Subgroups — #36573352

Order 22784.

812 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151729	C₂	1	2	yes	maximal
2	#26020976	C₂	1	4	yes	maximal
3	#26020971	C₂	1	2	yes	maximal
4	#26020369	C₂	1	1	yes	maximal, characteristic
5	#26020369	C₂	1	2	yes	maximal
6	#26020105	C₂	1	2	yes	maximal
7	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19151729	—	2	8	no	—
10	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
11	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
12	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
13	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
14	C₂×#15667923	C₂²	1	4	yes	—
15	C₂×C₄×#10623494	C₂²	1	2	yes	—
16	#19151353	C₂²	1	1	yes	characteristic
17	#19151353	C₂²	1	1	yes	characteristic
18	#19151353	C₂²	1	2	yes	—
19	#19151353	C₂²	1	4	yes	—
20	#19151353	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₈₉	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₈₉	C₄	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
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	2	yes	—
27	C₄×C₄⋊C₄×C₈₉	C₄	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
30	C₄³×C₈₉	C₂²	1	1	yes	characteristic
31	C₄³×C₈₉	C₂²	1	2	yes	—
32	C₂²×#10623494	C₂³	1	2	yes	—
33	C₂²×#10623494	C₂³	1	4	yes	—
34	C₂²×#10623494	C₂³	1	2	yes	—
35	C₄⋊#10623494	—	2	4	no	—
36	C₄⋊#10623494	—	2	8	no	—
37	C₄⋊#10623494	—	2	4	no	—
38	C₄⋊#10623494	—	2	8	no	—
39	#15667923	—	2	8	no	—
40	C₄×#10623494	—	2	4	no	—
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	D₄	1	4	yes	—
45	C₄×Q₈×C₈₉	Q₈	1	4	yes	—
46	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
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×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
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	2	yes	—
55	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
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	2	yes	—
59	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
65	#95×C₈₉	C₂×C₄	1	2	yes	—
66	#95×C₈₉	C₂×C₄	1	2	yes	—
67	#95×C₈₉	C₂×C₄	1	2	yes	—
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	2	no	—
73	C₂×#10623494	—	2	4	no	—
74	C₂×#10623494	—	2	2	no	—
75	C₂×#10623494	—	2	4	no	—
76	C₂×#10623494	#41	1	4	yes	—
77	C₂×Q₈×C₈₉	—	2	2	no	—
78	C₂×Q₈×C₈₉	—	2	2	no	—
79	C₂×Q₈×C₈₉	C₄⋊C₄	1	4	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	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
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	2	yes	—
87	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₈₉	—	2	4	no	—
92	C₄⋊C₄×C₈₉	—	2	4	no	—
93	C₄⋊C₄×C₈₉	—	2	2	no	—
94	C₄⋊C₄×C₈₉	C₄⋊C₄	1	8	yes	—
95	C₄⋊C₄×C₈₉	C₄⋊C₄	1	4	yes	—
96	C₄⋊C₄×C₈₉	M₁₆	1	4	yes	—
97	C₄²×C₈₉	—	2	4	no	—
98	C₄²×C₈₉	—	2	2	no	—
99	C₄²×C₈₉	—	2	4	no	—
100	C₄²×C₈₉	—	2	2	no	—
101	C₄²×C₈₉	—	2	4	no	—
102	C₄²×C₈₉	—	2	2	no	—
103	C₄²×C₈₉	—	2	4	no	—
104	C₄²×C₈₉	—	2	2	no	—
105	C₄²×C₈₉	—	4	2	no	—
106	C₄²×C₈₉	—	4	1	no	—
107	C₄²×C₈₉	M₁₆	1	4	yes	—
108	C₄²×C₈₉	C₂×D₄	1	2	yes	—
109	C₄²×C₈₉	C₂×D₄	1	4	yes	—
110	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
111	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
112	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
113	Q₈×C₈₉	—	2	8	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	2	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	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	—	2	2	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
