Subgroups — #26029170

Order 11392.

452 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668034	C₂	1	1	yes	maximal, characteristic
2	#19152364	C₂	1	2	yes	maximal, complemented
3	#19151721	C₂	1	4	yes	maximal, complemented
4	#19151716	C₂	1	4	yes	maximal, complemented
5	C₂×#15667948	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151616	C₂	1	2	yes	maximal, complemented
7	#405×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13083574	C₂²	1	2	yes	—
9	#15668034	C₄	1	2	yes	complemented
10	#15668034	C₄	1	2	yes	complemented
11	#15668034	C₂²	1	2	yes	—
12	#15668034	C₂²	1	2	yes	—
13	C₂×Dic₃₅₆	C₄	1	1	yes	characteristic, complemented
14	C₂×#13083563	C₄	1	2	yes	complemented
15	C₂×D₇₁₂	C₄	1	1	yes	characteristic, complemented
16	#15668020	—	2	4	no	—
17	#15668018	C₂²	1	8	yes	complemented
18	C₂×#13083558	C₂²	1	2	yes	—
19	M₁₆×D₈₉	—	2	4	no	—
20	#15667948	C₂²	1	2	yes	complemented
21	#15667948	C₂²	1	2	yes	complemented
22	#15667940	C₂²	1	8	yes	complemented
23	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	#117×C₈₉	C₂²	1	1	yes	characteristic
26	#97×C₈₉	C₂²	1	2	yes	complemented
27	C₄×C₈×C₈₉	C₂²	1	2	yes	complemented
28	#13083574	—	2	2	no	—
29	#13083574	—	2	2	no	—
30	#13083574	C₂×C₄	1	4	yes	—
31	#13083574	C₂³	1	4	yes	—
32	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
33	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊D₈₉	—	2	2	no	—
35	C₂×C₄×D₈₉	—	2	2	no	—
36	Dic₃₅₆	C₂×C₄	1	2	yes	—
37	Dic₃₅₆	C₂×C₄	1	2	yes	—
38	#13083563	C₂×C₄	1	4	yes	—
39	#13083563	C₂×C₄	1	4	yes	—
40	D₇₁₂	C₂×C₄	1	2	yes	—
41	D₇₁₂	C₂×C₄	1	2	yes	—
42	#13083558	—	2	2	no	—
43	#13083558	C₂³	1	4	yes	—
44	#13083557	—	2	8	no	—
45	C₂×#10623494	—	2	2	no	—
46	C₈×D₈₉	—	2	8	no	—
47	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
48	M₁₆×C₈₉	D₄	1	2	yes	—
49	M₁₆×C₈₉	D₄	1	2	yes	—
50	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₈₉	C₂³	1	2	yes	—
57	C₄⋊C₄×C₈₉	—	2	1	no	—
58	C₂²⋊C₄×C₈₉	—	2	1	no	—
59	C₄²×C₈₉	C₂³	1	2	yes	—
60	C₂³×C₈₉	#41	1	1	yes	characteristic
61	C₂²×D₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
69	C₂²⋊D₈₉	—	2	4	no	—
70	C₂²⋊D₈₉	—	2	4	no	—
71	C₂×Dic₈₉	—	2	2	no	—
72	D₃₅₆	—	2	2	no	—
73	D₃₅₆	C₂²×C₄	1	4	yes	—
74	C₄×D₈₉	—	2	4	no	—
75	C₄×D₈₉	—	2	4	no	—
76	Dic₁₇₈	—	2	2	no	—
77	Dic₁₇₈	C₂²×C₄	1	4	yes	—
78	C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
80	C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₈×C₈₉	C₂×D₄	1	4	yes	—
83	#10623494	—	2	4	no	—
84	C₂²×C₈₉	—	4	1	no	—
85	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
87	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
88	C₂×D₈₉	—	2	4	no	—
89	C₂×D₈₉	—	8	2	no	—
90	C₄×C₈₉	—	4	2	no	—
91	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
92	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
93	C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
94	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
95	Dic₈₉	—	2	4	no	—
96	C₂×C₈₉	—	2	1	no	—
97	C₂×C₈₉	—	2	1	no	—
98	C₂×C₈₉	—	2	1	no	—
99	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
100	D₈₉	—	4	4	no	—
101	#2957	—	89	1	no	maximal, Sylow
102	C₈₉	#2957	1	1	yes	characteristic, complemented, Sylow
103	C₂×#135	—	89	1	no	—
104	#568	—	89	2	no	—
105	#444	—	89	4	no	—
106	#443	—	89	4	no	—
107	C₂×#108	—	89	1	no	—
108	#421	—	89	2	no	—
109	#405	D₈₉	1	1	yes	characteristic, complemented
110	C₂×#41	—	89	2	no	—
111	#135	—	89	2	no	—
112	#135	—	89	2	no	—
113	#135	—	89	2	no	—
114	#135	—	89	2	no	—
115	C₂×Q₁₆	—	89	1	no	—
116	C₂×SD₁₆	—	89	2	no	—
117	C₂×D₈	—	89	1	no	—
118	#131	—	89	8	no	—
119	#131	—	178	4	no	—
120	C₂×M₁₆	—	89	2	no	—
121	C₂×M₁₆	—	178	4	no	—
