Subgroups — #26018481

Order 11392.

656 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667919	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₈₉	C₂²	1	2	yes	—
6	C₂³×Dic₈₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₉	—	2	2	no	—
8	C₂×C₄⋊Dic₈₉	—	2	2	no	—
9	C₂×C₄⋊Dic₈₉	C₄	1	4	yes	complemented
10	C₂×C₄×Dic₈₉	—	2	2	no	—
11	C₂×C₄×Dic₈₉	—	2	2	no	—
12	C₂×C₄×Dic₈₉	C₄	1	4	yes	complemented
13	#15667919	—	2	8	no	—
14	C₂³×C₄×C₈₉	C₂²	1	2	yes	—
15	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
17	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
18	C₂×C₄⋊C₄×C₈₉	C₄	1	4	yes	complemented
19	C₂²×Dic₈₉	—	2	2	no	—
20	C₂²×Dic₈₉	—	2	2	no	—
21	C₂²×Dic₈₉	—	2	2	no	—
22	C₂²×Dic₈₉	—	2	2	no	—
23	C₂²×Dic₈₉	—	2	4	no	—
24	C₂²×Dic₈₉	—	2	2	no	—
25	C₂²×Dic₈₉	—	2	2	no	—
26	C₂²×Dic₈₉	—	2	4	no	—
27	C₂²×Dic₈₉	C₂×C₄	1	4	yes	—
28	C₂²×Dic₈₉	C₂×C₄	1	4	yes	—
29	C₂²×Dic₈₉	D₄	1	4	yes	—
30	C₂²×Dic₈₉	Q₈	1	4	yes	—
31	C₄⋊Dic₈₉	—	2	8	no	—
32	C₄×Dic₈₉	—	2	8	no	—
33	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	—	2	2	no	—
35	C₂²×C₄×C₈₉	—	2	2	no	—
36	C₂²×C₄×C₈₉	—	2	4	no	—
37	C₂²×C₄×C₈₉	—	2	4	no	—
38	C₂²×C₄×C₈₉	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₈₉	D₄	1	2	yes	—
43	C₂²×C₄×C₈₉	D₄	1	2	yes	—
44	C₂²×C₄×C₈₉	D₄	1	2	yes	—
45	C₂²×C₄×C₈₉	Q₈	1	2	yes	—
46	C₄⋊C₄×C₈₉	—	2	8	no	—
47	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂³×C₈₉	C₂×D₄	1	2	yes	—
51	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂³×C₈₉	C₂×Q₈	1	2	yes	—
55	C₂³×C₈₉	C₂×Q₈	1	1	yes	characteristic
56	C₂³×C₈₉	#41	1	2	yes	—
57	C₂³×C₈₉	#41	1	2	yes	—
58	C₂×C₄×C₈₉	—	2	4	no	—
59	C₂×C₄×C₈₉	—	2	8	no	—
60	C₂×C₄×C₈₉	—	4	4	no	—
61	C₂×C₄×C₈₉	—	4	2	no	—
62	C₂×C₄×C₈₉	—	4	2	no	—
63	C₂×C₄×C₈₉	C₄²	1	4	yes	—
64	C₂×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
65	C₂×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
66	C₂×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
67	C₂×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
68	C₂×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
69	C₂×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
70	C₂×Dic₈₉	—	2	4	no	—
71	C₂×Dic₈₉	—	2	4	no	—
72	C₂×Dic₈₉	—	2	4	no	—
73	C₂×Dic₈₉	—	2	4	no	—
74	C₂×Dic₈₉	—	2	4	no	—
75	C₂×Dic₈₉	—	2	4	no	—
76	C₂×Dic₈₉	—	2	4	no	—
77	C₂×Dic₈₉	—	2	8	no	—
78	C₂×Dic₈₉	—	4	4	no	—
79	C₂×Dic₈₉	—	4	2	no	—
80	C₂×Dic₈₉	—	4	2	no	—
81	C₂×Dic₈₉	C₄⋊C₄	1	8	yes	complemented
82	C₂²×C₈₉	C₂×C₄²	1	1	yes	characteristic
83	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
90	C₂²×C₈₉	#117	1	2	yes	—
91	C₂²×C₈₉	C₄×D₄	1	2	yes	—
92	C₂²×C₈₉	C₄×D₄	1	2	yes	—
