Subgroups — #137×Dic₈₉ #26029073

Order 11392.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×Dic₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151705	C₂	1	2	yes	maximal, complemented
4	#19151699	C₂	1	2	yes	maximal, complemented
5	Q₁₆×Dic₈₉	C₂	1	2	yes	maximal, complemented
6	SD₁₆×Dic₈₉	C₂	1	2	yes	maximal, complemented
7	#19151640	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151607	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151599	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#137×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂×#13083558	C₂²	1	1	yes	characteristic
13	Q₈×Dic₈₉	—	2	1	no	—
14	Q₈×Dic₈₉	—	2	1	no	—
15	Q₈×Dic₈₉	C₂²	1	2	yes	—
16	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
17	Q₈×Dic₈₉	C₂²	1	2	yes	—
18	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
19	D₄×Dic₈₉	—	2	1	no	—
20	D₄×Dic₈₉	—	2	1	no	—
21	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
22	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
23	#15667971	—	2	1	no	—
24	#15667971	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₈₉	—	2	1	no	—
26	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₈₉	—	2	1	no	—
28	C₂×C₄×Dic₈₉	—	2	1	no	—
29	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
30	#15667946	C₂²	1	2	yes	—
31	#15667944	C₂²	1	2	yes	—
32	#15667936	C₂²	1	2	yes	—
33	#15667936	C₂²	1	2	yes	—
34	#15667936	C₂²	1	2	yes	—
35	#15667933	C₂²	1	2	yes	—
36	#15667922	C₂²	1	2	yes	—
37	C₈×Dic₈₉	C₂²	1	2	yes	—
38	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
40	#137×C₈₉	C₄	1	8	yes	complemented
41	C₂×Q₁₆×C₈₉	C₂²	1	2	yes	—
42	C₂×SD₁₆×C₈₉	C₂²	1	2	yes	—
43	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
44	C₂²×Dic₈₉	—	2	1	no	—
45	C₂²×Dic₈₉	—	2	1	no	—
46	C₂²×Dic₈₉	—	2	1	no	—
47	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
48	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
49	#13083558	—	2	2	no	—
50	C₂×#10623494	C₂³	1	2	yes	—
51	C₄⋊Dic₈₉	—	2	1	no	—
52	C₄⋊Dic₈₉	—	2	1	no	—
53	C₄⋊Dic₈₉	—	2	1	no	—
54	C₄⋊Dic₈₉	—	2	2	no	—
55	C₄⋊Dic₈₉	—	2	2	no	—
56	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
57	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₈₉	C₂³	1	2	yes	—
59	C₂²⋊Dic₈₉	—	2	1	no	—
60	C₂²⋊Dic₈₉	—	2	1	no	—
61	C₂²⋊Dic₈₉	—	2	1	no	—
62	C₄×Dic₈₉	—	2	1	no	—
63	C₄×Dic₈₉	—	2	1	no	—
64	C₄×Dic₈₉	—	2	1	no	—
65	C₄×Dic₈₉	—	2	2	no	—
66	C₄×Dic₈₉	—	2	2	no	—
67	C₄×Dic₈₉	D₄	1	1	yes	characteristic
68	C₄×Dic₈₉	D₄	1	1	yes	characteristic
69	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
70	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
71	#41×C₈₉	—	2	2	no	—
72	#41×C₈₉	C₂×C₄	1	4	yes	—
73	C₂×Q₈×C₈₉	—	2	1	no	—
74	C₂×Q₈×C₈₉	—	2	2	no	—
75	C₂×Q₈×C₈₉	C₂×C₄	1	4	yes	complemented
76	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
77	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×D₄×C₈₉	—	2	1	no	—
79	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	—	2	1	no	—
81	C₂²×C₄×C₈₉	—	2	1	no	—
82	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
83	Q₁₆×C₈₉	C₂×C₄	1	8	yes	complemented
84	SD₁₆×C₈₉	C₂×C₄	1	8	yes	complemented
85	M₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
