Subgroups — #36471374

Order 22784.

574 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151839	C₂	1	1	yes	maximal, characteristic
2	C₂²×#15667946	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151403	C₂	1	2	yes	maximal, complemented
4	#26018344	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
7	C₂×#15667991	—	2	1	no	—
8	C₂²×C₄⋊Dic₈₉	—	2	1	no	—
9	#19151839	—	2	2	no	—
10	#19151839	C₄	1	4	yes	complemented
11	C₂×#15667946	—	2	4	no	—
12	C₂×#15667946	—	2	1	no	—
13	C₂×#15667946	C₂²	1	1	yes	characteristic
14	C₂×#15667946	C₂²	1	2	yes	—
15	C₂×#15667937	—	2	1	no	—
16	#19151403	—	2	4	no	—
17	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
19	C₂²⋊C₁₆×C₈₉	—	2	2	no	—
20	C₂²×Dic₁₇₈	—	2	1	no	—
21	C₂³×Dic₈₉	—	4	1	no	—
22	C₂×Dic₃₅₆	—	2	2	no	—
23	C₂×Dic₃₅₆	—	2	1	no	—
24	C₂×Dic₃₅₆	—	2	2	no	—
25	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	complemented
26	#15667991	—	2	4	no	—
27	#15667991	—	4	2	no	—
28	C₂×C₄⋊Dic₈₉	—	2	4	no	—
29	C₂×C₄⋊Dic₈₉	—	2	2	no	—
30	C₂×C₄⋊Dic₈₉	—	2	1	no	—
31	C₂×C₄⋊Dic₈₉	—	4	2	no	—
32	C₂×C₄⋊Dic₈₉	—	4	1	no	—
33	C₂×#13083553	—	4	1	no	—
34	C₂×C₂²⋊Dic₈₉	—	4	1	no	—
35	#15667946	—	2	4	no	—
36	#15667946	—	2	2	no	—
37	#15667946	—	2	1	no	—
38	#15667946	C₂×C₄	1	2	yes	complemented
39	#15667946	D₄	1	4	yes	—
40	#15667937	—	2	4	no	—
41	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	—	2	2	no	—
43	C₂²×C₈×C₈₉	—	2	2	no	—
44	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	D₄	1	2	yes	—
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₈₉	—	2	2	no	—
49	C₂×C₁₆×C₈₉	—	2	2	no	—
50	C₂×C₁₆×C₈₉	—	2	2	no	—
51	C₂×Dic₁₇₈	—	2	2	no	—
52	C₂×Dic₁₇₈	—	4	1	no	—
53	C₂×Dic₁₇₈	—	4	1	no	—
54	C₂×Dic₁₇₈	—	4	1	no	—
55	C₂×Dic₁₇₈	—	4	1	no	—
56	C₂×Dic₁₇₈	—	4	1	no	—
57	C₂²×Dic₈₉	—	4	4	no	—
58	C₂²×Dic₈₉	—	4	2	no	—
59	C₂²×Dic₈₉	—	4	1	no	—
60	C₂²×Dic₈₉	—	4	1	no	—
61	C₂²×Dic₈₉	—	8	2	no	—
62	C₂²×Dic₈₉	—	8	2	no	—
63	Dic₃₅₆	—	2	4	no	—
64	Dic₃₅₆	—	4	2	no	—
65	C₄⋊Dic₈₉	—	2	2	no	—
66	C₄⋊Dic₈₉	—	2	4	no	—
67	C₄⋊Dic₈₉	—	4	4	no	—
68	C₄⋊Dic₈₉	—	4	1	no	—
69	C₄⋊Dic₈₉	—	4	2	no	—
70	#13083553	—	4	4	no	—
71	C₂²⋊Dic₈₉	—	4	4	no	—
72	C₂⁴×C₈₉	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	—	2	1	no	—
74	C₂²×C₄×C₈₉	—	2	1	no	—
75	C₂²×C₄×C₈₉	—	2	1	no	—
76	C₂²×C₄×C₈₉	—	2	1	no	—
77	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₉	—	2	1	no	—
82	C₂×C₈×C₈₉	—	2	1	no	—
83	C₂×C₈×C₈₉	—	2	4	no	—
84	C₂×C₈×C₈₉	—	2	4	no	—
85	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	#41	1	2	yes	—
90	C₁₆×C₈₉	—	4	2	no	—
91	C₂³×C₈₉	—	2	1	no	—
92	C₂³×C₈₉	—	2	1	no	—
93	C₂³×C₈₉	—	2	1	no	—
94	C₂³×C₈₉	—	2	1	no	—
95	C₂³×C₈₉	#102	1	2	yes	—
96	C₂³×C₈₉	SD₃₂	1	2	yes	—
97	C₂³×C₈₉	Q₃₂	1	2	yes	—
98	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₈₉	—	2	1	no	—
100	C₂×C₄×C₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	2	2	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	#102	1	2	yes	—
106	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
112	C₂×Dic₈₉	—	4	2	no	—
113	C₂×Dic₈₉	—	4	2	no	—
114	C₂×Dic₈₉	—	4	4	no	—
115	C₂×Dic₈₉	—	8	8	no	—
116	C₂×Dic₈₉	—	8	1	no	—
117	C₂×Dic₈₉	—	8	1	no	—
118	C₂×Dic₈₉	—	8	2	no	—
119	C₂×Dic₈₉	—	8	1	no	—
