Subgroups — #36572904

Order 22784.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 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	#26029137	C₂	1	2	yes	maximal
3	#26028844	C₂	1	2	yes	maximal, complemented
4	#26021032	C₂	1	2	yes	maximal, complemented
5	#26020479	C₂	1	4	yes	maximal, complemented
6	#26020409	C₂	1	1	yes	maximal, characteristic, complemented
7	#26020308	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	#19151711	—	2	2	no	—
14	#19151706	—	2	2	no	—
15	C₄×#13083563	—	2	4	no	—
16	C₂×#15667948	C₂²	1	1	yes	characteristic
17	#19151641	C₂²	1	4	yes	—
18	#19151639	—	2	4	no	—
19	#19151633	—	2	2	no	—
20	C₂×#15667944	C₂²	1	2	yes	—
21	C₂×#15667944	C₂²	1	2	yes	—
22	#106×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	#15667945	—	2	4	no	—
52	#15667944	—	2	2	no	—
53	#15667944	—	2	2	no	—
54	#15667944	—	2	2	no	—
55	#15667944	—	2	2	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	2	yes	—
67	#106×C₈₉	D₄	1	2	yes	—
68	#97×C₈₉	—	2	2	no	—
69	C₄×C₈×C₈₉	—	2	2	no	—
70	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	complemented
71	#13083574	—	2	2	no	—
72	#13083574	—	2	2	no	—
73	#13083574	C₂²×C₄	1	4	yes	—
74	#13083574	C₂×Q₈	1	4	yes	—
75	C₂×Dic₁₇₈	C₂×D₄	1	2	yes	—
76	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
77	C₂×C₂²⋊D₈₉	—	2	2	no	—
78	C₂²×Dic₈₉	—	2	4	no	—
79	C₂×C₄×D₈₉	—	2	2	no	—
80	C₂×C₄×D₈₉	—	4	4	no	—
81	Dic₃₅₆	C₄⋊C₄	1	4	yes	complemented
82	#13083563	C₄⋊C₄	1	8	yes	complemented
83	D₇₁₂	C₄⋊C₄	1	4	yes	complemented
84	#13083558	—	2	2	no	—
85	#13083558	C₂²×C₄	1	4	yes	—
86	#13083557	—	2	8	no	—
87	C₂×#10623494	—	2	2	no	—
88	C₈×D₈₉	—	2	8	no	—
89	C₄⋊Dic₈₉	—	2	4	no	—
90	C₄⋊Dic₈₉	—	2	4	no	—
91	C₄⋊Dic₈₉	—	8	4	no	—
92	#13083553	—	4	4	no	—
93	#13083551	—	4	4	no	—
94	C₄×Dic₈₉	—	4	4	no	—
95	C₂²×C₄×C₈₉	—	2	1	no	—
96	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
97	M₁₆×C₈₉	—	2	2	no	—
98	M₁₆×C₈₉	—	2	2	no	—
99	C₂×C₈×C₈₉	—	2	2	no	—
100	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₈₉	#41	1	2	yes	—
108	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
109	C₂²⋊C₄×C₈₉	—	2	2	no	—
110	C₄²×C₈₉	—	2	2	no	—
111	C₄²×C₈₉	—	2	1	no	—
112	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
113	C₂³×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×D₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	4	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	4	no	—
118	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂²⋊D₈₉	—	2	4	no	—
125	C₂²⋊D₈₉	—	2	4	no	—
126	C₂×Dic₈₉	—	2	2	no	—
127	C₂×Dic₈₉	—	4	4	no	—
128	C₂×Dic₈₉	—	8	4	no	—
129	D₃₅₆	—	2	2	no	—
130	D₃₅₆	C₂×C₄⋊C₄	1	4	yes	—
131	C₄×D₈₉	—	2	4	no	—
132	C₄×D₈₉	—	2	4	no	—
133	C₄×D₈₉	—	16	4	no	—
134	Dic₁₇₈	—	2	2	no	—
135	Dic₁₇₈	C₂×C₄⋊C₄	1	4	yes	—
136	C₈×C₈₉	—	4	2	no	—
137	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
138	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
139	C₈×C₈₉	C₄×D₄	1	2	yes	—
