Subgroups — #36440977

Order 22784.

468 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020232	C₂	1	1	yes	maximal, characteristic
2	#26018479	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10623494	—	2	1	no	—
5	C₂×C₂²⋊#10623494	—	2	1	no	—
6	C₂×#15667919	—	2	1	no	—
7	C₂×#15667919	—	2	1	no	—
8	C₂×#15667919	C₄	1	2	yes	—
9	#19151353	C₄	1	2	yes	complemented
10	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
12	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	complemented
13	#336×C₈₉	—	2	1	no	—
14	C₂³×Dic₈₉	—	2	2	no	—
15	C₂²×#10623494	—	2	2	no	—
16	C₄⋊#10623494	—	4	2	no	—
17	C₂²⋊#10623494	—	4	1	no	—
18	C₂²⋊#10623494	—	4	1	no	—
19	#15667919	—	2	2	no	—
20	#15667919	—	4	2	no	—
21	#15667919	—	4	1	no	—
22	#15667919	—	4	1	no	—
23	#15667919	C₈	1	4	yes	complemented
24	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	—	2	2	no	—
28	C₂×C₄²×C₈₉	—	4	2	no	—
29	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
33	C₂²⋊C₈×C₈₉	—	2	4	no	—
34	C₄×C₈×C₈₉	—	4	2	no	—
35	C₂²×Dic₈₉	—	2	4	no	—
36	C₂²×Dic₈₉	—	4	2	no	—
37	C₂²×Dic₈₉	—	4	2	no	—
38	C₂²×Dic₈₉	—	4	2	no	—
39	C₂²×Dic₈₉	—	4	2	no	—
40	C₂²×Dic₈₉	—	4	2	no	—
41	C₂²×Dic₈₉	—	4	2	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂×#10623494	—	4	2	no	—
44	C₂×#10623494	—	4	2	no	—
45	C₂⁴×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	—	2	2	no	—
47	C₂²×C₄×C₈₉	—	2	2	no	—
48	C₂²×C₄×C₈₉	—	4	1	no	—
49	C₂²×C₄×C₈₉	—	4	1	no	—
50	C₂²×C₄×C₈₉	—	4	1	no	—
51	C₂²×C₄×C₈₉	—	4	1	no	—
52	C₂²×C₄×C₈₉	—	4	1	no	—
53	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
61	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
65	C₂×C₈×C₈₉	—	2	4	no	—
66	C₂×C₈×C₈₉	—	4	2	no	—
67	C₂×C₈×C₈₉	—	4	2	no	—
68	C₄²×C₈₉	—	2	2	no	—
69	C₄²×C₈₉	—	2	2	no	—
70	C₄²×C₈₉	—	4	1	no	—
71	C₄²×C₈₉	—	4	1	no	—
72	C₂³×C₈₉	—	2	1	no	—
73	C₂³×C₈₉	—	2	1	no	—
74	C₂³×C₈₉	—	2	1	no	—
75	C₂³×C₈₉	—	2	1	no	—
76	C₂³×C₈₉	#95	1	1	yes	characteristic
77	C₂³×C₈₉	C₂²⋊C₈	1	2	yes	—
78	C₂³×C₈₉	#102	1	1	yes	characteristic
79	C₂³×C₈₉	#103	1	1	yes	characteristic
80	C₂³×C₈₉	#104	1	2	yes	—
81	C₂×C₄×C₈₉	—	2	1	no	—
82	C₂×C₄×C₈₉	—	2	1	no	—
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	1	no	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	4	2	no	—
95	C₂×C₄×C₈₉	—	4	2	no	—
96	C₂×C₄×C₈₉	—	4	1	no	—
97	C₂×C₄×C₈₉	—	4	1	no	—
98	C₂×C₄×C₈₉	—	4	1	no	—
99	C₂×C₄×C₈₉	—	4	1	no	—
100	C₂×C₄×C₈₉	—	4	4	no	—
101	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#99	1	2	yes	—
104	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
107	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
109	C₂×Dic₈₉	—	4	4	no	—
110	C₂×Dic₈₉	—	4	4	no	—
111	C₂×Dic₈₉	—	4	4	no	—
