Subgroups — #36572958

Order 22784.

808 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029169	C₂	1	2	yes	maximal
2	#26028863	C₂	1	2	yes	maximal, complemented
3	#26021043	C₂	1	2	yes	maximal, complemented
4	#26020494	C₂	1	4	yes	maximal, complemented
5	#26020427	C₂	1	1	yes	maximal, characteristic, complemented
6	#26020306	C₂	1	2	yes	maximal, complemented
7	#26020136	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#444×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19152365	C₂²	1	4	yes	—
10	#19151720	—	2	4	no	—
11	#19151715	—	2	4	no	—
12	#19151647	—	2	4	no	—
13	#19151647	C₂²	1	4	yes	—
14	C₂×#15667948	—	2	4	no	—
15	C₂×#15667948	C₂²	1	2	yes	—
16	C₂×#15667948	C₂²	1	2	yes	—
17	C₂×#15667948	C₂²	1	1	yes	characteristic
18	#19151636	C₂²	1	1	yes	characteristic
19	#19151622	C₂²	1	2	yes	—
20	#19151615	C₂²	1	4	yes	—
21	#19151553	C₂²	1	2	yes	—
22	#19151370	C₂²	1	4	yes	—
23	#19151358	C₂²	1	2	yes	—
24	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#131×C₈₉	C₂²	1	2	yes	—
26	#444×C₈₉	C₄	1	8	yes	complemented
27	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#104×C₈₉	C₂²	1	2	yes	—
29	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
30	#15668019	—	2	8	no	—
31	#15668019	—	4	4	no	—
32	C₂×#13083558	—	2	8	no	—
33	C₂×#13083558	C₂³	1	4	yes	—
34	C₂×#13083558	C₂³	1	2	yes	—
35	#15667971	C₂³	1	2	yes	—
36	#15667948	—	2	8	no	—
37	#15667948	—	2	2	no	—
38	#15667948	—	2	2	no	—
39	#15667948	—	2	2	no	—
40	#15667948	—	2	2	no	—
41	#15667948	D₄	1	4	yes	—
42	#15667946	—	2	1	no	—
43	#15667944	—	2	1	no	—
44	C₄⋊#10623494	—	4	2	no	—
45	#15667939	—	2	8	no	—
46	#15667922	—	2	2	no	—
47	C₈×Dic₈₉	—	2	2	no	—
48	C₂×#41×C₈₉	C₂³	1	2	yes	—
49	#135×C₈₉	C₂×C₄	1	4	yes	—
50	#135×C₈₉	D₄	1	4	yes	—
51	C₂×Q₁₆×C₈₉	Q₈	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₉	Q₈	1	2	yes	—
53	C₂×D₈×C₈₉	Q₈	1	1	yes	characteristic
54	#131×C₈₉	—	2	2	no	—
55	#131×C₈₉	—	2	2	no	—
56	#131×C₈₉	C₂×C₄	1	4	yes	—
57	#131×C₈₉	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₈₉	—	2	2	no	—
59	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
60	C₂²×C₈×C₈₉	—	2	2	no	—
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	#108×C₈₉	C₂×C₄	1	4	yes	—
65	#104×C₈₉	C₂×C₄	1	8	yes	—
66	#97×C₈₉	C₂×C₄	1	4	yes	—
67	#13083558	—	2	2	no	—
68	#13083558	—	2	4	no	—
69	#13083558	—	2	4	no	—
70	#13083558	—	4	8	no	—
71	#13083558	C₂×D₄	1	4	yes	—
72	C₂×#10623494	—	2	2	no	—
73	C₂×#10623494	—	2	4	no	—
74	C₂×#10623494	—	4	8	no	—
75	C₄⋊Dic₈₉	—	2	2	no	—
76	C₂²⋊Dic₈₉	—	2	2	no	—
77	C₄×Dic₈₉	—	2	2	no	—
78	#41×C₈₉	—	2	2	no	—
79	#41×C₈₉	—	2	2	no	—
80	#41×C₈₉	C₂²×C₄	1	4	yes	—
81	#41×C₈₉	C₂×D₄	1	4	yes	—
82	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	—
83	C₂×D₄×C₈₉	—	2	2	no	—
84	C₂×D₄×C₈₉	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₈₉	—	2	2	no	—
86	C₂²×C₄×C₈₉	—	2	1	no	—
87	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
88	Q₁₆×C₈₉	C₄⋊C₄	1	4	yes	—
89	SD₁₆×C₈₉	C₄⋊C₄	1	8	yes	—
90	D₈×C₈₉	C₄⋊C₄	1	4	yes	—
91	M₁₆×C₈₉	—	2	2	no	—
92	M₁₆×C₈₉	—	2	4	no	—
93	M₁₆×C₈₉	—	2	4	no	—
94	M₁₆×C₈₉	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	—	2	4	no	—
97	C₂×C₈×C₈₉	—	2	4	no	—
98	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
