Subgroups — #26030380

Order 11392.

380 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	#19152152	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151930	C₂	1	1	yes	maximal, characteristic
4	#19151927	C₂	1	1	yes	maximal, characteristic
5	#19151882	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151881	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	#19151706	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151703	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151703	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151653	C₂	1	1	yes	maximal, characteristic
15	#441×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	#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	—	2	1	no	—
29	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
31	#15667977	—	2	1	no	—
32	#15667977	C₂²	1	1	yes	characteristic
33	#15667969	—	2	1	no	—
34	C₄⋊C₄×D₈₉	—	2	1	no	—
35	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
36	C₄²×D₈₉	—	2	1	no	—
37	C₄²×D₈₉	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	#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	#106×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	Q₈×D₈₉	—	2	2	no	—
64	Q₈×D₈₉	—	2	2	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
68	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
70	Dic₃₅₆	—	2	2	no	—
71	#13083564	—	2	2	no	—
72	#13083564	—	2	2	no	—
73	C₂×#10623494	C₂³	1	1	yes	characteristic
74	C₂×#10623494	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₉	—	2	1	no	—
76	C₄⋊Dic₈₉	—	2	1	no	—
77	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	#13083553	—	2	1	no	—
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	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	C₄×Dic₈₉	—	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	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	2	no	—
129	C₄×D₈₉	—	2	1	no	—
130	C₄×D₈₉	—	2	1	no	—
131	C₄×D₈₉	—	2	2	no	—
132	C₄×D₈₉	—	4	2	no	—
133	Dic₁₇₈	—	2	4	no	—
134	Dic₁₇₈	—	2	1	no	—
135	Dic₁₇₈	—	2	1	no	—
136	Dic₁₇₈	—	2	1	no	—
137	Dic₁₇₈	—	4	2	no	—
138	Dic₁₇₈	#41	1	2	yes	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	#10623494	—	2	1	no	—
142	#10623494	—	2	1	no	—
143	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₈₉	—	2	1	no	—
145	C₂×D₈₉	—	2	1	no	—
146	C₂×D₈₉	#137	1	2	yes	—
147	C₄×C₈₉	—	2	2	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	4	1	no	—
151	C₄×C₈₉	—	4	1	no	—
152	C₄×C₈₉	—	4	1	no	—
153	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
154	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
155	Dic₈₉	—	2	1	no	—
156	Dic₈₉	—	2	2	no	—
157	Dic₈₉	—	2	1	no	—
158	Dic₈₉	—	4	1	no	—
159	Dic₈₉	—	4	1	no	—
160	Dic₈₉	—	4	1	no	—
161	Dic₈₉	#137	1	2	yes	—
162	C₂×C₈₉	#525	1	1	yes	characteristic
163	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
164	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
165	D₈₉	—	2	2	no	—
166	#3113	—	89	1	no	maximal, Sylow
167	C₈₉	#3113	1	1	yes	characteristic, complemented, Sylow
168	#533	—	89	1	no	—
169	C₂×C₄×Q₈	—	89	1	no	—
170	#483	—	89	1	no	—
171	#483	—	89	1	no	—
172	#475	—	89	1	no	—
173	#475	—	89	1	no	—
174	#462	—	89	1	no	—
175	#462	—	89	1	no	—
176	#451	—	89	1	no	—
177	#451	—	89	1	no	—
178	#441	—	89	1	no	—
179	#441	—	89	1	no	—
180	#441	—	89	1	no	—
181	#441	D₈₉	1	1	yes	characteristic, complemented
182	#432	—	89	1	no	—
183	C₂²×Q₈	—	89	1	no	—
184	C₂×Q₁₆	—	89	1	no	—
185	C₂×Q₁₆	—	89	1	no	—
186	C₂×Q₁₆	—	89	1	no	—
187	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
188	C₄⋊Q₈	—	89	1	no	—
189	C₄⋊Q₈	—	89	1	no	—
190	#125	—	178	1	no	—
