Subgroups — #26031145

Order 11392.

378 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	#19152172	C₂	1	1	yes	maximal, characteristic
2	#19152171	C₂	1	1	yes	maximal, characteristic
3	#19151940	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151938	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151937	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151936	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151855	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151853	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151838	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151836	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151754	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151752	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151752	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151563	C₂	1	1	yes	maximal, characteristic
15	#451×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083564	C₂²	1	1	yes	characteristic
19	C₂×#13083564	C₂²	1	1	yes	characteristic
20	C₂×#13083557	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
22	#15668011	—	2	1	no	—
23	#15668010	—	2	1	no	—
24	#15668001	—	2	1	no	—
25	#15667998	—	2	1	no	—
26	#15667997	—	2	1	no	—
27	#15667995	—	2	1	no	—
28	#15667995	—	2	1	no	—
29	#15667994	—	2	1	no	—
30	#15667992	C₂²	1	1	yes	characteristic
31	#15667991	C₂²	1	1	yes	characteristic
32	#15667988	C₂²	1	1	yes	characteristic
33	#15667988	C₂²	1	1	yes	characteristic
34	#15667988	C₂²	1	1	yes	characteristic
35	#15667988	C₂²	1	1	yes	characteristic
36	#15667988	C₂²	1	1	yes	characteristic
37	#15667971	C₂²	1	1	yes	characteristic
38	#15667970	C₂²	1	1	yes	characteristic
39	#15667946	C₂²	1	1	yes	characteristic
40	#15667945	C₂²	1	1	yes	characteristic
41	#15667944	C₂²	1	1	yes	characteristic
42	#15667943	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	#15667936	C₂²	1	1	yes	characteristic
48	#15667936	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667924	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	4	1	no	—
60	#13083574	—	4	1	no	—
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
63	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
64	Q₈×D₈₉	—	4	1	no	—
65	Q₈×D₈₉	—	4	1	no	—
66	C₂×D₃₅₆	D₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
72	#13083564	—	4	1	no	—
73	#13083564	—	4	1	no	—
74	#13083557	—	4	1	no	—
75	C₂×#10623494	C₂³	1	1	yes	characteristic
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₈×D₈₉	—	4	1	no	—
78	C₄⋊Dic₈₉	—	2	1	no	—
79	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₈₉	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	—	2	1	no	—
96	#13083551	—	2	1	no	—
97	C₄×Dic₈₉	—	2	1	no	—
98	C₄×Dic₈₉	—	2	1	no	—
99	C₂×Q₈×C₈₉	—	2	1	no	—
100	C₂×Q₈×C₈₉	—	4	1	no	—
101	C₂×Q₈×C₈₉	—	4	1	no	—
102	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	—	2	1	no	—
106	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
107	Q₁₆×C₈₉	—	4	1	no	—
108	Q₁₆×C₈₉	—	4	1	no	—
109	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	1	no	—
112	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
113	C₂²⋊C₄×C₈₉	—	2	1	no	—
114	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
117	Q₈×C₈₉	—	2	1	no	—
118	Q₈×C₈₉	—	2	1	no	—
119	Q₈×C₈₉	—	2	1	no	—
120	Q₈×C₈₉	—	2	1	no	—
