Subgroups — #563×D₈₉ #26034380

Order 11392.

1024 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152352	C₂	1	2	yes	maximal, complemented
2	#19152351	C₂	1	2	yes	maximal, complemented
3	#19152350	C₂	1	4	yes	maximal, complemented
4	#19152349	C₂	1	4	yes	maximal, complemented
5	#19152348	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152347	C₂	1	2	yes	maximal, complemented
7	C₄⋊Q₈×D₈₉	C₂	1	2	yes	maximal, complemented
8	#126×D₈₉	C₂	1	4	yes	maximal, complemented
9	#125×D₈₉	C₂	1	2	yes	maximal, complemented
10	#124×D₈₉	C₂	1	1	yes	maximal, characteristic
11	#123×D₈₉	C₂	1	2	yes	maximal
12	C₂²⋊Q₈×D₈₉	C₂	1	4	yes	maximal, complemented
13	#563×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
15	C₂×Q₈×D₈₉	C₂²	1	2	yes	—
16	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₈₉	C₂²	1	2	yes	—
18	#15668012	C₂²	1	2	yes	complemented
19	#15668011	C₂²	1	4	yes	complemented
20	#15668011	C₂²	1	4	yes	complemented
21	#15668010	C₂²	1	2	yes	complemented
22	#15668010	C₂²	1	2	yes	complemented
23	#15668007	C₂²	1	4	yes	complemented
24	#15668007	C₂²	1	4	yes	complemented
25	#15668007	C₂²	1	4	yes	complemented
26	#15668006	C₂²	1	4	yes	complemented
27	#15668005	C₂²	1	4	yes	complemented
28	#15668005	C₂²	1	4	yes	complemented
29	#15668005	C₂²	1	4	yes	complemented
30	#15668004	C₂²	1	2	yes	complemented
31	#15668003	C₂²	1	2	yes	complemented
32	#15668003	C₂²	1	2	yes	complemented
33	#15668002	C₂²	1	4	yes	complemented
34	#15668002	C₂²	1	4	yes	complemented
35	#15668001	C₂²	1	1	yes	characteristic
36	#15668000	C₂²	1	1	yes	characteristic
37	#15667999	C₂²	1	1	yes	characteristic
38	#15667998	C₂²	1	4	yes	—
39	#15667997	C₂²	1	4	yes	—
40	#15667996	C₂²	1	2	yes	—
41	#15667995	C₂²	1	2	yes	—
42	#15667994	C₂²	1	4	yes	—
43	#15667993	C₂²	1	2	yes	—
44	#15667992	C₂²	1	4	yes	complemented
45	#15667991	C₂²	1	4	yes	complemented
46	#15667990	C₂²	1	4	yes	complemented
47	#15667990	C₂²	1	4	yes	complemented
48	#15667989	C₂²	1	4	yes	complemented
49	#15667989	C₂²	1	4	yes	complemented
50	#15667988	C₂²	1	4	yes	complemented
51	C₄⋊C₄×D₈₉	C₂²	1	4	yes	—
52	C₄⋊C₄×D₈₉	C₂²	1	4	yes	—
53	C₄⋊C₄×D₈₉	C₂²	1	2	yes	—
54	C₄⋊C₄×D₈₉	C₂²	1	4	yes	—
55	C₄⋊C₄×D₈₉	C₂²	1	2	yes	—
56	C₂²⋊C₄×D₈₉	C₂²	1	4	yes	—
57	C₂²⋊C₄×D₈₉	C₂²	1	4	yes	—
58	C₂²⋊C₄×D₈₉	C₂²	1	2	yes	—
59	C₄²×D₈₉	C₂²	1	2	yes	—
60	C₄²×D₈₉	C₂²	1	1	yes	characteristic
61	C₄⋊Q₈×C₈₉	C₂²	1	2	yes	complemented
62	#126×C₈₉	C₂²	1	4	yes	complemented
63	#125×C₈₉	C₂²	1	2	yes	complemented
64	#124×C₈₉	C₂²	1	1	yes	characteristic
65	#123×C₈₉	C₂²	1	2	yes	—
66	C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	complemented
