Subgroups — #36520041

Order 22784.

826 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#13083574	C₂	1	1	yes	maximal, characteristic
2	#26029208	C₂	1	1	yes	maximal, characteristic
3	C₄×#15668020	C₂	1	1	yes	maximal, characteristic
4	#26018545	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018543	C₂	1	2	yes	maximal, complemented
6	#26018539	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018538	C₂	1	2	yes	maximal, complemented
8	#26018538	C₂	1	2	yes	maximal, complemented
9	C₂×#19151319	C₂	1	1	yes	maximal, characteristic, complemented
10	#322×D₈₉	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668020	C₄	1	2	yes	—
13	C₄×#13083574	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₇₈	C₄	1	2	yes	—
15	#19151722	C₄	1	2	yes	—
16	C₈×D₃₅₆	C₄	1	2	yes	—
17	#19151663	C₄	1	2	yes	—
18	C₈×C₂²⋊D₈₉	C₄	1	4	yes	—
19	#19151661	C₄	1	4	yes	—
20	#19151651	C₂²	1	1	yes	characteristic
21	#19151649	C₂²	1	2	yes	—
22	#19151649	C₂²	1	2	yes	—
23	#19151556	C₄	1	2	yes	—
24	#19151555	C₄	1	4	yes	—
25	C₄×#13083557	C₂²	1	2	yes	—
26	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
27	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
28	C₄×C₈×D₈₉	C₂²	1	2	yes	—
29	C₄×C₈×D₈₉	C₂²	1	2	yes	—
30	C₄×C₈×D₈₉	C₂²	1	2	yes	—
31	#19151320	C₂²	1	2	yes	—
32	#19151320	C₂²	1	2	yes	—
33	#19151320	C₂²	1	2	yes	—
34	#19151319	C₂²	1	1	yes	characteristic
35	#19151319	C₂²	1	1	yes	characteristic
36	#19151319	C₂²	1	1	yes	characteristic
37	#19151319	C₂²	1	1	yes	characteristic
38	#19151319	C₂²	1	2	yes	—
39	#431×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
42	#322×C₈₉	C₂²	1	2	yes	—
43	#322×C₈₉	C₂²	1	2	yes	—
44	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
45	#15668020	—	2	8	no	—
46	C₂×#13083557	C₂×C₄	1	4	yes	—
47	C₂²×#10623494	C₂×C₄	1	2	yes	—
48	C₂²×#10623494	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
50	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
51	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
52	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
54	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
55	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
56	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
57	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
58	#15667971	C₂×C₄	1	1	yes	characteristic
59	#15667969	C₂×C₄	1	2	yes	—
60	C₄²×D₈₉	C₂³	1	2	yes	—
61	C₄⋊#10623494	C₂×C₄	1	2	yes	—
62	#15667941	C₂×C₄	1	4	yes	—
63	#15667941	C₂×C₄	1	4	yes	—
64	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
65	#15667924	C₂×C₄	1	4	yes	—
66	#15667924	C₂×C₄	1	4	yes	—
67	#15667922	C₂×C₄	1	4	yes	—
68	C₄×#10623494	C₂×C₄	1	2	yes	—
69	C₄×#10623494	C₂×C₄	1	2	yes	—
70	C₄×#10623494	C₂³	1	1	yes	characteristic
71	C₄×#10623494	C₂³	1	2	yes	—
72	C₄×#10623494	C₂³	1	1	yes	characteristic
73	C₄×#10623494	C₂³	1	1	yes	characteristic
74	C₄×#10623494	C₂³	1	1	yes	characteristic
75	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
