Subgroups — #36444985

Order 22784.

542 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151663	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667928	C₂	1	1	yes	maximal, characteristic
3	#26018572	C₂	1	1	yes	maximal, characteristic
4	#26018533	C₂	1	1	yes	maximal, characteristic, complemented
5	M₁₆×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018102	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₈₉	C₄	1	2	yes	—
9	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
10	#19151663	—	2	1	no	—
11	#19151663	—	2	1	no	—
12	#19151663	—	2	1	no	—
13	#19151663	—	2	1	no	—
14	#19151651	C₂²	1	1	yes	characteristic
15	C₂×#15667928	C₄	1	2	yes	—
16	C₂×#15667924	C₄	1	2	yes	—
17	C₄×#13083558	C₂²	1	1	yes	characteristic
18	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
19	#19151366	C₄	1	2	yes	—
20	#19151321	C₄	1	4	yes	—
21	#19151320	—	2	1	no	—
22	#19151319	—	2	1	no	—
23	C₈×#10623494	—	2	1	no	—
24	C₈×#10623494	—	2	1	no	—
25	#431×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
28	C₂²×D₃₅₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₈₉	C₂×C₄	1	2	yes	—
30	C₂×#13083558	C₂×C₄	1	2	yes	—
31	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
32	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₈₉	—	2	2	no	—
34	C₂×C₈×D₈₉	—	2	2	no	—
35	C₂×C₈×D₈₉	—	4	4	no	—
36	C₄×D₃₅₆	—	2	1	no	—
37	C₄×D₃₅₆	—	2	1	no	—
38	C₄×D₃₅₆	—	2	1	no	—
39	C₄×D₃₅₆	—	2	1	no	—
40	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×#13083551	C₂×C₄	1	2	yes	—
42	C₄⋊#10623494	—	2	1	no	—
43	#15667928	C₈	1	8	yes	complemented
44	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
45	#15667924	—	2	2	no	—
46	#15667924	—	2	2	no	—
47	#15667923	—	2	1	no	—
48	C₄×#10623494	—	2	1	no	—
49	C₄×#10623494	—	2	1	no	—
50	C₄×#10623494	D₄	1	1	yes	characteristic
51	C₄×#10623494	D₄	1	1	yes	characteristic
52	C₄×#10623494	D₄	1	1	yes	characteristic
53	C₄×#10623494	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₈₉	—	2	1	no	—
58	C₄⋊C₈×C₈₉	—	2	1	no	—
59	C₄⋊C₈×C₈₉	—	2	1	no	—
60	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
61	#97×C₈₉	—	2	1	no	—
62	C₄×C₈×C₈₉	—	2	1	no	—
63	C₄×C₈×C₈₉	—	2	1	no	—
64	C₂³×D₈₉	C₄²	1	2	yes	—
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	—	2	2	no	—
68	C₂×D₃₅₆	C₂×C₈	1	4	yes	—
69	C₂²×Dic₈₉	C₄²	1	2	yes	—
70	C₂×C₄×D₈₉	—	2	2	no	—
71	C₂×C₄×D₈₉	—	2	2	no	—
72	C₂×C₄×D₈₉	—	4	2	no	—
73	C₂×C₄×D₈₉	—	4	2	no	—
74	#13083558	—	2	2	no	—
75	#13083558	C₂×C₈	1	4	yes	complemented
76	C₂×#10623494	—	2	2	no	—
77	C₂×#10623494	—	2	2	no	—
78	C₂×#10623494	—	2	2	no	—
79	C₂×#10623494	—	4	1	no	—
80	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
84	C₈×D₈₉	—	4	4	no	—
85	C₈×D₈₉	—	4	4	no	—
86	C₄⋊Dic₈₉	—	2	1	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	#13083551	—	2	2	no	—
89	#13083551	—	2	2	no	—
90	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₈₉	#41	1	1	yes	characteristic
102	C₄²×C₈₉	#41	1	1	yes	characteristic
103	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
104	C₂²×D₈₉	—	2	2	no	—
105	C₂²×D₈₉	—	2	2	no	—
106	C₂²×D₈₉	—	4	2	no	—
