Subgroups — #26031345

Order 11392.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152213	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152166	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151986	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151979	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151959	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151943	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151817	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151799	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151793	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151763	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151750	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151742	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151730	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151658	C₂	1	1	yes	maximal, characteristic, complemented
15	#463×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083564	C₂²	1	1	yes	characteristic, complemented
19	C₂×#13083563	C₂²	1	1	yes	characteristic
20	C₂×#13083563	C₂²	1	1	yes	characteristic, complemented
21	C₂×#13083562	C₂²	1	1	yes	characteristic
22	C₂×#13083559	C₂²	1	1	yes	characteristic
23	C₂×#13083559	C₂²	1	1	yes	characteristic, complemented
24	#15668011	C₂²	1	1	yes	characteristic, complemented
25	#15668009	C₂²	1	1	yes	characteristic
26	#15668008	C₂²	1	1	yes	characteristic
27	#15668005	—	2	1	no	—
28	#15668001	C₂²	1	1	yes	characteristic
29	#15667999	C₂²	1	1	yes	characteristic, complemented
30	#15667998	—	2	1	no	—
31	#15667995	—	2	1	no	—
32	#15667992	C₂²	1	1	yes	characteristic
33	#15667991	—	2	1	no	—
34	#15667985	—	2	1	no	—
35	#15667985	—	2	1	no	—
36	#15667983	C₂²	1	1	yes	characteristic
37	#15667982	—	2	1	no	—
38	#15667982	—	2	1	no	—
39	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic, complemented
40	#15667969	C₂²	1	1	yes	characteristic
41	C₄⋊#10623494	C₂²	1	1	yes	characteristic, complemented
42	#15667936	C₂²	1	1	yes	characteristic, complemented
43	#15667934	C₂²	1	1	yes	characteristic
44	#15667934	C₂²	1	1	yes	characteristic, complemented
45	#15667933	C₂²	1	1	yes	characteristic
46	#15667932	C₂²	1	1	yes	characteristic
47	#15667932	C₂²	1	1	yes	characteristic, complemented
48	#15667924	C₂²	1	1	yes	characteristic
49	#15667924	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
53	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
54	#124×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	C₂³×D₈₉	—	2	1	no	—
60	#13083575	—	4	1	no	—
61	#13083575	—	4	1	no	—
62	C₂×Dic₁₇₈	—	2	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	C₂³	1	1	yes	characteristic
68	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
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	D₄×D₈₉	—	4	1	no	—
74	D₄×D₈₉	—	4	1	no	—
75	C₂×C₄×D₈₉	—	2	1	no	—
76	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
77	#13083564	—	4	1	no	—
78	#13083563	—	4	1	no	—
79	#13083563	—	4	1	no	—
80	#13083562	—	4	1	no	—
81	#13083559	—	4	1	no	—
82	#13083559	—	4	1	no	—
