Subgroups — #26030258

Order 11392.

448 conjugacy classes of subgroups.

The subgroup lattice has 362 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×D₄×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151930	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151927	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151887	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151886	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151802	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151801	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151750	C₂	1	1	yes	maximal, characteristic
10	#19151747	C₂	1	1	yes	maximal, characteristic
11	C₄×#13083563	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13083561	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13083562	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151649	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₈₉	—	2	1	no	—
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	C₂×#13083561	C₂²	1	1	yes	characteristic
21	#15668011	C₂²	1	1	yes	characteristic, complemented
22	#15668010	C₂²	1	1	yes	characteristic, complemented
23	#15668002	—	2	1	no	—
24	#15667992	C₂²	1	1	yes	characteristic
25	#15667990	—	2	1	no	—
26	#15667988	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₇₈	—	2	1	no	—
28	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic, complemented
29	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
30	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
31	#15667974	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₈₉	—	2	1	no	—
33	#15667972	—	2	1	no	—
34	#15667969	—	2	1	no	—
35	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
36	C₂²⋊C₄×D₈₉	—	2	1	no	—
37	C₄²×D₈₉	C₂²	1	1	yes	characteristic
38	#15667944	C₂²	1	1	yes	characteristic, complemented
39	#15667943	C₂²	1	1	yes	characteristic, complemented
40	#15667943	C₂²	1	1	yes	characteristic, complemented
41	#15667941	C₂²	1	1	yes	characteristic, complemented
42	#15667941	C₂²	1	1	yes	characteristic, complemented
43	#15667937	C₂²	1	1	yes	characteristic, complemented
44	#15667936	C₂²	1	1	yes	characteristic, complemented
45	#15667935	C₂²	1	1	yes	characteristic, complemented
46	#15667934	C₂²	1	1	yes	characteristic
47	#15667933	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, complemented
52	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
54	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₈₉	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₉	—	2	1	no	—
64	D₄×D₈₉	—	2	2	no	—
65	D₄×D₈₉	—	2	2	no	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	4	4	no	—
71	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
72	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
74	#13083563	—	2	2	no	—
75	#13083562	—	2	2	no	—
76	#13083561	—	2	2	no	—
77	C₂×#10623494	C₂³	1	1	yes	characteristic
78	C₂×#10623494	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₉	—	2	1	no	—
80	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
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	#13083553	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	D₄	1	1	yes	characteristic
95	C₄×Dic₈₉	D₄	1	1	yes	characteristic
96	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	—	2	1	no	—
100	SD₁₆×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	C₄²×C₈₉	—	2	1	no	—
108	C₄²×C₈₉	C₂³	1	1	yes	characteristic
109	C₂³×C₈₉	—	2	1	no	—
110	C₂²×D₈₉	—	2	4	no	—
111	C₂²×D₈₉	—	2	1	no	—
112	C₂²×D₈₉	—	4	2	no	—
113	C₂²×D₈₉	—	4	2	no	—
114	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₈₉	—	2	1	no	—
116	Q₈×C₈₉	—	2	1	no	—
