Subgroups — #26032100

Order 11392.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152296	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152239	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151954	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151925	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151901	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151835	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151829	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151816	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151746	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151734	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151680	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151661	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151593	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#15667932	C₂	1	1	yes	maximal, characteristic, complemented
15	#484×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
18	C₂×#13083561	C₂²	1	1	yes	characteristic, complemented
19	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
20	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
21	C₂²×#10623494	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic, complemented
23	#15668007	—	2	1	no	—
24	#15668003	C₂²	1	1	yes	characteristic, complemented
25	#15668001	C₂²	1	1	yes	characteristic
26	#15667997	—	2	1	no	—
27	#15667997	—	2	1	no	—
28	#15667987	—	2	1	no	—
29	#15667987	C₂²	1	1	yes	characteristic
30	#15667987	C₂²	1	1	yes	characteristic, complemented
31	#15667986	—	2	1	no	—
32	#15667986	C₂²	1	1	yes	characteristic
33	#15667982	—	2	1	no	—
34	#15667974	C₂²	1	1	yes	characteristic
35	#15667974	C₂²	1	1	yes	characteristic, complemented
36	C₄×C₂²⋊D₈₉	—	2	1	no	—
37	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
38	#15667970	C₂²	1	1	yes	characteristic, complemented
39	C₂²⋊C₄×D₈₉	—	2	1	no	—
40	#15667943	C₂²	1	1	yes	characteristic, complemented
41	#15667941	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
45	#15667932	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
49	#15667932	C₂²	1	1	yes	characteristic, complemented
50	#15667924	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
52	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic, complemented
55	#107×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂³×D₈₉	—	2	1	no	—
60	#13083575	—	2	2	no	—
61	#13083575	—	4	1	no	—
62	C₂×D₃₅₆	—	2	2	no	—
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	—	2	1	no	—
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	—	4	1	no	—
68	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
69	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
70	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊D₈₉	—	2	1	no	—
72	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₈₉	—	2	1	no	—
74	C₂×C₄×D₈₉	—	2	1	no	—
75	C₂×C₄×D₈₉	—	2	1	no	—
76	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
77	#13083561	—	4	1	no	—
78	D₇₁₂	—	2	2	no	—
79	D₇₁₂	—	4	1	no	—
80	C₂×#10623494	—	4	1	no	—
81	C₂×#10623494	C₂³	1	1	yes	characteristic
82	C₂×#10623494	C₂³	1	1	yes	characteristic
83	C₈×D₈₉	—	4	1	no	—
84	C₄⋊Dic₈₉	—	2	1	no	—
85	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
86	#13083553	—	2	1	no	—
87	#13083553	D₄	1	1	yes	characteristic
88	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
89	#13083551	—	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	#13083551	D₄	1	1	yes	characteristic
98	C₄×Dic₈₉	—	2	1	no	—
99	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	—	2	1	no	—
102	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	—	2	1	no	—