121	Q₈×C₈₉	—	2	1	no	—
122	Q₈×C₈₉	—	2	1	no	—
123	Q₈×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
133	C₂²⋊D₈₉	—	4	1	no	—
134	C₂²⋊D₈₉	—	4	1	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	C₂×Dic₈₉	—	2	1	no	—
137	C₂×Dic₈₉	—	2	1	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
141	D₃₅₆	—	4	1	no	—
142	C₄×D₈₉	—	2	1	no	—
143	C₄×D₈₉	—	2	1	no	—
144	C₄×D₈₉	—	2	1	no	—
145	C₄×D₈₉	—	2	1	no	—
146	C₄×D₈₉	—	4	2	no	—
147	Dic₁₇₈	—	2	1	no	—
148	Dic₁₇₈	—	2	1	no	—
149	Dic₁₇₈	—	4	1	no	—
150	Dic₁₇₈	—	4	2	no	—
151	C₈×C₈₉	—	2	1	no	—
152	C₈×C₈₉	—	2	1	no	—
153	#10623494	—	2	1	no	—
154	#10623494	—	2	1	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₈₉	—	2	1	no	—
160	C₂×D₈₉	—	2	1	no	—
161	C₂×D₈₉	—	2	1	no	—
162	C₂×D₈₉	—	2	1	no	—
163	C₂×D₈₉	—	2	1	no	—
164	C₂×D₈₉	—	2	1	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	4	1	no	—
167	C₄×C₈₉	—	4	1	no	—
168	C₄×C₈₉	—	4	1	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
171	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₈₉	—	2	1	no	—
173	Dic₈₉	—	2	1	no	—
174	Dic₈₉	—	4	1	no	—
175	Dic₈₉	—	4	1	no	—
176	Dic₈₉	—	4	1	no	—
177	Dic₈₉	—	4	1	no	—
178	C₂×C₈₉	—	4	1	no	—
179	C₂×C₈₉	#534	1	1	yes	characteristic
180	C₂×C₈₉	#578	1	1	yes	characteristic
181	C₂×C₈₉	#578	1	1	yes	characteristic
182	D₈₉	—	4	1	no	—
183	D₈₉	—	4	1	no	—
184	#3202	—	89	1	no	maximal, Sylow
185	C₈₉	#3202	1	1	yes	characteristic, complemented, Sylow
186	#536	—	89	1	no	—
187	#536	—	89	1	no	—
188	#485	—	89	1	no	—
189	#485	—	89	1	no	—
190	#485	—	89	1	no	—
191	#485	—	89	1	no	—
192	#470	—	89	1	no	—
193	#470	—	89	1	no	—
194	#467	—	89	1	no	—
195	#467	—	89	1	no	—
196	#451	—	89	1	no	—
197	#451	—	89	1	no	—
198	#451	—	89	1	no	—
199	#451	D₈₉	1	1	yes	characteristic, complemented
200	#408	—	89	1	no	—
201	C₂×#41	—	89	1	no	—
202	C₂²×Q₈	—	89	1	no	—
203	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
204	C₂×Q₁₆	—	89	1	no	—
205	C₂×Q₁₆	—	89	1	no	—
206	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
207	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
208	C₂×M₁₆	—	89	1	no	—
209	C₂²×C₈	—	89	1	no	—
210	C₄⋊Q₈	—	178	1	no	—
211	C₄⋊Q₈	—	178	1	no	—
212	#124	—	178	1	no	—
213	#124	—	178	1	no	—
214	#123	—	178	1	no	—
215	#123	—	178	1	no	—
216	#123	—	178	1	no	—
217	#123	—	178	1	no	—
218	C₂²⋊Q₈	—	89	1	no	—
219	C₂²⋊Q₈	—	89	1	no	—
220	C₂²⋊Q₈	—	89	1	no	—
221	C₂²⋊Q₈	—	89	1	no	—
222	C₂²⋊Q₈	—	89	1	no	—
223	C₂²⋊Q₈	—	89	1	no	—
224	C₂²⋊Q₈	—	89	1	no	—
225	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
226	#117	—	89	1	no	—
227	#117	—	89	1	no	—
228	#107	—	89	1	no	—
229	#107	—	89	1	no	—
230	#106	—	89	1	no	—
231	#106	—	89	1	no	—
232	#103	—	89	1	no	—
233	#103	—	89	1	no	—
234	#103	—	89	1	no	—
235	#103	—	89	1	no	—
236	#103	—	89	1	no	—
237	#103	—	89	1	no	—
238	#103	C₂×D₈₉	1	1	yes	characteristic
239	#103	C₂×D₈₉	1	1	yes	characteristic
240	C₂²⋊C₈	—	89	1	no	—
241	C₂²⋊C₈	—	89	1	no	—
242	C₂²⋊C₈	—	89	1	no	—
243	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
244	#41	—	356	1	no	—
245	#41	—	356	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	89	1	no	—
250	C₂×Q₈	—	89	1	no	—
251	C₂×Q₈	—	356	1	no	—
252	C₂×Q₈	—	356	1	no	—
253	C₂×Q₈	—	178	1	no	—
254	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
255	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
