Subgroups — #36441010

Order 22784.

476 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020320	C₂	1	1	yes	maximal, characteristic
2	#26019946	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1434	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13083558	—	2	1	no	—
5	C₂×C₂²⋊#10623494	—	2	1	no	—
6	#19151353	—	2	1	no	—
7	#19151353	—	2	1	no	—
8	#19151353	C₄	1	2	yes	complemented
9	#19151353	C₄	1	2	yes	complemented
10	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×#95×C₈₉	—	2	1	no	—
12	C₂×#95×C₈₉	—	2	1	no	—
13	C₂×#95×C₈₉	C₄	1	2	yes	—
14	C₂×#13083558	—	2	4	no	—
15	C₂×#13083558	—	4	2	no	—
16	C₂×#13083558	—	4	1	no	—
17	C₂×#13083558	—	4	1	no	—
18	C₂²×#10623494	—	2	2	no	—
19	C₂²×#10623494	—	2	2	no	—
20	C₂²⋊#10623494	—	4	2	no	—
21	C₂³×C₄×C₈₉	—	2	2	no	—
22	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	—	4	2	no	—
26	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
30	#95×C₈₉	—	2	4	no	—
31	#95×C₈₉	—	4	2	no	—
32	#95×C₈₉	—	4	1	no	—
33	#95×C₈₉	—	4	1	no	—
34	#13083558	—	4	4	no	—
35	#13083558	—	4	4	no	—
36	C₂×#10623494	—	2	4	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₂×#10623494	—	4	2	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₂×#10623494	—	4	2	no	—
41	C₂×#10623494	—	4	2	no	—
42	C₂⁴×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	—	2	2	no	—
44	C₂²×C₄×C₈₉	—	2	2	no	—
45	C₂²×C₄×C₈₉	—	2	2	no	—
46	C₂²×C₄×C₈₉	—	2	2	no	—
47	C₂²×C₄×C₈₉	—	2	2	no	—
48	C₂²×C₄×C₈₉	—	2	4	no	—
49	C₂²×C₄×C₈₉	—	4	1	no	—
50	C₂²×C₄×C₈₉	—	4	1	no	—
51	C₂²×C₄×C₈₉	—	4	1	no	—
52	C₂²×C₄×C₈₉	—	4	2	no	—
53	C₂²×C₄×C₈₉	—	4	2	no	—
54	C₂²×C₄×C₈₉	—	4	2	no	—
55	C₂²×C₄×C₈₉	—	4	2	no	—
56	C₂²×C₄×C₈₉	—	4	2	no	—
57	C₂²×C₄×C₈₉	—	4	2	no	—
58	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
68	C₄²×C₈₉	—	4	1	no	—
69	C₄²×C₈₉	—	4	1	no	—
70	C₄²×C₈₉	—	4	1	no	—
71	C₄²×C₈₉	—	4	1	no	—
72	C₂³×C₈₉	—	2	1	no	—
73	C₂³×C₈₉	—	2	1	no	—
74	C₂³×C₈₉	—	2	1	no	—
75	C₂³×C₈₉	—	2	1	no	—
76	C₂³×C₈₉	#95	1	1	yes	characteristic
77	C₂³×C₈₉	#99	1	2	yes	—
78	C₂³×C₈₉	#102	1	1	yes	characteristic
79	C₂³×C₈₉	#103	1	1	yes	characteristic
80	C₂³×C₈₉	#104	1	2	yes	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	2	2	no	—
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	—	2	2	no	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	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	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	2	1	no	—
100	C₂×C₄×C₈₉	—	4	4	no	—
101	C₂×C₄×C₈₉	—	4	4	no	—
102	C₂×C₄×C₈₉	—	4	4	no	—
103	C₂×C₄×C₈₉	—	4	4	no	—
104	C₂×C₄×C₈₉	—	4	4	no	—
105	C₂×C₄×C₈₉	—	8	2	no	—
106	C₂×C₄×C₈₉	—	8	2	no	—
107	C₂×C₄×C₈₉	—	8	2	no	—
108	C₂×C₄×C₈₉	—	8	2	no	—
109	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#104	1	2	yes	—
112	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
114	#10623494	—	4	4	no	—
115	#10623494	—	8	2	no	—
116	C₂²×C₈₉	—	2	2	no	—
117	C₂²×C₈₉	—	2	1	no	—
118	C₂²×C₈₉	—	2	1	no	—
119	C₂²×C₈₉	—	2	2	no	—
120	C₂²×C₈₉	—	2	2	no	—
121	C₂²×C₈₉	—	2	1	no	—
122	C₂²×C₈₉	—	2	1	no	—
123	C₂²×C₈₉	—	2	2	no	—
124	C₂²×C₈₉	#327	1	2	yes	—
125	C₂²×C₈₉	#328	1	2	yes	—
126	C₂²×C₈₉	H(3,ℤ/4)	1	1	yes	characteristic
127	C₂²×C₈₉	#339	1	1	yes	characteristic
128	C₂²×C₈₉	#339	1	1	yes	characteristic
129	C₂²×C₈₉	#340	1	1	yes	characteristic
130	C₂²×C₈₉	#342	1	1	yes	characteristic
131	C₂²×C₈₉	#344	1	1	yes	characteristic
132	C₂²×C₈₉	#344	1	1	yes	characteristic
133	C₄×C₈₉	—	2	2	no	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	—	4	1	no	—
139	C₄×C₈₉	—	4	1	no	—
140	C₄×C₈₉	—	4	1	no	—
141	C₄×C₈₉	—	8	4	no	—
142	#7037	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	—	2	2	no	—
144	C₂×C₈₉	—	2	2	no	—
145	C₂×C₈₉	#1284	1	1	yes	characteristic
146	C₂×C₈₉	#1286	1	1	yes	characteristic
147	C₂×C₈₉	#1288	1	1	yes	characteristic
148	C₂×C₈₉	#1294	1	1	yes	characteristic
