Subgroups — #36708150

Order 22784.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029227	C₂	1	1	yes	maximal, characteristic
2	#26022160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022085	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022074	C₂	1	2	yes	maximal, complemented
6	C₁₆×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022069	C₂	1	2	yes	maximal, complemented
8	#26022058	C₂	1	2	yes	maximal, complemented
9	#26022009	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2162	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152132	C₄	1	1	yes	characteristic
13	C₂×#15667951	C₂²	1	2	yes	—
14	C₂²×#13083549	C₂²	1	2	yes	—
15	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
16	#19151728	C₄	1	1	yes	characteristic
17	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
18	#19151677	C₄	1	1	yes	characteristic
19	#19151669	C₄	1	2	yes	—
20	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
21	C₈×C₂²⋊D₈₉	C₂²	1	2	yes	—
22	#19151657	C₄	1	2	yes	—
23	#19151655	C₄	1	1	yes	characteristic
24	#19151552	C₂²	1	2	yes	—
25	C₄⋊#13083549	C₂²	1	1	yes	characteristic
26	C₄⋊#13083549	C₂²	1	1	yes	characteristic
27	C₄⋊#13083549	C₂²	1	2	yes	—
28	#19151416	C₂²	1	2	yes	—
29	#19151416	C₂²	1	2	yes	—
30	#19151377	C₂²	1	2	yes	—
31	#19151377	C₂²	1	2	yes	—
32	#19151373	C₂²	1	2	yes	—
33	C₁₆×Dic₈₉	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
37	C₂²⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	C₂²⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083557	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₈₉	C₂³	1	2	yes	—
43	#15668006	C₈	1	2	yes	—
44	#15668004	C₈	1	1	yes	characteristic
45	#15668000	C₈	1	1	yes	characteristic
46	#15667995	C₈	1	2	yes	—
47	#15667991	C₈	1	1	yes	characteristic
48	#15667986	C₈	1	1	yes	characteristic
49	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
50	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
51	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
52	#15667969	C₂×C₄	1	2	yes	—
53	#15667951	—	4	2	no	—
54	C₂×#13083549	—	4	2	no	—
55	C₂×#13083549	C₂³	1	2	yes	—
56	C₂×#13083549	C₂³	1	2	yes	—
57	C₁₆×D₈₉	—	4	2	no	—
58	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
59	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
60	#15667941	C₂×C₄	1	2	yes	—
61	#15667941	C₂×C₄	1	2	yes	—
62	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
63	#15667924	C₂×C₄	1	2	yes	—
64	#15667924	C₂×C₄	1	2	yes	—
65	#15667923	C₂×C₄	1	2	yes	—
66	#15667922	C₂×C₄	1	2	yes	—
67	C₈×Dic₈₉	C₂³	1	2	yes	—
68	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
80	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
81	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
82	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
83	#13083557	—	4	2	no	—
84	C₂×#10623494	C₂²×C₄	1	2	yes	—
85	C₂×#10623494	C₂²×C₄	1	2	yes	—
86	C₈×D₈₉	—	2	2	no	—
87	C₈×D₈₉	—	2	2	no	—
88	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
89	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
90	#13083553	C₂×C₈	1	2	yes	—
91	#13083553	C₂×C₈	1	2	yes	—
92	#13083553	C₂×C₈	1	2	yes	—
93	#13083553	C₂×C₈	1	2	yes	—
94	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
95	#13083551	C₂×C₈	1	2	yes	—
96	#13083551	C₂×C₈	1	2	yes	—
97	#13083551	C₂×C₈	1	2	yes	—
98	#13083551	C₂×C₈	1	2	yes	—
99	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
100	#13083549	—	2	2	no	—
101	#13083549	—	2	2	no	—
102	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	—	2	1	no	—
106	C₂×C₈×C₈₉	—	2	1	no	—
107	C₂×C₈×C₈₉	—	4	1	no	—
108	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
112	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
113	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
114	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	C₁₆×C₈₉	—	2	1	no	—
117	C₁₆×C₈₉	—	2	1	no	—
118	C₁₆×C₈₉	—	2	1	no	—
119	C₁₆×C₈₉	—	2	1	no	—
120	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
