Subgroups — #474×D₈₉ #26029993

Order 11392.

480 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151879	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151878	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151877	C₂	1	2	yes	maximal, complemented
6	#19151876	C₂	1	2	yes	maximal, complemented
7	#19151875	C₂	1	1	yes	maximal, characteristic, complemented
8	#107×D₈₉	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₈₉	C₂	1	2	yes	maximal
11	#474×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₈₉	—	2	1	no	—
13	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₈₉	—	2	1	no	—
15	C₂×C₈×D₈₉	C₂²	1	2	yes	—
16	#15668012	C₂²	1	1	yes	characteristic, complemented
17	#15668011	—	2	1	no	—
18	#15668011	C₂²	1	2	yes	complemented
19	#15668010	—	2	1	no	—
20	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
21	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
22	#15667974	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₈₉	—	2	1	no	—
24	#15667972	—	2	1	no	—
25	C₄⋊C₄×D₈₉	—	2	1	no	—
26	C₄⋊C₄×D₈₉	C₂²	1	2	yes	—
27	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×D₈₉	—	2	1	no	—
29	C₄²×D₈₉	C₂²	1	1	yes	characteristic
30	#15667946	C₂²	1	2	yes	complemented
31	#15667945	C₂²	1	2	yes	complemented
32	#15667945	C₂²	1	2	yes	complemented
33	C₄⋊#10623494	C₂²	1	1	yes	characteristic, complemented
34	#15667941	C₂²	1	2	yes	complemented
35	#15667934	C₂²	1	2	yes	—
36	#15667933	C₂²	1	2	yes	—
37	#15667932	C₂²	1	2	yes	—
38	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
39	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
40	#107×C₈₉	C₂²	1	2	yes	complemented
41	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
42	#102×C₈₉	C₂²	1	2	yes	—
43	C₂³×D₈₉	—	2	1	no	—
44	C₂×Dic₁₇₈	—	2	1	no	—
45	C₂×Dic₁₇₈	—	2	2	no	—
46	Q₈×D₈₉	—	4	4	no	—
47	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₈₉	—	2	1	no	—
49	D₄×D₈₉	—	2	2	no	—
50	D₄×D₈₉	Q₈	1	4	yes	complemented
51	C₂²×Dic₈₉	—	2	1	no	—
52	C₂×C₄×D₈₉	—	2	1	no	—
53	C₂×C₄×D₈₉	—	2	2	no	—
54	C₂×C₄×D₈₉	—	2	1	no	—
55	C₂×C₄×D₈₉	—	2	1	no	—
56	C₂×C₄×D₈₉	—	4	4	no	—
57	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
58	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
59	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
60	C₂×#10623494	C₂³	1	2	yes	—
61	C₈×D₈₉	—	2	4	no	—
62	C₄⋊Dic₈₉	—	2	1	no	—
63	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
64	C₄⋊Dic₈₉	C₂³	1	2	yes	—
65	#13083553	—	2	1	no	—
66	#13083553	—	2	2	no	—
67	#13083553	—	2	2	no	—
68	C₂²⋊Dic₈₉	—	2	1	no	—
69	#13083551	—	2	1	no	—
70	#13083551	—	2	1	no	—
71	C₄×Dic₈₉	D₄	1	1	yes	characteristic
72	C₄×Dic₈₉	D₄	1	1	yes	characteristic
73	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
74	C₂×Q₈×C₈₉	—	2	1	no	—
75	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	—	2	1	no	—
77	C₂×C₈×C₈₉	C₂³	1	2	yes	—
78	C₄⋊C₄×C₈₉	—	2	1	no	—
79	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
81	C₂²⋊C₄×C₈₉	—	2	1	no	—
82	C₄²×C₈₉	C₂³	1	1	yes	characteristic
83	C₂³×C₈₉	—	2	1	no	—
84	C₂²×D₈₉	—	2	1	no	—
85	C₂²×D₈₉	—	2	4	no	—
86	C₂²×D₈₉	—	4	2	no	—
87	C₂²×D₈₉	—	4	2	no	—
88	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₈₉	—	4	2	no	—
90	D₄×C₈₉	—	2	1	no	—
91	D₄×C₈₉	C₂×Q₈	1	2	yes	complemented
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	4	2	no	—
