Subgroups — #36523413

Order 22784.

734 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029361	C₂	1	2	yes	maximal
2	C₄×#15668018	C₂	1	1	yes	maximal, characteristic
3	#26019002	C₂	1	2	yes	maximal, complemented
4	#26019001	C₂	1	4	yes	maximal, complemented
5	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018997	C₂	1	2	yes	maximal, complemented
7	#334×D₈₉	C₂	1	2	yes	maximal, complemented
8	C₂×#334×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668018	C₄	1	1	yes	characteristic
10	C₂×#15668018	C₄	1	1	yes	characteristic
11	C₄×#13083574	C₂²	1	1	yes	characteristic
12	#19151722	—	2	2	no	—
13	#19151663	—	2	2	no	—
14	C₈×C₂²⋊D₈₉	—	2	4	no	—
15	#19151651	C₂²	1	2	yes	—
16	#19151650	C₂²	1	4	yes	—
17	C₄⋊C₈×D₈₉	C₂²	1	4	yes	—
18	#19151553	C₄	1	1	yes	characteristic
19	#19151553	C₄	1	1	yes	characteristic
20	#19151552	C₄	1	2	yes	—
21	#19151552	C₄	1	2	yes	—
22	C₄×#13083558	C₂²	1	1	yes	characteristic
23	C₄×#13083557	C₂²	1	2	yes	—
24	C₄×C₈×D₈₉	C₂²	1	2	yes	—
25	#19151351	C₂²	1	2	yes	—
26	#19151350	C₂²	1	4	yes	—
27	#19151349	C₂²	1	1	yes	characteristic
28	#19151349	C₂²	1	1	yes	characteristic
29	#19151348	C₂²	1	4	yes	—
30	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	#334×C₈₉	C₂²	1	2	yes	—
33	#334×C₈₉	C₂²	1	2	yes	—
34	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
35	#15668018	—	2	4	no	—
36	#15668018	C₈	1	8	yes	complemented
37	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
38	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
39	C₂×#13083557	C₂×C₄	1	2	yes	—
40	C₂×#13083557	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₈₉	—	2	4	no	—
42	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
45	C₄×Dic₁₇₈	Q₈	1	1	yes	characteristic
46	C₄×D₃₅₆	D₄	1	1	yes	characteristic
47	C₄×D₃₅₆	Q₈	1	1	yes	characteristic
48	C₄×C₂²⋊D₈₉	D₄	1	2	yes	—
49	C₄×C₂²⋊D₈₉	Q₈	1	2	yes	—
50	#15667971	C₂×C₄	1	1	yes	characteristic
51	#15667969	C₂×C₄	1	2	yes	—
52	C₄²×D₈₉	C₂³	1	2	yes	—
53	C₄⋊#10623494	C₂³	1	2	yes	—
54	C₄⋊#10623494	C₂³	1	2	yes	—
55	#15667941	—	2	4	no	—
56	C₂²⋊#10623494	—	2	2	no	—
57	#15667924	—	2	4	no	—
58	#15667923	C₂×C₄	1	2	yes	—
59	#15667922	C₂×C₄	1	2	yes	—
60	#15667922	C₂×C₄	1	2	yes	—
61	C₄×#10623494	—	2	2	no	—
62	C₄×#10623494	C₂³	1	2	yes	—
63	C₈×Dic₈₉	—	2	4	no	—
64	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
65	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₈₉	—	2	2	no	—
67	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₈₉	—	2	2	no	—
71	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
72	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	#13083574	C₂×C₈	1	4	yes	—
77	#13083574	M₁₆	1	4	yes	—
78	C₂×Dic₁₇₈	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₃₅₆	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₈₉	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
82	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
83	C₂×C₄×D₈₉	C₂×Q₈	1	2	yes	—
84	#13083558	—	2	2	no	—
85	#13083558	C₂×C₈	1	4	yes	complemented
86	#13083557	—	2	4	no	—
87	#13083557	C₂×C₈	1	8	yes	complemented
88	C₂×#10623494	—	2	2	no	—
89	C₂×#10623494	—	2	2	no	—
90	C₂×#10623494	C₂²×C₄	1	2	yes	—
91	C₂×#10623494	C₂²×C₄	1	2	yes	—
92	C₈×D₈₉	—	2	4	no	—
93	C₈×D₈₉	—	8	4	no	—
94	C₈×D₈₉	C₂×C₈	1	8	yes	complemented
95	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
96	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
97	#13083553	C₄⋊C₄	1	2	yes	—
98	#13083553	C₄⋊C₄	1	2	yes	—
99	C₂²⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
100	#13083551	C₄⋊C₄	1	2	yes	—
101	#13083551	C₄⋊C₄	1	2	yes	—
102	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
103	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
