Subgroups — #26028909

Order 11392.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#19151658	C₂	1	2	yes	maximal, complemented
3	#19151658	C₂	1	2	yes	maximal, complemented
4	#19151654	C₂	1	2	yes	maximal, complemented
5	#19151650	C₂	1	2	yes	maximal, complemented
6	#19151625	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151561	C₂	1	1	yes	maximal, characteristic
8	C₂×#15667924	C₂	1	1	yes	maximal, characteristic
9	#19151556	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151553	C₂	1	1	yes	maximal, characteristic, complemented
11	#424×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
13	C₂×#13083558	C₂²	1	1	yes	characteristic
14	C₂²×#10623494	C₂²	1	1	yes	characteristic
15	#15667971	C₂²	1	1	yes	characteristic
16	#15667970	C₄	1	1	yes	characteristic
17	#15667970	C₄	1	1	yes	characteristic
18	#15667969	C₂²	1	2	yes	—
19	#15667968	C₄	1	2	yes	—
20	C₄⋊C₄×D₈₉	C₄	1	1	yes	characteristic
21	C₄⋊C₄×D₈₉	C₄	1	1	yes	characteristic
22	C₂²⋊C₄×D₈₉	C₄	1	2	yes	—
23	C₄²×D₈₉	C₂²	1	2	yes	—
24	C₄⋊#10623494	C₂²	1	2	yes	complemented
25	C₄⋊#10623494	C₂²	1	2	yes	complemented
26	#15667924	C₂²	1	1	yes	characteristic
27	#15667924	C₂²	1	2	yes	—
28	#15667924	C₂²	1	1	yes	characteristic
29	#15667924	C₂²	1	1	yes	characteristic
30	#15667924	C₂²	1	1	yes	characteristic
31	#15667924	C₂²	1	2	yes	—
32	#15667922	C₂²	1	2	yes	complemented
33	#15667922	C₂²	1	2	yes	complemented
34	C₈×Dic₈₉	C₂²	1	2	yes	complemented
35	C₈×Dic₈₉	C₂²	1	2	yes	complemented
36	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
38	#117×C₈₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₈₉	C₂²	1	2	yes	complemented
40	C₄⋊C₈×C₈₉	C₂²	1	2	yes	complemented
41	C₂³×D₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₈₉	—	2	2	no	—
44	C₂×C₄×D₈₉	—	2	2	no	—
45	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₈₉	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₈₉	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
51	#13083558	—	2	2	no	—
52	C₂×#10623494	—	2	2	no	—
53	C₂×#10623494	C₂³	1	1	yes	characteristic
54	C₂×#10623494	C₂³	1	1	yes	characteristic
55	C₂×#10623494	C₂³	1	2	yes	—
56	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
58	#13083553	C₂×C₄	1	2	yes	—
59	#13083553	C₂×C₄	1	2	yes	—
60	C₂²⋊Dic₈₉	C₂×C₄	1	2	yes	—
61	#13083551	C₂×C₄	1	2	yes	—
62	#13083551	C₂×C₄	1	2	yes	—
63	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
64	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
65	C₄×Dic₈₉	C₂³	1	2	yes	—
66	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
67	M₁₆×C₈₉	—	2	2	no	—
68	C₂×C₈×C₈₉	—	2	2	no	—
69	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₈₉	C₂³	1	2	yes	—
72	C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
75	C₄²×C₈₉	C₂³	1	2	yes	—
76	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₈₉	—	2	2	no	—
78	C₂²×D₈₉	—	2	2	no	—
79	C₂²×D₈₉	—	4	1	no	—
80	C₂²×D₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₂²×D₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	#41	1	2	yes	—
