Subgroups — #26029938

Order 11392.

472 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	C₂×#15668041	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15668020	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152177	C₂	1	1	yes	maximal, characteristic
4	#19152171	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151856	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151855	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151843	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151840	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151837	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151825	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151824	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151639	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151591	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#468×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂×#13083573	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
19	#15668041	—	2	4	no	—
20	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
21	C₂×#13083564	C₂²	1	1	yes	characteristic
22	C₂×#13083562	C₂²	1	1	yes	characteristic
23	#15668020	—	2	4	no	—
24	C₂×#13083557	C₂²	1	1	yes	characteristic
25	C₂×#13083557	C₂²	1	1	yes	characteristic
26	C₂²×#10623494	C₂²	1	1	yes	characteristic
27	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
29	#15668011	—	2	1	no	—
30	#15667998	—	2	1	no	—
31	#15667997	—	2	1	no	—
32	#15667994	—	2	1	no	—
33	#15667992	C₂²	1	1	yes	characteristic
34	#15667992	C₂²	1	1	yes	characteristic
35	#15667991	C₂²	1	1	yes	characteristic
36	#15667991	C₂²	1	1	yes	characteristic
37	#15667989	—	2	1	no	—
38	#15667988	C₂²	1	1	yes	characteristic
39	#15667985	—	2	1	no	—
40	#15667984	—	2	1	no	—
41	#15667983	—	2	1	no	—
42	#15667983	C₂²	1	1	yes	characteristic
43	#15667970	C₂²	1	1	yes	characteristic
44	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
45	#15667945	C₂²	1	1	yes	characteristic
46	#15667944	C₂²	1	1	yes	characteristic
47	#15667943	C₂²	1	1	yes	characteristic
48	#15667937	C₂²	1	1	yes	characteristic
49	#15667937	C₂²	1	1	yes	characteristic
50	#15667936	C₂²	1	1	yes	characteristic
51	#15667935	C₂²	1	1	yes	characteristic
52	#15667935	C₂²	1	1	yes	characteristic
53	#15667933	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
55	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
56	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
57	#121×C₈₉	C₂²	1	1	yes	characteristic
58	#107×C₈₉	C₂²	1	1	yes	characteristic
59	#103×C₈₉	C₂²	1	1	yes	characteristic
60	#102×C₈₉	C₂²	1	1	yes	characteristic
61	#13083574	—	2	2	no	—
62	#13083574	—	2	2	no	—
63	#13083573	—	2	2	no	—
64	#13083573	—	2	2	no	—
65	C₂×Dic₁₇₈	—	2	1	no	—
66	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
67	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
69	Q₈×D₈₉	—	2	2	no	—
70	Q₈×D₈₉	—	2	2	no	—
71	C₂×D₃₅₆	D₄	1	1	yes	characteristic
72	C₂×C₂²⋊D₈₉	—	2	1	no	—
73	C₂×C₂²⋊D₈₉	—	2	1	no	—
