Subgroups — #36445142

Order 22784.

498 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	#26029208	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667938	C₂	1	1	yes	maximal, characteristic
3	#26019997	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018635	C₂	1	1	yes	maximal, characteristic
5	M₁₆×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018084	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13083574	C₂²	1	1	yes	characteristic
9	#19151722	C₄	1	2	yes	complemented
10	#19151663	C₄	1	2	yes	complemented
11	#19151661	—	2	2	no	—
12	#19151651	C₂²	1	1	yes	characteristic
13	#19151649	—	2	1	no	—
14	C₂×#15667938	C₄	1	2	yes	—
15	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₈₉	—	2	1	no	—
17	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
18	#19151359	C₄	1	2	yes	—
19	#19151352	—	2	2	no	—
20	#19151330	C₄	1	2	yes	complemented
21	#19151327	C₄	1	2	yes	complemented
22	#19151319	—	2	1	no	—
23	C₈×#10623494	—	2	1	no	—
24	#431×C₈₉	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
28	C₂²×#10623494	D₄	1	2	yes	—
29	C₂×C₈×D₈₉	—	2	2	no	—
30	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
32	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
33	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₈₉	—	2	1	no	—
35	C₄×C₂²⋊D₈₉	—	2	1	no	—
36	#15667971	C₂×C₄	1	1	yes	characteristic
37	#15667969	—	2	1	no	—
38	C₂×C₄×Dic₈₉	—	2	1	no	—
39	C₂×C₄×Dic₈₉	—	2	1	no	—
40	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
41	C₄²×D₈₉	—	2	1	no	—
42	C₄⋊#10623494	C₂×C₄	1	2	yes	—
43	#15667941	—	2	2	no	—
44	#15667938	C₈	1	8	yes	complemented
45	C₂²⋊#10623494	—	2	1	no	—
46	#15667924	—	2	2	no	—
47	C₄×#10623494	—	2	1	no	—
48	C₄×#10623494	C₂×C₄	1	2	yes	—
49	C₄×#10623494	D₄	1	1	yes	characteristic
50	C₄×#10623494	D₄	1	1	yes	characteristic
51	C₈×Dic₈₉	—	2	2	no	—
52	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₈₉	—	2	1	no	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₈₉	—	2	1	no	—
57	#97×C₈₉	—	2	1	no	—
58	C₄×C₈×C₈₉	—	2	1	no	—
59	C₄×C₈×C₈₉	—	2	1	no	—
60	C₄×C₈×C₈₉	—	2	1	no	—
61	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
62	#13083574	—	2	2	no	—
63	#13083574	C₂×C₈	1	4	yes	—
64	C₂×Dic₁₇₈	C₄²	1	1	yes	characteristic
65	C₂×D₃₅₆	C₄²	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂²×Dic₈₉	—	2	2	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×#10623494	—	2	2	no	—
72	C₂×#10623494	—	4	2	no	—
73	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
74	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
75	C₈×D₈₉	—	4	4	no	—
76	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
77	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
78	#13083553	—	2	1	no	—
79	#13083553	—	2	1	no	—
80	C₂²⋊Dic₈₉	—	2	1	no	—
81	#13083551	—	2	1	no	—
82	#13083551	—	2	1	no	—
83	C₄×Dic₈₉	—	2	1	no	—
84	C₄×Dic₈₉	—	2	1	no	—
85	C₄×Dic₈₉	—	2	1	no	—
86	C₄×Dic₈₉	—	2	2	no	—
87	C₄×Dic₈₉	—	4	2	no	—
88	C₄×Dic₈₉	—	4	2	no	—
89	C₄×Dic₈₉	C₂×C₈	1	4	yes	complemented
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	M₁₆×C₈₉	—	2	2	no	—
94	M₁₆×C₈₉	C₂×C₈	1	4	yes	complemented
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	—	2	2	no	—
97	C₂×C₈×C₈₉	—	2	2	no	—
98	C₂×C₈×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	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₈₉	#131	1	2	yes	—
115	C₂²⋊D₈₉	—	2	4	no	—
116	C₂×Dic₈₉	—	2	1	no	—
117	C₂×Dic₈₉	—	2	1	no	—
118	C₂×Dic₈₉	—	2	1	no	—
119	C₂×Dic₈₉	—	2	4	no	—
120	C₂×Dic₈₉	—	4	2	no	—
121	C₂×Dic₈₉	—	4	2	no	—
122	D₃₅₆	—	2	1	no	—
123	D₃₅₆	C₄×C₈	1	2	yes	—
124	C₄×D₈₉	—	2	4	no	—
125	C₄×D₈₉	—	4	2	no	—
126	C₄×D₈₉	—	4	2	no	—
127	Dic₁₇₈	—	2	1	no	—
128	Dic₁₇₈	C₄×C₈	1	2	yes	—
129	C₈×C₈₉	—	2	4	no	—
130	C₈×C₈₉	—	4	2	no	—
131	C₈×C₈₉	—	4	2	no	—
132	#10623494	—	2	2	no	—
133	#10623494	—	2	2	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	C₈×D₄	1	2	yes	—
139	C₂×D₈₉	—	2	2	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	C₂×C₄×C₈	1	1	yes	characteristic
145	C₄×C₈₉	#405	1	1	yes	characteristic
146	C₄×C₈₉	#421	1	2	yes	—
147	C₄×C₈₉	C₈×D₄	1	2	yes	—
148	C₄×C₈₉	#444	1	2	yes	—
149	Dic₈₉	—	2	2	no	—
150	Dic₈₉	—	4	4	no	—
151	Dic₈₉	—	4	1	no	—
152	Dic₈₉	—	4	1	no	—
153	#8191	—	89	1	no	maximal, Sylow
