Subgroups — #36472931

Order 22784.

482 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151982	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151419	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151413	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151410	C₂	1	1	yes	maximal, characteristic
5	#26021971	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021967	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
9	C₂²×#13083563	—	2	1	no	—
10	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
11	C₂×#15668000	—	2	1	no	—
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	#19151982	—	2	2	no	—
15	#19151982	C₄	1	4	yes	complemented
16	#19151419	—	2	2	no	—
17	#19151413	—	2	2	no	—
18	#19151410	—	2	2	no	—
19	#19151401	—	2	2	no	—
20	#19151398	—	2	2	no	—
21	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
22	#427×C₈₉	C₂²	1	1	yes	characteristic
23	#405×C₈₉	—	2	1	no	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	#339×C₈₉	—	2	1	no	—
26	C₂²×Dic₁₇₈	—	2	1	no	—
27	C₂²×D₃₅₆	—	2	1	no	—
28	C₂×Dic₃₅₆	—	2	1	no	—
29	C₂×Dic₃₅₆	—	2	1	no	—
30	C₂×Dic₃₅₆	—	2	1	no	—
31	C₂×Dic₃₅₆	—	2	1	no	—
32	C₂×Dic₃₅₆	—	2	1	no	—
33	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
34	C₂×#13083563	—	2	4	no	—
35	C₂×#13083563	—	4	1	no	—
36	C₂×#13083563	—	4	1	no	—
37	C₂×#13083563	—	4	2	no	—
38	C₂×D₇₁₂	—	2	1	no	—
39	C₂×D₇₁₂	—	2	1	no	—
40	C₂×D₇₁₂	—	2	1	no	—
41	C₂×D₇₁₂	—	2	1	no	—
42	C₂×D₇₁₂	—	2	1	no	—
43	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
44	#15668000	—	2	4	no	—
45	#15668000	—	4	2	no	—
46	C₂×#13083551	—	4	2	no	—
47	#15667952	—	2	1	no	—
48	#15667952	—	2	1	no	—
49	#15667952	—	2	1	no	—
50	#15667952	—	2	1	no	—
51	C₂×#13083549	—	2	1	no	—
52	C₂×#13083549	—	2	1	no	—
53	C₂×M₁₆×C₈₉	—	2	1	no	—
54	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	#117×C₈₉	—	2	1	no	—
57	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
58	#108×C₈₉	—	2	1	no	—
59	#108×C₈₉	D₄	1	2	yes	—
60	#107×C₈₉	D₄	1	2	yes	—
61	#106×C₈₉	—	2	1	no	—
62	#97×C₈₉	—	4	1	no	—
63	C₄×C₈×C₈₉	—	4	1	no	—
64	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₈₉	D₄	1	2	yes	—
66	C₂³×D₈₉	—	4	1	no	—
67	C₂×Dic₁₇₈	—	2	2	no	—
68	C₂×Dic₁₇₈	—	4	1	no	—
69	C₂×Dic₁₇₈	—	4	1	no	—
70	C₂×Dic₁₇₈	—	4	1	no	—
71	C₂×Dic₁₇₈	—	4	1	no	—
72	C₂×Dic₁₇₈	—	4	1	no	—
73	C₂×D₃₅₆	—	2	2	no	—
74	C₂×D₃₅₆	—	4	1	no	—
75	C₂×D₃₅₆	—	4	1	no	—
76	C₂×D₃₅₆	—	4	1	no	—
77	C₂×D₃₅₆	—	4	1	no	—
78	C₂×D₃₅₆	—	4	1	no	—
79	C₂²×Dic₈₉	—	4	1	no	—
80	Dic₃₅₆	—	2	2	no	—
81	Dic₃₅₆	—	2	2	no	—
82	Dic₃₅₆	—	4	1	no	—
83	Dic₃₅₆	—	4	1	no	—
84	#13083563	—	4	4	no	—
85	#13083563	—	4	4	no	—
86	D₇₁₂	—	2	2	no	—
87	D₇₁₂	—	2	2	no	—
88	D₇₁₂	—	4	1	no	—
89	D₇₁₂	—	4	1	no	—
90	#13083551	—	4	8	no	—
91	#13083549	—	4	1	no	—
92	#13083549	—	4	1	no	—
93	C₂²×C₄×C₈₉	—	2	1	no	—
94	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
95	M₁₆×C₈₉	—	2	2	no	—
96	M₁₆×C₈₉	—	4	1	no	—
97	C₂×C₈×C₈₉	—	4	1	no	—
98	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
99	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
100	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
101	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
102	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
107	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	—	2	2	no	—
109	C₂²⋊C₄×C₈₉	—	4	1	no	—
