Subgroups — #36707613

Order 22784.

690 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	C₂×#19152016	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19152015	C₂	1	1	yes	maximal, characteristic, complemented
3	#26028962	C₂	1	1	yes	maximal, characteristic
4	#26022081	C₂	1	4	yes	maximal, complemented
5	#26022069	C₂	1	4	yes	maximal, complemented
6	#26022007	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021865	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2113	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668018	C₂²	1	1	yes	characteristic
11	#19152144	C₄	1	1	yes	characteristic
12	#19152016	—	2	4	no	—
13	#19152015	—	2	4	no	—
14	C₂×#15667952	C₂²	1	1	yes	characteristic
15	C₂×#15667952	C₂²	1	2	yes	—
16	C₂×#15667951	C₂²	1	2	yes	—
17	C₂²×#13083549	C₂²	1	2	yes	—
18	C₂²×#13083549	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
20	#19151674	C₄	1	4	yes	—
21	C₈×C₂²⋊D₈₉	C₂²	1	4	yes	—
22	#19151621	C₄	1	1	yes	characteristic
23	#19151566	C₄	1	1	yes	characteristic
24	#19151565	C₄	1	1	yes	characteristic
25	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
26	#19151416	C₂²	1	4	yes	—
27	#19151377	C₂²	1	4	yes	—
28	C₁₆×Dic₈₉	C₂²	1	2	yes	—
29	C₁₆×Dic₈₉	C₂²	1	2	yes	—
30	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
32	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
33	C₂²⋊C₁₆×C₈₉	C₂²	1	4	yes	—
34	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
35	#15668018	—	2	2	no	—
36	#15668018	—	2	2	no	—
37	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
38	C₂×#13083557	C₂×C₄	1	2	yes	—
39	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₈₉	C₂³	1	2	yes	—
41	#15668010	C₈	1	1	yes	characteristic
42	#15668008	C₈	1	1	yes	characteristic
43	#15668001	C₈	1	1	yes	characteristic
44	#15667999	C₈	1	1	yes	characteristic
45	#15667984	C₈	1	4	yes	—
46	C₄×C₂²⋊D₈₉	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
48	#15667952	—	2	4	no	—
49	#15667952	—	2	2	no	—
50	#15667951	—	2	4	no	—
51	C₂×#13083549	—	2	4	no	—
52	C₂×#13083549	D₄	1	4	yes	complemented
53	C₂×#13083549	C₂³	1	2	yes	—
54	C₂×#13083549	C₂³	1	1	yes	characteristic
55	C₂×#13083549	C₂³	1	1	yes	characteristic
56	C₁₆×D₈₉	—	2	4	no	—
57	#15667941	C₂×C₄	1	4	yes	—
58	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
59	#15667924	C₂×C₄	1	4	yes	—
60	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
61	C₈×Dic₈₉	C₂³	1	2	yes	—
62	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
63	#131×C₈₉	—	2	2	no	—
64	#131×C₈₉	—	2	2	no	—
65	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
67	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
68	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	M₃₂×C₈₉	—	2	2	no	—
70	C₂×C₁₆×C₈₉	—	2	2	no	—
71	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
72	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
73	#13083574	—	2	2	no	—
74	#13083574	—	2	2	no	—
75	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
76	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
77	C₂×C₂²⋊D₈₉	C₂×C₈	1	4	yes	—
78	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
79	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
81	#13083558	—	2	2	no	—
82	#13083557	—	2	4	no	—
83	C₂×#10623494	—	2	2	no	—
84	C₂×#10623494	C₂²×C₄	1	2	yes	—
85	C₂×#10623494	C₂²×C₄	1	2	yes	—
86	C₈×D₈₉	—	2	2	no	—
87	C₈×D₈₉	—	2	2	no	—
88	#13083553	C₂×C₈	1	4	yes	—
89	C₂²⋊Dic₈₉	C₂×C₈	1	4	yes	—
90	#13083551	C₂×C₈	1	4	yes	—
91	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
92	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
94	#13083549	—	2	2	no	—
95	#13083549	C₂×D₄	1	2	yes	—
