Subgroups — #36470354

Order 22784.

486 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13083557	C₂	1	1	yes	maximal, characteristic
2	#26022008	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151379	C₂	1	2	yes	maximal
4	#26018335	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13083557	C₄	1	1	yes	characteristic
7	C₂²×#13083557	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₈₉	C₄	1	1	yes	characteristic
9	C₂×#15667952	C₂²	1	2	yes	—
10	C₄×#13083557	—	2	2	no	—
11	C₄×#13083557	—	2	2	no	—
12	C₂×#15667922	C₄	1	1	yes	characteristic
13	C₂×#15667922	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
15	C₄⋊#13083549	—	2	2	no	—
16	#19151379	—	2	4	no	—
17	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆×C₈₉	—	2	2	no	—
20	C₂²×C₄×D₈₉	C₈	1	2	yes	—
21	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×#13083557	—	2	2	no	—
23	C₂×#13083557	—	2	2	no	—
24	C₂×#13083557	—	2	2	no	—
25	C₂×#13083557	—	2	1	no	—
26	C₂×#13083557	—	2	1	no	—
27	C₂×#13083557	—	2	1	no	—
28	C₂×#13083557	—	2	1	no	—
29	C₂×#13083557	D₄	1	2	yes	—
30	C₂×#13083557	Q₈	1	2	yes	—
31	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
32	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₈₉	C₈	1	2	yes	—
34	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₈₉	—	2	2	no	—
36	C₄²×D₈₉	—	2	1	no	—
37	C₄²×D₈₉	—	2	1	no	—
38	#15667952	—	2	2	no	—
39	#15667952	—	2	2	no	—
40	C₂×#13083549	—	2	2	no	—
41	#15667922	—	2	2	no	—
42	#15667922	—	2	1	no	—
43	#15667922	—	2	1	no	—
44	C₄×#10623494	—	2	2	no	—
45	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₈₉	—	2	2	no	—
49	M₃₂×C₈₉	—	2	2	no	—
50	C₂×C₁₆×C₈₉	—	2	2	no	—
51	C₄×C₈×C₈₉	D₄	1	2	yes	—
52	C₄×C₈×C₈₉	D₄	1	2	yes	—
53	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
54	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₈₉	—	2	1	no	—
57	C₂×C₄×D₈₉	—	2	1	no	—
58	C₂×C₄×D₈₉	—	2	1	no	—
59	C₂×C₄×D₈₉	—	2	1	no	—
60	C₂×C₄×D₈₉	—	2	2	no	—
61	C₂×C₄×D₈₉	—	2	2	no	—
62	C₂×C₄×D₈₉	—	4	4	no	—
63	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×D₈₉	M₁₆	1	2	yes	—
65	#13083557	—	2	8	no	—
66	#13083557	—	2	2	no	—
67	#13083557	—	2	2	no	—
68	#13083557	—	2	2	no	—
69	#13083557	—	2	2	no	—
70	C₂×#10623494	—	2	1	no	—
71	C₂×#10623494	—	2	1	no	—
72	C₂×#10623494	—	2	1	no	—
73	C₂×#10623494	—	2	1	no	—
74	C₂×#10623494	C₄⋊C₄	1	2	yes	—
75	C₂×#10623494	M₁₆	1	2	yes	—
76	C₄×Dic₈₉	—	2	2	no	—
77	C₄×Dic₈₉	—	2	2	no	—
78	C₄×Dic₈₉	—	2	1	no	—
79	C₄×Dic₈₉	—	2	1	no	—
80	#13083549	—	4	2	no	—
81	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
82	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	—	2	1	no	—
84	C₂×C₈×C₈₉	—	2	1	no	—
85	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
91	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
92	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
94	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
95	C₁₆×C₈₉	—	4	2	no	—
96	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
97	C₂²×D₈₉	—	2	1	no	—
98	C₂²×D₈₉	—	2	1	no	—
99	C₂²×D₈₉	—	4	2	no	—
100	C₂²×D₈₉	C₄⋊C₈	1	2	yes	—
101	C₂×C₄×C₈₉	—	2	4	no	—