83	C₂×#10623494	C₂³	1	1	yes	characteristic
84	C₂×#10623494	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₈₉	—	2	1	no	—
86	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
87	#13083553	—	2	1	no	—
88	#13083553	—	2	1	no	—
89	#13083553	D₄	1	1	yes	characteristic
90	#13083553	D₄	1	1	yes	characteristic
91	C₂²⋊Dic₈₉	—	2	1	no	—
92	C₂²⋊Dic₈₉	—	2	1	no	—
93	C₂²⋊Dic₈₉	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	—	2	1	no	—
96	#13083551	—	2	1	no	—
97	#13083551	D₄	1	1	yes	characteristic
98	#13083551	D₄	1	1	yes	characteristic
99	C₄×Dic₈₉	—	2	1	no	—
100	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	—	2	1	no	—
105	SD₁₆×C₈₉	—	4	1	no	—
106	D₈×C₈₉	—	4	1	no	—
107	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₉	—	2	1	no	—
111	C₂²⋊C₄×C₈₉	—	2	1	no	—
112	C₂²⋊C₄×C₈₉	—	2	1	no	—
113	C₄²×C₈₉	C₂³	1	1	yes	characteristic
114	C₂³×C₈₉	—	2	1	no	—
115	C₂³×C₈₉	—	2	1	no	—
116	C₂²×D₈₉	—	2	1	no	—
117	C₂²×D₈₉	—	2	1	no	—
118	C₂²×D₈₉	—	4	2	no	—
119	C₂²×D₈₉	—	8	1	no	—
120	C₂²×D₈₉	—	8	1	no	—
121	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₈₉	—	2	1	no	—
123	Q₈×C₈₉	—	2	1	no	—
124	D₄×C₈₉	—	2	1	no	—
125	D₄×C₈₉	—	2	1	no	—
126	D₄×C₈₉	—	2	1	no	—
127	D₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	8	1	no	—
131	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
134	C₂²⋊D₈₉	—	4	2	no	—
135	C₂²⋊D₈₉	—	8	1	no	—
136	C₂²⋊D₈₉	—	8	1	no	—
137	C₂²⋊D₈₉	—	8	1	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	—	2	1	no	—
140	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
143	D₃₅₆	—	2	1	no	—
144	D₃₅₆	—	2	1	no	—
145	D₃₅₆	—	2	1	no	—
146	D₃₅₆	—	2	1	no	—
147	D₃₅₆	—	4	2	no	—
148	D₃₅₆	—	8	1	no	—
149	D₃₅₆	—	8	1	no	—
150	C₄×D₈₉	—	2	1	no	—
151	C₄×D₈₉	—	2	1	no	—
152	C₄×D₈₉	—	4	2	no	—
153	Dic₁₇₈	—	2	1	no	—
154	Dic₁₇₈	—	2	1	no	—
155	Dic₁₇₈	—	8	1	no	—
156	C₈×C₈₉	—	2	1	no	—
157	C₈×C₈₉	—	2	1	no	—
158	#10623494	—	2	1	no	—
159	#10623494	—	2	1	no	—
160	C₂²×C₈₉	—	4	1	no	—
161	C₂²×C₈₉	—	4	1	no	—
162	C₂²×C₈₉	—	4	1	no	—
163	C₂²×C₈₉	—	4	1	no	—
164	C₂²×C₈₉	—	4	1	no	—
165	C₂²×C₈₉	—	4	1	no	—
166	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
167	C₂×D₈₉	—	2	1	no	—
168	C₂×D₈₉	—	2	1	no	—
169	C₂×D₈₉	—	2	1	no	—
170	C₂×D₈₉	—	4	1	no	—
171	C₂×D₈₉	—	4	1	no	—
172	C₂×D₈₉	—	4	1	no	—
173	C₂×D₈₉	—	4	1	no	—
174	C₂×D₈₉	—	4	1	no	—
175	C₂×D₈₉	—	4	1	no	—
176	C₂×D₈₉	—	8	4	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	2	1	no	—
179	C₄×C₈₉	—	4	1	no	—
180	C₄×C₈₉	—	4	1	no	—
181	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
182	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
183	Dic₈₉	—	2	1	no	—
184	Dic₈₉	—	2	1	no	—
185	Dic₈₉	—	2	1	no	—
186	Dic₈₉	—	4	1	no	—
187	Dic₈₉	—	4	1	no	—
188	C₂×C₈₉	—	8	1	no	—
189	C₂×C₈₉	—	8	1	no	—
190	C₂×C₈₉	#535	1	1	yes	characteristic
191	C₂×C₈₉	#576	1	1	yes	characteristic
192	C₂×C₈₉	#577	1	1	yes	characteristic
193	D₈₉	—	4	1	no	—
194	D₈₉	—	8	1	no	—
195	D₈₉	—	8	1	no	—
196	#3224	—	89	1	no	maximal, Sylow
197	C₈₉	#3224	1	1	yes	characteristic, complemented, Sylow
198	#542	—	89	1	no	—
199	#535	—	89	1	no	—
200	#496	—	89	1	no	—
