Subgroups — #26032586

Order 11392.

398 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152303	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152267	C₂	1	1	yes	maximal, characteristic
3	#19151943	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151911	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151910	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151891	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151844	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151843	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151797	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151763	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151760	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151690	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151674	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151598	C₂	1	1	yes	maximal, characteristic, complemented
15	#417×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×#13083564	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	C₂×#13083557	C₂²	1	1	yes	characteristic
21	#15668011	C₂²	1	1	yes	characteristic
22	#15668008	—	2	1	no	—
23	#15668008	C₂²	1	1	yes	characteristic
24	#15668002	—	2	1	no	—
25	#15667992	C₂²	1	1	yes	characteristic
26	#15667991	—	2	1	no	—
27	#15667991	C₂²	1	1	yes	characteristic
28	#15667990	—	2	1	no	—
29	#15667986	—	2	1	no	—
30	#15667986	C₂²	1	1	yes	characteristic
31	#15667983	—	2	1	no	—
32	#15667983	C₂²	1	1	yes	characteristic
33	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
34	#15667977	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊D₈₉	—	2	1	no	—
36	C₄×C₂²⋊D₈₉	—	2	1	no	—
37	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
39	#15667946	C₂²	1	1	yes	characteristic
40	#15667945	C₂²	1	1	yes	characteristic
41	#15667945	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667937	C₂²	1	1	yes	characteristic
44	#15667936	C₂²	1	1	yes	characteristic
45	#15667934	C₂²	1	1	yes	characteristic
46	#15667933	C₂²	1	1	yes	characteristic
47	#15667932	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
55	#103×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	#13083575	—	2	2	no	—
60	#13083575	—	4	1	no	—
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₉	—	2	1	no	—
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
73	#13083564	—	2	2	no	—
74	#13083563	—	4	1	no	—
75	#13083562	—	4	1	no	—
76	#13083557	—	4	1	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	—	2	1	no	—
81	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	D₄	1	1	yes	characteristic
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	D₄	1	1	yes	characteristic
94	C₄×Dic₈₉	—	2	1	no	—
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
97	#41×C₈₉	—	2	2	no	—
98	#41×C₈₉	—	4	1	no	—
99	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₈₉	—	2	1	no	—
101	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	—	2	1	no	—
103	C₂²×C₄×C₈₉	—	2	1	no	—
104	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
105	M₁₆×C₈₉	—	4	1	no	—
106	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
111	C₂³×C₈₉	—	2	1	no	—
112	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₈₉	—	2	1	no	—
114	C₂²×D₈₉	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	—	4	2	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	—	2	1	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	8	1	no	—