67	C₂³×D₈₉	C₂³	1	2	yes	—
68	C₂×Dic₁₇₈	C₂³	1	2	yes	—
69	C₂×Dic₁₇₈	C₂³	1	4	yes	—
70	C₂×Dic₁₇₈	C₂³	1	2	yes	—
71	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
72	Q₈×D₈₉	—	4	4	no	—
73	Q₈×D₈₉	—	4	2	no	—
74	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
75	C₂×C₂²⋊D₈₉	C₂³	1	2	yes	—
76	D₄×D₈₉	—	4	2	no	—
77	C₂²×Dic₈₉	C₂³	1	2	yes	—
78	C₂×C₄×D₈₉	—	4	4	no	—
79	C₂×C₄×D₈₉	C₂³	1	4	yes	—
80	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
81	C₂×C₄×D₈₉	C₂³	1	2	yes	—
82	C₂×C₄×D₈₉	C₂³	1	4	yes	—
83	C₂×C₄×D₈₉	C₂³	1	2	yes	—
84	C₂×C₄×D₈₉	C₂³	1	4	yes	—
85	C₄⋊Dic₈₉	C₂³	1	4	yes	—
86	C₄⋊Dic₈₉	C₂³	1	2	yes	—
87	C₄⋊Dic₈₉	C₂³	1	4	yes	—
88	C₄⋊Dic₈₉	C₂³	1	4	yes	—
89	C₄⋊Dic₈₉	C₂³	1	2	yes	—
90	#13083553	C₂³	1	4	yes	—
91	#13083553	C₂³	1	4	yes	—
92	#13083553	C₂³	1	4	yes	—
93	#13083553	C₂³	1	4	yes	—
94	#13083553	C₂³	1	4	yes	—
95	#13083553	C₂³	1	4	yes	—
96	#13083553	C₂³	1	4	yes	—
97	#13083553	C₂³	1	4	yes	—
98	C₂²⋊Dic₈₉	C₂³	1	4	yes	—
99	C₂²⋊Dic₈₉	C₂³	1	4	yes	—
100	C₂²⋊Dic₈₉	C₂³	1	2	yes	—
101	#13083551	C₂³	1	4	yes	—
102	#13083551	C₂³	1	4	yes	—
103	#13083551	C₂³	1	4	yes	—
104	#13083551	C₂³	1	4	yes	—
105	#13083551	C₂³	1	4	yes	—
106	C₄×Dic₈₉	C₂³	1	2	yes	—
107	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
108	C₄×Dic₈₉	C₂³	1	2	yes	—
109	C₄×Dic₈₉	C₂³	1	4	yes	—
110	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
111	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
112	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
113	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
114	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
115	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
116	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
117	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
118	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
119	C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
120	C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
121	C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
122	C₄²×C₈₉	C₂³	1	1	yes	characteristic
123	C₄²×C₈₉	C₂³	1	2	yes	—
124	C₂³×C₈₉	C₂⁴	1	2	yes	—
125	C₂²×D₈₉	—	2	4	no	—
126	C₂²×D₈₉	—	2	8	no	—
127	C₂²×D₈₉	C₂⁴	1	2	yes	—
128	C₂²×D₈₉	C₂⁴	1	1	yes	characteristic
129	Q₈×C₈₉	—	4	1	no	—
130	Q₈×C₈₉	—	4	2	no	—
131	D₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
134	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
135	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
136	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
138	C₂²⋊D₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	4	2	no	—
