Subgroups — #36463054

Order 22784.

730 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667947	C₂	1	2	yes	maximal
2	C₂×#19151622	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151362	C₂	1	2	yes	maximal
4	C₂×#19151353	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	2	yes	—
7	C₂³×#10623494	C₂²	1	2	yes	—
8	C₂×#15667947	—	2	4	no	—
9	C₂×#15667947	—	2	4	no	—
10	C₂×#15667947	—	2	4	no	—
11	#19151622	—	2	4	no	—
12	C₂×C₄⋊#10623494	—	2	2	no	—
13	#19151362	—	2	4	no	—
14	#19151353	C₄	1	2	yes	—
15	#19151353	C₄	1	2	yes	—
16	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	#336×C₈₉	—	2	2	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	2	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	D₄	1	4	yes	—
26	C₂²×#10623494	—	2	2	no	—
27	C₂²×#10623494	—	2	2	no	—
28	C₂²×#10623494	—	2	2	no	—
29	C₂²×#10623494	—	2	2	no	—
30	C₂²×#10623494	—	2	2	no	—
31	C₂²×#10623494	C₂×C₄	1	4	yes	—
32	C₂²×#10623494	D₄	1	4	yes	—
33	C₂²×#10623494	Q₈	1	4	yes	—
34	#15667947	—	2	16	no	—
35	C₄⋊#10623494	—	2	8	no	—
36	C₂³×C₄×C₈₉	Q₈	1	2	yes	—
37	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₈₉	—	2	1	no	—
39	C₂×M₁₆×C₈₉	—	2	1	no	—
40	C₂×M₁₆×C₈₉	—	2	2	no	—
41	C₂×M₁₆×C₈₉	—	2	2	no	—
42	C₂×M₁₆×C₈₉	—	2	2	no	—
43	C₂×M₁₆×C₈₉	—	2	2	no	—
44	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
45	C₂×M₁₆×C₈₉	Q₈	1	2	yes	—
46	C₂²×C₈×C₈₉	—	2	1	no	—
47	C₂²×C₈×C₈₉	—	2	1	no	—
48	C₂×C₄²×C₈₉	—	2	2	no	—
49	C₂×C₄²×C₈₉	—	2	2	no	—
50	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	D₄	1	2	yes	—
53	#13083558	—	2	4	no	—
54	#13083558	—	2	4	no	—
55	#13083558	—	4	2	no	—
56	#13083558	—	4	2	no	—
57	C₂×#10623494	—	2	4	no	—
58	C₂×#10623494	—	2	4	no	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	—	2	4	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	—	2	4	no	—
63	C₂×#10623494	—	2	4	no	—
64	C₂×#10623494	—	4	2	no	—
65	C₂×#10623494	—	4	2	no	—
66	C₂×#10623494	C₄⋊C₄	1	8	yes	—
67	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	—	2	4	no	—
69	C₂²×C₄×C₈₉	—	2	4	no	—
70	C₂²×C₄×C₈₉	—	4	2	no	—
71	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	#41	1	2	yes	—
82	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
83	M₁₆×C₈₉	—	2	4	no	—
84	M₁₆×C₈₉	—	2	4	no	—
85	M₁₆×C₈₉	—	4	2	no	—
86	M₁₆×C₈₉	—	4	2	no	—
87	C₂×C₈×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	4	1	no	—
90	C₂×C₈×C₈₉	—	4	1	no	—
91	C₂×C₈×C₈₉	—	4	1	no	—
92	C₂×C₈×C₈₉	—	4	1	no	—
93	C₄²×C₈₉	—	2	2	no	—
94	C₄²×C₈₉	—	2	2	no	—
95	C₄²×C₈₉	—	2	2	no	—
96	C₄²×C₈₉	—	2	2	no	—
97	C₂³×C₈₉	—	2	2	no	—
98	C₂³×C₈₉	—	2	2	no	—
99	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₈₉	#125	1	1	yes	characteristic
102	C₂³×C₈₉	C₄⋊Q₈	1	2	yes	—
103	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	2	2	no	—
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	4	8	no	—
110	C₂×C₄×C₈₉	#108	1	4	yes	complemented