201	#495	—	89	1	no	—
202	#489	—	89	1	no	—
203	#486	—	89	1	no	—
204	#464	—	89	1	no	—
205	#463	D₈₉	1	1	yes	characteristic, complemented
206	#460	—	89	1	no	—
207	#459	—	89	1	no	—
208	#452	—	89	1	no	—
209	#450	—	89	1	no	—
210	#449	—	89	1	no	—
211	#447	—	89	1	no	—
212	#433	—	89	1	no	—
213	C₂×#41	—	89	1	no	—
214	C₂²×D₄	—	89	1	no	—
215	C₂×Q₁₆	—	89	1	no	—
216	C₂×SD₁₆	—	89	1	no	—
217	C₂×SD₁₆	—	89	1	no	—
218	C₂×SD₁₆	—	89	1	no	—
219	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
220	C₂×D₈	—	89	1	no	—
221	C₂×D₈	—	89	1	no	—
222	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
223	C₄⋊Q₈	—	89	1	no	—
224	#127	—	89	1	no	—
225	#127	—	89	1	no	—
226	#126	—	178	1	no	—
227	#124	—	89	1	no	—
228	#124	—	89	1	no	—
229	#124	—	178	1	no	—
230	#124	C₂×D₈₉	1	1	yes	characteristic
231	#123	—	178	1	no	—
232	C₂²⋊Q₈	—	89	1	no	—
233	C₂²⋊Q₈	—	178	1	no	—
234	#121	—	89	1	no	—
235	#121	—	178	1	no	—
236	#121	—	178	1	no	—
237	#120	—	178	1	no	—
238	#120	—	178	1	no	—
239	C₄×Q₈	—	89	1	no	—
240	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic, complemented
241	#117	—	89	1	no	—
242	C₄⋊C₈	—	89	1	no	—
243	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
244	#103	—	89	1	no	—
245	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
246	#102	—	89	1	no	—
247	#102	—	89	1	no	—
248	#102	—	89	1	no	—
249	#102	—	89	1	no	—
250	#102	—	89	1	no	—
251	#102	C₂×D₈₉	1	1	yes	characteristic
252	C₂²⋊C₈	—	89	1	no	—
253	C₂²⋊C₈	—	89	1	no	—
254	#97	—	89	1	no	—
255	C₄×C₈	—	89	1	no	—
256	C₂⁴	—	178	1	no	—
257	#41	—	356	1	no	—
258	#41	—	356	1	no	—
259	C₂×Q₈	—	89	1	no	—
260	C₂×Q₈	—	178	1	no	—
261	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
262	C₂×D₄	—	89	1	no	—
263	C₂×D₄	—	89	1	no	—
264	C₂×D₄	—	356	1	no	—
265	C₂×D₄	—	356	1	no	—
266	C₂×D₄	—	178	1	no	—
267	C₂×D₄	—	178	1	no	—
268	C₂×D₄	—	178	1	no	—
269	C₂×D₄	—	178	1	no	—
270	C₂×D₄	—	178	1	no	—
271	C₂×D₄	—	178	1	no	—
272	C₂×D₄	—	178	1	no	—
273	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
274	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	89	1	no	—
277	C₂²×C₄	—	178	1	no	—
278	Q₁₆	—	356	1	no	—
279	SD₁₆	—	4	1	no	—
280	SD₁₆	—	356	1	no	—
281	SD₁₆	—	356	1	no	—
282	SD₁₆	—	356	1	no	—
283	D₈	—	4	1	no	—
284	D₈	—	356	1	no	—
285	D₈	—	356	1	no	—
286	C₂×C₈	—	89	1	no	—
287	C₂×C₈	—	89	1	no	—
288	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
289	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
290	C₄⋊C₄	—	89	1	no	—
291	C₄⋊C₄	—	89	1	no	—
292	C₄⋊C₄	—	89	1	no	—
293	C₄⋊C₄	—	178	1	no	—
294	C₄⋊C₄	—	178	1	no	—
295	C₄⋊C₄	—	178	1	no	—
296	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
297	C₂²⋊C₄	—	2	1	no	—
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	89	1	no	—
301	C₂²⋊C₄	—	89	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	C₂²⋊C₄	—	178	1	no	—
307	C₂²⋊C₄	—	178	1	no	—
308	C₄²	—	89	1	no	—
309	C₄²	—	178	1	no	—
310	C₄²	C₂²×D₈₉	1	1	yes	characteristic
311	C₂³	—	2	1	no	—
312	C₂³	—	2	1	no	—
313	C₂³	—	89	1	no	—
314	C₂³	—	356	2	no	—
315	C₂³	—	178	1	no	—
316	C₂³	—	178	1	no	—
317	C₂³	—	712	1	no	—
