Subgroups — #26030321

Order 11392.

390 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152152	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13083575	C₂	1	1	yes	maximal, characteristic
3	#19151941	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151937	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151881	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151877	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151806	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151802	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151763	C₂	1	1	yes	maximal, characteristic
10	#19151759	C₂	1	1	yes	maximal, characteristic
11	C₄×#13083564	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13083563	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13083561	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151649	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×#13083564	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083561	C₂²	1	1	yes	characteristic
20	#15668011	C₂²	1	1	yes	characteristic
21	#15668010	C₂²	1	1	yes	characteristic
22	#15668002	—	2	1	no	—
23	#15667992	C₂²	1	1	yes	characteristic
24	#15667990	—	2	1	no	—
25	#15667988	C₂²	1	1	yes	characteristic
26	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₇₈	—	2	1	no	—
28	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
29	C₄×D₃₅₆	—	2	1	no	—
30	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
31	#15667974	—	2	1	no	—
32	#15667974	C₂²	1	1	yes	characteristic
33	#15667970	—	2	1	no	—
34	#15667970	C₂²	1	1	yes	characteristic
35	#15667969	—	2	1	no	—
36	C₄²×D₈₉	—	2	1	no	—
37	C₄²×D₈₉	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667944	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	#15667936	C₂²	1	1	yes	characteristic
45	#15667936	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	#15667924	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	C₄×#10623494	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	#13083575	—	2	2	no	—
60	#13083575	—	2	2	no	—
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
70	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
71	#13083564	—	2	2	no	—
72	#13083563	—	2	2	no	—
73	#13083561	—	2	2	no	—
74	C₂×#10623494	C₂³	1	1	yes	characteristic
75	C₂×#10623494	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₉	—	2	1	no	—
77	C₄⋊Dic₈₉	—	2	1	no	—
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	C₄×Dic₈₉	D₄	1	1	yes	characteristic
94	C₄×Dic₈₉	D₄	1	1	yes	characteristic
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	—	2	1	no	—
109	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₈₉	—	2	1	no	—
111	Q₈×C₈₉	—	2	1	no	—
112	Q₈×C₈₉	—	2	1	no	—
113	Q₈×C₈₉	#41	1	2	yes	—
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	—	2	1	no	—
118	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
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	D₃₅₆	—	2	4	no	—
128	D₃₅₆	—	2	1	no	—
129	D₃₅₆	#41	1	2	yes	—
130	C₄×D₈₉	—	2	4	no	—
131	C₄×D₈₉	—	2	2	no	—
132	C₄×D₈₉	—	2	1	no	—
133	C₄×D₈₉	—	2	1	no	—
134	C₄×D₈₉	—	2	2	no	—
135	C₄×D₈₉	—	4	2	no	—
136	C₄×D₈₉	—	4	2	no	—
137	Dic₁₇₈	—	2	1	no	—
138	Dic₁₇₈	—	2	1	no	—
139	Dic₁₇₈	—	4	2	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	#10623494	—	2	1	no	—
143	#10623494	—	2	1	no	—
144	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₈₉	—	2	1	no	—
146	C₂×D₈₉	—	2	2	no	—
147	C₂×D₈₉	—	2	1	no	—
148	C₂×D₈₉	—	2	1	no	—
149	C₂×D₈₉	—	4	1	no	—
150	C₂×D₈₉	—	4	1	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	4	1	no	—
154	C₄×C₈₉	—	4	1	no	—
155	C₄×C₈₉	—	4	1	no	—
156	C₄×C₈₉	#137	1	2	yes	—
157	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
158	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
159	Dic₈₉	—	2	1	no	—
160	Dic₈₉	—	2	1	no	—
161	Dic₈₉	—	4	1	no	—
162	Dic₈₉	—	4	1	no	—
163	Dic₈₉	—	4	1	no	—
164	Dic₈₉	#135	1	2	yes	—
165	C₂×C₈₉	#525	1	1	yes	characteristic
166	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
168	D₈₉	—	4	1	no	—
169	D₈₉	—	4	2	no	—
170	#3107	—	89	1	no	maximal, Sylow
171	C₈₉	#3107	1	1	yes	characteristic, complemented, Sylow
172	#533	—	89	1	no	—
173	C₄×#41	—	89	1	no	—
174	#485	—	89	1	no	—
175	#485	—	89	1	no	—
176	#475	—	89	1	no	—
177	#474	—	89	1	no	—
178	#462	—	89	1	no	—
179	#461	—	89	1	no	—
180	#452	—	89	1	no	—
181	#452	—	89	1	no	—
182	C₄×Q₁₆	—	89	1	no	—
183	C₄×Q₁₆	D₈₉	1	1	yes	characteristic, complemented
184	C₄×SD₁₆	—	89	1	no	—
185	C₄×SD₁₆	—	89	1	no	—
