Subgroups — #26031034

Order 11392.

420 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152111	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151941	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151937	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151927	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151913	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151837	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151829	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151827	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151763	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151759	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151752	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151738	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#15667924	C₂	1	1	yes	maximal, characteristic, complemented
15	#467×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
19	C₂×#13083564	C₂²	1	1	yes	characteristic, complemented
20	C₂×#13083563	C₂²	1	1	yes	characteristic, complemented
21	C₂×#13083561	C₂²	1	1	yes	characteristic, complemented
22	C₂²×#10623494	C₂²	1	1	yes	characteristic
23	#15667997	—	2	1	no	—
24	#15667992	C₂²	1	1	yes	characteristic
25	#15667991	C₂²	1	1	yes	characteristic
26	#15667990	—	2	1	no	—
27	#15667989	—	2	1	no	—
28	#15667988	C₂²	1	1	yes	characteristic
29	#15667988	C₂²	1	1	yes	characteristic
30	#15667987	C₂²	1	1	yes	characteristic
31	#15667986	C₂²	1	1	yes	characteristic
32	#15667982	—	2	1	no	—
33	#15667974	—	2	1	no	—
34	#15667972	—	2	1	no	—
35	#15667970	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₉	—	2	1	no	—
37	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
38	C₂²⋊C₄×D₈₉	—	2	1	no	—
39	#15667945	C₂²	1	1	yes	characteristic, complemented
40	#15667944	C₂²	1	1	yes	characteristic, complemented
41	#15667943	C₂²	1	1	yes	characteristic, complemented
42	#15667937	C₂²	1	1	yes	characteristic, complemented
43	#15667936	C₂²	1	1	yes	characteristic, complemented
44	#15667936	C₂²	1	1	yes	characteristic, complemented
45	#15667935	C₂²	1	1	yes	characteristic, complemented
46	#15667934	C₂²	1	1	yes	characteristic, complemented
47	#15667932	C₂²	1	1	yes	characteristic, complemented
48	#15667924	C₂²	1	1	yes	characteristic
49	#15667924	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667924	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
53	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	#107×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
58	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₈₉	D₄	1	1	yes	characteristic
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	Q₈×D₈₉	—	4	1	no	—
65	Q₈×D₈₉	—	4	1	no	—
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	—	2	1	no	—
68	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊D₈₉	—	2	1	no	—
70	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	—	2	1	no	—
74	C₂×C₄×D₈₉	—	4	1	no	—
75	C₂×C₄×D₈₉	—	4	1	no	—
76	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
77	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
78	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
80	#13083564	—	4	1	no	—
81	#13083563	—	4	1	no	—
82	#13083561	—	4	1	no	—
83	C₂×#10623494	—	4	1	no	—
84	C₂×#10623494	C₂³	1	1	yes	characteristic
85	C₂×#10623494	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₈₉	—	2	1	no	—
87	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
89	#13083553	—	2	1	no	—
90	#13083553	—	2	1	no	—
91	#13083553	—	2	1	no	—
92	#13083553	—	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	—	2	1	no	—
98	#13083551	—	2	1	no	—
99	#13083551	—	2	1	no	—
100	C₄×Dic₈₉	—	2	1	no	—
101	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
104	Q₁₆×C₈₉	—	4	1	no	—
105	C₂×C₈×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	—	2	1	no	—
110	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
112	C₂²⋊C₄×C₈₉	—	2	1	no	—
113	C₂²⋊C₄×C₈₉	—	2	1	no	—
114	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₉	—	2	2	no	—
116	C₂²×D₈₉	—	2	2	no	—
117	C₂²×D₈₉	—	2	1	no	—
118	C₂²×D₈₉	—	4	1	no	—
119	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
121	Q₈×C₈₉	—	2	1	no	—
122	Q₈×C₈₉	—	2	1	no	—
123	Q₈×C₈₉	—	2	1	no	—
124	Q₈×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
133	C₂²⋊D₈₉	—	4	2	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	C₂×Dic₈₉	—	2	1	no	—
137	C₂×Dic₈₉	—	4	1	no	—
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	D₃₅₆	—	4	2	no	—
144	C₄×D₈₉	—	2	1	no	—
145	C₄×D₈₉	—	2	1	no	—
146	C₄×D₈₉	—	2	1	no	—
147	C₄×D₈₉	—	2	1	no	—
148	C₄×D₈₉	—	4	1	no	—
149	C₄×D₈₉	—	4	1	no	—
150	C₄×D₈₉	—	4	2	no	—
151	C₄×D₈₉	—	4	2	no	—
152	C₄×D₈₉	—	4	2	no	—
153	Dic₁₇₈	—	2	1	no	—
154	Dic₁₇₈	—	2	1	no	—
155	Dic₁₇₈	—	4	2	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	—	2	1	no	—
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	#137	1	2	yes	—
163	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₈₉	—	2	1	no	—
165	C₂×D₈₉	—	2	1	no	—
166	C₂×D₈₉	—	2	1	no	—
167	C₂×D₈₉	—	2	1	no	—
168	C₂×D₈₉	—	2	1	no	—
169	C₂×D₈₉	—	4	4	no	—
170	C₂×D₈₉	—	4	1	no	—
171	C₂×D₈₉	—	4	1	no	—
172	C₂×D₈₉	—	4	1	no	—
173	C₂×D₈₉	#135	1	2	yes	—
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	—	4	1	no	—
