Subgroups — #26032378

Order 11392.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152288	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151962	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151919	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151914	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151905	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151842	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151825	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151795	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151760	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151760	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151703	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151670	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151599	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151594	C₂	1	1	yes	maximal, characteristic, complemented
15	#550×C₈₉	C₂	1	1	yes	maximal, characteristic
16	C₂×#13083564	C₂²	1	1	yes	characteristic
17	C₂×#13083562	C₂²	1	1	yes	characteristic
18	C₂×#13083562	C₂²	1	1	yes	characteristic
19	C₂×#13083558	C₂²	1	1	yes	characteristic
20	C₂²×#10623494	C₂²	1	1	yes	characteristic
21	#15668005	—	2	1	no	—
22	#15668004	C₂²	1	1	yes	characteristic
23	#15668002	—	2	1	no	—
24	#15667993	—	2	1	no	—
25	#15667991	C₂²	1	1	yes	characteristic
26	#15667991	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
28	#15667971	C₂²	1	1	yes	characteristic
29	#15667968	—	2	1	no	—
30	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
31	#15667945	C₂²	1	1	yes	characteristic
32	#15667943	C₂²	1	1	yes	characteristic
33	#15667943	C₂²	1	1	yes	characteristic
34	C₄⋊#10623494	C₂²	1	1	yes	characteristic
35	#15667936	C₂²	1	1	yes	characteristic
36	#15667936	C₂²	1	1	yes	characteristic
37	#15667936	C₂²	1	1	yes	characteristic
38	#15667935	C₂²	1	1	yes	characteristic
39	#15667933	C₂²	1	1	yes	characteristic
40	#15667933	C₂²	1	1	yes	characteristic
41	#15667933	C₂²	1	1	yes	characteristic
42	#15667933	C₂²	1	1	yes	characteristic
43	#15667933	C₂²	1	1	yes	characteristic
44	#15667933	C₂²	1	1	yes	characteristic
45	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
46	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
47	#15667923	C₂²	1	1	yes	characteristic
48	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
49	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
50	#127×C₈₉	—	2	1	no	—
51	#127×C₈₉	C₂²	1	1	yes	characteristic
52	#121×C₈₉	—	2	1	no	—
53	#121×C₈₉	—	2	1	no	—
54	#121×C₈₉	C₂²	1	1	yes	characteristic
55	#121×C₈₉	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₈₉	—	2	1	no	—
58	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
60	C₂²×Dic₈₉	—	2	1	no	—
61	#13083564	—	2	2	no	—
62	#13083562	—	4	1	no	—
63	#13083562	—	4	1	no	—
64	#13083558	—	4	1	no	—
65	C₂×#10623494	—	4	1	no	—
66	C₂×#10623494	C₂³	1	1	yes	characteristic
67	C₂×#10623494	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
74	#13083553	—	2	1	no	—
75	#13083553	—	2	1	no	—
76	#13083553	—	2	1	no	—
77	#13083553	—	2	1	no	—
78	#13083553	—	2	1	no	—
79	C₂²⋊Dic₈₉	—	2	1	no	—
80	C₂²⋊Dic₈₉	—	2	1	no	—
81	C₂²⋊Dic₈₉	—	2	1	no	—
82	C₄×Dic₈₉	—	2	1	no	—
83	C₄×Dic₈₉	—	2	1	no	—
84	#41×C₈₉	—	2	2	no	—
85	#41×C₈₉	—	2	2	no	—
86	#41×C₈₉	—	4	1	no	—
87	#41×C₈₉	—	4	1	no	—
88	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
89	C₂×D₄×C₈₉	—	2	1	no	—
90	C₂×D₄×C₈₉	—	2	1	no	—
91	C₂×D₄×C₈₉	—	2	1	no	—