111	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#120	1	2	yes	—
117	C₂×C₄×C₈₉	#121	1	2	yes	—
118	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#123	1	2	yes	—
125	C₂×C₄×C₈₉	#125	1	2	yes	—
126	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
129	C₈×C₈₉	—	4	2	no	—
130	C₈×C₈₉	—	4	2	no	—
131	#10623494	—	2	8	no	—
132	#10623494	—	4	4	no	—
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	—	2	4	no	—
136	C₂²×C₈₉	—	2	2	no	—
137	C₂²×C₈₉	—	2	2	no	—
138	C₂²×C₈₉	#384	1	1	yes	characteristic
139	C₂²×C₈₉	#385	1	1	yes	characteristic
140	C₂²×C₈₉	#389	1	1	yes	characteristic
141	C₂²×C₈₉	#395	1	2	yes	—
142	C₂²×C₈₉	#398	1	2	yes	—
143	C₂²×C₈₉	C₂×#108	1	2	yes	—
144	C₂²×C₈₉	#430	1	2	yes	—
145	C₄×C₈₉	—	2	4	no	—
146	C₄×C₈₉	—	4	2	no	—
147	C₄×C₈₉	#385	1	2	yes	—
148	C₄×C₈₉	#389	1	2	yes	—
149	C₄×C₈₉	#392	1	1	yes	characteristic
150	C₄×C₈₉	#395	1	1	yes	characteristic
151	C₄×C₈₉	#398	1	1	yes	characteristic
152	C₄×C₈₉	#399	1	1	yes	characteristic
153	#11326	—	89	1	no	maximal, Sylow
154	C₂×C₈₉	—	2	2	no	—
155	C₂×C₈₉	—	2	2	no	—
156	C₂×C₈₉	#1444	1	1	yes	characteristic
157	C₂×C₈₉	#1848	1	1	yes	characteristic
158	C₂×C₈₉	#1849	1	1	yes	characteristic
159	C₂×C₈₉	#1855	1	2	yes	—
160	C₂×C₈₉	#1950	1	2	yes	—
161	C₂²×#108	—	89	2	no	—
162	C₂×#423	—	89	1	no	—
163	C₂×#341	—	89	2	no	—
164	#1728	D₈₉	1	1	yes	characteristic
165	C₂×#336	—	89	1	no	—
166	C₈₉	#11326	1	1	yes	characteristic, complemented, Sylow
167	C₂²×M₁₆	—	89	2	no	—
168	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
169	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
170	C₂³×C₈	—	89	2	no	—
171	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
172	C₂×#108	—	178	4	no	—
173	C₂×#108	—	178	4	no	—
174	C₂×#108	—	178	4	no	—
175	#423	—	178	4	no	—
176	C₂×C₄⋊C₈	—	178	2	no	—
177	#341	—	178	4	no	—
178	#336	—	2	2	no	—
179	#336	—	89	2	no	—
180	#336	—	89	2	no	—
181	C₂³×C₄	Dic₁₇₈	1	2	yes	—
182	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	89	4	no	—
190	C₂×M₁₆	—	178	2	no	—
191	C₂×M₁₆	—	178	2	no	—
192	C₂×M₁₆	—	178	2	no	—
193	C₂×M₁₆	—	178	2	no	—
194	C₂×M₁₆	—	178	2	no	—
195	C₂×M₁₆	Dic₁₇₈	1	2	yes	—
196	C₂×M₁₆	D₃₅₆	1	2	yes	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	89	4	no	—
200	C₂²×C₈	—	89	4	no	—
201	C₂²×C₈	—	89	4	no	—
202	C₂²×C₈	—	178	2	no	—
203	C₂²×C₈	—	178	2	no	—
204	C₂²×C₈	—	178	2	no	—
205	C₂²×C₈	—	178	2	no	—
206	C₂²×C₈	—	178	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
212	#108	—	178	16	no	—
213	C₄⋊C₈	—	178	8	no	—
214	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	#13083553	1	2	yes	—
219	C₂²×C₄	#13083553	1	2	yes	—
220	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
221	C₂²×C₄	D₄×D₈₉	1	2	yes	—
222	C₂²×C₄	D₄×D₈₉	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
225	C₂²×C₄	Q₈×D₈₉	1	2	yes	—
226	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
