Subgroups — #26030445

Order 11392.

396 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	#19152078	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151914	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151909	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151902	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151795	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151790	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151741	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151741	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151708	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151698	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13083562	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13083559	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151659	C₂	1	1	yes	maximal, characteristic, complemented
15	#540×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂×#13083562	C₂²	1	1	yes	characteristic
18	C₂×#13083562	C₂²	1	1	yes	characteristic
19	C₂×#13083559	C₂²	1	1	yes	characteristic
20	C₂×#13083559	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
23	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
24	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
25	#15667972	—	2	1	no	—
26	#15667972	—	2	1	no	—
27	#15667970	—	2	1	no	—
28	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×D₈₉	—	2	1	no	—
30	#15667945	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	C₄⋊#10623494	C₂²	1	1	yes	characteristic
36	#15667935	C₂²	1	1	yes	characteristic
37	#15667935	C₂²	1	1	yes	characteristic
38	#15667933	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	#15667932	C₂²	1	1	yes	characteristic
43	#15667932	C₂²	1	1	yes	characteristic
44	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
45	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
46	#15667923	C₂²	1	1	yes	characteristic
47	C₄×#10623494	C₂²	1	1	yes	characteristic
48	#127×C₈₉	C₂²	1	1	yes	characteristic
49	#125×C₈₉	C₂²	1	1	yes	characteristic
50	#123×C₈₉	—	2	1	no	—
51	#123×C₈₉	—	2	1	no	—
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×D₄×C₈₉	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
58	#117×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	2	2	no	—
60	#13083574	—	2	2	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	C₂²×Dic₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	#13083562	—	2	2	no	—
69	#13083562	—	2	2	no	—
70	#13083559	—	2	2	no	—
71	#13083559	—	2	2	no	—
72	C₂×#10623494	C₂³	1	1	yes	characteristic
73	C₂×#10623494	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	#13083551	—	2	1	no	—
82	#13083551	—	2	1	no	—
83	C₄×Dic₈₉	—	2	1	no	—
84	C₄×Dic₈₉	—	2	1	no	—
85	C₂×D₄×C₈₉	—	2	1	no	—
86	C₂×D₄×C₈₉	—	2	1	no	—
87	C₂×D₄×C₈₉	—	2	1	no	—
88	C₂×D₄×C₈₉	—	2	1	no	—
89	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
90	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	—	2	1	no	—
92	C₂²×C₄×C₈₉	—	2	1	no	—
93	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₈₉	—	2	1	no	—
95	C₄⋊C₄×C₈₉	—	2	1	no	—
96	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₂²⋊C₄×C₈₉	—	2	1	no	—
101	C₂²⋊C₄×C₈₉	—	2	1	no	—
102	C₂²⋊C₄×C₈₉	—	2	1	no	—
103	C₂²⋊C₄×C₈₉	—	2	1	no	—