117	D₄×C₈₉	—	2	1	no	—
118	D₄×C₈₉	#41	1	2	yes	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₈₉	—	2	4	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	2	1	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	D₃₅₆	—	2	1	no	—
135	D₃₅₆	#41	1	2	yes	—
136	C₄×D₈₉	—	2	1	no	—
137	C₄×D₈₉	—	2	1	no	—
138	C₄×D₈₉	—	4	1	no	—
139	C₄×D₈₉	—	4	2	no	—
140	C₄×D₈₉	—	4	2	no	—
141	C₄×D₈₉	—	4	2	no	—
142	C₄×D₈₉	SD₁₆	1	4	yes	complemented
143	Dic₁₇₈	—	2	1	no	—
144	Dic₁₇₈	—	2	1	no	—
145	Dic₁₇₈	—	4	2	no	—
146	C₈×C₈₉	—	2	1	no	—
147	C₈×C₈₉	—	2	1	no	—
148	#10623494	—	2	1	no	—
149	#10623494	—	2	1	no	—
150	C₂²×C₈₉	—	2	2	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₈₉	—	2	1	no	—
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	2	2	no	—
157	C₂×D₈₉	—	4	8	no	—
158	C₂×D₈₉	—	4	1	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	C₂×SD₁₆	1	2	yes	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	C₄×C₈₉	—	4	1	no	—
165	C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
166	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
167	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
168	Dic₈₉	—	2	1	no	—
169	Dic₈₉	—	2	1	no	—
170	Dic₈₉	—	4	1	no	—
171	Dic₈₉	—	4	1	no	—
172	Dic₈₉	—	4	1	no	—
173	Dic₈₉	C₂×SD₁₆	1	2	yes	—
174	C₂×C₈₉	—	4	2	no	—
175	C₂×C₈₉	#525	1	1	yes	characteristic
176	C₂×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
177	C₂×C₈₉	#575	1	1	yes	characteristic
178	D₈₉	—	2	2	no	—
179	D₈₉	—	4	2	no	—
180	#3101	—	89	1	no	maximal, Sylow
181	C₈₉	#3101	1	1	yes	characteristic, complemented, Sylow
182	#533	—	89	1	no	—
183	C₂×C₄×D₄	—	89	1	no	—
184	#483	—	89	1	no	—
185	#483	—	89	1	no	—
186	#476	—	89	1	no	—
187	#476	—	89	1	no	—
188	#461	—	89	1	no	—
189	#461	—	89	1	no	—
190	#450	—	89	1	no	—
191	#450	—	89	1	no	—
192	C₄×SD₁₆	—	89	1	no	—
193	C₄×SD₁₆	—	89	1	no	—
194	C₄×SD₁₆	—	89	1	no	—
195	C₄×SD₁₆	D₈₉	1	1	yes	characteristic, complemented
196	#431	—	89	1	no	—
197	C₂²×D₄	—	89	1	no	—
198	C₂³×C₄	—	178	1	no	—
199	C₂×SD₁₆	—	89	1	no	—
200	C₂×SD₁₆	—	89	1	no	—
201	C₂×SD₁₆	—	89	1	no	—
202	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
203	C₄⋊Q₈	—	89	1	no	—
204	C₄⋊Q₈	—	89	1	no	—
205	#125	—	178	1	no	—
206	C₂²⋊Q₈	—	89	1	no	—
207	C₂²⋊Q₈	—	89	1	no	—
208	C₂²⋊Q₈	—	178	1	no	—
209	C₄×Q₈	—	89	1	no	—
210	C₄×Q₈	—	178	1	no	—
211	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
212	C₄×D₄	—	89	1	no	—
213	C₄×D₄	—	89	1	no	—
214	C₄×D₄	—	89	1	no	—
215	C₄×D₄	—	178	1	no	—
216	C₄×D₄	—	178	1	no	—
217	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
218	#117	—	178	1	no	—
219	C₂×C₄⋊C₄	—	89	1	no	—
220	C₂×C₂²⋊C₄	—	178	1	no	—
221	C₂×C₄²	—	89	1	no	—
222	#106	—	89	1	no	—
223	#106	—	89	1	no	—
224	#106	—	89	1	no	—
225	#106	C₂×D₈₉	1	1	yes	characteristic, complemented
226	C₄⋊C₈	—	89	1	no	—
227	C₄⋊C₈	—	89	1	no	—
228	#103	—	89	1	no	—
229	#103	—	89	1	no	—
230	#103	—	89	1	no	—
231	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
232	#102	—	89	1	no	—
233	#102	—	89	1	no	—
234	#102	—	89	1	no	—
235	#102	C₂×D₈₉	1	1	yes	characteristic
236	C₂²⋊C₈	—	89	1	no	—
237	C₂²⋊C₈	—	89	1	no	—
238	C₄×C₈	—	89	1	no	—
239	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
240	C₂⁴	—	178	1	no	—
241	C₂×Q₈	—	89	1	no	—
242	C₂×Q₈	—	178	1	no	—
243	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₂×D₄	—	89	1	no	—
245	C₂×D₄	—	178	1	no	—
246	C₂×D₄	—	178	2	no	—
247	C₂×D₄	—	178	2	no	—
248	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	89	1	no	—
251	C₂²×C₄	—	89	1	no	—
252	C₂²×C₄	—	89	1	no	—
253	C₂²×C₄	—	356	4	no	—
254	C₂²×C₄	—	178	1	no	—
255	C₂²×C₄	—	178	1	no	—
256	C₂²×C₄	—	178	1	no	—
257	C₂²×C₄	—	178	1	no	—
