Subgroups — #36459622

Order 22784.

926 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151553	C₂	1	2	yes	maximal
2	C₂×#19151363	C₂	1	1	yes	maximal, characteristic
3	#26019969	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151358	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	2	yes	—
7	C₂×#15667971	C₂²	1	2	yes	—
8	#19151553	—	2	4	no	—
9	#19151553	C₄	1	8	yes	complemented
10	C₂×#15667922	—	2	2	no	—
11	C₂×C₈×Dic₈₉	—	2	2	no	—
12	#19151363	C₄	1	4	yes	complemented
13	#19151361	C₄	1	4	yes	complemented
14	#19151358	C₄	1	8	yes	complemented
15	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₉	—	2	4	no	—
19	C₂×C₄×C₈×C₈₉	C₄	1	4	yes	complemented
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	2	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	C₂×C₄	1	4	yes	—
24	C₂×#13083558	C₂×C₄	1	4	yes	—
25	C₂×#13083558	C₂×C₄	1	4	yes	—
26	C₂²×#10623494	—	2	2	no	—
27	#15667971	—	2	2	no	—
28	#15667971	—	2	2	no	—
29	#15667971	C₂×C₄	1	4	yes	—
30	#15667971	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊Dic₈₉	—	2	2	no	—
33	C₂×C₄×Dic₈₉	—	2	2	no	—
34	#15667922	—	2	8	no	—
35	C₈×Dic₈₉	—	2	8	no	—
36	C₂³×C₄×C₈₉	—	2	1	no	—
37	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	—	2	2	no	—
39	C₂²×C₈×C₈₉	—	2	2	no	—
40	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	D₄	1	4	yes	—
48	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
49	C₂×C₄²×C₈₉	—	2	2	no	—
50	C₂×C₄²×C₈₉	—	2	1	no	—
51	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₈₉	—	2	8	no	—
54	C₂²×Dic₈₉	—	2	4	no	—
55	#13083558	—	2	4	no	—
56	#13083558	—	2	4	no	—
57	#13083558	—	2	4	no	—
58	#13083558	C₄²	1	8	yes	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	—	2	4	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₄⋊Dic₈₉	—	2	4	no	—
63	C₄⋊Dic₈₉	C₄²	1	8	yes	—
64	C₂²⋊Dic₈₉	—	2	4	no	—
65	C₂²⋊Dic₈₉	—	2	4	no	—
66	C₄×Dic₈₉	—	2	4	no	—
67	C₄×Dic₈₉	—	2	4	no	—
68	C₂⁴×C₈₉	#41	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	—	2	8	no	—
70	C₂²×C₄×C₈₉	—	2	2	no	—
71	C₂²×C₄×C₈₉	—	2	4	no	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	#41	1	2	yes	—
81	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
82	C₂×C₈×C₈₉	—	2	2	no	—
83	C₂×C₈×C₈₉	—	2	2	no	—
84	C₂×C₈×C₈₉	—	2	4	no	—
85	C₂×C₈×C₈₉	—	2	4	no	—
86	C₂×C₈×C₈₉	C₄²	1	4	yes	—
87	C₂×C₈×C₈₉	C₄²	1	4	yes	—
88	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
89	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
90	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
91	C₂×C₈×C₈₉	C₄⋊C₄	1	8	yes	—
92	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
93	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
94	C₄²×C₈₉	—	2	2	no	—
95	C₄²×C₈₉	—	2	4	no	—
96	C₄²×C₈₉	C₄²	1	4	yes	—
97	C₂³×C₈₉	—	2	4	no	—
98	C₂³×C₈₉	#117	1	2	yes	—
99	C₂³×C₈₉	#117	1	1	yes	characteristic
100	C₂³×C₈₉	C₄×D₄	1	2	yes	—
101	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
102	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	—	2	16	no	—
104	C₂×C₄×C₈₉	—	2	8	no	—
105	C₂×C₄×C₈₉	—	2	4	no	—
106	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#117	1	2	yes	—
116	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
124	C₂×Dic₈₉	—	2	8	no	—
125	C₂×Dic₈₉	—	4	8	no	—
126	C₈×C₈₉	—	2	4	no	—
127	C₈×C₈₉	#95	1	4	yes	—
128	C₈×C₈₉	#95	1	4	yes	—
129	#10623494	—	2	8	no	—
130	C₂²×C₈₉	—	2	4	no	—
131	C₂²×C₈₉	—	2	4	no	—
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	#376	1	1	yes	characteristic
135	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₈₉	C₄×C₂²⋊C₄	1	2	yes	—
137	C₂²×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
138	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₈₉	#443	1	4	yes	—
140	C₄×C₈₉	—	2	8	no	—
141	C₄×C₈₉	C₄³	1	1	yes	characteristic
142	C₄×C₈₉	C₂×#95	1	1	yes	characteristic
143	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₈₉	C₄×C₂²⋊C₄	1	2	yes	—
145	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
147	Dic₈₉	—	4	8	no	—
148	#10706	—	89	1	no	maximal, Sylow
149	C₂×C₈₉	—	2	1	no	—
150	C₂×C₈₉	—	2	1	no	—
151	C₂×C₈₉	—	2	2	no	—
152	C₂×C₈₉	C₄×#95	1	1	yes	characteristic
153	C₂×C₈₉	#1764	1	4	yes	—
154	C₂×C₈₉	#1772	1	2	yes	—
155	C₂×#405	—	89	2	no	—
156	C₂²×C₄×C₈	D₈₉	1	1	yes	characteristic
157	C₂×#341	—	89	1	no	—
158	#1735	—	89	1	no	—
159	C₂×#339	—	89	2	no	—
160	C₈₉	#10706	1	1	yes	characteristic, complemented, Sylow
161	C₂²×M₁₆	—	89	2	no	—
162	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
163	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
164	C₂×#117	—	89	2	no	—
165	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
166	#405	—	89	8	no	—
167	#405	—	178	4	no	—
168	C₂×#97	—	178	2	no	—
169	C₂×C₄×C₈	—	2	4	no	—
170	C₂×C₄×C₈	—	178	2	no	—
171	C₂×C₄×C₈	Dic₈₉	1	4	yes	complemented
172	#341	—	89	4	no	—
173	#340	—	89	4	no	—
174	#339	—	89	8	no	—
175	C₂³×C₄	—	2	1	no	—
176	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
