Subgroups — #26033295

Order 11392.

418 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152331	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152269	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151987	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151986	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151973	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151813	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151812	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151797	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151793	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151792	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151792	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151726	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151709	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151685	C₂	1	1	yes	maximal, characteristic, complemented
15	#442×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	—	2	1	no	—
17	C₂×#13083563	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
21	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
22	C₂×#13083559	C₂²	1	1	yes	characteristic
23	C₂×#13083559	C₂²	1	1	yes	characteristic
24	C₂×#13083559	C₂²	1	1	yes	characteristic
25	#15668009	C₂²	1	1	yes	characteristic
26	#15668008	C₂²	1	1	yes	characteristic
27	#15668008	C₂²	1	1	yes	characteristic
28	#15668008	C₂²	1	1	yes	characteristic
29	#15668000	C₂²	1	1	yes	characteristic
30	#15667999	C₂²	1	1	yes	characteristic
31	#15667998	—	2	1	no	—
32	#15667995	—	2	1	no	—
33	#15667994	—	2	1	no	—
34	#15667986	—	2	1	no	—
35	#15667985	—	2	1	no	—
36	#15667985	—	2	1	no	—
37	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
38	#15667977	C₂²	1	1	yes	characteristic
39	#15667974	C₂²	1	1	yes	characteristic
40	#15667974	C₂²	1	1	yes	characteristic
41	#15667974	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊D₈₉	—	2	1	no	—
43	C₄⋊#10623494	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667941	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667932	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic
56	#102×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	4	4	no	—
60	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×D₃₅₆	—	2	1	no	—
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
65	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
66	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂×C₂²⋊D₈₉	—	2	1	no	—
70	C₂×C₂²⋊D₈₉	—	2	1	no	—
71	C₂×C₂²⋊D₈₉	—	2	1	no	—
72	C₂×C₂²⋊D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	—	2	1	no	—
74	C₂×C₄×D₈₉	—	2	1	no	—
75	C₂×C₄×D₈₉	—	2	1	no	—
76	#13083563	—	2	2	no	—
77	#13083563	—	4	1	no	—
78	#13083562	—	4	1	no	—
79	D₇₁₂	—	2	2	no	—
80	D₇₁₂	—	4	1	no	—
81	#13083559	—	4	1	no	—
82	#13083559	—	4	1	no	—
83	#13083559	—	4	1	no	—
84	C₂×#10623494	C₂³	1	1	yes	characteristic
85	C₂×#10623494	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
87	#13083553	—	2	1	no	—
88	#13083553	D₄	1	1	yes	characteristic
89	#13083553	D₄	1	1	yes	characteristic
90	C₂²⋊Dic₈₉	—	2	1	no	—
91	C₂²⋊Dic₈₉	—	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	C₂³	1	1	yes	characteristic
99	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	—	2	1	no	—
103	C₂²×C₄×C₈₉	—	2	1	no	—
104	D₈×C₈₉	—	2	2	no	—
105	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₂²⋊C₄×C₈₉	—	2	1	no	—
110	C₂²⋊C₄×C₈₉	—	2	1	no	—
111	C₄²×C₈₉	C₂³	1	1	yes	characteristic
112	C₂³×C₈₉	—	2	1	no	—
113	C₂³×C₈₉	—	2	1	no	—
114	C₂²×D₈₉	—	2	1	no	—
115	C₂²×D₈₉	—	2	1	no	—
116	C₂²×D₈₉	—	2	1	no	—
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	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	8	1	no	—
