Subgroups — #26031997

Order 11392.

472 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	C₂²×#13083564	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×Q₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₂²⋊D₈₉	C₂	1	1	yes	maximal, characteristic
4	#19152261	C₂	1	1	yes	maximal, characteristic
5	#19151976	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151838	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151837	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151807	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151806	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151756	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151755	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151752	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151690	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151661	C₂	1	1	yes	maximal, characteristic
15	#451×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
20	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
21	C₂×#13083564	—	2	4	no	—
22	C₂×#13083564	C₂²	1	1	yes	characteristic
23	C₂×#13083564	C₂²	1	1	yes	characteristic
24	C₂×#13083564	C₂²	1	1	yes	characteristic
25	C₂×#13083564	C₂²	1	1	yes	characteristic
26	C₂×#13083564	C₂²	1	1	yes	characteristic
27	Q₁₆×D₈₉	—	2	4	no	—
28	C₂²×#10623494	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
30	#15668011	—	2	1	no	—
31	#15668010	—	2	1	no	—
32	#15668010	C₂²	1	1	yes	characteristic
33	#15668010	C₂²	1	1	yes	characteristic
34	#15667992	C₂²	1	1	yes	characteristic
35	#15667991	—	2	1	no	—
36	#15667991	C₂²	1	1	yes	characteristic
37	#15667990	—	2	1	no	—
38	#15667989	—	2	1	no	—
39	#15667988	—	2	1	no	—
40	#15667988	C₂²	1	1	yes	characteristic
41	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
42	#15667977	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊D₈₉	—	2	1	no	—
44	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
45	#15667972	—	2	1	no	—
46	#15667945	C₂²	1	1	yes	characteristic
47	#15667941	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	#15667924	C₂²	1	1	yes	characteristic
53	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
54	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
56	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
57	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	#103×C₈₉	C₂²	1	1	yes	characteristic
60	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
61	C₂×Dic₁₇₈	—	2	2	no	—
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×Dic₁₇₈	—	2	1	no	—
64	C₂×Dic₁₇₈	—	2	1	no	—
65	C₂×Dic₁₇₈	—	4	1	no	—
66	C₂×Dic₁₇₈	C₂³	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	Q₈×D₈₉	—	4	1	no	—
72	Q₈×D₈₉	—	4	1	no	—
73	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
74	C₂²×Dic₈₉	—	2	1	no	—
75	C₂×C₄×D₈₉	—	2	1	no	—
76	C₂×C₄×D₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
78	Dic₃₅₆	—	2	2	no	—
79	Dic₃₅₆	—	4	1	no	—
80	#13083564	—	2	2	no	—
81	#13083564	—	2	2	no	—
82	#13083564	—	2	2	no	—
