Subgroups — #36707542

Order 22784.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668018	C₂	1	1	yes	maximal, characteristic
2	#26022058	C₂	1	4	yes	maximal, complemented
3	#26022054	C₂	1	2	yes	maximal, complemented
4	#26022050	C₂	1	2	yes	maximal, complemented
5	#26022008	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021865	C₂	1	2	yes	maximal
7	#26021848	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668018	C₄	1	1	yes	characteristic
10	C₂×#15668018	C₂²	1	1	yes	characteristic
11	C₄×#13083574	C₄	1	1	yes	characteristic
12	C₂×#15667952	C₂²	1	2	yes	—
13	#19151553	C₄	1	1	yes	characteristic
14	#19151553	C₂²	1	1	yes	characteristic
15	#19151552	C₄	1	2	yes	—
16	#19151552	C₂²	1	2	yes	—
17	C₄×#13083558	C₄	1	1	yes	characteristic
18	C₄×#13083557	C₄	1	2	yes	—
19	C₄×C₈×D₈₉	C₂²	1	2	yes	—
20	C₄⋊#13083549	C₂²	1	2	yes	—
21	C₄⋊#13083549	C₂²	1	2	yes	—
22	#19151377	C₂²	1	4	yes	—
23	#19151377	C₂²	1	4	yes	—
24	#19151373	C₂²	1	4	yes	—
25	C₁₆×Dic₈₉	C₂²	1	4	yes	—
26	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
29	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
30	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
31	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
32	#15668018	—	2	4	no	—
33	#15668018	—	2	2	no	—
34	#15668018	—	2	2	no	—
35	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
36	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
37	C₂×#13083557	C₂×C₄	1	2	yes	—
38	C₂×#13083557	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₈₉	C₂³	1	2	yes	—
41	C₄×Dic₁₇₈	C₈	1	1	yes	characteristic
42	C₄×Dic₁₇₈	C₈	1	1	yes	characteristic
43	C₄×D₃₅₆	C₈	1	1	yes	characteristic
44	C₄×D₃₅₆	C₈	1	1	yes	characteristic
45	C₄×C₂²⋊D₈₉	C₈	1	2	yes	—
46	C₄×C₂²⋊D₈₉	C₈	1	2	yes	—
47	#15667971	C₂×C₄	1	1	yes	characteristic
48	#15667969	C₂×C₄	1	2	yes	—
49	C₄²×D₈₉	C₂×C₄	1	2	yes	—
50	#15667952	—	2	4	no	—
51	C₂×#13083549	C₂³	1	4	yes	—
52	#15667923	C₂×C₄	1	2	yes	—
53	#15667922	C₂×C₄	1	2	yes	—
54	#15667922	C₂×C₄	1	2	yes	—
55	C₄×#10623494	C₂×C₄	1	2	yes	—
56	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
57	C₈×Dic₈₉	C₂³	1	2	yes	—
58	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
61	C₂×C₁₆×C₈₉	—	2	4	no	—
62	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
63	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
64	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	#13083574	—	2	2	no	—
69	#13083574	—	2	2	no	—
70	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
71	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
72	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
75	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
76	#13083558	—	2	2	no	—
77	#13083558	M₁₆	1	4	yes	—
78	#13083557	—	2	4	no	—
79	#13083557	—	2	4	no	—
80	C₂×#10623494	C₂²×C₄	1	2	yes	—
81	C₂×#10623494	C₂²×C₄	1	2	yes	—
82	C₈×D₈₉	—	2	2	no	—
83	C₈×D₈₉	—	2	2	no	—
84	C₈×D₈₉	—	2	4	no	—
85	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
86	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
87	#13083553	C₂×C₈	1	2	yes	—
88	#13083553	C₂×C₈	1	2	yes	—
89	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
90	#13083551	C₂×C₈	1	2	yes	—
91	#13083551	C₂×C₈	1	2	yes	—
92	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
93	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
94	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
95	#13083549	—	2	4	no	—
96	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	—	2	2	no	—
100	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	#41	1	2	yes	—
