Subgroups — #36709638

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	#26029121	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022440	C₂	1	2	yes	maximal, complemented
3	#26022413	C₂	1	2	yes	maximal, complemented
4	#26022289	C₂	1	2	yes	maximal, complemented
5	#26022168	C₂	1	2	yes	maximal, complemented
6	#26022116	C₂	1	2	yes	maximal, complemented
7	C₄×#15667956	C₂	1	2	yes	maximal, complemented
8	#26022059	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3176	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
11	#19152075	—	2	1	no	—
12	C₂×#15667956	C₂²	1	2	yes	—
13	C₂×#15667956	C₂²	1	2	yes	—
14	#19151706	—	2	1	no	—
15	#19151706	—	2	1	no	—
16	C₄×Dic₃₅₆	C₂²	1	2	yes	—
17	C₄×Dic₃₅₆	C₂²	1	2	yes	—
18	#19151636	C₂²	1	1	yes	characteristic
19	#19151608	—	2	1	no	—
20	#19151425	C₂²	1	2	yes	—
21	#19151423	C₂²	1	2	yes	—
22	C₄⋊#13083549	C₂²	1	2	yes	—
23	#19151401	C₂²	1	2	yes	—
24	#19151401	C₂²	1	2	yes	—
25	#19151399	C₂²	1	2	yes	—
26	#19151399	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	#540×C₈₉	—	2	1	no	—
31	#499×C₈₉	C₂²	1	1	yes	characteristic
32	#493×C₈₉	C₂²	1	1	yes	characteristic
33	#490×C₈₉	—	2	1	no	—
34	#469×C₈₉	—	2	1	no	—
35	#466×C₈₉	C₂²	1	2	yes	—
36	#442×C₈₉	—	2	1	no	—
37	C₄×D₈×C₈₉	C₂²	1	2	yes	—
38	#405×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²×Dic₁₇₈	—	2	1	no	—
40	C₂×Dic₃₅₆	—	2	4	no	—
41	C₂×Dic₃₅₆	—	2	1	no	—
42	C₂×Dic₃₅₆	C₂³	1	2	yes	—
43	C₄×Dic₁₇₈	—	2	2	no	—
44	C₄×Dic₁₇₈	—	2	2	no	—
45	#15667977	—	4	2	no	—
46	#15667971	—	2	1	no	—
47	#15667968	—	4	2	no	—
48	#15667956	—	2	4	no	—
49	#15667956	—	2	4	no	—
50	C₂×#13083549	C₂³	1	2	yes	—
51	C₂×#13083549	C₂³	1	2	yes	—
52	#15667946	C₂³	1	2	yes	—
53	#15667944	—	2	1	no	—
54	#15667937	—	2	2	no	—
55	#15667937	—	2	2	no	—
56	C₂×D₈×C₈₉	—	2	1	no	—
57	C₂×D₈×C₈₉	C₂³	1	2	yes	—
58	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	#127×C₈₉	—	2	1	no	—
61	#125×C₈₉	—	2	1	no	—
62	#123×C₈₉	—	4	2	no	—
63	#121×C₈₉	—	2	2	no	—
64	#121×C₈₉	—	4	2	no	—
65	C₄×D₄×C₈₉	—	2	2	no	—
66	#117×C₈₉	D₄	1	1	yes	characteristic
67	#107×C₈₉	C₂³	1	2	yes	—
68	#106×C₈₉	—	2	1	no	—
69	#102×C₈₉	—	2	2	no	—
70	#102×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×Dic₁₇₈	—	2	4	no	—
76	C₂×Dic₁₇₈	—	2	2	no	—
77	C₂×Dic₁₇₈	—	4	2	no	—
78	C₂×Dic₁₇₈	—	4	1	no	—
79	C₂²×Dic₈₉	—	4	1	no	—
80	Dic₃₅₆	—	2	4	no	—
81	Dic₃₅₆	—	2	2	no	—
82	Dic₃₅₆	#41	1	4	yes	—
83	C₄⋊Dic₈₉	—	2	2	no	—
84	#13083553	—	4	2	no	—
85	#13083553	—	4	2	no	—
86	C₂²⋊Dic₈₉	—	4	1	no	—
87	C₄×Dic₈₉	—	4	2	no	—
88	C₄×Dic₈₉	—	4	2	no	—
89	C₄×Dic₈₉	—	4	1	no	—
90	#13083549	—	2	2	no	—
91	#13083549	—	2	2	no	—
92	C₂×D₄×C₈₉	—	2	2	no	—
93	C₂×D₄×C₈₉	—	4	2	no	—
94	C₂×D₄×C₈₉	—	4	2	no	—
95	C₂²×C₄×C₈₉	—	4	2	no	—
96	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
97	D₈×C₈₉	—	2	2	no	—
98	D₈×C₈₉	—	2	2	no	—
99	D₈×C₈₉	—	4	2	no	—
100	M₁₆×C₈₉	—	4	1	no	—
101	C₂×C₈×C₈₉	—	2	2	no	—
102	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₈₉	—	2	2	no	—
107	C₄⋊C₄×C₈₉	—	4	2	no	—
108	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
109	C₂²⋊C₄×C₈₉	—	4	2	no	—
110	C₂²⋊C₄×C₈₉	—	4	2	no	—
111	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
112	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₈₉	—	4	2	no	—
115	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
116	D₄×C₈₉	—	4	2	no	—
117	D₄×C₈₉	—	4	2	no	—
118	D₄×C₈₉	—	16	2	no	—
119	D₄×C₈₉	—	16	2	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	16	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	C₂×Dic₈₉	—	4	4	no	—
129	C₂×Dic₈₉	—	4	2	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	Dic₁₇₈	—	2	4	no	—
132	Dic₁₇₈	—	4	4	no	—
133	Dic₁₇₈	—	4	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	—	8	2	no	—
141	C₂²×C₈₉	—	8	2	no	—
142	C₂²×C₈₉	—	8	2	no	—