136	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
137	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
140	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
141	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
142	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
143	C₂×C₄×C₈₉	#131	1	2	yes	—
144	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
147	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
148	#10623494	—	2	4	no	—
149	#10623494	—	4	2	no	—
150	#10623494	—	4	4	no	—
151	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₈₉	C₄×#41	1	1	yes	characteristic
153	C₂²×C₈₉	#519	1	1	yes	characteristic
154	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
155	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
156	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
157	C₂²×C₈₉	#568	1	1	yes	characteristic
158	C₄×C₈₉	—	2	4	no	—
159	C₄×C₈₉	—	2	2	no	—
160	C₄×C₈₉	—	2	4	no	—
161	C₄×C₈₉	—	2	2	no	—
162	C₄×C₈₉	—	2	4	no	—
163	C₄×C₈₉	—	2	2	no	—
164	C₄×C₈₉	—	2	2	no	—
165	C₄×C₈₉	—	4	2	no	—
166	C₄×C₈₉	—	4	1	no	—
167	C₄×C₈₉	#423	1	4	yes	—
168	C₄×C₈₉	#548	1	2	yes	—
169	C₄×C₈₉	#557	1	2	yes	—
170	#22736	—	89	1	no	maximal, Sylow
171	C₂×C₈₉	#2351	1	1	yes	characteristic
172	C₂×C₈₉	C₂×#423	1	1	yes	characteristic
173	C₂×C₈₉	#2906	1	1	yes	characteristic
174	C₂×C₈₉	#2965	1	1	yes	characteristic
175	C₂×C₈₉	#2970	1	1	yes	characteristic
176	C₂×C₈₉	#2983	1	1	yes	characteristic
177	C₂×C₈₉	#2984	1	1	yes	characteristic
178	C₂×#446	—	89	2	no	—
179	C₄²×Q₈	D₈₉	1	1	yes	characteristic
180	#1940	—	89	4	no	—
181	#1939	—	89	2	no	—
182	#1831	—	89	2	no	—
183	#1831	—	89	1	no	—
184	#1767	—	89	2	no	—
185	C₄×C₄⋊C₈	—	89	1	no	—
186	C₈₉	#22736	1	1	yes	characteristic, complemented, Sylow
187	C₂×C₄×Q₈	Dic₈₉	1	1	yes	characteristic
188	C₂×C₄×Q₈	Dic₈₉	1	1	yes	characteristic
189	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
190	#446	—	178	8	no	—
191	C₂×C₄⋊C₈	—	89	2	no	—
192	C₂×C₄⋊C₈	—	89	4	no	—
193	C₂×C₄⋊C₈	—	89	4	no	—
194	C₂×C₄⋊C₈	—	89	2	no	—
195	C₂×#97	—	89	4	no	—
196	C₂×C₄×C₈	—	89	2	no	—
197	C₄×C₄⋊C₄	Dic₈₉	1	1	yes	characteristic
198	C₄×C₄⋊C₄	Dic₈₉	1	2	yes	—
199	C₄×C₄⋊C₄	Dic₈₉	1	2	yes	—
200	C₄×C₄⋊C₄	Dic₈₉	1	1	yes	characteristic
201	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
202	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
203	C₄³	C₂×D₈₉	1	1	yes	characteristic
204	C₄³	C₂×D₈₉	1	2	yes	—
205	#336	—	89	1	no	—
206	#336	—	89	2	no	—
207	#336	—	89	4	no	—
208	#336	—	89	2	no	—
209	#336	—	89	1	no	—
210	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
211	C₂²×C₈	—	89	2	no	—
212	C₂²×C₈	—	89	2	no	—
213	C₂²×C₈	—	89	4	no	—
214	C₄×Q₈	—	2	4	no	—
215	C₄×Q₈	—	2	4	no	—
216	C₄×Q₈	Dic₁₇₈	1	4	yes	—
217	C₄×Q₈	D₃₅₆	1	4	yes	—
218	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
220	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
221	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
222	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
223	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
224	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
225	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
226	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
227	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
228	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
229	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
230	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