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	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₈₉	—	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	Q₈×C₈₉	—	2	1	no	—
120	Q₈×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	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	8	1	no	—
128	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₈₉	—	2	2	no	—
131	C₂²⋊D₈₉	—	4	2	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
136	D₃₅₆	—	2	1	no	—
137	D₃₅₆	—	2	1	no	—
138	D₃₅₆	—	2	1	no	—
139	D₃₅₆	—	2	1	no	—
140	D₃₅₆	—	2	1	no	—
141	D₃₅₆	—	4	2	no	—
142	D₃₅₆	—	4	2	no	—
143	D₃₅₆	—	8	1	no	—
144	D₃₅₆	—	8	1	no	—
145	D₃₅₆	#41	1	2	yes	—
146	C₄×D₈₉	—	2	1	no	—
147	C₄×D₈₉	—	2	1	no	—
148	C₄×D₈₉	—	4	2	no	—
149	C₄×D₈₉	—	8	1	no	—
150	C₄×D₈₉	—	8	1	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	#135	1	2	yes	—
158	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₈₉	—	2	1	no	—
160	C₂×D₈₉	—	2	2	no	—
161	C₂×D₈₉	—	2	1	no	—
162	C₂×D₈₉	—	2	1	no	—
163	C₂×D₈₉	—	4	1	no	—
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	—	8	4	no	—
172	C₄×C₈₉	—	2	1	no	—
173	C₄×C₈₉	—	4	1	no	—
174	C₄×C₈₉	—	4	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	C₂×C₈₉	—	2	2	no	—
184	C₂×C₈₉	#546	1	1	yes	characteristic
185	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
186	C₂×C₈₉	#576	1	1	yes	characteristic
187	D₈₉	—	4	1	no	—
188	D₈₉	—	4	2	no	—
189	D₈₉	—	8	1	no	—
190	D₈₉	—	8	1	no	—
191	#3297	—	89	1	no	maximal, Sylow
192	C₈₉	#3297	1	1	yes	characteristic, complemented, Sylow
193	#553	—	89	1	no	—
194	#546	—	89	1	no	—
195	#488	—	89	1	no	—
196	#484	D₈₉	1	1	yes	characteristic, complemented
197	#482	—	89	1	no	—
198	#478	—	89	1	no	—
199	#466	—	89	1	no	—
200	#465	—	89	1	no	—
201	#463	—	89	1	no	—
202	#449	—	89	1	no	—
203	#447	—	89	1	no	—
204	C₄×D₈	—	89	1	no	—
205	C₈×D₄	—	89	1	no	—
206	#416	—	89	1	no	—
207	C₂×#102	—	89	1	no	—
208	C₂×#41	—	89	1	no	—
209	C₂²×D₄	—	89	1	no	—
210	C₂×SD₁₆	—	89	1	no	—
211	C₂×D₈	—	89	1	no	—
212	C₂×D₈	—	89	1	no	—
213	C₂²×C₈	—	89	1	no	—
214	C₂²×C₈	—	89	1	no	—
215	#126	—	178	1	no	—
216	#125	—	89	1	no	—
217	#124	—	89	1	no	—
218	#123	—	178	1	no	—
219	#123	—	178	1	no	—
220	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
221	#121	—	89	1	no	—
222	#121	—	89	1	no	—
223	#121	—	89	1	no	—
224	#121	—	178	1	no	—
225	#121	—	178	1	no	—
226	#120	—	178	1	no	—
227	C₄×D₄	—	89	1	no	—
228	C₄×D₄	—	89	1	no	—
229	C₄×D₄	—	89	1	no	—
230	C₄×D₄	—	178	1	no	—
231	#117	—	89	1	no	—
232	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
233	C₂×C₂²⋊C₄	—	178	1	no	—
234	#107	C₂×D₈₉	1	1	yes	characteristic, complemented
235	#107	C₂×D₈₉	1	1	yes	characteristic, complemented
236	#106	—	89	1	no	—
237	C₄⋊C₈	—	89	1	no	—
238	#103	—	89	1	no	—
239	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
240	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
241	#102	—	89	1	no	—
242	#102	—	89	1	no	—
243	#102	—	89	1	no	—
244	#102	—	89	1	no	—
245	#102	—	89	1	no	—
246	#102	—	89	1	no	—
247	#102	—	89	1	no	—
248	C₂²⋊C₈	—	89	1	no	—
249	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
250	C₄×C₈	—	89	1	no	—
251	C₂⁴	—	178	1	no	—
252	#41	—	356	1	no	—
253	#41	—	178	2	no	—
254	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
255	C₂×D₄	—	89	1	no	—
256	C₂×D₄	—	89	1	no	—
257	C₂×D₄	—	89	1	no	—
258	C₂×D₄	—	89	1	no	—
259	C₂×D₄	—	356	1	no	—
260	C₂×D₄	—	178	2	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	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	89	1	no	—
268	C₂²×C₄	—	178	1	no	—
269	C₂²×C₄	—	178	1	no	—
270	C₂²×C₄	—	178	1	no	—
271	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
272	SD₁₆	—	356	1	no	—
273	D₈	—	356	1	no	—
274	D₈	—	178	2	no	—
275	C₂×C₈	—	89	1	no	—
276	C₂×C₈	—	89	1	no	—