318	C₂³	—	712	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	178	1	no	—
322	Q₈	—	178	1	no	—
323	Q₈	—	712	1	no	—
324	D₄	—	2	1	no	—
325	D₄	—	2	1	no	—
326	D₄	—	2	1	no	—
327	D₄	—	2	1	no	—
328	D₄	—	356	2	no	—
329	D₄	—	356	2	no	—
330	D₄	—	178	1	no	—
331	D₄	—	178	1	no	—
332	D₄	—	178	1	no	—
333	D₄	—	178	1	no	—
334	D₄	—	712	1	no	—
335	D₄	—	712	1	no	—
336	D₄	—	712	1	no	—
337	D₄	—	712	1	no	—
338	D₄	—	712	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	8	1	no	—
342	C₂×C₄	—	89	1	no	—
343	C₂×C₄	—	89	1	no	—
344	C₂×C₄	—	89	1	no	—
345	C₂×C₄	—	356	2	no	—
346	C₂×C₄	—	178	1	no	—
347	C₂×C₄	—	178	1	no	—
348	C₂×C₄	—	178	1	no	—
349	C₂×C₄	—	178	1	no	—
350	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
351	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
352	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	178	1	no	—
356	C₈	—	178	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	4	1	no	—
363	C₂²	—	356	1	no	—
364	C₂²	—	356	1	no	—
365	C₂²	—	356	1	no	—
366	C₂²	—	356	1	no	—
367	C₂²	—	356	1	no	—
368	C₂²	—	356	1	no	—
369	C₂²	—	178	1	no	—
370	C₂²	—	178	1	no	—
371	C₂²	—	178	1	no	—
372	C₂²	—	712	4	no	—
373	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	4	1	no	—
378	C₄	—	356	1	no	—
379	C₄	—	356	1	no	—
380	C₄	—	178	1	no	—
381	C₄	—	178	1	no	—
382	C₄	—	178	1	no	—
383	C₄	#15668057	1	1	yes	characteristic
384	C₄	#15668058	1	1	yes	characteristic
385	C₂	—	8	1	no	—
386	C₂	—	8	1	no	—
387	C₂	—	356	1	no	—
388	C₂	—	712	1	no	—
389	C₂	—	712	1	no	—
390	C₂	#19152164	1	1	yes	characteristic
391	C₂	#19152402	1	1	yes	characteristic
392	C₂	#19152405	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#463×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#463	D₈₉
hypocenter	C₈₉	#3224
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #19152213	C₂	semidirect
2. #19152166	C₂	semidirect
3. #19151986	C₂	semidirect
4. #19151979	C₂	semidirect
5. #19151959	C₂	semidirect
6. #19151943	C₂	semidirect
7. #19151817	C₂	semidirect
8. #19151799	C₂	semidirect
9. #19151793	C₂	semidirect
10. #19151763	C₂	semidirect
11. #19151750	C₂	semidirect
12. #19151742	C₂	semidirect
13. #19151730	C₂	semidirect
14. #19151658	C₂	semidirect
15. #463×C₈₉	C₂	semidirect
18. C₂×#13083564	C₂²	semidirect
20. C₂×#13083563	C₂²	semidirect
23. C₂×#13083559	C₂²	semidirect
24. #15668011	C₂²	semidirect
29. #15667999	C₂²	semidirect
39. C₄×Dic₁₇₈	C₂²	semidirect
41. C₄⋊#10623494	C₂²	semidirect
42. #15667936	C₂²	semidirect
44. #15667934	C₂²	semidirect
47. #15667932	C₂²	semidirect
50. #15667922	C₂²	semidirect
51. C₈×Dic₈₉	C₂²	semidirect
52. C₂×SD₁₆×C₈₉	C₂²	semidirect
55. C₄×D₄×C₈₉	C₂²	semidirect
56. C₄⋊C₈×C₈₉	C₂²	semidirect
57. #103×C₈₉	C₂²	semidirect
197. C₈₉	#3224	semidirect
205. #463	D₈₉	semidirect
219. C₂×SD₁₆	C₂×D₈₉	semidirect
240. C₄×D₄	C₂×D₈₉	semidirect
243. C₄⋊C₈	C₂×D₈₉	semidirect
245. #103	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152213	2
#19152166	2
#19151986	2
#19151979	2
#19151959	2
#19151943	2
#19151817	2
#19151799	2
#19151793	2
#19151763	2
#19151750	2
#19151742	2
#19151730	2
#19151658	2
#463×C₈₉	2
#3224	89