102	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
110	C₂×Dic₈₉	—	2	1	no	—
111	C₂×Dic₈₉	—	2	1	no	—
112	C₂×Dic₈₉	—	2	2	no	—
113	C₂×Dic₈₉	—	4	2	no	—
114	C₂×Dic₈₉	C₄⋊C₈	1	2	yes	—
115	C₄×D₈₉	—	2	2	no	—
116	C₄×D₈₉	—	2	2	no	—
117	C₄×D₈₉	—	2	2	no	—
118	C₄×D₈₉	—	2	2	no	—
119	C₄×D₈₉	—	4	4	no	—
120	C₄×D₈₉	—	4	4	no	—
121	C₈×C₈₉	—	2	2	no	—
122	C₈×C₈₉	#121	1	1	yes	characteristic
123	C₈×C₈₉	#121	1	1	yes	characteristic
124	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
126	#10623494	—	2	2	no	—
127	#10623494	—	2	2	no	—
128	C₂²×C₈₉	—	2	2	no	—
129	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂²×C₈₉	#431	1	1	yes	characteristic
132	C₂×D₈₉	—	2	2	no	—
133	C₂×D₈₉	—	4	2	no	—
134	C₂×D₈₉	—	4	2	no	—
135	C₂×D₈₉	—	4	2	no	—
136	C₄×C₈₉	—	2	2	no	—
137	C₄×C₈₉	#349	1	4	yes	complemented
138	C₄×C₈₉	#380	1	1	yes	characteristic
139	C₄×C₈₉	#407	1	1	yes	characteristic
140	C₄×C₈₉	#423	1	1	yes	characteristic
141	C₄×C₈₉	#432	1	1	yes	characteristic
142	Dic₈₉	—	2	2	no	—
143	Dic₈₉	—	4	2	no	—
144	#12785	—	89	1	no	maximal, Sylow
145	C₂×C₈₉	—	2	1	no	—
146	C₂×C₈₉	—	2	1	no	—
147	C₂×C₈₉	#1797	1	1	yes	characteristic
148	C₂×C₈₉	C₂×#349	1	2	yes	—
149	D₈₉	—	4	2	no	—
150	C₂×C₄×M₁₆	—	89	1	no	—
151	#2150	—	89	1	no	—
152	#2150	D₈₉	1	1	yes	characteristic, complemented
153	C₂×#349	—	89	2	no	—
154	#1377	—	89	2	no	—
155	C₈₉	#12785	1	1	yes	characteristic, complemented, Sylow
156	C₂²×M₁₆	—	89	1	no	—
157	C₂²×M₁₆	—	89	1	no	—
158	C₂²×C₄²	—	89	1	no	—
159	C₂×M₃₂	—	89	2	no	—
160	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
161	C₄×M₁₆	—	178	2	no	—
162	C₄×M₁₆	—	178	2	no	—
163	C₂×#97	—	89	1	no	—
164	C₂×#97	—	89	1	no	—
165	C₂×C₄×C₈	—	89	1	no	—
166	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
167	C₄⋊C₁₆	—	2	2	no	—
168	C₄⋊C₁₆	—	178	2	no	—
169	#349	—	178	4	no	—
170	C₂³×C₄	—	89	1	no	—
171	C₂³×C₄	—	89	2	no	—
172	C₂×M₁₆	—	89	2	no	—
173	C₂×M₁₆	—	89	2	no	—
174	C₂×M₁₆	—	178	2	no	—
175	C₂×M₁₆	—	178	2	no	—
176	C₂×M₁₆	—	178	2	no	—
177	C₂×M₁₆	—	178	1	no	—
178	C₂×M₁₆	—	178	1	no	—
179	C₂×M₁₆	—	178	1	no	—
180	C₂×M₁₆	—	178	1	no	—
181	C₂²×C₈	—	89	1	no	—
182	C₂²×C₈	—	89	1	no	—
183	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
185	C₂×C₄²	—	89	2	no	—
186	C₂×C₄²	—	89	1	no	—
187	C₂×C₄²	—	178	2	no	—
188	C₂×C₄²	—	178	1	no	—
189	C₂×C₄²	—	178	1	no	—
190	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
191	M₃₂	—	2	2	no	—
192	M₃₂	—	2	2	no	—
193	M₃₂	—	178	2	no	—
194	M₃₂	—	178	2	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	—	178	2	no	—
197	#97	—	178	2	no	—
198	#97	—	178	1	no	—
199	#97	—	178	1	no	—
200	C₄×C₈	—	178	2	no	—
201	C₄×C₈	D₃₅₆	1	2	yes	—
202	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
203	C₂⁴	—	89	1	no	—
204	C₂²×C₄	—	89	1	no	—
205	C₂²×C₄	—	89	2	no	—
206	C₂²×C₄	—	89	2	no	—
207	C₂²×C₄	—	89	2	no	—
208	C₂²×C₄	—	356	4	no	—
209	C₂²×C₄	—	178	2	no	—
210	C₂²×C₄	—	178	2	no	—
211	C₂²×C₄	—	178	1	no	—
212	C₂²×C₄	—	178	1	no	—
213	C₂²×C₄	—	178	1	no	—
214	C₂²×C₄	—	178	1	no	—
215	C₂²×C₄	C₈×D₈₉	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
217	M₁₆	—	178	8	no	—
218	M₁₆	—	178	2	no	—