191	C₂²⋊Q₈	—	89	1	no	—
192	C₂²⋊Q₈	—	89	1	no	—
193	C₂²⋊Q₈	—	178	1	no	—
194	C₄×Q₈	—	89	1	no	—
195	C₄×Q₈	—	89	1	no	—
196	C₄×Q₈	—	89	1	no	—
197	C₄×Q₈	—	89	1	no	—
198	C₄×Q₈	—	178	1	no	—
199	C₄×Q₈	—	178	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	#117	—	178	1	no	—
204	C₂×C₄⋊C₄	—	89	1	no	—
205	C₂×C₄⋊C₄	—	178	1	no	—
206	C₂×C₄²	—	89	1	no	—
207	C₂×C₄²	—	178	1	no	—
208	#106	—	89	1	no	—
209	#106	—	89	1	no	—
210	#106	—	89	1	no	—
211	#106	C₂×D₈₉	1	1	yes	characteristic
212	C₄⋊C₈	—	89	1	no	—
213	C₄⋊C₈	—	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	—	89	1	no	—
220	#103	C₂×D₈₉	1	1	yes	characteristic
221	#103	C₂×D₈₉	1	1	yes	characteristic
222	C₂²⋊C₈	—	89	1	no	—
223	C₂²⋊C₈	—	89	1	no	—
224	#97	—	89	1	no	—
225	#97	C₂×D₈₉	1	1	yes	characteristic
226	C₂×Q₈	—	89	1	no	—
227	C₂×Q₈	—	89	1	no	—
228	C₂×Q₈	—	178	1	no	—
229	C₂×Q₈	—	178	1	no	—
230	C₂×Q₈	—	178	2	no	—
231	C₂×Q₈	—	178	2	no	—
232	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
234	C₂²×C₄	—	89	1	no	—
235	C₂²×C₄	—	89	1	no	—
236	C₂²×C₄	—	89	1	no	—
237	C₂²×C₄	—	178	1	no	—
238	C₂²×C₄	—	178	1	no	—
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	178	2	no	—
241	Q₁₆	—	178	2	no	—
242	Q₁₆	—	178	2	no	—
243	C₂×C₈	—	89	1	no	—
244	C₂×C₈	—	89	1	no	—
245	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	89	1	no	—
250	C₄⋊C₄	—	89	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	—	178	1	no	—
260	C₄⋊C₄	—	178	1	no	—
261	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
263	C₂²⋊C₄	—	178	1	no	—
264	C₂²⋊C₄	—	178	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	89	1	no	—
268	C₄²	—	89	1	no	—
269	C₄²	—	89	1	no	—
270	C₄²	—	178	1	no	—
271	C₄²	—	178	1	no	—
272	C₄²	—	178	1	no	—
273	C₄²	—	178	1	no	—
274	C₄²	C₂²×D₈₉	1	1	yes	characteristic
275	C₂³	—	89	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	89	2	no	—
280	Q₈	—	356	2	no	—
281	Q₈	—	178	4	no	—
282	Q₈	—	178	1	no	—
283	Q₈	—	178	1	no	—
284	Q₈	—	178	1	no	—
285	Q₈	#13083574	1	2	yes	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	89	1	no	—
291	C₂×C₄	—	89	1	no	—
292	C₂×C₄	—	89	1	no	—
293	C₂×C₄	—	356	2	no	—
294	C₂×C₄	—	178	2	no	—
295	C₂×C₄	—	178	2	no	—
296	C₂×C₄	—	178	4	no	—
297	C₂×C₄	—	178	1	no	—
298	C₂×C₄	—	178	1	no	—
299	C₂×C₄	—	178	1	no	—
300	C₂×C₄	—	178	1	no	—
301	C₂×C₄	—	178	1	no	—
302	C₂×C₄	—	178	1	no	—
303	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
304	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	178	1	no	—
309	C₈	—	178	1	no	—
310	C₂²	—	89	2	no	—
311	C₂²	—	178	1	no	—
312	C₂²	—	178	1	no	—
313	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	89	2	no	—
321	C₄	—	356	1	no	—
322	C₄	—	356	1	no	—
323	C₄	—	356	1	no	—
324	C₄	—	178	1	no	—
325	C₄	—	178	1	no	—
326	C₄	—	178	2	no	—
327	C₄	#41×D₈₉	1	1	yes	characteristic
328	C₄	C₂×#13083574	1	1	yes	characteristic
329	C₂	—	178	2	no	—
330	C₂	#19152107	1	1	yes	characteristic
331	C₂	#137×D₈₉	1	1	yes	characteristic
332	C₂	#137×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#441×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂²×D₈₉	C₂×D₄
hypercenter	#441	D₈₉
hypocenter	C₈₉	#3113
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
5. #19151882	C₂	semidirect
6. #19151881	C₂	semidirect
7. #19151807	C₂	semidirect
8. #19151806	C₂	semidirect
11. #19151706	C₂	semidirect
12. #19151703	C₂	semidirect
13. #19151703	C₂	semidirect
15. #441×C₈₉	C₂	semidirect
167. C₈₉	#3113	semidirect
181. #441	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×Q₈×D₈₉	2
#19151930	2
#19151927	2
#19151882	2
#19151881	2
#19151807	2
#19151806	2
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