76	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
77	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
78	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
79	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
80	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
82	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
83	#97×C₈₉	C₂×C₄	1	2	yes	—
84	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₈₉	C₂³	1	2	yes	—
88	C₄×C₈×C₈₉	C₂³	1	2	yes	—
89	#13083574	M₁₆	1	8	yes	—
90	C₂×Dic₁₇₈	C₄²	1	1	yes	characteristic
91	C₂×D₃₅₆	C₄²	1	1	yes	characteristic
92	C₂×C₂²⋊D₈₉	C₄²	1	2	yes	—
93	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
96	#13083557	—	2	8	no	—
97	C₂×#10623494	—	2	4	no	—
98	C₂×#10623494	—	4	2	no	—
99	C₂×#10623494	C₂²×C₄	1	2	yes	—
100	C₂×#10623494	C₂²×C₄	1	2	yes	—
101	C₂×#10623494	C₂²×C₄	1	4	yes	—
102	C₂×#10623494	C₂²×C₄	1	2	yes	—
103	C₂×#10623494	C₂²×C₄	1	2	yes	—
104	C₈×D₈₉	—	2	8	no	—
105	C₈×D₈₉	—	2	8	no	—
106	C₈×D₈₉	—	4	4	no	—
107	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
108	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
109	#13083553	C₄²	1	2	yes	—
110	#13083553	C₄²	1	2	yes	—
111	C₂²⋊Dic₈₉	C₄²	1	2	yes	—
112	#13083551	C₄²	1	2	yes	—
113	#13083551	C₄²	1	2	yes	—
114	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
115	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
116	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
120	M₁₆×C₈₉	—	2	4	no	—
121	C₂×C₈×C₈₉	—	2	4	no	—
122	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
123	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
124	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
126	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
131	C₂²×D₈₉	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
138	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
141	C₂²⋊D₈₉	#97	1	8	yes	—
142	C₂×Dic₈₉	C₂×C₄²	1	2	yes	—
143	C₂×Dic₈₉	C₂×C₄²	1	2	yes	—
144	C₂×Dic₈₉	C₂×C₄²	1	2	yes	—
145	D₃₅₆	#97	1	4	yes	—
146	C₄×D₈₉	—	2	4	no	—
147	C₄×D₈₉	—	2	4	no	—
148	C₄×D₈₉	C₂×M₁₆	1	8	yes	—
149	Dic₁₇₈	#97	1	4	yes	—
150	C₈×C₈₉	—	2	4	no	—
151	C₈×C₈₉	—	2	2	no	—
152	C₈×C₈₉	—	2	4	no	—
153	C₈×C₈₉	#131	1	4	yes	complemented
154	#10623494	—	2	2	no	—
155	#10623494	—	2	2	no	—
156	#10623494	—	2	4	no	—
157	#10623494	—	2	2	no	—
158	#10623494	—	2	2	no	—
159	C₂²×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	C₂×#97	1	2	yes	—
162	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
163	C₂×D₈₉	—	2	2	no	—
164	C₂×D₈₉	—	2	2	no	—
165	C₂×D₈₉	C₂×#97	1	4	yes	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	C₂×#97	1	2	yes	—
169	C₄×C₈₉	C₂²×M₁₆	1	2	yes	—
170	C₄×C₈₉	C₂×#131	1	2	yes	—
171	C₄×C₈₉	#568	1	2	yes	—
172	Dic₈₉	—	2	2	no	—
173	Dic₈₉	—	2	2	no	—
174	Dic₈₉	C₂×#97	1	4	yes	—
175	#17320	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	—	2	2	no	—
177	C₂×C₈₉	C₂²×#97	1	1	yes	characteristic
