Subgroups — #26029158

Order 11392.

458 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152396	C₂	1	1	yes	maximal, characteristic
2	#19152365	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152364	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151717	C₂	1	4	yes	maximal, complemented
5	#19151712	C₂	1	4	yes	maximal, complemented
6	#19151646	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15667940	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#104×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13083574	C₂²	1	1	yes	characteristic
11	#15668043	C₄	1	2	yes	complemented
12	#15668042	C₄	1	2	yes	complemented
13	#15668038	C₄	1	2	yes	complemented
14	#15668037	C₄	1	2	yes	complemented
15	#15668032	C₂²	1	4	yes	—
16	#15668020	C₂²	1	4	yes	complemented
17	#15668019	—	2	2	no	—
18	#15668019	C₂²	1	4	yes	complemented
19	#15668018	—	2	2	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	C₂²	1	1	yes	characteristic
22	C₂×#13083558	C₂²	1	1	yes	characteristic
23	C₂×#13083558	C₂²	1	1	yes	characteristic
24	M₁₆×D₈₉	—	2	2	no	—
25	#15667947	C₂²	1	4	yes	complemented
26	#15667940	C₂²	1	2	yes	complemented
27	#15667940	C₂²	1	2	yes	complemented
28	#15667923	C₂²	1	2	yes	complemented
29	C₄×#10623494	C₂²	1	2	yes	complemented
30	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
33	#104×C₈₉	C₂²	1	2	yes	complemented
34	#104×C₈₉	C₂²	1	2	yes	complemented
35	#13083574	—	2	1	no	—
36	#13083574	—	2	1	no	—
37	#13083574	C₂×C₄	1	2	yes	—
38	#13083574	C₂³	1	2	yes	—
39	C₂×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
40	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₈₉	—	2	1	no	—
42	C₂×C₄×D₈₉	—	2	1	no	—
43	#13083564	C₂×C₄	1	4	yes	complemented
44	#13083562	C₂×C₄	1	4	yes	complemented
45	#13083561	C₂×C₄	1	4	yes	complemented
46	#13083559	C₂×C₄	1	4	yes	complemented
47	#13083558	—	2	1	no	—
48	#13083558	—	2	4	no	—
49	#13083558	C₂×C₄	1	2	yes	—
50	#13083558	C₂×C₄	1	2	yes	—
51	#13083558	D₄	1	2	yes	—
52	#13083558	D₄	1	2	yes	—
53	#13083558	C₂³	1	2	yes	—
54	#13083557	—	2	4	no	—
55	C₂×#10623494	—	2	1	no	—
56	C₂×#10623494	—	2	4	no	—
57	C₂×#10623494	C₂³	1	2	yes	—
58	C₂×#10623494	C₂³	1	2	yes	—
59	C₈×D₈₉	—	2	4	no	—
60	#41×C₈₉	—	2	1	no	—
61	#41×C₈₉	—	2	1	no	—
62	#41×C₈₉	C₂×C₄	1	2	yes	—
63	#41×C₈₉	C₂³	1	2	yes	—
64	C₂×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂×D₄×C₈₉	—	2	1	no	—
66	C₂×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	—	2	1	no	—
68	C₂²×C₄×C₈₉	—	2	1	no	—
69	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
70	M₁₆×C₈₉	—	2	1	no	—
71	M₁₆×C₈₉	C₂³	1	2	yes	—
72	C₂×C₈×C₈₉	—	2	1	no	—
73	C₄²×C₈₉	—	2	1	no	—
74	C₄²×C₈₉	C₂³	1	1	yes	characteristic
75	C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	C₂³×C₈₉	—	2	1	no	—
77	C₂³×C₈₉	#41	1	1	yes	characteristic
78	C₂²×D₈₉	—	2	1	no	—
79	Q₈×C₈₉	—	2	1	no	—
80	Q₈×C₈₉	C₂²×C₄	1	2	yes	—
