Subgroups — #26028955

Order 11392.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668018	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152109	C₂	1	1	yes	maximal, characteristic
3	#19151674	C₂	1	4	yes	maximal, complemented
4	#19151667	C₂	1	4	yes	maximal, complemented
5	#19151624	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151564	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151562	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151553	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13083574	C₂²	1	1	yes	characteristic
11	#15668018	—	2	4	no	—
12	C₂×#13083558	C₂²	1	1	yes	characteristic
13	C₂×#13083558	C₂²	1	1	yes	characteristic
14	C₂×#13083557	C₂²	1	2	yes	complemented
15	C₂×C₈×D₈₉	C₂²	1	2	yes	complemented
16	#15667995	C₄	1	2	yes	—
17	#15667991	C₄	1	2	yes	—
18	#15667986	C₄	1	2	yes	—
19	#15667981	C₄	1	2	yes	—
20	C₄×C₂²⋊D₈₉	C₂²	1	4	yes	—
21	#15667971	C₂²	1	1	yes	characteristic
22	#15667941	C₂²	1	4	yes	complemented
23	C₂²⋊#10623494	C₂²	1	4	yes	—
24	#15667924	C₂²	1	4	yes	complemented
25	#15667922	C₂²	1	2	yes	complemented
26	C₈×Dic₈₉	C₂²	1	2	yes	complemented
27	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₈₉	—	2	4	no	—
29	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
30	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
31	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	#13083574	—	2	2	no	—
34	#13083574	—	2	2	no	—
35	C₂×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
36	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊D₈₉	C₂×C₄	1	4	yes	—
39	C₂×C₄×D₈₉	C₂³	1	2	yes	—
40	#13083558	—	2	2	no	—
41	#13083558	—	2	2	no	—
42	#13083557	—	2	4	no	—
43	C₂×#10623494	C₂³	1	2	yes	—
44	C₂×#10623494	C₂³	1	2	yes	—
45	C₈×D₈₉	—	2	4	no	—
46	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
47	#13083553	C₂×C₄	1	4	yes	—
48	C₂²⋊Dic₈₉	C₂×C₄	1	2	yes	—
49	C₂²⋊Dic₈₉	C₂×C₄	1	4	yes	—
50	#13083551	C₂×C₄	1	4	yes	—
51	C₄×Dic₈₉	C₂³	1	2	yes	—
52	C₂⁴×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	—	2	2	no	—
54	C₂²×C₄×C₈₉	—	2	2	no	—
55	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
59	M₁₆×C₈₉	—	2	4	no	—
60	M₁₆×C₈₉	—	2	4	no	—
61	C₂×C₈×C₈₉	—	2	2	no	—
62	C₂×C₈×C₈₉	D₄	1	4	yes	complemented
63	C₂×C₈×C₈₉	C₂³	1	2	yes	—
64	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₂³×C₈₉	—	2	2	no	—
67	C₂³×C₈₉	—	4	2	no	—
68	C₂³×C₈₉	C₂²×C₄	1	2	yes	—
69	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	—	2	4	no	—
73	C₂×C₄×C₈₉	—	2	2	no	—
74	C₂×C₄×C₈₉	—	2	2	no	—
75	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₈₉	#41	1	2	yes	—
79	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₈₉	—	2	4	no	—
81	C₂²⋊D₈₉	—	4	4	no	—
82	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
83	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
84	D₃₅₆	—	2	2	no	—
85	C₄×D₈₉	—	2	2	no	—
86	C₄×D₈₉	—	2	2	no	—
87	Dic₁₇₈	—	2	2	no	—
88	C₈×C₈₉	—	2	2	no	—
89	C₈×C₈₉	C₂×D₄	1	2	yes	—
90	C₈×C₈₉	C₂×D₄	1	2	yes	—
91	#10623494	—	2	2	no	—
92	#10623494	—	2	2	no	—
93	C₂²×C₈₉	—	2	2	no	—
94	C₂²×C₈₉	—	2	2	no	—
95	C₂²×C₈₉	—	2	2	no	—
96	C₂²×C₈₉	—	2	2	no	—
97	C₂²×C₈₉	—	4	4	no	—
98	C₂²×C₈₉	C₄×D₄	1	2	yes	—
99	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×D₈₉	—	2	2	no	—
101	C₂×D₈₉	—	2	4	no	—
102	C₄×C₈₉	—	2	2	no	—
103	C₄×C₈₉	C₄×D₄	1	2	yes	—
104	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
106	Dic₈₉	—	2	2	no	—
107	Dic₈₉	—	2	4	no	—
108	C₂×C₈₉	—	2	2	no	—
109	C₂×C₈₉	—	4	2	no	—
110	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₈₉	#568	1	2	yes	—
112	D₈₉	—	4	2	no	—
113	#2930	—	89	1	no	maximal, Sylow
114	C₈₉	#2930	1	1	yes	characteristic, complemented, Sylow
115	C₂×#131	—	89	1	no	—
116	C₂²×M₁₆	D₈₉	1	1	yes	characteristic, complemented
117	#525	—	89	1	no	—
118	#436	—	89	4	no	—
