Subgroups — #36709643

Order 22784.

544 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030902	C₂	1	1	yes	maximal, characteristic
2	#26022443	C₂	1	2	yes	maximal, complemented
3	#26022416	C₂	1	2	yes	maximal, complemented
4	#26022292	C₂	1	2	yes	maximal, complemented
5	#26022169	C₂	1	2	yes	maximal, complemented
6	#26022114	C₂	1	2	yes	maximal, complemented
7	C₄×#15667955	C₂	1	2	yes	maximal, complemented
8	#26022059	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2950	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152203	—	2	1	no	—
11	C₂×#15667955	C₂²	1	2	yes	—
12	C₂×#15667955	C₂²	1	2	yes	—
13	#19152000	C₂²	1	1	yes	characteristic
14	#19151975	C₂²	1	1	yes	characteristic
15	#19151963	—	2	1	no	—
16	#19151851	—	2	1	no	—
17	#19151834	C₂²	1	2	yes	—
18	#19151711	—	2	1	no	—
19	C₄×D₇₁₂	C₂²	1	2	yes	—
20	#19151425	C₂²	1	2	yes	—
21	#19151423	C₂²	1	2	yes	—
22	C₄⋊#13083549	C₂²	1	2	yes	—
23	#19151402	C₂²	1	2	yes	—
24	#19151402	C₂²	1	2	yes	—
25	#19151398	C₂²	1	2	yes	—
26	#19151398	C₂²	1	2	yes	—
27	C₂²⋊#13083549	C₂²	1	2	yes	—
28	#19151374	C₂²	1	1	yes	characteristic
29	C₄×#13083549	C₂²	1	1	yes	characteristic
30	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
31	#519×C₈₉	—	2	1	no	—
32	#441×C₈₉	—	2	1	no	—
33	#441×C₈₉	—	2	1	no	—
34	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
35	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
36	#427×C₈₉	C₂²	1	1	yes	characteristic
37	#419×C₈₉	—	2	1	no	—
38	#405×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×D₇₁₂	—	2	1	no	—
40	C₂×D₇₁₂	C₂³	1	2	yes	—
41	#15668009	—	2	1	no	—
42	#15668004	—	2	1	no	—
43	#15667997	—	4	2	no	—
44	#15667986	—	2	2	no	—
45	#15667985	—	4	2	no	—
46	C₄×D₃₅₆	—	2	2	no	—
47	#15667955	—	2	4	no	—
48	#15667955	—	2	4	no	—
49	C₂×#13083549	C₂³	1	2	yes	—
50	C₂×#13083549	C₂³	1	2	yes	—
51	#15667946	C₂³	1	2	yes	—
52	#15667944	—	2	1	no	—
53	#15667934	—	2	2	no	—
54	#15667934	—	2	2	no	—
55	C₂²×Q₈×C₈₉	—	2	1	no	—
56	C₂×Q₁₆×C₈₉	—	2	4	no	—
57	C₂×Q₁₆×C₈₉	—	2	1	no	—
58	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
59	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₄×Q₈×C₈₉	—	2	2	no	—
62	C₄×Q₈×C₈₉	—	2	2	no	—
63	C₄×Q₈×C₈₉	—	4	2	no	—
64	#117×C₈₉	—	2	1	no	—
65	#117×C₈₉	—	4	2	no	—
66	#117×C₈₉	D₄	1	1	yes	characteristic
67	#107×C₈₉	C₂³	1	2	yes	—
68	#106×C₈₉	—	2	1	no	—
69	#103×C₈₉	—	2	2	no	—
70	#103×C₈₉	—	2	2	no	—
71	#97×C₈₉	D₄	1	1	yes	characteristic
72	#97×C₈₉	D₄	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×D₃₅₆	—	2	2	no	—
76	C₂×D₃₅₆	—	4	2	no	—
77	C₂×C₂²⋊D₈₉	—	4	2	no	—
78	C₂×C₄×D₈₉	—	4	2	no	—
79	D₇₁₂	—	2	2	no	—
80	D₇₁₂	—	2	2	no	—
81	D₇₁₂	—	4	2	no	—
82	C₄⋊Dic₈₉	—	2	2	no	—
83	#13083553	—	4	2	no	—
84	#13083551	—	4	2	no	—
85	#13083551	—	4	2	no	—
86	#13083549	—	2	2	no	—
87	#13083549	—	2	2	no	—
88	C₂×Q₈×C₈₉	—	2	4	no	—
89	C₂×Q₈×C₈₉	—	2	2	no	—
90	C₂×Q₈×C₈₉	—	4	2	no	—
91	C₂×Q₈×C₈₉	—	4	1	no	—
92	C₂²×C₄×C₈₉	—	4	1	no	—