93	C₂²×C₈₉	C₄×Q₈	1	2	yes	—
94	C₂²×C₈₉	C₄×Q₈	1	2	yes	—
95	C₂²×C₈₉	#121	1	2	yes	—
96	C₂²×C₈₉	#121	1	2	yes	—
97	C₂²×C₈₉	C₂²⋊Q₈	1	2	yes	—
98	C₂²×C₈₉	C₂²⋊Q₈	1	2	yes	—
99	C₂²×C₈₉	#124	1	1	yes	characteristic
100	C₂²×C₈₉	#124	1	1	yes	characteristic
101	C₂²×C₈₉	#125	1	2	yes	—
102	C₂²×C₈₉	#127	1	1	yes	characteristic
103	C₂²×C₈₉	C₄⋊Q₈	1	2	yes	—
104	C₂²×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
105	C₄×C₈₉	—	4	4	no	—
106	C₄×C₈₉	#95	1	8	yes	complemented
107	Dic₈₉	—	2	8	no	—
108	Dic₈₉	—	4	4	no	—
109	C₂×C₈₉	C₂×#95	1	1	yes	characteristic
110	C₂×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
111	C₂×C₈₉	#380	1	2	yes	—
112	C₂×C₈₉	#382	1	1	yes	characteristic
113	C₂×C₈₉	#384	1	1	yes	characteristic
114	C₂×C₈₉	#389	1	2	yes	—
115	C₂×C₈₉	#389	1	2	yes	—
116	C₂×C₈₉	#390	1	1	yes	characteristic
117	C₂×C₈₉	#390	1	1	yes	characteristic
118	C₂×C₈₉	#391	1	1	yes	characteristic
119	C₂×C₈₉	#391	1	1	yes	characteristic
120	C₄⋊#95	—	89	1	no	maximal, Sylow
121	C₈₉	C₄⋊#95	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₄⋊C₄	—	89	1	no	—
123	C₂²×C₄⋊C₄	D₈₉	1	1	yes	characteristic
124	C₂²×C₄²	—	89	1	no	—
125	C₂×#95	—	89	4	no	—
126	C₂³×C₄	—	89	2	no	—
127	C₂³×C₄	—	89	2	no	—
128	C₂³×C₄	C₂×D₈₉	1	2	yes	—
129	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	89	4	no	—
133	C₂×C₄⋊C₄	—	178	2	no	—
134	C₂×C₄⋊C₄	—	178	2	no	—
135	C₂×C₄⋊C₄	Dic₈₉	1	4	yes	complemented
136	C₂×C₄²	—	89	4	no	—
137	C₂×C₄²	—	178	2	no	—
138	C₂×C₄²	—	178	2	no	—
139	#95	—	178	8	no	—
140	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	89	4	no	—
146	C₂²×C₄	—	89	4	no	—
147	C₂²×C₄	—	89	4	no	—
148	C₂²×C₄	—	89	4	no	—
149	C₂²×C₄	—	178	4	no	—
150	C₂²×C₄	—	178	2	no	—
151	C₂²×C₄	—	178	2	no	—
152	C₂²×C₄	—	178	4	no	—
153	C₂²×C₄	—	178	2	no	—
154	C₂²×C₄	—	178	2	no	—
155	C₂²×C₄	—	178	2	no	—
156	C₂²×C₄	—	178	2	no	—
157	C₂²×C₄	Dic₁₇₈	1	2	yes	—
158	C₂²×C₄	C₄×D₈₉	1	2	yes	—
159	C₂²×C₄	C₄×D₈₉	1	2	yes	—
160	C₂²×C₄	D₃₅₆	1	2	yes	—
161	C₂²×C₄	C₂×Dic₈₉	1	4	yes	—
162	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
163	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
164	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	178	8	no	—
167	C₄²	—	178	8	no	—
168	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
169	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
170	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
171	C₂³	D₄×D₈₉	1	2	yes	—
172	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
173	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
174	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
175	C₂³	Q₈×D₈₉	1	2	yes	—