83	#13083564	—	2	2	no	—
84	#13083564	D₄	1	4	yes	complemented
85	C₂×#10623494	—	2	2	no	—
86	C₂×#10623494	C₂³	1	1	yes	characteristic
87	C₂×#10623494	C₂³	1	1	yes	characteristic
88	C₈×D₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	—	2	1	no	—
90	C₄⋊Dic₈₉	—	2	1	no	—
91	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
92	#13083553	—	2	1	no	—
93	#13083553	—	2	1	no	—
94	#13083553	—	2	1	no	—
95	#13083553	—	2	1	no	—
96	#13083553	—	2	1	no	—
97	#13083553	—	2	1	no	—
98	#13083553	—	2	1	no	—
99	#13083553	D₄	1	1	yes	characteristic
100	C₂²⋊Dic₈₉	—	2	1	no	—
101	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
102	#13083551	—	2	1	no	—
103	#13083551	—	2	1	no	—
104	#13083551	D₄	1	1	yes	characteristic
105	C₄×Dic₈₉	—	2	1	no	—
106	C₄×Dic₈₉	—	2	1	no	—
107	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₈₉	—	2	1	no	—
109	C₂×Q₈×C₈₉	—	2	2	no	—
110	C₂×Q₈×C₈₉	—	4	1	no	—
111	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
112	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
113	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
114	C₂²×C₄×C₈₉	—	2	1	no	—
115	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
116	Q₁₆×C₈₉	—	2	2	no	—
117	Q₁₆×C₈₉	—	4	1	no	—
118	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
120	C₄⋊C₄×C₈₉	—	2	1	no	—
121	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
122	C₂²⋊C₄×C₈₉	—	2	1	no	—
123	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
125	Q₈×C₈₉	—	2	1	no	—
126	Q₈×C₈₉	—	2	1	no	—
127	Q₈×C₈₉	—	2	1	no	—
128	Q₈×C₈₉	—	2	1	no	—
129	Q₈×C₈₉	—	2	1	no	—
130	Q₈×C₈₉	—	4	2	no	—
131	Q₈×C₈₉	C₂×D₄	1	2	yes	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	—	2	1	no	—
135	C₂×C₄×C₈₉	—	2	1	no	—
136	C₂×C₄×C₈₉	—	2	1	no	—
137	C₂×C₄×C₈₉	—	2	1	no	—
138	C₂×C₄×C₈₉	—	4	2	no	—
139	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
141	C₂²⋊D₈₉	Q₁₆	1	4	yes	complemented
142	C₂×Dic₈₉	—	2	1	no	—
143	C₂×Dic₈₉	—	2	1	no	—
144	C₂×Dic₈₉	—	2	1	no	—
145	C₂×Dic₈₉	—	2	1	no	—
146	C₂×Dic₈₉	—	4	2	no	—
147	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
148	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
149	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
150	C₄×D₈₉	—	2	1	no	—
151	C₄×D₈₉	—	2	1	no	—
152	C₄×D₈₉	—	4	2	no	—
153	C₄×D₈₉	—	4	2	no	—
154	Dic₁₇₈	—	2	1	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	Dic₁₇₈	—	4	2	no	—
161	Dic₁₇₈	—	4	2	no	—
162	Dic₁₇₈	C₂×D₄	1	2	yes	—
163	C₈×C₈₉	—	2	1	no	—
164	C₈×C₈₉	—	2	1	no	—
165	#10623494	—	2	1	no	—
166	#10623494	C₂×D₄	1	2	yes	—
167	C₂²×C₈₉	—	2	1	no	—
168	C₂²×C₈₉	—	2	1	no	—
169	C₂²×C₈₉	C₂×Q₁₆	1	2	yes	complemented
170	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
171	C₂×D₈₉	—	2	1	no	—
172	C₂×D₈₉	—	2	1	no	—
173	C₂×D₈₉	C₂×Q₁₆	1	2	yes	complemented
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	—	2	2	no	—
176	C₄×C₈₉	—	4	1	no	—
177	C₄×C₈₉	—	4	1	no	—
178	C₄×C₈₉	—	4	1	no	—
179	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
180	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
