Subgroups — #26028997

Order 11392.

492 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668020	C₂	1	2	yes	maximal, complemented
2	#19152145	C₂	1	1	yes	maximal, characteristic
3	#19151675	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151667	C₂	1	2	yes	maximal, complemented
5	#19151666	C₂	1	2	yes	maximal, complemented
6	C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151563	C₂	1	2	yes	maximal, complemented
10	C₄×#13083557	C₂	1	1	yes	maximal, characteristic, complemented
11	#436×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13083574	C₂²	1	2	yes	—
13	#15668020	—	2	8	no	—
14	C₂×#13083557	C₂²	1	2	yes	—
15	C₂×#13083557	C₂²	1	1	yes	characteristic
16	C₂×#13083557	C₂²	1	1	yes	characteristic
17	C₂²×#10623494	C₂²	1	2	yes	—
18	C₂×C₈×D₈₉	C₂²	1	2	yes	—
19	C₂×C₈×D₈₉	C₂²	1	2	yes	—
20	#15668011	C₄	1	1	yes	characteristic
21	#15668008	C₄	1	1	yes	characteristic
22	#15667998	C₄	1	2	yes	—
23	#15667987	C₄	1	1	yes	characteristic
24	#15667985	C₄	1	2	yes	—
25	#15667983	C₄	1	1	yes	characteristic
26	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₈₉	C₂²	1	2	yes	—
28	C₄²×D₈₉	C₂²	1	1	yes	characteristic
29	C₄⋊#10623494	C₂²	1	1	yes	characteristic
30	#15667941	C₂²	1	2	yes	—
31	C₂²⋊#10623494	C₂²	1	2	yes	—
32	#15667924	C₂²	1	2	yes	—
33	#15667924	C₂²	1	2	yes	—
34	#15667922	C₂²	1	1	yes	characteristic
35	#15667922	C₂²	1	1	yes	characteristic
36	C₄×#10623494	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
38	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
41	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
42	#13083574	—	2	4	no	—
43	#13083574	—	2	4	no	—
44	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
45	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
46	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₈₉	C₂³	1	2	yes	—
53	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
54	#13083557	—	2	2	no	—
55	#13083557	—	2	4	no	—
56	#13083557	D₄	1	4	yes	complemented
57	C₂×#10623494	—	2	4	no	—
58	C₂×#10623494	C₂³	1	2	yes	—
59	C₂×#10623494	C₂³	1	1	yes	characteristic
60	C₂×#10623494	C₂³	1	1	yes	characteristic
61	C₈×D₈₉	—	2	4	no	—
62	C₈×D₈₉	—	2	4	no	—
63	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
64	#13083553	C₂×C₄	1	2	yes	—
65	C₂²⋊Dic₈₉	C₂×C₄	1	2	yes	—
66	#13083551	C₂×C₄	1	2	yes	—
67	#13083551	C₂×C₄	1	2	yes	—
68	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
73	M₁₆×C₈₉	—	2	4	no	—
74	C₂×C₈×C₈₉	C₂³	1	2	yes	—
75	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
79	C₄²×C₈₉	C₂³	1	1	yes	characteristic
80	C₂³×C₈₉	C₂²×C₄	1	2	yes	—
81	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
82	C₂²×D₈₉	C₂²×C₄	1	1	yes	characteristic
83	D₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	—	2	2	no	—
