Subgroups — #26028995

Order 11392.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 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	2	yes	maximal, complemented
2	#19152148	C₂	1	1	yes	maximal, characteristic
3	#19151675	C₂	1	2	yes	maximal, complemented
4	#19151668	C₂	1	4	yes	maximal, complemented
5	#19151663	C₂	1	2	yes	maximal, complemented
6	#19151564	C₂	1	2	yes	maximal, complemented
7	C₄×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13083574	C₂²	1	2	yes	—
10	#15668018	—	2	8	no	—
11	C₂×#13083558	C₂²	1	1	yes	characteristic
12	C₂×#13083558	C₂²	1	1	yes	characteristic
13	C₂×#13083557	C₂²	1	4	yes	—
14	C₂×C₈×D₈₉	C₂²	1	4	yes	—
15	#15668012	C₄	1	1	yes	characteristic
16	#15668009	C₄	1	1	yes	characteristic
17	#15668000	C₄	1	2	yes	—
18	#15667986	C₄	1	2	yes	—
19	#15667986	C₄	1	2	yes	—
20	C₄×D₃₅₆	C₂²	1	2	yes	—
21	C₄×D₃₅₆	C₂²	1	2	yes	—
22	#15667924	C₂²	1	4	yes	—
23	#15667924	C₂²	1	4	yes	—
24	#15667923	C₂²	1	2	yes	—
25	C₄×#10623494	C₂²	1	2	yes	—
26	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
30	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#13083574	—	2	4	no	—
32	#13083574	—	2	4	no	—
33	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
34	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
35	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
36	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊D₈₉	C₂×C₄	1	4	yes	—
38	C₂×C₄×D₈₉	C₂³	1	4	yes	—
39	#13083558	—	2	2	no	—
40	#13083558	D₄	1	4	yes	complemented
41	#13083557	—	2	8	no	—
42	C₂×#10623494	C₂³	1	2	yes	—
43	C₂×#10623494	C₂³	1	2	yes	—
44	C₈×D₈₉	—	2	8	no	—
45	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
47	#13083551	C₂×C₄	1	4	yes	—
48	#13083551	C₂×C₄	1	4	yes	—
49	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₈₉	—	2	4	no	—
53	C₂×C₈×C₈₉	C₂³	1	2	yes	—
54	C₂×C₈×C₈₉	C₂³	1	2	yes	—
55	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₈₉	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	4	1	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	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₈₉	#41	1	2	yes	—
70	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₈₉	—	2	8	no	—
72	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
73	D₃₅₆	—	2	4	no	—
74	D₃₅₆	—	2	4	no	—
75	D₃₅₆	—	4	2	no	—
76	C₄×D₈₉	—	2	4	no	—
77	C₄×D₈₉	—	2	4	no	—
78	Dic₁₇₈	—	2	4	no	—
79	C₈×C₈₉	—	2	2	no	—
80	C₈×C₈₉	—	2	2	no	—
81	#10623494	—	2	2	no	—
82	#10623494	C₂×D₄	1	4	yes	—
83	C₂²×C₈₉	—	2	1	no	—
84	C₂²×C₈₉	—	2	1	no	—
85	C₂²×C₈₉	C₄×D₄	1	2	yes	—
86	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₈₉	—	2	4	no	—
88	C₂×D₈₉	—	2	4	no	—
89	C₂×D₈₉	—	2	4	no	—
90	C₄×C₈₉	—	2	1	no	—
91	C₄×C₈₉	—	2	1	no	—
92	C₄×C₈₉	—	2	1	no	—
93	C₄×C₈₉	C₄×D₄	1	2	yes	—
94	C₄×C₈₉	#131	1	2	yes	complemented
95	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
97	Dic₈₉	—	2	4	no	—
98	C₂×C₈₉	—	2	2	no	—
99	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
101	C₂×C₈₉	#568	1	1	yes	characteristic
102	D₈₉	—	4	4	no	—
103	#2935	—	89	1	no	maximal, Sylow
104	C₈₉	#2935	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	—	89	2	no	—
106	#532	—	89	1	no	—
107	#436	—	89	2	no	—
108	#435	—	89	4	no	—
109	C₈×D₄	—	89	2	no	—