258	SD₁₆	—	2	2	no	—
259	SD₁₆	—	178	2	no	—
260	SD₁₆	—	178	2	no	—
261	SD₁₆	—	178	2	no	—
262	C₂×C₈	—	89	1	no	—
263	C₂×C₈	—	89	1	no	—
264	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	89	1	no	—
268	C₄⋊C₄	—	89	1	no	—
269	C₄⋊C₄	—	178	1	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
277	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	178	1	no	—
280	C₂²⋊C₄	—	178	1	no	—
281	C₂²⋊C₄	—	178	1	no	—
282	C₂²⋊C₄	—	178	1	no	—
283	C₂²⋊C₄	—	178	1	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	89	1	no	—
286	C₄²	—	89	1	no	—
287	C₄²	—	89	1	no	—
288	C₄²	—	178	1	no	—
289	C₄²	C₂²×D₈₉	1	1	yes	characteristic
290	C₂³	—	2	1	no	—
291	C₂³	—	89	1	no	—
292	C₂³	—	356	2	no	—
293	C₂³	—	356	2	no	—
294	C₂³	—	178	4	no	—
295	C₂³	—	178	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	356	2	no	—
299	Q₈	—	178	1	no	—
300	Q₈	—	178	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	89	2	no	—
303	D₄	—	178	4	no	—
304	D₄	—	178	1	no	—
305	D₄	#13083574	1	2	yes	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	89	1	no	—
311	C₂×C₄	—	89	4	no	—
312	C₂×C₄	—	89	1	no	—
313	C₂×C₄	—	89	1	no	—
314	C₂×C₄	—	356	1	no	—
315	C₂×C₄	—	356	2	no	—
316	C₂×C₄	—	356	2	no	—
317	C₂×C₄	—	356	2	no	—
318	C₂×C₄	—	356	2	no	—
319	C₂×C₄	—	178	1	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	178	1	no	—
323	C₂×C₄	—	178	1	no	—
324	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
325	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	178	1	no	—
330	C₈	—	178	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	89	2	no	—
335	C₂²	—	356	8	no	—
336	C₂²	—	356	1	no	—
337	C₂²	—	356	1	no	—
338	C₂²	—	178	2	no	—
339	C₂²	—	178	1	no	—
340	C₂²	—	178	1	no	—
341	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	89	2	no	—
347	C₄	—	356	1	no	—
348	C₄	—	356	1	no	—
349	C₄	—	356	1	no	—
350	C₄	—	178	1	no	—
351	C₄	—	178	1	no	—
352	C₄	SD₁₆×D₈₉	1	2	yes	—
353	C₄	#41×D₈₉	1	1	yes	characteristic
354	C₄	C₂×#13083574	1	1	yes	characteristic
355	C₂	—	4	2	no	—
356	C₂	—	356	2	no	—
357	C₂	—	178	2	no	—
358	C₂	#19152107	1	1	yes	characteristic
359	C₂	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
360	C₂	#19152395	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×SD₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₄×D₈₉	C₂²
hypercenter	C₄×SD₁₆	D₈₉
hypocenter	C₈₉	#3101
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #19152152	C₂	semidirect
3. #19151930	C₂	semidirect
4. #19151927	C₂	semidirect
5. #19151887	C₂	semidirect
6. #19151886	C₂	semidirect
7. #19151802	C₂	semidirect
8. #19151801	C₂	semidirect
11. C₄×#13083563	C₂	semidirect
12. C₄×#13083561	C₂	semidirect
13. C₄×#13083562	C₂	semidirect
14. #19151649	C₂	semidirect
15. C₄×SD₁₆×C₈₉	C₂	semidirect
21. #15668011	C₂²	semidirect
22. #15668010	C₂²	semidirect
28. C₄×Dic₁₇₈	C₂²	semidirect
38. #15667944	C₂²	semidirect
39. #15667943	C₂²	semidirect
40. #15667943	C₂²	semidirect
41. #15667941	C₂²	semidirect
42. #15667941	C₂²	semidirect
43. #15667937	C₂²	semidirect
44. #15667936	C₂²	semidirect
45. #15667935	C₂²	semidirect
51. C₄×#10623494	C₂²	semidirect
53. C₄×Q₈×C₈₉	C₂²	semidirect
55. #106×C₈₉	C₂²	semidirect
56. #103×C₈₉	C₂²	semidirect
58. C₄×C₈×C₈₉	C₂²	semidirect
142. C₄×D₈₉	SD₁₆	semidirect
181. C₈₉	#3101	semidirect
195. C₄×SD₁₆	D₈₉	semidirect
211. C₄×Q₈	C₂×D₈₉	semidirect
225. #106	C₂×D₈₉	semidirect
231. #103	C₂×D₈₉	semidirect
239. C₄×C₈	C₂×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×D₄×D₈₉	2
#19151930	2
#19151927	2
#19151887	2
#19151886	2
#19151802	2
#19151801	2
#19151750	2
#19151747	2
C₄×#13083563	2
C₄×#13083561	2
C₄×#13083562	2
#19151649	2
C₄×SD₁₆×C₈₉	2
#3101	89