140	C₂×Dic₈₉	C₂⁴	1	2	yes	—
141	C₂×Dic₈₉	C₂⁴	1	4	yes	—
142	C₂×Dic₈₉	C₂⁴	1	2	yes	—
143	C₂×Dic₈₉	C₂⁴	1	1	yes	characteristic
144	C₂×Dic₈₉	C₂⁴	1	4	yes	—
145	D₃₅₆	—	4	1	no	—
146	C₄×D₈₉	—	2	8	no	—
147	C₄×D₈₉	—	2	2	no	—
148	C₄×D₈₉	—	2	4	no	—
149	C₄×D₈₉	—	2	8	no	—
150	C₄×D₈₉	—	2	4	no	—
151	C₄×D₈₉	—	4	4	no	—
152	Dic₁₇₈	—	4	1	no	—
153	Dic₁₇₈	—	4	4	no	—
154	Dic₁₇₈	—	4	2	no	—
155	Dic₁₇₈	—	4	2	no	—
156	C₂²×C₈₉	—	2	2	no	—
157	C₂²×C₈₉	—	2	4	no	—
158	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
159	C₂×D₈₉	—	2	2	no	—
160	C₂×D₈₉	—	2	4	no	—
161	C₂×D₈₉	—	4	8	no	—
162	C₂×D₈₉	H(5,𝔽₂)	1	2	yes	—
163	C₂×D₈₉	2₋¹⁺⁴	1	4	yes	—
164	C₄×C₈₉	—	2	2	no	—
165	C₄×C₈₉	—	2	4	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	—	2	4	no	—
169	Dic₈₉	—	2	2	no	—
170	Dic₈₉	—	2	4	no	—
171	Dic₈₉	—	2	1	no	—
172	Dic₈₉	—	2	2	no	—
173	Dic₈₉	—	2	4	no	—
174	C₂×C₈₉	—	4	2	no	—
175	C₂×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
176	C₂×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
177	D₈₉	—	4	2	no	—
178	D₈₉	#563	1	4	yes	direct product factor
179	C₂×#563	—	89	1	no	maximal, Sylow
180	C₈₉	C₂×#563	1	1	yes	characteristic, complemented, Sylow
181	#563	—	89	2	no	—
182	#563	—	89	4	no	—
183	#563	—	89	2	no	—
184	#563	—	89	1	no	—
185	#563	—	89	2	no	—
186	#563	—	89	4	no	—
187	#563	D₈₉	1	1	yes	characteristic, direct product factor
188	C₂×C₄⋊Q₈	—	89	2	no	—
189	C₂×#126	—	89	4	no	—
190	C₂×#125	—	89	2	no	—
191	C₂×#124	—	89	1	no	—
192	C₂×#123	—	89	2	no	—
193	C₂×C₂²⋊Q₈	—	89	4	no	—
194	C₂²×Q₈	—	89	2	no	—
195	C₂²×Q₈	—	89	1	no	—
196	C₂²×D₄	—	89	1	no	—
197	C₂³×C₄	—	89	2	no	—
198	C₄⋊Q₈	—	89	2	no	—
199	C₄⋊Q₈	—	89	2	no	—
200	C₄⋊Q₈	—	89	4	no	—
201	C₄⋊Q₈	—	89	4	no	—
202	C₄⋊Q₈	—	89	2	no	—
203	C₄⋊Q₈	C₂×D₈₉	1	2	yes	complemented
204	#126	—	89	4	no	—
205	#126	—	89	4	no	—
206	#126	—	89	4	no	—
207	#126	—	89	4	no	—
208	#126	—	89	4	no	—
209	#126	—	89	4	no	—
210	#126	—	89	4	no	—
211	#126	C₂×D₈₉	1	4	yes	complemented
212	#125	—	89	4	no	—
213	#125	—	89	4	no	—
214	#125	—	89	2	no	—
215	#125	—	89	2	no	—
216	#125	—	89	2	no	—
217	#125	C₂×D₈₉	1	2	yes	complemented
218	#124	—	89	1	no	—
219	#124	—	89	1	no	—
220	#124	—	89	4	no	—
221	#124	—	89	1	no	—
222	#124	C₂×D₈₉	1	1	yes	characteristic
223	#123	—	89	2	no	—
224	#123	—	89	2	no	—
225	#123	—	89	4	no	—
226	#123	—	89	4	no	—
227	#123	—	89	2	no	—
228	#123	C₂×D₈₉	1	2	yes	—
229	C₂²⋊Q₈	—	89	4	no	—
230	C₂²⋊Q₈	—	89	4	no	—
231	C₂²⋊Q₈	—	89	4	no	—
232	C₂²⋊Q₈	—	89	4	no	—
233	C₂²⋊Q₈	—	89	4	no	—
234	C₂²⋊Q₈	—	89	4	no	—
235	C₂²⋊Q₈	—	89	4	no	—
236	C₂²⋊Q₈	C₂×D₈₉	1	4	yes	complemented
237	C₂×C₄⋊C₄	—	89	2	no	—