143	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	2	2	no	—
146	C₄×C₈₉	—	2	2	no	—
147	C₄×C₈₉	—	8	2	no	—
148	C₄×C₈₉	#525	1	1	yes	characteristic
149	C₄×C₈₉	#575	1	1	yes	characteristic
150	Dic₈₉	—	4	4	no	—
151	Dic₈₉	—	8	2	no	—
152	#33686	—	89	1	no	maximal, Sylow
153	C₂×C₈₉	—	4	1	no	—
154	C₂×C₈₉	—	16	2	no	—
155	C₂×C₈₉	#3100	1	1	yes	characteristic
156	C₂×C₈₉	#3416	1	2	yes	—
157	#3176	D₈₉	1	1	yes	characteristic
158	#2950	—	89	1	no	—
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	—	89	1	no	—
168	#540	—	2	1	no	—
169	#519	—	178	1	no	—
170	C₂×SD₃₂	—	89	2	no	—
171	C₂×SD₃₂	—	89	2	no	—
172	#499	C₂×D₈₉	1	1	yes	characteristic
173	#493	C₂×D₈₉	1	1	yes	characteristic
174	#490	—	2	1	no	—
175	#469	—	2	1	no	—
176	#466	C₂×D₈₉	1	2	yes	—
177	#442	—	2	1	no	—
178	#441	—	178	1	no	—
179	#441	—	178	1	no	—
180	C₄×Q₁₆	—	89	2	no	—
181	C₄×Q₁₆	—	89	2	no	—
182	C₄×D₈	C₂×D₈₉	1	2	yes	—
183	#427	—	89	1	no	—
184	#419	—	178	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	—	178	1	no	—
197	C₂×Q₁₆	—	89	2	no	—
198	C₂×Q₁₆	—	178	4	no	—
199	C₂×Q₁₆	—	178	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	C₂²×D₈₉	1	2	yes	—
202	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
204	#127	—	2	1	no	—
205	#125	—	2	1	no	—
206	#123	—	4	2	no	—
207	#121	—	2	2	no	—
208	#121	—	4	2	no	—
209	C₄×Q₈	—	356	2	no	—
210	C₄×Q₈	—	178	2	no	—
211	C₄×Q₈	—	178	2	no	—
212	C₄×D₄	—	2	2	no	—
213	#117	—	356	2	no	—
214	#117	—	178	1	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	—	178	2	no	—
225	#103	—	178	2	no	—
226	#102	—	2	2	no	—
227	#102	—	2	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	—	356	2	no	—
233	C₂×Q₈	—	356	1	no	—
234	C₂×Q₈	—	178	4	no	—
235	C₂×Q₈	—	178	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	356	1	no	—
241	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
242	Q₁₆	—	89	4	no	—
243	Q₁₆	—	178	4	no	—
244	Q₁₆	—	178	2	no	—
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	D₈	—	4	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	—	356	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	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	356	1	no	—
263	C₂²⋊C₄	#13083559	1	2	yes	—
264	C₄²	—	356	2	no	—
265	C₄²	—	356	2	no	—
266	C₄²	—	356	1	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	—	4	2	no	—
272	C₂³	C₂×#13083559	1	1	yes	characteristic
273	Q₈	—	356	4	no	—
274	Q₈	—	356	2	no	—
275	Q₈	—	178	4	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	16	2	no	—
279	D₄	—	16	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	16	2	no	—
284	C₂×C₄	—	356	4	no	—
285	C₂×C₄	—	356	2	no	—
286	C₂×C₄	—	356	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	#41×D₈₉	1	2	yes	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	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	—	8	2	no	—
305	C₄	—	356	4	no	—
306	C₄	—	712	2	no	—
307	C₄	#19152105	1	1	yes	characteristic
308	C₄	#19152396	1	1	yes	characteristic
309	C₂	—	4	1	no	—
310	C₂	—	16	2	no	—
311	C₂	#26030248	1	1	yes	characteristic
312	C₂	#26033399	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting, prime radical	C₈₉×#3176	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#3176	D₈₉
hypocenter	C₈₉	#33686
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3176

Complements

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

N	H ≅ G/N	type
1. #26029121	C₂	semidirect
2. #26022440	C₂	semidirect
3. #26022413	C₂	semidirect
4. #26022289	C₂	semidirect
5. #26022168	C₂	semidirect
6. #26022116	C₂	semidirect
7. C₄×#15667956	C₂	semidirect
8. #26022059	C₂	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
#26029121	2
#26022440	2
#26022413	2
#26022289	2
#26022168	2
#26022116	2
C₄×#15667956	2
#26022059	2
C₈₉×#3176	2
#33686	89