112	C₂×Dic₈₉	—	8	2	no	—
113	C₂×Dic₈₉	—	8	2	no	—
114	C₂×Dic₈₉	—	8	2	no	—
115	C₂×Dic₈₉	—	8	2	no	—
116	C₈×C₈₉	—	4	4	no	—
117	#10623494	—	8	2	no	—
118	C₂²×C₈₉	—	2	2	no	—
119	C₂²×C₈₉	—	2	2	no	—
120	C₂²×C₈₉	—	2	1	no	—
121	C₂²×C₈₉	—	2	1	no	—
122	C₂²×C₈₉	—	2	2	no	—
123	C₂²×C₈₉	—	2	1	no	—
124	C₂²×C₈₉	—	2	1	no	—
125	C₂²×C₈₉	—	2	2	no	—
126	C₂²×C₈₉	D₄⋊C₈	1	2	yes	—
127	C₂²×C₈₉	Q₈⋊C₈	1	2	yes	—
128	C₂²×C₈₉	#336	1	1	yes	characteristic
129	C₂²×C₈₉	H(3,ℤ/4)	1	1	yes	characteristic
130	C₂²×C₈₉	H(3,ℤ/4)	1	1	yes	characteristic
131	C₂²×C₈₉	#339	1	1	yes	characteristic
132	C₂²×C₈₉	#340	1	1	yes	characteristic
133	C₂²×C₈₉	#342	1	1	yes	characteristic
134	C₂²×C₈₉	#344	1	1	yes	characteristic
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	—	2	2	no	—
138	C₄×C₈₉	—	2	2	no	—
139	C₄×C₈₉	—	2	1	no	—
140	C₄×C₈₉	—	2	1	no	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	4	1	no	—
144	Dic₈₉	—	8	4	no	—
145	#7024	—	89	1	no	maximal, Sylow
146	C₂×C₈₉	—	2	2	no	—
147	C₂×C₈₉	—	2	2	no	—
148	C₂×C₈₉	#1274	1	1	yes	characteristic
149	C₂×C₈₉	#1279	1	1	yes	characteristic
150	C₂×C₈₉	#1282	1	1	yes	characteristic
151	C₂×C₈₉	#1284	1	1	yes	characteristic
152	C₂×C₈₉	#1286	1	1	yes	characteristic
153	C₂×C₈₉	#1288	1	1	yes	characteristic
154	C₂×C₈₉	#1289	1	1	yes	characteristic
155	#1797	—	89	1	no	—
156	C₄×C₂²⋊C₈	D₈₉	1	1	yes	characteristic
157	#1434	—	89	1	no	—
158	C₈₉	#7024	1	1	yes	characteristic, complemented, Sylow
159	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	178	1	no	—
161	C₂×C₂²⋊C₈	—	2	1	no	—
162	C₂×C₂²⋊C₈	—	178	1	no	—
163	C₂×C₄×C₈	Dic₈₉	1	2	yes	complemented
164	C₂×#95	—	89	2	no	—
165	C₂×#95	—	178	1	no	—
166	C₂×#95	—	178	1	no	—
167	#336	—	2	1	no	—
168	#336	—	89	2	no	—
169	C₂³×C₄	—	178	2	no	—
170	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
171	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
172	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	178	2	no	—
175	C₂×C₄²	—	4	2	no	—
176	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
180	C₄⋊C₈	—	356	2	no	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	356	1	no	—
183	C₂²⋊C₈	—	356	1	no	—
184	C₄×C₈	—	4	2	no	—
185	#95	—	89	4	no	—
186	#95	—	356	2	no	—
187	#95	—	356	1	no	—
188	#95	—	356	1	no	—
189	#95	—	178	2	no	—
190	C₂⁴	#13083551	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	356	2	no	—
199	C₂²×C₄	—	356	2	no	—
200	C₂²×C₄	—	356	2	no	—
201	C₂²×C₄	—	356	2	no	—
202	C₂²×C₄	—	356	2	no	—
203	C₂²×C₄	—	356	2	no	—
204	C₂²×C₄	—	178	4	no	—
205	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
206	C₂²×C₄	#13083551	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
208	C₂²×C₄	#13083553	1	1	yes	characteristic