105	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
106	C₄²×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
108	C₂³×C₈₉	—	2	2	no	—
109	C₂³×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
110	Q₈×C₈₉	—	2	2	no	—
111	Q₈×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
112	D₄×C₈₉	—	2	4	no	—
113	D₄×C₈₉	—	2	4	no	—
114	D₄×C₈₉	—	2	2	no	—
115	D₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	4	no	—
118	C₂×C₄×C₈₉	—	2	4	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	8	1	no	—
121	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×Dic₈₉	—	4	2	no	—
128	C₈×C₈₉	—	2	4	no	—
129	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
130	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
131	C₈×C₈₉	C₄×D₄	1	2	yes	—
132	C₈×C₈₉	C₄×D₄	1	2	yes	—
133	#10623494	—	2	4	no	—
134	#10623494	—	4	4	no	—
135	C₂²×C₈₉	—	2	4	no	—
136	C₂²×C₈₉	—	4	1	no	—
137	C₂²×C₈₉	—	8	2	no	—
138	C₂²×C₈₉	#520	1	1	yes	characteristic
139	C₂²×C₈₉	#547	1	1	yes	characteristic
140	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
141	C₄×C₈₉	—	2	4	no	—
142	C₄×C₈₉	—	4	2	no	—
143	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
145	C₄×C₈₉	D₄²	1	1	yes	characteristic
146	C₄×C₈₉	#554	1	1	yes	characteristic
147	Dic₈₉	—	8	2	no	—
148	#22695	—	89	1	no	maximal, Sylow
149	C₂×C₈₉	—	2	1	no	—
150	C₂×C₈₉	—	2	1	no	—
151	C₂×C₈₉	—	2	1	no	—
152	C₂×C₈₉	—	4	4	no	—
153	C₂×C₈₉	D₄×C₄⋊C₄	1	1	yes	characteristic
154	#2957	—	89	2	no	—
155	C₂×#444	D₈₉	1	1	yes	characteristic
156	#2916	—	89	2	no	—
157	#1953	—	89	2	no	—
158	#1861	—	89	4	no	—
159	#1849	—	89	1	no	—
160	#1814	—	89	2	no	—
161	#1773	—	89	1	no	—
162	C₈₉	#22695	1	1	yes	characteristic, complemented, Sylow
163	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
164	#568	—	89	4	no	—
165	C₂×#131	C₂×D₈₉	1	2	yes	—
166	#444	—	178	4	no	—
167	#444	Dic₈₉	1	8	yes	complemented
168	#443	—	178	4	no	—
169	#430	—	89	4	no	—
170	#430	—	178	4	no	—
171	C₂×#108	—	89	1	no	—
172	C₂×#108	—	89	2	no	—
173	C₂×#108	—	89	2	no	—
174	C₂×#108	—	178	4	no	—
175	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
176	#427	—	89	1	no	—
177	#423	—	89	2	no	—
178	#421	—	89	4	no	—
179	C₂×#104	C₂×D₈₉	1	2	yes	—
180	#405	—	89	2	no	—
181	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
182	#344	—	89	4	no	—
183	#339	—	89	2	no	—
184	C₂×#41	C₂²×D₈₉	1	2	yes	—
185	#135	D₃₅₆	1	4	yes	—
186	#135	C₂×Dic₈₉	1	4	yes	—
187	C₂×Q₁₆	Dic₁₇₈	1	1	yes	characteristic
188	C₂×SD₁₆	Dic₁₇₈	1	2	yes	—
189	C₂×D₈	Dic₁₇₈	1	1	yes	characteristic
190	#131	—	2	2	no	—
191	#131	—	2	2	no	—
192	#131	—	356	4	no	—
193	#131	—	178	8	no	—
194	#131	C₂×Dic₈₉	1	4	yes	—
195	#131	C₂×Dic₈₉	1	4	yes	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	89	2	no	—
198	C₂×M₁₆	—	89	4	no	—
199	C₂×M₁₆	—	178	8	no	—
200	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
204	#117	—	89	2	no	—
205	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
206	#108	—	89	4	no	—
207	#108	—	178	8	no	—
208	#108	—	178	2	no	—
209	#108	—	178	2	no	—
210	#108	—	178	2	no	—
211	#108	—	178	2	no	—
212	#108	C₂×Dic₈₉	1	4	yes	—
213	#107	—	178	1	no	—
214	#106	—	178	1	no	—
215	C₄⋊C₈	—	356	2	no	—
216	#104	—	178	8	no	—