81	D₄×C₈₉	—	2	2	no	—
82	D₄×C₈₉	—	2	2	no	—
83	D₄×C₈₉	—	2	1	no	—
84	D₄×C₈₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	4	2	no	—
91	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₈₉	—	2	2	no	—
98	C₂²⋊D₈₉	—	2	2	no	—
99	C₂×Dic₈₉	—	2	1	no	—
100	D₃₅₆	—	2	1	no	—
101	D₃₅₆	C₂²×C₄	1	2	yes	—
102	C₄×D₈₉	—	2	2	no	—
103	C₄×D₈₉	—	2	2	no	—
104	Dic₁₇₈	—	2	1	no	—
105	Dic₁₇₈	C₂²×C₄	1	2	yes	—
106	C₈×C₈₉	—	2	2	no	—
107	#10623494	—	2	2	no	—
108	#10623494	C₂²×C₄	1	4	yes	—
109	#10623494	C₂×D₄	1	4	yes	—
110	C₂²×C₈₉	—	2	2	no	—
111	C₂²×C₈₉	—	4	1	no	—
112	C₂²×C₈₉	—	8	1	no	—
113	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
114	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
116	C₂×D₈₉	—	2	2	no	—
117	C₂×D₈₉	—	8	1	no	—
118	C₄×C₈₉	—	2	2	no	—
119	C₄×C₈₉	—	2	2	no	—
120	C₄×C₈₉	—	4	1	no	—
121	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
122	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
123	C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
125	Dic₈₉	—	2	2	no	—
126	C₂×C₈₉	—	2	1	no	—
127	C₂×C₈₉	—	2	1	no	—
128	C₂×C₈₉	—	2	1	no	—
129	C₂×C₈₉	—	4	2	no	—
130	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
131	D₈₉	—	4	2	no	—
132	#2956	—	89	1	no	maximal, Sylow
133	C₈₉	#2956	1	1	yes	characteristic, complemented, Sylow
134	#575	—	89	1	no	—
135	#568	—	89	1	no	—
136	#568	—	89	1	no	—
137	#444	—	89	4	no	—
138	#443	—	89	4	no	—
139	#430	—	89	1	no	—
140	C₂×#104	—	89	1	no	—
141	C₂×#104	D₈₉	1	1	yes	characteristic, complemented
142	C₄×M₁₆	—	89	1	no	—
143	C₂×#41	—	89	1	no	—
144	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
145	#137	—	89	2	no	—
146	#137	—	89	2	no	—
147	AGL(1,ℤ/8)	—	89	2	no	—
148	AGL(1,ℤ/8)	—	89	2	no	—
149	#135	—	89	4	no	—
150	#131	—	89	4	no	—
151	#131	—	89	4	no	—
152	#131	—	178	2	no	—
153	#131	—	178	2	no	—
154	C₂×M₁₆	—	89	1	no	—
155	C₂×M₁₆	—	89	1	no	—
156	C₂×M₁₆	—	89	1	no	—
157	C₂×M₁₆	—	178	2	no	—
158	C₂×M₁₆	—	178	2	no	—
159	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
160	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
161	#108	—	89	4	no	—
162	#104	—	89	2	no	—
163	#104	—	89	2	no	—
164	#104	C₂×D₈₉	1	2	yes	complemented
165	#104	C₂×D₈₉	1	2	yes	complemented
166	#97	—	89	2	no	—
167	C₄×C₈	—	89	2	no	—
168	#41	—	2	1	no	—
169	#41	—	2	1	no	—
170	#41	—	89	2	no	—
171	#41	—	89	2	no	—
172	#41	—	178	1	no	—
173	#41	—	178	1	no	—
174	#41	C₄×D₈₉	1	2	yes	—
175	#41	C₂²×D₈₉	1	2	yes	—
176	C₂×Q₈	—	89	1	no	—
177	C₂×Q₈	C₄×D₈₉	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	89	1	no	—
180	C₂×D₄	—	178	1	no	—
181	C₂×D₄	C₄×D₈₉	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	178	1	no	—
185	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
186	Q₁₆	—	89	4	no	—
187	SD₁₆	—	89	4	no	—
188	SD₁₆	—	89	4	no	—
189	D₈	—	89	4	no	—
190	M₁₆	—	2	1	no	—
191	M₁₆	—	89	2	no	—
192	M₁₆	—	89	2	no	—
193	M₁₆	—	89	2	no	—
194	M₁₆	—	89	2	no	—
