Subgroups — #26032210

Order 11392.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152251	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152243	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151960	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151930	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151918	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151852	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151828	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151814	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151744	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151733	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151708	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151668	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151596	C₂	1	1	yes	maximal, characteristic, complemented
14	#102×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#476×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	—	2	1	no	—
17	C₂×#13083573	C₂²	1	1	yes	characteristic
18	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
19	C₂×#13083563	C₂²	1	1	yes	characteristic, complemented
20	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
21	C₂×#13083559	C₂²	1	1	yes	characteristic
22	C₂×#13083557	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
24	#15668000	C₂²	1	1	yes	characteristic
25	#15667997	—	2	1	no	—
26	#15667997	—	2	1	no	—
27	#15667996	—	2	1	no	—
28	#15667992	C₂²	1	1	yes	characteristic
29	#15667988	—	2	1	no	—
30	#15667987	C₂²	1	1	yes	characteristic
31	#15667987	C₂²	1	1	yes	characteristic
32	#15667987	C₂²	1	1	yes	characteristic, complemented
33	#15667986	—	2	1	no	—
34	#15667982	—	2	1	no	—
35	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
36	C₄×D₃₅₆	C₂²	1	1	yes	characteristic, complemented
37	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic, complemented
39	C₂×#13083551	—	2	1	no	—
40	#15667943	C₂²	1	1	yes	characteristic, complemented
41	#15667937	C₂²	1	1	yes	characteristic, complemented
42	#15667935	C₂²	1	1	yes	characteristic, complemented
43	#15667935	C₂²	1	1	yes	characteristic, complemented
44	#15667934	C₂²	1	1	yes	characteristic
45	#15667933	C₂²	1	1	yes	characteristic
46	#15667932	C₂²	1	1	yes	characteristic
47	#15667932	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic, complemented
49	#15667924	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic, complemented
51	#15667923	C₂²	1	1	yes	characteristic, complemented
52	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
53	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
54	#106×C₈₉	C₂²	1	1	yes	characteristic, complemented
55	#106×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	C₂³×D₈₉	—	2	1	no	—
60	#13083575	—	4	4	no	—
61	#13083573	—	2	2	no	—
62	#13083573	—	4	1	no	—
63	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
64	C₂×D₃₅₆	—	2	1	no	—
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	D₄	1	1	yes	characteristic
68	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
69	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊D₈₉	—	2	1	no	—