86	C₂×C₈×C₈₉	C₂³	1	2	yes	—
87	C₂³×C₈₉	—	2	1	no	—
88	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₈₉	—	2	4	no	—
90	Q₈×C₈₉	—	2	2	no	—
91	Q₈×C₈₉	—	2	1	no	—
92	Q₈×C₈₉	C₂²×C₄	1	4	yes	—
93	Q₈×C₈₉	C₂²×C₄	1	2	yes	—
94	D₄×C₈₉	—	2	4	no	—
95	D₄×C₈₉	—	2	1	no	—
96	D₄×C₈₉	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₈₉	—	2	4	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	2	4	no	—
101	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#41	1	2	yes	—
104	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
105	C₂×Dic₈₉	—	2	2	no	—
106	C₂×Dic₈₉	—	2	1	no	—
107	C₂×Dic₈₉	—	2	1	no	—
108	C₂×Dic₈₉	—	2	2	no	—
109	C₂×Dic₈₉	—	4	1	no	—
110	C₂×Dic₈₉	—	4	2	no	—
111	C₂×Dic₈₉	—	4	2	no	—
112	C₂×Dic₈₉	—	4	2	no	—
113	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
117	C₈×C₈₉	C₂²×C₄	1	4	yes	—
118	#10623494	—	2	2	no	—
119	C₂²×C₈₉	—	2	1	no	—
120	C₂²×C₈₉	—	2	1	no	—
121	C₂²×C₈₉	—	2	2	no	—
122	C₂²×C₈₉	—	4	1	no	—
123	C₂²×C₈₉	—	4	1	no	—
124	C₂²×C₈₉	C₄×D₄	1	2	yes	—
125	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	2	4	no	—
128	C₄×C₈₉	C₄×D₄	1	2	yes	—
129	C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
131	Dic₈₉	—	2	1	no	—
132	Dic₈₉	—	2	1	no	—
133	Dic₈₉	—	2	1	no	—
134	Dic₈₉	—	4	1	no	—
135	Dic₈₉	—	4	1	no	—
136	Dic₈₉	—	4	2	no	—
137	Dic₈₉	#137	1	2	yes	direct product factor
138	C₂×C₈₉	—	2	2	no	—
139	C₂×C₈₉	—	4	2	no	—
140	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
142	C₂×C₈₉	#575	1	1	yes	characteristic
143	C₄×#137	—	89	1	no	maximal, Sylow
144	C₈₉	C₄×#137	1	1	yes	characteristic, complemented, Sylow
145	C₂×#137	D₈₉	1	1	yes	characteristic
146	C₄×#41	—	89	1	no	—
147	C₂×C₄×Q₈	—	89	1	no	—
148	#441	—	89	2	no	—
149	#440	—	89	2	no	—
150	C₄×Q₁₆	—	89	2	no	—
151	C₄×SD₁₆	—	89	2	no	—
152	#428	—	89	1	no	—
153	#419	—	89	1	no	—
154	#417	—	89	1	no	—
155	C₄×M₁₆	—	89	1	no	—
156	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
157	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
158	#137	Dic₈₉	1	8	yes	direct product factor
159	C₂×Q₁₆	C₂×D₈₉	1	2	yes	—
160	C₂×SD₁₆	C₂×D₈₉	1	2	yes	—
161	C₂×M₁₆	—	89	1	no	—
162	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
163	C₄×Q₈	—	89	2	no	—
164	C₄×Q₈	—	89	2	no	—
165	C₄×Q₈	—	89	1	no	—
166	C₄×Q₈	—	89	1	no	—
167	C₄×Q₈	—	178	1	no	—
168	C₄×Q₈	—	178	1	no	—
169	C₄×D₄	—	89	1	no	—
170	C₄×D₄	—	89	1	no	—
171	C₄×D₄	—	178	1	no	—
172	C₄×D₄	—	178	1	no	—
173	#117	—	89	1	no	—
174	#117	—	178	1	no	—
175	C₂×C₄⋊C₄	—	89	1	no	—
176	C₂×C₄⋊C₄	—	178	1	no	—
177	C₂×C₄²	—	89	1	no	—
178	C₂×C₄²	—	178	1	no	—
179	C₂×C₄²	—	178	1	no	—
180	#107	—	89	2	no	—
181	#106	—	89	2	no	—
182	#103	—	89	2	no	—
183	#103	—	89	2	no	—
184	#103	—	89	2	no	—
185	#102	—	89	2	no	—
186	#97	—	89	2	no	—
187	C₄×C₈	—	89	2	no	—
188	#41	—	2	2	no	—
189	#41	C₂×Dic₈₉	1	4	yes	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	C₂×Dic₈₉	1	4	yes	complemented
193	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
194	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	89	1	no	—
200	C₂²×C₄	—	89	1	no	—
201	C₂²×C₄	—	178	1	no	—
202	C₂²×C₄	—	178	1	no	—
203	C₂²×C₄	—	178	1	no	—
204	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
