Subgroups — #26030913

Order 11392.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152215	C₂	1	1	yes	maximal, characteristic
2	#19152208	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152006	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151989	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151951	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151947	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151856	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151837	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151825	C₂	1	1	yes	maximal, characteristic
10	#19151702	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151695	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151692	C₂	1	1	yes	maximal, characteristic, complemented
13	SD₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151556	C₂	1	1	yes	maximal, characteristic, complemented
15	#468×C₈₉	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
17	C₂×#13083564	C₂²	1	1	yes	characteristic
18	C₂×#13083562	C₂²	1	1	yes	characteristic
19	C₂²×#10623494	C₂²	1	1	yes	characteristic
20	#15668011	C₂²	1	1	yes	characteristic
21	#15668010	C₂²	1	1	yes	characteristic
22	#15668005	—	2	1	no	—
23	#15668005	—	2	1	no	—
24	#15668005	—	2	1	no	—
25	#15668003	—	2	1	no	—
26	#15668003	C₂²	1	1	yes	characteristic
27	#15668002	—	2	1	no	—
28	#15667999	—	2	1	no	—
29	#15667999	C₂²	1	1	yes	characteristic
30	#15667993	—	2	1	no	—
31	#15667991	C₂²	1	1	yes	characteristic
32	#15667991	C₂²	1	1	yes	characteristic
33	#15667989	—	2	1	no	—
34	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
35	#15667977	C₂²	1	1	yes	characteristic
36	#15667977	C₂²	1	1	yes	characteristic
37	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
38	#15667971	C₂²	1	1	yes	characteristic
39	#15667945	C₂²	1	1	yes	characteristic
40	#15667944	C₂²	1	1	yes	characteristic
41	#15667943	C₂²	1	1	yes	characteristic
42	#15667937	C₂²	1	1	yes	characteristic
43	#15667937	C₂²	1	1	yes	characteristic
44	#15667936	C₂²	1	1	yes	characteristic
45	#15667935	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic
48	#15667922	C₂²	1	1	yes	characteristic
49	#15667922	C₂²	1	1	yes	characteristic
50	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	#121×C₈₉	C₂²	1	1	yes	characteristic
56	#107×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂²×Dic₈₉	—	2	1	no	—
63	Dic₃₅₆	—	2	2	no	—
64	#13083564	—	2	2	no	—
65	#13083562	—	2	2	no	—
66	C₂×#10623494	—	2	2	no	—
67	C₂×#10623494	C₂³	1	1	yes	characteristic
68	C₂×#10623494	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₈₉	—	2	1	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
72	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
73	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
75	#13083553	—	2	1	no	—
76	#13083553	—	2	1	no	—
77	#13083553	—	2	1	no	—
78	#13083553	—	2	1	no	—
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	C₂²⋊Dic₈₉	—	2	1	no	—
84	C₂²⋊Dic₈₉	—	2	1	no	—
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	C₂²⋊Dic₈₉	—	2	1	no	—
88	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