181	Dic₈₉	—	2	2	no	—
182	Dic₈₉	—	2	1	no	—
183	Dic₈₉	—	2	1	no	—
184	Dic₈₉	—	4	1	no	—
185	Dic₈₉	—	4	1	no	—
186	Dic₈₉	—	4	1	no	—
187	Dic₈₉	C₂×Q₁₆	1	2	yes	complemented
188	C₂×C₈₉	—	2	2	no	—
189	C₂×C₈₉	D₄²	1	1	yes	characteristic
190	C₂×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
191	C₂×C₈₉	#578	1	1	yes	characteristic
192	D₈₉	—	2	2	no	—
193	D₄×Q₁₆	—	89	1	no	maximal, Sylow
194	C₈₉	D₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₁₆	—	89	1	no	—
196	C₂²×Q₁₆	—	89	1	no	—
197	D₄×Q₈	—	89	1	no	—
198	D₄×Q₈	—	89	1	no	—
199	#494	—	89	1	no	—
200	#467	—	89	1	no	—
201	#467	—	89	1	no	—
202	#462	—	89	1	no	—
203	#462	—	89	1	no	—
204	#451	—	89	1	no	—
205	#451	—	89	1	no	—
206	#451	—	89	1	no	—
207	#451	D₈₉	1	1	yes	characteristic, complemented
208	C₄×Q₁₆	—	89	1	no	—
209	C₈×D₄	—	89	1	no	—
210	C₂²×Q₈	—	89	1	no	—
211	C₂²×Q₈	—	89	1	no	—
212	C₂²×Q₈	—	89	1	no	—
213	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
214	C₂×Q₁₆	—	89	1	no	—
215	C₂×Q₁₆	—	89	1	no	—
216	C₂×Q₁₆	—	89	1	no	—
217	C₂×Q₁₆	—	89	1	no	—
218	C₂×Q₁₆	—	89	1	no	—
219	C₂×Q₁₆	—	89	1	no	—
220	C₂×Q₁₆	—	89	1	no	—
221	C₂×Q₁₆	—	178	4	no	—
222	C₂×Q₁₆	—	178	4	no	—
223	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
224	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
225	C₂²×C₈	—	89	1	no	—
226	C₂²×C₈	—	89	1	no	—
227	C₄⋊Q₈	—	89	1	no	—
228	C₄⋊Q₈	—	89	1	no	—
229	C₄⋊Q₈	—	178	1	no	—
230	C₄⋊Q₈	—	178	1	no	—
231	C₂²⋊Q₈	—	89	1	no	—
232	C₂²⋊Q₈	—	89	1	no	—
233	C₂²⋊Q₈	—	89	1	no	—
234	C₂²⋊Q₈	—	178	1	no	—
235	C₂²⋊Q₈	—	178	1	no	—
236	C₂²⋊Q₈	—	178	1	no	—
237	C₂²⋊Q₈	—	178	1	no	—
238	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
239	C₄×Q₈	—	89	1	no	—
240	C₄×Q₈	—	89	1	no	—
241	C₄×D₄	—	89	1	no	—
242	C₄×D₄	—	178	1	no	—
243	C₄×D₄	—	178	1	no	—
244	#107	—	89	1	no	—
245	C₄⋊C₈	—	89	1	no	—
246	#103	—	89	1	no	—
247	#103	—	89	1	no	—
248	#103	—	89	1	no	—
249	#103	—	89	1	no	—
250	#103	C₂×D₈₉	1	1	yes	characteristic
251	#103	C₂×D₈₉	1	1	yes	characteristic
252	C₂²⋊C₈	—	89	1	no	—
253	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
254	C₄×C₈	—	89	1	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	89	1	no	—
259	C₂×Q₈	—	89	1	no	—
260	C₂×Q₈	—	89	1	no	—
261	C₂×Q₈	—	356	1	no	—
262	C₂×Q₈	—	356	1	no	—
263	C₂×Q₈	—	356	1	no	—
264	C₂×Q₈	—	178	1	no	—
265	C₂×Q₈	—	178	1	no	—
266	C₂×Q₈	—	178	2	no	—
267	C₂×Q₈	—	178	2	no	—
268	C₂×Q₈	—	178	2	no	—
269	C₂×Q₈	—	178	1	no	—
270	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
271	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
272	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
273	C₂×D₄	—	89	1	no	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	89	1	no	—
276	C₂²×C₄	—	178	1	no	—
277	C₂²×C₄	—	178	1	no	—
278	C₂²×C₄	—	178	1	no	—
279	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
280	Q₁₆	—	2	2	no	—
281	Q₁₆	—	4	1	no	—
282	Q₁₆	—	89	4	no	—
283	Q₁₆	—	356	1	no	—
284	Q₁₆	—	178	2	no	—
285	Q₁₆	—	178	2	no	—
286	Q₁₆	—	178	2	no	—
287	Q₁₆	—	178	2	no	—