178	C₂×C₈₉	C₄×#131	1	1	yes	characteristic
179	C₂×C₈₉	#2875	1	1	yes	characteristic
180	D₈₉	—	2	4	no	—
181	C₈×#41	—	89	1	no	—
182	C₈×#41	—	89	1	no	—
183	C₄×#131	—	89	1	no	—
184	#1452	—	89	1	no	—
185	#1452	—	89	2	no	—
186	#1451	—	89	1	no	—
187	#1451	—	89	2	no	—
188	#1451	—	89	2	no	—
189	#1451	D₈₉	1	1	yes	characteristic, complemented
190	C₂×#322	—	89	1	no	—
191	C₂×#322	—	89	2	no	—
192	C₈₉	#17320	1	1	yes	characteristic, complemented, Sylow
193	C₂×#131	—	89	2	no	—
194	C₄×#41	—	89	1	no	—
195	C₈×Q₈	—	89	2	no	—
196	C₈×Q₈	—	89	2	no	—
197	C₈×D₄	—	89	4	no	—
198	C₈×D₄	—	89	2	no	—
199	C₈×D₄	—	89	4	no	—
200	C₈×D₄	—	89	2	no	—
201	#431	—	89	1	no	—
202	#431	—	89	2	no	—
203	#431	—	89	2	no	—
204	#431	C₂×D₈₉	1	1	yes	characteristic
205	#405	—	89	2	no	—
206	#405	—	89	4	no	—
207	C₄×M₁₆	—	89	2	no	—
208	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
209	C₂×C₄×C₈	—	89	1	no	—
210	C₂×C₄×C₈	—	89	1	no	—
211	C₂×C₄×C₈	—	89	2	no	—
212	C₂×C₄×C₈	—	89	2	no	—
213	C₂×C₄×C₈	—	89	2	no	—
214	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
215	#322	—	89	1	no	—
216	#322	—	89	1	no	—
217	#322	—	89	1	no	—
218	#322	—	89	2	no	—
219	#322	—	89	1	no	—
220	#322	—	89	2	no	—
221	#322	—	89	2	no	—
222	#322	—	89	2	no	—
223	#322	C₂×D₈₉	1	2	yes	—
224	#322	C₂×D₈₉	1	2	yes	—
225	C₂×#41	—	89	1	no	—
226	#131	—	178	8	no	—
227	C₂×M₁₆	—	89	4	no	—
228	C₂×M₁₆	C₄×D₈₉	1	2	yes	—
229	C₂²×C₈	—	89	2	no	—
230	C₂²×C₈	—	89	2	no	—
231	C₂²×C₈	—	89	4	no	—
232	C₂²×C₈	—	89	4	no	—
233	C₂²×C₈	—	89	4	no	—
234	C₂²×C₈	C₄×D₈₉	1	2	yes	—
235	C₄×Q₈	—	89	1	no	—
236	C₄×Q₈	—	89	1	no	—
237	C₄×D₄	—	89	2	no	—
238	C₄×D₄	—	89	2	no	—
239	C₄×D₄	—	89	1	no	—
240	C₄×D₄	—	89	1	no	—
241	#117	—	89	1	no	—
242	#117	—	89	2	no	—
243	C₂×C₄²	—	89	2	no	—
244	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
245	C₄⋊C₈	—	89	4	no	—
246	C₄⋊C₈	—	89	4	no	—
247	C₄⋊C₈	—	89	2	no	—
248	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
249	C₂²⋊C₈	—	89	2	no	—
250	C₂²⋊C₈	—	89	4	no	—
251	C₂²⋊C₈	—	89	4	no	—
252	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
253	#97	—	89	4	no	—
254	#97	C₄×D₈₉	1	2	yes	—
255	C₄×C₈	—	89	4	no	—
256	C₄×C₈	—	89	4	no	—
257	C₄×C₈	—	89	4	no	—
258	C₄×C₈	—	89	2	no	—
259	C₄×C₈	—	89	2	no	—
260	C₄×C₈	—	89	1	no	—
261	C₄×C₈	—	89	1	no	—
262	C₄×C₈	—	89	1	no	—
263	C₄×C₈	—	89	1	no	—
264	C₄×C₈	—	89	2	no	—
265	C₄×C₈	C₄×D₈₉	1	2	yes	—
266	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₈₉	1	2	yes	—
268	C₄×C₈	C₂²×D₈₉	1	2	yes	—
269	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
270	#41	—	89	8	no	—
271	C₂×Q₈	—	89	1	no	—
272	C₂×D₄	—	89	2	no	—
273	C₂×D₄	—	89	1	no	—
274	C₂²×C₄	—	89	2	no	—
275	C₂²×C₄	—	89	2	no	—
276	C₂²×C₄	—	89	2	no	—
277	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
280	M₁₆	—	2	4	no	—
281	M₁₆	—	178	8	no	—
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	89	2	no	—
284	C₂×C₈	—	89	2	no	—
285	C₂×C₈	—	89	2	no	—
286	C₂×C₈	—	89	4	no	—
287	C₂×C₈	—	89	2	no	—
288	C₂×C₈	—	356	4	no	—