186	#431	—	89	1	no	—
187	C₂×#41	—	89	1	no	—
188	C₂×Q₁₆	—	89	1	no	—
189	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
190	C₂×SD₁₆	—	89	1	no	—
191	C₂×SD₁₆	—	89	1	no	—
192	C₄⋊Q₈	—	89	1	no	—
193	C₄⋊Q₈	—	89	1	no	—
194	#125	—	178	1	no	—
195	C₂²⋊Q₈	—	89	1	no	—
196	C₂²⋊Q₈	—	89	1	no	—
197	C₂²⋊Q₈	—	178	1	no	—
198	C₄×Q₈	—	89	1	no	—
199	C₄×Q₈	—	89	1	no	—
200	C₄×Q₈	—	178	1	no	—
201	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
202	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
203	C₄×D₄	—	89	1	no	—
204	C₄×D₄	—	89	1	no	—
205	C₄×D₄	—	178	1	no	—
206	C₄×D₄	—	178	1	no	—
207	#117	—	89	1	no	—
208	#117	—	178	1	no	—
209	#117	—	178	1	no	—
210	C₂×C₄²	—	89	1	no	—
211	C₂×C₄²	—	178	1	no	—
212	#107	—	89	1	no	—
213	#107	C₂×D₈₉	1	1	yes	characteristic
214	#106	—	89	1	no	—
215	#106	—	89	1	no	—
216	C₄⋊C₈	—	89	1	no	—
217	C₄⋊C₈	—	89	1	no	—
218	#103	—	89	1	no	—
219	#103	—	89	1	no	—
220	#103	—	89	1	no	—
221	#103	—	89	1	no	—
222	#103	C₂×D₈₉	1	1	yes	characteristic
223	#103	C₂×D₈₉	1	1	yes	characteristic
224	#102	—	89	1	no	—
225	#102	—	89	1	no	—
226	C₂²⋊C₈	—	89	1	no	—
227	C₂²⋊C₈	—	89	1	no	—
228	C₄×C₈	—	89	1	no	—
229	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
230	#41	—	178	2	no	—
231	#41	—	178	2	no	—
232	C₂×Q₈	—	89	1	no	—
233	C₂×Q₈	—	178	1	no	—
234	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
236	C₂×D₄	—	89	1	no	—
237	C₂×D₄	—	178	1	no	—
238	C₂²×C₄	—	89	1	no	—
239	C₂²×C₄	—	89	1	no	—
240	C₂²×C₄	—	89	1	no	—
241	C₂²×C₄	—	178	1	no	—
242	C₂²×C₄	—	178	1	no	—
243	C₂²×C₄	—	178	1	no	—
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	178	2	no	—
246	SD₁₆	—	178	2	no	—
247	SD₁₆	—	178	2	no	—
248	C₂×C₈	—	89	1	no	—
249	C₂×C₈	—	89	1	no	—
250	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	89	1	no	—
255	C₄⋊C₄	—	89	1	no	—
256	C₄⋊C₄	—	178	1	no	—
257	C₄⋊C₄	—	178	1	no	—
258	C₄⋊C₄	—	178	1	no	—
259	C₄⋊C₄	—	178	1	no	—
260	C₄⋊C₄	—	178	1	no	—
261	C₄⋊C₄	—	178	1	no	—
262	C₄⋊C₄	—	178	1	no	—
263	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
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	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	89	1	no	—
273	C₄²	—	89	1	no	—
274	C₄²	—	89	1	no	—
275	C₄²	—	178	1	no	—
276	C₄²	—	178	1	no	—
277	C₄²	—	178	1	no	—
278	C₄²	C₂²×D₈₉	1	1	yes	characteristic
279	C₂³	—	89	1	no	—
280	C₂³	—	178	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	356	2	no	—
285	Q₈	—	178	1	no	—
286	Q₈	—	178	1	no	—
287	Q₈	#13083574	1	2	yes	—
288	D₄	—	89	2	no	—
289	D₄	—	178	4	no	—
290	D₄	—	178	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	89	1	no	—
296	C₂×C₄	—	89	1	no	—
297	C₂×C₄	—	89	1	no	—
298	C₂×C₄	—	356	2	no	—
299	C₂×C₄	—	356	2	no	—
300	C₂×C₄	—	178	2	no	—
301	C₂×C₄	—	178	4	no	—
302	C₂×C₄	—	178	1	no	—
303	C₂×C₄	—	178	1	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	—	178	2	no	—
306	C₂×C₄	—	178	1	no	—
307	C₂×C₄	—	178	1	no	—
308	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
309	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	178	1	no	—
314	C₈	—	178	1	no	—
315	C₂²	—	356	1	no	—
316	C₂²	—	356	1	no	—
317	C₂²	—	178	1	no	—
318	C₂²	—	178	2	no	—
319	C₂²	—	178	1	no	—
320	C₂²	—	178	1	no	—
321	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	89	2	no	—
328	C₄	—	356	1	no	—
329	C₄	—	356	1	no	—
330	C₄	—	356	1	no	—
331	C₄	—	178	1	no	—
332	C₄	—	178	1	no	—
333	C₄	#15668044	1	2	yes	—
334	C₄	#41×D₈₉	1	1	yes	characteristic
335	C₄	C₂×#13083574	1	1	yes	characteristic
336	C₂	—	356	2	no	—
337	C₂	—	356	1	no	—
338	C₂	#19152107	1	1	yes	characteristic
339	C₂	#135×D₈₉	1	1	yes	characteristic
340	C₂	C₂×#15668044	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	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	C₂×#13083575	C₂²
hypercenter	C₄×Q₁₆	D₈₉
hypocenter	C₈₉	#3107
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #19152152	C₂	semidirect
3. #19151941	C₂	semidirect
4. #19151937	C₂	semidirect
5. #19151881	C₂	semidirect
6. #19151877	C₂	semidirect
7. #19151806	C₂	semidirect
8. #19151802	C₂	semidirect
11. C₄×#13083564	C₂	semidirect
12. C₄×#13083563	C₂	semidirect
13. C₄×#13083561	C₂	semidirect
14. #19151649	C₂	semidirect
15. C₄×Q₁₆×C₈₉	C₂	semidirect
171. C₈₉	#3107	semidirect
183. 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
#19152152	2
C₄×#13083575	2
#19151941	2
#19151937	2
#19151881	2
#19151877	2
#19151806	2
#19151802	2
#19151763	2
#19151759	2
C₄×#13083564	2
C₄×#13083563	2
C₄×#13083561	2
#19151649	2
C₄×Q₁₆×C₈₉	2
#3107	89