Subgroups — #36445182

Order 22784.

468 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029213	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667940	C₂	1	1	yes	maximal, characteristic
3	#26019998	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018632	C₂	1	1	yes	maximal, characteristic
5	#26018533	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018088	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13083574	C₂²	1	1	yes	characteristic
9	#19151725	C₄	1	2	yes	complemented
10	#19151675	C₄	1	2	yes	complemented
11	#19151666	—	2	2	no	—
12	#19151649	—	2	1	no	—
13	C₂×#15667940	C₄	1	2	yes	—
14	C₄×#13083558	C₂²	1	1	yes	characteristic
15	C₄×#13083557	—	2	1	no	—
16	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
17	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
18	#19151358	C₄	1	2	yes	—
19	#19151353	—	2	2	no	—
20	#19151328	C₄	1	2	yes	complemented
21	#19151325	C₄	1	2	yes	complemented
22	#19151320	—	2	1	no	—
23	C₈×#10623494	—	2	1	no	—
24	#431×C₈₉	C₂²	1	1	yes	characteristic
25	#431×C₈₉	C₂²	1	1	yes	characteristic
26	C₄³×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
28	C₂×#13083558	C₂×C₄	1	2	yes	—
29	C₂×#13083557	—	2	2	no	—
30	C₂²×#10623494	—	2	1	no	—
31	C₂²×#10623494	D₄	1	2	yes	—
32	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
34	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
35	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
36	C₄×C₂²⋊D₈₉	—	2	1	no	—
37	C₄×C₂²⋊D₈₉	—	2	1	no	—
38	#15667971	C₂×C₄	1	1	yes	characteristic
39	#15667969	—	2	1	no	—
40	C₄²×D₈₉	—	2	1	no	—
41	#15667941	—	2	2	no	—
42	#15667940	—	2	4	no	—
43	#15667924	—	2	2	no	—
44	#15667923	—	2	1	no	—
45	#15667922	—	2	2	no	—
46	C₄×#10623494	—	2	1	no	—
47	C₄×#10623494	—	2	1	no	—
48	C₄×#10623494	C₂×C₄	1	2	yes	—
49	C₄×#10623494	C₂×C₄	1	2	yes	—
50	C₄×#10623494	D₄	1	1	yes	characteristic
51	C₄×#10623494	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	—	2	1	no	—
53	C₂×C₄²×C₈₉	—	2	1	no	—
54	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
57	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
58	C₂²⋊C₈×C₈₉	—	2	1	no	—
59	C₂²⋊C₈×C₈₉	—	2	1	no	—
60	C₄×C₈×C₈₉	—	2	1	no	—
61	C₄×C₈×C₈₉	—	2	1	no	—
62	#13083574	—	2	2	no	—
63	#13083574	—	2	2	no	—
64	C₂×Dic₁₇₈	C₄²	1	1	yes	characteristic
65	C₂×D₃₅₆	C₄²	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	#13083558	—	2	2	no	—
71	#13083558	M₁₆	1	4	yes	complemented
72	#13083557	—	4	4	no	—
73	C₂×#10623494	—	2	2	no	—
74	C₂×#10623494	—	2	2	no	—
75	C₂×#10623494	—	4	2	no	—
76	C₂×#10623494	—	4	2	no	—
77	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
78	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
79	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
80	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	C₂²⋊Dic₈₉	—	2	1	no	—
84	#13083551	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	C₄×Dic₈₉	—	2	1	no	—
87	C₄×Dic₈₉	—	2	1	no	—
88	C₄×Dic₈₉	—	2	1	no	—
89	C₂²×C₄×C₈₉	—	2	2	no	—
90	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
93	C₂×C₈×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	—	2	2	no	—
95	C₄²×C₈₉	—	2	4	no	—
96	C₄²×C₈₉	—	2	2	no	—
