Subgroups — #36710853

Order 22784.

488 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	#26030911	C₂	1	1	yes	maximal, characteristic, complemented
2	#26030902	C₂	1	1	yes	maximal, characteristic
3	#26022417	C₂	1	2	yes	maximal, complemented
4	#26022410	C₂	1	2	yes	maximal, complemented
5	#26022365	C₂	1	2	yes	maximal, complemented
6	#26022348	C₂	1	2	yes	maximal, complemented
7	#26022298	C₂	1	2	yes	maximal, complemented
8	#26022293	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152210	—	2	1	no	—
11	#19152203	—	2	1	no	—
12	C₂×#15667957	C₂²	1	2	yes	—
13	C₂×#15667957	C₂²	1	2	yes	—
14	#19152000	C₂²	1	1	yes	characteristic
15	#19152000	C₂²	1	1	yes	characteristic
16	#19151978	C₂²	1	1	yes	characteristic
17	#19151975	C₂²	1	1	yes	characteristic
18	#19151967	—	2	1	no	—
19	#19151963	—	2	1	no	—
20	#19151856	—	2	1	no	—
21	#19151851	—	2	1	no	—
22	#19151839	C₂²	1	2	yes	—
23	#19151834	C₂²	1	2	yes	—
24	#19151711	—	2	1	no	—
25	#19151706	—	2	1	no	—
26	C₄×Dic₃₅₆	C₂²	1	2	yes	—
27	C₄×D₇₁₂	C₂²	1	2	yes	—
28	#19151426	C₂²	1	2	yes	—
29	#19151424	C₂²	1	2	yes	—
30	#19151403	C₂²	1	2	yes	—
31	#19151403	C₂²	1	2	yes	—
32	#19151400	C₂²	1	2	yes	—
33	#19151400	C₂²	1	2	yes	—
34	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	#405×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
38	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
39	C₂×Dic₃₅₆	—	2	1	no	—
40	C₂×Dic₃₅₆	C₂³	1	2	yes	—
41	C₂×D₇₁₂	—	2	1	no	—
42	C₂×D₇₁₂	C₂³	1	2	yes	—
43	#15668012	—	2	1	no	—
44	#15668009	—	2	1	no	—
45	#15668005	—	4	2	no	—
46	#15668004	—	2	1	no	—
47	#15668004	—	2	1	no	—
48	#15668002	—	4	2	no	—
49	#15667997	—	4	2	no	—
50	#15667991	—	2	2	no	—
51	#15667986	—	2	2	no	—
52	#15667985	—	4	2	no	—
53	C₄×Dic₁₇₈	—	2	2	no	—
54	C₄×D₃₅₆	—	2	2	no	—
55	#15667957	—	4	2	no	—
56	#15667957	—	4	2	no	—
57	#15667946	C₂³	1	2	yes	—
58	#15667946	C₂³	1	2	yes	—
59	#15667944	—	2	1	no	—
60	#15667944	—	2	1	no	—
61	#15667937	—	2	2	no	—
62	#15667937	—	2	2	no	—
63	#15667934	—	2	2	no	—
64	#15667934	—	2	2	no	—
65	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	#117×C₈₉	D₄	1	1	yes	characteristic
68	M₃₂×C₈₉	—	4	1	no	—
69	C₂×C₁₆×C₈₉	—	4	1	no	—
70	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
71	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
72	#97×C₈₉	D₄	1	1	yes	characteristic
73	#97×C₈₉	D₄	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×Dic₁₇₈	—	2	2	no	—
77	C₂×D₃₅₆	—	2	2	no	—
78	C₂×D₃₅₆	—	4	2	no	—
79	C₂×C₂²⋊D₈₉	—	4	2	no	—
80	C₂×C₄×D₈₉	—	4	2	no	—
81	Dic₃₅₆	—	2	2	no	—
82	Dic₃₅₆	—	2	2	no	—
83	Dic₃₅₆	—	4	2	no	—
84	D₇₁₂	—	2	2	no	—
85	D₇₁₂	—	2	2	no	—
86	D₇₁₂	—	4	2	no	—
87	C₄⋊Dic₈₉	—	2	2	no	—
88	C₄⋊Dic₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	—	4	2	no	—
90	#13083553	—	4	2	no	—
91	#13083553	—	4	2	no	—
92	#13083553	—	4	2	no	—
93	#13083553	—	4	2	no	—
94	C₂²⋊Dic₈₉	—	4	2	no	—
95	#13083551	—	4	2	no	—
96	#13083551	—	4	2	no	—
97	C₄×Dic₈₉	—	4	2	no	—
98	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
99	M₁₆×C₈₉	—	4	1	no	—
100	C₂×C₈×C₈₉	—	2	1	no	—
101	C₂×C₈×C₈₉	—	2	1	no	—