120	C₂×Dic₈₉	—	8	2	no	—
121	C₂×Dic₈₉	—	8	1	no	—
122	Dic₁₇₈	—	4	2	no	—
123	Dic₁₇₈	—	4	2	no	—
124	Dic₁₇₈	—	8	1	no	—
125	Dic₁₇₈	—	8	1	no	—
126	C₈×C₈₉	—	2	2	no	—
127	C₈×C₈₉	C₄×D₄	1	2	yes	—
128	C₈×C₈₉	#121	1	2	yes	—
129	C₂²×C₈₉	—	2	2	no	—
130	C₂²×C₈₉	—	2	2	no	—
131	C₂²×C₈₉	—	2	2	no	—
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	#358	1	4	yes	complemented
136	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
137	C₂²×C₈₉	#447	1	1	yes	characteristic
138	C₂²×C₈₉	#480	1	1	yes	characteristic
139	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
140	C₂²×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
141	C₂²×C₈₉	#509	1	1	yes	characteristic
142	C₂²×C₈₉	#510	1	1	yes	characteristic
143	C₄×C₈₉	—	2	2	no	—
144	C₄×C₈₉	#386	1	1	yes	characteristic
145	C₄×C₈₉	#417	1	1	yes	characteristic
146	C₄×C₈₉	#448	1	1	yes	characteristic
147	C₄×C₈₉	#481	1	1	yes	characteristic
148	Dic₈₉	—	8	2	no	—
149	Dic₈₉	—	8	2	no	—
150	Dic₈₉	—	8	4	no	—
151	#12993	—	89	1	no	maximal, Sylow
152	C₂×C₈₉	—	2	4	no	—
153	C₂×C₈₉	#1874	1	1	yes	characteristic
154	C₂×C₈₉	C₂×#358	1	1	yes	characteristic
155	C₂×C₈₉	#2139	1	1	yes	characteristic
156	C₂×C₈₉	#2187	1	1	yes	characteristic
157	C₂×C₈₉	#2188	1	1	yes	characteristic
158	C₂×C₈₉	#2235	1	1	yes	characteristic
159	C₂×C₈₉	#2236	1	1	yes	characteristic
160	C₂×#467	—	89	1	no	—
161	C₂²×#107	—	89	1	no	—
162	C₂×#358	—	89	2	no	—
163	C₂×C₂²⋊C₁₆	D₈₉	1	1	yes	characteristic
164	#1380	—	89	2	no	—
165	C₈₉	#12993	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₁₆	—	89	1	no	—
167	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
168	C₂×C₂²⋊Q₈	—	178	1	no	—
169	C₂²×C₄⋊C₄	—	178	1	no	—
170	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
171	#467	—	89	4	no	—
172	#467	—	178	2	no	—
173	C₂×#107	—	89	2	no	—
174	C₂×#107	—	89	1	no	—
175	C₂×#107	—	178	4	no	—
176	C₂×#107	—	178	1	no	—
177	C₂×#103	—	178	1	no	—
178	#358	—	178	4	no	—
179	C₂²⋊C₁₆	—	2	2	no	—
180	C₂²×Q₈	—	178	1	no	—
181	C₂³×C₄	—	356	1	no	—
182	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
183	C₂×Q₁₆	—	89	2	no	—
184	C₂×Q₁₆	—	178	2	no	—
185	C₂×Q₁₆	—	178	1	no	—
186	C₂×Q₁₆	—	178	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₄×D₈₉	1	2	yes	—
190	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
192	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
193	C₂²⋊Q₈	—	356	2	no	—
194	C₂²⋊Q₈	—	178	4	no	—
195	C₂×C₄⋊C₄	—	356	2	no	—
196	C₂×C₄⋊C₄	—	356	1	no	—
197	C₂×C₄⋊C₄	—	356	1	no	—
198	C₂×C₄⋊C₄	—	178	4	no	—
199	C₂×C₄⋊C₄	—	178	2	no	—
200	C₂×C₄⋊C₄	—	178	1	no	—
201	C₂×C₂²⋊C₄	—	356	1	no	—
202	C₂×C₁₆	—	2	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	2	2	no	—
205	#107	—	89	2	no	—
206	#107	—	89	4	no	—
207	#107	—	178	4	no	—
208	#107	—	178	2	no	—
209	#107	—	178	1	no	—
210	#103	—	178	4	no	—
211	C₂⁴	D₇₁₂	1	1	yes	characteristic
212	C₂×Q₈	—	356	1	no	—
213	C₂×Q₈	—	356	1	no	—
214	C₂×Q₈	—	356	1	no	—
215	C₂×Q₈	—	356	1	no	—
216	C₂×Q₈	—	356	1	no	—
217	C₂×Q₈	—	178	2	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	356	4	no	—
223	C₂²×C₄	—	356	1	no	—
224	C₂²×C₄	—	356	1	no	—
225	C₂²×C₄	—	356	2	no	—
226	C₂²×C₄	—	712	2	no	—
227	C₂²×C₄	—	712	2	no	—