110	C₂×C₄⋊C₈	D₈₉	1	1	yes	characteristic, complemented
111	#408	—	89	2	no	—
112	C₄×M₁₆	—	89	1	no	—
113	C₂×#41	—	89	2	no	—
114	#131	—	178	8	no	—
115	C₂×M₁₆	—	89	1	no	—
116	C₂×M₁₆	—	89	1	no	—
117	C₂×M₁₆	—	89	4	no	—
118	C₂²×C₈	—	89	4	no	—
119	C₂²×C₈	C₂×D₈₉	1	2	yes	—
120	C₄⋊Q₈	—	89	1	no	—
121	#127	—	89	1	no	—
122	#124	—	89	2	no	—
123	#121	—	89	2	no	—
124	#121	—	89	2	no	—
125	C₄×D₄	—	89	2	no	—
126	C₄×D₄	—	89	2	no	—
127	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
128	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
129	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
130	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
131	C₂²⋊C₈	—	89	4	no	—
132	C₂²⋊C₈	—	89	4	no	—
133	#97	—	89	2	no	—
134	C₄×C₈	—	89	2	no	—
135	#41	—	178	4	no	—
136	#41	—	178	4	no	—
137	C₂×Q₈	—	89	2	no	—
138	C₂×D₄	—	89	4	no	—
139	C₂×D₄	—	89	2	no	—
140	C₂×D₄	—	89	2	no	—
141	C₂×D₄	—	89	2	no	—
142	C₂²×C₄	—	89	4	no	—
143	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
146	M₁₆	—	89	4	no	—
147	M₁₆	—	178	8	no	—
148	M₁₆	—	178	2	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	89	2	no	—
151	C₂×C₈	—	89	2	no	—
152	C₂×C₈	—	178	8	no	—
153	C₂×C₈	C₂²×D₈₉	1	2	yes	—
154	C₂×C₈	C₂²×D₈₉	1	2	yes	—
155	C₄⋊C₄	—	89	2	no	—
156	C₄⋊C₄	—	89	2	no	—
157	C₂²⋊C₄	—	89	4	no	—
158	C₂²⋊C₄	—	89	4	no	—
159	C₄²	C₄×D₈₉	1	1	yes	characteristic
160	C₄²	C₄×D₈₉	1	1	yes	characteristic
161	C₄²	C₂²×D₈₉	1	1	yes	characteristic
162	C₄²	C₂²×D₈₉	1	1	yes	characteristic
163	C₂³	—	89	4	no	—
164	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
165	Q₈	—	178	4	no	—
166	D₄	—	356	2	no	—
167	D₄	—	178	8	no	—
168	D₄	—	178	4	no	—
169	D₄	—	178	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	89	4	no	—
173	C₂×C₄	—	178	4	no	—
174	C₂×C₄	—	178	4	no	—
175	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
180	C₂×C₄	D₄×D₈₉	1	2	yes	—
181	C₂×C₄	#13083575	1	2	yes	—
182	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	2	2	no	—
185	C₈	—	89	4	no	—
186	C₈	—	178	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	178	4	no	—
190	C₂²	—	178	4	no	—
191	C₂²	—	178	4	no	—
192	C₂²	#15667974	1	2	yes	—
193	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	178	4	no	—
198	C₄	#15667974	1	2	yes	—
199	C₄	#15668018	1	2	yes	complemented
200	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
201	C₄	C₂×#13083575	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	356	4	no	—
204	C₂	C₂×#15667974	1	1	yes	characteristic
205	C₂	C₂×#15668018	1	1	yes	characteristic
206	C₂	#19152364	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	C₂×C₄⋊C₈×C₈₉	C₂
prime radical	#19152148	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#2935
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15668018	C₂	semidirect
3. #19151675	C₂	semidirect
4. #19151668	C₂	semidirect
5. #19151663	C₂	semidirect
6. #19151564	C₂	semidirect
7. C₄×#13083558	C₂	semidirect
8. C₂×C₄⋊C₈×C₈₉	C₂	semidirect
40. #13083558	D₄	semidirect
94. C₄×C₈₉	#131	semidirect
104. C₈₉	#2935	semidirect
110. C₂×C₄⋊C₈	D₈₉	semidirect
199. C₄	#15668018	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
#19152148	2
#19151675	2
#19151668	2
#19151663	2
#19151564	2
C₄×#13083558	2
C₂×C₄⋊C₈×C₈₉	2
#2935	89