Subgroups — #36445177

Order 22784.

500 conjugacy classes of subgroups.

The subgroup lattice has 320 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×#15667938	C₂	1	1	yes	maximal, characteristic
3	C₄×#15667922	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	#19151725	C₄	1	2	yes	complemented
10	#19151675	C₄	1	2	yes	complemented
11	#19151666	—	2	2	no	—
12	#19151651	C₂²	1	1	yes	characteristic
13	#19151649	—	2	1	no	—
14	C₂×#15667938	C₄	1	2	yes	—
15	C₄×#13083557	—	2	1	no	—
16	C₂×#15667922	—	2	2	no	—
17	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
18	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
19	#19151359	C₄	1	2	yes	—
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	C₂×#13083557	—	2	2	no	—
29	C₂²×#10623494	D₄	1	2	yes	—
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	—	2	4	no	—
45	C₂²⋊#10623494	—	2	1	no	—
46	#15667924	—	2	2	no	—
47	#15667922	—	2	8	no	—
48	#15667922	—	2	2	no	—
49	C₄×#10623494	—	2	1	no	—
50	C₄×#10623494	C₂×C₄	1	2	yes	—
51	C₄×#10623494	D₄	1	1	yes	characteristic
52	C₄×#10623494	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₈₉	—	2	1	no	—
55	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₈₉	—	2	1	no	—
58	#97×C₈₉	—	2	1	no	—
59	C₄×C₈×C₈₉	—	2	1	no	—
60	C₄×C₈×C₈₉	—	2	1	no	—
61	C₄×C₈×C₈₉	—	2	1	no	—
62	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
63	#13083574	—	2	2	no	—
64	#13083574	—	2	2	no	—
65	C₂×Dic₁₇₈	C₄²	1	1	yes	characteristic
66	C₂×D₃₅₆	C₄²	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂²×Dic₈₉	—	2	2	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	#13083557	—	4	4	no	—
73	C₂×#10623494	—	2	4	no	—
74	C₂×#10623494	—	2	2	no	—
75	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
76	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
77	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	C₂²⋊Dic₈₉	—	2	1	no	—
82	#13083551	—	2	1	no	—
83	#13083551	—	2	1	no	—
84	C₄×Dic₈₉	—	2	4	no	—
85	C₄×Dic₈₉	—	2	1	no	—
86	C₄×Dic₈₉	—	2	1	no	—
87	C₄×Dic₈₉	—	2	1	no	—
88	C₄×Dic₈₉	—	2	2	no	—
89	C₄×Dic₈₉	—	2	2	no	—
90	C₄×Dic₈₉	—	4	2	no	—
91	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
94	M₁₆×C₈₉	—	2	2	no	—
95	M₁₆×C₈₉	M₁₆	1	4	yes	complemented
96	C₂×C₈×C₈₉	—	2	4	no	—
97	C₂×C₈×C₈₉	—	2	2	no	—
98	C₂×C₈×C₈₉	—	2	2	no	—
99	C₂×C₈×C₈₉	—	2	2	no	—
100	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₈₉	#41	1	1	yes	characteristic
104	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
105	C₂²×D₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	2	no	—
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#117	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	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₈₉	#131	1	2	yes	—
116	C₂²⋊D₈₉	—	4	2	no	—
117	C₂×Dic₈₉	—	2	4	no	—
118	C₂×Dic₈₉	—	2	1	no	—
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	—	4	2	no	—
122	C₂×Dic₈₉	—	4	2	no	—