256	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
257	C₂×D₄	—	89	1	no	—
258	C₂×D₄	—	89	1	no	—
259	C₂×D₄	—	89	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	89	1	no	—
262	C₂²×C₄	—	89	1	no	—
263	C₂²×C₄	—	178	1	no	—
264	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
265	Q₁₆	—	4	1	no	—
266	Q₁₆	—	4	1	no	—
267	Q₁₆	—	356	1	no	—
268	Q₁₆	—	356	1	no	—
269	M₁₆	—	356	1	no	—
270	C₂×C₈	—	89	1	no	—
271	C₂×C₈	—	89	1	no	—
272	C₂×C₈	—	356	1	no	—
273	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
274	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	89	1	no	—
277	C₄⋊C₄	—	89	1	no	—
278	C₄⋊C₄	—	89	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	—	178	1	no	—
284	C₄⋊C₄	—	178	1	no	—
285	C₄⋊C₄	—	178	1	no	—
286	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
287	C₂²⋊C₄	—	2	1	no	—
288	C₂²⋊C₄	—	178	1	no	—
289	C₂²⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	—	178	1	no	—
291	C₂²⋊C₄	—	178	1	no	—
292	C₂²⋊C₄	—	178	1	no	—
293	C₂²⋊C₄	—	178	1	no	—
294	C₂²⋊C₄	—	178	1	no	—
295	C₂²⋊C₄	—	178	1	no	—
296	C₂²⋊C₄	—	178	1	no	—
297	C₄²	—	178	1	no	—
298	C₄²	—	178	1	no	—
299	C₂³	—	89	1	no	—
300	C₂³	—	89	1	no	—
301	C₂³	D₄×D₈₉	1	1	yes	characteristic
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	356	1	no	—
310	Q₈	—	356	2	no	—
311	Q₈	—	178	1	no	—
312	Q₈	—	178	1	no	—
313	D₄	—	356	1	no	—
314	D₄	—	356	1	no	—
315	D₄	—	356	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	—	2	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	89	1	no	—
324	C₂×C₄	—	89	1	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	178	1	no	—
327	C₂×C₄	—	178	1	no	—
328	C₂×C₄	—	178	1	no	—
329	C₂×C₄	—	178	1	no	—
330	C₂×C₄	—	178	1	no	—
331	C₂×C₄	—	178	1	no	—
332	C₂×C₄	—	178	1	no	—
333	C₂×C₄	—	178	1	no	—
334	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
335	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	178	1	no	—
339	C₈	—	178	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	178	1	no	—
344	C₂²	—	178	1	no	—
345	C₂²	—	178	1	no	—
346	C₂²	—	178	1	no	—
347	C₂²	—	178	1	no	—
348	C₂²	—	178	1	no	—
349	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	356	1	no	—
356	C₄	—	356	1	no	—
357	C₄	—	356	1	no	—
358	C₄	—	356	1	no	—
359	C₄	—	178	1	no	—
360	C₄	—	178	1	no	—
361	C₄	#15668057	1	1	yes	characteristic
362	C₄	#15668057	1	1	yes	characteristic
363	C₂	—	4	1	no	—
364	C₂	—	356	1	no	—
365	C₂	—	356	1	no	—
366	C₂	#19152158	1	1	yes	characteristic
367	C₂	#19152411	1	1	yes	characteristic
368	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#451×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083574	C₂²
hypercenter	#451	D₈₉
hypocenter	C₈₉	#3202
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
3. #19151940	C₂	semidirect
4. #19151938	C₂	semidirect
5. #19151937	C₂	semidirect
6. #19151936	C₂	semidirect
7. #19151855	C₂	semidirect
8. #19151853	C₂	semidirect
9. #19151838	C₂	semidirect
10. #19151836	C₂	semidirect
11. #19151754	C₂	semidirect
12. #19151752	C₂	semidirect
13. #19151752	C₂	semidirect
15. #451×C₈₉	C₂	semidirect
185. C₈₉	#3202	semidirect
199. #451	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152172	2
#19152171	2
#19151940	2
#19151938	2
#19151937	2
#19151936	2
#19151855	2
#19151853	2
#19151838	2
#19151836	2
#19151754	2
#19151752	2
#19151563	2
#451×C₈₉	2
#3202	89