176	C₂³	C₂×Dic₁₇₈	1	1	yes	characteristic
177	C₂³	#13083573	1	2	yes	—
178	C₂³	#13083575	1	2	yes	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	89	8	no	—
185	C₂×C₄	—	356	4	no	—
186	C₂×C₄	—	356	2	no	—
187	C₂×C₄	—	356	2	no	—
188	C₂×C₄	—	178	8	no	—
189	C₂×C₄	—	178	4	no	—
190	C₂×C₄	—	178	4	no	—
191	C₂×C₄	—	178	4	no	—
192	C₂×C₄	—	178	4	no	—
193	C₂×C₄	—	178	4	no	—
194	C₂×C₄	—	178	4	no	—
195	C₂×C₄	—	178	4	no	—
196	C₂×C₄	C₄×Dic₈₉	1	4	yes	—
197	C₂×C₄	#13083551	1	4	yes	—
198	C₂×C₄	#13083551	1	4	yes	—
199	C₂×C₄	C₂²⋊Dic₈₉	1	4	yes	—
200	C₂×C₄	#13083553	1	4	yes	—
201	C₂×C₄	#13083553	1	4	yes	—
202	C₂×C₄	C₄⋊Dic₈₉	1	4	yes	—
203	C₂²	C₂×C₄×Dic₈₉	1	1	yes	characteristic
204	C₂²	C₂×#13083551	1	1	yes	characteristic
205	C₂²	C₂×#13083551	1	1	yes	characteristic
206	C₂²	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
207	C₂²	C₄⋊C₄×D₈₉	1	2	yes	—
208	C₂²	C₂×#13083553	1	1	yes	characteristic
209	C₂²	C₂×#13083553	1	1	yes	characteristic
210	C₂²	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
211	C₂²	#15667970	1	2	yes	—
212	C₂²	#15667974	1	2	yes	—
213	C₂²	D₄×Dic₈₉	1	2	yes	—
214	C₂²	#15667977	1	2	yes	—
215	C₂²	Q₈×Dic₈₉	1	2	yes	—
216	C₂²	#15667983	1	2	yes	—
217	C₂²	#15667987	1	2	yes	—
218	C₂²	#15667988	1	2	yes	—
219	C₂²	#15667992	1	2	yes	—
220	C₂²	#15667999	1	1	yes	characteristic
221	C₂²	#15668001	1	1	yes	characteristic
222	C₂²	#15668003	1	2	yes	—
223	C₂²	#15668008	1	1	yes	characteristic
224	C₂²	#15668010	1	1	yes	characteristic
225	C₂²	#15668011	1	2	yes	—
226	C₄	—	4	4	no	—
227	C₄	—	356	4	no	—
228	C₄	—	178	8	no	—
229	C₄	#15667919	1	8	yes	complemented
230	C₂	C₂×#15667919	1	1	yes	characteristic
231	C₂	C₄⋊C₄×Dic₈₉	1	2	yes	—
232	C₂	#19151455	1	2	yes	—
233	C₂	#19151463	1	1	yes	characteristic
234	C₂	#19151467	1	1	yes	characteristic
235	C₂	#19151494	1	2	yes	—
236	C₂	#19151497	1	2	yes	—
237	C₂	#19151499	1	1	yes	characteristic
238	C₂	#19151501	1	1	yes	characteristic
239	C₂	#19151503	1	1	yes	characteristic
240	C₂	#19151505	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂×C₄²
Fitting	C₂²×C₄⋊C₄×C₈₉	C₂
socle, prime radical, cosocle	C₂⁴×C₈₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₈₉
hypocenter	C₈₉	C₄⋊#95
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊Dic₈₉	C₄	semidirect
12. C₂×C₄×Dic₈₉	C₄	semidirect
18. C₂×C₄⋊C₄×C₈₉	C₄	semidirect
81. C₂×Dic₈₉	C₄⋊C₄	semidirect
106. C₄×C₈₉	#95	semidirect
121. C₈₉	C₄⋊#95	semidirect
135. C₂×C₄⋊C₄	Dic₈₉	semidirect
229. C₄	#15667919	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₈₉	2
C₂²×C₄×Dic₈₉	2
C₂×#15667919	2
C₂²×C₄⋊C₄×C₈₉	2
C₄⋊#95	89