Subgroups — #26033124

Order 11392.

338 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152325	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152304	C₂	1	1	yes	maximal, characteristic
3	#19151998	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151995	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151904	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151902	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151901	C₂	1	1	yes	maximal, characteristic
8	#19151895	C₂	1	1	yes	maximal, characteristic
9	#19151892	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151877	C₂	1	1	yes	maximal, characteristic
11	C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13083563	C₂	1	1	yes	maximal, characteristic
13	#19151685	C₂	1	1	yes	maximal, characteristic
14	#19151685	C₂	1	1	yes	maximal, characteristic
15	#499×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083563	C₂²	1	1	yes	characteristic
17	#15668011	—	2	1	no	—
18	#15668011	C₂²	1	1	yes	characteristic
19	#15668011	C₂²	1	1	yes	characteristic
20	#15668003	C₂²	1	1	yes	characteristic
21	#15668003	C₂²	1	1	yes	characteristic
22	#15668002	—	2	1	no	—
23	#15668002	—	2	1	no	—
24	#15667992	—	2	1	no	—
25	#15667990	—	2	1	no	—
26	#15667990	—	2	1	no	—
27	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
29	#15667977	—	2	1	no	—
30	#15667977	C₂²	1	1	yes	characteristic
31	#15667977	C₂²	1	1	yes	characteristic
32	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
33	#15667974	C₂²	1	1	yes	characteristic
34	#15667974	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₈₉	—	2	1	no	—
36	#15667945	C₂²	1	1	yes	characteristic
37	#15667945	C₂²	1	1	yes	characteristic
38	#15667944	C₂²	1	1	yes	characteristic
39	#15667943	C₂²	1	1	yes	characteristic
40	#15667943	C₂²	1	1	yes	characteristic
41	C₄⋊#10623494	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667941	C₂²	1	1	yes	characteristic
44	#15667937	C₂²	1	1	yes	characteristic
45	#15667935	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667934	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	#125×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#106×C₈₉	C₂²	1	1	yes	characteristic
57	#106×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	#13083563	Q₈	1	4	yes	complemented
68	C₂×#10623494	C₂³	1	1	yes	characteristic
69	C₂×#10623494	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	—	2	1	no	—
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	D₄	1	1	yes	characteristic
83	#13083553	D₄	1	1	yes	characteristic
84	#13083551	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	C₄×Dic₈₉	—	2	1	no	—
88	C₄×Dic₈₉	—	2	1	no	—
89	C₄×Dic₈₉	—	2	1	no	—
90	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
91	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₈₉	—	2	1	no	—
95	C₄⋊C₄×C₈₉	—	2	1	no	—
96	C₄⋊C₄×C₈₉	—	2	1	no	—
97	C₄⋊C₄×C₈₉	—	2	1	no	—
