Subgroups — #36553045

Order 22784.

886 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	C₂×#19151729	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#10623494	C₂	1	1	yes	maximal, characteristic
3	#26020127	C₂	1	2	yes	maximal
4	#26020121	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×#10623494	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×#10623494	C₂	1	2	yes	maximal
7	#26020109	C₂	1	2	yes	maximal
8	#26020105	C₂	1	1	yes	maximal, characteristic
9	#26019939	C₂	1	2	yes	maximal
10	C₄²×#10623494	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2295	C₂	1	1	yes	maximal, characteristic
12	#19151729	C₄	1	8	yes	complemented
13	Q₈×#10623494	C₄	1	8	yes	complemented
14	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
17	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
18	C₂×#15667923	C₂²	1	2	yes	—
19	C₂×#15667923	C₂²	1	2	yes	—
20	C₂×#15667923	C₂²	1	2	yes	—
21	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
22	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
23	C₂×C₄×#10623494	C₂²	1	2	yes	—
24	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
25	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
26	C₂×C₄×#10623494	C₂²	1	2	yes	—
27	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
28	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
29	#19151353	C₂²	1	2	yes	—
30	#19151353	C₂²	1	2	yes	—
31	#19151353	C₂²	1	2	yes	—
32	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
33	#391×C₈₉	C₄	1	2	yes	—
34	#389×C₈₉	C₄	1	2	yes	—
35	#387×C₈₉	C₄	1	4	yes	—
36	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
38	#376×C₈₉	C₂²	1	2	yes	—
39	C₄³×C₈₉	C₂²	1	1	yes	characteristic
40	C₂²×#10623494	C₂³	1	1	yes	characteristic
41	C₂²×#10623494	C₂³	1	1	yes	characteristic
42	C₂²×#10623494	C₂³	1	2	yes	—
43	C₂²×#10623494	C₂³	1	2	yes	—
44	C₂²×#10623494	C₂³	1	1	yes	characteristic
45	C₂²×#10623494	C₂³	1	1	yes	characteristic
46	C₄⋊#10623494	C₂×C₄	1	4	yes	—
47	C₄⋊#10623494	C₂×C₄	1	8	yes	—
48	C₄⋊#10623494	C₂×C₄	1	4	yes	—
49	C₄⋊#10623494	C₂×C₄	1	8	yes	—
50	#15667923	—	2	4	no	—
51	#15667923	—	2	4	no	—
52	#15667923	C₂×C₄	1	8	yes	—
53	C₄×#10623494	—	2	2	no	—
54	C₄×#10623494	—	2	4	no	—
55	C₄×#10623494	—	2	2	no	—
56	C₄×#10623494	—	2	2	no	—
57	C₄×#10623494	C₂×C₄	1	4	yes	—
58	C₄×#10623494	C₂×C₄	1	8	yes	—
59	C₄×#10623494	C₂×C₄	1	4	yes	—
60	C₄×#10623494	Q₈	1	4	yes	complemented
61	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
63	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
65	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
67	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
72	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
74	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
75	#95×C₈₉	C₂×C₄	1	4	yes	—
76	#95×C₈₉	C₂×C₄	1	4	yes	—
77	#95×C₈₉	C₂×C₄	1	2	yes	—
78	#95×C₈₉	C₂×C₄	1	4	yes	—
79	C₂×#10623494	—	2	1	no	—
80	C₂×#10623494	—	2	1	no	—
81	C₂×#10623494	—	2	2	no	—
82	C₂×#10623494	—	2	1	no	—
83	C₂×#10623494	—	2	1	no	—
84	C₂×#10623494	—	2	2	no	—
85	C₂×#10623494	—	2	2	no	—
86	C₂×#10623494	—	2	2	no	—
87	C₂×#10623494	C₂²×C₄	1	2	yes	—
88	C₂×#10623494	C₂²×C₄	1	2	yes	—
89	C₂×#10623494	C₂²×C₄	1	4	yes	—
90	C₂×#10623494	C₂²×C₄	1	4	yes	—
91	C₂×#10623494	C₂²×C₄	1	2	yes	—
92	C₂×#10623494	C₂²×C₄	1	2	yes	—
93	C₂×#10623494	C₂×Q₈	1	2	yes	—
94	C₂×#10623494	C₂×Q₈	1	2	yes	—
95	C₂×#10623494	#41	1	2	yes	—
96	C₂×#10623494	#41	1	2	yes	—
97	C₂×Q₈×C₈₉	—	2	4	no	—
98	C₂×Q₈×C₈₉	C₄²	1	4	yes	—
99	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
100	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	—	2	8	no	—
108	C₄⋊C₄×C₈₉	—	2	4	no	—
109	C₄⋊C₄×C₈₉	C₄²	1	8	yes	—
110	C₄⋊C₄×C₈₉	C₄²	1	4	yes	—
111	C₄²×C₈₉	—	2	4	no	—
112	C₄²×C₈₉	—	2	4	no	—
113	C₄²×C₈₉	—	2	1	no	—
114	C₄²×C₈₉	—	2	1	no	—
115	C₄²×C₈₉	—	2	2	no	—
116	C₄²×C₈₉	—	2	2	no	—
117	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