104	C₂×C₈×C₈₉	#41	1	2	yes	—
105	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
106	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
107	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
108	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₁₆×C₈₉	—	2	2	no	—
111	C₁₆×C₈₉	—	2	2	no	—
112	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
113	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	4	1	no	—
116	C₂×C₄×C₈₉	#117	1	2	yes	—
117	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
124	C₂²⋊D₈₉	—	2	4	no	—
125	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
126	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
127	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
128	D₃₅₆	—	2	2	no	—
129	C₄×D₈₉	—	2	2	no	—
130	C₄×D₈₉	—	2	2	no	—
131	C₄×D₈₉	—	2	4	no	—
132	C₄×D₈₉	—	4	2	no	—
133	Dic₁₇₈	—	2	2	no	—
134	C₈×C₈₉	—	2	1	no	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	#117	1	2	yes	—
137	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
138	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
139	#10623494	—	2	2	no	—
140	#10623494	C₂×M₁₆	1	4	yes	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	#431	1	2	yes	—
144	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
145	C₂×D₈₉	—	2	2	no	—
146	C₂×D₈₉	—	2	2	no	—
147	C₂×D₈₉	—	2	2	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	#431	1	2	yes	—
152	C₄×C₈₉	#504	1	2	yes	—
153	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
154	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
155	Dic₈₉	—	2	2	no	—
156	Dic₈₉	—	2	2	no	—
157	Dic₈₉	—	2	2	no	—
158	#33423	—	89	1	no	maximal, Sylow
159	C₂×C₈₉	—	2	2	no	—
160	C₂×C₈₉	C₂×#431	1	1	yes	characteristic
161	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
162	C₂×C₈₉	#3407	1	1	yes	characteristic
163	D₈₉	—	4	2	no	—
164	C₄×#131	—	89	1	no	—
165	#2164	—	89	4	no	—
166	#2163	—	89	2	no	—
167	#2162	—	89	2	no	—
168	#2150	—	89	1	no	—
169	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic, complemented
170	#2113	—	89	2	no	—
171	#2108	—	89	2	no	—
172	C₈₉	#33423	1	1	yes	characteristic, complemented, Sylow
173	C₂×#131	—	89	1	no	—
174	C₂×#131	—	89	1	no	—
175	C₄×#41	—	89	1	no	—
176	C₂×M₃₂	—	89	2	no	—
177	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
178	#405	—	89	1	no	—
179	#405	—	89	1	no	—
180	#405	—	89	2	no	—
181	#405	—	89	2	no	—
182	C₄×M₁₆	—	89	1	no	—
183	C₄×M₁₆	—	89	2	no	—
184	C₂×C₄×C₈	—	89	2	no	—
185	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
186	C₄⋊C₁₆	—	89	2	no	—
187	C₄⋊C₁₆	—	89	2	no	—
188	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
189	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
190	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
191	C₂²⋊C₁₆	—	89	4	no	—
192	C₂²⋊C₁₆	—	89	4	no	—
193	#346	—	89	4	no	—
194	C₄×C₁₆	—	89	4	no	—
195	C₂×#41	—	89	1	no	—
196	#131	—	178	4	no	—
197	#131	—	178	2	no	—
198	#131	—	178	2	no	—
199	C₂×M₁₆	—	89	1	no	—
200	C₂×M₁₆	—	89	1	no	—
201	C₂×M₁₆	—	89	2	no	—
202	C₂×M₁₆	—	89	2	no	—
203	C₂²×C₈	—	89	2	no	—
204	C₂²×C₈	—	89	2	no	—
205	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
207	C₄×Q₈	—	89	1	no	—
208	C₄×Q₈	—	89	1	no	—
209	C₄×D₄	—	89	2	no	—
210	C₄×D₄	—	89	2	no	—
211	C₄×D₄	—	89	1	no	—
212	C₄×D₄	—	89	1	no	—
213	#117	—	89	1	no	—
214	#117	—	89	2	no	—
215	C₂×C₄²	—	89	2	no	—
216	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
217	M₃₂	—	178	4	no	—
218	C₂×C₁₆	—	2	4	no	—
219	C₂×C₁₆	—	89	4	no	—
220	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