71	C₂×C₂²⋊D₈₉	—	2	1	no	—
72	D₄×D₈₉	—	2	2	no	—
73	D₄×D₈₉	—	4	1	no	—
74	C₂²×Dic₈₉	—	2	1	no	—
75	C₂×C₄×D₈₉	—	2	1	no	—
76	C₂×C₄×D₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	—	2	1	no	—
78	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
80	#13083563	—	4	1	no	—
81	D₇₁₂	—	4	1	no	—
82	#13083559	—	2	2	no	—
83	#13083557	—	4	1	no	—
84	C₂×#10623494	C₂³	1	1	yes	characteristic
85	C₂×#10623494	C₂³	1	1	yes	characteristic
86	C₈×D₈₉	—	4	1	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
89	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
90	#13083553	—	2	1	no	—
91	#13083553	—	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	#13083551	—	2	1	no	—
98	#13083551	—	2	1	no	—
99	#13083551	D₄	1	1	yes	characteristic
100	#13083551	D₄	1	1	yes	characteristic
101	C₂×Q₈×C₈₉	—	2	1	no	—
102	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	—	2	1	no	—
104	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₈₉	—	2	1	no	—
107	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₉	—	2	1	no	—
111	C₄²×C₈₉	C₂³	1	1	yes	characteristic
112	C₂³×C₈₉	—	2	1	no	—
113	C₂²×D₈₉	—	2	1	no	—
114	C₂²×D₈₉	—	2	1	no	—
115	C₂²×D₈₉	—	4	2	no	—
116	C₂²×D₈₉	—	4	2	no	—
117	C₂²×D₈₉	—	8	1	no	—
118	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
120	Q₈×C₈₉	—	4	2	no	—
121	D₄×C₈₉	—	2	1	no	—
122	D₄×C₈₉	#41	1	2	yes	complemented
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	8	1	no	—
127	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₈₉	—	4	2	no	—
131	C₂²⋊D₈₉	—	4	2	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	4	2	no	—
135	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	D₃₅₆	—	2	2	no	—
138	D₃₅₆	—	2	1	no	—
139	D₃₅₆	—	2	1	no	—
140	D₃₅₆	—	2	1	no	—
141	D₃₅₆	—	2	1	no	—
142	D₃₅₆	—	4	2	no	—
143	D₃₅₆	—	4	2	no	—
144	D₃₅₆	—	8	1	no	—
145	C₄×D₈₉	—	2	1	no	—
146	C₄×D₈₉	—	2	1	no	—
147	C₄×D₈₉	—	2	1	no	—
148	C₄×D₈₉	—	2	1	no	—
149	C₄×D₈₉	—	4	2	no	—
150	C₄×D₈₉	—	4	2	no	—
151	C₄×D₈₉	—	4	2	no	—
152	Dic₁₇₈	—	2	1	no	—
153	Dic₁₇₈	—	2	1	no	—
154	C₈×C₈₉	—	2	1	no	—
155	C₈×C₈₉	—	2	1	no	—
156	#10623494	—	2	1	no	—
157	#10623494	—	2	1	no	—
158	C₂²×C₈₉	—	2	2	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	—	4	1	no	—
161	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×D₈₉	—	2	1	no	—
163	C₂×D₈₉	—	2	1	no	—
164	C₂×D₈₉	—	2	1	no	—
165	C₂×D₈₉	—	2	1	no	—
166	C₂×D₈₉	—	2	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	C₂×D₈₉	—	8	4	no	—
174	C₂×D₈₉	AGL(1,ℤ/8)	1	2	yes	—
175	C₄×C₈₉	—	2	1	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	4	1	no	—
178	C₄×C₈₉	—	4	1	no	—
179	C₄×C₈₉	—	4	1	no	—
180	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
181	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
182	Dic₈₉	—	2	1	no	—
183	Dic₈₉	—	2	1	no	—
184	Dic₈₉	—	4	1	no	—
185	Dic₈₉	—	4	1	no	—
186	C₂×C₈₉	—	4	2	no	—
187	C₂×C₈₉	#546	1	1	yes	characteristic
188	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₂×C₈₉	#577	1	1	yes	characteristic
190	D₈₉	—	2	2	no	—
191	D₈₉	—	4	1	no	—
192	D₈₉	—	8	1	no	—
193	D₈₉	—	8	1	no	—
194	#3306	—	89	1	no	maximal, Sylow
195	C₈₉	#3306	1	1	yes	characteristic, complemented, Sylow
196	#547	—	89	1	no	—
