Subgroups — #36441032

Order 22784.

402 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020232	C₂	1	1	yes	maximal, characteristic
2	#26020232	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1728	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10623494	—	2	1	no	—
5	C₂×C₄⋊#10623494	C₄	1	2	yes	—
6	C₂×C₂²⋊#10623494	—	2	1	no	—
7	C₂×C₂²⋊#10623494	C₄	1	2	yes	—
8	#19151353	—	2	1	no	—
9	#19151353	—	2	1	no	—
10	C₂²×M₁₆×C₈₉	—	2	1	no	—
11	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
12	#336×C₈₉	—	2	1	no	—
13	#336×C₈₉	C₄	1	2	yes	—
14	C₂²×#10623494	—	2	2	no	—
15	C₂²×#10623494	—	2	2	no	—
16	C₄⋊#10623494	—	4	2	no	—
17	C₄⋊#10623494	—	4	2	no	—
18	C₂²⋊#10623494	—	4	2	no	—
19	C₂²⋊#10623494	—	4	1	no	—
20	C₂²⋊#10623494	—	4	1	no	—
21	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
24	C₂×M₁₆×C₈₉	—	2	4	no	—
25	C₂×M₁₆×C₈₉	—	4	2	no	—
26	C₂×M₁₆×C₈₉	—	4	1	no	—
27	C₂×M₁₆×C₈₉	—	4	1	no	—
28	C₂²×C₈×C₈₉	—	2	2	no	—
29	C₂×C₄²×C₈₉	—	4	2	no	—
30	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
34	C₂×#10623494	—	4	2	no	—
35	C₂×#10623494	—	4	2	no	—
36	C₂×#10623494	—	4	2	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₂×#10623494	—	4	2	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	—	2	2	no	—
42	C₂²×C₄×C₈₉	—	2	2	no	—
43	C₂²×C₄×C₈₉	—	2	1	no	—
44	C₂²×C₄×C₈₉	—	2	1	no	—
45	C₂²×C₄×C₈₉	—	2	2	no	—
46	C₂²×C₄×C₈₉	—	4	1	no	—
47	C₂²×C₄×C₈₉	—	4	1	no	—
48	C₂²×C₄×C₈₉	—	4	1	no	—
49	C₂²×C₄×C₈₉	—	4	1	no	—
50	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
60	M₁₆×C₈₉	—	4	4	no	—
61	M₁₆×C₈₉	—	4	4	no	—
62	C₂×C₈×C₈₉	—	2	4	no	—
63	C₂×C₈×C₈₉	—	4	2	no	—
64	C₂×C₈×C₈₉	—	4	2	no	—
65	C₄²×C₈₉	—	2	2	no	—
66	C₄²×C₈₉	—	2	2	no	—
67	C₄²×C₈₉	—	4	1	no	—
68	C₄²×C₈₉	—	4	1	no	—
69	C₂³×C₈₉	—	2	1	no	—
70	C₂³×C₈₉	—	2	1	no	—
71	C₂³×C₈₉	—	2	1	no	—
72	C₂³×C₈₉	—	2	1	no	—
73	C₂³×C₈₉	—	2	1	no	—
74	C₂³×C₈₉	—	2	1	no	—
75	C₂³×C₈₉	#95	1	1	yes	characteristic
76	C₂³×C₈₉	#106	1	1	yes	characteristic
77	C₂³×C₈₉	#107	1	1	yes	characteristic
78	C₂×C₄×C₈₉	—	2	1	no	—
79	C₂×C₄×C₈₉	—	2	1	no	—
80	C₂×C₄×C₈₉	—	2	1	no	—
81	C₂×C₄×C₈₉	—	2	1	no	—
82	C₂×C₄×C₈₉	—	2	1	no	—
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	2	no	—
85	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	4	4	no	—
95	C₂×C₄×C₈₉	—	4	2	no	—
96	C₂×C₄×C₈₉	—	4	2	no	—
97	C₂×C₄×C₈₉	#99	1	2	yes	—
98	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#101	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
102	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#104	1	2	yes	—
105	C₂×C₄×C₈₉	#108	1	2	yes	—
106	C₈×C₈₉	—	4	4	no	—
107	#10623494	—	8	2	no	—
108	#10623494	—	8	2	no	—
109	C₂²×C₈₉	—	2	1	no	—
110	C₂²×C₈₉	—	2	1	no	—
111	C₂²×C₈₉	—	2	1	no	—
112	C₂²×C₈₉	—	2	1	no	—
113	C₂²×C₈₉	—	2	1	no	—
114	C₂²×C₈₉	—	2	1	no	—
115	C₂²×C₈₉	—	4	1	no	—
116	C₂²×C₈₉	—	4	1	no	—
117	C₂²×C₈₉	—	4	1	no	—
118	C₂²×C₈₉	—	4	1	no	—
119	C₂²×C₈₉	#338	1	1	yes	characteristic
120	C₂²×C₈₉	#339	1	1	yes	characteristic
121	C₂²×C₈₉	#340	1	1	yes	characteristic
122	C₂²×C₈₉	#341	1	1	yes	characteristic
123	C₂²×C₈₉	#342	1	1	yes	characteristic
124	C₂²×C₈₉	#343	1	1	yes	characteristic
125	C₂²×C₈₉	#344	1	1	yes	characteristic
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	4	1	no	—
132	C₄×C₈₉	—	4	1	no	—
133	C₄×C₈₉	#327	1	2	yes	—
134	C₄×C₈₉	#328	1	2	yes	—
135	#7045	—	89	1	no	maximal, Sylow
136	C₂×C₈₉	—	4	1	no	—
137	C₂×C₈₉	—	4	1	no	—
138	C₂×C₈₉	#1291	1	1	yes	characteristic
139	C₂×C₈₉	#1292	1	1	yes	characteristic
140	C₂×C₈₉	#1293	1	1	yes	characteristic
141	C₂×C₈₉	#1294	1	1	yes	characteristic
142	C₂×C₈₉	#1299	1	1	yes	characteristic
143	C₂×C₈₉	#1301	1	1	yes	characteristic
144	C₂×C₈₉	#1305	1	1	yes	characteristic
