Subgroups — #26030549

Order 11392.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152206	C₂	1	1	yes	maximal, characteristic
2	#19152074	C₂	1	1	yes	maximal, characteristic
3	#19151941	C₂	1	1	yes	maximal, characteristic
4	#19151938	C₂	1	1	yes	maximal, characteristic
5	#19151907	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151904	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151807	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151806	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151755	C₂	1	1	yes	maximal, characteristic
10	#19151752	C₂	1	1	yes	maximal, characteristic
11	#19151703	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151703	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151657	C₂	1	1	yes	maximal, characteristic
15	C₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
18	C₂×#13083564	C₂²	1	1	yes	characteristic
19	C₂×#13083564	C₂²	1	1	yes	characteristic
20	#15668010	C₂²	1	1	yes	characteristic
21	#15668007	—	2	1	no	—
22	#15668006	—	2	1	no	—
23	#15668004	—	2	1	no	—
24	#15668003	C₂²	1	1	yes	characteristic
25	#15668002	—	2	1	no	—
26	#15667992	C₂²	1	1	yes	characteristic
27	#15667988	C₂²	1	1	yes	characteristic
28	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₇₈	—	2	1	no	—
30	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
31	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
32	#15667977	—	2	1	no	—
33	#15667977	C₂²	1	1	yes	characteristic
34	#15667970	—	2	1	no	—
35	#15667970	C₂²	1	1	yes	characteristic
36	#15667969	—	2	1	no	—
37	#15667969	C₂²	1	1	yes	characteristic
38	#15667946	C₂²	1	1	yes	characteristic
39	#15667943	C₂²	1	1	yes	characteristic
40	#15667943	C₂²	1	1	yes	characteristic
41	#15667941	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667937	C₂²	1	1	yes	characteristic
44	#15667937	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667935	C₂²	1	1	yes	characteristic
48	#15667935	C₂²	1	1	yes	characteristic
49	#15667924	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667923	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×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	Q₈×D₈₉	—	2	2	no	—
63	Q₈×D₈₉	—	2	2	no	—
64	C₂×C₄×D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
66	Dic₃₅₆	—	2	2	no	—
67	#13083564	—	2	2	no	—
68	#13083564	—	2	2	no	—
69	C₂×#10623494	C₂³	1	1	yes	characteristic
70	C₂×#10623494	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₄⋊Dic₈₉	—	2	1	no	—
73	C₄⋊Dic₈₉	—	2	1	no	—
74	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
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	#13083553	D₄	1	1	yes	characteristic
84	#13083553	D₄	1	1	yes	characteristic
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	D₄	1	1	yes	characteristic
89	#13083551	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	—	2	1	no	—
94	C₄×Dic₈₉	—	2	1	no	—
95	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
98	Q₁₆×C₈₉	—	2	2	no	—
99	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	—	2	1	no	—
102	C₄⋊C₄×C₈₉	—	2	1	no	—
103	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₄²×C₈₉	—	2	1	no	—
106	C₄²×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₈₉	—	2	1	no	—
110	Q₈×C₈₉	—	2	1	no	—
111	Q₈×C₈₉	—	2	1	no	—
112	Q₈×C₈₉	#41	1	2	yes	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄×D₈₉	—	2	4	no	—
128	C₄×D₈₉	—	2	1	no	—
129	C₄×D₈₉	—	2	1	no	—
130	Dic₁₇₈	—	2	4	no	—
131	Dic₁₇₈	—	2	1	no	—
132	Dic₁₇₈	—	2	1	no	—
133	Dic₁₇₈	—	2	1	no	—
134	Dic₁₇₈	#41	1	2	yes	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	—	2	1	no	—
137	#10623494	—	2	1	no	—
138	#10623494	—	2	1	no	—
139	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₈₉	—	2	1	no	—
141	C₂×D₈₉	—	2	1	no	—
142	C₂×D₈₉	—	2	1	no	—
143	C₄×C₈₉	—	2	2	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	2	1	no	—
146	C₄×C₈₉	—	4	1	no	—
147	C₄×C₈₉	—	4	1	no	—
148	C₄×C₈₉	—	4	1	no	—
149	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
150	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
151	Dic₈₉	—	2	1	no	—
152	Dic₈₉	—	2	2	no	—
153	Dic₈₉	—	2	1	no	—
154	Dic₈₉	—	2	1	no	—
155	Dic₈₉	—	4	1	no	—
156	Dic₈₉	—	4	1	no	—
157	Dic₈₉	—	4	1	no	—
158	C₂×C₈₉	#525	1	1	yes	characteristic
159	C₂×C₈₉	#578	1	1	yes	characteristic
160	C₂×C₈₉	#578	1	1	yes	characteristic
161	D₈₉	—	4	1	no	—
162	#3127	—	89	1	no	maximal, Sylow
163	C₈₉	#3127	1	1	yes	characteristic, complemented, Sylow
164	#541	—	89	1	no	—