85	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₈₉	—	2	4	no	—
91	C₂²⋊D₈₉	—	2	4	no	—
92	C₂²⋊D₈₉	—	4	2	no	—
93	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
96	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
97	D₃₅₆	—	2	4	no	—
98	D₃₅₆	—	2	2	no	—
99	C₄×D₈₉	—	2	2	no	—
100	C₄×D₈₉	—	2	2	no	—
101	C₄×D₈₉	—	2	2	no	—
102	C₄×D₈₉	—	4	1	no	—
103	C₄×D₈₉	C₂×D₄	1	2	yes	—
104	C₄×D₈₉	#41	1	2	yes	—
105	Dic₁₇₈	—	2	4	no	—
106	C₈×C₈₉	—	2	2	no	—
107	C₈×C₈₉	—	2	1	no	—
108	C₈×C₈₉	C₂×D₄	1	2	yes	—
109	#10623494	—	2	2	no	—
110	#10623494	—	2	1	no	—
111	#10623494	C₂×D₄	1	2	yes	—
112	C₂²×C₈₉	—	2	2	no	—
113	C₂²×C₈₉	—	2	2	no	—
114	C₂²×C₈₉	—	2	2	no	—
115	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₈₉	—	2	1	no	—
117	C₂×D₈₉	—	2	1	no	—
118	C₂×D₈₉	—	2	2	no	—
119	C₂×D₈₉	—	2	2	no	—
120	C₂×D₈₉	—	2	2	no	—
121	C₂×D₈₉	C₄×D₄	1	2	yes	—
122	C₄×C₈₉	—	2	2	no	—
123	C₄×C₈₉	—	2	1	no	—
124	C₄×C₈₉	#131	1	2	yes	complemented
125	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
127	Dic₈₉	—	2	2	no	—
128	Dic₈₉	—	2	1	no	—
129	Dic₈₉	—	2	1	no	—
130	Dic₈₉	C₄×D₄	1	2	yes	—
131	C₂×C₈₉	—	4	2	no	—
132	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
134	C₂×C₈₉	#568	1	1	yes	characteristic
135	D₈₉	—	2	2	no	—
136	D₈₉	—	4	2	no	—
137	#2935	—	89	1	no	maximal, Sylow
138	C₈₉	#2935	1	1	yes	characteristic, complemented, Sylow
139	C₂×#131	—	89	2	no	—
140	#532	—	89	1	no	—
141	#436	—	89	1	no	—
142	#436	D₈₉	1	1	yes	characteristic, complemented
143	#435	—	89	2	no	—
144	#435	—	89	2	no	—
145	C₈×D₄	—	89	1	no	—
146	C₈×D₄	—	89	1	no	—
147	C₂×C₄⋊C₈	—	89	1	no	—
148	#408	—	89	2	no	—
149	C₄×M₁₆	—	89	1	no	—
150	C₂×#41	—	89	2	no	—
151	#131	—	178	8	no	—
152	C₂×M₁₆	—	89	1	no	—
153	C₂×M₁₆	—	89	2	no	—
154	C₂×M₁₆	—	89	1	no	—
155	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
156	C₂²×C₈	—	89	2	no	—
157	C₂²×C₈	—	89	2	no	—
158	C₂²×C₈	—	89	2	no	—
159	C₄⋊Q₈	—	89	1	no	—
160	#127	—	89	1	no	—
161	#124	—	89	2	no	—
162	#121	—	89	1	no	—
163	#121	—	89	1	no	—
164	#121	—	89	2	no	—
165	C₄×D₄	—	89	2	no	—
166	C₄×D₄	—	89	1	no	—
167	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
168	C₂×C₄²	—	89	1	no	—
169	C₄⋊C₈	—	89	2	no	—
170	C₄⋊C₈	—	89	1	no	—
171	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
172	C₂²⋊C₈	—	89	2	no	—
173	C₂²⋊C₈	—	89	2	no	—
174	C₂²⋊C₈	—	89	2	no	—
175	C₂²⋊C₈	C₂×D₈₉	1	2	yes	—
176	#97	—	89	1	no	—
177	#97	—	89	1	no	—
178	C₄×C₈	—	89	1	no	—
179	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
180	#41	—	178	4	no	—
181	#41	—	178	4	no	—
182	C₂×Q₈	—	89	2	no	—
183	C₂×D₄	—	89	2	no	—
184	C₂×D₄	—	89	2	no	—
185	C₂×D₄	—	89	2	no	—
186	C₂×D₄	—	89	1	no	—
187	C₂×D₄	—	89	2	no	—
188	C₂×D₄	C₄×D₈₉	1	1	yes	characteristic
189	C₂²×C₄	—	89	1	no	—
190	C₂²×C₄	—	89	1	no	—
191	C₂²×C₄	—	89	1	no	—
192	C₂²×C₄	—	89	2	no	—
193	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	89	4	no	—