149	C₂×C₈₉	#1300	1	1	yes	characteristic
150	C₂×C₈₉	#1303	1	1	yes	characteristic
151	C₂×C₈₉	#1306	1	1	yes	characteristic
152	#1817	—	89	1	no	—
153	#1728	—	89	1	no	—
154	#1434	D₈₉	1	1	yes	characteristic
155	C₈₉	#7037	1	1	yes	characteristic, complemented, Sylow
156	C₂²×M₁₆	—	178	1	no	—
157	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
158	C₂×C₂²⋊C₈	—	178	1	no	—
159	C₂×#95	—	2	1	no	—
160	C₂×#95	—	2	1	no	—
161	C₂×#95	Dic₈₉	1	2	yes	—
162	#336	—	89	2	no	—
163	#336	—	89	2	no	—
164	#336	—	178	1	no	—
165	#336	—	178	1	no	—
166	C₂³×C₄	—	2	2	no	—
167	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
168	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
169	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
170	C₂×M₁₆	—	356	2	no	—
171	C₂×M₁₆	—	356	1	no	—
172	C₂×M₁₆	—	356	1	no	—
173	C₂×M₁₆	—	178	4	no	—
174	C₂²×C₈	—	178	2	no	—
175	C₂²×C₈	—	178	2	no	—
176	C₂×C₄²	—	4	2	no	—
177	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
179	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
180	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
181	C₂²⋊C₈	—	356	2	no	—
182	#95	—	2	4	no	—
183	#95	—	4	2	no	—
184	#95	—	4	1	no	—
185	#95	—	4	1	no	—
186	C₂⁴	C₂²⋊Dic₈₉	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
203	C₂²×C₄	#13083551	1	1	yes	characteristic
204	C₂²×C₄	#13083551	1	1	yes	characteristic
205	C₂²×C₄	#13083553	1	1	yes	characteristic
206	C₂²×C₄	#13083553	1	1	yes	characteristic
207	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
208	C₂²×C₄	#13083559	1	1	yes	characteristic
209	C₂²×C₄	#13083561	1	1	yes	characteristic
210	C₂²×C₄	#13083562	1	1	yes	characteristic
211	C₂²×C₄	#13083564	1	1	yes	characteristic
212	M₁₆	—	356	4	no	—
213	M₁₆	—	356	4	no	—
214	C₂×C₈	—	356	2	no	—
215	C₂×C₈	—	356	2	no	—
216	C₂×C₈	—	356	2	no	—
217	C₂×C₈	—	356	2	no	—
218	C₂×C₈	—	356	2	no	—
219	C₂×C₈	—	178	4	no	—
220	C₄²	—	4	1	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	#15667919	1	1	yes	characteristic
229	C₂³	#15667927	1	2	yes	—
230	C₂³	#15667933	1	1	yes	characteristic
231	C₂³	#15667936	1	1	yes	characteristic
232	C₂³	#15667939	1	2	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	—	8	2	no	—
261	C₂×C₄	#15667932	1	1	yes	characteristic
262	C₂×C₄	#15667935	1	1	yes	characteristic
263	C₂×C₄	#15667940	1	2	yes	—
264	C₂×C₄	#15667943	1	1	yes	characteristic
265	C₂×C₄	#15667945	1	1	yes	characteristic
266	C₈	—	356	4	no	—
267	C₈	—	712	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	2	no	—
276	C₂²	#19151332	1	2	yes	—
277	C₂²	#19151335	1	2	yes	—
278	C₂²	#19151354	1	1	yes	characteristic
279	C₂²	#19151357	1	1	yes	characteristic
280	C₂²	#19151358	1	1	yes	characteristic
281	C₂²	#19151360	1	1	yes	characteristic
282	C₂²	#19151365	1	1	yes	characteristic
283	C₂²	#19151368	1	1	yes	characteristic
284	C₂²	#19151368	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	8	4	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	#26018105	1	1	yes	characteristic
297	C₂	#26018110	1	1	yes	characteristic
298	C₂	#26018116	1	1	yes	characteristic
299	C₂	#26018130	1	1	yes	characteristic
300	C₂	#26018145	1	1	yes	characteristic
301	C₂	#26018155	1	1	yes	characteristic
302	C₂	#26018160	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667919
commutator	C₂²×C₄×C₈₉	C₄²
Frattini	C₂²×C₄²	C₂×D₈₉
Fitting	C₈₉×#1434	C₂
socle	C₂³×C₈₉	#95
prime radical	C₂⁴×C₈₉	C₂²⋊C₄
cosocle	C₂²×C₄²×C₈₉	C₂²
hypercenter	#1434	D₈₉
hypocenter	C₈₉	#7037
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1434

Complements

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

N	H ≅ G/N	type
8. #19151353	C₄	semidirect
9. #19151353	C₄	semidirect
155. C₈₉	#7037	semidirect

Maximal subgroups

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

subgroup	index
#26020320	2
#26019946	2
C₈₉×#1434	2
#7037	89