104	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
105	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₈×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	—	2	4	no	—
110	C₂×C₈×C₈₉	—	8	2	no	—
111	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	complemented
112	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
119	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	4	1	no	—
124	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
132	C₂²⋊D₈₉	C₄⋊C₈	1	8	yes	—
133	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
136	D₃₅₆	C₄⋊C₈	1	4	yes	—
137	C₄×D₈₉	—	2	4	no	—
138	C₄×D₈₉	—	4	2	no	—
139	C₄×D₈₉	C₂²×C₈	1	4	yes	—
140	C₄×D₈₉	C₂×M₁₆	1	4	yes	—
141	Dic₁₇₈	C₄⋊C₈	1	4	yes	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	—	2	1	no	—
144	C₈×C₈₉	—	4	4	no	—
145	C₈×C₈₉	C₂²×C₈	1	2	yes	—
146	C₈×C₈₉	C₂²×C₈	1	2	yes	—
147	#10623494	—	2	2	no	—
148	#10623494	—	4	2	no	—
149	#10623494	—	4	2	no	—
150	#10623494	C₂²×C₈	1	4	yes	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
154	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂×D₈₉	—	2	2	no	—
156	C₂×D₈₉	—	2	2	no	—
157	C₂×D₈₉	C₂×C₄⋊C₈	1	4	yes	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
162	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
163	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
164	C₄×C₈₉	#577	1	2	yes	—
165	Dic₈₉	—	2	2	no	—
166	Dic₈₉	—	2	2	no	—
167	Dic₈₉	C₂×C₄⋊C₈	1	4	yes	—
168	#17788	—	89	1	no	maximal, Sylow
169	C₂×C₈₉	—	2	2	no	—
170	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
171	C₂×C₈₉	#2912	1	1	yes	characteristic
172	C₂×C₈₉	#2916	1	1	yes	characteristic
173	D₈₉	—	2	4	no	—
174	#2988	—	89	2	no	—
175	C₄×#131	—	89	1	no	—
176	#1565	—	89	2	no	—
177	#1565	—	89	4	no	—
178	#1564	—	89	1	no	—
179	#1564	—	89	2	no	—
180	C₂×#334	—	89	2	no	—
181	C₂×#334	D₈₉	1	1	yes	characteristic, complemented
182	C₈₉	#17788	1	1	yes	characteristic, complemented, Sylow
183	C₂×#131	—	89	1	no	—
184	C₂×#131	—	89	1	no	—
185	C₄×#41	—	89	1	no	—
186	C₈×Q₈	—	178	2	no	—
187	C₈×D₄	—	178	2	no	—
188	C₈×D₄	—	178	4	no	—
189	#431	—	89	2	no	—
190	#431	—	89	4	no	—
191	C₂×C₄⋊C₈	—	89	4	no	—
192	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
193	#405	—	89	1	no	—
194	#405	—	89	1	no	—
195	#405	—	89	2	no	—
196	#405	—	89	2	no	—
197	C₄×M₁₆	—	89	1	no	—
198	C₄×M₁₆	—	89	2	no	—
199	C₂×C₄×C₈	—	89	2	no	—
200	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
201	#335	—	89	4	no	—
202	#335	—	89	2	no	—
203	#334	—	89	4	no	—
204	#334	—	89	1	no	—
205	#334	—	89	1	no	—
206	#334	C₂×D₈₉	1	2	yes	—
207	#334	C₂×D₈₉	1	2	yes	—
208	C₂×#41	—	89	1	no	—
209	#131	—	89	8	no	—
210	#131	—	178	4	no	—
211	C₂×M₁₆	—	89	1	no	—
212	C₂×M₁₆	—	89	1	no	—
213	C₂×M₁₆	—	89	2	no	—
214	C₂×M₁₆	—	89	2	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	—	89	2	no	—
217	C₂²×C₈	—	89	2	no	—
218	C₂²×C₈	—	178	4	no	—
219	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
220	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
221	C₄×Q₈	—	89	1	no	—
222	C₄×Q₈	—	89	1	no	—
223	C₄×D₄	—	89	2	no	—
224	C₄×D₄	—	89	2	no	—
225	C₄×D₄	—	89	1	no	—
226	C₄×D₄	—	89	1	no	—
227	#117	—	89	1	no	—
228	#117	—	89	2	no	—
229	C₂×C₄²	—	89	2	no	—
230	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	89	2	no	—
233	C₄⋊C₈	—	89	2	no	—
234	C₄⋊C₈	—	178	4	no	—
235	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
236	C₂²⋊C₈	—	178	4	no	—
237	C₂²⋊C₈	—	178	2	no	—
238	#97	—	89	2	no	—
239	#97	—	89	2	no	—
240	#97	—	89	2	no	—
241	C₄×C₈	—	89	2	no	—
242	C₄×C₈	—	89	2	no	—
243	C₄×C₈	—	89	2	no	—
244	C₄×C₈	—	178	4	no	—
245	C₄×C₈	—	178	2	no	—
246	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