118	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
119	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
120	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
121	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
122	C₄²×C₈₉	#41	1	2	yes	—
123	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
124	Q₈×C₈₉	—	2	8	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	—	2	2	no	—
129	C₂×C₄×C₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	—	2	4	no	—
131	C₂×C₄×C₈₉	—	2	4	no	—
132	C₂×C₄×C₈₉	—	2	4	no	—
133	C₂×C₄×C₈₉	—	2	4	no	—
134	C₂×C₄×C₈₉	—	2	4	no	—
135	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
136	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
137	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
138	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
139	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
140	C₂×C₄×C₈₉	#131	1	4	yes	—
141	C₂×C₄×C₈₉	#131	1	4	yes	—
142	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
148	#10623494	—	2	2	no	—
149	#10623494	—	2	2	no	—
150	#10623494	—	2	4	no	—
151	#10623494	—	2	4	no	—
152	#10623494	C₄×Q₈	1	4	yes	—
153	#10623494	C₄×Q₈	1	4	yes	—
154	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
155	C₂²×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₂²×C₈₉	C₄×#41	1	1	yes	characteristic
157	C₂²×C₈₉	C₂×#131	1	2	yes	—
158	C₂²×C₈₉	C₂×#131	1	2	yes	—
159	C₄×C₈₉	—	2	4	no	—
160	C₄×C₈₉	—	2	2	no	—
161	C₄×C₈₉	—	2	4	no	—
162	C₄×C₈₉	—	2	4	no	—
163	C₄×C₈₉	—	2	4	no	—
164	C₄×C₈₉	—	4	4	no	—
165	C₄×C₈₉	#405	1	4	yes	—
166	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
169	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
170	#20516	—	89	1	no	maximal, Sylow
171	C₂×C₈₉	C₄²×Q₈	1	1	yes	characteristic
172	C₂×C₈₉	C₂×#405	1	1	yes	characteristic
173	C₂×C₈₉	C₄×#131	1	1	yes	characteristic
174	C₂×C₈₉	#2960	1	2	yes	—
175	C₂×C₈₉	#2966	1	2	yes	—
176	C₂×#446	—	89	1	no	—
177	C₂×C₈×Q₈	—	89	1	no	—
178	#2295	D₈₉	1	1	yes	characteristic
179	#1771	—	89	2	no	—
180	#1770	—	89	1	no	—
181	C₈×C₄⋊C₄	—	89	2	no	—
182	C₈×C₄⋊C₄	—	89	1	no	—
183	#1768	—	89	2	no	—
184	#1767	—	89	1	no	—
185	#1726	—	89	2	no	—
186	C₄²×C₈	—	89	1	no	—
187	C₈₉	#20516	1	1	yes	characteristic, complemented, Sylow
188	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
189	#446	—	89	8	no	—
190	C₈×Q₈	—	89	8	no	—
191	C₂×C₄⋊C₈	—	89	1	no	—
192	C₂×C₄⋊C₈	—	89	2	no	—
193	C₂×C₄⋊C₈	—	89	1	no	—
194	C₂×C₄⋊C₈	—	89	2	no	—
195	C₂×#97	—	89	2	no	—
196	C₂×#97	—	89	2	no	—
197	C₂×#97	—	89	2	no	—
198	C₂×C₄×C₈	—	89	1	no	—
199	C₂×C₄×C₈	—	89	2	no	—
200	C₂×C₄×C₈	—	89	1	no	—
201	C₂×C₄×C₈	—	89	1	no	—
202	C₂×C₄×C₈	—	89	1	no	—
203	C₂×C₄×C₈	—	89	1	no	—
204	C₂×C₄×C₈	—	89	1	no	—
205	C₂×C₄×C₈	—	89	2	no	—
206	#391	Dic₈₉	1	2	yes	—
207	#389	Dic₈₉	1	2	yes	—
208	#387	Dic₈₉	1	4	yes	—
209	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
210	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
211	#376	C₂×D₈₉	1	2	yes	—
212	C₄³	C₂×D₈₉	1	1	yes	characteristic
213	#336	—	89	2	no	—
214	#336	—	89	2	no	—
215	#336	—	89	2	no	—
216	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
217	C₂²×C₈	—	89	1	no	—
218	C₂²×C₈	—	89	1	no	—
219	C₂²×C₈	—	89	1	no	—
220	C₂²×C₈	—	89	1	no	—
221	C₂²×C₈	—	89	2	no	—
222	C₂²×C₈	—	89	2	no	—
223	C₄×Q₈	C₄×D₈₉	1	4	yes	—
224	C₄×Q₈	C₄×D₈₉	1	4	yes	—
225	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
226	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
227	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
228	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
229	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
230	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
231	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
233	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
234	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