209	C₂²×C₄	#13083553	1	1	yes	characteristic
210	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
211	C₂²×C₄	C₈×D₈₉	1	2	yes	—
212	C₂²×C₄	#13083557	1	2	yes	—
213	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
214	C₂²×C₄	#13083563	1	1	yes	characteristic
215	C₂²×C₄	#13083563	1	1	yes	characteristic
216	C₂²×C₄	Dic₃₅₆	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	356	2	no	—
221	C₂×C₈	—	356	2	no	—
222	C₂×C₈	—	356	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	4	1	no	—
226	C₄²	—	4	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	#15667919	1	1	yes	characteristic
232	C₂³	#15667924	1	2	yes	—
233	C₂³	#15667934	1	1	yes	characteristic
234	C₂³	#15667937	1	1	yes	characteristic
235	C₂³	#15667940	1	2	yes	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	356	4	no	—
257	C₂×C₄	—	356	4	no	—
258	C₂×C₄	—	356	4	no	—
259	C₂×C₄	—	712	2	no	—
260	C₂×C₄	—	712	2	no	—
261	C₂×C₄	—	712	2	no	—
262	C₂×C₄	—	712	2	no	—
263	C₂×C₄	C₈×Dic₈₉	1	1	yes	characteristic
264	C₂×C₄	#15667922	1	1	yes	characteristic
265	C₂×C₄	#15667927	1	2	yes	—
266	C₂×C₄	#15667934	1	1	yes	characteristic
267	C₂×C₄	#15667937	1	1	yes	characteristic
268	C₂×C₄	#15667941	1	2	yes	—
269	C₂×C₄	#15667944	1	1	yes	characteristic
270	C₂×C₄	#15667946	1	1	yes	characteristic
271	C₈	—	4	4	no	—
272	C₈	—	712	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	#19151327	1	2	yes	—
282	C₂²	#19151330	1	2	yes	—
283	C₂²	#19151352	1	1	yes	characteristic
284	C₂²	#19151354	1	1	yes	characteristic
285	C₂²	#19151355	1	1	yes	characteristic
286	C₂²	#19151358	1	1	yes	characteristic
287	C₂²	#19151361	1	1	yes	characteristic
288	C₂²	#19151365	1	1	yes	characteristic
289	C₂²	#19151369	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	712	4	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	#26018080	1	1	yes	characteristic
303	C₂	#26018092	1	1	yes	characteristic
304	C₂	#26018099	1	1	yes	characteristic
305	C₂	#26018105	1	1	yes	characteristic
306	C₂	#26018111	1	1	yes	characteristic
307	C₂	#26018117	1	1	yes	characteristic
308	C₂	#26018120	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667919
commutator	C₂×C₄×C₈₉	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₈₉
Fitting	C₄×C₂²⋊C₈×C₈₉	C₂
socle	C₂³×C₈₉	#95
prime radical	C₂⁴×C₈₉	C₂²⋊C₄
cosocle	C₂²×C₄²×C₈₉	C₂²
hypercenter	C₄×C₂²⋊C₈	D₈₉
hypocenter	C₈₉	#7024
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
9. #19151353	C₄	semidirect
12. C₂×C₄×C₈×C₈₉	C₄	semidirect
23. #15667919	C₈	semidirect
158. C₈₉	#7024	semidirect
163. C₂×C₄×C₈	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26020232	2
#26018479	2
C₄×C₂²⋊C₈×C₈₉	2
#7024	89