92	C₂×D₄×C₈₉	—	2	1	no	—
93	C₂×D₄×C₈₉	—	2	1	no	—
94	C₂×D₄×C₈₉	—	2	1	no	—
95	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
96	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	—	2	1	no	—
99	C₂²×C₄×C₈₉	—	2	1	no	—
100	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	—	2	1	no	—
103	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
108	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
109	C₄²×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂³×C₈₉	—	2	1	no	—
112	C₂³×C₈₉	—	2	1	no	—
113	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₈₉	—	2	1	no	—
116	Q₈×C₈₉	#41	1	2	yes	—
117	D₄×C₈₉	—	2	1	no	—
118	D₄×C₈₉	—	2	1	no	—
119	D₄×C₈₉	—	2	1	no	—
120	D₄×C₈₉	—	2	1	no	—
121	D₄×C₈₉	—	4	1	no	—
122	D₄×C₈₉	—	4	2	no	—
123	D₄×C₈₉	—	4	2	no	—
124	D₄×C₈₉	—	4	2	no	—
125	D₄×C₈₉	—	8	1	no	—
126	D₄×C₈₉	—	8	1	no	—
127	D₄×C₈₉	—	8	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	—	4	2	no	—
136	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
141	C₂×Dic₈₉	—	2	1	no	—
142	C₂×Dic₈₉	—	2	1	no	—
143	C₂×Dic₈₉	—	2	1	no	—
144	C₂×Dic₈₉	—	2	1	no	—
145	C₂×Dic₈₉	—	4	2	no	—
146	Dic₁₇₈	—	2	1	no	—
147	Dic₁₇₈	—	2	1	no	—
148	#10623494	—	2	1	no	—
149	#10623494	—	2	1	no	—
150	#10623494	—	2	1	no	—
151	#10623494	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	—	2	1	no	—
154	C₂²×C₈₉	—	2	1	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	—	4	1	no	—
161	C₂²×C₈₉	—	4	1	no	—
162	C₂²×C₈₉	—	4	1	no	—
163	C₂²×C₈₉	—	4	1	no	—
164	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	4	1	no	—
171	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
172	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
173	Dic₈₉	—	4	1	no	—
174	Dic₈₉	—	4	1	no	—
175	Dic₈₉	—	4	1	no	—
176	Dic₈₉	—	4	1	no	—
177	C₂×C₈₉	—	4	1	no	—
178	C₂×C₈₉	—	4	1	no	—
179	C₂×C₈₉	—	8	1	no	—
180	C₂×C₈₉	—	8	1	no	—
181	C₂×C₈₉	#546	1	1	yes	characteristic
182	C₂×C₈₉	#575	1	1	yes	characteristic
183	C₂×C₈₉	#577	1	1	yes	characteristic
184	#3317	—	89	1	no	maximal, Sylow
185	C₈₉	#3317	1	1	yes	characteristic, complemented, Sylow
186	#552	—	89	1	no	—
187	#550	D₈₉	1	1	yes	characteristic
188	#490	—	89	1	no	—
189	#482	—	89	1	no	—
190	#481	—	89	1	no	—
191	#479	—	89	1	no	—
192	#468	—	89	1	no	—
193	#465	—	89	1	no	—
194	#460	—	89	1	no	—
195	#452	—	89	1	no	—
196	#452	—	89	1	no	—
197	#441	—	89	1	no	—
198	#435	—	89	1	no	—
199	#417	—	89	1	no	—
200	#416	—	89	1	no	—
201	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
202	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
203	C₂×Q₁₆	—	89	1	no	—
204	C₂×SD₁₆	—	89	1	no	—
205	C₂×SD₁₆	—	89	1	no	—
206	C₂×M₁₆	—	89	1	no	—
207	C₂²×C₈	—	89	1	no	—
208	#127	—	2	1	no	—
209	#127	C₂×D₈₉	1	1	yes	characteristic
210	#126	—	178	1	no	—
211	#125	—	89	1	no	—
212	#125	—	178	1	no	—
213	#123	—	178	1	no	—
214	C₂²⋊Q₈	—	89	1	no	—
215	C₂²⋊Q₈	—	89	1	no	—
216	#121	—	2	1	no	—
217	#121	—	2	1	no	—
218	#121	C₂×D₈₉	1	1	yes	characteristic
219	#121	C₂×D₈₉	1	1	yes	characteristic
220	C₄×Q₈	—	89	1	no	—
221	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
222	C₄×D₄	—	2	1	no	—
223	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
224	#117	—	89	1	no	—
225	#117	—	178	1	no	—
226	C₂×C₄⋊C₄	—	89	1	no	—
227	#107	—	89	1	no	—
228	#106	—	89	1	no	—
229	#106	—	89	1	no	—
230	C₄⋊C₈	—	89	1	no	—
231	#103	—	89	1	no	—
232	#103	—	89	1	no	—
233	#103	—	89	1	no	—
234	#103	—	89	1	no	—
235	#102	—	89	1	no	—
236	#102	—	89	1	no	—
237	#102	—	89	1	no	—
238	#102	—	89	1	no	—
239	#102	—	89	1	no	—