104	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₂³	1	1	yes	characteristic
107	C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	C₂³×C₈₉	—	2	1	no	—
109	C₂³×C₈₉	—	2	1	no	—
110	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×D₈₉	—	2	1	no	—
112	D₄×C₈₉	—	2	1	no	—
113	D₄×C₈₉	—	2	1	no	—
114	D₄×C₈₉	—	2	1	no	—
115	D₄×C₈₉	—	2	1	no	—
116	D₄×C₈₉	—	8	1	no	—
117	D₄×C₈₉	—	8	1	no	—
118	D₄×C₈₉	—	8	1	no	—
119	D₄×C₈₉	—	8	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	8	1	no	—
125	C₂×C₄×C₈₉	—	8	1	no	—
126	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₈₉	—	2	4	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	C₂×Dic₈₉	—	8	1	no	—
137	D₃₅₆	—	2	1	no	—
138	D₃₅₆	#41	1	2	yes	—
139	C₄×D₈₉	—	2	4	no	—
140	C₄×D₈₉	—	8	1	no	—
141	C₄×D₈₉	—	8	1	no	—
142	Dic₁₇₈	—	2	1	no	—
143	Dic₁₇₈	#41	1	2	yes	—
144	#10623494	—	2	1	no	—
145	#10623494	—	2	1	no	—
146	#10623494	—	2	1	no	—
147	#10623494	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	—	4	1	no	—
154	C₂²×C₈₉	—	4	1	no	—
155	C₂²×C₈₉	—	4	1	no	—
156	C₂²×C₈₉	—	4	1	no	—
157	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₈₉	—	2	2	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	4	1	no	—
167	C₄×C₈₉	—	4	1	no	—
168	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
169	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
170	Dic₈₉	—	2	2	no	—
171	Dic₈₉	—	4	1	no	—
172	Dic₈₉	—	4	1	no	—
173	C₂×C₈₉	—	4	1	no	—
174	C₂×C₈₉	—	8	1	no	—
175	C₂×C₈₉	—	8	1	no	—
176	C₂×C₈₉	#525	1	1	yes	characteristic
177	C₂×C₈₉	#576	1	1	yes	characteristic
178	C₂×C₈₉	#577	1	1	yes	characteristic
179	D₈₉	—	4	2	no	—
180	#3119	—	89	1	no	maximal, Sylow
181	C₈₉	#3119	1	1	yes	characteristic, complemented, Sylow
182	#540	D₈₉	1	1	yes	characteristic, complemented
183	#520	—	89	1	no	—
184	#481	—	89	1	no	—
185	#480	—	89	1	no	—
186	#479	—	89	1	no	—
187	#478	—	89	1	no	—
188	#460	—	89	1	no	—
189	#459	—	89	1	no	—
190	#449	—	89	1	no	—
191	#449	—	89	1	no	—
192	#442	—	89	1	no	—
193	#440	—	89	1	no	—
194	C₄×SD₁₆	—	89	1	no	—
195	C₄×D₈	—	89	1	no	—
196	#433	—	89	1	no	—
197	C₂×#41	—	89	1	no	—
198	C₂×SD₁₆	—	89	1	no	—
199	C₂×SD₁₆	—	89	1	no	—
200	C₂×D₈	—	89	1	no	—
201	C₂×D₈	—	89	1	no	—
202	#127	C₂×D₈₉	1	1	yes	characteristic
203	#125	C₂×D₈₉	1	1	yes	characteristic
204	#123	—	2	1	no	—
205	#123	—	2	1	no	—
206	#121	—	2	1	no	—
207	#121	—	2	1	no	—
208	#121	C₂×D₈₉	1	1	yes	characteristic
209	#121	C₂×D₈₉	1	1	yes	characteristic
210	C₄×Q₈	—	89	1	no	—
211	C₄×Q₈	—	89	1	no	—
212	C₄×D₄	—	89	1	no	—
213	C₄×D₄	—	89	1	no	—
214	C₄×D₄	—	178	1	no	—
215	C₄×D₄	—	178	1	no	—
216	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
217	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
218	#117	—	178	1	no	—
219	#117	C₂×D₈₉	1	1	yes	characteristic
220	C₂×C₄⋊C₄	—	89	1	no	—
221	C₂×C₄⋊C₄	—	178	1	no	—
222	#107	—	89	1	no	—
223	#107	—	89	1	no	—
224	#106	—	89	1	no	—
225	#106	—	89	1	no	—
226	C₄⋊C₈	—	89	1	no	—
227	C₄⋊C₈	—	89	1	no	—
228	#103	—	89	1	no	—
229	#103	—	89	1	no	—
230	#102	—	89	1	no	—
231	#102	—	89	1	no	—
232	#102	—	89	1	no	—
233	#102	—	89	1	no	—
234	#102	—	89	1	no	—
235	#102	—	89	1	no	—
236	C₂²⋊C₈	—	89	1	no	—
237	C₂²⋊C₈	—	89	1	no	—
238	#97	—	89	1	no	—
239	C₄×C₈	—	89	1	no	—
240	#41	—	178	2	no	—
241	#41	—	178	2	no	—
