Subgroups — #36526246

Order 22784.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030838	C₂	1	1	yes	maximal, characteristic
2	#26029199	C₂	1	2	yes	maximal
3	#26019162	C₂	1	2	yes	maximal, complemented
4	#26019125	C₂	1	4	yes	maximal, complemented
5	#26019072	C₂	1	2	yes	maximal, complemented
6	C₂×#19151349	C₂	1	1	yes	maximal, characteristic
7	#26018656	C₂	1	2	yes	maximal
8	C₈₉×#1451	C₂	1	1	yes	maximal, characteristic
9	#19152155	C₂²	1	2	yes	—
10	#19152003	C₄	1	1	yes	characteristic
11	#19151978	C₄	1	1	yes	characteristic
12	#19151949	C₄	1	2	yes	—
13	#19151839	C₄	1	2	yes	—
14	#19151839	C₄	1	2	yes	—
15	#19151728	C₂²	1	4	yes	—
16	#19151726	—	2	4	no	—
17	C₈×Dic₁₇₈	C₂²	1	4	yes	—
18	C₄×Dic₃₅₆	C₂²	1	2	yes	—
19	C₄×Dic₃₅₆	C₂²	1	2	yes	—
20	#19151626	—	2	2	no	—
21	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
22	#19151556	—	2	2	no	—
23	#19151349	C₂²	1	1	yes	characteristic
24	#19151349	C₂²	1	2	yes	—
25	#19151349	C₂²	1	1	yes	characteristic
26	#19151330	C₂²	1	4	yes	—
27	#19151330	C₂²	1	4	yes	—
28	#431×C₈₉	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#322×C₈₉	C₂²	1	2	yes	—
32	#322×C₈₉	C₂²	1	2	yes	—
33	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
34	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
35	C₂²×#10623494	—	2	2	no	—
36	#15668012	C₂×C₄	1	2	yes	—
37	#15668012	C₂×C₄	1	2	yes	—
38	#15668004	—	2	2	no	—
39	#15667991	—	2	2	no	—
40	#15667991	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₇₈	—	2	2	no	—
42	C₄×Dic₁₇₈	C₂³	1	4	yes	—
43	#15667971	—	2	2	no	—
44	#15667946	C₂×C₄	1	2	yes	—
45	#15667946	C₂×C₄	1	2	yes	—
46	C₄⋊#10623494	—	2	2	no	—
47	C₄⋊#10623494	C₂³	1	2	yes	—
48	C₄⋊#10623494	C₂³	1	2	yes	—
49	#15667941	—	2	4	no	—
50	#15667941	—	2	4	no	—
51	#15667937	C₂×C₄	1	4	yes	—
52	#15667937	C₂×C₄	1	4	yes	—
53	#15667922	—	2	4	no	—
54	C₈×Dic₈₉	—	2	4	no	—
55	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
56	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₈₉	—	2	1	no	—
60	C₂²⋊C₈×C₈₉	—	2	1	no	—
61	#97×C₈₉	D₄	1	2	yes	—
62	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₈₉	C₂³	1	2	yes	—
65	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₈₉	C₂³	1	2	yes	—
68	C₂×Dic₁₇₈	—	2	2	no	—
69	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
70	Dic₃₅₆	—	2	4	no	—
71	Dic₃₅₆	—	4	2	no	—
72	C₂×#10623494	—	2	2	no	—
73	C₂×#10623494	—	2	2	no	—
74	C₂×#10623494	—	4	4	no	—
75	C₄⋊Dic₈₉	—	2	2	no	—
76	C₄⋊Dic₈₉	—	2	2	no	—
77	C₄⋊Dic₈₉	—	2	2	no	—
78	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
79	#13083553	—	2	4	no	—
80	#13083553	—	2	4	no	—
81	C₂²⋊Dic₈₉	—	2	4	no	—
82	C₄×Dic₈₉	—	2	4	no	—
83	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
86	M₁₆×C₈₉	—	2	4	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	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
95	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₈₉	#41	1	1	yes	characteristic
98	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
99	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	4	1	no	—
103	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#137	1	4	yes	—
107	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₈₉	—	2	4	no	—
111	C₂×Dic₈₉	—	2	4	no	—
112	Dic₁₇₈	—	2	4	no	—
113	Dic₁₇₈	—	2	4	no	—
114	Dic₁₇₈	—	8	2	no	—
115	C₈×C₈₉	—	2	1	no	—
116	C₈×C₈₉	—	2	1	no	—
117	C₈×C₈₉	—	2	1	no	—
118	C₈×C₈₉	—	2	4	no	—
119	C₈×C₈₉	—	4	2	no	—
120	C₈×C₈₉	#131	1	2	yes	—
121	#10623494	—	4	2	no	—
122	#10623494	—	4	2	no	—
123	C₂²×C₈₉	—	2	1	no	—
124	C₂²×C₈₉	—	2	1	no	—
125	C₂²×C₈₉	#441	1	2	yes	—
126	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	#441	1	2	yes	—
131	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
132	C₄×C₈₉	#568	1	1	yes	characteristic
133	C₄×C₈₉	C₂×#137	1	2	yes	—
134	Dic₈₉	—	4	4	no	—
135	Dic₈₉	—	4	4	no	—
136	#18072	—	89	1	no	maximal, Sylow
137	C₂×C₈₉	—	2	2	no	—
138	C₂×C₈₉	#2935	1	1	yes	characteristic
139	C₂×C₈₉	C₂×#441	1	1	yes	characteristic
140	C₂×C₈₉	#2957	1	1	yes	characteristic
141	#3170	—	89	1	no	—
142	#2962	—	89	2	no	—
143	#1593	—	89	2	no	—
144	#1587	—	89	4	no	—
145	#1579	—	89	2	no	—
146	C₂×#334	—	89	1	no	—