98	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₄²×C₈₉	C₂³	1	1	yes	characteristic
103	C₂²×D₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₈₉	—	2	1	no	—
112	C₂×Dic₈₉	—	2	1	no	—
113	C₂×Dic₈₉	—	2	1	no	—
114	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
118	D₃₅₆	—	2	1	no	—
119	D₃₅₆	C₂×Q₈	1	2	yes	—
120	C₄×D₈₉	—	4	2	no	—
121	C₄×D₈₉	—	8	1	no	—
122	C₄×D₈₉	—	8	1	no	—
123	Dic₁₇₈	—	2	1	no	—
124	Dic₁₇₈	—	2	2	no	—
125	Dic₁₇₈	—	4	2	no	—
126	Dic₁₇₈	—	8	1	no	—
127	Dic₁₇₈	C₂×Q₈	1	2	yes	—
128	C₈×C₈₉	—	2	1	no	—
129	C₈×C₈₉	C₂×Q₈	1	2	yes	—
130	#10623494	—	2	1	no	—
131	#10623494	—	2	1	no	—
132	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₈₉	—	2	2	no	—
134	C₂×D₈₉	—	4	1	no	—
135	C₂×D₈₉	—	4	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	—	4	1	no	—
139	C₄×C₈₉	—	4	1	no	—
140	C₄×C₈₉	—	4	1	no	—
141	C₄×C₈₉	—	4	1	no	—
142	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
143	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
144	Dic₈₉	—	2	2	no	—
145	Dic₈₉	—	2	1	no	—
146	Dic₈₉	—	2	1	no	—
147	Dic₈₉	—	2	1	no	—
148	Dic₈₉	—	4	1	no	—
149	Dic₈₉	—	4	1	no	—
150	Dic₈₉	#135	1	2	yes	—
151	C₂×C₈₉	D₄×Q₈	1	1	yes	characteristic
152	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
153	C₂×C₈₉	#577	1	1	yes	characteristic
154	D₈₉	—	4	2	no	—
155	#3381	—	89	1	no	maximal, Sylow
156	C₈₉	#3381	1	1	yes	characteristic, complemented, Sylow
157	#557	—	89	1	no	—
158	#554	—	89	1	no	—
159	#499	—	89	1	no	—
160	#499	D₈₉	1	1	yes	characteristic, complemented
161	#498	—	89	1	no	—
162	#479	—	89	1	no	—
163	#479	—	89	1	no	—
164	#478	—	89	1	no	—
165	#478	—	89	1	no	—
166	#477	—	89	1	no	—
167	#474	—	89	1	no	—
168	C₈×Q₈	—	89	1	no	—
169	C₄×SD₁₆	—	89	1	no	—
170	C₄×SD₁₆	—	89	1	no	—
171	C₄×SD₁₆	—	89	1	no	—
172	C₂×SD₁₆	—	89	1	no	—
173	C₄⋊Q₈	—	89	1	no	—
174	C₄⋊Q₈	—	89	1	no	—
175	C₄⋊Q₈	—	178	1	no	—
176	#125	—	89	1	no	—
177	#125	—	89	1	no	—
178	#125	—	178	1	no	—
179	#125	—	178	1	no	—
180	#125	C₂×D₈₉	1	1	yes	characteristic
181	#125	C₂×D₈₉	1	1	yes	characteristic
182	C₂²⋊Q₈	—	178	1	no	—
183	C₂²⋊Q₈	—	178	1	no	—
184	C₂²⋊Q₈	—	178	1	no	—
185	C₄×Q₈	—	89	1	no	—
186	C₄×Q₈	—	89	1	no	—
187	C₄×Q₈	—	89	1	no	—
188	C₄×Q₈	—	89	1	no	—
189	C₄×Q₈	—	178	1	no	—
190	C₄×D₄	—	89	1	no	—
191	C₄×D₄	—	89	1	no	—
192	C₄×D₄	—	89	1	no	—
193	C₂×C₄⋊C₄	—	178	1	no	—
194	#107	—	89	1	no	—
195	#107	—	89	1	no	—
196	#107	C₂×D₈₉	1	1	yes	characteristic
197	#107	C₂×D₈₉	1	1	yes	characteristic
198	#106	—	89	1	no	—
199	#106	—	89	1	no	—
200	#106	—	89	1	no	—
201	#106	C₂×D₈₉	1	1	yes	characteristic
202	#106	C₂×D₈₉	1	1	yes	characteristic
203	C₄⋊C₈	—	89	1	no	—
204	C₄⋊C₈	—	89	1	no	—
205	C₄⋊C₈	—	89	1	no	—
206	#103	—	89	1	no	—
207	#103	—	89	1	no	—
208	#103	—	89	1	no	—
209	#102	—	89	1	no	—
210	#102	—	89	1	no	—
211	#102	—	89	1	no	—
212	C₄×C₈	—	89	1	no	—
213	C₄×C₈	—	89	1	no	—
214	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