110	C₄²×C₈₉	—	2	1	no	—
111	C₄²×C₈₉	—	4	1	no	—
112	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
113	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂²×D₈₉	—	4	2	no	—
115	C₂²×D₈₉	—	8	4	no	—
116	C₂²×D₈₉	—	8	1	no	—
117	C₂²×D₈₉	—	8	1	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×Dic₈₉	—	4	2	no	—
128	C₂×Dic₈₉	—	8	1	no	—
129	C₂×Dic₈₉	—	8	1	no	—
130	D₃₅₆	—	4	2	no	—
131	D₃₅₆	—	4	2	no	—
132	D₃₅₆	—	8	1	no	—
133	D₃₅₆	—	8	1	no	—
134	Dic₁₇₈	—	4	2	no	—
135	Dic₁₇₈	—	4	2	no	—
136	Dic₁₇₈	—	8	1	no	—
137	Dic₁₇₈	—	8	1	no	—
138	C₈×C₈₉	—	4	2	no	—
139	C₈×C₈₉	#102	1	2	yes	—
140	C₈×C₈₉	#102	1	2	yes	—
141	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
142	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
143	C₈×C₈₉	#124	1	1	yes	characteristic
144	C₈×C₈₉	#127	1	1	yes	characteristic
145	C₂²×C₈₉	—	2	2	no	—
146	C₂²×C₈₉	#418	1	1	yes	characteristic
147	C₂²×C₈₉	#459	1	1	yes	characteristic
148	C₂²×C₈₉	#460	1	1	yes	characteristic
149	C₂×D₈₉	—	8	4	no	—
150	C₂×D₈₉	—	8	2	no	—
151	C₂×D₈₉	—	16	2	no	—
152	C₂×D₈₉	—	16	2	no	—
153	C₄×C₈₉	—	4	2	no	—
154	C₄×C₈₉	—	8	2	no	—
155	C₄×C₈₉	#390	1	1	yes	characteristic
156	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
157	C₄×C₈₉	#459	1	1	yes	characteristic
158	C₄×C₈₉	#460	1	1	yes	characteristic
159	Dic₈₉	—	8	2	no	—
160	#13260	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	—	2	1	no	—
162	C₂×C₈₉	—	2	1	no	—
163	C₂×C₈₉	#1982	1	1	yes	characteristic
164	C₂×C₈₉	#2212	1	2	yes	—
165	D₈₉	—	16	2	no	—
166	C₂×#495	—	89	1	no	—
167	C₂×#364	—	89	1	no	—
168	C₂×#362	—	89	1	no	—
169	C₂×#361	—	89	1	no	—
170	#2141	—	89	1	no	—
171	#2140	—	89	1	no	—
172	#1772	D₈₉	1	1	yes	characteristic, complemented
173	C₈₉	#13260	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₁₆	—	89	1	no	—
175	C₂²×SD₁₆	—	178	1	no	—
176	C₂²×D₈	—	89	1	no	—
177	C₂×#124	—	178	1	no	—
178	C₂×M₃₂	—	89	1	no	—
179	C₂×M₃₂	—	89	1	no	—
180	#495	—	89	4	no	—
181	#495	—	178	2	no	—
182	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
183	#427	C₂×D₈₉	1	1	yes	characteristic
184	#405	—	2	1	no	—
185	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
186	#364	—	178	2	no	—
187	#362	—	178	2	no	—
188	#361	—	178	2	no	—
189	#358	—	178	2	no	—
190	#357	—	178	2	no	—
191	#339	—	2	1	no	—
192	C₂²×Q₈	—	178	1	no	—
193	C₂²×D₄	—	178	1	no	—
194	C₂×Q₁₆	—	89	2	no	—
195	C₂×Q₁₆	—	178	1	no	—
196	C₂×Q₁₆	—	178	1	no	—
197	C₂×Q₁₆	—	178	1	no	—
198	C₂×Q₁₆	—	178	1	no	—
199	C₂×Q₁₆	—	178	1	no	—
200	C₂×SD₁₆	—	356	1	no	—
201	C₂×SD₁₆	—	356	1	no	—
202	C₂×SD₁₆	—	356	2	no	—
203	C₂×SD₁₆	—	178	4	no	—
204	C₂×D₈	—	89	2	no	—
205	C₂×D₈	—	178	1	no	—
206	C₂×D₈	—	178	1	no	—
207	C₂×D₈	—	178	1	no	—
208	C₂×D₈	—	178	1	no	—
209	C₂×D₈	—	178	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
213	#124	—	356	2	no	—
214	#124	—	178	4	no	—
215	#117	—	2	1	no	—
216	C₂×C₂²⋊C₄	—	356	2	no	—
217	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
218	M₃₂	—	178	1	no	—
219	M₃₂	—	178	1	no	—
220	M₃₂	—	178	1	no	—
221	M₃₂	—	178	1	no	—
222	C₂×C₁₆	—	178	1	no	—
223	C₂×C₁₆	—	178	1	no	—
224	#108	—	2	1	no	—
225	#108	D₃₅₆	1	2	yes	—
226	#107	D₃₅₆	1	2	yes	—
227	#106	—	2	1	no	—
228	#97	—	4	1	no	—
229	C₄×C₈	—	4	1	no	—
230	C₄×C₈	C₄×D₈₉	1	2	yes	—
231	C₄×C₈	D₃₅₆	1	2	yes	—
232	C₂⁴	—	356	1	no	—
233	C₂×Q₈	—	356	1	no	—
234	C₂×Q₈	—	356	1	no	—
235	C₂×Q₈	—	356	1	no	—
