Subgroups — #26029086

Order 11392.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668040	C₂	1	1	yes	maximal, characteristic
2	#19152078	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152072	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151711	C₂	1	2	yes	maximal, complemented
5	#19151700	C₂	1	2	yes	maximal, complemented
6	C₄×#13083563	C₂	1	2	yes	maximal, complemented
7	C₄×D₇₁₂	C₂	1	2	yes	maximal, complemented
8	#19151640	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151595	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15667934	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13083574	C₂²	1	1	yes	characteristic
13	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
14	#15668040	C₄	1	8	yes	complemented
15	C₂×#13083563	C₂²	1	2	yes	—
16	C₂×D₇₁₂	C₂²	1	2	yes	—
17	C₄×Dic₁₇₈	C₂²	1	2	yes	complemented
18	C₄×D₃₅₆	C₂²	1	2	yes	—
19	C₄×D₃₅₆	C₂²	1	2	yes	—
20	C₄×D₃₅₆	C₂²	1	2	yes	complemented
21	#15667974	—	2	1	no	—
22	#15667974	—	2	1	no	—
23	#15667972	—	2	2	no	—
24	#15667971	C₂²	1	1	yes	characteristic
25	#15667970	—	2	1	no	—
26	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₈₉	—	2	1	no	—
28	C₂²⋊C₄×D₈₉	—	2	2	no	—
29	#15667946	C₂²	1	2	yes	complemented
30	#15667944	C₂²	1	2	yes	complemented
31	#15667937	C₂²	1	1	yes	characteristic, complemented
32	#15667937	C₂²	1	1	yes	characteristic, complemented
33	#15667934	C₂²	1	2	yes	—
34	#15667934	C₂²	1	2	yes	—
35	#15667934	C₂²	1	1	yes	characteristic, complemented
36	#15667934	C₂²	1	1	yes	characteristic, complemented
37	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
39	#117×C₈₉	C₂²	1	1	yes	characteristic
40	#97×C₈₉	C₂²	1	2	yes	complemented
41	C₄×C₈×C₈₉	C₂²	1	2	yes	complemented
42	C₂³×D₈₉	—	2	1	no	—
43	#13083574	—	2	2	no	—
44	#13083574	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
46	C₂×D₃₅₆	—	2	1	no	—
47	C₂×D₃₅₆	—	2	2	no	—
48	C₂×D₃₅₆	C₂×C₄	1	4	yes	complemented
49	C₂×D₃₅₆	C₂³	1	2	yes	—
50	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₈₉	—	2	1	no	—
52	C₂²×Dic₈₉	—	2	1	no	—
53	C₂×C₄×D₈₉	—	2	1	no	—
54	C₂×C₄×D₈₉	—	2	2	no	—
55	C₂×C₄×D₈₉	—	2	2	no	—
56	C₂×C₄×D₈₉	—	2	2	no	—
57	#13083563	C₂×C₄	1	8	yes	complemented
58	D₇₁₂	C₂×C₄	1	8	yes	complemented
59	C₄⋊Dic₈₉	—	2	1	no	—
60	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
61	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
62	C₄⋊Dic₈₉	C₂³	1	2	yes	—
63	#13083553	—	2	2	no	—
64	C₂²⋊Dic₈₉	—	2	1	no	—
65	#13083551	—	2	2	no	—
66	#13083551	—	2	2	no	—
67	#13083551	—	2	2	no	—
68	C₄×Dic₈₉	—	2	2	no	—
69	C₄×Dic₈₉	—	2	1	no	—
70	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
71	M₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
72	C₂×C₈×C₈₉	—	2	2	no	—
73	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₈₉	C₂³	1	2	yes	—
76	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
79	C₄²×C₈₉	C₂³	1	2	yes	—
80	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₈₉	—	2	1	no	—
82	C₂²×D₈₉	—	2	2	no	—
83	C₂²×D₈₉	—	2	4	no	—
84	C₂²×D₈₉	—	8	1	no	—
85	C₂²×D₈₉	—	8	1	no	—
86	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#41	1	2	yes	—
91	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