165	#519	—	89	1	no	—
166	#485	—	89	1	no	—
167	#485	—	89	1	no	—
168	#479	—	89	1	no	—
169	#479	—	89	1	no	—
170	#462	—	89	1	no	—
171	#462	—	89	1	no	—
172	#451	—	89	1	no	—
173	#451	—	89	1	no	—
174	#441	—	89	1	no	—
175	#441	—	89	1	no	—
176	C₄×Q₁₆	—	89	1	no	—
177	C₄×Q₁₆	D₈₉	1	1	yes	characteristic, complemented
178	#433	—	89	1	no	—
179	C₂²×Q₈	—	89	1	no	—
180	C₂×Q₁₆	—	89	1	no	—
181	C₂×Q₁₆	—	89	1	no	—
182	C₂×Q₁₆	—	89	1	no	—
183	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
184	C₄⋊Q₈	—	89	1	no	—
185	#126	—	178	1	no	—
186	#126	—	178	1	no	—
187	#125	—	89	1	no	—
188	#125	—	178	1	no	—
189	#125	—	178	1	no	—
190	C₂²⋊Q₈	—	89	1	no	—
191	C₂²⋊Q₈	—	89	1	no	—
192	C₄×Q₈	—	89	1	no	—
193	C₄×Q₈	—	89	1	no	—
194	C₄×Q₈	—	89	1	no	—
195	C₄×Q₈	—	89	1	no	—
196	C₄×Q₈	—	178	1	no	—
197	C₄×Q₈	—	178	1	no	—
198	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
199	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
200	#117	—	89	1	no	—
201	#117	—	89	1	no	—
202	#117	—	178	1	no	—
203	#117	—	178	1	no	—
204	#107	—	89	1	no	—
205	#107	C₂×D₈₉	1	1	yes	characteristic
206	#106	—	89	1	no	—
207	#106	—	89	1	no	—
208	C₄⋊C₈	—	89	1	no	—
209	C₄⋊C₈	—	89	1	no	—
210	#103	—	89	1	no	—
211	#103	—	89	1	no	—
212	#103	—	89	1	no	—
213	#103	—	89	1	no	—
214	#103	—	89	1	no	—
215	#103	—	89	1	no	—
216	#103	C₂×D₈₉	1	1	yes	characteristic
217	#103	C₂×D₈₉	1	1	yes	characteristic
218	C₂²⋊C₈	—	89	1	no	—
219	C₂²⋊C₈	—	89	1	no	—
220	#97	—	89	1	no	—
221	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
222	C₂×Q₈	—	89	1	no	—
223	C₂×Q₈	—	89	1	no	—
224	C₂×Q₈	—	178	1	no	—
225	C₂×Q₈	—	178	2	no	—
226	C₂×Q₈	—	178	2	no	—
227	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
228	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
229	C₂²×C₄	—	89	1	no	—
230	C₂²×C₄	—	178	1	no	—
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	178	2	no	—
233	Q₁₆	—	178	2	no	—
234	Q₁₆	—	178	2	no	—
235	C₂×C₈	—	89	1	no	—
236	C₂×C₈	—	89	1	no	—
237	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	89	1	no	—
242	C₄⋊C₄	—	89	1	no	—
243	C₄⋊C₄	—	89	1	no	—
244	C₄⋊C₄	—	89	1	no	—
245	C₄⋊C₄	—	178	1	no	—
246	C₄⋊C₄	—	178	1	no	—
247	C₄⋊C₄	—	178	1	no	—
248	C₄⋊C₄	—	178	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	C₂²×D₈₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	89	1	no	—
258	C₂²⋊C₄	—	89	1	no	—
259	C₂²⋊C₄	—	178	1	no	—
260	C₂²⋊C₄	—	178	1	no	—
261	C₂²⋊C₄	—	178	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	89	1	no	—
265	C₄²	—	178	1	no	—
266	C₄²	—	178	1	no	—
267	C₄²	—	178	1	no	—
268	C₄²	—	178	1	no	—
269	C₄²	—	178	1	no	—
270	C₄²	C₂²×D₈₉	1	1	yes	characteristic
271	C₂³	—	89	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	89	2	no	—
276	Q₈	—	178	4	no	—
277	Q₈	—	178	1	no	—
278	Q₈	—	178	1	no	—
279	Q₈	—	178	1	no	—
280	Q₈	#13083574	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	89	1	no	—
286	C₂×C₄	—	89	1	no	—
287	C₂×C₄	—	89	1	no	—
288	C₂×C₄	—	178	4	no	—
289	C₂×C₄	—	178	1	no	—
290	C₂×C₄	—	178	1	no	—
291	C₂×C₄	—	178	1	no	—
292	C₂×C₄	—	178	1	no	—
293	C₂×C₄	—	178	1	no	—
294	C₂×C₄	—	178	1	no	—
295	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
296	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	178	1	no	—
301	C₈	—	178	1	no	—
302	C₂²	—	178	1	no	—
303	C₂²	—	178	1	no	—
304	C₂²	—	178	1	no	—
305	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	356	1	no	—
313	C₄	—	356	1	no	—
314	C₄	—	356	1	no	—
315	C₄	—	178	1	no	—
316	C₄	—	178	1	no	—
317	C₄	—	178	1	no	—
318	C₄	—	178	2	no	—
319	C₄	#41×D₈₉	1	1	yes	characteristic
320	C₄	C₂×#13083574	1	1	yes	characteristic
321	C₂	—	356	1	no	—
322	C₂	#19152107	1	1	yes	characteristic
323	C₂	#19152411	1	1	yes	characteristic
324	C₂	#19152412	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₄×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂²×D₈₉	C₂×D₄
hypercenter	C₄×Q₁₆	D₈₉
hypocenter	C₈₉	#3127
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
5. #19151907	C₂	semidirect
6. #19151904	C₂	semidirect
7. #19151807	C₂	semidirect
8. #19151806	C₂	semidirect
11. #19151703	C₂	semidirect
12. #19151703	C₂	semidirect
13. C₄×Dic₃₅₆	C₂	semidirect
15. C₄×Q₁₆×C₈₉	C₂	semidirect
163. C₈₉	#3127	semidirect
177. C₄×Q₁₆	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152206	2
#19152074	2
#19151941	2
#19151938	2
#19151907	2
#19151904	2
#19151807	2
#19151806	2
#19151755	2
#19151752	2
#19151703	2
C₄×Dic₃₅₆	2
#19151657	2
C₄×Q₁₆×C₈₉	2
#3127	89