Subgroups — #36559115

Order 22784.

814 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26021189	C₂	1	2	yes	maximal
2	#26020971	C₂	1	2	yes	maximal
3	#26020811	C₂	1	4	yes	maximal
4	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊#10623494	C₂	1	2	yes	maximal
6	#26019998	C₂	1	2	yes	maximal
7	C₄²×#10623494	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
10	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
11	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
12	C₂×C₄×#10623494	C₂²	1	2	yes	—
13	C₂×C₄×#10623494	C₂²	1	2	yes	—
14	C₂×C₄×#10623494	C₂²	1	2	yes	—
15	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
16	#19151353	C₂²	1	4	yes	—
17	#19151353	C₂²	1	2	yes	—
18	#19151353	C₂²	1	4	yes	—
19	#19151353	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₈₉	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₈₉	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₈₉	C₄	1	2	yes	—
24	C₄×C₄⋊C₄×C₈₉	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₈₉	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₈₉	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
29	C₄³×C₈₉	C₂²	1	1	yes	characteristic
30	C₄³×C₈₉	C₂²	1	2	yes	—
31	C₂²×#10623494	C₂³	1	4	yes	—
32	C₂²×#10623494	C₂³	1	2	yes	—
33	C₂²×#10623494	C₂³	1	2	yes	—
34	C₄⋊#10623494	—	2	8	no	—
35	C₄⋊#10623494	—	2	4	no	—
36	C₄×#10623494	—	2	4	no	—
37	C₄×#10623494	—	2	4	no	—
38	C₄×#10623494	—	2	2	no	—
39	C₄×#10623494	—	2	4	no	—
40	C₄×#10623494	Q₈	1	4	yes	complemented
41	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₄×Q₈×C₈₉	—	2	4	no	—
43	C₄×Q₈×C₈₉	—	2	4	no	—
44	C₄×Q₈×C₈₉	C₈	1	8	yes	complemented
45	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
60	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
62	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	#95×C₈₉	C₂×C₄	1	2	yes	—
65	#95×C₈₉	C₂×C₄	1	2	yes	—
66	#95×C₈₉	C₂×C₄	1	2	yes	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	—	2	2	no	—
69	C₂×#10623494	—	2	2	no	—
70	C₂×#10623494	—	2	2	no	—
71	C₂×#10623494	—	2	4	no	—
72	C₂×#10623494	—	2	4	no	—
73	C₂×#10623494	—	2	4	no	—
74	C₂×#10623494	C₂×Q₈	1	4	yes	—
75	C₂×#10623494	#41	1	4	yes	—
76	C₂×Q₈×C₈₉	—	2	2	no	—
77	C₂×Q₈×C₈₉	—	2	2	no	—
78	C₂×Q₈×C₈₉	C₂×C₈	1	4	yes	—
79	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₈₉	—	2	4	no	—
91	C₄⋊C₄×C₈₉	—	2	2	no	—
92	C₄⋊C₄×C₈₉	—	2	4	no	—
93	C₄⋊C₄×C₈₉	—	2	2	no	—
94	C₄⋊C₄×C₈₉	C₂×C₈	1	8	yes	—
95	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
96	C₄²×C₈₉	—	2	4	no	—
97	C₄²×C₈₉	—	2	2	no	—
98	C₄²×C₈₉	—	2	2	no	—
99	C₄²×C₈₉	—	2	4	no	—
100	C₄²×C₈₉	—	2	4	no	—
101	C₄²×C₈₉	—	2	4	no	—
102	C₄²×C₈₉	—	2	2	no	—
103	C₄²×C₈₉	—	2	4	no	—
104	C₄²×C₈₉	C₂×C₈	1	8	yes	—
105	C₄²×C₈₉	C₂×C₈	1	4	yes	—
106	C₄²×C₈₉	M₁₆	1	4	yes	—
107	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
108	C₄²×C₈₉	#41	1	4	yes	—
109	C₄²×C₈₉	#41	1	2	yes	—
110	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
111	Q₈×C₈₉	—	2	8	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	—	2	2	no	—
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	#117	1	2	yes	—
129	C₂×C₄×C₈₉	#117	1	2	yes	—
130	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
133	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
135	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
140	C₂×C₄×C₈₉	#131	1	2	yes	—
141	C₂×C₄×C₈₉	#131	1	2	yes	—
142	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