123	D₃₅₆	—	2	1	no	—
124	D₃₅₆	—	2	1	no	—
125	C₄×D₈₉	—	4	2	no	—
126	C₄×D₈₉	—	4	2	no	—
127	C₄×D₈₉	—	4	2	no	—
128	Dic₁₇₈	—	2	1	no	—
129	Dic₁₇₈	—	2	1	no	—
130	C₈×C₈₉	—	2	4	no	—
131	C₈×C₈₉	—	2	4	no	—
132	C₈×C₈₉	—	4	2	no	—
133	#10623494	—	2	2	no	—
134	#10623494	—	4	2	no	—
135	#10623494	#104	1	4	yes	complemented
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₈₉	#436	1	2	yes	—
140	C₂×D₈₉	—	4	1	no	—
141	C₂×D₈₉	—	4	1	no	—
142	C₂×D₈₉	—	4	1	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
146	C₄×C₈₉	#405	1	1	yes	characteristic
147	C₄×C₈₉	C₂×#104	1	2	yes	—
148	C₄×C₈₉	#435	1	2	yes	—
149	C₄×C₈₉	#444	1	2	yes	—
150	Dic₈₉	—	4	4	no	—
151	Dic₈₉	—	4	1	no	—
152	Dic₈₉	—	4	1	no	—
153	Dic₈₉	—	4	1	no	—
154	#8196	—	89	1	no	maximal, Sylow
155	C₂×C₈₉	—	2	2	no	—
156	C₂×C₈₉	#1753	1	1	yes	characteristic
157	C₂×C₈₉	C₄×#104	1	1	yes	characteristic
158	C₂×C₈₉	#1765	1	1	yes	characteristic
159	D₈₉	—	8	1	no	—
160	#2965	—	89	1	no	—
161	C₄×#104	—	89	1	no	—
162	C₄×#97	—	89	1	no	—
163	#1476	—	89	1	no	—
164	C₈×M₁₆	—	89	1	no	—
165	#1278	D₈₉	1	1	yes	characteristic, complemented
166	#1276	—	89	1	no	—
167	C₈₉	#8196	1	1	yes	characteristic, complemented, Sylow
168	C₄×#41	—	89	1	no	—
169	#446	—	89	2	no	—
170	#436	—	89	2	no	—
171	#435	—	178	2	no	—
172	#431	—	89	1	no	—
173	#431	—	178	1	no	—
174	#431	C₂×D₈₉	1	1	yes	characteristic
175	C₂×#104	—	89	2	no	—
176	C₄×M₁₆	—	178	1	no	—
177	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
178	C₂×#97	—	178	2	no	—
179	C₂×C₄×C₈	—	89	1	no	—
180	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
181	C₄³	—	89	1	no	—
182	#339	—	89	2	no	—
183	Q₈⋊C₈	—	89	2	no	—
184	D₄⋊C₈	—	89	2	no	—
185	#322	—	178	1	no	—
186	C₈²	—	178	1	no	—
187	C₂×#41	—	89	1	no	—
188	C₂×M₁₆	—	178	2	no	—
189	C₂×M₁₆	C₄×D₈₉	1	2	yes	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	89	2	no	—
192	C₄×Q₈	—	89	1	no	—
193	C₄×Q₈	—	89	1	no	—
194	C₄×D₄	—	89	1	no	—
195	C₄×D₄	—	89	1	no	—
196	C₄×D₄	—	178	1	no	—
197	C₄×D₄	—	178	1	no	—
198	#117	—	89	1	no	—
199	#117	—	178	1	no	—
200	C₂×C₄²	—	89	2	no	—
201	C₂×C₄²	—	178	1	no	—
202	C₂×C₄²	—	178	1	no	—
203	C₂×C₄²	—	178	1	no	—
204	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
205	C₄⋊C₈	—	89	2	no	—
206	C₄⋊C₈	—	178	2	no	—
207	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
208	#104	—	178	4	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	178	2	no	—
211	C₂²⋊C₈	—	178	1	no	—
212	#97	—	2	1	no	—
213	#97	—	178	2	no	—
214	#97	—	178	8	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	89	2	no	—
219	C₄×C₈	—	89	1	no	—
220	C₄×C₈	—	89	1	no	—
221	C₄×C₈	—	178	1	no	—
222	C₄×C₈	C₄×D₈₉	1	2	yes	—
223	#41	—	178	2	no	—
224	#41	—	178	2	no	—
225	C₂×Q₈	—	89	1	no	—
226	C₂×D₄	—	89	1	no	—
227	C₂×D₄	—	178	1	no	—
228	C₂²×C₄	—	178	2	no	—
229	C₂²×C₄	—	178	1	no	—
230	C₂²×C₄	—	178	1	no	—
231	C₂²×C₄	—	178	1	no	—
232	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
