Subgroups — #26029703

Order 11392.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668032	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152144	C₂	1	1	yes	maximal, characteristic, complemented
3	#41×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151822	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151818	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151815	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151811	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151806	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151801	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151796	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151791	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15667941	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151595	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151594	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#135×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	#15668032	—	2	4	no	—
18	C₂×#13083564	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	C₂×#13083561	C₂²	1	1	yes	characteristic
21	C₂×#13083559	C₂²	1	1	yes	characteristic
22	C₂²×#10623494	C₂²	1	1	yes	characteristic
23	#15668010	C₂²	1	1	yes	characteristic
24	#15668008	C₂²	1	1	yes	characteristic
25	#15668001	C₂²	1	1	yes	characteristic
26	#15667999	C₂²	1	1	yes	characteristic
27	#15667984	—	2	1	no	—
28	#15667984	—	2	1	no	—
29	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
30	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₈₉	—	2	1	no	—
32	D₄×Dic₈₉	—	2	1	no	—
33	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
34	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊D₈₉	—	2	1	no	—
36	C₄×C₂²⋊D₈₉	—	2	1	no	—
37	#15667971	—	2	1	no	—
38	#15667971	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₈₉	—	2	1	no	—
40	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
41	#15667941	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667941	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667937	C₂²	1	1	yes	characteristic
46	#15667937	C₂²	1	1	yes	characteristic
47	#15667936	C₂²	1	1	yes	characteristic
48	#15667936	C₂²	1	1	yes	characteristic
49	#15667934	C₂²	1	1	yes	characteristic
50	#15667934	C₂²	1	1	yes	characteristic
51	#15667933	C₂²	1	1	yes	characteristic
52	#15667933	C₂²	1	1	yes	characteristic
53	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
55	#135×C₈₉	—	2	4	no	—
56	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
57	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
58	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
60	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
61	#13083574	—	2	2	no	—
62	#13083574	—	2	2	no	—
63	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
64	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₉	—	2	1	no	—
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂²×Dic₈₉	—	2	1	no	—
69	C₂²×Dic₈₉	—	2	1	no	—
70	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
71	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
72	C₂×C₄×D₈₉	—	2	1	no	—