205	Q₁₆	C₂×Dic₈₉	1	8	yes	complemented
206	SD₁₆	C₂×Dic₈₉	1	8	yes	complemented
207	M₁₆	—	178	2	no	—
208	M₁₆	C₂×Dic₈₉	1	4	yes	complemented
209	C₂×C₈	—	89	2	no	—
210	C₂×C₈	C₂²×D₈₉	1	2	yes	—
211	C₄⋊C₄	—	89	2	no	—
212	C₄⋊C₄	—	89	1	no	—
213	C₄⋊C₄	—	89	1	no	—
214	C₄⋊C₄	—	178	2	no	—
215	C₄⋊C₄	—	178	2	no	—
216	C₄⋊C₄	—	178	1	no	—
217	C₄⋊C₄	—	178	1	no	—
218	C₄⋊C₄	—	178	1	no	—
219	C₂²⋊C₄	—	178	1	no	—
220	C₂²⋊C₄	—	178	1	no	—
221	C₂²⋊C₄	—	178	1	no	—
222	C₄²	—	89	1	no	—
223	C₄²	—	89	1	no	—
224	C₄²	—	89	1	no	—
225	C₄²	—	89	1	no	—
226	C₄²	—	178	1	no	—
227	C₄²	—	178	1	no	—
228	C₄²	—	178	2	no	—
229	C₄²	—	178	2	no	—
230	C₄²	—	178	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	D₄×D₈₉	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	2	1	no	—
236	Q₈	C₂²×Dic₈₉	1	4	yes	—
237	Q₈	C₂²×Dic₈₉	1	2	yes	—
238	D₄	—	2	4	no	—
239	D₄	—	2	1	no	—
240	D₄	C₂²×Dic₈₉	1	2	yes	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	89	1	no	—
246	C₂×C₄	—	89	1	no	—
247	C₂×C₄	—	89	1	no	—
248	C₂×C₄	—	89	1	no	—
249	C₂×C₄	—	356	2	no	—
250	C₂×C₄	—	356	2	no	—
251	C₂×C₄	—	356	2	no	—
252	C₂×C₄	—	356	1	no	—
253	C₂×C₄	—	178	2	no	—
254	C₂×C₄	—	178	1	no	—
255	C₂×C₄	—	178	1	no	—
256	C₂×C₄	—	178	2	no	—
257	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
258	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
259	C₂×C₄	#13083573	1	2	yes	—
260	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
261	C₈	—	178	2	no	—
262	C₈	C₂²×Dic₈₉	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	D₄×Dic₈₉	1	2	yes	—
269	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	89	2	no	—
273	C₄	—	356	1	no	—
274	C₄	—	356	2	no	—
275	C₄	—	356	1	no	—
276	C₄	—	178	1	no	—
277	C₄	—	178	1	no	—
278	C₄	—	178	1	no	—
279	C₄	D₄×Dic₈₉	1	2	yes	—
280	C₄	C₂³×Dic₈₉	1	1	yes	characteristic
281	C₄	C₂×#13083573	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×D₄×Dic₈₉	1	1	yes	characteristic
285	C₂	#137×D₈₉	1	1	yes	characteristic
286	C₂	#19152397	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#137×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#41×C₈₉	C₂²
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₂×#137	D₈₉
hypocenter	C₈₉	C₄×#137
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#137

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₈₉	C₂	semidirect
3. #19151705	C₂	semidirect
4. #19151699	C₂	semidirect
5. Q₁₆×Dic₈₉	C₂	semidirect
6. SD₁₆×Dic₈₉	C₂	semidirect
7. #19151640	C₂	semidirect
8. #19151607	C₂	semidirect
10. M₁₆×Dic₈₉	C₂	semidirect
40. #137×C₈₉	C₄	semidirect
75. C₂×Q₈×C₈₉	C₂×C₄	semidirect
83. Q₁₆×C₈₉	C₂×C₄	semidirect
84. SD₁₆×C₈₉	C₂×C₄	semidirect
85. M₁₆×C₈₉	C₂×C₄	semidirect
137. Dic₈₉	#137	direct (N × H)
144. C₈₉	C₄×#137	semidirect
158. #137	Dic₈₉	direct (N × H)
192. C₂×Q₈	C₂×Dic₈₉	semidirect
205. Q₁₆	C₂×Dic₈₉	semidirect
206. SD₁₆	C₂×Dic₈₉	semidirect
208. M₁₆	C₂×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#41×Dic₈₉	2
C₂×Q₈×Dic₈₉	2
#19151705	2
#19151699	2
Q₁₆×Dic₈₉	2
SD₁₆×Dic₈₉	2
#19151640	2
#19151607	2
#19151599	2
M₁₆×Dic₈₉	2
C₂×#137×C₈₉	2
C₄×#137	89

Subgroups — #137×Dic89 #26029073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #137×Dic₈₉ #26029073