144	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
145	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
146	#10623494	—	2	4	no	—
147	#10623494	—	2	4	no	—
148	#10623494	—	4	4	no	—
149	C₂²×C₈₉	C₂×#117	1	1	yes	characteristic
150	C₂²×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₂²×C₈₉	C₄×#41	1	1	yes	characteristic
152	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
153	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
154	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
155	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
156	C₄×C₈₉	—	2	4	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	2	4	no	—
160	C₄×C₈₉	—	2	2	no	—
161	C₄×C₈₉	—	4	2	no	—
162	C₄×C₈₉	—	4	2	no	—
163	C₄×C₈₉	#431	1	4	yes	—
164	C₄×C₈₉	C₈×Q₈	1	4	yes	—
165	C₄×C₈₉	#446	1	4	yes	—
166	C₄×C₈₉	#529	1	2	yes	—
167	C₄×C₈₉	#533	1	2	yes	—
168	#21339	—	89	1	no	maximal, Sylow
169	C₂×C₈₉	#2295	1	1	yes	characteristic
170	C₂×C₈₉	C₂×#431	1	1	yes	characteristic
171	C₂×C₈₉	#2922	1	1	yes	characteristic
172	C₂×C₈₉	C₂×C₈×Q₈	1	1	yes	characteristic
173	C₂×C₈₉	C₂×#446	1	1	yes	characteristic
174	C₂×C₈₉	C₈×#41	1	1	yes	characteristic
175	C₂×C₈₉	#2965	1	1	yes	characteristic
176	C₄²×Q₈	D₈₉	1	1	yes	characteristic
177	#1984	—	89	2	no	—
178	#1939	—	89	2	no	—
179	#1916	—	89	4	no	—
180	C₄×C₄⋊C₈	—	89	1	no	—
181	C₄×C₄⋊C₈	—	89	2	no	—
182	#1744	—	89	2	no	—
183	C₄²×C₈	—	89	1	no	—
184	C₈₉	#21339	1	1	yes	characteristic, complemented, Sylow
185	C₂×C₄×Q₈	Dic₈₉	1	1	yes	characteristic
186	C₂×C₄×Q₈	Dic₈₉	1	1	yes	characteristic
187	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
188	C₂×C₄⋊C₈	—	89	4	no	—
189	C₂×C₄⋊C₈	—	89	2	no	—
190	C₂×C₄×C₈	—	89	1	no	—
191	C₂×C₄×C₈	—	89	2	no	—
192	C₂×C₄×C₈	—	89	2	no	—
193	C₂×C₄×C₈	—	89	1	no	—
194	C₂×C₄×C₈	—	89	2	no	—
195	C₄×C₄⋊C₄	Dic₈₉	1	2	yes	—
196	C₄×C₄⋊C₄	Dic₈₉	1	1	yes	characteristic
197	C₄×C₄⋊C₄	Dic₈₉	1	2	yes	—
198	C₄×C₄⋊C₄	Dic₈₉	1	1	yes	characteristic
199	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
200	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
201	C₄³	C₂×D₈₉	1	1	yes	characteristic
202	C₄³	C₂×D₈₉	1	2	yes	—
203	#336	—	89	2	no	—
204	#336	—	89	4	no	—
205	#336	—	89	4	no	—
206	#336	—	89	4	no	—
207	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
208	C₂²×C₈	—	89	2	no	—
209	C₂²×C₈	—	89	2	no	—
210	C₂²×C₈	—	89	4	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	—	2	4	no	—
213	C₄×Q₈	#10623494	1	8	yes	complemented
214	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
216	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
218	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
219	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
220	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
221	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
222	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
223	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
224	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
226	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
227	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
229	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
231	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
233	C₄⋊C₈	—	178	8	no	—
234	C₄⋊C₈	—	178	4	no	—
235	C₄×C₈	—	89	4	no	—
236	C₄×C₈	—	178	4	no	—
237	C₄×C₈	—	178	4	no	—
238	C₄×C₈	—	178	4	no	—
239	C₄×C₈	—	178	2	no	—
