Subgroups — #36572928

Order 22784.

862 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151716	C₂	1	1	yes	maximal, characteristic
2	#26029129	C₂	1	2	yes	maximal
3	#26028849	C₂	1	2	yes	maximal, complemented
4	#26021037	C₂	1	2	yes	maximal, complemented
5	#26020479	C₂	1	4	yes	maximal, complemented
6	#26020412	C₂	1	1	yes	maximal, characteristic, complemented
7	#26020313	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₂²	1	1	yes	characteristic
10	C₂×#15668018	C₂²	1	2	yes	—
11	#19152078	C₂²	1	4	yes	—
12	#19151716	C₄	1	8	yes	complemented
13	#19151700	—	2	4	no	—
14	C₄×Dic₃₅₆	—	2	2	no	—
15	C₄×D₇₁₂	—	2	2	no	—
16	C₂×#15667948	C₂²	1	1	yes	characteristic
17	#19151641	C₂²	1	4	yes	—
18	#19151637	—	2	4	no	—
19	#19151635	—	2	2	no	—
20	C₂×#15667946	C₂²	1	2	yes	—
21	C₂×#15667946	C₂²	1	2	yes	—
22	#107×D₈₉	—	2	2	no	—
23	C₂×#15667940	C₂²	1	2	yes	—
24	#19151600	C₂²	1	4	yes	—
25	#19151468	C₂²	1	2	yes	—
26	#19151366	C₂²	1	4	yes	—
27	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
28	#427×C₈₉	C₂²	1	1	yes	characteristic
29	#405×C₈₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#339×C₈₉	C₂²	1	2	yes	—
32	C₂×#13083574	C₂³	1	2	yes	—
33	#15668034	C₂×C₄	1	4	yes	—
34	#15668034	Q₈	1	4	yes	—
35	C₂×Dic₃₅₆	D₄	1	1	yes	characteristic
36	C₂×#13083563	D₄	1	2	yes	—
37	C₂×D₇₁₂	D₄	1	1	yes	characteristic
38	#15668018	—	2	4	no	—
39	#15668018	C₂×C₄	1	8	yes	complemented
40	C₂×#13083558	C₂³	1	2	yes	—
41	C₂×#13083557	—	2	2	no	—
42	C₂×C₈×D₈₉	—	2	2	no	—
43	C₄×Dic₁₇₈	—	2	4	no	—
44	C₄×D₃₅₆	—	2	4	no	—
45	#15667972	—	4	4	no	—
46	#15667970	—	2	4	no	—
47	C₂×C₄⋊Dic₈₉	—	4	2	no	—
48	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
49	C₄⋊C₄×D₈₉	—	2	4	no	—
50	#15667948	C₂×C₄	1	4	yes	complemented
51	#15667946	—	2	2	no	—
52	#15667946	—	2	2	no	—
53	#15667946	—	2	2	no	—
54	#15667946	—	2	2	no	—
55	#15667945	—	2	4	no	—
56	#15667943	—	2	4	no	—
57	#15667940	C₂×C₄	1	8	yes	complemented
58	#15667937	—	2	4	no	—
59	#15667934	—	2	4	no	—
60	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	#117×C₈₉	C₂³	1	2	yes	—
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	#108×C₈₉	—	2	2	no	—
66	#107×C₈₉	D₄	1	1	yes	characteristic
67	#107×C₈₉	D₄	1	1	yes	characteristic
68	#106×C₈₉	D₄	1	2	yes	—
69	#97×C₈₉	—	2	2	no	—
70	C₄×C₈×C₈₉	—	2	2	no	—
71	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	complemented
72	#13083574	—	2	2	no	—
73	#13083574	—	2	2	no	—
74	#13083574	C₂²×C₄	1	4	yes	—
75	#13083574	C₂×Q₈	1	4	yes	—
76	C₂×Dic₁₇₈	C₂×D₄	1	2	yes	—
77	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
78	C₂×C₂²⋊D₈₉	—	2	2	no	—
79	C₂²×Dic₈₉	—	2	4	no	—
80	C₂×C₄×D₈₉	—	2	2	no	—
81	C₂×C₄×D₈₉	—	4	4	no	—
82	Dic₃₅₆	C₄⋊C₄	1	4	yes	complemented
83	#13083563	C₄⋊C₄	1	8	yes	complemented
84	D₇₁₂	C₄⋊C₄	1	4	yes	complemented
85	#13083558	—	2	2	no	—
86	#13083558	C₂²×C₄	1	4	yes	—
87	#13083557	—	2	8	no	—
88	C₂×#10623494	—	2	2	no	—
89	C₈×D₈₉	—	2	8	no	—
90	C₄⋊Dic₈₉	—	2	4	no	—
91	C₄⋊Dic₈₉	—	2	4	no	—
92	C₄⋊Dic₈₉	—	8	4	no	—
93	#13083553	—	4	4	no	—
94	#13083551	—	4	4	no	—
95	C₄×Dic₈₉	—	4	4	no	—
96	C₂²×C₄×C₈₉	—	2	1	no	—
97	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
98	M₁₆×C₈₉	—	2	2	no	—
99	M₁₆×C₈₉	—	2	2	no	—
100	C₂×C₈×C₈₉	—	2	2	no	—
101	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
108	C₂×C₈×C₈₉	#41	1	2	yes	—
109	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₈₉	—	2	2	no	—
112	C₄²×C₈₉	—	2	2	no	—
113	C₄²×C₈₉	—	2	1	no	—