92	C₂²⋊D₈₉	—	2	4	no	—
93	C₂×Dic₈₉	—	2	1	no	—
94	C₂×Dic₈₉	—	2	1	no	—
95	C₂×Dic₈₉	—	2	1	no	—
96	C₂×Dic₈₉	—	2	2	no	—
97	C₂×Dic₈₉	—	8	1	no	—
98	D₃₅₆	—	2	4	no	—
99	D₃₅₆	—	2	2	no	—
100	D₃₅₆	—	2	1	no	—
101	D₃₅₆	C₂²×C₄	1	2	yes	—
102	D₃₅₆	C₂²×C₄	1	4	yes	—
103	C₄×D₈₉	—	2	4	no	—
104	C₄×D₈₉	—	8	2	no	—
105	C₄×D₈₉	—	8	2	no	—
106	C₄×D₈₉	—	8	2	no	—
107	Dic₁₇₈	—	2	1	no	—
108	Dic₁₇₈	C₂²×C₄	1	2	yes	—
109	C₈×C₈₉	—	2	1	no	—
110	C₈×C₈₉	—	2	1	no	—
111	C₈×C₈₉	C₂²×C₄	1	4	yes	—
112	C₂²×C₈₉	—	2	1	no	—
113	C₂²×C₈₉	—	2	1	no	—
114	C₂²×C₈₉	C₄×D₄	1	2	yes	—
115	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₈₉	—	2	2	no	—
117	C₂×D₈₉	—	2	4	no	—
118	C₂×D₈₉	—	4	1	no	—
119	C₂×D₈₉	—	4	2	no	—
120	C₂×D₈₉	—	4	2	no	—
121	C₂×D₈₉	—	4	1	no	—
122	C₂×D₈₉	—	8	4	no	—
123	C₄×C₈₉	—	2	2	no	—
124	C₄×C₈₉	—	2	2	no	—
125	C₄×C₈₉	C₄×D₄	1	2	yes	—
126	C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
127	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
128	Dic₈₉	—	2	2	no	—
129	Dic₈₉	—	4	1	no	—
130	Dic₈₉	—	4	2	no	—
131	Dic₈₉	—	4	1	no	—
132	C₂×C₈₉	—	2	2	no	—
133	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂×C₈₉	#576	1	1	yes	characteristic
135	C₂×C₈₉	#577	1	1	yes	characteristic
136	D₈₉	—	4	2	no	—
137	D₈₉	—	4	4	no	—
138	#2946	—	89	1	no	maximal, Sylow
139	C₈₉	#2946	1	1	yes	characteristic, complemented, Sylow
140	C₂×AGL(1,ℤ/8)	—	89	1	no	—
141	#520	—	89	1	no	—
142	#518	—	89	1	no	—
143	#442	—	89	2	no	—
144	#440	—	89	2	no	—
145	C₄×SD₁₆	—	89	2	no	—
146	C₄×D₈	—	89	2	no	—
147	#428	—	89	1	no	—
148	#416	—	89	1	no	—
149	C₂×#102	—	89	1	no	—
150	#405	D₈₉	1	1	yes	characteristic, complemented
151	C₂×#41	—	89	1	no	—
152	C₂²×D₄	—	89	1	no	—
153	AGL(1,ℤ/8)	—	89	8	no	—
154	C₂×SD₁₆	—	89	2	no	—
155	C₂×D₈	—	89	2	no	—
156	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
157	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
158	C₄×Q₈	—	89	2	no	—
159	C₄×D₄	—	89	2	no	—
160	C₄×D₄	—	89	2	no	—
161	C₄×D₄	—	89	2	no	—
162	C₄×D₄	—	178	1	no	—
163	C₄×D₄	—	178	1	no	—
164	C₄×D₄	—	178	2	no	—
165	#117	—	89	1	no	—
166	#117	—	178	1	no	—
167	#117	C₂×D₈₉	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	89	1	no	—
169	C₂×C₄⋊C₄	—	178	1	no	—
170	C₂×C₂²⋊C₄	—	178	2	no	—
171	#107	—	89	2	no	—
172	#106	—	89	2	no	—
173	#103	—	89	1	no	—
174	#103	—	89	1	no	—
175	#102	—	89	2	no	—
176	#102	—	89	2	no	—
177	#102	—	89	1	no	—
178	#102	—	89	1	no	—
179	#97	C₂×D₈₉	1	2	yes	complemented
180	C₄×C₈	C₂×D₈₉	1	2	yes	complemented
181	C₂⁴	—	178	1	no	—
182	#41	—	89	4	no	—
183	#41	—	178	2	no	—
184	C₂×Q₈	—	89	1	no	—
185	C₂×D₄	—	89	4	no	—
186	C₂×D₄	—	89	1	no	—
187	C₂×D₄	—	89	2	no	—
188	C₂×D₄	—	178	1	no	—
189	C₂×D₄	—	178	1	no	—
190	C₂×D₄	—	178	2	no	—
191	C₂²×C₄	—	178	1	no	—
192	C₂²×C₄	—	178	1	no	—
193	C₂²×C₄	—	178	2	no	—
194	C₂²×C₄	—	178	2	no	—
195	C₂²×C₄	—	178	2	no	—
196	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
197	SD₁₆	—	89	8	no	—
198	D₈	—	89	8	no	—
199	M₁₆	C₄×D₈₉	1	4	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₈₉	1	2	yes	—
204	C₄⋊C₄	—	89	2	no	—
205	C₄⋊C₄	—	89	1	no	—
206	C₄⋊C₄	—	89	1	no	—
207	C₄⋊C₄	—	178	1	no	—
208	C₄⋊C₄	—	178	2	no	—
209	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
