Subgroups — #36709111

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	#26029105	C₂	1	1	yes	maximal, characteristic
2	#26022406	C₂	1	2	yes	maximal, complemented
3	#26022406	C₂	1	2	yes	maximal, complemented
4	#26022396	C₂	1	2	yes	maximal, complemented
5	#26022386	C₂	1	2	yes	maximal, complemented
6	#26022038	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022034	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022011	C₂	1	1	yes	maximal, characteristic, complemented
9	#26021967	C₂	1	1	yes	maximal, characteristic
10	C₂×#19151398	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
13	#19152072	—	2	1	no	—
14	C₂×#15667952	C₂²	1	1	yes	characteristic
15	C₂²×#13083549	C₂²	1	1	yes	characteristic
16	#19151711	—	2	2	no	—
17	C₄×D₇₁₂	C₂²	1	2	yes	—
18	C₄×D₇₁₂	C₂²	1	2	yes	—
19	#19151636	C₂²	1	1	yes	characteristic
20	#19151604	—	2	1	no	—
21	#19151425	C₂²	1	1	yes	characteristic, complemented
22	#19151425	C₂²	1	2	yes	complemented
23	#19151425	C₂²	1	1	yes	characteristic, complemented
24	#19151423	C₂²	1	1	yes	characteristic, complemented
25	#19151423	C₂²	1	2	yes	complemented
26	#19151423	C₂²	1	1	yes	characteristic, complemented
27	C₄⋊#13083549	C₂²	1	2	yes	complemented
28	C₄⋊#13083549	C₂²	1	2	yes	complemented
29	#19151398	C₂²	1	2	yes	—
30	#19151398	C₂²	1	2	yes	—
31	#19151398	C₂²	1	2	yes	—
32	#19151398	C₂²	1	2	yes	—
33	#544×C₈₉	—	2	1	no	—
34	#501×C₈₉	—	2	2	no	—
35	#500×C₈₉	C₂²	1	2	yes	complemented
36	#499×C₈₉	C₂²	1	2	yes	complemented
37	#428×C₈₉	—	2	1	no	—
38	#427×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
40	#405×C₈₉	C₂²	1	1	yes	characteristic
41	C₂²×D₃₅₆	—	2	1	no	—
42	C₂×D₇₁₂	—	2	1	no	—
43	C₂×D₇₁₂	—	2	2	no	—
44	C₂×D₇₁₂	Q₈	1	4	yes	complemented
45	C₂×D₇₁₂	C₂³	1	2	yes	—
46	C₄×D₃₅₆	—	2	2	no	—
47	C₄×D₃₅₆	—	2	2	no	—
48	#15667974	—	4	1	no	—
49	#15667974	—	4	1	no	—
50	#15667971	—	2	1	no	—
51	C₂²⋊C₄×D₈₉	—	4	2	no	—
52	#15667952	—	2	2	no	—
53	C₂×#13083549	—	2	2	no	—
54	C₂×#13083549	C₂³	1	1	yes	characteristic
55	C₂×#13083549	C₂³	1	2	yes	—
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	#15667946	C₂³	1	1	yes	characteristic
58	#15667946	C₂³	1	1	yes	characteristic
59	#15667944	—	2	1	no	—
60	#15667934	—	2	2	no	—
61	#15667934	—	2	2	no	—
62	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₈₉	—	2	2	no	—
65	C₄⋊Q₈×C₈₉	—	4	1	no	—
66	#125×C₈₉	—	2	2	no	—
67	#125×C₈₉	—	4	1	no	—
68	C₂²⋊Q₈×C₈₉	—	4	2	no	—
69	#117×C₈₉	—	2	1	no	—
70	#117×C₈₉	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
72	#107×C₈₉	—	2	1	no	—
73	#107×C₈₉	—	2	2	no	—
74	#107×C₈₉	C₂³	1	2	yes	—
75	#107×C₈₉	C₂³	1	1	yes	characteristic
76	#107×C₈₉	C₂³	1	1	yes	characteristic
77	#106×C₈₉	—	2	1	no	—
78	#106×C₈₉	—	2	2	no	—
79	#97×C₈₉	D₄	1	2	yes	—
80	C₄×C₈×C₈₉	C₂³	1	2	yes	—
81	C₂³×D₈₉	—	4	1	no	—
82	C₂×D₃₅₆	—	2	4	no	—
83	C₂×D₃₅₆	—	2	2	no	—
84	C₂×D₃₅₆	—	4	1	no	—
85	C₂×D₃₅₆	—	4	2	no	—
86	C₂×C₄×D₈₉	—	4	2	no	—
87	C₂×C₄×D₈₉	—	4	2	no	—
88	D₇₁₂	—	2	4	no	—
89	D₇₁₂	—	2	2	no	—
90	D₇₁₂	C₂×Q₈	1	4	yes	—
91	C₄⋊Dic₈₉	—	2	1	no	—
92	C₄⋊Dic₈₉	—	2	1	no	—
93	C₂²⋊Dic₈₉	—	4	1	no	—
94	#13083551	—	4	2	no	—
95	#13083551	—	4	2	no	—
96	C₄×Dic₈₉	—	4	1	no	—
97	#13083549	—	2	1	no	—
98	#13083549	—	2	1	no	—
99	#13083549	—	2	2	no	—
100	C₂×Q₈×C₈₉	—	4	2	no	—
101	C₂²×C₄×C₈₉	—	4	1	no	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	M₁₆×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₈₉	#41	1	2	yes	—
108	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	—	2	1	no	—