196	M₁₆	—	178	4	no	—
197	M₁₆	—	178	2	no	—
198	C₂×C₈	—	89	1	no	—
199	C₂×C₈	—	89	1	no	—
200	C₂×C₈	—	89	2	no	—
201	C₂×C₈	—	178	4	no	—
202	C₂×C₈	—	178	4	no	—
203	C₂×C₈	—	178	4	no	—
204	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
205	C₂×C₈	C₂²×D₈₉	1	2	yes	—
206	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
207	C₄⋊C₄	—	89	2	no	—
208	C₄⋊C₄	—	89	1	no	—
209	C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
210	C₂²⋊C₄	—	89	2	no	—
211	C₂²⋊C₄	—	89	2	no	—
212	C₂²⋊C₄	—	89	2	no	—
213	C₂²⋊C₄	C₄×D₈₉	1	2	yes	—
214	C₄²	—	89	1	no	—
215	C₄²	—	89	1	no	—
216	C₄²	—	89	1	no	—
217	C₄²	C₂²×D₈₉	1	1	yes	characteristic
218	C₂³	—	89	1	no	—
219	C₂³	—	89	2	no	—
220	C₂³	C₂×C₄×D₈₉	1	2	yes	—
221	Q₈	—	178	4	no	—
222	D₄	—	2	2	no	—
223	D₄	—	356	2	no	—
224	D₄	—	178	4	no	—
225	D₄	—	178	4	no	—
226	D₄	—	178	4	no	—
227	D₄	—	178	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	89	1	no	—
231	C₂×C₄	—	89	2	no	—
232	C₂×C₄	—	89	2	no	—
233	C₂×C₄	—	89	1	no	—
234	C₂×C₄	—	89	1	no	—
235	C₂×C₄	—	89	2	no	—
236	C₂×C₄	—	356	1	no	—
237	C₂×C₄	—	178	2	no	—
238	C₂×C₄	—	178	2	no	—
239	C₂×C₄	—	178	2	no	—
240	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
242	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	89	2	no	—
247	C₈	—	178	2	no	—
248	C₈	—	178	1	no	—
249	C₈	D₄×D₈₉	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	89	2	no	—
254	C₂²	—	178	2	no	—
255	C₂²	—	178	2	no	—
256	C₂²	—	178	2	no	—
257	C₂²	—	178	1	no	—
258	C₂²	—	178	1	no	—
259	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	89	2	no	—
263	C₄	—	178	2	no	—
264	C₄	—	178	1	no	—
265	C₄	—	178	1	no	—
266	C₄	#15668020	1	2	yes	complemented
267	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
268	C₄	#41×D₈₉	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	356	2	no	—
271	C₂	—	178	2	no	—
272	C₂	C₄×D₄×D₈₉	1	1	yes	characteristic
273	C₂	C₂×#15668020	1	1	yes	characteristic
274	C₂	#19152363	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#436×C₈₉	C₂
prime radical	#19152145	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#436	D₈₉
hypocenter	C₈₉	#2935
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×#15668020	C₂	semidirect
3. #19151675	C₂	semidirect
4. #19151667	C₂	semidirect
5. #19151666	C₂	semidirect
6. C₈×D₃₅₆	C₂	semidirect
7. D₄×#10623494	C₂	semidirect
8. C₄⋊C₈×D₈₉	C₂	semidirect
9. #19151563	C₂	semidirect
10. C₄×#13083557	C₂	semidirect
11. #436×C₈₉	C₂	semidirect
56. #13083557	D₄	semidirect
124. C₄×C₈₉	#131	semidirect
138. C₈₉	#2935	semidirect
142. #436	D₈₉	semidirect
266. C₄	#15668020	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668020	2
#19152145	2
#19151675	2
#19151667	2
#19151666	2
C₈×D₃₅₆	2
D₄×#10623494	2
C₄⋊C₈×D₈₉	2
#19151563	2
C₄×#13083557	2
#436×C₈₉	2
#2935	89