176	C₄×C₈₉	—	4	1	no	—
177	C₄×C₈₉	—	4	1	no	—
178	C₄×C₈₉	—	4	1	no	—
179	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
180	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
181	Dic₈₉	—	2	1	no	—
182	Dic₈₉	—	2	1	no	—
183	Dic₈₉	—	4	1	no	—
184	Dic₈₉	—	4	1	no	—
185	Dic₈₉	—	4	1	no	—
186	C₂×C₈₉	—	2	2	no	—
187	C₂×C₈₉	#534	1	1	yes	characteristic
188	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
189	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
190	D₈₉	—	2	2	no	—
191	D₈₉	—	4	1	no	—
192	D₈₉	—	8	1	no	—
193	#3190	—	89	1	no	maximal, Sylow
194	C₈₉	#3190	1	1	yes	characteristic, complemented, Sylow
195	#525	—	89	1	no	—
196	C₂×C₂²⋊Q₈	—	89	1	no	—
197	#485	—	89	1	no	—
198	#485	—	89	1	no	—
199	#483	—	89	1	no	—
200	#480	—	89	1	no	—
201	#467	—	89	1	no	—
202	#467	D₈₉	1	1	yes	characteristic, complemented
203	#465	—	89	1	no	—
204	#465	—	89	1	no	—
205	#452	—	89	1	no	—
206	#452	—	89	1	no	—
207	#451	—	89	1	no	—
208	#448	—	89	1	no	—
209	C₂×C₂²⋊C₈	—	89	1	no	—
210	C₂×#41	—	89	1	no	—
211	C₂²×Q₈	—	89	1	no	—
212	C₂³×C₄	—	89	1	no	—
213	C₂×Q₁₆	—	89	1	no	—
214	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
215	C₂×SD₁₆	—	89	1	no	—
216	C₂×SD₁₆	—	89	1	no	—
217	C₂²×C₈	—	89	1	no	—
218	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
219	#123	—	178	1	no	—
220	C₂²⋊Q₈	—	89	1	no	—
221	C₂²⋊Q₈	—	89	1	no	—
222	C₂²⋊Q₈	—	89	1	no	—
223	C₂²⋊Q₈	—	89	1	no	—
224	C₂²⋊Q₈	—	178	1	no	—
225	C₂²⋊Q₈	—	178	1	no	—
226	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
227	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
228	#121	—	89	1	no	—
229	#121	—	89	1	no	—
230	#120	—	178	1	no	—
231	C₄×D₄	—	178	1	no	—
232	C₄×D₄	—	178	1	no	—
233	#117	—	89	1	no	—
234	C₂×C₄⋊C₄	—	89	1	no	—
235	C₂×C₄⋊C₄	—	178	1	no	—
236	C₂×C₂²⋊C₄	—	178	1	no	—
237	#107	—	89	1	no	—
238	#107	C₂×D₈₉	1	1	yes	characteristic, complemented
239	#106	—	89	1	no	—
240	#106	—	89	1	no	—
241	#103	—	89	1	no	—
242	#103	—	89	1	no	—
243	#103	—	89	1	no	—
244	#103	—	89	1	no	—
245	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
246	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
247	#102	—	89	1	no	—
248	#102	—	89	1	no	—
249	C₂²⋊C₈	—	89	1	no	—
250	C₂²⋊C₈	—	89	1	no	—
251	C₂²⋊C₈	—	89	1	no	—
252	C₂²⋊C₈	—	89	1	no	—
253	C₂⁴	—	89	1	no	—
254	#41	—	356	1	no	—
255	#41	—	356	1	no	—
256	C₂×Q₈	—	89	1	no	—
257	C₂×Q₈	—	356	1	no	—
258	C₂×Q₈	—	356	1	no	—
259	C₂×Q₈	—	178	1	no	—
260	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
261	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
262	C₂×D₄	—	89	1	no	—
263	C₂×D₄	—	178	1	no	—
264	C₂×D₄	—	178	1	no	—
265	C₂×D₄	—	178	1	no	—
266	C₂²×C₄	—	89	1	no	—
267	C₂²×C₄	—	89	1	no	—
268	C₂²×C₄	—	89	1	no	—
269	C₂²×C₄	—	89	1	no	—
270	C₂²×C₄	—	89	1	no	—
271	C₂²×C₄	—	356	1	no	—
272	C₂²×C₄	—	356	1	no	—
273	C₂²×C₄	—	178	1	no	—
274	C₂²×C₄	—	178	1	no	—
275	C₂²×C₄	—	178	1	no	—
276	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
277	Q₁₆	—	4	1	no	—
278	Q₁₆	—	356	1	no	—
279	SD₁₆	—	356	1	no	—
280	SD₁₆	—	356	1	no	—
281	C₂×C₈	—	4	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	—	2	1	no	—
289	C₄⋊C₄	—	89	1	no	—
290	C₄⋊C₄	—	89	1	no	—
291	C₄⋊C₄	—	178	1	no	—
292	C₄⋊C₄	—	178	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	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	178	1	no	—
307	C₄²	—	178	1	no	—
308	C₂³	—	89	1	no	—
309	C₂³	—	89	1	no	—
310	C₂³	—	356	1	no	—
311	C₂³	—	178	2	no	—
312	C₂³	—	178	1	no	—
313	C₂³	—	178	2	no	—
314	C₂³	D₄×D₈₉	1	1	yes	characteristic
315	Q₈	—	2	1	no	—
316	Q₈	—	2	1	no	—
317	Q₈	—	2	1	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	356	2	no	—
320	Q₈	—	178	1	no	—
321	Q₈	—	178	1	no	—
322	D₄	—	356	2	no	—
323	D₄	—	356	2	no	—
324	D₄	—	356	2	no	—
325	D₄	—	178	1	no	—
326	D₄	—	178	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	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	89	1	no	—
334	C₂×C₄	—	89	1	no	—
335	C₂×C₄	—	356	1	no	—
336	C₂×C₄	—	356	1	no	—
337	C₂×C₄	—	356	2	no	—
338	C₂×C₄	—	356	2	no	—
339	C₂×C₄	—	356	2	no	—
340	C₂×C₄	—	356	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	—	178	1	no	—
347	C₂×C₄	—	178	1	no	—
348	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
349	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	178	1	no	—
353	C₈	—	178	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	89	2	no	—
357	C₂²	—	356	4	no	—
358	C₂²	—	356	1	no	—
359	C₂²	—	356	1	no	—
360	C₂²	—	356	1	no	—
361	C₂²	—	178	1	no	—
362	C₂²	—	178	1	no	—
363	C₂²	—	178	1	no	—
364	C₂²	—	178	1	no	—
365	C₂²	—	178	1	no	—
366	C₂²	#15668044	1	2	yes	—
367	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	4	1	no	—
373	C₄	—	356	1	no	—
374	C₄	—	356	1	no	—
375	C₄	—	356	1	no	—
376	C₄	—	178	1	no	—
377	C₄	—	178	1	no	—
378	C₄	#15668057	1	1	yes	characteristic
379	C₄	#15668058	1	1	yes	characteristic
380	C₂	—	2	2	no	—
381	C₂	—	356	1	no	—
382	C₂	—	178	2	no	—
383	C₂	—	712	1	no	—
384	C₂	#19152156	1	1	yes	characteristic
385	C₂	#135×D₈₉	1	1	yes	characteristic
386	C₂	C₂×#15668044	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#467×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152111	C₂
hypercenter	#467	D₈₉
hypocenter	C₈₉	#3190
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