96	#13083549	C₂×D₄	1	2	yes	—
97	#41×C₈₉	—	2	2	no	—
98	#41×C₈₉	—	2	2	no	—
99	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
100	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
101	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	M₁₆×C₈₉	—	2	2	no	—
105	M₁₆×C₈₉	—	2	4	no	—
106	C₂×C₈×C₈₉	—	2	2	no	—
107	C₂×C₈×C₈₉	—	2	2	no	—
108	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₈₉	#41	1	2	yes	—
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₁₆×C₈₉	—	2	2	no	—
114	C₁₆×C₈₉	—	2	2	no	—
115	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
116	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
117	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
118	Q₈×C₈₉	—	2	2	no	—
119	D₄×C₈₉	—	2	4	no	—
120	D₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	#131	1	2	yes	—
127	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
128	C₂²⋊D₈₉	—	2	4	no	—
129	C₂²⋊D₈₉	—	2	8	no	—
130	C₂×Dic₈₉	—	2	4	no	—
131	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
132	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
133	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
134	D₃₅₆	—	2	2	no	—
135	C₄×D₈₉	—	2	2	no	—
136	C₄×D₈₉	—	2	2	no	—
137	Dic₁₇₈	—	2	2	no	—
138	C₈×C₈₉	—	2	2	no	—
139	C₈×C₈₉	C₄×D₄	1	2	yes	—
140	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
142	#10623494	—	2	2	no	—
143	#10623494	—	2	2	no	—
144	C₂²×C₈₉	—	2	2	no	—
145	C₂²×C₈₉	—	2	4	no	—
146	C₂²×C₈₉	—	2	2	no	—
147	C₂²×C₈₉	C₈×D₄	1	2	yes	—
148	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
149	C₂×D₈₉	—	2	4	no	—
150	C₂×D₈₉	—	2	2	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	C₈×D₄	1	2	yes	—
153	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
155	Dic₈₉	—	2	2	no	—
156	Dic₈₉	—	2	2	no	—
157	Dic₈₉	#504	1	4	yes	complemented
158	#33434	—	89	1	no	maximal, Sylow
159	C₂×C₈₉	—	2	2	no	—
160	C₂×C₈₉	—	4	2	no	—
161	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
162	C₂×C₈₉	C₂×#504	1	2	yes	—
163	D₈₉	—	4	2	no	—
164	C₂×#504	—	89	1	no	—
165	C₂×#504	—	89	1	no	—
166	#2931	—	89	1	no	—
167	#2169	—	89	4	no	—
168	D₄×C₁₆	—	89	4	no	—
169	#2150	—	89	1	no	—
170	#2113	—	89	1	no	—
171	#2113	D₈₉	1	1	yes	characteristic, complemented
172	C₂×C₄×C₁₆	—	89	1	no	—
173	C₈₉	#33434	1	1	yes	characteristic, complemented, Sylow
174	C₂×#131	—	89	1	no	—
175	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
176	#532	—	89	1	no	—
177	#504	—	178	4	no	—
178	#504	—	178	4	no	—
179	C₂×M₃₂	—	89	1	no	—
180	C₂×M₃₂	—	89	2	no	—
181	C₂×M₃₂	—	89	2	no	—
182	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
183	C₂²×C₁₆	—	89	1	no	—
184	C₂²×C₁₆	—	89	2	no	—
185	C₂²×C₁₆	—	89	2	no	—
186	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
187	#436	—	89	4	no	—
188	C₈×D₄	—	89	4	no	—
189	#423	—	89	1	no	—
190	#408	—	89	1	no	—
191	#408	—	89	1	no	—
192	C₂×C₄×C₈	—	89	1	no	—
193	C₄⋊C₁₆	—	89	4	no	—
194	C₂²⋊C₁₆	—	89	4	no	—
195	C₂²⋊C₁₆	C₂×D₈₉	1	4	yes	—
196	C₄×C₁₆	—	89	2	no	—
197	C₄×C₁₆	—	89	2	no	—
198	C₂×#41	—	89	1	no	—
199	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
200	#131	—	2	2	no	—
201	#131	—	2	2	no	—
202	#131	—	178	2	no	—
203	#131	—	178	2	no	—
204	C₂×M₁₆	—	89	1	no	—
205	C₂×M₁₆	—	89	2	no	—
206	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
207	C₂×M₁₆	C₄×D₈₉	1	2	yes	—
208	C₂²×C₈	—	89	1	no	—
209	C₂²×C₈	—	89	2	no	—
210	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
211	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
212	C₄⋊Q₈	—	89	1	no	—
213	#127	—	89	1	no	—
214	#124	—	89	1	no	—
215	#124	—	89	1	no	—
216	#121	—	89	4	no	—
217	C₄×D₄	—	89	4	no	—
218	C₂×C₄²	—	89	1	no	—
219	M₃₂	—	2	2	no	—
220	M₃₂	—	178	4	no	—