125	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₈₉	—	4	2	no	—
129	C₂²⋊D₈₉	—	4	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	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
139	D₃₅₆	—	2	1	no	—
140	D₃₅₆	—	2	1	no	—
141	D₃₅₆	—	2	1	no	—
142	D₃₅₆	—	2	1	no	—
143	D₃₅₆	—	2	1	no	—
144	D₃₅₆	—	2	1	no	—
145	D₃₅₆	—	4	2	no	—
146	D₃₅₆	—	8	1	no	—
147	C₄×D₈₉	—	4	2	no	—
148	C₄×D₈₉	—	4	2	no	—
149	C₄×D₈₉	—	8	1	no	—
150	Dic₁₇₈	—	2	2	no	—
151	Dic₁₇₈	—	2	1	no	—
152	Dic₁₇₈	—	2	1	no	—
153	C₈×C₈₉	—	2	1	no	—
154	C₈×C₈₉	#41	1	2	yes	complemented
155	#10623494	—	2	1	no	—
156	#10623494	—	2	1	no	—
157	C₂²×C₈₉	—	4	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	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₈₉	—	4	1	no	—
165	C₂×D₈₉	—	4	1	no	—
166	C₂×D₈₉	—	4	1	no	—
167	C₂×D₈₉	—	4	1	no	—
168	C₂×D₈₉	—	4	1	no	—
169	C₂×D₈₉	—	4	1	no	—
170	C₂×D₈₉	—	4	1	no	—
171	C₂×D₈₉	—	4	1	no	—
172	C₂×D₈₉	—	4	1	no	—
173	C₄×C₈₉	—	2	1	no	—
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	—	4	1	no	—
176	C₄×C₈₉	—	4	1	no	—
177	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
178	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
179	Dic₈₉	—	2	1	no	—
180	Dic₈₉	—	2	1	no	—
181	Dic₈₉	—	2	1	no	—
182	Dic₈₉	—	4	1	no	—
183	Dic₈₉	AGL(1,ℤ/8)	1	2	yes	—
184	C₂×C₈₉	—	8	1	no	—
185	C₂×C₈₉	—	8	1	no	—
186	C₂×C₈₉	#550	1	1	yes	characteristic
187	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₂×C₈₉	#576	1	1	yes	characteristic
189	D₈₉	—	8	1	no	—
190	D₈₉	—	8	1	no	—
191	D₈₉	—	8	1	no	—
192	#3399	—	89	1	no	maximal, Sylow
193	C₈₉	#3399	1	1	yes	characteristic, complemented, Sylow
194	#559	—	89	1	no	—
195	#550	—	89	1	no	—
196	#496	—	89	1	no	—
197	#496	—	89	1	no	—
198	#493	—	89	1	no	—
199	#463	—	89	1	no	—
200	#463	—	89	1	no	—
201	#460	—	89	1	no	—
202	#459	—	89	1	no	—
203	#459	—	89	1	no	—
204	#459	—	89	1	no	—
205	#446	—	89	1	no	—
206	#442	—	89	1	no	—
207	#442	D₈₉	1	1	yes	characteristic, complemented
208	C₄×SD₁₆	—	89	1	no	—
209	C₂×#41	—	178	1	no	—
210	C₂×SD₁₆	—	89	1	no	—
211	C₂×SD₁₆	—	89	1	no	—
212	C₂×SD₁₆	—	89	1	no	—
213	C₂×D₈	—	89	1	no	—
214	C₂×D₈	—	89	1	no	—
215	C₂×D₈	—	89	1	no	—
216	C₂×D₈	—	89	1	no	—
217	C₂×D₈	—	89	1	no	—
218	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
219	#127	—	89	1	no	—
220	#127	—	89	1	no	—
221	#127	—	89	1	no	—
222	#127	—	89	1	no	—
223	#124	—	89	1	no	—
224	#124	—	89	1	no	—
225	#124	—	178	1	no	—
226	#123	—	178	1	no	—
227	#123	—	178	1	no	—
228	#121	—	178	1	no	—
229	#121	—	178	1	no	—
230	#121	—	178	1	no	—
231	C₄×Q₈	—	89	1	no	—
232	C₄×Q₈	—	89	1	no	—
233	C₄×D₄	—	89	1	no	—
234	C₄×D₄	—	89	1	no	—
235	C₄×D₄	—	89	1	no	—
236	C₄×D₄	—	178	1	no	—
237	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
238	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
239	#106	C₂×D₈₉	1	1	yes	characteristic
240	C₄⋊C₈	—	89	1	no	—
241	C₄⋊C₈	—	89	1	no	—
242	C₄⋊C₈	—	89	1	no	—
243	#103	—	89	1	no	—
244	#102	—	89	1	no	—
245	#102	—	89	1	no	—
246	#102	—	89	1	no	—
247	#102	C₂×D₈₉	1	1	yes	characteristic
248	#102	C₂×D₈₉	1	1	yes	characteristic
249	#97	—	89	1	no	—
250	#97	C₂×D₈₉	1	1	yes	characteristic
251	C₄×C₈	—	89	1	no	—
252	#41	—	356	4	no	—
253	C₂×Q₈	—	89	1	no	—
254	C₂×Q₈	—	89	1	no	—
255	C₂×D₄	—	89	1	no	—
256	C₂×D₄	—	89	1	no	—
257	C₂×D₄	—	89	1	no	—
258	C₂×D₄	—	178	1	no	—
259	C₂×D₄	—	178	1	no	—
260	C₂×D₄	—	178	1	no	—
261	C₂×D₄	—	178	1	no	—
262	C₂×D₄	—	178	1	no	—
263	C₂×D₄	—	178	1	no	—
264	C₂×D₄	—	178	1	no	—
265	C₂×D₄	—	178	1	no	—
266	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
267	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	178	1	no	—
271	C₂²×C₄	—	178	1	no	—
272	C₂²×C₄	—	178	1	no	—
273	SD₁₆	—	356	1	no	—