Complements

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

N	H ≅ G/N	type
2. C₂²⋊Q₈×D₈₉	C₂	semidirect
3. #19151941	C₂	semidirect
4. #19151937	C₂	semidirect
5. #19151927	C₂	semidirect
6. #19151913	C₂	semidirect
7. #19151837	C₂	semidirect
8. #19151829	C₂	semidirect
9. #19151827	C₂	semidirect
10. #19151763	C₂	semidirect
11. #19151759	C₂	semidirect
12. #19151752	C₂	semidirect
13. #19151738	C₂	semidirect
14. C₂×#15667924	C₂	semidirect
15. #467×C₈₉	C₂	semidirect
19. C₂×#13083564	C₂²	semidirect
20. C₂×#13083563	C₂²	semidirect
21. C₂×#13083561	C₂²	semidirect
39. #15667945	C₂²	semidirect
40. #15667944	C₂²	semidirect
41. #15667943	C₂²	semidirect
42. #15667937	C₂²	semidirect
43. #15667936	C₂²	semidirect
44. #15667936	C₂²	semidirect
45. #15667935	C₂²	semidirect
46. #15667934	C₂²	semidirect
47. #15667932	C₂²	semidirect
52. C₂×Q₁₆×C₈₉	C₂²	semidirect
56. #107×C₈₉	C₂²	semidirect
57. #103×C₈₉	C₂²	semidirect
58. #103×C₈₉	C₂²	semidirect
194. C₈₉	#3190	semidirect
202. #467	D₈₉	semidirect
214. C₂×Q₁₆	C₂×D₈₉	semidirect
238. #107	C₂×D₈₉	semidirect
245. #103	C₂×D₈₉	semidirect
246. #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
#19152111	2
C₂²⋊Q₈×D₈₉	2
#19151941	2
#19151937	2
#19151927	2
#19151913	2
#19151837	2
#19151829	2
#19151827	2
#19151763	2
#19151759	2
#19151752	2
#19151738	2
C₂×#15667924	2
#467×C₈₉	2
#3190	89