289	C₂×C₈	—	356	2	no	—
290	C₂×C₈	—	178	8	no	—
291	C₂×C₈	—	178	4	no	—
292	C₂×C₈	—	178	8	no	—
293	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
294	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
295	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
296	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
297	C₄⋊C₄	—	89	2	no	—
298	C₄⋊C₄	—	89	2	no	—
299	C₄⋊C₄	—	89	1	no	—
300	C₄⋊C₄	—	89	1	no	—
301	C₂²⋊C₄	—	89	2	no	—
302	C₂²⋊C₄	—	89	2	no	—
303	C₂²⋊C₄	—	89	2	no	—
304	C₄²	—	89	2	no	—
305	C₄²	—	89	2	no	—
306	C₄²	—	89	2	no	—
307	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
308	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
309	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
310	C₄²	C₂³×D₈₉	1	1	yes	characteristic
311	C₂³	—	89	2	no	—
312	C₂³	C₄²×D₈₉	1	1	yes	characteristic
313	Q₈	—	89	4	no	—
314	D₄	—	89	8	no	—
315	D₄	—	89	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	89	2	no	—
319	C₂×C₄	—	89	8	no	—
320	C₂×C₄	—	89	2	no	—
321	C₂×C₄	—	89	2	no	—
322	C₂×C₄	—	178	4	no	—
323	C₂×C₄	—	178	4	no	—
324	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
325	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
326	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
327	C₂×C₄	M₁₆×D₈₉	1	4	yes	—
328	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
331	C₈	—	2	4	no	—
332	C₈	—	2	2	no	—
333	C₈	—	2	4	no	—
334	C₈	—	178	2	no	—
335	C₈	—	178	2	no	—
336	C₈	—	178	4	no	—
337	C₈	—	178	2	no	—
338	C₈	—	178	2	no	—
339	C₈	#15668020	1	4	yes	complemented
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	89	4	no	—
343	C₂²	—	178	2	no	—
344	C₂²	—	178	2	no	—
345	C₂²	#97×D₈₉	1	2	yes	—
346	C₂²	C₂×C₄²×D₈₉	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	89	4	no	—
350	C₄	—	178	2	no	—
351	C₄	—	178	2	no	—
352	C₄	#97×D₈₉	1	2	yes	—
353	C₄	C₂×M₁₆×D₈₉	1	2	yes	—
354	C₄	C₂×#15668020	1	2	yes	—
355	C₄	#19152363	1	2	yes	—
356	C₂	—	2	2	no	—
357	C₂	—	178	4	no	—
358	C₂	C₂×#97×D₈₉	1	1	yes	characteristic
359	C₂	C₄×#15668020	1	1	yes	characteristic
360	C₂	#26028679	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₄²
Fitting	C₈₉×#1451	C₂
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1451	D₈₉
hypocenter	C₈₉	#17320
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ #1451

Complements

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

N	H ≅ G/N	type
4. #26018545	C₂	semidirect
5. #26018543	C₂	semidirect
6. #26018539	C₂	semidirect
7. #26018538	C₂	semidirect
8. #26018538	C₂	semidirect
9. C₂×#19151319	C₂	semidirect
10. #322×D₈₉	C₂	semidirect
11. C₈₉×#1451	C₂	semidirect
153. C₈×C₈₉	#131	semidirect
189. #1451	D₈₉	semidirect
192. C₈₉	#17320	semidirect
339. C₈	#15668020	semidirect

Maximal subgroups

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

subgroup	index
C₈×#13083574	2
#26029208	2
C₄×#15668020	2
#26018545	2
#26018543	2
#26018539	2
#26018538	2
C₂×#19151319	2
#322×D₈₉	2
C₈₉×#1451	2
#17320	89