238	C₂×C₄⋊C₄	—	89	4	no	—
239	C₂×C₄⋊C₄	—	89	4	no	—
240	C₂×C₄⋊C₄	—	89	4	no	—
241	C₂×C₄⋊C₄	—	89	2	no	—
242	C₂×C₂²⋊C₄	—	89	4	no	—
243	C₂×C₂²⋊C₄	—	89	4	no	—
244	C₂×C₂²⋊C₄	—	89	2	no	—
245	C₂×C₄²	—	89	1	no	—
246	C₂×C₄²	—	89	2	no	—
247	C₂⁴	—	89	2	no	—
248	C₂×Q₈	—	89	4	no	—
249	C₂×Q₈	—	89	2	no	—
250	C₂×Q₈	—	89	1	no	—
251	C₂×Q₈	—	89	2	no	—
252	C₂×Q₈	—	356	4	no	—
253	C₂×Q₈	—	356	2	no	—
254	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
255	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
256	C₂×D₄	—	89	2	no	—
257	C₂×D₄	—	89	1	no	—
258	C₂×D₄	—	356	2	no	—
259	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
260	C₂²×C₄	—	89	2	no	—
261	C₂²×C₄	—	89	4	no	—
262	C₂²×C₄	—	89	2	no	—
263	C₂²×C₄	—	89	4	no	—
264	C₂²×C₄	—	89	2	no	—
265	C₂²×C₄	—	89	1	no	—
266	C₂²×C₄	—	89	4	no	—
267	C₂²×C₄	—	356	4	no	—
268	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
269	C₄⋊C₄	—	89	4	no	—
270	C₄⋊C₄	—	89	2	no	—
271	C₄⋊C₄	—	89	4	no	—
272	C₄⋊C₄	—	89	4	no	—
273	C₄⋊C₄	—	89	4	no	—
274	C₄⋊C₄	—	89	4	no	—
275	C₄⋊C₄	—	89	4	no	—
276	C₄⋊C₄	—	89	4	no	—
277	C₄⋊C₄	—	89	4	no	—
278	C₄⋊C₄	—	89	4	no	—
279	C₄⋊C₄	—	89	4	no	—
280	C₄⋊C₄	—	89	4	no	—
281	C₄⋊C₄	—	89	2	no	—
282	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
283	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
284	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
285	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
286	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
287	C₂²⋊C₄	—	89	4	no	—
288	C₂²⋊C₄	—	89	4	no	—
289	C₂²⋊C₄	—	89	2	no	—
290	C₂²⋊C₄	—	89	4	no	—
291	C₂²⋊C₄	—	89	4	no	—
292	C₂²⋊C₄	—	89	4	no	—
293	C₂²⋊C₄	—	89	4	no	—
294	C₂²⋊C₄	—	89	4	no	—
295	C₂²⋊C₄	C₂²×D₈₉	1	4	yes	—
296	C₂²⋊C₄	C₂²×D₈₉	1	4	yes	—
297	C₂²⋊C₄	C₂²×D₈₉	1	2	yes	—
298	C₄²	—	89	1	no	—
299	C₄²	—	89	2	no	—
300	C₄²	—	89	2	no	—
301	C₄²	—	89	4	no	—
302	C₄²	C₂²×D₈₉	1	1	yes	characteristic
303	C₄²	C₂²×D₈₉	1	2	yes	—
304	C₂³	—	89	1	no	—
305	C₂³	—	89	2	no	—
306	C₂³	—	178	4	no	—
307	C₂³	—	178	8	no	—
308	C₂³	C₂³×D₈₉	1	2	yes	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	356	4	no	—
312	Q₈	—	356	1	no	—
313	Q₈	—	356	2	no	—
314	Q₈	—	356	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	356	1	no	—
317	D₄	—	356	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	89	2	no	—
320	C₂×C₄	—	89	4	no	—
321	C₂×C₄	—	89	2	no	—
322	C₂×C₄	—	89	1	no	—
323	C₂×C₄	—	89	4	no	—
324	C₂×C₄	—	356	4	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	178	4	no	—
327	C₂×C₄	—	178	8	no	—
328	C₂×C₄	—	178	8	no	—
329	C₂×C₄	—	178	2	no	—
330	C₂×C₄	—	178	4	no	—
331	C₂×C₄	C₂³×D₈₉	1	2	yes	—
332	C₂×C₄	C₂³×D₈₉	1	4	yes	—