102	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
107	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
108	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₈₉	—	2	2	no	—
111	C₁₆×C₈₉	—	2	2	no	—
112	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
113	C₂²×D₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
120	C₂²⋊D₈₉	—	16	2	no	—
121	C₂×Dic₈₉	—	4	2	no	—
122	C₂×Dic₈₉	—	4	2	no	—
123	C₂×Dic₈₉	—	4	2	no	—
124	D₃₅₆	—	4	2	no	—
125	D₃₅₆	—	4	2	no	—
126	D₃₅₆	—	16	2	no	—
127	C₄×D₈₉	—	16	2	no	—
128	Dic₁₇₈	—	4	2	no	—
129	Dic₁₇₈	—	4	2	no	—
130	C₈×C₈₉	—	2	1	no	—
131	C₈×C₈₉	—	2	1	no	—
132	C₈×C₈₉	—	2	1	no	—
133	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
134	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂²×C₈₉	—	2	2	no	—
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
138	C₂×D₈₉	—	8	2	no	—
139	C₂×D₈₉	—	8	2	no	—
140	C₂×D₈₉	—	8	2	no	—
141	C₄×C₈₉	—	2	2	no	—
142	C₄×C₈₉	—	2	2	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	#537	1	1	yes	characteristic
145	C₄×C₈₉	#575	1	1	yes	characteristic
146	Dic₈₉	—	8	2	no	—
147	Dic₈₉	—	8	2	no	—
148	Dic₈₉	—	8	2	no	—
149	#33826	—	89	1	no	maximal, Sylow
150	C₂×C₈₉	—	4	1	no	—
151	C₂×C₈₉	#3245	1	1	yes	characteristic
152	C₂×C₈₉	#3416	1	2	yes	—
153	D₈₉	—	16	2	no	—
154	#3177	—	89	1	no	—
155	#3176	—	89	1	no	—
156	#2229	—	89	2	no	—
157	#2228	—	89	2	no	—
158	#2220	—	89	2	no	—
159	#2217	—	89	2	no	—
160	#2208	—	89	2	no	—
161	#2207	—	89	2	no	—
162	#2151	D₈₉	1	1	yes	characteristic, complemented
163	C₈₉	#33826	1	1	yes	characteristic, complemented, Sylow
164	#541	—	178	1	no	—
165	#540	—	178	1	no	—
166	C₂×SD₃₂	—	89	2	no	—
167	C₂×SD₃₂	—	89	2	no	—
168	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
169	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
170	#499	—	89	1	no	—
171	#499	—	89	1	no	—
172	#494	—	89	1	no	—
173	#493	—	89	1	no	—
174	#491	—	178	1	no	—
175	#490	—	178	1	no	—
176	#470	—	178	1	no	—
177	#469	—	178	1	no	—
178	#467	—	89	2	no	—
179	#466	—	89	2	no	—
180	#442	—	178	1	no	—
181	#441	—	178	1	no	—
182	C₄×Q₁₆	—	89	2	no	—
183	C₄×D₈	—	89	2	no	—
184	#405	C₂×D₈₉	1	1	yes	characteristic
185	#367	—	89	2	no	—
186	#366	—	89	2	no	—
187	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
188	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
189	#358	—	89	2	no	—
190	#358	—	89	2	no	—
191	#357	—	89	2	no	—
192	#357	—	89	2	no	—
193	C₂×Q₁₆	—	89	2	no	—
194	C₂×Q₁₆	—	178	1	no	—
195	C₂×D₈	—	89	2	no	—
196	C₂×D₈	—	178	1	no	—
197	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
199	C₄⋊Q₈	—	178	1	no	—
200	#127	—	178	1	no	—
201	#126	—	356	2	no	—
202	#125	—	356	2	no	—
203	#125	—	178	1	no	—
204	#125	—	178	1	no	—
205	#123	—	356	2	no	—
206	C₂²⋊Q₈	—	178	2	no	—
207	#121	—	356	2	no	—
208	#121	—	178	2	no	—
209	C₄×Q₈	—	178	2	no	—
210	C₄×D₄	—	178	2	no	—
211	#117	D₃₅₆	1	1	yes	characteristic
212	SD₃₂	—	356	2	no	—
213	SD₃₂	—	356	2	no	—
214	M₃₂	—	4	1	no	—
215	C₂×C₁₆	—	4	1	no	—
216	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
217	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