234	C₂²×C₄	#13083575	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	356	4	no	—
237	M₁₆	#13083557	1	4	yes	complemented
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	89	2	no	—
243	C₂×C₈	—	89	2	no	—
244	C₂×C₈	—	178	4	no	—
245	C₂×C₈	—	178	2	no	—
246	C₄⋊C₄	—	89	1	no	—
247	C₄⋊C₄	—	89	1	no	—
248	C₄⋊C₄	—	178	1	no	—
249	C₄⋊C₄	—	178	1	no	—
250	C₂²⋊C₄	—	178	1	no	—
251	C₂²⋊C₄	—	178	1	no	—
252	C₂²⋊C₄	—	178	1	no	—
253	C₄²	—	356	2	no	—
254	C₄²	—	178	4	no	—
255	C₄²	—	178	2	no	—
256	C₄²	—	178	2	no	—
257	C₄²	—	178	1	no	—
258	C₄²	—	178	1	no	—
259	C₄²	—	178	1	no	—
260	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
262	C₄²	D₄×D₈₉	1	1	yes	characteristic
263	C₄²	#13083573	1	1	yes	characteristic
264	C₂³	—	178	1	no	—
265	C₂³	#15667974	1	1	yes	characteristic
266	Q₈	—	178	1	no	—
267	Q₈	—	178	1	no	—
268	D₄	—	356	2	no	—
269	D₄	—	178	1	no	—
270	D₄	—	178	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	356	2	no	—
274	C₂×C₄	—	356	2	no	—
275	C₂×C₄	—	356	2	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	356	2	no	—
278	C₂×C₄	—	178	4	no	—
279	C₂×C₄	—	178	1	no	—
280	C₂×C₄	—	178	1	no	—
281	C₂×C₄	—	178	1	no	—
282	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
284	C₂×C₄	#15667970	1	1	yes	characteristic
285	C₂×C₄	#15667972	1	1	yes	characteristic
286	C₂×C₄	#15667972	1	1	yes	characteristic
287	C₂×C₄	#15667974	1	1	yes	characteristic
288	C₂×C₄	C₂×#13083557	1	2	yes	—
289	C₂×C₄	#15668020	1	2	yes	—
290	C₈	—	2	4	no	—
291	C₈	—	2	4	no	—
292	C₈	—	4	2	no	—
293	C₈	—	89	4	no	—
294	C₈	—	356	2	no	—
295	C₈	—	178	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	356	1	no	—
299	C₂²	—	356	1	no	—
300	C₂²	—	356	1	no	—
301	C₂²	#19151445	1	1	yes	characteristic
302	C₂²	#19151675	1	2	yes	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	356	4	no	—
306	C₄	—	356	1	no	—
307	C₄	—	356	1	no	—
308	C₄	—	356	1	no	—
309	C₄	C₄×#13083557	1	1	yes	characteristic
310	C₄	#19151555	1	1	yes	characteristic
311	C₄	#104×D₈₉	1	2	yes	—
312	C₄	#19151666	1	2	yes	—
313	C₄	#19151719	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	712	1	no	—
316	C₂	#26020032	1	1	yes	characteristic
317	C₂	#26020056	1	1	yes	characteristic
318	C₂	#26020095	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
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	#8196
LCS	G ▷ C₂×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. C₄×#15667922	C₂	semidirect
5. M₁₆×#10623494	C₂	semidirect
6. #26018084	C₂	semidirect
7. C₈₉×#1278	C₂	semidirect
9. #19151725	C₄	semidirect
10. #19151675	C₄	semidirect
20. #19151330	C₄	semidirect
21. #19151327	C₄	semidirect
95. M₁₆×C₈₉	M₁₆	semidirect
135. #10623494	#104	semidirect
165. #1278	D₈₉	semidirect
167. C₈₉	#8196	semidirect
237. M₁₆	#13083557	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×#15667938	2
C₄×#15667922	2
#26018635	2
M₁₆×#10623494	2
#26018084	2
C₈₉×#1278	2
#8196	89