114	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
115	C₂³×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²×D₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂²⋊D₈₉	—	2	4	no	—
129	C₂²⋊D₈₉	—	2	4	no	—
130	C₂×Dic₈₉	—	2	2	no	—
131	C₂×Dic₈₉	—	4	4	no	—
132	C₂×Dic₈₉	—	8	4	no	—
133	D₃₅₆	—	2	2	no	—
134	D₃₅₆	C₂×C₄⋊C₄	1	4	yes	—
135	C₄×D₈₉	—	2	4	no	—
136	C₄×D₈₉	—	2	4	no	—
137	C₄×D₈₉	—	16	4	no	—
138	Dic₁₇₈	—	2	2	no	—
139	Dic₁₇₈	C₂×C₄⋊C₄	1	4	yes	—
140	C₈×C₈₉	—	4	2	no	—
141	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
142	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
143	C₈×C₈₉	C₄×D₄	1	2	yes	—
144	C₈×C₈₉	C₄×D₄	1	2	yes	—
145	#10623494	—	2	4	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	#520	1	1	yes	characteristic
149	C₂²×C₈₉	D₄²	1	1	yes	characteristic
150	C₂²×C₈₉	#554	1	1	yes	characteristic
151	C₂×D₈₉	—	2	4	no	—
152	C₂×D₈₉	—	4	2	no	—
153	C₂×D₈₉	—	4	2	no	—
154	C₄×C₈₉	—	2	4	no	—
155	C₄×C₈₉	—	4	2	no	—
156	C₄×C₈₉	—	4	2	no	—
157	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₈₉	#547	1	1	yes	characteristic
160	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
161	Dic₈₉	—	2	4	no	—
162	Dic₈₉	—	8	4	no	—
163	#22692	—	89	1	no	maximal, Sylow
164	C₂×C₈₉	—	2	1	no	—
165	C₂×C₈₉	—	2	1	no	—
166	C₂×C₈₉	D₄×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₈₉	#3292	1	1	yes	characteristic
168	C₂×C₈₉	#3293	1	1	yes	characteristic
169	D₈₉	—	4	4	no	—
170	C₂×#443	—	89	1	no	—
171	#2951	—	89	2	no	—
172	#2913	—	89	2	no	—
173	#1952	—	89	2	no	—
174	#1859	—	89	4	no	—
175	#1845	—	89	1	no	—
176	#1816	—	89	2	no	—
177	#1772	D₈₉	1	1	yes	characteristic, complemented
178	C₈₉	#22692	1	1	yes	characteristic, complemented, Sylow
179	C₂×#135	—	89	1	no	—
180	C₂×#131	—	89	2	no	—
181	#520	—	89	4	no	—
182	#443	—	89	8	no	—
183	#440	—	178	4	no	—
184	C₄×Q₁₆	—	178	2	no	—
185	C₄×D₈	—	178	2	no	—
186	C₂×#108	—	89	1	no	—
187	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
188	#428	—	89	4	no	—
189	#428	—	178	4	no	—
190	#427	—	178	2	no	—
191	#427	C₂×D₈₉	1	1	yes	characteristic
192	C₂×#107	—	89	2	no	—
193	C₂×#107	—	89	2	no	—
194	C₂×#107	—	178	2	no	—
195	C₂×#104	—	89	2	no	—
196	#417	—	89	4	no	—
197	#405	C₂×D₈₉	1	2	yes	—
198	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
199	#384	—	89	2	no	—
200	#343	—	89	4	no	—
201	#339	C₂×D₈₉	1	2	yes	—
202	C₂×#41	—	89	2	no	—
203	#135	—	89	4	no	—
204	#135	—	89	4	no	—
205	C₂×Q₁₆	—	89	1	no	—
206	C₂×SD₁₆	—	89	2	no	—
207	C₂×D₈	—	89	1	no	—
208	#131	—	89	8	no	—
209	#131	—	178	4	no	—
210	C₂×M₁₆	—	89	2	no	—
211	C₂×M₁₆	—	178	2	no	—
212	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
213	C₂²×C₈	—	178	2	no	—
214	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
216	C₄×Q₈	—	178	4	no	—
217	C₄×D₄	—	356	4	no	—
218	C₄×D₄	—	178	4	no	—
219	#117	—	178	4	no	—
220	#117	C₂²×D₈₉	1	2	yes	—
221	C₂×C₄⋊C₄	—	89	4	no	—
222	C₂×C₄⋊C₄	—	356	2	no	—
223	C₂×C₄⋊C₄	—	178	4	no	—
224	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
225	#108	—	2	2	no	—
226	#108	—	89	4	no	—
227	#107	—	178	2	no	—
228	#107	—	178	2	no	—
229	#107	—	178	2	no	—
230	#107	—	178	2	no	—
231	#107	—	178	4	no	—
232	#107	D₃₅₆	1	1	yes	characteristic
233	#107	D₃₅₆	1	1	yes	characteristic
234	#106	—	178	4	no	—
235	#106	D₃₅₆	1	2	yes	—
236	#104	—	89	8	no	—
237	#103	—	178	4	no	—
238	#102	—	178	4	no	—
239	#97	—	2	2	no	—
240	C₄×C₈	—	2	2	no	—
241	C₄×C₈	C₄×D₈₉	1	4	yes	complemented
242	#41	—	89	4	no	—
243	#41	—	89	4	no	—