74	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
76	C₂²×Dic₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	—	2	1	no	—
78	C₂×C₄×D₈₉	—	2	1	no	—
79	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
81	Dic₃₅₆	—	2	2	no	—
82	#13083564	—	2	2	no	—
83	#13083562	—	2	2	no	—
84	#13083557	—	2	2	no	—
85	#13083557	D₄	1	4	yes	complemented
86	C₂×#10623494	—	2	2	no	—
87	C₂×#10623494	C₂³	1	1	yes	characteristic
88	C₂×#10623494	C₂³	1	1	yes	characteristic
89	C₈×D₈₉	—	2	2	no	—
90	C₈×D₈₉	—	2	2	no	—
91	C₄⋊Dic₈₉	—	2	1	no	—
92	C₄⋊Dic₈₉	—	2	1	no	—
93	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
95	#13083553	—	2	1	no	—
96	#13083553	—	2	1	no	—
97	#13083553	—	2	1	no	—
98	#13083553	—	2	1	no	—
99	C₂²⋊Dic₈₉	—	2	1	no	—
100	C₂²⋊Dic₈₉	—	2	1	no	—
101	C₂²⋊Dic₈₉	—	2	1	no	—
102	#13083551	—	2	1	no	—
103	#13083551	—	2	1	no	—
104	#13083551	—	2	1	no	—
105	#13083551	—	2	1	no	—
106	C₄×Dic₈₉	—	2	1	no	—
107	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₂×D₄×C₈₉	—	2	1	no	—
109	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
111	SD₁₆×C₈₉	—	2	2	no	—
112	M₁₆×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
115	C₄⋊C₄×C₈₉	—	2	1	no	—
116	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
118	C₂²⋊C₄×C₈₉	—	2	1	no	—
119	C₂²⋊C₄×C₈₉	—	2	1	no	—
120	C₂³×C₈₉	—	2	1	no	—
121	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
124	Q₈×C₈₉	—	2	1	no	—
125	Q₈×C₈₉	—	2	1	no	—
126	D₄×C₈₉	—	2	1	no	—
127	D₄×C₈₉	—	2	1	no	—
128	D₄×C₈₉	—	8	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
136	C₂²⋊D₈₉	—	2	2	no	—
137	C₂²⋊D₈₉	—	2	2	no	—
138	C₂²⋊D₈₉	—	4	2	no	—
139	C₂²⋊D₈₉	—	8	1	no	—
140	C₂×Dic₈₉	—	2	1	no	—
141	C₂×Dic₈₉	—	2	1	no	—
142	C₂×Dic₈₉	—	2	1	no	—
143	C₂×Dic₈₉	—	2	1	no	—
144	C₂×Dic₈₉	—	4	2	no	—
145	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
146	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
147	D₃₅₆	—	2	2	no	—
148	C₄×D₈₉	—	2	1	no	—
149	C₄×D₈₉	—	2	1	no	—
150	C₄×D₈₉	—	4	2	no	—
151	C₄×D₈₉	—	8	1	no	—
152	C₄×D₈₉	C₂×D₄	1	2	yes	—
153	C₄×D₈₉	#41	1	2	yes	—
154	Dic₁₇₈	—	2	2	no	—
155	Dic₁₇₈	—	2	1	no	—
156	Dic₁₇₈	—	2	1	no	—
157	Dic₁₇₈	—	2	1	no	—
158	Dic₁₇₈	—	2	1	no	—
159	Dic₁₇₈	—	4	2	no	—
160	C₈×C₈₉	—	2	1	no	—
161	C₈×C₈₉	C₂×D₄	1	2	yes	—
162	#10623494	—	2	1	no	—
163	#10623494	C₂×D₄	1	2	yes	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	—	2	1	no	—
166	C₂²×C₈₉	—	2	1	no	—
167	C₂²×C₈₉	—	4	1	no	—
168	C₂²×C₈₉	—	4	1	no	—
169	C₂²×C₈₉	—	4	1	no	—
170	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
171	C₂×D₈₉	—	2	1	no	—
172	C₂×D₈₉	—	2	1	no	—
173	C₂×D₈₉	—	2	1	no	—
174	C₂×D₈₉	—	2	1	no	—
175	C₂×D₈₉	—	2	1	no	—
176	C₂×D₈₉	#121	1	2	yes	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	4	1	no	—
179	C₄×C₈₉	—	4	1	no	—
180	C₄×C₈₉	—	4	1	no	—
181	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
182	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
183	Dic₈₉	—	2	1	no	—
184	Dic₈₉	—	4	1	no	—
185	Dic₈₉	—	4	1	no	—
186	Dic₈₉	—	4	1	no	—
187	Dic₈₉	—	4	1	no	—
188	Dic₈₉	#121	1	2	yes	—
189	C₂×C₈₉	—	4	1	no	—