242	C₂×Q₈	—	89	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	89	1	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
250	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	178	1	no	—
254	C₂²×C₄	—	178	1	no	—
255	C₂²×C₄	—	178	1	no	—
256	C₂²×C₄	—	178	1	no	—
257	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
258	SD₁₆	—	178	2	no	—
259	SD₁₆	—	178	2	no	—
260	D₈	—	178	2	no	—
261	D₈	—	178	2	no	—
262	C₂×C₈	—	89	1	no	—
263	C₂×C₈	—	89	1	no	—
264	C₂×C₈	—	89	1	no	—
265	C₂×C₈	—	89	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	89	1	no	—
269	C₄⋊C₄	—	89	1	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	178	1	no	—
282	C₂²⋊C₄	—	178	1	no	—
283	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
284	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
285	C₄²	—	178	1	no	—
286	C₄²	—	178	1	no	—
287	C₄²	C₂²×D₈₉	1	1	yes	characteristic
288	C₄²	C₂²×D₈₉	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	178	1	no	—
292	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
293	Q₈	—	89	2	no	—
294	Q₈	—	178	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	2	1	no	—
299	D₄	—	8	1	no	—
300	D₄	—	8	1	no	—
301	D₄	—	8	1	no	—
302	D₄	—	8	1	no	—
303	D₄	—	89	2	no	—
304	D₄	—	178	4	no	—
305	D₄	—	178	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	178	4	no	—
313	C₂×C₄	—	178	1	no	—
314	C₂×C₄	—	178	1	no	—
315	C₂×C₄	—	178	1	no	—
316	C₂×C₄	—	712	1	no	—
317	C₂×C₄	—	712	1	no	—
318	C₂×C₄	—	712	1	no	—
319	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
320	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
321	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
322	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
323	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
325	C₈	—	178	1	no	—
326	C₈	—	178	1	no	—
327	C₈	—	178	1	no	—
328	C₈	—	178	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	356	1	no	—
339	C₂²	—	356	1	no	—
340	C₂²	—	178	2	no	—
341	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	356	1	no	—
350	C₄	—	356	1	no	—
351	C₄	—	178	2	no	—
352	C₄	#41×D₈₉	1	1	yes	characteristic
353	C₄	#41×D₈₉	1	1	yes	characteristic
354	C₂	—	4	1	no	—
355	C₂	—	8	1	no	—
356	C₂	—	8	1	no	—
357	C₂	—	356	2	no	—
358	C₂	#19152105	1	1	yes	characteristic
359	C₂	#19152401	1	1	yes	characteristic
360	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	#540×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#540	D₈₉
hypocenter	C₈₉	#3119
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #540

Complements

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

N	H ≅ G/N	type
1. #19152078	C₂	semidirect
2. #19151914	C₂	semidirect
3. #19151909	C₂	semidirect
4. #19151902	C₂	semidirect
5. #19151895	C₂	semidirect
6. #19151795	C₂	semidirect
7. #19151790	C₂	semidirect
8. #19151741	C₂	semidirect
9. #19151741	C₂	semidirect
10. #19151708	C₂	semidirect
11. #19151698	C₂	semidirect
12. C₄×#13083562	C₂	semidirect
13. C₄×#13083559	C₂	semidirect
14. #19151659	C₂	semidirect
15. #540×C₈₉	C₂	semidirect
181. C₈₉	#3119	semidirect
182. #540	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152078	2
#19151914	2
#19151909	2
#19151902	2
#19151895	2
#19151795	2
#19151790	2
#19151741	2
#19151708	2
#19151698	2
C₄×#13083562	2
C₄×#13083559	2
#19151659	2
#540×C₈₉	2
#3119	89