177	C₂×M₁₆	—	89	4	no	—
178	C₂×M₁₆	—	89	4	no	—
179	C₂×M₁₆	—	89	4	no	—
180	C₂×M₁₆	—	178	2	no	—
181	C₂×M₁₆	—	178	2	no	—
182	C₂×M₁₆	—	178	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	178	2	no	—
186	C₂²×C₈	Dic₁₇₈	1	2	yes	—
187	C₂²×C₈	C₄×D₈₉	1	2	yes	—
188	C₂²×C₈	C₄×D₈₉	1	2	yes	—
189	C₂²×C₈	C₄×D₈₉	1	4	yes	—
190	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
191	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
192	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
193	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
194	C₂²×C₈	C₂²⋊D₈₉	1	4	yes	—
195	#117	—	89	4	no	—
196	#117	—	89	4	no	—
197	#117	—	178	2	no	—
198	#117	—	178	2	no	—
199	C₂×C₄⋊C₄	—	89	4	no	—
200	C₂×C₂²⋊C₄	—	178	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	—	178	2	no	—
204	C₂×C₄²	C₄×D₈₉	1	4	yes	—
205	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
206	#97	—	178	8	no	—
207	C₄×C₈	—	2	8	no	—
208	C₄×C₈	—	178	8	no	—
209	C₂⁴	#13083574	1	1	yes	characteristic
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	178	4	no	—
214	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
215	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
220	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
221	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
222	C₂²×C₄	#13083574	1	1	yes	characteristic
223	C₂²×C₄	#13083574	1	2	yes	—
224	M₁₆	—	89	8	no	—
225	M₁₆	—	178	4	no	—
226	M₁₆	—	178	4	no	—
227	M₁₆	—	178	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	178	4	no	—
233	C₂×C₈	—	178	4	no	—
234	C₂×C₈	—	178	4	no	—
235	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
236	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
237	C₂×C₈	#13083551	1	4	yes	—
238	C₂×C₈	#13083551	1	4	yes	—
239	C₂×C₈	C₂²⋊Dic₈₉	1	4	yes	—
240	C₂×C₈	#13083553	1	8	yes	—
241	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
242	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
243	C₄⋊C₄	—	89	8	no	—
244	C₄⋊C₄	—	178	4	no	—
245	C₂²⋊C₄	—	178	4	no	—
246	C₂²⋊C₄	—	178	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	178	4	no	—
250	C₄²	—	178	4	no	—
251	C₄²	C₄×Dic₈₉	1	4	yes	—
252	C₂³	—	2	4	no	—
253	C₂³	#15667969	1	2	yes	—
254	C₂³	#15667971	1	1	yes	characteristic
255	C₂³	C₄×C₂²⋊D₈₉	1	2	yes	—
256	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
257	C₂³	C₄×Dic₁₇₈	1	1	yes	characteristic
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	16	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	356	8	no	—
262	C₂×C₄	—	178	8	no	—
263	C₂×C₄	C₄²×D₈₉	1	2	yes	—
264	C₂×C₄	C₄²×D₈₉	1	2	yes	—
265	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
268	C₂×C₄	C₂×#13083551	1	2	yes	—
269	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
270	C₂×C₄	C₂×#13083553	1	2	yes	—
271	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
272	C₂×C₄	#15667969	1	2	yes	—
273	C₂×C₄	#15667971	1	1	yes	characteristic
274	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
275	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
276	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
277	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
278	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
279	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
280	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	178	8	no	—
283	C₈	#15667919	1	4	yes	—
284	C₈	#15667919	1	4	yes	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	#19151444	1	1	yes	characteristic
290	C₂²	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
291	C₂²	C₄×#13083551	1	2	yes	—
292	C₂²	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
293	C₂²	C₄×#13083553	1	2	yes	—
294	C₂²	#19151716	1	4	yes	—
295	C₄	—	2	8	no	—
296	C₄	—	356	8	no	—
297	C₄	C₄²×Dic₈₉	1	1	yes	characteristic
298	C₄	C₂×#15667919	1	1	yes	characteristic
299	C₄	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
300	C₄	C₄×#13083551	1	2	yes	—
301	C₄	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
302	C₄	C₄×#13083553	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	2	no	—
306	C₂	C₄×#15667919	1	1	yes	characteristic
307	C₂	#26020089	1	4	yes	—
308	C₂	#26020131	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	D₃₅₆
commutator	C₄×C₈₉	C₄³
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂²×C₄×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₄×D₄
prime radical	C₂⁴×C₈₉	#41
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂²×C₄×C₈	D₈₉
hypocenter	C₈₉	#10706
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. #19151553	C₄	semidirect
12. #19151363	C₄	semidirect
13. #19151361	C₄	semidirect
14. #19151358	C₄	semidirect
19. C₂×C₄×C₈×C₈₉	C₄	semidirect
160. C₈₉	#10706	semidirect
171. C₂×C₄×C₈	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151553	2
C₂×#19151363	2
#26019969	2
C₂×#19151358	2
C₂²×C₄×C₈×C₈₉	2
#10706	89