221	M₃₂	—	178	4	no	—
222	M₃₂	—	178	2	no	—
223	C₂×C₁₆	—	2	2	no	—
224	C₂×C₁₆	—	89	4	no	—
225	C₂×C₁₆	—	89	1	no	—
226	C₂×C₁₆	—	89	1	no	—
227	C₂×C₁₆	—	89	2	no	—
228	C₂×C₁₆	—	178	4	no	—
229	C₂×C₁₆	—	178	4	no	—
230	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
231	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
232	C₄⋊C₈	—	89	4	no	—
233	C₂²⋊C₈	—	89	4	no	—
234	C₂²⋊C₈	—	89	4	no	—
235	C₄×C₈	—	89	2	no	—
236	C₄×C₈	—	89	2	no	—
237	#41	—	2	2	no	—
238	#41	—	2	2	no	—
239	#41	—	178	2	no	—
240	#41	—	178	2	no	—
241	C₂×Q₈	—	89	1	no	—
242	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
243	C₂×D₄	—	89	4	no	—
244	C₂×D₄	—	89	2	no	—
245	C₂×D₄	—	89	1	no	—
246	C₂×D₄	C₈×D₈₉	1	2	yes	—
247	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
248	C₂²×C₄	—	89	2	no	—
249	C₂²×C₄	—	89	2	no	—
250	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
251	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	4	no	—
254	M₁₆	—	178	2	no	—
255	M₁₆	—	178	4	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	89	2	no	—
259	C₂×C₈	—	89	2	no	—
260	C₂×C₈	—	178	2	no	—
261	C₂×C₈	—	178	2	no	—
262	C₂×C₈	—	178	2	no	—
263	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
264	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
265	C₂×C₈	D₄×D₈₉	1	2	yes	—
266	C₂×C₈	#13083573	1	2	yes	—
267	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
268	C₄⋊C₄	—	89	4	no	—
269	C₂²⋊C₄	—	89	4	no	—
270	C₂²⋊C₄	—	89	4	no	—
271	C₄²	—	89	1	no	—
272	C₄²	—	89	1	no	—
273	C₄²	—	89	2	no	—
274	C₁₆	—	2	2	no	—
275	C₁₆	—	2	2	no	—
276	C₁₆	—	89	2	no	—
277	C₁₆	—	89	2	no	—
278	C₁₆	—	178	2	no	—
279	C₂³	—	89	2	no	—
280	C₂³	C₂×C₈×D₈₉	1	2	yes	—
281	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	178	2	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	2	no	—
286	D₄	—	178	8	no	—
287	D₄	—	178	4	no	—
288	D₄	—	178	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	89	2	no	—
292	C₂×C₄	—	89	2	no	—
293	C₂×C₄	—	89	2	no	—
294	C₂×C₄	—	178	4	no	—
295	C₂×C₄	—	178	2	no	—
296	C₂×C₄	—	178	2	no	—
297	C₂×C₄	#15667972	1	2	yes	—
298	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
299	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
300	C₂×C₄	#15668020	1	2	yes	—
301	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	178	2	no	—
304	C₈	—	178	2	no	—
305	C₈	#15667972	1	2	yes	—
306	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
307	C₈	C₂×#13083573	1	1	yes	characteristic
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	178	2	no	—
312	C₂²	—	178	4	no	—
313	C₂²	#19151661	1	2	yes	—
314	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	89	4	no	—
317	C₄	—	178	2	no	—
318	C₄	—	178	2	no	—
319	C₄	#19151661	1	2	yes	—
320	C₄	C₂×#15667972	1	1	yes	characteristic
321	C₄	C₂×#15668020	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	—	356	2	no	—
325	C₂	C₂×#19151661	1	1	yes	characteristic
326	C₂	#504×D₈₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2113	C₂
prime radical	#19152144	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2113	D₈₉
hypocenter	C₈₉	#33434
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
1. C₂×#19152016	C₂	semidirect
2. C₂×#19152015	C₂	semidirect
4. #26022081	C₂	semidirect
5. #26022069	C₂	semidirect
6. #26022007	C₂	semidirect
7. #26021865	C₂	semidirect
8. C₂×C₁₆×Dic₈₉	C₂	semidirect
9. C₈₉×#2113	C₂	semidirect
52. C₂×#13083549	D₄	semidirect
157. Dic₈₉	#504	semidirect
171. #2113	D₈₉	semidirect
173. C₈₉	#33434	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152016	2
C₂×#19152015	2
#26028962	2
#26022081	2
#26022069	2
#26022007	2
#26021865	2
C₂×C₁₆×Dic₈₉	2
C₈₉×#2113	2
#33434	89