219	M₁₆	—	178	2	no	—
220	M₁₆	—	178	2	no	—
221	M₁₆	—	178	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	89	2	no	—
225	C₂×C₈	—	89	2	no	—
226	C₂×C₈	—	178	1	no	—
227	C₂×C₈	—	178	1	no	—
228	C₂×C₈	—	178	1	no	—
229	C₂×C₈	—	178	1	no	—
230	C₂×C₈	#13083551	1	2	yes	—
231	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
232	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
233	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
234	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
235	C₂×C₈	#13083573	1	1	yes	characteristic
236	C₂×C₈	#13083575	1	1	yes	characteristic
237	C₄²	—	178	1	no	—
238	C₄²	—	178	2	no	—
239	C₄²	—	178	2	no	—
240	C₄²	—	178	1	no	—
241	C₄²	#13083551	1	2	yes	—
242	C₄²	#13083557	1	1	yes	characteristic
243	C₄²	#13083557	1	1	yes	characteristic
244	C₁₆	—	4	2	no	—
245	C₁₆	—	356	2	no	—
246	C₂³	—	89	2	no	—
247	C₂³	—	356	2	no	—
248	C₂³	—	178	1	no	—
249	C₂³	—	178	1	no	—
250	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	89	2	no	—
253	C₂×C₄	—	356	4	no	—
254	C₂×C₄	—	356	4	no	—
255	C₂×C₄	—	356	2	no	—
256	C₂×C₄	—	178	2	no	—
257	C₂×C₄	—	178	1	no	—
258	C₂×C₄	—	178	1	no	—
259	C₂×C₄	—	178	2	no	—
260	C₂×C₄	—	178	2	no	—
261	C₂×C₄	—	178	2	no	—
262	C₂×C₄	—	178	2	no	—
263	C₂×C₄	#15667924	1	2	yes	—
264	C₂×C₄	#15667924	1	2	yes	—
265	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
266	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₄	#15667970	1	1	yes	characteristic
268	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
269	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
270	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	178	2	no	—
273	C₈	—	178	2	no	—
274	C₈	#15667983	1	1	yes	characteristic
275	C₈	#15667987	1	1	yes	characteristic
276	C₈	#15667988	1	1	yes	characteristic
277	C₈	#15667992	1	1	yes	characteristic
278	C₂²	—	2	2	no	—
279	C₂²	—	356	2	no	—
280	C₂²	—	356	2	no	—
281	C₂²	—	356	2	no	—
282	C₂²	—	178	2	no	—
283	C₂²	C₂×#15667924	1	1	yes	characteristic
284	C₂²	C₄⋊C₈×D₈₉	1	1	yes	characteristic
285	C₂²	#19151650	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	356	2	no	—
288	C₄	—	178	2	no	—
289	C₄	#19151379	1	4	yes	complemented
290	C₄	#19151455	1	1	yes	characteristic
291	C₄	#19151561	1	1	yes	characteristic
292	C₄	#19151620	1	1	yes	characteristic
293	C₄	#19151654	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	356	2	no	—
297	C₂	#26020229	1	1	yes	characteristic
298	C₂	C₂×#19151379	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#13083551
commutator	C₂³×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2150	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂³×D₈₉	C₂×C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2150	D₈₉
hypocenter	C₈₉	#12785
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #26022008	C₂	semidirect
4. #26018335	C₂	semidirect
5. C₈₉×#2150	C₂	semidirect
137. C₄×C₈₉	#349	semidirect
152. #2150	D₈₉	semidirect
155. C₈₉	#12785	semidirect
289. C₄	#19151379	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#13083557	2
#26022008	2
C₂×#19151379	2
#26018335	2
C₈₉×#2150	2
#12785	89