93	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
94	Q₁₆×C₈₉	—	2	4	no	—
95	Q₁₆×C₈₉	—	2	2	no	—
96	Q₁₆×C₈₉	#41	1	4	yes	—
97	M₁₆×C₈₉	—	4	1	no	—
98	C₂×C₈×C₈₉	—	2	2	no	—
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	—	2	2	no	—
104	C₄⋊C₄×C₈₉	—	4	2	no	—
105	C₄⋊C₄×C₈₉	—	4	2	no	—
106	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
107	C₂²⋊C₄×C₈₉	—	4	1	no	—
108	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
109	C₄²×C₈₉	—	4	1	no	—
110	C₄²×C₈₉	—	4	2	no	—
111	C₄²×C₈₉	—	4	2	no	—
112	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₈₉	—	4	2	no	—
116	Q₈×C₈₉	—	2	4	no	—
117	Q₈×C₈₉	—	4	4	no	—
118	Q₈×C₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	4	4	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
128	C₂²⋊D₈₉	—	16	2	no	—
129	C₂×Dic₈₉	—	4	2	no	—
130	D₃₅₆	—	4	2	no	—
131	D₃₅₆	—	4	2	no	—
132	D₃₅₆	—	16	2	no	—
133	C₄×D₈₉	—	16	2	no	—
134	C₈×C₈₉	—	2	1	no	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	C₂×#41	1	2	yes	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	—	2	2	no	—
140	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₈₉	—	8	2	no	—
142	C₂×D₈₉	—	8	2	no	—
143	C₂×D₈₉	—	8	2	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	2	2	no	—
146	C₄×C₈₉	—	2	2	no	—
147	C₄×C₈₉	—	4	4	no	—
148	C₄×C₈₉	—	8	2	no	—
149	C₄×C₈₉	#525	1	1	yes	characteristic
150	C₄×C₈₉	#575	1	1	yes	characteristic
151	Dic₈₉	—	8	2	no	—
152	#33686	—	89	1	no	maximal, Sylow
153	C₂×C₈₉	—	4	1	no	—
154	C₂×C₈₉	#3100	1	1	yes	characteristic
155	C₂×C₈₉	#3416	1	2	yes	—
156	D₈₉	—	16	2	no	—
157	#3176	—	89	1	no	—
158	#2950	D₈₉	1	1	yes	characteristic, complemented
159	#2234	—	89	2	no	—
160	#2229	—	89	2	no	—
161	#2207	—	89	2	no	—
162	#2185	—	89	2	no	—
163	#2175	—	89	2	no	—
164	C₄×SD₃₂	—	89	2	no	—
165	#2164	—	89	1	no	—
166	C₈₉	#33686	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
168	#540	—	178	1	no	—
169	#519	—	2	1	no	—
170	C₂×SD₃₂	—	89	2	no	—
171	C₂×SD₃₂	—	89	2	no	—
172	#499	—	89	1	no	—
173	#493	—	89	1	no	—
174	#490	—	178	1	no	—
175	#469	—	178	1	no	—
176	#466	—	89	2	no	—
177	#442	—	178	1	no	—
178	#441	—	2	1	no	—
179	#441	—	2	1	no	—
180	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
181	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
182	C₄×D₈	—	89	2	no	—
183	#427	C₂×D₈₉	1	1	yes	characteristic
184	#419	—	2	1	no	—
185	#405	C₂×D₈₉	1	1	yes	characteristic
186	#367	—	89	2	no	—
187	#366	—	89	2	no	—
188	C₄⋊C₁₆	—	89	2	no	—
189	#358	—	89	2	no	—
190	#358	—	89	2	no	—
191	#357	—	89	2	no	—
192	#357	—	89	2	no	—
193	C₂²⋊C₁₆	—	89	2	no	—
194	#346	—	89	1	no	—
195	C₄×C₁₆	—	89	1	no	—
196	C₂²×Q₈	—	2	1	no	—
197	C₂×Q₁₆	—	2	4	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
200	C₂×D₈	—	89	2	no	—
201	C₂×D₈	—	178	1	no	—
202	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
204	#127	—	178	1	no	—
205	#125	—	178	1	no	—
206	#123	—	356	2	no	—
207	#121	—	356	2	no	—