86	C₂×C₄×C₈₉	#41	1	2	yes	—
87	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₈₉	—	2	2	no	—
89	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
91	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
92	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₈₉	—	2	4	no	—
94	C₄×D₈₉	—	2	2	no	—
95	C₄×D₈₉	—	2	2	no	—
96	C₄×D₈₉	—	2	1	no	—
97	C₄×D₈₉	—	2	1	no	—
98	C₄×D₈₉	—	2	1	no	—
99	C₄×D₈₉	—	2	1	no	—
100	C₄×D₈₉	—	4	2	no	—
101	C₈×C₈₉	—	2	1	no	—
102	C₈×C₈₉	—	2	1	no	—
103	C₈×C₈₉	—	2	2	no	—
104	#10623494	—	2	1	no	—
105	#10623494	—	2	1	no	—
106	#10623494	—	2	2	no	—
107	C₂²×C₈₉	—	2	1	no	—
108	C₂²×C₈₉	—	2	1	no	—
109	C₂²×C₈₉	#117	1	2	yes	—
110	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
111	C₂×D₈₉	—	2	1	no	—
112	C₂×D₈₉	—	2	1	no	—
113	C₂×D₈₉	—	2	1	no	—
114	C₂×D₈₉	—	2	1	no	—
115	C₂×D₈₉	—	2	1	no	—
116	C₂×D₈₉	—	4	4	no	—
117	C₂×D₈₉	#131	1	2	yes	—
118	C₄×C₈₉	—	2	2	no	—
119	C₄×C₈₉	—	2	2	no	—
120	C₄×C₈₉	#117	1	2	yes	—
121	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
122	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
123	Dic₈₉	—	2	1	no	—
124	Dic₈₉	—	2	1	no	—
125	Dic₈₉	—	2	2	no	—
126	Dic₈₉	—	2	2	no	—
127	C₂×C₈₉	—	2	2	no	—
128	C₂×C₈₉	C₂×#117	1	1	yes	characteristic
129	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
130	C₂×C₈₉	#568	1	1	yes	characteristic
131	D₈₉	—	2	2	no	—
132	D₈₉	—	4	1	no	—
133	#2924	—	89	1	no	maximal, Sylow
134	C₈₉	#2924	1	1	yes	characteristic, complemented, Sylow
135	C₂×#117	—	89	1	no	—
136	#433	—	89	2	no	—
137	#433	—	89	2	no	—
138	#432	—	89	2	no	—
139	#431	—	89	2	no	—
140	#424	—	89	1	no	—
141	#424	D₈₉	1	1	yes	characteristic, complemented
142	#407	—	89	1	no	—
143	C₂×C₂²⋊C₈	—	89	1	no	—
144	#405	—	89	1	no	—
145	#405	—	89	1	no	—
146	C₂³×C₄	—	89	1	no	—
147	C₂×M₁₆	—	89	1	no	—
148	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
149	C₂²×C₈	—	89	1	no	—
150	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
151	#117	—	89	1	no	—
152	#117	—	89	1	no	—
153	#117	—	89	1	no	—
154	#117	—	89	2	no	—
155	#117	—	89	2	no	—
156	#117	C₂×D₈₉	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	89	1	no	—
158	C₂×C₄⋊C₄	—	89	1	no	—
159	C₂×C₂²⋊C₄	—	89	2	no	—
160	C₂×C₄²	—	89	2	no	—
161	C₄⋊C₈	—	89	2	no	—
162	C₄⋊C₈	—	89	2	no	—
163	C₄⋊C₈	C₂×D₈₉	1	2	yes	complemented
164	C₄⋊C₈	C₂×D₈₉	1	2	yes	complemented
165	C₂²⋊C₈	—	89	1	no	—
166	C₂²⋊C₈	—	89	1	no	—
167	C₂²⋊C₈	—	89	1	no	—
168	C₂²⋊C₈	—	89	1	no	—
169	C₂²⋊C₈	—	89	2	no	—
170	C₂²⋊C₈	—	89	2	no	—
171	#97	—	89	2	no	—
172	#97	—	89	2	no	—
173	C₄×C₈	—	89	2	no	—
174	C₄×C₈	—	89	2	no	—
175	C₂⁴	—	89	1	no	—
176	C₂²×C₄	—	89	1	no	—
177	C₂²×C₄	—	89	2	no	—
178	C₂²×C₄	—	89	2	no	—
179	C₂²×C₄	—	89	1	no	—
180	C₂²×C₄	—	89	1	no	—
181	C₂²×C₄	—	89	1	no	—
182	C₂²×C₄	—	89	1	no	—
183	C₂²×C₄	—	178	2	no	—
184	C₂²×C₄	—	178	2	no	—
185	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	178	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	89	1	no	—
190	C₂×C₈	—	89	2	no	—
191	C₂×C₈	—	89	1	no	—
192	C₂×C₈	—	178	2	no	—