197	#546	—	89	1	no	—
198	#489	—	89	1	no	—
199	#483	—	89	1	no	—
200	#481	—	89	1	no	—
201	#476	D₈₉	1	1	yes	characteristic, complemented
202	#469	—	89	1	no	—
203	#465	—	89	1	no	—
204	#463	—	89	1	no	—
205	#449	—	89	1	no	—
206	#447	—	89	1	no	—
207	#442	—	89	1	no	—
208	#435	—	89	1	no	—
209	#417	—	89	1	no	—
210	C₂×#102	—	89	1	no	—
211	C₂×#41	—	89	1	no	—
212	C₂×#41	—	178	1	no	—
213	C₂²×D₄	—	89	1	no	—
214	C₂×SD₁₆	—	89	1	no	—
215	C₂×D₈	—	89	1	no	—
216	C₂×D₈	—	89	1	no	—
217	C₂×M₁₆	—	89	1	no	—
218	C₂²×C₈	—	89	1	no	—
219	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
220	#124	—	89	1	no	—
221	#123	—	178	1	no	—
222	#123	—	178	1	no	—
223	#123	—	178	1	no	—
224	C₂²⋊Q₈	—	89	1	no	—
225	C₂²⋊Q₈	—	178	1	no	—
226	#121	—	89	1	no	—
227	#121	—	89	1	no	—
228	#121	—	89	1	no	—
229	#121	—	178	1	no	—
230	#120	—	178	1	no	—
231	C₄×D₄	—	89	1	no	—
232	C₄×D₄	—	89	1	no	—
233	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
234	C₂×C₄⋊C₄	—	89	1	no	—
235	C₂×C₄⋊C₄	—	89	1	no	—
236	C₂×C₂²⋊C₄	—	178	1	no	—
237	#106	—	89	1	no	—
238	#106	C₂×D₈₉	1	1	yes	characteristic, complemented
239	#106	C₂×D₈₉	1	1	yes	characteristic, complemented
240	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
241	#103	—	89	1	no	—
242	#103	—	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	—	89	1	no	—
248	#102	—	89	1	no	—
249	#102	C₂×D₈₉	1	1	yes	characteristic
250	#102	C₂×D₈₉	1	1	yes	characteristic
251	C₂²⋊C₈	—	89	1	no	—
252	C₂²⋊C₈	—	89	1	no	—
253	#97	—	89	1	no	—
254	C₂⁴	—	178	1	no	—
255	#41	—	356	4	no	—
256	#41	—	356	1	no	—
257	#41	—	178	2	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	89	1	no	—
260	C₂×D₄	—	89	1	no	—
261	C₂×D₄	—	89	1	no	—
262	C₂×D₄	—	89	1	no	—
263	C₂×D₄	—	356	1	no	—
264	C₂×D₄	—	178	1	no	—
265	C₂×D₄	—	178	1	no	—
266	C₂×D₄	—	178	1	no	—
267	C₂×D₄	—	178	1	no	—
268	C₂×D₄	—	178	1	no	—
269	C₂×D₄	—	178	2	no	—
270	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	89	1	no	—
273	C₂²×C₄	—	89	1	no	—
274	C₂²×C₄	—	178	1	no	—
275	C₂²×C₄	—	178	1	no	—
276	C₂²×C₄	—	178	1	no	—
277	C₂²×C₄	—	178	1	no	—
278	SD₁₆	—	356	1	no	—
279	D₈	—	356	1	no	—
280	D₈	—	178	2	no	—
281	M₁₆	—	356	1	no	—
282	C₂×C₈	—	89	1	no	—
283	C₂×C₈	—	89	1	no	—
284	C₂×C₈	—	356	1	no	—
285	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
286	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	89	1	no	—
289	C₄⋊C₄	—	89	1	no	—
290	C₄⋊C₄	—	178	1	no	—
291	C₄⋊C₄	—	178	1	no	—
292	C₄⋊C₄	—	178	1	no	—
293	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
294	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
295	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
296	C₂²⋊C₄	—	2	1	no	—
297	C₂²⋊C₄	—	89	1	no	—
298	C₂²⋊C₄	—	89	1	no	—
299	C₂²⋊C₄	—	178	1	no	—
300	C₂²⋊C₄	—	178	1	no	—
301	C₂²⋊C₄	—	178	1	no	—
302	C₂²⋊C₄	—	178	1	no	—
303	C₂²⋊C₄	—	178	1	no	—
304	C₂²⋊C₄	—	178	1	no	—
305	C₂²⋊C₄	—	178	1	no	—
306	C₄²	C₂²×D₈₉	1	1	yes	characteristic
307	C₂³	—	2	1	no	—
308	C₂³	—	89	1	no	—
309	C₂³	—	89	1	no	—
310	C₂³	—	356	2	no	—
311	C₂³	—	356	2	no	—
312	C₂³	—	178	1	no	—
313	C₂³	—	178	1	no	—
314	C₂³	—	712	1	no	—
315	Q₈	—	4	2	no	—
316	Q₈	—	178	1	no	—
317	Q₈	—	178	1	no	—
318	D₄	—	2	1	no	—