140	C₈×C₈₉	C₄×D₄	1	2	yes	—
141	#10623494	—	2	4	no	—
142	C₂²×C₈₉	—	2	2	no	—
143	C₂²×C₈₉	#520	1	1	yes	characteristic
144	C₂²×C₈₉	D₄²	1	1	yes	characteristic
145	C₂²×C₈₉	#554	1	1	yes	characteristic
146	C₂×D₈₉	—	2	4	no	—
147	C₂×D₈₉	—	4	4	no	—
148	C₄×C₈₉	—	2	4	no	—
149	C₄×C₈₉	—	4	4	no	—
150	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₈₉	#547	1	1	yes	characteristic
153	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
154	Dic₈₉	—	2	4	no	—
155	Dic₈₉	—	8	4	no	—
156	#22689	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	—	2	1	no	—
158	C₂×C₈₉	—	2	1	no	—
159	C₂×C₈₉	D₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₈₉	#3291	1	2	yes	—
161	D₈₉	—	4	4	no	—
162	C₂×#443	—	89	1	no	—
163	#2952	—	89	2	no	—
164	#2912	—	89	2	no	—
165	#1951	—	89	2	no	—
166	#1859	—	89	4	no	—
167	#1844	—	89	1	no	—
168	#1815	—	89	2	no	—
169	#1772	D₈₉	1	1	yes	characteristic, complemented
170	C₈₉	#22689	1	1	yes	characteristic, complemented, Sylow
171	C₂×#135	—	89	1	no	—
172	C₂×#131	—	89	2	no	—
173	#520	—	89	4	no	—
174	#443	—	89	8	no	—
175	#442	—	178	2	no	—
176	#441	—	178	2	no	—
177	C₄×SD₁₆	—	178	4	no	—
178	C₂×#108	—	89	1	no	—
179	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
180	#428	—	89	4	no	—
181	#428	—	178	4	no	—
182	#427	—	178	2	no	—
183	#427	C₂×D₈₉	1	1	yes	characteristic
184	C₂×#106	—	89	2	no	—
185	C₂×#106	—	89	2	no	—
186	C₂×#106	—	178	2	no	—
187	C₂×#104	—	89	2	no	—
188	#417	—	89	4	no	—
189	#405	C₂×D₈₉	1	2	yes	—
190	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
191	#384	—	89	2	no	—
192	#343	—	89	4	no	—
193	#339	C₂×D₈₉	1	2	yes	—
194	C₂×#41	—	89	2	no	—
195	#135	—	89	4	no	—
196	#135	—	89	4	no	—
197	C₂×Q₁₆	—	89	1	no	—
198	C₂×SD₁₆	—	89	2	no	—
199	C₂×D₈	—	89	1	no	—
200	#131	—	89	8	no	—
201	#131	—	178	4	no	—
202	C₂×M₁₆	—	89	2	no	—
203	C₂×M₁₆	—	178	2	no	—
204	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
205	C₂²×C₈	—	178	2	no	—
206	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
208	C₄×Q₈	—	178	4	no	—
209	C₄×D₄	—	356	4	no	—
210	C₄×D₄	—	178	4	no	—
211	#117	—	178	4	no	—
212	#117	C₂²×D₈₉	1	2	yes	—
213	C₂×C₄⋊C₄	—	89	4	no	—
214	C₂×C₄⋊C₄	—	356	2	no	—
215	C₂×C₄⋊C₄	—	178	4	no	—
216	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
217	#108	—	2	2	no	—
218	#108	—	89	4	no	—
219	#107	—	178	4	no	—
220	#107	D₃₅₆	1	2	yes	—
221	#106	—	178	2	no	—
222	#106	—	178	2	no	—
223	#106	—	178	2	no	—
224	#106	—	178	2	no	—
225	#106	—	178	4	no	—
226	#106	D₃₅₆	1	2	yes	—
227	#104	—	89	8	no	—
228	#103	—	178	4	no	—
229	#102	—	178	4	no	—
230	#97	—	2	2	no	—
231	C₄×C₈	—	2	2	no	—
232	C₄×C₈	C₄×D₈₉	1	4	yes	complemented
233	#41	—	89	4	no	—
234	#41	—	89	4	no	—
235	#41	—	178	2	no	—
236	#41	—	178	2	no	—
237	C₂×Q₈	—	89	2	no	—
238	C₂×D₄	—	89	2	no	—
239	C₂×D₄	—	178	2	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	356	4	no	—
242	C₂²×C₄	—	178	4	no	—
243	C₂²×C₄	—	178	2	no	—
244	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