195	M₁₆	—	89	2	no	—
196	M₁₆	—	178	4	no	—
197	M₁₆	—	178	4	no	—
198	M₁₆	—	178	1	no	—
199	M₁₆	C₂²×D₈₉	1	2	yes	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	89	2	no	—
202	C₂×C₈	—	89	2	no	—
203	C₂×C₈	—	178	4	no	—
204	C₂×C₈	—	178	4	no	—
205	C₂×C₈	—	178	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	C₂²×D₈₉	1	1	yes	characteristic
208	C₄²	C₂²×D₈₉	1	1	yes	characteristic
209	C₂³	—	2	1	no	—
210	C₂³	—	178	1	no	—
211	C₂³	#13083573	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	89	2	no	—
214	Q₈	—	178	1	no	—
215	Q₈	C₂×C₄×D₈₉	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	89	2	no	—
220	D₄	—	178	2	no	—
221	D₄	—	178	2	no	—
222	D₄	—	178	1	no	—
223	D₄	C₂×C₄×D₈₉	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	178	1	no	—
231	C₂×C₄	—	178	2	no	—
232	C₂×C₄	—	178	2	no	—
233	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
235	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
236	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
237	C₂×C₄	#13083573	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	89	4	no	—
241	C₈	—	89	4	no	—
242	C₈	—	178	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	178	2	no	—
247	C₂²	—	712	1	no	—
248	C₂²	#15667972	1	1	yes	characteristic
249	C₂²	#15667972	1	1	yes	characteristic
250	C₂²	C₂×#13083573	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	178	2	no	—
255	C₄	#15667972	1	1	yes	characteristic
256	C₄	#15667972	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
258	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	356	2	no	—
264	C₂	C₂×#15667972	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#104×C₈₉	C₂
socle	C₂×C₈₉	C₂×C₄×D₄
prime radical	#19152396	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₂×#104	D₈₉
hypocenter	C₈₉	#2956
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
2. #19152365	C₂	semidirect
3. #19152364	C₂	semidirect
4. #19151717	C₂	semidirect
5. #19151712	C₂	semidirect
6. #19151646	C₂	semidirect
7. C₂×#15667940	C₂	semidirect
8. C₄×#13083558	C₂	semidirect
9. C₂×#104×C₈₉	C₂	semidirect
11. #15668043	C₄	semidirect
12. #15668042	C₄	semidirect
13. #15668038	C₄	semidirect
14. #15668037	C₄	semidirect
16. #15668020	C₂²	semidirect
18. #15668019	C₂²	semidirect
25. #15667947	C₂²	semidirect
26. #15667940	C₂²	semidirect
27. #15667940	C₂²	semidirect
28. #15667923	C₂²	semidirect
29. C₄×#10623494	C₂²	semidirect
33. #104×C₈₉	C₂²	semidirect
34. #104×C₈₉	C₂²	semidirect
43. #13083564	C₂×C₄	semidirect
44. #13083562	C₂×C₄	semidirect
45. #13083561	C₂×C₄	semidirect
46. #13083559	C₂×C₄	semidirect
133. C₈₉	#2956	semidirect
141. C₂×#104	D₈₉	semidirect
164. #104	C₂×D₈₉	semidirect
165. #104	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152396	2
#19152365	2
#19152364	2
#19151717	2
#19151712	2
#19151646	2
C₂×#15667940	2
C₄×#13083558	2
C₂×#104×C₈₉	2
#2956	89