94	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₈₉	—	2	1	no	—
97	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
99	SD₁₆×C₈₉	—	2	2	no	—
100	M₁₆×C₈₉	—	2	2	no	—
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	—	2	1	no	—
104	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₂²⋊C₄×C₈₉	—	2	1	no	—
108	C₂³×C₈₉	—	2	1	no	—
109	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₈₉	—	2	1	no	—
111	Q₈×C₈₉	—	2	1	no	—
112	D₄×C₈₉	—	2	1	no	—
113	D₄×C₈₉	—	2	1	no	—
114	D₄×C₈₉	—	8	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	8	1	no	—
127	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
131	Dic₁₇₈	—	2	1	no	—
132	Dic₁₇₈	—	2	1	no	—
133	Dic₁₇₈	—	2	1	no	—
134	Dic₁₇₈	—	2	1	no	—
135	Dic₁₇₈	—	8	1	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	#41	1	2	yes	—
138	#10623494	—	2	1	no	—
139	#10623494	#41	1	2	yes	—
140	C₂²×C₈₉	—	2	1	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	—	4	1	no	—
144	C₂²×C₈₉	—	4	1	no	—
145	C₂²×C₈₉	—	4	1	no	—
146	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	4	1	no	—
149	C₄×C₈₉	—	4	1	no	—
150	C₄×C₈₉	—	4	1	no	—
151	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
152	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
153	Dic₈₉	—	2	1	no	—
154	Dic₈₉	—	2	1	no	—
155	Dic₈₉	—	2	1	no	—
156	Dic₈₉	—	2	1	no	—
157	Dic₈₉	—	4	1	no	—
158	Dic₈₉	—	4	1	no	—
159	Dic₈₉	—	4	1	no	—
160	Dic₈₉	—	4	1	no	—
161	C₂×C₈₉	—	4	1	no	—
162	C₂×C₈₉	—	8	1	no	—
163	C₂×C₈₉	#532	1	1	yes	characteristic
164	C₂×C₈₉	#577	1	1	yes	characteristic
165	C₂×C₈₉	#578	1	1	yes	characteristic
166	#3178	—	89	1	no	maximal, Sylow
167	C₈₉	#3178	1	1	yes	characteristic, complemented, Sylow
168	#542	—	89	1	no	—
169	#541	—	89	1	no	—
170	#501	—	89	1	no	—
171	#497	—	89	1	no	—
172	#488	—	89	1	no	—
173	#487	—	89	1	no	—
174	#470	—	89	1	no	—
175	#468	D₈₉	1	1	yes	characteristic
176	#467	—	89	1	no	—
177	#465	—	89	1	no	—
178	#441	—	89	1	no	—
179	#440	—	89	1	no	—
180	C₄×Q₁₆	—	89	1	no	—
181	C₄×SD₁₆	—	89	1	no	—
182	#405	—	89	1	no	—
183	C₂×Q₁₆	—	89	1	no	—
184	C₂×Q₁₆	—	89	1	no	—
185	C₂×SD₁₆	—	89	1	no	—
186	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
187	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
188	C₂²×C₈	—	89	1	no	—
189	C₄⋊Q₈	—	89	1	no	—
190	C₄⋊Q₈	—	89	1	no	—
191	#126	—	178	1	no	—
192	#126	—	178	1	no	—
193	#126	—	178	1	no	—
194	#125	—	89	1	no	—
195	#125	—	178	1	no	—
196	#125	—	178	1	no	—
197	#124	—	89	1	no	—
198	#124	—	178	1	no	—
199	#123	—	178	1	no	—
200	C₂²⋊Q₈	—	89	1	no	—
201	C₂²⋊Q₈	—	89	1	no	—
202	C₂²⋊Q₈	—	178	1	no	—
203	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
204	#121	C₂×D₈₉	1	1	yes	characteristic
205	C₄×Q₈	—	89	1	no	—
206	C₄×Q₈	—	89	1	no	—
207	C₄×Q₈	—	89	1	no	—
208	C₄×D₄	—	89	1	no	—
209	#117	—	89	1	no	—
210	#107	—	89	1	no	—
211	#107	C₂×D₈₉	1	1	yes	characteristic
212	#106	—	89	1	no	—
213	#106	—	89	1	no	—
214	#103	—	89	1	no	—
215	#103	—	89	1	no	—
216	#103	—	89	1	no	—
217	#103	—	89	1	no	—
218	#103	—	89	1	no	—
219	#103	C₂×D₈₉	1	1	yes	characteristic
220	#102	—	89	1	no	—
221	#102	C₂×D₈₉	1	1	yes	characteristic
222	#97	—	89	1	no	—
223	#97	—	89	1	no	—