122	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
123	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
124	#117	C₂×D₈₉	1	1	yes	characteristic
125	#108	—	89	2	no	—
126	#108	—	89	2	no	—
127	#104	—	89	8	no	—
128	#97	C₂×D₈₉	1	2	yes	complemented
129	C₄×C₈	C₂×D₈₉	1	2	yes	complemented
130	#41	—	89	4	no	—
131	#41	—	89	4	no	—
132	#41	—	178	2	no	—
133	#41	—	178	2	no	—
134	C₂×Q₈	—	89	2	no	—
135	C₂×D₄	—	89	2	no	—
136	C₂×D₄	—	178	2	no	—
137	C₂²×C₄	—	178	2	no	—
138	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
139	Q₁₆	—	89	2	no	—
140	Q₁₆	—	89	2	no	—
141	SD₁₆	—	89	4	no	—
142	SD₁₆	—	89	4	no	—
143	D₈	—	89	2	no	—
144	D₈	—	89	2	no	—
145	M₁₆	—	89	4	no	—
146	M₁₆	—	178	2	no	—
147	M₁₆	—	178	8	no	—
148	M₁₆	D₃₅₆	1	2	yes	—
149	M₁₆	D₃₅₆	1	2	yes	—
150	C₂×C₈	—	178	2	no	—
151	C₂×C₈	—	178	8	no	—
152	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
153	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
154	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
155	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
156	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₈₉	1	2	yes	—
159	C₄⋊C₄	—	2	1	no	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₄²	C₂²×D₈₉	1	2	yes	—
162	C₂³	—	178	2	no	—
163	C₂³	#13083574	1	1	yes	characteristic
164	Q₈	—	89	4	no	—
165	Q₈	—	178	2	no	—
166	D₄	—	89	4	no	—
167	D₄	—	178	4	no	—
168	D₄	—	178	4	no	—
169	D₄	—	178	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	178	2	no	—
172	C₂×C₄	—	178	4	no	—
173	C₂×C₄	—	178	4	no	—
174	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
176	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
177	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
178	C₂×C₄	#13083574	1	1	yes	characteristic
179	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
180	C₈	—	178	4	no	—
181	C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
182	C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
183	C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
184	C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
185	C₈	C₂×D₃₅₆	1	4	yes	—
186	C₂²	—	4	1	no	—
187	C₂²	—	178	4	no	—
188	C₂²	—	712	2	no	—
189	C₂²	C₄×D₃₅₆	1	1	yes	characteristic
190	C₂²	C₄×D₃₅₆	1	1	yes	characteristic
191	C₂²	C₂×#13083574	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	178	4	no	—
194	C₄	C₄×D₃₅₆	1	1	yes	characteristic
195	C₄	C₄×D₃₅₆	1	1	yes	characteristic
196	C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
197	C₄	C₂²×D₃₅₆	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	356	4	no	—
202	C₂	C₂×C₄×D₃₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₃₅₆
commutator	C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#405×C₈₉	C₂
socle	C₂×C₈₉	C₂×C₄×D₄
prime radical	C₂×#15668034	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#405	D₈₉
hypocenter	C₈₉	#2957
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #19152364	C₂	semidirect
3. #19151721	C₂	semidirect
4. #19151716	C₂	semidirect
5. C₂×#15667948	C₂	semidirect
6. #19151616	C₂	semidirect
7. #405×C₈₉	C₂	semidirect
9. #15668034	C₄	semidirect
10. #15668034	C₄	semidirect
13. C₂×Dic₃₅₆	C₄	semidirect
14. C₂×#13083563	C₄	semidirect
15. C₂×D₇₁₂	C₄	semidirect
17. #15668018	C₂²	semidirect
20. #15667948	C₂²	semidirect
21. #15667948	C₂²	semidirect
22. #15667940	C₂²	semidirect
26. #97×C₈₉	C₂²	semidirect
27. C₄×C₈×C₈₉	C₂²	semidirect
102. C₈₉	#2957	semidirect
109. #405	D₈₉	semidirect
128. #97	C₂×D₈₉	semidirect
129. C₄×C₈	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668034	2
#19152364	2
#19151721	2
#19151716	2
C₂×#15667948	2
#19151616	2
#405×C₈₉	2
#2957	89