73	#13083564	—	2	2	no	—
74	#13083562	—	2	2	no	—
75	#13083561	—	2	2	no	—
76	#13083559	—	2	2	no	—
77	C₂×#10623494	—	2	2	no	—
78	C₂×#10623494	C₂³	1	1	yes	characteristic
79	C₂×#10623494	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₉	—	2	1	no	—
81	C₄⋊Dic₈₉	—	2	1	no	—
82	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	C₂²⋊Dic₈₉	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	C₂²⋊Dic₈₉	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	—	2	1	no	—
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	D₄	1	1	yes	characteristic
97	C₄×Dic₈₉	D₄	1	1	yes	characteristic
98	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
99	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
100	#41×C₈₉	—	2	2	no	—
101	#41×C₈₉	—	2	2	no	—
102	#41×C₈₉	—	2	2	no	—
103	#41×C₈₉	D₄	1	4	yes	complemented
104	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×D₄×C₈₉	—	2	1	no	—
107	C₂×D₄×C₈₉	—	2	1	no	—
108	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	—	2	1	no	—
111	C₂²×C₄×C₈₉	—	2	1	no	—
112	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
113	Q₁₆×C₈₉	—	2	2	no	—
114	SD₁₆×C₈₉	—	2	2	no	—
115	SD₁₆×C₈₉	—	2	2	no	—
116	D₈×C₈₉	—	2	2	no	—
117	C₂×C₈×C₈₉	—	2	2	no	—
118	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
120	C₂³×C₈₉	—	2	1	no	—
121	C₂³×C₈₉	—	2	1	no	—
122	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂²×D₈₉	—	2	1	no	—
124	Q₈×C₈₉	—	2	1	no	—
125	Q₈×C₈₉	—	2	1	no	—
126	Q₈×C₈₉	—	2	1	no	—
127	Q₈×C₈₉	C₂×D₄	1	2	yes	—
128	D₄×C₈₉	—	2	4	no	—
129	D₄×C₈₉	—	2	1	no	—
130	D₄×C₈₉	—	2	1	no	—
131	D₄×C₈₉	—	2	1	no	—
132	D₄×C₈₉	—	4	2	no	—
133	D₄×C₈₉	C₂×D₄	1	2	yes	—
134	C₂×C₄×C₈₉	—	2	1	no	—
135	C₂×C₄×C₈₉	—	2	4	no	—
136	C₂×C₄×C₈₉	—	2	1	no	—
137	C₂×C₄×C₈₉	—	4	2	no	—
138	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
140	C₂×C₄×C₈₉	#41	1	2	yes	—
141	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
142	C₂²⋊D₈₉	—	4	2	no	—
143	C₂²⋊D₈₉	—	4	2	no	—
144	C₂²⋊D₈₉	—	4	2	no	—
145	C₂×Dic₈₉	—	2	2	no	—
146	C₂×Dic₈₉	—	2	1	no	—
147	C₂×Dic₈₉	—	2	1	no	—
148	C₂×Dic₈₉	—	2	1	no	—
149	C₂×Dic₈₉	—	4	1	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	C₂×Dic₈₉	—	4	2	no	—
152	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
153	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
154	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
155	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
156	D₃₅₆	—	2	1	no	—
157	D₃₅₆	—	2	1	no	—
158	C₄×D₈₉	—	4	2	no	—
159	Dic₁₇₈	—	2	1	no	—
160	Dic₁₇₈	—	2	1	no	—
161	C₈×C₈₉	—	2	1	no	—
162	C₈×C₈₉	—	2	1	no	—
163	#10623494	—	2	1	no	—
164	#10623494	—	2	1	no	—
165	C₂²×C₈₉	—	2	1	no	—
166	C₂²×C₈₉	—	2	1	no	—
167	C₂²×C₈₉	—	2	2	no	—
168	C₂²×C₈₉	—	4	1	no	—
169	C₂²×C₈₉	—	4	1	no	—
170	C₂²×C₈₉	—	4	1	no	—
171	C₂²×C₈₉	—	4	1	no	—
172	C₂²×C₈₉	—	4	1	no	—
173	C₂²×C₈₉	#121	1	2	yes	—
174	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
175	C₂×D₈₉	—	4	1	no	—
176	C₂×D₈₉	—	4	1	no	—
177	C₂×D₈₉	—	4	1	no	—
178	C₄×C₈₉	—	2	2	no	—
179	C₄×C₈₉	—	4	1	no	—
180	C₄×C₈₉	#121	1	2	yes	—
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	—	2	1	no	—
185	Dic₈₉	—	2	1	no	—