245	Q₁₆	—	89	4	no	—
246	SD₁₆	—	89	8	no	—
247	D₈	—	89	4	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	89	4	no	—
251	M₁₆	—	178	2	no	—
252	M₁₆	—	178	8	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	178	2	no	—
255	C₂×C₈	—	178	8	no	—
256	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
258	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
259	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
260	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
261	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
262	C₂×C₈	Q₈×D₈₉	1	2	yes	—
263	C₂×C₈	#13083574	1	2	yes	—
264	C₄⋊C₄	—	356	4	no	—
265	C₄⋊C₄	—	178	4	no	—
266	C₄⋊C₄	—	178	4	no	—
267	C₄⋊C₄	—	712	4	no	—
268	C₄⋊C₄	C₂×D₃₅₆	1	4	yes	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	356	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	356	4	no	—
274	C₄²	C₂×C₄×D₈₉	1	2	yes	—
275	C₂³	—	178	2	no	—
276	C₂³	#15668058	1	1	yes	characteristic
277	Q₈	—	89	4	no	—
278	Q₈	—	178	2	no	—
279	D₄	—	89	4	no	—
280	D₄	—	178	4	no	—
281	D₄	—	178	4	no	—
282	D₄	—	178	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	356	4	no	—
287	C₂×C₄	—	1424	4	no	—
288	C₂×C₄	—	178	2	no	—
289	C₂×C₄	—	178	4	no	—
290	C₂×C₄	—	178	4	no	—
291	C₂×C₄	—	712	4	no	—
292	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
295	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
296	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
297	C₂×C₄	#15668059	1	1	yes	characteristic
298	C₈	—	4	2	no	—
299	C₈	—	178	4	no	—
300	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
301	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
302	C₈	C₄×D₃₅₆	1	2	yes	—
303	C₈	C₄×D₃₅₆	1	2	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	356	4	no	—
306	C₂²	—	178	4	no	—
307	C₂²	#19152078	1	1	yes	characteristic
308	C₂²	D₄×D₃₅₆	1	1	yes	characteristic
309	C₂²	#19152308	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	4	4	no	—
312	C₄	—	178	4	no	—
313	C₄	—	712	4	no	—
314	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
315	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
316	C₄	#19152253	1	1	yes	characteristic
317	C₄	Q₈×D₃₅₆	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	356	4	no	—
321	C₂	C₄⋊C₄×D₃₅₆	1	1	yes	characteristic
322	C₂	#26032046	1	2	yes	—

	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	#22689
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. #26028844	C₂	semidirect
4. #26021032	C₂	semidirect
5. #26020479	C₂	semidirect
6. #26020409	C₂	semidirect
7. #26020308	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
70. C₄×C₈×C₈₉	C₂×C₄	semidirect
81. Dic₃₅₆	C₄⋊C₄	semidirect
82. #13083563	C₄⋊C₄	semidirect
83. D₇₁₂	C₄⋊C₄	semidirect
169. #1772	D₈₉	semidirect
170. C₈₉	#22689	semidirect
232. 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
#26029137	2
#26028844	2
#26021032	2
#26020479	2
#26020409	2
#26020308	2
C₈₉×#1772	2
#22689	89