224	C₄×C₈	—	89	1	no	—
225	C₄×C₈	—	89	1	no	—
226	C₂×Q₈	—	89	1	no	—
227	C₂×Q₈	—	89	1	no	—
228	C₂×Q₈	—	178	1	no	—
229	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
232	C₂²×C₄	—	178	1	no	—
233	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
234	Q₁₆	—	178	2	no	—
235	Q₁₆	—	178	2	no	—
236	SD₁₆	—	2	2	no	—
237	SD₁₆	—	178	2	no	—
238	M₁₆	—	2	2	no	—
239	C₂×C₈	—	89	1	no	—
240	C₂×C₈	—	89	1	no	—
241	C₂×C₈	—	178	2	no	—
242	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	89	1	no	—
246	C₄⋊C₄	—	89	1	no	—
247	C₄⋊C₄	—	89	1	no	—
248	C₄⋊C₄	—	89	1	no	—
249	C₄⋊C₄	—	178	1	no	—
250	C₄⋊C₄	—	178	1	no	—
251	C₄⋊C₄	—	178	1	no	—
252	C₄⋊C₄	—	178	1	no	—
253	C₄⋊C₄	—	178	1	no	—
254	C₄⋊C₄	—	178	1	no	—
255	C₄⋊C₄	—	178	1	no	—
256	C₄⋊C₄	—	178	1	no	—
257	C₄⋊C₄	—	178	1	no	—
258	C₄⋊C₄	—	178	1	no	—
259	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	89	1	no	—
264	C₂²⋊C₄	—	89	1	no	—
265	C₂²⋊C₄	—	178	1	no	—
266	C₂²⋊C₄	—	178	1	no	—
267	C₂²⋊C₄	—	178	1	no	—
268	C₂²⋊C₄	—	178	1	no	—
269	C₂²⋊C₄	—	178	1	no	—
270	C₄²	—	89	1	no	—
271	C₄²	—	89	1	no	—
272	C₄²	—	178	1	no	—
273	C₄²	—	178	1	no	—
274	C₄²	—	178	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	D₄×D₈₉	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	178	1	no	—
280	Q₈	—	178	1	no	—
281	Q₈	—	178	1	no	—
282	Q₈	—	178	1	no	—
283	Q₈	—	712	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	8	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	89	1	no	—
293	C₂×C₄	—	89	1	no	—
294	C₂×C₄	—	89	1	no	—
295	C₂×C₄	—	89	1	no	—
296	C₂×C₄	—	178	1	no	—
297	C₂×C₄	—	178	1	no	—
298	C₂×C₄	—	178	1	no	—
299	C₂×C₄	—	178	1	no	—
300	C₂×C₄	—	712	1	no	—
301	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	89	2	no	—
305	C₈	—	178	1	no	—
306	C₈	#13083573	1	2	yes	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	356	1	no	—
319	C₄	—	356	1	no	—
320	C₄	—	356	1	no	—
321	C₄	—	356	1	no	—
322	C₄	—	178	1	no	—
323	C₄	—	178	1	no	—
324	C₄	—	178	1	no	—
325	C₄	—	178	1	no	—
326	C₄	#41×D₈₉	1	1	yes	characteristic
327	C₄	C₂×#13083573	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	8	1	no	—
330	C₂	#19152143	1	1	yes	characteristic
331	C₂	#19152408	1	1	yes	characteristic
332	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#468×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#121×C₈₉	C₂²
hypercenter	#468	D₈₉
hypocenter	C₈₉	#3178
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
2. #19152208	C₂	semidirect
3. #19152006	C₂	semidirect
4. #19151989	C₂	semidirect
5. #19151951	C₂	semidirect
6. #19151947	C₂	semidirect
7. #19151856	C₂	semidirect
8. #19151837	C₂	semidirect
10. #19151702	C₂	semidirect
11. #19151695	C₂	semidirect
12. #19151692	C₂	semidirect
13. SD₁₆×Dic₈₉	C₂	semidirect
14. #19151556	C₂	semidirect
167. C₈₉	#3178	semidirect

Maximal subgroups

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

subgroup	index
#19152215	2
#19152208	2
#19152006	2
#19151989	2
#19151951	2
#19151947	2
#19151856	2
#19151837	2
#19151825	2
#19151702	2
#19151695	2
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SD₁₆×Dic₈₉	2
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#3178	89