215	C₂×Q₈	—	89	1	no	—
216	C₂×Q₈	—	89	1	no	—
217	C₂×Q₈	—	178	1	no	—
218	C₂×Q₈	—	178	1	no	—
219	C₂×D₄	—	89	1	no	—
220	C₂²×C₄	—	178	1	no	—
221	C₂²×C₄	—	178	1	no	—
222	C₂²×C₄	—	178	1	no	—
223	SD₁₆	—	89	4	no	—
224	C₂×C₈	—	89	1	no	—
225	C₂×C₈	—	89	1	no	—
226	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
227	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	89	1	no	—
233	C₄⋊C₄	—	89	1	no	—
234	C₄⋊C₄	—	89	1	no	—
235	C₄⋊C₄	—	89	1	no	—
236	C₄⋊C₄	—	89	1	no	—
237	C₄⋊C₄	—	178	1	no	—
238	C₄⋊C₄	—	178	1	no	—
239	C₄⋊C₄	—	178	1	no	—
240	C₄⋊C₄	—	178	1	no	—
241	C₄⋊C₄	—	178	1	no	—
242	C₄⋊C₄	—	178	1	no	—
243	C₄⋊C₄	—	178	1	no	—
244	C₄⋊C₄	—	178	1	no	—
245	C₄⋊C₄	—	178	1	no	—
246	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
247	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
250	C₂²⋊C₄	—	178	1	no	—
251	C₂²⋊C₄	—	178	1	no	—
252	C₂²⋊C₄	—	178	1	no	—
253	C₄²	—	89	1	no	—
254	C₄²	—	89	1	no	—
255	C₄²	—	178	1	no	—
256	C₄²	—	178	1	no	—
257	C₄²	—	178	1	no	—
258	C₄²	C₂²×D₈₉	1	1	yes	characteristic
259	C₂³	—	178	1	no	—
260	Q₈	—	89	2	no	—
261	Q₈	—	356	2	no	—
262	Q₈	—	178	2	no	—
263	Q₈	—	178	1	no	—
264	Q₈	—	712	1	no	—
265	D₄	—	89	2	no	—
266	D₄	—	178	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	89	1	no	—
272	C₂×C₄	—	89	1	no	—
273	C₂×C₄	—	89	1	no	—
274	C₂×C₄	—	89	1	no	—
275	C₂×C₄	—	356	2	no	—
276	C₂×C₄	—	178	1	no	—
277	C₂×C₄	—	178	1	no	—
278	C₂×C₄	—	178	1	no	—
279	C₂×C₄	—	712	1	no	—
280	C₂×C₄	—	712	1	no	—
281	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
282	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	178	1	no	—
286	C₈	—	178	1	no	—
287	C₈	Q₈×D₈₉	1	2	yes	—
288	C₂²	—	356	1	no	—
289	C₂²	—	356	1	no	—
290	C₂²	—	178	2	no	—
291	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	4	1	no	—
298	C₄	—	89	2	no	—
299	C₄	—	356	1	no	—
300	C₄	—	356	1	no	—
301	C₄	—	178	1	no	—
302	C₄	—	178	1	no	—
303	C₄	—	178	2	no	—
304	C₄	—	178	1	no	—
305	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
306	C₄	#15668060	1	1	yes	characteristic
307	C₂	—	356	2	no	—
308	C₂	#19152263	1	1	yes	characteristic
309	C₂	#135×D₈₉	1	1	yes	characteristic
310	C₂	#19152405	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#499×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₃₅₆	C₂³
hypercenter	#499	D₈₉
hypocenter	C₈₉	#3381
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
1. #19152325	C₂	semidirect
3. #19151998	C₂	semidirect
4. #19151995	C₂	semidirect
5. #19151904	C₂	semidirect
6. #19151902	C₂	semidirect
9. #19151892	C₂	semidirect
11. C₈×Dic₁₇₈	C₂	semidirect
15. #499×C₈₉	C₂	semidirect
67. #13083563	Q₈	semidirect
156. C₈₉	#3381	semidirect
160. #499	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152325	2
#19152304	2
#19151998	2
#19151995	2
#19151904	2
#19151902	2
#19151901	2
#19151895	2
#19151892	2
#19151877	2
C₈×Dic₁₇₈	2
C₄×#13083563	2
#19151685	2
#499×C₈₉	2
#3381	89