235	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
236	C₄⋊C₈	—	89	4	no	—
237	C₄⋊C₈	—	89	8	no	—
238	C₄⋊C₈	—	89	4	no	—
239	C₄⋊C₈	—	89	8	no	—
240	#97	—	89	8	no	—
241	#97	—	178	4	no	—
242	#97	—	178	4	no	—
243	C₄×C₈	—	89	4	no	—
244	C₄×C₈	—	89	8	no	—
245	C₄×C₈	—	89	4	no	—
246	C₄×C₈	—	89	4	no	—
247	C₄×C₈	—	178	2	no	—
248	C₄×C₈	—	178	4	no	—
249	C₄×C₈	—	178	2	no	—
250	C₄×C₈	—	178	2	no	—
251	#95	C₂×Dic₈₉	1	4	yes	—
252	#95	C₂×Dic₈₉	1	4	yes	—
253	#95	C₂×Dic₈₉	1	2	yes	—
254	#95	C₂×Dic₈₉	1	4	yes	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	C₄×Dic₈₉	1	4	yes	—
257	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
258	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
259	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
260	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
262	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
263	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
264	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
265	C₂×C₈	—	89	2	no	—
266	C₂×C₈	—	89	2	no	—
267	C₂×C₈	—	89	2	no	—
268	C₂×C₈	—	89	2	no	—
269	C₂×C₈	—	89	2	no	—
270	C₂×C₈	—	89	2	no	—
271	C₂×C₈	—	89	2	no	—
272	C₂×C₈	—	89	4	no	—
273	C₂×C₈	—	89	2	no	—
274	C₂×C₈	—	89	4	no	—
275	C₂×C₈	—	178	1	no	—
276	C₂×C₈	—	178	1	no	—
277	C₂×C₈	—	178	2	no	—
278	C₂×C₈	—	178	2	no	—
279	C₂×C₈	—	178	1	no	—
280	C₂×C₈	—	178	1	no	—
281	C₂×C₈	—	178	2	no	—
282	C₂×C₈	—	178	2	no	—
283	C₄⋊C₄	—	2	8	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	C₄×Dic₈₉	1	8	yes	—
286	C₄⋊C₄	C₄×Dic₈₉	1	4	yes	—
287	C₄²	—	2	4	no	—
288	C₄²	—	2	4	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	C₂×C₄×D₈₉	1	4	yes	—
294	C₄²	C₂×C₄×D₈₉	1	2	yes	—
295	C₄²	C₂×C₄×D₈₉	1	2	yes	—
296	C₄²	C₂×C₄×D₈₉	1	4	yes	—
297	C₄²	Q₈×D₈₉	1	2	yes	—
298	C₄²	#13083575	1	2	yes	—
299	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
300	Q₈	—	2	8	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
312	C₂×C₄	C₂×C₄×Dic₈₉	1	4	yes	—
313	C₂×C₄	C₂×C₄×Dic₈₉	1	4	yes	—
314	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
315	C₂×C₄	Q₈×Dic₈₉	1	4	yes	—
316	C₂×C₄	#15668019	1	4	yes	—
317	C₂×C₄	#15668019	1	4	yes	—
318	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
320	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
321	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
322	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
323	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
324	C₈	—	89	4	no	—
325	C₈	—	89	4	no	—
326	C₈	—	178	2	no	—
327	C₈	—	178	2	no	—
328	C₈	—	178	4	no	—
329	C₈	—	178	4	no	—
330	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
331	C₂²	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
332	C₂²	#41×Dic₈₉	1	1	yes	characteristic
333	C₂²	C₂×#15668019	1	2	yes	—
334	C₂²	C₂×#15668019	1	2	yes	—
335	C₄	—	2	4	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	4	no	—
339	C₄	—	2	4	no	—
340	C₄	—	4	4	no	—
341	C₄	#19151554	1	4	yes	—
342	C₄	C₄×Q₈×D₈₉	1	1	yes	characteristic
343	C₄	C₄×Q₈×D₈₉	1	1	yes	characteristic
344	C₄	C₄×#13083575	1	1	yes	characteristic
345	C₄	C₄×#13083575	1	1	yes	characteristic
346	C₂	C₄×Q₈×Dic₈₉	1	1	yes	characteristic
347	C₂	C₂×#19151554	1	1	yes	characteristic
348	C₂	C₄×#15668019	1	1	yes	characteristic
349	C₂	#26029187	1	2	yes	—
350	C₂	#26029228	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂²×C₄²
Fitting	C₈₉×#2295	C₂
socle, prime radical	C₂³×C₈₉	C₂³×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#2295	D₈₉
hypocenter	C₈₉	#20516
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
12. #19151729	C₄	semidirect
13. Q₈×#10623494	C₄	semidirect
60. C₄×#10623494	Q₈	semidirect
187. C₈₉	#20516	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
C₂×Q₈×#10623494	2
#26020127	2
#26020121	2
C₄⋊C₄×#10623494	2
#26020109	2
#26020105	2
#26019939	2
C₄²×#10623494	2
C₈₉×#2295	2
#20516	89