288	Q₁₆	—	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	C₂²×D₈₉	1	1	yes	characteristic
294	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	89	1	no	—
297	C₄⋊C₄	—	89	1	no	—
298	C₄⋊C₄	—	178	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	—	178	1	no	—
306	C₄⋊C₄	—	178	1	no	—
307	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
308	C₂²⋊C₄	—	2	1	no	—
309	C₂²⋊C₄	—	89	1	no	—
310	C₂²⋊C₄	—	89	1	no	—
311	C₂²⋊C₄	—	178	1	no	—
312	C₂²⋊C₄	—	178	1	no	—
313	C₂²⋊C₄	—	178	1	no	—
314	C₄²	—	89	1	no	—
315	C₄²	—	178	1	no	—
316	C₄²	—	178	1	no	—
317	C₂³	—	89	1	no	—
318	C₂³	D₄×D₈₉	1	1	yes	characteristic
319	Q₈	—	2	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	2	1	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	89	2	no	—
326	Q₈	—	356	2	no	—
327	Q₈	—	356	2	no	—
328	Q₈	—	356	2	no	—
329	Q₈	—	178	1	no	—
330	Q₈	—	178	1	no	—
331	Q₈	—	178	1	no	—
332	Q₈	—	178	1	no	—
333	Q₈	—	178	1	no	—
334	Q₈	D₄×D₈₉	1	2	yes	—
335	D₄	—	89	4	no	—
336	C₂×C₄	—	2	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	—	4	2	no	—
343	C₂×C₄	—	89	1	no	—
344	C₂×C₄	—	89	1	no	—
345	C₂×C₄	—	89	1	no	—
346	C₂×C₄	—	356	2	no	—
347	C₂×C₄	—	356	2	no	—
348	C₂×C₄	—	356	2	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	—	178	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	—	2	1	no	—
359	C₈	—	89	2	no	—
360	C₈	—	178	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	89	2	no	—
364	C₂²	—	178	1	no	—
365	C₂²	—	178	1	no	—
366	C₂²	Q₁₆×D₈₉	1	2	yes	complemented
367	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	4	1	no	—
373	C₄	—	89	2	no	—
374	C₄	—	356	1	no	—
375	C₄	—	356	1	no	—
376	C₄	—	356	1	no	—
377	C₄	—	178	1	no	—
378	C₄	—	178	1	no	—
379	C₄	—	178	2	no	—
380	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
381	C₄	#15668057	1	1	yes	characteristic
382	C₂	—	2	2	no	—
383	C₂	—	178	2	no	—
384	C₂	#19152230	1	1	yes	characteristic
385	C₂	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
386	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	#451×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×C₂²⋊D₈₉	D₄
hypercenter	#451	D₈₉
hypocenter	C₈₉	D₄×Q₁₆
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
1. C₂²×#13083564	C₂	semidirect
2. C₂×Q₁₆×D₈₉	C₂	semidirect
5. #19151976	C₂	semidirect
6. #19151838	C₂	semidirect
7. #19151837	C₂	semidirect
8. #19151807	C₂	semidirect
9. #19151806	C₂	semidirect
10. #19151756	C₂	semidirect
11. #19151755	C₂	semidirect
12. #19151752	C₂	semidirect
13. #19151690	C₂	semidirect
15. #451×C₈₉	C₂	semidirect
84. #13083564	D₄	semidirect
141. C₂²⋊D₈₉	Q₁₆	semidirect
169. C₂²×C₈₉	C₂×Q₁₆	semidirect
173. C₂×D₈₉	C₂×Q₁₆	semidirect
187. Dic₈₉	C₂×Q₁₆	semidirect
194. C₈₉	D₄×Q₁₆	semidirect
207. #451	D₈₉	semidirect
366. C₂²	Q₁₆×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13083564	2
C₂×Q₁₆×D₈₉	2
Q₈×C₂²⋊D₈₉	2
#19152261	2
#19151976	2
#19151838	2
#19151837	2
#19151807	2
#19151806	2
#19151756	2
#19151755	2
#19151752	2
#19151690	2
#19151661	2
#451×C₈₉	2
D₄×Q₁₆	89