127	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₈₉	—	2	2	no	—
130	C₂²⋊D₈₉	—	4	2	no	—
131	C₂²⋊D₈₉	—	8	1	no	—
132	C₂²⋊D₈₉	—	8	1	no	—
133	C₂²⋊D₈₉	—	8	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	C₂×Dic₈₉	—	2	1	no	—
137	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
140	D₃₅₆	—	2	1	no	—
141	D₃₅₆	—	2	1	no	—
142	D₃₅₆	—	4	2	no	—
143	C₄×D₈₉	—	2	1	no	—
144	C₄×D₈₉	—	2	1	no	—
145	C₄×D₈₉	—	4	2	no	—
146	C₄×D₈₉	—	8	1	no	—
147	Dic₁₇₈	—	2	1	no	—
148	Dic₁₇₈	—	2	1	no	—
149	Dic₁₇₈	—	8	1	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	—	2	1	no	—
152	#10623494	—	2	1	no	—
153	#10623494	—	2	1	no	—
154	C₂²×C₈₉	—	2	1	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	4	1	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₈₉	—	2	1	no	—
162	C₂×D₈₉	—	2	1	no	—
163	C₂×D₈₉	—	2	1	no	—
164	C₂×D₈₉	—	4	1	no	—
165	C₂×D₈₉	—	4	1	no	—
166	C₂×D₈₉	—	4	1	no	—
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	4	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	—	2	1	no	—
174	Dic₈₉	—	2	1	no	—
175	Dic₈₉	—	4	1	no	—
176	Dic₈₉	—	4	1	no	—
177	Dic₈₉	—	4	1	no	—
178	Dic₈₉	#137	1	2	yes	—
179	C₂×C₈₉	—	4	1	no	—
180	C₂×C₈₉	—	8	1	no	—
181	C₂×C₈₉	#546	1	1	yes	characteristic
182	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
183	C₂×C₈₉	#576	1	1	yes	characteristic
184	D₈₉	—	4	1	no	—
185	D₈₉	—	8	1	no	—
186	#3333	—	89	1	no	maximal, Sylow
187	C₈₉	#3333	1	1	yes	characteristic, complemented, Sylow
188	#554	—	89	1	no	—
189	#550	—	89	1	no	—
190	#486	—	89	1	no	—
191	#480	—	89	1	no	—
192	#480	—	89	1	no	—
193	#477	—	89	1	no	—
194	#468	—	89	1	no	—
195	#468	—	89	1	no	—
196	#460	—	89	1	no	—
197	#452	—	89	1	no	—
198	#452	—	89	1	no	—
199	C₄×Q₁₆	—	89	1	no	—
200	#436	—	89	1	no	—
201	#417	—	89	1	no	—
202	#417	D₈₉	1	1	yes	characteristic, complemented
203	C₂×#41	—	89	1	no	—
204	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
205	C₂×Q₁₆	—	89	1	no	—
206	C₂×SD₁₆	—	89	1	no	—
207	C₂×SD₁₆	—	89	1	no	—
208	C₂×M₁₆	—	89	1	no	—
209	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
210	C₄⋊Q₈	—	89	1	no	—
211	#127	—	89	1	no	—
212	#127	—	178	1	no	—
213	#125	—	178	1	no	—
214	C₂²⋊Q₈	—	89	1	no	—
215	C₂²⋊Q₈	—	89	1	no	—
216	C₂²⋊Q₈	—	178	1	no	—
217	C₂²⋊Q₈	—	178	1	no	—
218	#121	—	89	1	no	—
219	#121	—	89	1	no	—
220	#121	—	178	1	no	—
221	#121	—	178	1	no	—
222	C₄×Q₈	—	89	1	no	—
223	C₄×Q₈	—	89	1	no	—
224	C₄×D₄	—	89	1	no	—
225	C₄×D₄	—	178	1	no	—
226	C₄×D₄	—	178	1	no	—
227	C₂×C₄⋊C₄	—	89	1	no	—
228	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
229	#107	—	89	1	no	—
230	#107	—	89	1	no	—
231	#107	—	89	1	no	—
232	C₄⋊C₈	—	89	1	no	—
233	#103	—	89	1	no	—
234	#103	—	89	1	no	—
235	#103	C₂×D₈₉	1	1	yes	characteristic
236	#103	C₂×D₈₉	1	1	yes	characteristic
237	#102	—	89	1	no	—
238	#102	—	89	1	no	—
239	#102	—	89	1	no	—
240	#102	—	89	1	no	—
241	#102	C₂×D₈₉	1	1	yes	characteristic
242	#102	C₂×D₈₉	1	1	yes	characteristic
243	C₂²⋊C₈	—	89	1	no	—
244	C₂²⋊C₈	—	89	1	no	—
245	C₄×C₈	—	89	1	no	—
246	#41	—	2	2	no	—
247	#41	—	4	1	no	—
248	#41	—	356	1	no	—
249	#41	—	178	2	no	—
250	C₂×Q₈	—	89	1	no	—
251	C₂×Q₈	—	178	1	no	—
252	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	89	1	no	—
255	C₂×D₄	—	89	1	no	—
256	C₂×D₄	—	178	1	no	—
257	C₂×D₄	—	178	1	no	—
258	C₂×D₄	—	178	1	no	—