186	Dic₈₉	—	4	1	no	—
187	Dic₈₉	—	4	1	no	—
188	Dic₈₉	—	4	1	no	—
189	Dic₈₉	#135	1	2	yes	complemented
190	C₂×C₈₉	—	2	2	no	—
191	C₂×C₈₉	—	4	2	no	—
192	C₂×C₈₉	—	8	1	no	—
193	C₂×C₈₉	C₂×#121	1	1	yes	characteristic
194	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
195	C₂×C₈₉	#575	1	1	yes	characteristic
196	D₈₉	—	8	1	no	—
197	#3035	—	89	1	no	maximal, Sylow
198	C₈₉	#3035	1	1	yes	characteristic, complemented, Sylow
199	C₂×#135	—	89	1	no	—
200	C₂×#135	D₈₉	1	1	yes	characteristic, complemented
201	#532	—	89	1	no	—
202	C₄×#41	—	89	1	no	—
203	#464	—	89	1	no	—
204	#464	—	89	1	no	—
205	#463	—	89	1	no	—
206	#463	—	89	1	no	—
207	#462	—	89	1	no	—
208	#461	—	89	1	no	—
209	#460	—	89	1	no	—
210	#459	—	89	1	no	—
211	C₂×C₄⋊C₈	—	89	1	no	—
212	#416	—	89	1	no	—
213	#416	—	89	1	no	—
214	C₂×#41	—	89	1	no	—
215	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
216	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
217	#135	—	2	4	no	—
218	#135	—	178	4	no	—
219	C₂×Q₁₆	—	89	1	no	—
220	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
221	C₂×SD₁₆	—	89	1	no	—
222	C₂×SD₁₆	—	89	1	no	—
223	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
224	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
225	C₂×D₈	—	89	1	no	—
226	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
227	C₂²×C₈	—	89	1	no	—
228	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
229	C₄⋊Q₈	—	89	1	no	—
230	#127	—	89	1	no	—
231	#124	—	89	1	no	—
232	#124	—	89	1	no	—
233	#121	—	178	1	no	—
234	#121	—	178	1	no	—
235	C₄×Q₈	—	89	1	no	—
236	C₄×Q₈	—	89	1	no	—
237	C₄×D₄	—	89	1	no	—
238	C₄×D₄	—	89	1	no	—
239	C₄×D₄	—	178	1	no	—
240	C₄×D₄	—	178	1	no	—
241	C₄×D₄	—	178	1	no	—
242	C₄×D₄	—	178	1	no	—
243	#117	—	89	1	no	—
244	#117	—	178	1	no	—
245	C₂×C₄²	—	89	1	no	—
246	C₂×C₄²	—	178	1	no	—
247	C₄⋊C₈	—	89	1	no	—
248	C₄⋊C₈	—	89	1	no	—
249	C₄⋊C₈	—	89	1	no	—
250	C₄⋊C₈	—	89	1	no	—
251	#103	—	89	1	no	—
252	#103	—	89	1	no	—
253	#103	—	89	1	no	—
254	#103	—	89	1	no	—
255	#102	—	89	1	no	—
256	#102	—	89	1	no	—
257	#102	—	89	1	no	—
258	#102	—	89	1	no	—
259	#41	—	2	2	no	—
260	#41	—	2	2	no	—
261	#41	—	2	2	no	—
262	#41	—	178	2	no	—
263	#41	—	178	2	no	—
264	#41	C₂²⋊D₈₉	1	4	yes	complemented
265	C₂×Q₈	—	89	1	no	—
266	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
267	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	89	1	no	—
271	C₂×D₄	—	178	1	no	—
272	C₂×D₄	—	178	1	no	—
273	C₂×D₄	—	178	1	no	—
274	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
275	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	2	1	no	—
278	C₂²×C₄	—	89	1	no	—
279	C₂²×C₄	—	89	1	no	—
280	C₂²×C₄	—	178	1	no	—
281	C₂²×C₄	—	178	1	no	—
282	C₂²×C₄	—	178	1	no	—
283	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
284	Q₁₆	—	2	2	no	—
285	Q₁₆	—	178	2	no	—
286	SD₁₆	—	2	2	no	—
287	SD₁₆	—	2	2	no	—
288	SD₁₆	—	178	2	no	—
289	SD₁₆	—	178	2	no	—
290	D₈	—	2	2	no	—
291	D₈	—	178	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	89	1	no	—
294	C₂×C₈	—	89	1	no	—
295	C₂×C₈	—	178	2	no	—
296	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
297	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