110	C₄⋊C₄×C₈₉	—	2	2	no	—
111	C₄⋊C₄×C₈₉	—	2	1	no	—
112	C₄⋊C₄×C₈₉	—	4	1	no	—
113	C₄⋊C₄×C₈₉	—	4	2	no	—
114	C₄⋊C₄×C₈₉	—	4	2	no	—
115	C₄⋊C₄×C₈₉	—	4	2	no	—
116	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
118	C₂²⋊C₄×C₈₉	—	4	1	no	—
119	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
120	C₄²×C₈₉	—	4	1	no	—
121	C₄²×C₈₉	C₂×D₄	1	2	yes	—
122	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₈₉	—	4	2	no	—
124	C₂²×D₈₉	—	4	4	no	—
125	C₂²×D₈₉	—	16	1	no	—
126	C₂²×D₈₉	—	16	1	no	—
127	Q₈×C₈₉	—	16	2	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	1	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	—	16	1	no	—
132	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	#137	1	2	yes	—
137	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₈₉	—	4	1	no	—
139	C₂×Dic₈₉	—	4	1	no	—
140	D₃₅₆	—	2	4	no	—
141	D₃₅₆	—	4	2	no	—
142	D₃₅₆	—	4	4	no	—
143	C₄×D₈₉	—	16	2	no	—
144	C₄×D₈₉	—	16	2	no	—
145	C₈×C₈₉	—	2	2	no	—
146	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
147	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
148	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	#474	1	2	yes	—
152	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₈₉	—	4	4	no	—
154	C₂×D₈₉	—	8	2	no	—
155	C₂×D₈₉	—	8	2	no	—
156	C₂×D₈₉	—	16	4	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	8	2	no	—
160	C₄×C₈₉	—	8	1	no	—
161	C₄×C₈₉	—	8	1	no	—
162	C₄×C₈₉	#474	1	2	yes	—
163	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
164	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
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	C₂×#474	1	1	yes	characteristic
170	C₂×C₈₉	#3415	1	1	yes	characteristic
171	C₂×C₈₉	#3416	1	1	yes	characteristic
172	D₈₉	—	8	4	no	—
173	#3164	D₈₉	1	1	yes	characteristic, complemented
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	—	89	1	no	—
183	C₂×#357	—	89	1	no	—
184	C₈₉	#33631	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₈	—	89	1	no	—
186	#544	—	2	1	no	—
187	#518	—	178	1	no	—
188	C₂×M₃₂	—	89	1	no	—
189	C₂²×C₁₆	—	89	1	no	—
190	#501	—	2	2	no	—
191	#500	C₂×D₈₉	1	2	yes	complemented
192	#499	C₂×D₈₉	1	2	yes	complemented
193	#442	—	178	2	no	—
194	C₄×D₈	—	89	2	no	—
195	C₄×D₈	—	89	2	no	—
196	#428	—	2	1	no	—
197	#427	—	89	1	no	—
198	#427	C₂×D₈₉	1	1	yes	characteristic
199	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
200	#418	—	178	1	no	—
201	#405	C₂×D₈₉	1	1	yes	characteristic
202	#367	—	89	1	no	—
203	#367	—	89	1	no	—
204	#367	—	89	2	no	—
205	#366	—	89	1	no	—
206	#366	—	89	1	no	—
207	#366	—	89	2	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	—	89	2	no	—
213	#357	—	89	2	no	—
214	C₂²×D₄	—	178	1	no	—
215	C₂×D₈	—	89	4	no	—
216	C₂×D₈	—	89	2	no	—
217	C₂×D₈	—	178	1	no	—
218	C₂×D₈	—	178	2	no	—
219	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	2	no	—
222	C₄⋊Q₈	—	4	1	no	—
223	#125	—	2	2	no	—
224	#125	—	4	1	no	—
225	C₂²⋊Q₈	—	4	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	—	178	1	no	—
232	#117	C₂²⋊D₈₉	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₂²⋊C₄	—	356	2	no	—
235	M₃₂	—	178	2	no	—
236	C₂×C₁₆	—	89	1	no	—
237	C₂×C₁₆	—	89	2	no	—
238	C₂×C₁₆	—	89	1	no	—
239	C₂×C₁₆	—	178	2	no	—
240	#107	—	2	1	no	—
241	#107	—	2	2	no	—
242	#107	—	89	1	no	—
243	#107	—	89	1	no	—
244	#107	C₂²×D₈₉	1	2	yes	—
245	#107	C₂²×D₈₉	1	1	yes	characteristic
246	#107	C₂²×D₈₉	1	1	yes	characteristic
247	#106	—	2	1	no	—
248	#106	—	2	2	no	—
249	#106	—	178	1	no	—
250	#102	—	178	2	no	—