107	C₂²×D₈₉	—	4	2	no	—
108	C₂²×D₈₉	C₄×C₈	1	4	yes	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	4	1	no	—
112	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#117	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	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₈₉	#131	1	2	yes	—
120	C₂×Dic₈₉	—	2	2	no	—
121	C₂×Dic₈₉	—	4	2	no	—
122	D₃₅₆	—	2	4	no	—
123	D₃₅₆	—	2	4	no	—
124	C₄×D₈₉	—	4	4	no	—
125	C₄×D₈₉	—	4	2	no	—
126	C₄×D₈₉	—	4	2	no	—
127	C₈×C₈₉	—	2	4	no	—
128	C₈×C₈₉	—	4	2	no	—
129	C₈×C₈₉	—	4	2	no	—
130	#10623494	—	2	1	no	—
131	#10623494	—	2	1	no	—
132	#10623494	—	2	1	no	—
133	#10623494	—	2	4	no	—
134	#10623494	—	4	2	no	—
135	#10623494	#100	1	2	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	C₂×C₄×C₈	1	1	yes	characteristic
140	C₂²×C₈₉	#405	1	1	yes	characteristic
141	C₂×D₈₉	—	2	4	no	—
142	C₂×D₈₉	—	4	4	no	—
143	C₂×D₈₉	—	4	2	no	—
144	C₂×D₈₉	—	4	2	no	—
145	C₂×D₈₉	—	4	4	no	—
146	C₄×C₈₉	—	2	1	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
150	C₄×C₈₉	#413	1	1	yes	characteristic
151	C₄×C₈₉	C₈×D₄	1	2	yes	—
152	C₄×C₈₉	C₈×D₄	1	2	yes	—
153	Dic₈₉	—	4	2	no	—
154	C₈×#100	—	89	1	no	maximal, Sylow
155	C₂×C₈₉	—	2	1	no	—
156	C₂×C₈₉	—	2	1	no	—
157	C₂×C₈₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₈₉	C₄×#100	1	1	yes	characteristic
159	C₂×C₈₉	#1757	1	1	yes	characteristic
160	D₈₉	—	4	4	no	—
161	C₂×C₈×D₄	—	89	1	no	—
162	C₄×#100	—	89	1	no	—
163	#1460	—	89	1	no	—
164	C₈×M₁₆	—	89	1	no	—
165	C₈×M₁₆	—	89	1	no	—
166	#1283	—	89	1	no	—
167	#1283	D₈₉	1	1	yes	characteristic, complemented
168	C₈₉	C₈×#100	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₈	—	89	2	no	—
170	C₂×C₄×D₄	—	89	1	no	—
171	C₈×D₄	—	178	1	no	—
172	C₈×D₄	—	178	1	no	—
173	C₈×D₄	—	178	1	no	—
174	C₈×D₄	—	178	1	no	—
175	#431	—	89	1	no	—
176	#431	C₂×D₈₉	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
178	C₂×#100	—	89	2	no	—
179	C₂×C₂²⋊C₈	—	89	2	no	—
180	C₄×M₁₆	—	89	1	no	—
181	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
182	C₂×C₄×C₈	—	89	1	no	—
183	#343	—	89	2	no	—
184	#323	—	89	4	no	—
185	#322	—	178	1	no	—
186	#322	—	178	1	no	—
187	C₈²	—	178	1	no	—
188	C₈²	—	178	1	no	—
189	C₂²×D₄	—	89	1	no	—
190	C₂³×C₄	—	89	2	no	—
191	C₂×M₁₆	—	89	2	no	—
192	C₂×M₁₆	C₄×D₈₉	1	2	yes	—
193	C₂²×C₈	—	89	1	no	—
194	C₂²×C₈	—	89	1	no	—
195	C₂²×C₈	—	356	4	no	—
196	C₂²×C₈	—	178	2	no	—
197	C₂²×C₈	—	178	2	no	—
198	C₂²×C₈	C₄×D₈₉	1	2	yes	—
199	C₄×D₄	—	178	1	no	—
200	C₄×D₄	—	178	1	no	—
201	C₄×D₄	—	178	1	no	—
202	C₄×D₄	—	178	1	no	—
203	C₂×C₄⋊C₄	—	89	1	no	—
204	C₂×C₂²⋊C₄	—	89	2	no	—
205	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	178	1	no	—
210	#100	—	89	8	no	—
211	C₂²⋊C₈	—	89	2	no	—
212	C₂²⋊C₈	—	178	2	no	—
213	C₂²⋊C₈	—	178	2	no	—
214	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
215	#97	—	2	1	no	—
216	#97	—	178	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	89	1	no	—
220	C₄×C₈	—	89	1	no	—
221	C₄×C₈	—	89	1	no	—
222	C₄×C₈	—	89	1	no	—
223	C₄×C₈	—	178	1	no	—
224	C₄×C₈	—	178	1	no	—
225	C₂⁴	—	89	2	no	—
226	C₂×D₄	—	89	4	no	—
227	C₂×D₄	—	178	1	no	—