259	C₂×D₄	—	178	1	no	—
260	C₂×D₄	—	178	1	no	—
261	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	89	1	no	—
265	C₂²×C₄	—	178	1	no	—
266	C₂²×C₄	—	178	1	no	—
267	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
268	Q₁₆	—	178	2	no	—
269	SD₁₆	—	356	1	no	—
270	SD₁₆	—	356	1	no	—
271	M₁₆	—	4	1	no	—
272	M₁₆	—	356	1	no	—
273	C₂×C₈	—	89	1	no	—
274	C₂×C₈	—	89	1	no	—
275	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
276	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	89	1	no	—
279	C₄⋊C₄	—	89	1	no	—
280	C₄⋊C₄	—	89	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	—	178	1	no	—
287	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
288	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
289	C₂²⋊C₄	—	89	1	no	—
290	C₂²⋊C₄	—	89	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	C₄²	—	89	1	no	—
296	C₄²	—	178	1	no	—
297	C₄²	—	178	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	89	1	no	—
300	C₂³	—	178	1	no	—
301	C₂³	D₄×D₈₉	1	1	yes	characteristic
302	Q₈	—	2	1	no	—
303	Q₈	—	178	1	no	—
304	Q₈	—	178	1	no	—
305	Q₈	—	712	1	no	—
306	Q₈	#13083574	1	2	yes	—
307	D₄	—	2	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	356	2	no	—
311	D₄	—	356	2	no	—
312	D₄	—	178	2	no	—
313	D₄	—	178	1	no	—
314	D₄	—	178	1	no	—
315	D₄	—	712	1	no	—
316	D₄	—	712	1	no	—
317	D₄	—	712	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	—	2	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	1	no	—
325	C₂×C₄	—	89	1	no	—
326	C₂×C₄	—	89	1	no	—
327	C₂×C₄	—	89	1	no	—
328	C₂×C₄	—	356	2	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	—	712	1	no	—
335	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
336	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	178	1	no	—
340	C₈	—	178	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	356	1	no	—
348	C₂²	—	356	1	no	—
349	C₂²	—	356	1	no	—
350	C₂²	—	178	1	no	—
351	C₂²	—	178	1	no	—
352	C₂²	—	178	1	no	—
353	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	2	2	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	89	2	no	—
359	C₄	—	356	1	no	—
360	C₄	—	356	1	no	—
361	C₄	—	356	1	no	—
362	C₄	—	178	1	no	—
363	C₄	—	178	1	no	—
364	C₄	C₂×#13083574	1	1	yes	characteristic
365	C₄	#15668057	1	1	yes	characteristic
366	C₂	—	4	1	no	—
367	C₂	—	8	1	no	—
368	C₂	—	356	1	no	—
369	C₂	—	712	1	no	—
370	C₂	#19152238	1	1	yes	characteristic
371	C₂	#137×D₈₉	1	1	yes	characteristic
372	C₂	#19152402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#417×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152267	C₂
hypercenter	#417	D₈₉
hypocenter	C₈₉	#3333
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
1. #19152303	C₂	semidirect
3. #19151943	C₂	semidirect
4. #19151911	C₂	semidirect
5. #19151910	C₂	semidirect
6. #19151891	C₂	semidirect
7. #19151844	C₂	semidirect
8. #19151843	C₂	semidirect
9. #19151797	C₂	semidirect
10. #19151763	C₂	semidirect
11. #19151760	C₂	semidirect
12. #19151690	C₂	semidirect
13. #19151674	C₂	semidirect
14. #19151598	C₂	semidirect
15. #417×C₈₉	C₂	semidirect
187. C₈₉	#3333	semidirect
202. #417	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152303	2
#19152267	2
#19151943	2
#19151911	2
#19151910	2
#19151891	2
#19151844	2
#19151843	2
#19151797	2
#19151763	2
#19151760	2
#19151690	2
#19151674	2
#19151598	2
#417×C₈₉	2
#3333	89