251	#102	—	178	2	no	—
252	#97	C₂²⋊D₈₉	1	2	yes	—
253	C₄×C₈	C₂²×D₈₉	1	2	yes	—
254	C₂⁴	—	356	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×D₄	—	356	1	no	—
257	C₂×D₄	—	356	2	no	—
258	C₂×D₄	—	178	4	no	—
259	C₂×D₄	—	178	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	356	2	no	—
262	C₂²×C₄	—	356	2	no	—
263	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
264	D₈	—	89	4	no	—
265	D₈	—	178	4	no	—
266	D₈	—	178	2	no	—
267	M₁₆	—	2	2	no	—
268	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
269	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
270	C₂×C₈	Q₈×D₈₉	1	2	yes	—
271	C₂×C₈	#13083575	1	2	yes	—
272	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	178	1	no	—
281	C₄⋊C₄	—	178	1	no	—
282	C₄⋊C₄	#13083559	1	1	yes	characteristic
283	C₄⋊C₄	#13083559	1	1	yes	characteristic
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	356	1	no	—
286	C₂²⋊C₄	—	356	2	no	—
287	C₂²⋊C₄	—	356	2	no	—
288	C₂²⋊C₄	#13083559	1	2	yes	—
289	C₄²	—	4	1	no	—
290	C₄²	—	356	1	no	—
291	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
292	C₁₆	—	178	1	no	—
293	C₁₆	—	178	1	no	—
294	C₁₆	—	178	2	no	—
295	C₂³	—	356	4	no	—
296	C₂³	—	356	2	no	—
297	C₂³	—	1424	1	no	—
298	C₂³	—	1424	1	no	—
299	C₂³	C₂×#13083559	1	1	yes	characteristic
300	Q₈	—	16	2	no	—
301	D₄	—	356	4	no	—
302	D₄	—	356	2	no	—
303	D₄	—	178	4	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	16	1	no	—
308	C₂×C₄	—	356	1	no	—
309	C₂×C₄	—	356	1	no	—
310	C₂×C₄	—	1424	2	no	—
311	C₂×C₄	—	1424	2	no	—
312	C₂×C₄	#15667988	1	2	yes	—
313	C₂×C₄	C₂×#13083559	1	2	yes	—
314	C₂×C₄	C₂×#13083559	1	2	yes	—
315	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
316	C₂×C₄	#15668043	1	2	yes	—
317	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	#15667988	1	2	yes	—
320	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
321	C₈	C₂×#13083575	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	356	4	no	—
325	C₂²	—	1424	4	no	—
326	C₂²	—	712	2	no	—
327	C₂²	—	712	2	no	—
328	C₂²	#19151875	1	2	yes	—
329	C₂²	C₂²×#13083559	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	712	1	no	—
336	C₄	—	712	1	no	—
337	C₄	#19151875	1	2	yes	—
338	C₄	C₂×#15667988	1	1	yes	characteristic
339	C₄	C₂×#15668043	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	712	4	no	—
342	C₂	C₂×#19151875	1	1	yes	characteristic
343	C₂	#26033391	1	1	yes	characteristic
344	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×#3164	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₇₁₂	C₂²
hypercenter	#3164	D₈₉
hypocenter	C₈₉	#33631
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #26022406	C₂	semidirect
3. #26022406	C₂	semidirect
4. #26022396	C₂	semidirect
5. #26022386	C₂	semidirect
6. #26022038	C₂	semidirect
7. #26022034	C₂	semidirect
8. #26022011	C₂	semidirect
11. C₈₉×#3164	C₂	semidirect
21. #19151425	C₂²	semidirect
22. #19151425	C₂²	semidirect
23. #19151425	C₂²	semidirect
24. #19151423	C₂²	semidirect
25. #19151423	C₂²	semidirect
26. #19151423	C₂²	semidirect
27. C₄⋊#13083549	C₂²	semidirect
28. C₄⋊#13083549	C₂²	semidirect
35. #500×C₈₉	C₂²	semidirect
36. #499×C₈₉	C₂²	semidirect
44. C₂×D₇₁₂	Q₈	semidirect
173. #3164	D₈₉	semidirect
184. C₈₉	#33631	semidirect
191. #500	C₂×D₈₉	semidirect
192. #499	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029105	2
#26022406	2
#26022396	2
#26022386	2
#26022038	2
#26022034	2
#26022011	2
#26021967	2
C₂×#19151398	2
C₈₉×#3164	2
#33631	89