240	#102	—	89	1	no	—
241	C₂²⋊C₈	—	89	1	no	—
242	C₂²⋊C₈	—	89	1	no	—
243	#97	—	89	1	no	—
244	#41	—	2	2	no	—
245	#41	—	2	2	no	—
246	#41	—	4	1	no	—
247	#41	—	4	1	no	—
248	C₂×Q₈	—	89	1	no	—
249	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
257	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
258	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	178	1	no	—
262	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
264	Q₁₆	—	178	2	no	—
265	SD₁₆	—	356	1	no	—
266	SD₁₆	—	356	1	no	—
267	M₁₆	—	356	1	no	—
268	C₂×C₈	—	89	1	no	—
269	C₂×C₈	—	89	1	no	—
270	C₂×C₈	—	89	1	no	—
271	C₂×C₈	—	89	1	no	—
272	C₂×C₈	—	356	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	89	1	no	—
275	C₄⋊C₄	—	89	1	no	—
276	C₄⋊C₄	—	89	1	no	—
277	C₄⋊C₄	—	178	1	no	—
278	C₄⋊C₄	—	178	1	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₄⋊C₄	—	178	1	no	—
281	C₄⋊C₄	—	178	1	no	—
282	C₄⋊C₄	—	178	1	no	—
283	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
284	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	2	1	no	—
287	C₂²⋊C₄	—	178	1	no	—
288	C₂²⋊C₄	—	178	1	no	—
289	C₂²⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
291	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
292	C₄²	—	2	1	no	—
293	C₄²	—	178	1	no	—
294	C₄²	—	178	1	no	—
295	C₄²	C₂²×D₈₉	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
299	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
300	Q₈	—	2	1	no	—
301	Q₈	—	178	1	no	—
302	Q₈	—	178	1	no	—
303	Q₈	#13083573	1	2	yes	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	4	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	8	1	no	—
313	D₄	—	8	1	no	—
314	D₄	—	8	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	356	2	no	—
324	C₂×C₄	—	178	1	no	—
325	C₂×C₄	—	178	1	no	—
326	C₂×C₄	—	178	1	no	—
327	C₂×C₄	—	178	1	no	—
328	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
330	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
331	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
333	C₈	—	178	1	no	—
334	C₈	—	178	1	no	—
335	C₈	—	178	1	no	—
336	C₈	—	178	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	356	1	no	—
357	C₄	—	356	1	no	—
358	C₄	—	356	1	no	—
359	C₄	—	356	1	no	—
360	C₄	C₂×#13083573	1	1	yes	characteristic
361	C₄	#15668057	1	1	yes	characteristic
362	C₂	—	4	1	no	—
363	C₂	—	4	1	no	—
364	C₂	—	8	1	no	—
365	C₂	—	8	1	no	—
366	C₂	#19152242	1	1	yes	characteristic
367	C₂	#19152396	1	1	yes	characteristic
368	C₂	#19152409	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting, prime radical	#550×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#550	D₈₉
hypocenter	C₈₉	#3317
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #550

Complements

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

N	H ≅ G/N	type
1. #19152288	C₂	semidirect
2. #19151962	C₂	semidirect
3. #19151919	C₂	semidirect
4. #19151914	C₂	semidirect
5. #19151905	C₂	semidirect
6. #19151842	C₂	semidirect
7. #19151825	C₂	semidirect
8. #19151795	C₂	semidirect
9. #19151760	C₂	semidirect
10. #19151760	C₂	semidirect
11. #19151703	C₂	semidirect
12. #19151670	C₂	semidirect
13. #19151599	C₂	semidirect
14. #19151594	C₂	semidirect
185. C₈₉	#3317	semidirect

Maximal subgroups

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

subgroup	index
#19152288	2
#19151962	2
#19151919	2
#19151914	2
#19151905	2
#19151842	2
#19151825	2
#19151795	2
#19151760	2
#19151703	2
#19151670	2
#19151599	2
#19151594	2
#550×C₈₉	2
#3317	89