218	#107	—	89	2	no	—
219	#107	—	89	2	no	—
220	#106	—	178	1	no	—
221	#106	—	178	1	no	—
222	#103	—	178	2	no	—
223	#103	—	178	2	no	—
224	#102	—	178	2	no	—
225	#102	—	178	2	no	—
226	#97	D₃₅₆	1	1	yes	characteristic
227	#97	D₃₅₆	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
230	C₂×Q₈	—	178	2	no	—
231	C₂×D₄	—	356	2	no	—
232	C₂×D₄	—	356	2	no	—
233	C₂×D₄	—	178	2	no	—
234	C₂²×C₄	—	356	2	no	—
235	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
236	Q₁₆	—	356	2	no	—
237	Q₁₆	—	178	2	no	—
238	Q₁₆	—	178	2	no	—
239	D₈	—	356	2	no	—
240	D₈	—	178	2	no	—
241	D₈	—	178	2	no	—
242	M₁₆	—	4	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
246	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
247	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
248	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
249	C₄⋊C₄	—	356	2	no	—
250	C₄⋊C₄	—	356	2	no	—
251	C₄⋊C₄	—	356	2	no	—
252	C₄⋊C₄	—	356	2	no	—
253	C₄⋊C₄	—	356	2	no	—
254	C₄⋊C₄	—	178	2	no	—
255	C₄⋊C₄	—	178	2	no	—
256	C₄⋊C₄	D₇₁₂	1	2	yes	—
257	C₂²⋊C₄	—	356	2	no	—
258	C₂²⋊C₄	—	356	2	no	—
259	C₂²⋊C₄	—	356	2	no	—
260	C₂²⋊C₄	D₇₁₂	1	2	yes	—
261	C₄²	—	356	2	no	—
262	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
263	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
264	C₁₆	—	2	2	no	—
265	C₁₆	—	2	2	no	—
266	C₂³	—	356	2	no	—
267	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
268	Q₈	—	356	2	no	—
269	Q₈	—	356	2	no	—
270	D₄	—	356	2	no	—
271	D₄	—	356	2	no	—
272	D₄	—	1424	2	no	—
273	D₄	—	1424	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	356	2	no	—
278	C₂×C₄	—	356	2	no	—
279	C₂×C₄	—	1424	2	no	—
280	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
281	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
282	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	#15668057	1	1	yes	characteristic
288	C₈	#15668060	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	712	2	no	—
292	C₂²	—	712	2	no	—
293	C₂²	—	712	2	no	—
294	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	712	2	no	—
299	C₄	—	712	2	no	—
300	C₄	—	712	2	no	—
301	C₄	#19152179	1	1	yes	characteristic
302	C₄	#19152398	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	1424	2	no	—
305	C₂	#26031562	1	1	yes	characteristic
306	C₂	#26033400	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2151	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26030902	C₂
hypercenter	#2151	D₈₉
hypocenter	C₈₉	#33826
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2151

Complements

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

N	H ≅ G/N	type
1. #26030911	C₂	semidirect
3. #26022417	C₂	semidirect
4. #26022410	C₂	semidirect
5. #26022365	C₂	semidirect
6. #26022348	C₂	semidirect
7. #26022298	C₂	semidirect
8. #26022293	C₂	semidirect
9. C₈₉×#2151	C₂	semidirect
162. #2151	D₈₉	semidirect
163. C₈₉	#33826	semidirect

Maximal subgroups

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

subgroup	index
#26030911	2
#26030902	2
#26022417	2
#26022410	2
#26022365	2
#26022348	2
#26022298	2
#26022293	2
C₈₉×#2151	2
#33826	89