274	SD₁₆	—	356	1	no	—
275	SD₁₆	—	178	2	no	—
276	D₈	—	2	2	no	—
277	D₈	—	356	1	no	—
278	D₈	—	356	1	no	—
279	D₈	—	356	1	no	—
280	D₈	—	356	1	no	—
281	D₈	—	178	2	no	—
282	C₂×C₈	—	89	1	no	—
283	C₂×C₈	—	89	1	no	—
284	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
285	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
286	C₄⋊C₄	—	89	1	no	—
287	C₄⋊C₄	—	89	1	no	—
288	C₄⋊C₄	—	89	1	no	—
289	C₄⋊C₄	—	178	1	no	—
290	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
291	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
292	C₂²⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	178	1	no	—
295	C₂²⋊C₄	—	178	1	no	—
296	C₂²⋊C₄	—	178	1	no	—
297	C₂²⋊C₄	—	178	1	no	—
298	C₂²⋊C₄	—	178	1	no	—
299	C₂²⋊C₄	—	178	1	no	—
300	C₂²⋊C₄	—	178	1	no	—
301	C₄²	—	89	1	no	—
302	C₄²	—	89	1	no	—
303	C₄²	—	178	1	no	—
304	C₄²	C₂²×D₈₉	1	1	yes	characteristic
305	C₂³	—	2	1	no	—
306	C₂³	—	2	1	no	—
307	C₂³	—	178	1	no	—
308	C₂³	—	178	1	no	—
309	C₂³	—	178	1	no	—
310	Q₈	—	178	2	no	—
311	Q₈	—	178	1	no	—
312	Q₈	—	178	1	no	—
313	D₄	—	2	1	no	—
314	D₄	—	2	1	no	—
315	D₄	—	2	1	no	—
316	D₄	—	2	1	no	—
317	D₄	—	356	2	no	—
318	D₄	—	356	2	no	—
319	D₄	—	356	2	no	—
320	D₄	—	356	2	no	—
321	D₄	—	178	1	no	—
322	D₄	—	178	1	no	—
323	D₄	—	178	1	no	—
324	D₄	—	178	1	no	—
325	D₄	—	178	1	no	—
326	D₄	—	178	1	no	—
327	D₄	—	712	1	no	—
328	D₄	—	712	1	no	—
329	D₄	—	712	1	no	—
330	D₄	—	712	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	1	no	—
335	C₂×C₄	—	89	1	no	—
336	C₂×C₄	—	89	1	no	—
337	C₂×C₄	—	89	1	no	—
338	C₂×C₄	—	89	1	no	—
339	C₂×C₄	—	356	2	no	—
340	C₂×C₄	—	356	2	no	—
341	C₂×C₄	—	178	1	no	—
342	C₂×C₄	—	712	1	no	—
343	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
344	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
345	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
346	C₈	—	2	1	no	—
347	C₈	—	178	1	no	—
348	C₈	—	178	1	no	—
349	C₈	#13083574	1	2	yes	complemented
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	356	1	no	—
357	C₂²	—	356	1	no	—
358	C₂²	—	356	1	no	—
359	C₂²	—	356	1	no	—
360	C₂²	—	356	1	no	—
361	C₂²	—	356	1	no	—
362	C₂²	—	356	1	no	—
363	C₂²	—	356	1	no	—
364	C₂²	—	356	1	no	—
365	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	89	2	no	—
371	C₄	—	356	1	no	—
372	C₄	—	178	1	no	—
373	C₄	—	178	1	no	—
374	C₄	—	178	1	no	—
375	C₄	C₂×#13083574	1	1	yes	characteristic
376	C₄	#15668058	1	1	yes	characteristic
377	C₂	—	8	1	no	—
378	C₂	—	8	1	no	—
379	C₂	—	712	1	no	—
380	C₂	—	712	1	no	—
381	C₂	—	712	1	no	—
382	C₂	#19152269	1	1	yes	characteristic
383	C₂	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
384	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	#442×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#442	D₈₉
hypocenter	C₈₉	#3399
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #442

Complements

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

N	H ≅ G/N	type
1. #19152331	C₂	semidirect
2. #19152269	C₂	semidirect
3. #19151987	C₂	semidirect
4. #19151986	C₂	semidirect
5. #19151973	C₂	semidirect
6. #19151813	C₂	semidirect
7. #19151812	C₂	semidirect
8. #19151797	C₂	semidirect
9. #19151793	C₂	semidirect
10. #19151792	C₂	semidirect
11. #19151792	C₂	semidirect
12. #19151726	C₂	semidirect
13. #19151709	C₂	semidirect
14. #19151685	C₂	semidirect
15. #442×C₈₉	C₂	semidirect
154. C₈×C₈₉	#41	semidirect
193. C₈₉	#3399	semidirect
207. #442	D₈₉	semidirect
349. C₈	#13083574	semidirect

Maximal subgroups

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

subgroup	index
#19152331	2
#19152269	2
#19151987	2
#19151986	2
#19151973	2
#19151813	2
#19151812	2
#19151797	2
#19151793	2
#19151792	2
#19151726	2
#19151709	2
#19151685	2
#442×C₈₉	2
#3399	89