147	#1481	—	89	2	no	—
148	#1451	D₈₉	1	1	yes	characteristic
149	C₈₉	#18072	1	1	yes	characteristic, complemented, Sylow
150	#533	—	89	2	no	—
151	#500	—	89	1	no	—
152	#494	—	89	1	no	—
153	#487	—	89	2	no	—
154	#467	—	89	2	no	—
155	#467	—	89	2	no	—
156	#446	—	89	4	no	—
157	#446	—	178	4	no	—
158	C₈×Q₈	—	89	4	no	—
159	C₄×Q₁₆	—	89	2	no	—
160	C₄×Q₁₆	—	89	2	no	—
161	#431	C₂×D₈₉	1	1	yes	characteristic
162	#424	—	178	2	no	—
163	C₂×C₄⋊C₈	—	89	2	no	—
164	#405	—	178	2	no	—
165	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
167	#334	—	89	1	no	—
168	#334	—	89	2	no	—
169	#334	—	89	1	no	—
170	Q₈⋊C₈	—	89	4	no	—
171	Q₈⋊C₈	—	89	4	no	—
172	#322	C₂×D₈₉	1	2	yes	—
173	#322	C₂×D₈₉	1	2	yes	—
174	C₂×Q₁₆	—	89	2	no	—
175	C₂×Q₁₆	—	89	2	no	—
176	C₂×M₁₆	D₃₅₆	1	2	yes	—
177	C₂²×C₈	—	178	2	no	—
178	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
180	C₄⋊Q₈	—	89	2	no	—
181	C₄⋊Q₈	—	89	2	no	—
182	#125	—	178	2	no	—
183	C₂²⋊Q₈	—	89	4	no	—
184	C₂²⋊Q₈	—	178	2	no	—
185	C₄×Q₈	—	89	4	no	—
186	C₄×Q₈	—	178	2	no	—
187	#117	—	178	2	no	—
188	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
189	#107	—	89	2	no	—
190	#107	—	89	2	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	89	2	no	—
193	C₄⋊C₈	—	89	2	no	—
194	C₄⋊C₈	—	178	4	no	—
195	C₄⋊C₈	—	178	4	no	—
196	C₄⋊C₈	—	178	2	no	—
197	#103	—	89	4	no	—
198	#103	—	89	4	no	—
199	C₂²⋊C₈	—	2	1	no	—
200	#97	—	178	4	no	—
201	#97	D₃₅₆	1	2	yes	—
202	C₄×C₈	—	178	4	no	—
203	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
204	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₈₉	1	2	yes	—
206	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₈₉	1	2	yes	—
209	C₂×Q₈	—	89	4	no	—
210	C₂×Q₈	—	178	2	no	—
211	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
213	C₂²×C₄	#13083574	1	1	yes	characteristic
214	Q₁₆	—	356	2	no	—
215	Q₁₆	—	178	4	no	—
216	M₁₆	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	356	4	no	—
221	C₂×C₈	—	178	2	no	—
222	C₂×C₈	—	178	2	no	—
223	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
227	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
228	C₄⋊C₄	—	89	4	no	—
229	C₄⋊C₄	—	178	2	no	—
230	C₄⋊C₄	—	178	2	no	—
231	C₄⋊C₄	—	178	4	no	—
232	C₄⋊C₄	—	178	2	no	—
233	C₄⋊C₄	—	178	4	no	—
234	C₂²⋊C₄	—	178	4	no	—
235	C₄²	—	178	4	no	—
236	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
237	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
238	C₄²	#13083574	1	1	yes	characteristic
239	C₄²	C₂³×D₈₉	1	1	yes	characteristic
240	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
241	Q₈	—	178	4	no	—
242	Q₈	—	178	4	no	—
243	Q₈	—	712	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	178	4	no	—
248	C₂×C₄	—	178	4	no	—
249	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
250	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
251	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
252	C₂×C₄	#15668045	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
255	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
256	C₈	—	2	4	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	4	2	no	—
261	C₈	—	356	2	no	—
262	C₈	—	356	2	no	—
263	C₈	#15668020	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#19151706	1	2	yes	—
267	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	356	4	no	—
272	C₄	—	356	4	no	—
273	C₄	#19151706	1	2	yes	—
274	C₄	C₂×#15668020	1	1	yes	characteristic
275	C₄	#19152364	1	1	yes	characteristic
276	C₄	C₂×#15668045	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	#26028998	1	1	yes	characteristic
279	C₂	C₂×#19151706	1	1	yes	characteristic
280	C₂	#26029170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1451	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1451	D₈₉
hypocenter	C₈₉	#18072
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #26019162	C₂	semidirect
4. #26019125	C₂	semidirect
5. #26019072	C₂	semidirect
149. C₈₉	#18072	semidirect

Maximal subgroups

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

subgroup	index
#26030838	2
#26029199	2
#26019162	2
#26019125	2
#26019072	2
C₂×#19151349	2
#26018656	2
C₈₉×#1451	2
#18072	89