154	C₂×C₈₉	—	2	2	no	—
155	C₂×C₈₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₈₉	#1759	1	1	yes	characteristic
157	C₂×C₈₉	#1765	1	1	yes	characteristic
158	D₈₉	—	4	2	no	—
159	C₈×#41	—	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	D₈₉	1	1	yes	characteristic, complemented
165	#1276	—	89	1	no	—
166	C₈₉	#8191	1	1	yes	characteristic, complemented, Sylow
167	C₄×#41	—	89	1	no	—
168	C₈×Q₈	—	89	2	no	—
169	C₈×D₄	—	89	2	no	—
170	C₈×D₄	—	178	2	no	—
171	#431	—	89	1	no	—
172	#431	—	178	1	no	—
173	#431	C₂×D₈₉	1	1	yes	characteristic
174	C₂×#104	—	89	2	no	—
175	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
176	C₂×C₄×C₈	—	89	1	no	—
177	C₂×C₄×C₈	—	178	1	no	—
178	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
179	C₄³	—	89	1	no	—
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	C₄×D₈₉	1	2	yes	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	89	2	no	—
190	C₂²×C₈	—	178	2	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	—	89	2	no	—
200	C₂×C₄²	—	178	1	no	—
201	C₂×C₄²	—	178	1	no	—
202	C₂×C₄²	—	178	1	no	—
203	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
204	C₄⋊C₈	—	89	2	no	—
205	C₄⋊C₈	—	178	2	no	—
206	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
207	#104	—	89	8	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	178	2	no	—
210	C₂²⋊C₈	—	178	1	no	—
211	#97	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	89	2	no	—
216	C₄×C₈	—	89	1	no	—
217	C₄×C₈	—	89	1	no	—
218	C₄×C₈	—	178	2	no	—
219	C₄×C₈	—	178	1	no	—
220	C₄×C₈	C₄×D₈₉	1	2	yes	—
221	#41	—	89	4	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	—	178	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	#13083575	1	1	yes	characteristic
233	M₁₆	—	2	2	no	—
234	M₁₆	C₈×D₈₉	1	4	yes	complemented
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	89	2	no	—
240	C₂×C₈	—	89	2	no	—
241	C₂×C₈	—	356	4	no	—
242	C₂×C₈	—	356	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	—	89	4	no	—
252	C₄²	—	356	2	no	—
253	C₄²	—	356	2	no	—
254	C₄²	—	178	1	no	—
255	C₄²	—	178	1	no	—
256	C₄²	—	178	2	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	#13083573	1	1	yes	characteristic
262	C₂³	—	178	1	no	—
263	C₂³	#15667974	1	1	yes	characteristic
264	Q₈	—	89	2	no	—
265	Q₈	—	178	1	no	—
266	D₄	—	89	2	no	—
267	D₄	—	178	4	no	—
268	D₄	—	178	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	356	2	no	—
272	C₂×C₄	—	356	2	no	—
273	C₂×C₄	—	356	2	no	—
274	C₂×C₄	—	356	2	no	—
275	C₂×C₄	—	178	4	no	—
276	C₂×C₄	—	178	1	no	—
277	C₂×C₄	—	178	1	no	—
278	C₂×C₄	—	178	1	no	—
279	C₂×C₄	—	178	4	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	#15667972	1	1	yes	characteristic
285	C₂×C₄	#15667974	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
287	C₂×C₄	#15668020	1	2	yes	—
288	C₈	—	2	4	no	—
289	C₈	—	4	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	356	2	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	—	178	2	no	—
299	C₂²	#19151445	1	1	yes	characteristic
300	C₂²	#19151663	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	356	4	no	—
304	C₄	—	356	1	no	—
305	C₄	—	356	1	no	—
306	C₄	—	178	2	no	—
307	C₄	C₄×C₈×D₈₉	1	1	yes	characteristic
308	C₄	#19151555	1	1	yes	characteristic
309	C₄	#19151613	1	2	yes	—
310	C₄	#19151661	1	2	yes	—
311	C₄	#19151719	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	356	2	no	—
314	C₂	#26020022	1	1	yes	characteristic
315	C₂	#26020061	1	1	yes	characteristic
316	C₂	#26020095	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₉
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1278	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1278	D₈₉
hypocenter	C₈₉	#8191
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
3. #26019997	C₂	semidirect
5. M₁₆×#10623494	C₂	semidirect
6. #26018084	C₂	semidirect
7. C₈₉×#1278	C₂	semidirect
9. #19151722	C₄	semidirect
10. #19151663	C₄	semidirect
20. #19151330	C₄	semidirect
21. #19151327	C₄	semidirect
44. #15667938	C₈	semidirect
89. C₄×Dic₈₉	C₂×C₈	semidirect
94. M₁₆×C₈₉	C₂×C₈	semidirect
164. #1278	D₈₉	semidirect
166. C₈₉	#8191	semidirect
234. M₁₆	C₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029208	2
C₄×#15667938	2
#26019997	2
#26018635	2
M₁₆×#10623494	2
#26018084	2
C₈₉×#1278	2
#8191	89