227	C₂²×C₄	#13083573	1	2	yes	—
228	C₂²×C₄	#13083574	1	1	yes	characteristic
229	C₂²×C₄	#13083574	1	1	yes	characteristic
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	356	2	no	—
235	M₁₆	—	356	2	no	—
236	M₁₆	—	178	4	no	—
237	M₁₆	—	178	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	89	8	no	—
245	C₂×C₈	—	356	2	no	—
246	C₂×C₈	—	356	2	no	—
247	C₂×C₈	—	178	4	no	—
248	C₂×C₈	—	178	4	no	—
249	C₂×C₈	—	178	4	no	—
250	C₂×C₈	—	178	4	no	—
251	C₂×C₈	—	178	4	no	—
252	C₂×C₈	—	178	4	no	—
253	C₂×C₈	—	178	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×#13083553	1	1	yes	characteristic
261	C₂³	#15667989	1	2	yes	—
262	C₂³	#15668004	1	1	yes	characteristic
263	C₂³	#15668011	1	2	yes	—
264	C₂³	#15668012	1	1	yes	characteristic
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	—	2	2	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	#15667947	1	4	yes	complemented
272	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
273	C₂×C₄	#15667972	1	2	yes	—
274	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
275	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
276	C₂×C₄	#15667980	1	1	yes	characteristic
277	C₂×C₄	#15667982	1	2	yes	—
278	C₂×C₄	#15667985	1	2	yes	—
279	C₂×C₄	#15667986	1	1	yes	characteristic
280	C₂×C₄	#15667989	1	2	yes	—
281	C₂×C₄	#15667990	1	2	yes	—
282	C₂×C₄	#15667991	1	1	yes	characteristic
283	C₂×C₄	#15667992	1	2	yes	—
284	C₂×C₄	#15667993	1	1	yes	characteristic
285	C₂×C₄	#15667994	1	2	yes	—
286	C₂×C₄	#15668002	1	2	yes	—
287	C₂×C₄	#15668008	1	1	yes	characteristic
288	C₂×C₄	#15668010	1	1	yes	characteristic
289	C₂×C₄	#15668011	1	2	yes	—
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	356	4	no	—
293	C₈	—	178	8	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	#19151467	1	1	yes	characteristic
300	C₂²	#19151470	1	1	yes	characteristic
301	C₂²	#19151496	1	1	yes	characteristic
302	C₂²	#19151519	1	2	yes	—
303	C₂²	#19151532	1	2	yes	—
304	C₂²	C₂×#15667947	1	2	yes	—
305	C₂²	#19151646	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	#19151472	1	2	yes	—
309	C₄	#19151495	1	2	yes	—
310	C₄	#19151507	1	1	yes	characteristic
311	C₄	#19151518	1	1	yes	characteristic
312	C₄	#19151528	1	1	yes	characteristic
313	C₄	#19151534	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	#26018516	1	1	yes	characteristic
317	C₂	#26020423	1	1	yes	characteristic
318	C₂	#26020426	1	1	yes	characteristic
319	C₂	#26020456	1	2	yes	—
320	C₂	#26021027	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₈₉×#1728	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	#1728	D₈₉
hypocenter	C₈₉	#11326
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
110. C₂×C₄×C₈₉	#108	semidirect
166. C₈₉	#11326	semidirect
271. C₂×C₄	#15667947	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667947	2
C₂×#19151622	2
C₂×#19151362	2
C₂×#19151353	2
C₈₉×#1728	2
#11326	89