244	#41	—	178	2	no	—
245	#41	—	178	2	no	—
246	C₂×Q₈	—	89	2	no	—
247	C₂×D₄	—	89	2	no	—
248	C₂×D₄	—	178	2	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	356	4	no	—
251	C₂²×C₄	—	178	4	no	—
252	C₂²×C₄	—	178	2	no	—
253	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
254	Q₁₆	—	89	4	no	—
255	SD₁₆	—	89	8	no	—
256	D₈	—	89	4	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	89	4	no	—
260	M₁₆	—	178	2	no	—
261	M₁₆	—	178	8	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	178	2	no	—
264	C₂×C₈	—	178	8	no	—
265	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
266	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
267	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
270	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
271	C₂×C₈	Q₈×D₈₉	1	2	yes	—
272	C₂×C₈	#13083574	1	2	yes	—
273	C₄⋊C₄	—	356	4	no	—
274	C₄⋊C₄	—	178	4	no	—
275	C₄⋊C₄	—	178	4	no	—
276	C₄⋊C₄	—	712	4	no	—
277	C₄⋊C₄	C₂×D₃₅₆	1	2	yes	—
278	C₄⋊C₄	C₂×D₃₅₆	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	356	4	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	356	4	no	—
284	C₄²	C₂×C₄×D₈₉	1	2	yes	—
285	C₂³	—	178	2	no	—
286	C₂³	#15668058	1	1	yes	characteristic
287	Q₈	—	89	4	no	—
288	Q₈	—	178	2	no	—
289	D₄	—	89	4	no	—
290	D₄	—	178	4	no	—
291	D₄	—	178	4	no	—
292	D₄	—	178	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	356	4	no	—
299	C₂×C₄	—	1424	4	no	—
300	C₂×C₄	—	178	2	no	—
301	C₂×C₄	—	178	4	no	—
302	C₂×C₄	—	178	4	no	—
303	C₂×C₄	—	712	4	no	—
304	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
307	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
309	C₂×C₄	#15668059	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	178	4	no	—
312	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
313	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
314	C₈	C₄×D₃₅₆	1	2	yes	—
315	C₈	C₄×D₃₅₆	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	356	2	no	—
319	C₂²	—	356	2	no	—
320	C₂²	—	178	4	no	—
321	C₂²	#19152078	1	1	yes	characteristic
322	C₂²	D₄×D₃₅₆	1	1	yes	characteristic
323	C₂²	#19152308	1	1	yes	characteristic
324	C₄	—	2	4	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	178	4	no	—
328	C₄	—	712	4	no	—
329	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
330	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
331	C₄	#19152253	1	1	yes	characteristic
332	C₄	Q₈×D₃₅₆	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	356	4	no	—
336	C₂	C₄⋊C₄×D₃₅₆	1	1	yes	characteristic
337	C₂	#26032052	1	1	yes	characteristic
338	C₂	#26032057	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1772	C₂
socle	C₂²×C₈₉	D₄²
prime radical	C₂×#15668034	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1772	D₈₉
hypocenter	C₈₉	#22692
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
3. #26028849	C₂	semidirect
4. #26021037	C₂	semidirect
5. #26020479	C₂	semidirect
6. #26020412	C₂	semidirect
7. #26020313	C₂	semidirect
8. C₈₉×#1772	C₂	semidirect
12. #19151716	C₄	semidirect
39. #15668018	C₂×C₄	semidirect
50. #15667948	C₂×C₄	semidirect
57. #15667940	C₂×C₄	semidirect
71. C₄×C₈×C₈₉	C₂×C₄	semidirect
82. Dic₃₅₆	C₄⋊C₄	semidirect
83. #13083563	C₄⋊C₄	semidirect
84. D₇₁₂	C₄⋊C₄	semidirect
177. #1772	D₈₉	semidirect
178. C₈₉	#22692	semidirect
241. C₄×C₈	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151716	2
#26029129	2
#26028849	2
#26021037	2
#26020479	2
#26020412	2
#26020313	2
C₈₉×#1772	2
#22692	89