190	C₂×C₈₉	—	8	1	no	—
191	C₂×C₈₉	C₂×#121	1	1	yes	characteristic
192	C₂×C₈₉	#577	1	1	yes	characteristic
193	C₂×C₈₉	#578	1	1	yes	characteristic
194	D₈₉	—	2	2	no	—
195	D₈₉	—	4	1	no	—
196	#3063	—	89	1	no	maximal, Sylow
197	C₈₉	#3063	1	1	yes	characteristic, complemented, Sylow
198	C₂×#137	—	89	1	no	—
199	C₂×#131	—	89	1	no	—
200	#537	—	89	1	no	—
201	#536	—	89	1	no	—
202	#470	—	89	1	no	—
203	#470	—	89	1	no	—
204	#468	—	89	1	no	—
205	#468	D₈₉	1	1	yes	characteristic, complemented
206	#467	—	89	1	no	—
207	#467	—	89	1	no	—
208	#465	—	89	1	no	—
209	#465	—	89	1	no	—
210	#428	—	89	1	no	—
211	#416	—	89	1	no	—
212	C₂×#103	—	89	1	no	—
213	C₂×#41	—	89	1	no	—
214	C₂×#41	—	89	1	no	—
215	C₂²×Q₈	—	89	1	no	—
216	#137	—	178	4	no	—
217	C₂×Q₁₆	—	89	1	no	—
218	C₂×Q₁₆	—	89	1	no	—
219	C₂×SD₁₆	—	89	1	no	—
220	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
221	#131	—	178	4	no	—
222	C₂×M₁₆	—	89	1	no	—
223	C₂×M₁₆	—	89	1	no	—
224	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
225	C₂²×C₈	—	89	1	no	—
226	C₂²×C₈	—	89	1	no	—
227	C₂²×C₈	—	89	1	no	—
228	C₄⋊Q₈	—	178	1	no	—
229	#124	—	178	1	no	—
230	#123	—	178	1	no	—
231	#123	—	178	1	no	—
232	C₂²⋊Q₈	—	89	1	no	—
233	C₂²⋊Q₈	—	89	1	no	—
234	C₂²⋊Q₈	—	89	1	no	—
235	C₂²⋊Q₈	—	89	1	no	—
236	C₂²⋊Q₈	—	89	1	no	—
237	C₂²⋊Q₈	—	178	1	no	—
238	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
239	#121	—	89	1	no	—
240	#121	—	178	1	no	—
241	#121	—	178	1	no	—
242	#121	—	178	1	no	—
243	#121	C₂×D₈₉	1	1	yes	characteristic
244	#117	—	89	1	no	—
245	C₂×C₄⋊C₄	—	89	1	no	—
246	#107	—	89	1	no	—
247	#107	C₂×D₈₉	1	1	yes	characteristic
248	#106	—	89	1	no	—
249	#106	—	89	1	no	—
250	#103	—	89	1	no	—
251	#103	—	89	1	no	—
252	#103	—	89	1	no	—
253	#103	—	89	1	no	—
254	#103	—	89	1	no	—
255	#103	C₂×D₈₉	1	1	yes	characteristic
256	#102	—	89	1	no	—
257	#102	C₂×D₈₉	1	1	yes	characteristic
258	#41	—	178	2	no	—
259	#41	—	178	2	no	—
260	#41	—	178	2	no	—
261	#41	—	178	2	no	—
262	C₂×Q₈	—	89	1	no	—
263	C₂×Q₈	—	89	1	no	—
264	C₂×Q₈	—	89	1	no	—
265	C₂×Q₈	—	178	1	no	—
266	C₂×Q₈	—	178	2	no	—
267	C₂×Q₈	—	178	2	no	—
268	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	89	1	no	—
271	C₂×D₄	—	89	1	no	—
272	C₂×D₄	—	89	1	no	—
273	C₂×D₄	—	178	1	no	—
274	C₂×D₄	—	178	1	no	—
275	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
276	C₂²×C₄	—	89	1	no	—
277	C₂²×C₄	—	89	1	no	—
278	C₂²×C₄	—	178	1	no	—
279	C₂²×C₄	—	178	1	no	—
280	C₂²×C₄	—	178	1	no	—
281	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
282	Q₁₆	—	178	2	no	—
283	Q₁₆	—	178	2	no	—
284	SD₁₆	—	2	2	no	—
285	SD₁₆	—	178	2	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	89	4	no	—
288	M₁₆	—	178	2	no	—
289	C₂×C₈	—	89	1	no	—
290	C₂×C₈	—	89	1	no	—
291	C₂×C₈	—	178	2	no	—
292	C₂×C₈	—	178	2	no	—
293	C₂×C₈	—	178	2	no	—
294	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
295	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	89	1	no	—
298	C₄⋊C₄	—	89	1	no	—
299	C₄⋊C₄	—	178	1	no	—
300	C₄⋊C₄	—	178	1	no	—
301	C₄⋊C₄	—	178	1	no	—
302	C₄⋊C₄	—	178	1	no	—