333	C₂×C₄	C₂³×D₈₉	1	2	yes	—
334	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
335	C₂×C₄	C₂³×D₈₉	1	4	yes	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	4	no	—
338	C₂²	—	89	2	no	—
339	C₂²	—	89	4	no	—
340	C₂²	—	356	8	no	—
341	C₂²	—	178	2	no	—
342	C₂²	—	178	4	no	—
343	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
344	C₄	—	2	4	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	4	no	—
349	C₄	—	178	4	no	—
350	C₄	—	178	1	no	—
351	C₄	—	178	2	no	—
352	C₄	—	178	2	no	—
353	C₄	—	178	4	no	—
354	C₂	—	4	2	no	—
355	C₂	—	89	4	no	—
356	C₂	—	356	2	no	—
357	C₂	H(5,𝔽₂)×D₈₉	1	1	yes	characteristic
358	C₂	2₋¹⁺⁴×D₈₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	#563×C₈₉	C₂
prime radical	C₂×D₄×D₈₉	C₂²
hypercenter	#563	D₈₉
hypocenter	C₈₉	C₂×#563
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #563

Complements

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

N	H ≅ G/N	type
1. #19152352	C₂	semidirect
2. #19152351	C₂	semidirect
3. #19152350	C₂	semidirect
4. #19152349	C₂	semidirect
5. #19152348	C₂	semidirect
6. #19152347	C₂	semidirect
7. C₄⋊Q₈×D₈₉	C₂	semidirect
8. #126×D₈₉	C₂	semidirect
9. #125×D₈₉	C₂	semidirect
12. C₂²⋊Q₈×D₈₉	C₂	semidirect
13. #563×C₈₉	C₂	semidirect
18. #15668012	C₂²	semidirect
19. #15668011	C₂²	semidirect
20. #15668011	C₂²	semidirect
21. #15668010	C₂²	semidirect
22. #15668010	C₂²	semidirect
23. #15668007	C₂²	semidirect
24. #15668007	C₂²	semidirect
25. #15668007	C₂²	semidirect
26. #15668006	C₂²	semidirect
27. #15668005	C₂²	semidirect
28. #15668005	C₂²	semidirect
29. #15668005	C₂²	semidirect
30. #15668004	C₂²	semidirect
31. #15668003	C₂²	semidirect
32. #15668003	C₂²	semidirect
33. #15668002	C₂²	semidirect
34. #15668002	C₂²	semidirect
44. #15667992	C₂²	semidirect
45. #15667991	C₂²	semidirect
46. #15667990	C₂²	semidirect
47. #15667990	C₂²	semidirect
48. #15667989	C₂²	semidirect
49. #15667989	C₂²	semidirect
50. #15667988	C₂²	semidirect
61. C₄⋊Q₈×C₈₉	C₂²	semidirect
62. #126×C₈₉	C₂²	semidirect
63. #125×C₈₉	C₂²	semidirect
66. C₂²⋊Q₈×C₈₉	C₂²	semidirect
178. D₈₉	#563	direct (N × H)
180. C₈₉	C₂×#563	semidirect
187. #563	D₈₉	direct (N × H)
203. C₄⋊Q₈	C₂×D₈₉	semidirect
211. #126	C₂×D₈₉	semidirect
217. #125	C₂×D₈₉	semidirect
236. C₂²⋊Q₈	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152352	2
#19152351	2
#19152350	2
#19152349	2
#19152348	2
#19152347	2
C₄⋊Q₈×D₈₉	2
#126×D₈₉	2
#125×D₈₉	2
#124×D₈₉	2
#123×D₈₉	2
C₂²⋊Q₈×D₈₉	2
#563×C₈₉	2
C₂×#563	89

Subgroups — #563×D89 #26034380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #563×D₈₉ #26034380