319	D₄	—	356	2	no	—
320	D₄	—	356	2	no	—
321	D₄	—	356	2	no	—
322	D₄	—	356	2	no	—
323	D₄	—	178	2	no	—
324	D₄	—	178	1	no	—
325	D₄	—	178	1	no	—
326	D₄	—	178	1	no	—
327	D₄	—	178	1	no	—
328	D₄	—	712	1	no	—
329	D₄	#13083575	1	2	yes	complemented
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	8	1	no	—
334	C₂×C₄	—	89	1	no	—
335	C₂×C₄	—	89	1	no	—
336	C₂×C₄	—	356	2	no	—
337	C₂×C₄	—	356	2	no	—
338	C₂×C₄	—	356	2	no	—
339	C₂×C₄	—	356	2	no	—
340	C₂×C₄	—	178	1	no	—
341	C₂×C₄	—	178	1	no	—
342	C₂×C₄	—	178	1	no	—
343	C₂×C₄	—	178	1	no	—
344	C₂×C₄	—	178	1	no	—
345	C₂×C₄	—	178	1	no	—
346	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
347	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
348	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	178	1	no	—
352	C₈	—	178	1	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	89	2	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	—	178	1	no	—
364	C₂²	—	178	1	no	—
365	C₂²	—	178	1	no	—
366	C₂²	—	178	1	no	—
367	C₂²	—	178	1	no	—
368	C₂²	—	712	4	no	—
369	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	4	1	no	—
373	C₄	—	4	1	no	—
374	C₄	—	4	1	no	—
375	C₄	—	356	1	no	—
376	C₄	—	356	1	no	—
377	C₄	—	178	1	no	—
378	C₄	—	178	1	no	—
379	C₄	C₂×#13083575	1	1	yes	characteristic
380	C₄	#15668058	1	1	yes	characteristic
381	C₂	—	4	2	no	—
382	C₂	—	356	1	no	—
383	C₂	—	178	2	no	—
384	C₂	—	712	1	no	—
385	C₂	—	712	1	no	—
386	C₂	#19152239	1	1	yes	characteristic
387	C₂	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
388	C₂	#19152404	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#476×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#476	D₈₉
hypocenter	C₈₉	#3306
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #19152251	C₂	semidirect
2. #19152243	C₂	semidirect
3. #19151960	C₂	semidirect
4. #19151930	C₂	semidirect
5. #19151918	C₂	semidirect
6. #19151852	C₂	semidirect
7. #19151828	C₂	semidirect
8. #19151814	C₂	semidirect
9. #19151744	C₂	semidirect
10. #19151733	C₂	semidirect
11. #19151708	C₂	semidirect
12. #19151668	C₂	semidirect
13. #19151596	C₂	semidirect
14. #102×D₈₉	C₂	semidirect
15. #476×C₈₉	C₂	semidirect
19. C₂×#13083563	C₂²	semidirect
22. C₂×#13083557	C₂²	semidirect
32. #15667987	C₂²	semidirect
36. C₄×D₃₅₆	C₂²	semidirect
38. C₄⋊C₄×D₈₉	C₂²	semidirect
40. #15667943	C₂²	semidirect
41. #15667937	C₂²	semidirect
42. #15667935	C₂²	semidirect
43. #15667935	C₂²	semidirect
48. #15667932	C₂²	semidirect
50. #15667924	C₂²	semidirect
51. #15667923	C₂²	semidirect
52. C₄⋊Q₈×C₈₉	C₂²	semidirect
54. #106×C₈₉	C₂²	semidirect
55. #106×C₈₉	C₂²	semidirect
56. C₄⋊C₈×C₈₉	C₂²	semidirect
122. D₄×C₈₉	#41	semidirect
195. C₈₉	#3306	semidirect
201. #476	D₈₉	semidirect
219. C₄⋊Q₈	C₂×D₈₉	semidirect
238. #106	C₂×D₈₉	semidirect
239. #106	C₂×D₈₉	semidirect
240. C₄⋊C₈	C₂×D₈₉	semidirect
329. D₄	#13083575	semidirect

Maximal subgroups

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

subgroup	index
#19152251	2
#19152243	2
#19151960	2
#19151930	2
#19151918	2
#19151852	2
#19151828	2
#19151814	2
#19151744	2
#19151733	2
#19151708	2
#19151668	2
#19151596	2
#102×D₈₉	2
#476×C₈₉	2
#3306	89