97	C₄²×C₈₉	—	2	2	no	—
98	C₄²×C₈₉	—	4	2	no	—
99	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₈₉	#41	1	1	yes	characteristic
103	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
104	C₂²×D₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	2	2	no	—
107	C₂×C₄×C₈₉	—	2	4	no	—
108	C₂×C₄×C₈₉	—	4	2	no	—
109	C₂×C₄×C₈₉	—	4	2	no	—
110	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₈₉	#131	1	2	yes	—
118	C₂²⋊D₈₉	—	4	2	no	—
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	—	2	1	no	—
122	D₃₅₆	—	2	1	no	—
123	D₃₅₆	—	2	1	no	—
124	C₄×D₈₉	—	4	2	no	—
125	C₄×D₈₉	—	4	2	no	—
126	C₄×D₈₉	—	4	2	no	—
127	Dic₁₇₈	—	2	1	no	—
128	Dic₁₇₈	—	2	1	no	—
129	C₈×C₈₉	—	4	2	no	—
130	C₈×C₈₉	—	4	2	no	—
131	#10623494	—	2	2	no	—
132	#10623494	—	2	2	no	—
133	#10623494	—	2	4	no	—
134	#10623494	—	4	2	no	—
135	C₂²×C₈₉	—	2	1	no	—
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₈₉	#436	1	2	yes	—
139	C₂×D₈₉	—	4	1	no	—
140	C₂×D₈₉	—	4	1	no	—
141	C₂×D₈₉	—	4	1	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	4	4	no	—
145	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
146	C₄×C₈₉	#405	1	1	yes	characteristic
147	C₄×C₈₉	#421	1	2	yes	—
148	C₄×C₈₉	#435	1	2	yes	—
149	C₄×C₈₉	#443	1	2	yes	—
150	Dic₈₉	—	4	1	no	—
151	Dic₈₉	—	4	1	no	—
152	Dic₈₉	—	4	1	no	—
153	#8197	—	89	1	no	maximal, Sylow
154	C₂×C₈₉	—	2	2	no	—
155	C₂×C₈₉	#1753	1	1	yes	characteristic
156	C₂×C₈₉	#1759	1	1	yes	characteristic
157	C₂×C₈₉	#1764	1	1	yes	characteristic
158	D₈₉	—	8	1	no	—
159	#2965	—	89	1	no	—
160	C₄×#104	—	89	1	no	—
161	#1744	—	89	1	no	—
162	#1476	—	89	1	no	—
163	C₈×M₁₆	—	89	1	no	—
164	#1278	—	89	1	no	—
165	#1276	D₈₉	1	1	yes	characteristic, complemented
166	C₈₉	#8197	1	1	yes	characteristic, complemented, Sylow
167	C₄×#41	—	89	1	no	—
168	#446	—	89	2	no	—
169	#436	—	89	2	no	—
170	#435	—	178	2	no	—
171	#431	—	178	1	no	—
172	#431	C₂×D₈₉	1	1	yes	characteristic
173	#431	C₂×D₈₉	1	1	yes	characteristic
174	C₂×#104	—	89	2	no	—
175	C₄×M₁₆	—	89	1	no	—
176	C₄×M₁₆	—	178	1	no	—
177	C₂×C₄×C₈	—	89	1	no	—
178	C₂×C₄×C₈	—	89	1	no	—
179	C₄³	C₂×D₈₉	1	1	yes	characteristic
180	#339	—	89	2	no	—
181	#336	—	178	2	no	—
182	Q₈⋊C₈	—	89	2	no	—
183	D₄⋊C₈	—	89	2	no	—
184	#322	—	178	1	no	—
185	C₈²	—	178	1	no	—
186	C₂×#41	—	89	1	no	—
187	C₂×M₁₆	—	89	2	no	—
188	C₂×M₁₆	—	178	2	no	—
189	C₂²×C₈	—	89	2	no	—
190	C₂²×C₈	—	178	1	no	—
191	C₄×Q₈	—	89	1	no	—
192	C₄×Q₈	—	89	1	no	—
193	C₄×D₄	—	89	1	no	—
194	C₄×D₄	—	89	1	no	—
195	C₄×D₄	—	178	1	no	—
196	C₄×D₄	—	178	1	no	—
197	#117	—	89	1	no	—
198	#117	—	178	1	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	178	1	no	—
202	C₂×C₄²	C₄×D₈₉	1	2	yes	—
203	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
204	C₄⋊C₈	—	178	2	no	—
205	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
206	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
207	#104	—	178	4	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	178	2	no	—
211	#97	—	178	2	no	—
212	#97	—	178	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	89	2	no	—
216	C₄×C₈	—	89	2	no	—
217	C₄×C₈	—	89	1	no	—
218	C₄×C₈	—	89	1	no	—
219	C₄×C₈	—	178	1	no	—
220	C₄×C₈	—	178	1	no	—
221	#41	—	178	2	no	—