236	C₂×Q₈	—	356	1	no	—
237	C₂×Q₈	—	356	1	no	—
238	C₂×Q₈	—	178	2	no	—
239	C₂×D₄	—	356	1	no	—
240	C₂×D₄	—	356	1	no	—
241	C₂×D₄	—	356	1	no	—
242	C₂×D₄	—	356	1	no	—
243	C₂×D₄	—	356	1	no	—
244	C₂×D₄	—	178	2	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	356	1	no	—
247	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
248	Q₁₆	—	356	1	no	—
249	Q₁₆	—	356	1	no	—
250	Q₁₆	—	178	2	no	—
251	Q₁₆	—	178	2	no	—
252	SD₁₆	—	356	4	no	—
253	SD₁₆	—	356	4	no	—
254	D₈	—	356	1	no	—
255	D₈	—	356	1	no	—
256	D₈	—	178	2	no	—
257	D₈	—	178	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	#13083551	1	2	yes	—
262	C₂×C₈	#13083559	1	1	yes	characteristic
263	C₂×C₈	#13083559	1	1	yes	characteristic
264	C₂×C₈	#13083561	1	2	yes	—
265	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
267	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
268	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
269	C₂×C₈	#13083574	1	1	yes	characteristic
270	C₂×C₈	#13083574	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	356	8	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	#13083551	1	2	yes	—
277	C₁₆	—	356	1	no	—
278	C₁₆	—	356	1	no	—
279	C₂³	—	356	2	no	—
280	C₂³	—	712	4	no	—
281	C₂³	—	712	1	no	—
282	C₂³	—	712	1	no	—
283	C₂³	#15668037	1	1	yes	characteristic
284	Q₈	—	356	2	no	—
285	Q₈	—	356	2	no	—
286	Q₈	—	712	1	no	—
287	Q₈	—	712	1	no	—
288	D₄	—	356	2	no	—
289	D₄	—	356	2	no	—
290	D₄	—	712	1	no	—
291	D₄	—	712	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	356	2	no	—
296	C₂×C₄	—	712	1	no	—
297	C₂×C₄	—	712	1	no	—
298	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
299	C₂×C₄	#15667986	1	1	yes	characteristic
300	C₂×C₄	#15667986	1	1	yes	characteristic
301	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
302	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
303	C₂×C₄	#15668037	1	1	yes	characteristic
304	C₈	—	4	2	no	—
305	C₈	#15667932	1	2	yes	—
306	C₈	#15667932	1	2	yes	—
307	C₈	C₄×D₃₅₆	1	1	yes	characteristic
308	C₈	C₄×D₃₅₆	1	1	yes	characteristic
309	C₈	#15668000	1	1	yes	characteristic
310	C₈	#15668009	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	1424	2	no	—
313	C₂²	—	1424	2	no	—
314	C₂²	—	712	4	no	—
315	C₂²	—	712	2	no	—
316	C₂²	#19151602	1	1	yes	characteristic
317	C₂²	#19151793	1	1	yes	characteristic
318	C₂²	#19151798	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	712	2	no	—
322	C₄	#19151500	1	1	yes	characteristic
323	C₄	C₂×#15667932	1	1	yes	characteristic
324	C₄	#19151793	1	1	yes	characteristic
325	C₄	#19151798	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	1424	2	no	—
329	C₂	#26021179	1	1	yes	characteristic
330	C₂	#26022321	1	2	yes	—

	subgroup	G/H
center	C₂²	#19151793
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1772	C₂
socle	C₂²×C₈₉	#459
prime radical	C₂²×D₇₁₂	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1772	D₈₉
hypocenter	C₈₉	#13260
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
2. C₂×#19151419	C₂	semidirect
3. C₂×#19151413	C₂	semidirect
5. #26021971	C₂	semidirect
7. C₈₉×#1772	C₂	semidirect
15. #19151982	C₄	semidirect
172. #1772	D₈₉	semidirect
173. C₈₉	#13260	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151982	2
C₂×#19151419	2
C₂×#19151413	2
C₂×#19151410	2
#26021971	2
#26021967	2
C₈₉×#1772	2
#13260	89