277	C₂×C₈	—	356	1	no	—
278	C₂×C₈	—	356	1	no	—
279	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
280	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	89	1	no	—
284	C₄⋊C₄	—	89	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	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
290	C₂²⋊C₄	—	2	1	no	—
291	C₂²⋊C₄	—	89	1	no	—
292	C₂²⋊C₄	—	89	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	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	—	178	1	no	—
303	C₂³	—	89	1	no	—
304	C₂³	—	356	2	no	—
305	C₂³	—	356	2	no	—
306	C₂³	—	178	1	no	—
307	C₂³	—	178	1	no	—
308	C₂³	—	178	1	no	—
309	C₂³	—	712	1	no	—
310	C₂³	D₄×D₈₉	1	1	yes	characteristic
311	Q₈	—	2	1	no	—
312	Q₈	—	2	1	no	—
313	D₄	—	89	2	no	—
314	D₄	—	356	2	no	—
315	D₄	—	356	2	no	—
316	D₄	—	356	2	no	—
317	D₄	—	178	2	no	—
318	D₄	—	178	1	no	—
319	D₄	—	178	1	no	—
320	D₄	—	178	1	no	—
321	D₄	—	178	1	no	—
322	D₄	—	178	1	no	—
323	D₄	—	712	1	no	—
324	D₄	—	712	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	8	1	no	—
332	C₂×C₄	—	89	1	no	—
333	C₂×C₄	—	89	1	no	—
334	C₂×C₄	—	89	1	no	—
335	C₂×C₄	—	356	2	no	—
336	C₂×C₄	—	178	1	no	—
337	C₂×C₄	—	178	1	no	—
338	C₂×C₄	—	178	1	no	—
339	C₂×C₄	—	712	1	no	—
340	C₂×C₄	—	712	1	no	—
341	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
342	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	178	1	no	—
346	C₈	—	178	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	356	1	no	—
350	C₂²	—	356	1	no	—
351	C₂²	—	356	1	no	—
352	C₂²	—	356	1	no	—
353	C₂²	—	356	1	no	—
354	C₂²	—	356	1	no	—
355	C₂²	—	356	1	no	—
356	C₂²	—	356	1	no	—
357	C₂²	—	178	1	no	—
358	C₂²	—	178	2	no	—
359	C₂²	—	178	1	no	—
360	C₂²	—	178	1	no	—
361	C₂²	—	712	4	no	—
362	C₂²	#15668035	1	2	yes	—
363	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	4	1	no	—
366	C₄	—	4	1	no	—
367	C₄	—	4	1	no	—
368	C₄	—	4	1	no	—
369	C₄	—	356	1	no	—
370	C₄	—	178	1	no	—
371	C₄	—	178	1	no	—
372	C₄	—	178	1	no	—
373	C₄	#41×D₈₉	1	1	yes	characteristic
374	C₄	#15668058	1	1	yes	characteristic
375	C₂	—	2	2	no	—
376	C₂	—	356	2	no	—
377	C₂	—	356	1	no	—
378	C₂	—	712	1	no	—
379	C₂	—	712	1	no	—
380	C₂	#19152240	1	1	yes	characteristic
381	C₂	C₂×#15668035	1	1	yes	characteristic
382	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	#484×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#484	D₈₉
hypocenter	C₈₉	#3297
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
1. #19152296	C₂	semidirect
2. #19152239	C₂	semidirect
3. #19151954	C₂	semidirect
4. #19151925	C₂	semidirect
5. #19151901	C₂	semidirect
6. #19151835	C₂	semidirect
7. #19151829	C₂	semidirect
8. #19151816	C₂	semidirect
9. #19151746	C₂	semidirect
10. #19151734	C₂	semidirect
11. #19151680	C₂	semidirect
12. #19151661	C₂	semidirect
13. #19151593	C₂	semidirect
14. C₂×#15667932	C₂	semidirect
15. #484×C₈₉	C₂	semidirect
18. C₂×#13083561	C₂²	semidirect
22. C₂×C₈×D₈₉	C₂²	semidirect
24. #15668003	C₂²	semidirect
30. #15667987	C₂²	semidirect
35. #15667974	C₂²	semidirect
38. #15667970	C₂²	semidirect
40. #15667943	C₂²	semidirect
41. #15667941	C₂²	semidirect
42. #15667936	C₂²	semidirect
49. #15667932	C₂²	semidirect
50. #15667924	C₂²	semidirect
51. C₈×Dic₈₉	C₂²	semidirect
54. #107×C₈₉	C₂²	semidirect
55. #107×C₈₉	C₂²	semidirect
56. #103×C₈₉	C₂²	semidirect
57. #103×C₈₉	C₂²	semidirect
192. C₈₉	#3297	semidirect
196. #484	D₈₉	semidirect
234. #107	C₂×D₈₉	semidirect
235. #107	C₂×D₈₉	semidirect
239. #103	C₂×D₈₉	semidirect
240. #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
#19152296	2
#19152239	2
#19151954	2
#19151925	2
#19151901	2
#19151835	2
#19151829	2
#19151816	2
#19151746	2
#19151734	2
#19151680	2
#19151661	2
#19151593	2
C₂×#15667932	2
#484×C₈₉	2
#3297	89