228	C₂×D₄	—	178	1	no	—
229	C₂×D₄	—	178	2	no	—
230	C₂²×C₄	—	89	2	no	—
231	C₂²×C₄	—	356	2	no	—
232	C₂²×C₄	—	356	2	no	—
233	C₂²×C₄	—	178	2	no	—
234	C₂²×C₄	—	178	2	no	—
235	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	89	4	no	—
240	M₁₆	—	178	2	no	—
241	M₁₆	C₈×D₈₉	1	4	yes	complemented
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	89	1	no	—
247	C₂×C₈	—	89	1	no	—
248	C₂×C₈	—	89	1	no	—
249	C₂×C₈	—	89	1	no	—
250	C₂×C₈	—	356	4	no	—
251	C₂×C₈	—	356	4	no	—
252	C₂×C₈	—	356	1	no	—
253	C₂×C₈	—	178	2	no	—
254	C₂×C₈	—	178	2	no	—
255	C₂×C₈	—	178	2	no	—
256	C₄⋊C₄	—	178	1	no	—
257	C₄⋊C₄	—	178	1	no	—
258	C₂²⋊C₄	—	178	2	no	—
259	C₂²⋊C₄	—	178	2	no	—
260	C₄²	D₄×D₈₉	1	1	yes	characteristic
261	C₄²	D₄×D₈₉	1	1	yes	characteristic
262	C₄²	#13083573	1	1	yes	characteristic
263	C₄²	#13083575	1	1	yes	characteristic
264	C₂³	—	89	4	no	—
265	C₂³	—	356	2	no	—
266	C₂³	—	356	2	no	—
267	C₂³	—	178	2	no	—
268	C₂³	—	178	2	no	—
269	C₂³	C₄²×D₈₉	1	1	yes	characteristic
270	D₄	—	178	4	no	—
271	D₄	—	178	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	356	4	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	356	2	no	—
278	C₂×C₄	—	356	2	no	—
279	C₂×C₄	—	178	2	no	—
280	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
281	C₂×C₄	#15667970	1	1	yes	characteristic
282	C₂×C₄	#15667972	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	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	—	89	2	no	—
292	C₈	—	356	2	no	—
293	C₈	—	178	4	no	—
294	C₈	—	178	1	no	—
295	C₈	—	178	1	no	—
296	C₈	—	178	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	356	4	no	—
300	C₂²	—	356	4	no	—
301	C₂²	—	356	2	no	—
302	C₂²	—	356	2	no	—
303	C₂²	—	178	4	no	—
304	C₂²	#19151445	1	1	yes	characteristic
305	C₂²	C₄×C₈×D₈₉	1	1	yes	characteristic
306	C₂²	#19151555	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	356	2	no	—
311	C₄	#100×D₈₉	1	1	yes	characteristic
312	C₄	#19151581	1	1	yes	characteristic
313	C₄	#19151661	1	2	yes	—
314	C₄	#19151663	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	356	4	no	—
318	C₂	#26020022	1	1	yes	characteristic
319	C₂	#26020043	1	1	yes	characteristic
320	C₂	#26020051	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₈₉
commutator	C₂²×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1283	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂²×D₃₅₆	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1283	D₈₉
hypocenter	C₈₉	C₈×#100
LCS	G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1283

Complements

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

N	H ≅ G/N	type
4. #26018533	C₂	semidirect
5. M₁₆×#10623494	C₂	semidirect
6. #26018102	C₂	semidirect
7. C₈₉×#1283	C₂	semidirect
43. #15667928	C₈	semidirect
75. #13083558	C₂×C₈	semidirect
94. M₁₆×C₈₉	C₂×C₈	semidirect
135. #10623494	#100	semidirect
167. #1283	D₈₉	semidirect
168. C₈₉	C₈×#100	semidirect
241. 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
C₂×#19151663	2
C₄×#15667928	2
#26018572	2
#26018533	2
M₁₆×#10623494	2
#26018102	2
C₈₉×#1283	2
C₈×#100	89