240	#95	C₂×Dic₈₉	1	2	yes	—
241	#95	C₂×Dic₈₉	1	2	yes	—
242	#95	C₂×Dic₈₉	1	2	yes	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	C₂×#10623494	1	4	yes	—
246	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
247	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
248	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
252	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
255	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
257	C₂×C₈	—	89	4	no	—
258	C₂×C₈	—	89	4	no	—
259	C₂×C₈	—	178	4	no	—
260	C₂×C₈	—	178	4	no	—
261	C₂×C₈	—	178	4	no	—
262	C₂×C₈	—	178	2	no	—
263	C₂×C₈	—	178	2	no	—
264	C₂×C₈	—	178	2	no	—
265	C₂×C₈	—	178	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	C₂×#10623494	1	8	yes	—
271	C₄⋊C₄	C₂×#10623494	1	4	yes	—
272	C₄²	—	2	4	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	4	no	—
276	C₄²	—	2	4	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	4	no	—
279	C₄²	—	2	4	no	—
280	C₄²	C₂×#10623494	1	8	yes	—
281	C₄²	C₂×#10623494	1	4	yes	—
282	C₄²	#13083558	1	4	yes	—
283	C₄²	Q₈×D₈₉	1	2	yes	—
284	C₄²	#13083574	1	4	yes	—
285	C₄²	#13083575	1	2	yes	—
286	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
287	Q₈	—	2	8	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	#15667971	1	2	yes	—
305	C₂×C₄	#15667971	1	2	yes	—
306	C₂×C₄	Q₈×Dic₈₉	1	2	yes	—
307	C₂×C₄	Q₈×Dic₈₉	1	2	yes	—
308	C₂×C₄	C₂²×#10623494	1	4	yes	—
309	C₂×C₄	C₂²×#10623494	1	2	yes	—
310	C₂×C₄	C₂²×#10623494	1	4	yes	—
311	C₂×C₄	C₂²×#10623494	1	2	yes	—
312	C₂×C₄	C₂²×#10623494	1	2	yes	—
313	C₂×C₄	C₂×#13083558	1	2	yes	—
314	C₂×C₄	C₂×#13083558	1	2	yes	—
315	C₂×C₄	C₂×#13083558	1	2	yes	—
316	C₂×C₄	#15668019	1	2	yes	—
317	C₂×C₄	#15668019	1	2	yes	—
318	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
319	C₂×C₄	#41×D₈₉	1	2	yes	—
320	C₂×C₄	C₂×#13083574	1	2	yes	—
321	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
322	C₈	—	356	4	no	—
323	C₈	—	178	4	no	—
324	C₈	—	178	4	no	—
325	C₂²	C₂×#15667971	1	1	yes	characteristic
326	C₂²	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
327	C₂²	#41×Dic₈₉	1	1	yes	characteristic
328	C₂²	C₂³×#10623494	1	1	yes	characteristic
329	C₂²	C₂²×#13083558	1	1	yes	characteristic
330	C₂²	C₂×#15668019	1	1	yes	characteristic
331	C₂²	C₂×#15668019	1	1	yes	characteristic
332	C₄	—	2	4	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	4	no	—
336	C₄	—	2	2	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	2	no	—
339	C₄	#19151651	1	4	yes	—
340	C₄	Q₈×#10623494	1	4	yes	—
341	C₄	#19151729	1	4	yes	—
342	C₄	#19152135	1	2	yes	—
343	C₄	#19152152	1	2	yes	—
344	C₂	#26022730	1	1	yes	characteristic
345	C₂	C₂×#19151651	1	1	yes	characteristic
346	C₂	#26028896	1	1	yes	characteristic
347	C₂	C₂×Q₈×#10623494	1	1	yes	characteristic
348	C₂	C₂×#19151729	1	1	yes	characteristic
349	C₂	#41×#10623494	1	1	yes	characteristic
350	C₂	#26029219	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄²×Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂³×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄²×Q₈	D₈₉
hypocenter	C₈₉	#21339
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
40. C₄×#10623494	Q₈	semidirect
44. C₄×Q₈×C₈₉	C₈	semidirect
184. C₈₉	#21339	semidirect
213. C₄×Q₈	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26021189	2
#26020971	2
#26020811	2
C₄×C₄⋊#10623494	2
#26019998	2
C₄²×#10623494	2
C₄²×Q₈×C₈₉	2
#21339	89