303	C₄⋊C₄	—	178	1	no	—
304	C₄⋊C₄	—	178	1	no	—
305	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
306	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
307	C₂²⋊C₄	—	2	1	no	—
308	C₂²⋊C₄	—	2	1	no	—
309	C₂²⋊C₄	—	178	1	no	—
310	C₂²⋊C₄	—	178	1	no	—
311	C₂²⋊C₄	—	178	1	no	—
312	C₂²⋊C₄	—	178	1	no	—
313	C₂²⋊C₄	—	178	1	no	—
314	C₂²⋊C₄	—	178	1	no	—
315	C₂²⋊C₄	—	178	1	no	—
316	C₄²	—	178	1	no	—
317	C₂³	—	2	1	no	—
318	C₂³	—	89	1	no	—
319	C₂³	—	89	1	no	—
320	C₂³	D₄×D₈₉	1	1	yes	characteristic
321	Q₈	—	2	1	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	356	2	no	—
324	Q₈	—	178	2	no	—
325	Q₈	—	178	1	no	—
326	Q₈	—	178	1	no	—
327	Q₈	—	178	1	no	—
328	Q₈	—	178	1	no	—
329	D₄	—	2	1	no	—
330	D₄	—	2	1	no	—
331	D₄	—	8	1	no	—
332	D₄	—	356	2	no	—
333	D₄	—	178	2	no	—
334	D₄	—	178	2	no	—
335	D₄	—	178	2	no	—
336	D₄	—	712	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	89	1	no	—
343	C₂×C₄	—	89	2	no	—
344	C₂×C₄	—	89	2	no	—
345	C₂×C₄	—	89	1	no	—
346	C₂×C₄	—	356	2	no	—
347	C₂×C₄	—	356	2	no	—
348	C₂×C₄	—	178	1	no	—
349	C₂×C₄	—	178	1	no	—
350	C₂×C₄	—	178	1	no	—
351	C₂×C₄	—	178	1	no	—
352	C₂×C₄	—	178	1	no	—
353	C₂×C₄	—	178	1	no	—
354	C₂×C₄	—	712	1	no	—
355	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
356	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
357	C₈	—	2	1	no	—
358	C₈	—	89	2	no	—
359	C₈	—	178	1	no	—
360	C₈	D₄×D₈₉	1	2	yes	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	4	1	no	—
365	C₂²	—	4	1	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	89	2	no	—
368	C₂²	—	178	1	no	—
369	C₂²	—	178	1	no	—
370	C₂²	—	178	1	no	—
371	C₂²	—	178	1	no	—
372	C₂²	—	178	1	no	—
373	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	4	1	no	—
378	C₄	—	89	2	no	—
379	C₄	—	356	1	no	—
380	C₄	—	356	1	no	—
381	C₄	—	356	1	no	—
382	C₄	—	356	1	no	—
383	C₄	—	178	1	no	—
384	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
385	C₄	#41×D₈₉	1	1	yes	characteristic
386	C₂	—	4	1	no	—
387	C₂	—	8	1	no	—
388	C₂	—	356	1	no	—
389	C₂	—	178	2	no	—
390	C₂	#121×D₈₉	1	1	yes	characteristic
391	C₂	#19152408	1	1	yes	characteristic
392	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#468×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152177	C₂
hypercenter	#468	D₈₉
hypocenter	C₈₉	#3063
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
1. C₂×#15668041	C₂	semidirect
2. C₂×#15668020	C₂	semidirect
4. #19152171	C₂	semidirect
5. #19151856	C₂	semidirect
6. #19151855	C₂	semidirect
7. #19151843	C₂	semidirect
8. #19151840	C₂	semidirect
9. #19151837	C₂	semidirect
10. #19151825	C₂	semidirect
11. #19151824	C₂	semidirect
12. #19151639	C₂	semidirect
13. #19151591	C₂	semidirect
14. #103×D₈₉	C₂	semidirect
15. #468×C₈₉	C₂	semidirect
85. #13083557	D₄	semidirect
197. C₈₉	#3063	semidirect
205. #468	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668041	2
C₂×#15668020	2
#19152177	2
#19152171	2
#19151856	2
#19151855	2
#19151843	2
#19151840	2
#19151837	2
#19151825	2
#19151824	2
#19151639	2
#19151591	2
#103×D₈₉	2
#468×C₈₉	2
#3063	89