247	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
250	#41	—	89	4	no	—
251	#41	—	89	4	no	—
252	C₂×Q₈	—	89	1	no	—
253	C₂×D₄	—	89	2	no	—
254	C₂×D₄	—	89	1	no	—
255	C₂²×C₄	—	89	2	no	—
256	C₂²×C₄	—	89	2	no	—
257	C₂²×C₄	—	89	2	no	—
258	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
259	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
260	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
261	M₁₆	—	89	4	no	—
262	M₁₆	—	89	8	no	—
263	M₁₆	—	178	4	no	—
264	M₁₆	—	178	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	8	2	no	—
268	C₂×C₈	—	89	2	no	—
269	C₂×C₈	—	89	8	no	—
270	C₂×C₈	—	89	2	no	—
271	C₂×C₈	—	178	4	no	—
272	C₂×C₈	—	178	2	no	—
273	C₂×C₈	—	178	2	no	—
274	C₂×C₈	—	712	4	no	—
275	C₂×C₈	C₈×D₈₉	1	4	yes	complemented
276	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
277	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
278	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
279	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
280	C₄⋊C₄	—	89	2	no	—
281	C₄⋊C₄	—	89	2	no	—
282	C₄⋊C₄	—	89	1	no	—
283	C₄⋊C₄	—	89	1	no	—
284	C₂²⋊C₄	—	89	2	no	—
285	C₂²⋊C₄	—	89	2	no	—
286	C₂²⋊C₄	—	89	2	no	—
287	C₄²	—	89	2	no	—
288	C₄²	—	89	2	no	—
289	C₄²	—	89	2	no	—
290	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
291	C₄²	D₄×D₈₉	1	1	yes	characteristic
292	C₄²	Q₈×D₈₉	1	1	yes	characteristic
293	C₄²	C₂³×D₈₉	1	1	yes	characteristic
294	C₂³	—	89	2	no	—
295	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
296	Q₈	—	89	4	no	—
297	D₄	—	89	8	no	—
298	D₄	—	89	4	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	89	2	no	—
302	C₂×C₄	—	89	4	no	—
303	C₂×C₄	—	89	4	no	—
304	C₂×C₄	—	89	2	no	—
305	C₂×C₄	—	89	2	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	178	4	no	—
308	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
309	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
310	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
311	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
312	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
313	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
315	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	4	4	no	—
319	C₈	—	89	4	no	—
320	C₈	—	356	2	no	—
321	C₈	—	356	2	no	—
322	C₈	—	178	2	no	—
323	C₈	C₂×C₈×D₈₉	1	2	yes	—
324	C₈	C₂×C₈×D₈₉	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	89	4	no	—
328	C₂²	—	178	2	no	—
329	C₂²	—	178	2	no	—
330	C₂²	C₄⋊C₈×D₈₉	1	2	yes	—
331	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	89	4	no	—
336	C₄	—	178	2	no	—
337	C₄	—	178	2	no	—
338	C₄	C₄⋊C₈×D₈₉	1	2	yes	—
339	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
340	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
341	C₄	#19152404	1	2	yes	—
342	C₂	—	2	2	no	—
343	C₂	—	178	4	no	—
344	C₂	C₂×C₄⋊C₈×D₈₉	1	1	yes	characteristic
345	C₂	#26028843	1	1	yes	characteristic
346	C₂	#26028860	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#334×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#334	D₈₉
hypocenter	C₈₉	#17788
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #26019002	C₂	semidirect
4. #26019001	C₂	semidirect
5. #26019000	C₂	semidirect
6. #26018997	C₂	semidirect
7. #334×D₈₉	C₂	semidirect
8. C₂×#334×C₈₉	C₂	semidirect
36. #15668018	C₈	semidirect
85. #13083558	C₂×C₈	semidirect
87. #13083557	C₂×C₈	semidirect
94. C₈×D₈₉	C₂×C₈	semidirect
111. C₂×C₈×C₈₉	C₂×C₈	semidirect
181. C₂×#334	D₈₉	semidirect
182. C₈₉	#17788	semidirect
275. C₂×C₈	C₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029361	2
C₄×#15668018	2
#26019002	2
#26019001	2
#26019000	2
#26018997	2
#334×D₈₉	2
C₂×#334×C₈₉	2
#17788	89