231	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
234	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
237	C₄⋊C₈	—	178	8	no	—
238	C₄⋊C₈	—	178	4	no	—
239	C₄⋊C₈	—	178	4	no	—
240	C₄⋊C₈	—	178	8	no	—
241	#97	—	178	8	no	—
242	C₄×C₈	—	178	4	no	—
243	#95	C₂×Dic₈₉	1	2	yes	—
244	#95	C₂×Dic₈₉	1	2	yes	—
245	#95	C₂×Dic₈₉	1	2	yes	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	C₄⋊Dic₈₉	1	4	yes	—
249	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
250	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
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	C₂²×Dic₈₉	1	2	yes	—
257	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
258	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
259	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
260	C₂×C₈	—	89	4	no	—
261	C₂×C₈	—	178	2	no	—
262	C₂×C₈	—	178	4	no	—
263	C₂×C₈	—	178	2	no	—
264	C₂×C₈	—	178	4	no	—
265	C₂×C₈	—	178	2	no	—
266	C₂×C₈	—	178	4	no	—
267	C₂×C₈	—	178	2	no	—
268	C₂×C₈	—	178	2	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	C₄⋊Dic₈₉	1	8	yes	—
273	C₄⋊C₄	C₄⋊Dic₈₉	1	4	yes	—
274	C₄⋊C₄	#13083558	1	4	yes	—
275	C₄²	—	2	4	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	4	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	4	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	—	4	1	no	—
285	C₄²	#13083558	1	4	yes	—
286	C₄²	C₂×D₃₅₆	1	2	yes	—
287	C₄²	C₂×D₃₅₆	1	4	yes	—
288	C₄²	C₂×Dic₁₇₈	1	4	yes	—
289	C₄²	C₂×Dic₁₇₈	1	2	yes	—
290	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
291	Q₈	—	2	8	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
313	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
314	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
315	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
316	C₂×C₄	C₂×#13083558	1	2	yes	—
317	C₂×C₄	C₂×#13083558	1	4	yes	—
318	C₂×C₄	C₂×#13083558	1	2	yes	—
319	C₂×C₄	C₂×#13083558	1	2	yes	—
320	C₂×C₄	C₂×#13083558	1	2	yes	—
321	C₂×C₄	#15668019	1	2	yes	—
322	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
323	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
324	C₂×C₄	#41×D₈₉	1	2	yes	—
325	C₂×C₄	#15668060	1	2	yes	—
326	C₈	—	356	2	no	—
327	C₈	—	356	4	no	—
328	C₈	—	178	4	no	—
329	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
330	C₂²	#41×Dic₈₉	1	1	yes	characteristic
331	C₂²	#19152076	1	1	yes	characteristic
332	C₂²	C₂²×#13083558	1	1	yes	characteristic
333	C₂²	C₂²×#13083558	1	1	yes	characteristic
334	C₂²	C₂×#15668019	1	1	yes	characteristic
335	C₂²	#19152365	1	1	yes	characteristic
336	C₄	—	2	4	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	4	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	4	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	1	no	—
345	C₄	#19151622	1	4	yes	—
346	C₄	#19152260	1	2	yes	—
347	C₄	#19152324	1	2	yes	—
348	C₂	#26023173	1	1	yes	characteristic
349	C₂	C₂×#19151622	1	1	yes	characteristic
350	C₂	#26028820	1	1	yes	characteristic
351	C₂	#26029219	1	1	yes	characteristic
352	C₂	#26029266	1	1	yes	characteristic
353	C₂	#26029337	1	1	yes	characteristic
354	C₂	#26029341	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂³×C₄
Fitting	C₄²×Q₈×C₈₉	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄²×Q₈	D₈₉
hypocenter	C₈₉	#22736
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
186. C₈₉	#22736	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151729	2
#26020976	2
#26020971	2
#26020369	2
#26020105	2
C₄×C₄⋊#10623494	2
C₄²×Q₈×C₈₉	2
#22736	89