect
217. C₂×D₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030792	2
#26029104	2
#26022411	2
#26022407	2
#26022397	2
#26022387	2
#26022040	2
#26022034	2
#26022012	2
C₂×#19151399	2
C₈₉×#2140	2
#33631	89