217	#104	C₂×Dic₈₉	1	8	yes	—
218	#97	—	178	2	no	—
219	#97	C₂×Dic₈₉	1	4	yes	—
220	C₄×C₈	—	178	2	no	—
221	#41	—	2	2	no	—
222	#41	—	2	2	no	—
223	#41	C₂²×Dic₈₉	1	4	yes	—
224	#41	C₂×D₃₅₆	1	4	yes	—
225	C₂×Q₈	C₂×Dic₁₇₈	1	2	yes	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	C₂×Dic₁₇₈	1	2	yes	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
231	Q₁₆	C₄⋊Dic₈₉	1	4	yes	—
232	SD₁₆	C₄⋊Dic₈₉	1	8	yes	—
233	D₈	C₄⋊Dic₈₉	1	4	yes	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	89	4	no	—
238	M₁₆	—	356	8	no	—
239	M₁₆	—	178	4	no	—
240	M₁₆	—	178	4	no	—
241	M₁₆	—	178	2	no	—
242	M₁₆	C₂²×Dic₈₉	1	4	yes	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	356	8	no	—
247	C₂×C₈	—	178	4	no	—
248	C₂×C₈	—	178	2	no	—
249	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
250	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
251	C₂×C₈	D₄×D₈₉	1	2	yes	—
252	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
253	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
254	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
255	C₂×C₈	#13083573	1	1	yes	characteristic
256	C₂×C₈	#13083573	1	1	yes	characteristic
257	C₄⋊C₄	—	178	2	no	—
258	C₂²⋊C₄	—	178	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	178	2	no	—
261	C₄²	C₂²×Dic₈₉	1	2	yes	—
262	C₂³	—	2	2	no	—
263	C₂³	#15668059	1	1	yes	characteristic
264	Q₈	—	2	2	no	—
265	Q₈	C₂×C₄⋊Dic₈₉	1	4	yes	—
266	D₄	—	2	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	2	2	no	—
269	D₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	356	2	no	—
276	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
279	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
280	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
281	C₂×C₄	#15668058	1	1	yes	characteristic
282	C₈	—	2	4	no	—
283	C₈	—	356	4	no	—
284	C₈	—	178	4	no	—
285	C₈	C₂×C₄⋊Dic₈₉	1	2	yes	—
286	C₈	C₂×C₄⋊Dic₈₉	1	2	yes	—
287	C₈	D₄×Dic₈₉	1	2	yes	—
288	C₈	D₄×Dic₈₉	1	2	yes	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	#19152081	1	1	yes	characteristic
293	C₂²	#19152252	1	1	yes	characteristic
294	C₂²	D₄×Dic₁₇₈	1	1	yes	characteristic
295	C₄	—	2	4	no	—
296	C₄	—	4	2	no	—
297	C₄	—	712	2	no	—
298	C₄	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
299	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
300	C₄	D₄×D₃₅₆	1	1	yes	characteristic
301	C₄	#19152305	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	4	no	—
306	C₂	D₄×C₄⋊Dic₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#444×C₈₉	C₂
socle	C₂×C₈₉	D₄×C₄⋊C₄
prime radical	C₂×#135×C₈₉	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×#444	D₈₉
hypocenter	C₈₉	#22695
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#444

Complements

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

N	H ≅ G/N	type
2. #26028863	C₂	semidirect
3. #26021043	C₂	semidirect
4. #26020494	C₂	semidirect
5. #26020427	C₂	semidirect
6. #26020306	C₂	semidirect
7. #26020136	C₂	semidirect
26. #444×C₈₉	C₄	semidirect
162. C₈₉	#22695	semidirect
167. #444	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029169	2
#26028863	2
#26021043	2
#26020494	2
#26020427	2
#26020306	2
#26020136	2
C₂×#444×C₈₉	2
#22695	89