95	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₈₉	—	2	4	no	—
99	C₂×Dic₈₉	—	2	1	no	—
100	C₂×Dic₈₉	—	2	2	no	—
101	C₂×Dic₈₉	—	4	2	no	—
102	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
105	D₃₅₆	—	2	1	no	—
106	D₃₅₆	C₂×Q₈	1	2	yes	complemented
107	C₄×D₈₉	—	2	2	no	—
108	C₄×D₈₉	—	4	2	no	—
109	C₄×D₈₉	—	4	4	no	—
110	C₄×D₈₉	—	4	2	no	—
111	C₄×D₈₉	—	4	2	no	—
112	C₄×D₈₉	D₈	1	4	yes	—
113	C₄×D₈₉	C₂×Q₈	1	2	yes	—
114	C₄×D₈₉	#41	1	2	yes	—
115	Dic₁₇₈	—	4	2	no	—
116	Dic₁₇₈	—	4	4	no	—
117	C₈×C₈₉	—	2	2	no	—
118	#10623494	—	2	2	no	—
119	C₂²×C₈₉	—	2	2	no	—
120	C₂²×C₈₉	—	4	1	no	—
121	C₂²×C₈₉	—	4	1	no	—
122	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₈₉	—	2	2	no	—
124	C₂×D₈₉	—	4	1	no	—
125	C₂×D₈₉	—	4	1	no	—
126	C₂×D₈₉	—	4	8	no	—
127	C₂×D₈₉	C₂²⋊Q₈	1	2	yes	—
128	C₂×D₈₉	C₂×D₈	1	2	yes	—
129	C₂×D₈₉	#137	1	2	yes	—
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	4	2	no	—
132	C₄×C₈₉	—	4	1	no	—
133	C₄×C₈₉	C₂×D₈	1	2	yes	—
134	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
135	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
136	Dic₈₉	—	2	1	no	—
137	Dic₈₉	—	4	2	no	—
138	Dic₈₉	—	4	1	no	—
139	Dic₈₉	C₂²⋊Q₈	1	2	yes	—
140	Dic₈₉	C₂×D₈	1	2	yes	—
141	C₂×C₈₉	—	4	2	no	—
142	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₈₉	C₂²×D₈	1	1	yes	characteristic
144	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
145	D₈₉	—	4	2	no	—
146	D₈₉	#474	1	4	yes	direct product factor
147	C₂×#474	—	89	1	no	maximal, Sylow
148	C₈₉	C₂×#474	1	1	yes	characteristic, complemented, Sylow
149	C₂×C₄⋊Q₈	—	89	1	no	—
150	C₂×C₄×D₄	—	89	1	no	—
151	#474	—	89	1	no	—
152	#474	—	89	1	no	—
153	#474	—	89	2	no	—
154	#474	—	89	1	no	—
155	#474	—	89	2	no	—
156	#474	D₈₉	1	1	yes	characteristic, direct product factor
157	C₂×#107	—	89	2	no	—
158	C₂×C₄⋊C₈	—	89	1	no	—
159	C₂×#102	—	89	2	no	—
160	C₂²×Q₈	—	178	1	no	—
161	C₂²×D₄	—	89	1	no	—
162	C₂³×C₄	—	178	1	no	—
163	C₂²×C₈	—	89	2	no	—
164	C₄⋊Q₈	—	89	2	no	—
165	C₄⋊Q₈	—	89	1	no	—
166	C₄⋊Q₈	—	178	1	no	—
167	C₄⋊Q₈	—	178	1	no	—
168	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
169	C₄×D₄	—	89	1	no	—
170	C₄×D₄	—	89	1	no	—
171	C₄×D₄	—	89	1	no	—
172	C₄×D₄	—	178	1	no	—
173	C₄×D₄	—	178	1	no	—
174	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	89	1	no	—
176	C₂×C₄⋊C₄	—	89	2	no	—
177	C₂×C₄⋊C₄	—	178	1	no	—
178	C₂×C₂²⋊C₄	—	178	1	no	—
179	C₂×C₄²	—	89	1	no	—
180	#107	—	89	2	no	—
181	#107	—	89	2	no	—
182	#107	—	89	2	no	—
183	#107	C₂×D₈₉	1	2	yes	complemented
184	C₄⋊C₈	—	89	2	no	—
185	C₄⋊C₈	—	89	1	no	—
186	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
187	#102	—	89	2	no	—
188	#102	—	89	2	no	—
189	#102	—	89	2	no	—
190	#102	C₂×D₈₉	1	2	yes	—
191	C₂⁴	—	178	1	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂×Q₈	—	356	4	no	—
194	C₂×Q₈	—	178	2	no	—
195	C₂×Q₈	—	178	1	no	—
196	C₂×D₄	—	89	4	no	—
197	C₂×D₄	—	89	1	no	—
198	C₂×D₄	—	178	2	no	—
199	C₂×D₄	—	178	1	no	—
200	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	89	1	no	—
203	C₂²×C₄	—	89	1	no	—
204	C₂²×C₄	—	89	1	no	—
205	C₂²×C₄	—	356	4	no	—
206	C₂²×C₄	—	178	1	no	—
207	C₂²×C₄	—	178	1	no	—
208	C₂²×C₄	—	178	2	no	—
209	C₂²×C₄	—	178	1	no	—
210	C₂²×C₄	—	178	1	no	—
211	C₂×C₈	—	89	2	no	—
212	C₂×C₈	—	178	4	no	—
213	C₂×C₈	C₂²×D₈₉	1	2	yes	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	89	2	no	—