208	#121	—	178	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	4	2	no	—
212	C₄×D₄	—	178	2	no	—
213	#117	—	2	1	no	—
214	#117	—	4	2	no	—
215	#117	C₂²⋊D₈₉	1	1	yes	characteristic
216	SD₃₂	—	178	4	no	—
217	SD₃₂	—	178	4	no	—
218	C₂×C₁₆	—	89	2	no	—
219	C₂×C₁₆	—	89	2	no	—
220	#107	—	89	2	no	—
221	#107	C₂²×D₈₉	1	2	yes	—
222	#106	—	2	1	no	—
223	#106	—	178	1	no	—
224	#103	—	2	2	no	—
225	#103	—	2	2	no	—
226	#102	—	178	2	no	—
227	#102	—	178	2	no	—
228	#97	C₂²⋊D₈₉	1	1	yes	characteristic
229	#97	C₂²⋊D₈₉	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×D₄	—	356	2	no	—
237	C₂×D₄	—	356	2	no	—
238	C₂×D₄	—	178	2	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	356	2	no	—
241	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
242	Q₁₆	—	2	4	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	#13083574	1	4	yes	—
245	D₈	—	356	2	no	—
246	D₈	—	178	2	no	—
247	D₈	—	178	2	no	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
252	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	356	2	no	—
258	C₄⋊C₄	—	178	2	no	—
259	C₄⋊C₄	#13083559	1	2	yes	—
260	C₂²⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	356	2	no	—
262	C₂²⋊C₄	—	356	2	no	—
263	C₂²⋊C₄	#13083559	1	2	yes	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
269	C₁₆	—	178	2	no	—
270	C₁₆	—	178	2	no	—
271	C₂³	—	356	2	no	—
272	C₂³	C₂×#13083559	1	1	yes	characteristic
273	Q₈	—	2	4	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	4	2	no	—
276	D₄	—	356	2	no	—
277	D₄	—	356	2	no	—
278	D₄	—	1424	2	no	—
279	D₄	—	1424	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	356	2	no	—
286	C₂×C₄	—	1424	2	no	—
287	C₂×C₄	C₂×#13083559	1	2	yes	—
288	C₂×C₄	C₂×#13083559	1	2	yes	—
289	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	C₂×#13083574	1	2	yes	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	712	2	no	—
298	C₂²	—	712	2	no	—
299	C₂²	—	712	2	no	—
300	C₂²	C₂²×#13083559	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	8	2	no	—
306	C₄	—	712	2	no	—
307	C₄	#19152109	1	1	yes	characteristic
308	C₄	#19152396	1	1	yes	characteristic
309	C₂	—	4	1	no	—
310	C₂	—	1424	2	no	—
311	C₂	#26030252	1	1	yes	characteristic
312	C₂	#26033396	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2950	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26030902	C₂
hypercenter	#2950	D₈₉
hypocenter	C₈₉	#33686
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
2. #26022443	C₂	semidirect
3. #26022416	C₂	semidirect
4. #26022292	C₂	semidirect
5. #26022169	C₂	semidirect
6. #26022114	C₂	semidirect
7. C₄×#15667955	C₂	semidirect
8. #26022059	C₂	semidirect
9. C₈₉×#2950	C₂	semidirect
158. #2950	D₈₉	semidirect
166. C₈₉	#33686	semidirect

Maximal subgroups

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

subgroup	index
#26030902	2
#26022443	2
#26022416	2
#26022292	2
#26022169	2
#26022114	2
C₄×#15667955	2
#26022059	2
C₈₉×#2950	2
#33686	89