145	#1797	—	89	1	no	—
146	#1797	—	89	1	no	—
147	#1728	D₈₉	1	1	yes	characteristic
148	C₈₉	#7045	1	1	yes	characteristic, complemented, Sylow
149	C₂²×M₁₆	—	2	1	no	—
150	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
151	C₂×C₄⋊C₈	—	89	2	no	—
152	C₂×C₄⋊C₈	—	178	1	no	—
153	C₂×C₂²⋊C₈	—	89	2	no	—
154	C₂×C₂²⋊C₈	—	178	1	no	—
155	#336	—	2	1	no	—
156	#336	—	178	1	no	—
157	#336	—	178	1	no	—
158	#336	Dic₈₉	1	2	yes	—
159	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
160	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
161	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	—	4	2	no	—
164	C₂×M₁₆	—	4	1	no	—
165	C₂×M₁₆	—	4	1	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	178	2	no	—
168	C₂²×C₈	—	178	2	no	—
169	C₂×C₄²	—	4	2	no	—
170	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
171	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
172	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
174	C₄⋊C₈	—	356	2	no	—
175	C₄⋊C₈	—	356	2	no	—
176	C₂²⋊C₈	—	356	1	no	—
177	C₂²⋊C₈	—	356	1	no	—
178	C₂²⋊C₈	—	356	2	no	—
179	C₂⁴	#13083553	1	1	yes	characteristic
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
190	C₂²×C₄	#13083551	1	1	yes	characteristic
191	C₂²×C₄	#13083551	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
193	C₂²×C₄	#13083553	1	1	yes	characteristic
194	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
195	C₂²×C₄	#13083559	1	1	yes	characteristic
196	C₂²×C₄	#13083561	1	1	yes	characteristic
197	C₂²×C₄	#13083562	1	1	yes	characteristic
198	C₂²×C₄	#13083564	1	1	yes	characteristic
199	M₁₆	—	4	4	no	—
200	M₁₆	—	4	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	356	2	no	—
205	C₂×C₈	—	356	2	no	—
206	C₂×C₈	—	356	2	no	—
207	C₂×C₈	—	356	2	no	—
208	C₂×C₈	—	356	2	no	—
209	C₂×C₈	—	356	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	#15667919	1	1	yes	characteristic
221	C₂³	#15667943	1	1	yes	characteristic
222	C₂³	#15667945	1	1	yes	characteristic
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	#15667926	1	2	yes	—
243	C₂×C₄	#15667928	1	1	yes	characteristic
244	C₂×C₄	#15667930	1	1	yes	characteristic
245	C₂×C₄	#15667932	1	1	yes	characteristic
246	C₂×C₄	#15667933	1	1	yes	characteristic
247	C₂×C₄	#15667935	1	1	yes	characteristic
248	C₂×C₄	#15667936	1	1	yes	characteristic
249	C₂×C₄	#15667940	1	2	yes	—
250	C₂×C₄	#15667947	1	2	yes	—
251	C₈	—	4	4	no	—
252	C₈	—	712	2	no	—
253	C₈	—	712	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	#19151356	1	1	yes	characteristic
265	C₂²	#19151358	1	1	yes	characteristic
266	C₂²	#19151360	1	1	yes	characteristic
267	C₂²	#19151362	1	1	yes	characteristic
268	C₂²	#19151364	1	1	yes	characteristic
269	C₂²	#19151366	1	1	yes	characteristic
270	C₂²	#19151369	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	#19151331	1	2	yes	—
279	C₄	#19151334	1	2	yes	—
280	C₂	—	4	1	no	—
281	C₂	—	4	1	no	—
282	C₂	#26018124	1	1	yes	characteristic
283	C₂	#26018126	1	1	yes	characteristic
284	C₂	#26018128	1	1	yes	characteristic
285	C₂	#26018131	1	1	yes	characteristic
286	C₂	#26018142	1	1	yes	characteristic
287	C₂	#26018147	1	1	yes	characteristic
288	C₂	#26018159	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667919
commutator	C₂²×C₄×C₈₉	C₄²
Frattini	C₂²×C₄²	C₂×D₈₉
Fitting	C₈₉×#1728	C₂
socle	C₂³×C₈₉	#95
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₂²×C₄²×C₈₉	C₂²
hypercenter	#1728	D₈₉
hypocenter	C₈₉	#7045
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
148. C₈₉	#7045	semidirect

Maximal subgroups

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

subgroup	index
#26020232	2
C₈₉×#1728	2
#7045	89