193	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
194	C₂×C₈	C₂²×D₈₉	1	2	yes	—
195	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
196	C₄⋊C₄	—	89	2	no	—
197	C₄⋊C₄	—	89	2	no	—
198	C₄⋊C₄	—	89	1	no	—
199	C₄⋊C₄	—	89	1	no	—
200	C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
201	C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
202	C₂²⋊C₄	—	89	2	no	—
203	C₂²⋊C₄	—	89	2	no	—
204	C₂²⋊C₄	—	89	2	no	—
205	C₂²⋊C₄	C₄×D₈₉	1	2	yes	—
206	C₄²	—	89	2	no	—
207	C₄²	—	89	2	no	—
208	C₄²	—	89	2	no	—
209	C₄²	C₂²×D₈₉	1	2	yes	—
210	C₂³	—	89	1	no	—
211	C₂³	—	89	1	no	—
212	C₂³	—	356	1	no	—
213	C₂³	—	178	2	no	—
214	C₂³	—	178	2	no	—
215	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
216	C₂×C₄	—	89	1	no	—
217	C₂×C₄	—	89	2	no	—
218	C₂×C₄	—	89	2	no	—
219	C₂×C₄	—	89	1	no	—
220	C₂×C₄	—	356	2	no	—
221	C₂×C₄	—	178	4	no	—
222	C₂×C₄	—	178	2	no	—
223	C₂×C₄	—	178	2	no	—
224	C₂×C₄	—	178	2	no	—
225	C₂×C₄	—	178	1	no	—
226	C₂×C₄	—	178	1	no	—
227	C₂×C₄	—	178	1	no	—
228	C₂×C₄	—	178	1	no	—
229	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
230	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
231	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
232	C₂×C₄	#13083573	1	2	yes	—
233	C₂×C₄	#13083575	1	2	yes	—
234	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	178	1	no	—
239	C₈	—	178	1	no	—
240	C₈	—	178	2	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	89	2	no	—
244	C₂²	—	356	4	no	—
245	C₂²	—	178	1	no	—
246	C₂²	—	178	1	no	—
247	C₂²	—	178	1	no	—
248	C₂²	—	178	1	no	—
249	C₂²	—	178	1	no	—
250	C₂²	#15667970	1	2	yes	—
251	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	178	1	no	—
255	C₄	—	178	1	no	—
256	C₄	—	178	2	no	—
257	C₄	—	178	2	no	—
258	C₄	#15667970	1	2	yes	—
259	C₄	C₂×#13083573	1	1	yes	characteristic
260	C₄	C₂×#13083575	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	356	1	no	—
263	C₂	—	178	2	no	—
264	C₂	C₂×#15667970	1	1	yes	characteristic
265	C₂	#131×D₈₉	1	1	yes	characteristic
266	C₂	#19152363	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#424×C₈₉	C₂
prime radical	C₂³×D₈₉	C₂×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#424	D₈₉
hypocenter	C₈₉	#2924
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #19151658	C₂	semidirect
3. #19151658	C₂	semidirect
4. #19151654	C₂	semidirect
5. #19151650	C₂	semidirect
6. #19151625	C₂	semidirect
9. #19151556	C₂	semidirect
10. #19151553	C₂	semidirect
11. #424×C₈₉	C₂	semidirect
24. C₄⋊#10623494	C₂²	semidirect
25. C₄⋊#10623494	C₂²	semidirect
32. #15667922	C₂²	semidirect
33. #15667922	C₂²	semidirect
34. C₈×Dic₈₉	C₂²	semidirect
35. C₈×Dic₈₉	C₂²	semidirect
39. C₄⋊C₈×C₈₉	C₂²	semidirect
40. C₄⋊C₈×C₈₉	C₂²	semidirect
134. C₈₉	#2924	semidirect
141. #424	D₈₉	semidirect
163. C₄⋊C₈	C₂×D₈₉	semidirect
164. 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
#117×D₈₉	2
#19151658	2
#19151654	2
#19151650	2
#19151625	2
#19151561	2
C₂×#15667924	2
#19151556	2
#19151553	2
#424×C₈₉	2
#2924	89