221	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
222	#97	—	89	2	no	—
223	#97	—	89	2	no	—
224	#97	—	89	2	no	—
225	C₄×C₈	—	89	2	no	—
226	C₄×C₈	—	89	2	no	—
227	C₄×C₈	—	89	2	no	—
228	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
229	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
232	#41	—	178	2	no	—
233	#41	—	178	2	no	—
234	C₂×Q₈	—	89	1	no	—
235	C₂×D₄	—	89	2	no	—
236	C₂×D₄	—	89	1	no	—
237	C₂²×C₄	—	89	2	no	—
238	C₂²×C₄	—	89	2	no	—
239	C₂²×C₄	—	89	2	no	—
240	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
241	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
243	M₁₆	—	89	4	no	—
244	M₁₆	—	178	4	no	—
245	M₁₆	—	178	2	no	—
246	M₁₆	—	178	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	89	2	no	—
249	C₂×C₈	—	89	2	no	—
250	C₂×C₈	—	178	4	no	—
251	C₂×C₈	—	178	2	no	—
252	C₂×C₈	—	178	2	no	—
253	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
254	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
255	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
256	C₂×C₈	#13083573	1	2	yes	—
257	C₂×C₈	#13083575	1	2	yes	—
258	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
259	C₄⋊C₄	—	89	2	no	—
260	C₄⋊C₄	—	89	2	no	—
261	C₄⋊C₄	—	89	1	no	—
262	C₄⋊C₄	—	89	1	no	—
263	C₂²⋊C₄	—	89	2	no	—
264	C₂²⋊C₄	—	89	2	no	—
265	C₂²⋊C₄	—	89	2	no	—
266	C₄²	—	89	2	no	—
267	C₄²	—	89	2	no	—
268	C₄²	—	89	2	no	—
269	C₄²	C₈×D₈₉	1	1	yes	characteristic
270	C₄²	C₈×D₈₉	1	1	yes	characteristic
271	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
272	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
273	C₁₆	—	2	2	no	—
274	C₁₆	—	2	2	no	—
275	C₁₆	—	178	4	no	—
276	C₂³	—	89	2	no	—
277	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
278	Q₈	—	178	2	no	—
279	D₄	—	178	4	no	—
280	D₄	—	178	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	89	2	no	—
284	C₂×C₄	—	89	2	no	—
285	C₂×C₄	—	89	2	no	—
286	C₂×C₄	—	356	2	no	—
287	C₂×C₄	—	178	4	no	—
288	C₂×C₄	—	178	2	no	—
289	C₂×C₄	—	178	2	no	—
290	C₂×C₄	#15667970	1	2	yes	—
291	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
292	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
293	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
295	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
296	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
297	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	89	4	no	—
301	C₈	—	178	2	no	—
302	C₈	#15667970	1	2	yes	—
303	C₈	C₂×#13083573	1	1	yes	characteristic
304	C₈	C₂×#13083575	1	1	yes	characteristic
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	178	2	no	—
308	C₂²	—	178	2	no	—
309	C₂²	—	178	2	no	—
310	C₂²	#19151650	1	2	yes	—
311	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	178	2	no	—
316	C₄	—	178	2	no	—
317	C₄	—	178	2	no	—
318	C₄	#19151650	1	2	yes	—
319	C₄	#19152014	1	2	yes	—
320	C₄	C₂×#15667970	1	1	yes	characteristic
321	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	356	2	no	—
324	C₂	C₂×#19151650	1	1	yes	characteristic
325	C₂	C₂×#19152014	1	1	yes	characteristic
326	C₂	#26033354	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	#33423
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #26022058	C₂	semidirect
3. #26022054	C₂	semidirect
4. #26022050	C₂	semidirect
5. #26022008	C₂	semidirect
7. #26021848	C₂	semidirect
8. C₂×C₄⋊C₁₆×C₈₉	C₂	semidirect
169. C₂×C₄⋊C₁₆	D₈₉	semidirect
172. C₈₉	#33423	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668018	2
#26022058	2
#26022054	2
#26022050	2
#26022008	2
#26021865	2
#26021848	2
C₂×C₄⋊C₁₆×C₈₉	2
#33423	89