119	#435	—	89	4	no	—
120	#424	—	89	1	no	—
121	#408	—	89	1	no	—
122	#407	—	89	1	no	—
123	#405	—	89	1	no	—
124	C₂×#41	—	89	1	no	—
125	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
126	#131	—	178	4	no	—
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	—	89	1	no	—
129	C₂×M₁₆	—	89	1	no	—
130	C₂×M₁₆	—	89	2	no	—
131	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
132	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
133	C₂²×C₈	—	89	2	no	—
134	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
136	#123	—	89	2	no	—
137	C₂²⋊Q₈	—	89	2	no	—
138	#121	—	89	2	no	—
139	#120	—	89	2	no	—
140	C₄×D₄	—	89	4	no	—
141	#117	—	89	1	no	—
142	C₄⋊C₈	—	89	4	no	—
143	C₂²⋊C₈	—	89	4	no	—
144	C₂²⋊C₈	—	89	4	no	—
145	#97	—	89	2	no	—
146	C₄×C₈	—	89	2	no	—
147	C₂⁴	C₄×D₈₉	1	1	yes	characteristic
148	#41	—	178	2	no	—
149	#41	—	178	2	no	—
150	C₂×Q₈	—	89	1	no	—
151	C₂×D₄	—	89	4	no	—
152	C₂×D₄	—	89	2	no	—
153	C₂×D₄	—	89	1	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	89	2	no	—
157	C₂²×C₄	C₄×D₈₉	1	2	yes	—
158	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
160	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
161	M₁₆	—	2	4	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	178	2	no	—
164	M₁₆	—	178	2	no	—
165	M₁₆	—	178	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	89	2	no	—
168	C₂×C₈	—	89	2	no	—
169	C₂×C₈	—	178	4	no	—
170	C₂×C₈	C₂²⋊D₈₉	1	4	yes	complemented
171	C₂×C₈	C₂²×D₈₉	1	2	yes	—
172	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
174	C₄⋊C₄	—	89	2	no	—
175	C₄⋊C₄	—	89	4	no	—
176	C₂²⋊C₄	—	89	2	no	—
177	C₂²⋊C₄	—	89	4	no	—
178	C₂²⋊C₄	—	89	4	no	—
179	C₄²	—	89	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	4	2	no	—
182	C₂³	—	89	2	no	—
183	C₂³	C₂×C₄×D₈₉	1	2	yes	—
184	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
185	Q₈	—	178	2	no	—
186	D₄	—	356	4	no	—
187	D₄	—	178	4	no	—
188	D₄	—	178	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	89	4	no	—
194	C₂×C₄	—	89	2	no	—
195	C₂×C₄	—	178	2	no	—
196	C₂×C₄	—	178	2	no	—
197	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
199	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
200	C₂×C₄	#13083574	1	2	yes	—
201	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	178	2	no	—
204	C₈	—	178	2	no	—
205	C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
206	C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	—	178	2	no	—
213	C₂²	—	178	4	no	—
214	C₂²	C₄×C₂²⋊D₈₉	1	2	yes	—
215	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	178	2	no	—
218	C₄	—	178	4	no	—
219	C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
220	C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
221	C₄	C₂×#13083574	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	356	2	no	—
225	C₂	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
226	C₂	#19152364	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	C₂²×M₁₆×C₈₉	C₂
prime radical	#19152109	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₂²×M₁₆	D₈₉
hypocenter	C₈₉	#2930
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#15668018	C₂	semidirect
3. #19151674	C₂	semidirect
4. #19151667	C₂	semidirect
5. #19151624	C₂	semidirect
6. #19151564	C₂	semidirect
7. #19151562	C₂	semidirect
8. #19151553	C₂	semidirect
9. C₂²×M₁₆×C₈₉	C₂	semidirect
14. C₂×#13083557	C₂²	semidirect
15. C₂×C₈×D₈₉	C₂²	semidirect
22. #15667941	C₂²	semidirect
24. #15667924	C₂²	semidirect
25. #15667922	C₂²	semidirect
26. C₈×Dic₈₉	C₂²	semidirect
62. C₂×C₈×C₈₉	D₄	semidirect
114. C₈₉	#2930	semidirect
116. C₂²×M₁₆	D₈₉	semidirect
170. C₂×C₈	C₂²⋊D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668018	2
#19152109	2
#19151674	2
#19151667	2
#19151624	2
#19151564	2
#19151562	2
#19151553	2
C₂²×M₁₆×C₈₉	2
#2930	89