228	C₂²×C₄	#13083551	1	2	yes	—
229	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
230	C₂²×C₄	#13083563	1	2	yes	—
231	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
232	Q₁₆	—	356	2	no	—
233	Q₁₆	—	178	4	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	#13083551	1	2	yes	—
239	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
240	C₂×C₈	D₄×D₈₉	1	2	yes	—
241	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
242	C₂×C₈	#13083573	1	2	yes	—
243	C₄⋊C₄	—	356	4	no	—
244	C₄⋊C₄	—	356	1	no	—
245	C₄⋊C₄	—	356	2	no	—
246	C₄⋊C₄	—	356	4	no	—
247	C₄⋊C₄	—	178	4	no	—
248	C₄⋊C₄	—	178	2	no	—
249	C₂²⋊C₄	—	356	4	no	—
250	C₁₆	—	4	2	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	#15667934	1	2	yes	—
256	C₂³	#15667957	1	2	yes	—
257	C₂³	Dic₇₁₂	1	2	yes	—
258	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
259	Q₈	—	356	2	no	—
260	Q₈	—	356	2	no	—
261	Q₈	—	712	1	no	—
262	Q₈	—	712	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	356	4	no	—
270	C₂×C₄	—	356	2	no	—
271	C₂×C₄	—	356	2	no	—
272	C₂×C₄	—	712	8	no	—
273	C₂×C₄	—	712	1	no	—
274	C₂×C₄	—	712	1	no	—
275	C₂×C₄	—	712	2	no	—
276	C₂×C₄	—	712	1	no	—
277	C₂×C₄	—	712	2	no	—
278	C₂×C₄	—	712	1	no	—
279	C₂×C₄	#15667934	1	2	yes	—
280	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
281	C₂×C₄	#15667982	1	1	yes	characteristic
282	C₂×C₄	#15667996	1	1	yes	characteristic
283	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
284	C₂×C₄	#15668040	1	1	yes	characteristic
285	C₂×C₄	#15668045	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	#15667972	1	2	yes	—
288	C₈	#15667984	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#19151403	1	4	yes	complemented
296	C₂²	C₂×#15667934	1	1	yes	characteristic
297	C₂²	#19151734	1	1	yes	characteristic
298	C₂²	#19151912	1	1	yes	characteristic
299	C₂²	C₂×#15667957	1	1	yes	characteristic
300	C₂²	C₂×Dic₇₁₂	1	1	yes	characteristic
301	C₂²	#19152036	1	1	yes	characteristic
302	C₂²	#19152041	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	712	4	no	—
305	C₄	—	712	2	no	—
306	C₄	—	712	2	no	—
307	C₄	#19151475	1	1	yes	characteristic
308	C₄	#19151600	1	1	yes	characteristic
309	C₄	#19151739	1	1	yes	characteristic
310	C₄	#19151917	1	1	yes	characteristic
311	C₂	—	2	4	no	—
312	C₂	#26020567	1	1	yes	characteristic
313	C₂	C₂×#19151403	1	1	yes	characteristic
314	C₂	#26021965	1	1	yes	characteristic
315	C₂	#26022183	1	1	yes	characteristic
316	C₂	#26022188	1	1	yes	characteristic
317	C₂	#26022450	1	1	yes	characteristic
318	C₂	#26022455	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₂²⋊C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂⁴×C₈₉	D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₈₉
hypocenter	C₈₉	#12993
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#19151403	C₂	semidirect
4. #26018344	C₂	semidirect
10. #19151839	C₄	semidirect
25. C₂×Dic₃₅₆	C₂×C₄	semidirect
38. #15667946	C₂×C₄	semidirect
135. C₂²×C₈₉	#358	semidirect
165. C₈₉	#12993	semidirect
295. C₂²	#19151403	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151839	2
C₂²×#15667946	2
C₂×#19151403	2
#26018344	2
C₂×C₂²⋊C₁₆×C₈₉	2
#12993	89