298	C₄⋊C₄	—	89	1	no	—
299	C₄⋊C₄	—	89	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	C₂²⋊C₄	—	178	1	no	—
305	C₂²⋊C₄	—	178	1	no	—
306	C₂²⋊C₄	—	178	1	no	—
307	C₂²⋊C₄	—	178	1	no	—
308	C₂²⋊C₄	—	178	1	no	—
309	C₂²⋊C₄	—	178	1	no	—
310	C₂²⋊C₄	—	178	1	no	—
311	C₄²	—	89	1	no	—
312	C₄²	—	89	1	no	—
313	C₄²	—	89	1	no	—
314	C₄²	—	89	1	no	—
315	C₄²	—	178	1	no	—
316	C₄²	—	178	1	no	—
317	C₄²	—	178	1	no	—
318	C₂³	—	2	1	no	—
319	C₂³	—	2	1	no	—
320	C₂³	—	178	1	no	—
321	C₂³	D₄×D₈₉	1	1	yes	characteristic
322	Q₈	—	2	1	no	—
323	Q₈	—	2	1	no	—
324	Q₈	—	2	1	no	—
325	Q₈	—	178	1	no	—
326	Q₈	—	178	1	no	—
327	Q₈	C₂×C₂²⋊D₈₉	1	2	yes	—
328	D₄	—	2	4	no	—
329	D₄	—	2	1	no	—
330	D₄	—	2	1	no	—
331	D₄	—	2	1	no	—
332	D₄	—	4	2	no	—
333	D₄	—	356	2	no	—
334	D₄	—	356	2	no	—
335	D₄	—	356	2	no	—
336	D₄	—	178	1	no	—
337	D₄	—	178	1	no	—
338	D₄	C₂×C₂²⋊D₈₉	1	2	yes	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	89	1	no	—
344	C₂×C₄	—	89	1	no	—
345	C₂×C₄	—	89	1	no	—
346	C₂×C₄	—	89	1	no	—
347	C₂×C₄	—	356	2	no	—
348	C₂×C₄	—	356	2	no	—
349	C₂×C₄	—	356	2	no	—
350	C₂×C₄	—	356	1	no	—
351	C₂×C₄	—	178	2	no	—
352	C₂×C₄	—	178	1	no	—
353	C₂×C₄	—	178	1	no	—
354	C₂×C₄	—	178	1	no	—
355	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
356	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
357	C₂×C₄	#13083573	1	2	yes	—
358	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	178	1	no	—
362	C₈	—	178	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	4	1	no	—
370	C₂²	—	4	1	no	—
371	C₂²	—	356	1	no	—
372	C₂²	—	356	1	no	—
373	C₂²	—	356	1	no	—
374	C₂²	#15667984	1	2	yes	—
375	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
376	C₄	—	2	2	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	—	178	1	no	—
383	C₄	—	178	1	no	—
384	C₄	—	178	1	no	—
385	C₄	#15667984	1	2	yes	—
386	C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
387	C₄	C₂×#13083573	1	1	yes	characteristic
388	C₂	—	2	2	no	—
389	C₂	—	4	2	no	—
390	C₂	—	8	1	no	—
391	C₂	—	712	1	no	—
392	C₂	C₂×#15667984	1	1	yes	characteristic
393	C₂	#135×D₈₉	1	1	yes	characteristic
394	C₂	#19152395	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#135×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	C₂×#135	D₈₉
hypocenter	C₈₉	#3035
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#15668032	C₂	semidirect
2. #19152144	C₂	semidirect
3. #41×Dic₈₉	C₂	semidirect
4. #19151822	C₂	semidirect
5. #19151818	C₂	semidirect
6. #19151815	C₂	semidirect
7. #19151811	C₂	semidirect
8. #19151806	C₂	semidirect
9. #19151801	C₂	semidirect
10. #19151796	C₂	semidirect
11. #19151791	C₂	semidirect
12. C₂×#15667941	C₂	semidirect
13. #19151595	C₂	semidirect
14. #19151594	C₂	semidirect
15. C₂×#135×C₈₉	C₂	semidirect
103. #41×C₈₉	D₄	semidirect
189. Dic₈₉	#135	semidirect
198. C₈₉	#3035	semidirect
200. C₂×#135	D₈₉	semidirect
264. #41	C₂²⋊D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668032	2
#19152144	2
#41×Dic₈₉	2
#19151822	2
#19151818	2
#19151815	2
#19151811	2
#19151806	2
#19151801	2
#19151796	2
#19151791	2
C₂×#15667941	2
#19151595	2
#19151594	2
C₂×#135×C₈₉	2
#3035	89