210	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
211	C₂²⋊C₄	—	178	1	no	—
212	C₂²⋊C₄	—	178	2	no	—
213	C₂²⋊C₄	—	178	2	no	—
214	C₂²⋊C₄	—	178	2	no	—
215	C₂²⋊C₄	D₃₅₆	1	2	yes	—
216	C₄²	—	178	1	no	—
217	C₄²	—	178	2	no	—
218	C₄²	C₂²×D₈₉	1	2	yes	—
219	C₂³	—	178	1	no	—
220	C₂³	—	178	2	no	—
221	C₂³	—	178	4	no	—
222	C₂³	—	712	1	no	—
223	C₂³	—	712	1	no	—
224	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
225	Q₈	—	89	2	no	—
226	Q₈	—	178	1	no	—
227	D₄	—	89	2	no	—
228	D₄	—	89	4	no	—
229	D₄	—	178	4	no	—
230	D₄	—	178	4	no	—
231	D₄	—	178	2	no	—
232	D₄	—	178	1	no	—
233	C₂×C₄	—	178	1	no	—
234	C₂×C₄	—	178	1	no	—
235	C₂×C₄	—	178	2	no	—
236	C₂×C₄	—	178	1	no	—
237	C₂×C₄	—	178	4	no	—
238	C₂×C₄	—	712	2	no	—
239	C₂×C₄	—	712	2	no	—
240	C₂×C₄	—	712	2	no	—
241	C₂×C₄	—	712	1	no	—
242	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
243	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
244	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
245	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
246	C₂×C₄	#13083574	1	2	yes	—
247	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	C₂×C₄×D₈₉	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	356	1	no	—
254	C₂²	—	356	2	no	—
255	C₂²	—	356	2	no	—
256	C₂²	—	356	1	no	—
257	C₂²	—	178	2	no	—
258	C₂²	—	178	4	no	—
259	C₂²	—	712	4	no	—
260	C₂²	C₄×D₃₅₆	1	2	yes	—
261	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	356	1	no	—
265	C₄	—	356	2	no	—
266	C₄	—	356	1	no	—
267	C₄	—	178	2	no	—
268	C₄	C₄×D₃₅₆	1	2	yes	—
269	C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
270	C₄	C₂×#13083574	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	356	2	no	—
273	C₂	—	356	4	no	—
274	C₂	C₂×C₄×D₃₅₆	1	1	yes	characteristic
275	C₂	#19152403	1	1	yes	characteristic
276	C₂	#19152410	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator	C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#405×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#15668040	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#405	D₈₉
hypocenter	C₈₉	#2946
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #19152078	C₂	semidirect
3. #19152072	C₂	semidirect
4. #19151711	C₂	semidirect
5. #19151700	C₂	semidirect
6. C₄×#13083563	C₂	semidirect
7. C₄×D₇₁₂	C₂	semidirect
8. #19151640	C₂	semidirect
9. #19151595	C₂	semidirect
10. C₂×#15667934	C₂	semidirect
11. #405×C₈₉	C₂	semidirect
14. #15668040	C₄	semidirect
17. C₄×Dic₁₇₈	C₂²	semidirect
20. C₄×D₃₅₆	C₂²	semidirect
29. #15667946	C₂²	semidirect
30. #15667944	C₂²	semidirect
31. #15667937	C₂²	semidirect
32. #15667937	C₂²	semidirect
35. #15667934	C₂²	semidirect
36. #15667934	C₂²	semidirect
40. #97×C₈₉	C₂²	semidirect
41. C₄×C₈×C₈₉	C₂²	semidirect
44. #13083574	C₂×C₄	semidirect
48. C₂×D₃₅₆	C₂×C₄	semidirect
57. #13083563	C₂×C₄	semidirect
58. D₇₁₂	C₂×C₄	semidirect
71. M₁₆×C₈₉	C₂×C₄	semidirect
139. C₈₉	#2946	semidirect
150. #405	D₈₉	semidirect
179. #97	C₂×D₈₉	semidirect
180. C₄×C₈	C₂×D₈₉	semidirect
199. M₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668040	2
#19152078	2
#19152072	2
#19151711	2
#19151700	2
C₄×#13083563	2
C₄×D₇₁₂	2
#19151640	2
#19151595	2
C₂×#15667934	2
#405×C₈₉	2
#2946	89