216	C₄⋊C₄	—	89	1	no	—
217	C₄⋊C₄	—	178	1	no	—
218	C₄⋊C₄	—	178	2	no	—
219	C₄⋊C₄	—	178	2	no	—
220	C₄⋊C₄	—	178	1	no	—
221	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
222	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	178	1	no	—
225	C₂²⋊C₄	—	178	1	no	—
226	C₂²⋊C₄	—	178	1	no	—
227	C₄²	—	89	1	no	—
228	C₄²	—	89	1	no	—
229	C₄²	—	89	1	no	—
230	C₄²	C₂²×D₈₉	1	1	yes	characteristic
231	C₂³	—	2	1	no	—
232	C₂³	—	89	1	no	—
233	C₂³	—	356	2	no	—
234	C₂³	—	356	2	no	—
235	C₂³	—	178	4	no	—
236	C₂³	—	178	1	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	356	4	no	—
239	Q₈	—	356	2	no	—
240	D₄	—	2	1	no	—
241	D₄	—	89	2	no	—
242	D₄	—	178	4	no	—
243	D₄	—	178	1	no	—
244	D₄	Q₈×D₈₉	1	2	yes	complemented
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	89	1	no	—
249	C₂×C₄	—	89	1	no	—
250	C₂×C₄	—	89	4	no	—
251	C₂×C₄	—	89	2	no	—
252	C₂×C₄	—	89	2	no	—
253	C₂×C₄	—	89	1	no	—
254	C₂×C₄	—	356	2	no	—
255	C₂×C₄	—	356	2	no	—
256	C₂×C₄	—	356	4	no	—
257	C₂×C₄	—	356	2	no	—
258	C₂×C₄	—	356	2	no	—
259	C₂×C₄	—	178	2	no	—
260	C₂×C₄	—	178	1	no	—
261	C₂×C₄	—	178	2	no	—
262	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
263	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
264	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	178	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	89	2	no	—
271	C₂²	—	89	2	no	—
272	C₂²	—	89	2	no	—
273	C₂²	—	356	8	no	—
274	C₂²	—	356	1	no	—
275	C₂²	—	356	1	no	—
276	C₂²	—	178	2	no	—
277	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	2	no	—
281	C₄	—	89	2	no	—
282	C₄	—	89	2	no	—
283	C₄	—	356	1	no	—
284	C₄	—	356	2	no	—
285	C₄	—	178	1	no	—
286	C₄	D₈×D₈₉	1	2	yes	—
287	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
288	C₄	#41×D₈₉	1	1	yes	characteristic
289	C₂	—	4	2	no	—
290	C₂	—	89	4	no	—
291	C₂	—	356	2	no	—
292	C₂	C₂²⋊Q₈×D₈₉	1	1	yes	characteristic
293	C₂	C₂×D₈×D₈₉	1	1	yes	characteristic
294	C₂	#137×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#474×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₄×D₈₉	C₂²
hypercenter	#474	D₈₉
hypocenter	C₈₉	C₂×#474
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×D₈₉	C₂	semidirect
3. #19151879	C₂	semidirect
4. #19151878	C₂	semidirect
5. #19151877	C₂	semidirect
6. #19151876	C₂	semidirect
7. #19151875	C₂	semidirect
8. #107×D₈₉	C₂	semidirect
9. C₄⋊C₈×D₈₉	C₂	semidirect
11. #474×C₈₉	C₂	semidirect
16. #15668012	C₂²	semidirect
18. #15668011	C₂²	semidirect
30. #15667946	C₂²	semidirect
31. #15667945	C₂²	semidirect
32. #15667945	C₂²	semidirect
33. C₄⋊#10623494	C₂²	semidirect
34. #15667941	C₂²	semidirect
38. C₄⋊Q₈×C₈₉	C₂²	semidirect
40. #107×C₈₉	C₂²	semidirect
41. C₄⋊C₈×C₈₉	C₂²	semidirect
50. D₄×D₈₉	Q₈	semidirect
91. D₄×C₈₉	C₂×Q₈	semidirect
106. D₃₅₆	C₂×Q₈	semidirect
146. D₈₉	#474	direct (N × H)
148. C₈₉	C₂×#474	semidirect
156. #474	D₈₉	direct (N × H)
168. C₄⋊Q₈	C₂×D₈₉	semidirect
183. #107	C₂×D₈₉	semidirect
186. C₄⋊C₈	C₂×D₈₉	semidirect
244. D₄	Q₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×D₈₉	2
C₄×D₄×D₈₉	2
#19151879	2
#19151878	2
#19151877	2
#19151876	2
#19151875	2
#107×D₈₉	2
C₄⋊C₈×D₈₉	2
#102×D₈₉	2
#474×C₈₉	2
C₂×#474	89

Subgroups — #474×D89 #26029993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #474×D₈₉ #26029993