222	#41	—	178	2	no	—
223	C₂×Q₈	—	89	1	no	—
224	C₂×D₄	—	89	1	no	—
225	C₂×D₄	—	178	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	178	1	no	—
228	C₂²×C₄	—	178	1	no	—
229	C₂²×C₄	—	178	1	no	—
230	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
232	C₂²×C₄	#13083573	1	1	yes	characteristic
233	M₁₆	—	89	4	no	—
234	M₁₆	—	356	4	no	—
235	M₁₆	—	178	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	89	2	no	—
239	C₂×C₈	—	89	2	no	—
240	C₂×C₈	—	356	2	no	—
241	C₂×C₈	—	356	2	no	—
242	C₂×C₈	—	178	2	no	—
243	C₂×C₈	—	178	2	no	—
244	C₄⋊C₄	—	89	1	no	—
245	C₄⋊C₄	—	89	1	no	—
246	C₄⋊C₄	—	178	1	no	—
247	C₄⋊C₄	—	178	1	no	—
248	C₂²⋊C₄	—	178	1	no	—
249	C₂²⋊C₄	—	178	1	no	—
250	C₂²⋊C₄	—	178	1	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	—	178	1	no	—
256	C₄²	—	178	1	no	—
257	C₄²	—	178	1	no	—
258	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
260	C₄²	D₄×D₈₉	1	1	yes	characteristic
261	C₄²	#13083575	1	1	yes	characteristic
262	C₂³	—	178	1	no	—
263	C₂³	#15667972	1	1	yes	characteristic
264	Q₈	—	178	1	no	—
265	Q₈	—	178	1	no	—
266	D₄	—	356	2	no	—
267	D₄	—	178	1	no	—
268	D₄	—	178	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	356	2	no	—
275	C₂×C₄	—	356	2	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	178	1	no	—
278	C₂×C₄	—	178	1	no	—
279	C₂×C₄	—	178	1	no	—
280	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
282	C₂×C₄	#15667970	1	1	yes	characteristic
283	C₂×C₄	#15667972	1	1	yes	characteristic
284	C₂×C₄	#15667974	1	1	yes	characteristic
285	C₂×C₄	#15667974	1	1	yes	characteristic
286	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
287	C₂×C₄	#15668018	1	2	yes	—
288	C₈	—	4	2	no	—
289	C₈	—	4	2	no	—
290	C₈	—	356	2	no	—
291	C₈	—	178	4	no	—
292	C₈	—	178	2	no	—
293	C₈	—	178	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	356	1	no	—
297	C₂²	—	356	1	no	—
298	C₂²	—	356	1	no	—
299	C₂²	#19151445	1	1	yes	characteristic
300	C₂²	#19151673	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	4	no	—
304	C₄	—	356	1	no	—
305	C₄	—	356	1	no	—
306	C₄	—	356	1	no	—
307	C₄	#19151550	1	1	yes	characteristic
308	C₄	#19151552	1	1	yes	characteristic
309	C₄	#19151613	1	2	yes	—
310	C₄	#19151668	1	2	yes	—
311	C₄	#19151712	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	712	1	no	—
314	C₂	#26020034	1	1	yes	characteristic
315	C₂	#26020061	1	1	yes	characteristic
316	C₂	#26020086	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1276	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1276	D₈₉
hypocenter	C₈₉	#8197
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

Complements

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

N	H ≅ G/N	type
3. #26019998	C₂	semidirect
5. #26018533	C₂	semidirect
6. #26018088	C₂	semidirect
7. C₈₉×#1276	C₂	semidirect
9. #19151725	C₄	semidirect
10. #19151675	C₄	semidirect
20. #19151328	C₄	semidirect
21. #19151325	C₄	semidirect
71. #13083558	M₁₆	semidirect
165. #1276	D₈₉	semidirect
166. C₈₉	#8197	semidirect

Maximal subgroups

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

subgroup	index
#26029213	2
C₄×#15667940	2
#26019998	2
#26018632	2
#26018533	2
#26018088	2
C₈₉×#1276	2
#8197	89