121	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
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	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
132	C₂²⋊D₈₉	—	2	4	no	—
133	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
134	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
135	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
136	D₃₅₆	—	2	2	no	—
137	C₄×D₈₉	—	2	2	no	—
138	C₄×D₈₉	—	2	2	no	—
139	Dic₁₇₈	—	2	2	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
145	#10623494	—	2	2	no	—
146	#10623494	—	2	2	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	#504	1	2	yes	—
150	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
151	C₂×D₈₉	—	2	2	no	—
152	C₂×D₈₉	—	2	2	no	—
153	C₂×D₈₉	—	2	2	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	#504	1	2	yes	—
159	C₄×C₈₉	#520	1	1	yes	characteristic
160	C₄×C₈₉	#568	1	1	yes	characteristic
161	Dic₈₉	—	2	2	no	—
162	Dic₈₉	—	2	2	no	—
163	Dic₈₉	—	2	2	no	—
164	#33508	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	—	2	2	no	—
166	C₂×C₈₉	#2988	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
168	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
169	D₈₉	—	4	2	no	—
170	#2966	—	89	1	no	—
171	#2183	—	89	1	no	—
172	Q₈×C₁₆	—	89	1	no	—
173	#2169	—	89	1	no	—
174	#2168	—	89	2	no	—
175	D₄×C₁₆	—	89	2	no	—
176	D₄×C₁₆	—	89	1	no	—
177	#2164	—	89	2	no	—
178	#2162	D₈₉	1	1	yes	characteristic, complemented
179	#2151	—	89	2	no	—
180	C₂×C₄⋊C₁₆	—	89	1	no	—
181	C₈₉	#33508	1	1	yes	characteristic, complemented, Sylow
182	#529	—	89	1	no	—
183	C₂×M₃₂	—	89	2	no	—
184	C₂²×C₁₆	—	89	2	no	—
185	C₂²×C₁₆	—	89	2	no	—
186	#446	—	89	1	no	—
187	C₈×Q₈	—	89	1	no	—
188	#436	—	89	1	no	—
189	#435	—	89	2	no	—
190	C₈×D₄	—	89	2	no	—
191	C₈×D₄	—	89	1	no	—
192	#433	—	89	2	no	—
193	#432	—	89	1	no	—
194	#405	—	89	2	no	—
195	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
196	C₄⋊C₁₆	—	89	2	no	—
197	C₄⋊C₁₆	—	89	2	no	—
198	C₄⋊C₁₆	—	89	2	no	—
199	C₄⋊C₁₆	—	89	1	no	—
200	C₄⋊C₁₆	—	89	1	no	—
201	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
202	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
203	C₂²⋊C₁₆	—	89	2	no	—
204	C₂²⋊C₁₆	—	89	2	no	—
205	C₂²⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
206	C₂²⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
207	#346	—	89	2	no	—
208	C₄×C₁₆	—	89	2	no	—
209	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
210	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
211	C₂×M₁₆	—	89	2	no	—
212	C₂²×C₈	—	89	2	no	—
213	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
215	#126	—	89	2	no	—
216	#125	—	89	1	no	—
217	#124	—	89	1	no	—
218	#123	—	89	2	no	—
219	C₂²⋊Q₈	—	89	1	no	—
220	#121	—	89	1	no	—
221	C₄×Q₈	—	89	1	no	—
222	C₄×D₄	—	89	2	no	—
223	C₄×D₄	—	89	1	no	—
224	#117	—	89	2	no	—
225	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
226	M₃₂	—	356	2	no	—
227	C₂×C₁₆	—	89	2	no	—
228	C₂×C₁₆	—	89	2	no	—
229	C₂×C₁₆	—	356	2	no	—
230	C₂×C₁₆	—	356	2	no	—
231	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
235	C₄⋊C₈	—	89	2	no	—
236	C₄⋊C₈	—	89	2	no	—
237	C₄⋊C₈	—	89	1	no	—
238	C₄⋊C₈	—	89	1	no	—
239	C₂²⋊C₈	—	89	2	no	—
240	C₂²⋊C₈	—	89	2	no	—
241	C₂²⋊C₈	—	89	2	no	—
242	#97	—	89	2	no	—
243	#97	—	89	2	no	—
244	C₄×C₈	—	89	2	no	—
245	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
246	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₂×Q₈	—	89	1	no	—
250	C₂×D₄	—	89	2	no	—
251	C₂×D₄	—	89	1	no	—
252	C₂²×C₄	—	89	2	no	—
253	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
256	M₁₆	—	356	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	89	2	no	—
261	C₂×C₈	—	89	2	no	—
262	C₂×C₈	—	178	2	no	—
263	C₂×C₈	—	178	2	no	—
264	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
268	C₄⋊C₄	—	89	2	no	—
269	C₄⋊C₄	—	89	2	no	—
270	C₄⋊C₄	—	89	1	no	—
271	C₄⋊C₄	—	89	2	no	—
272	C₄⋊C₄	—	89	1	no	—
273	C₄⋊C₄	—	89	2	no	—
274	C₂²⋊C₄	—	89	2	no	—
275	C₂²⋊C₄	—	89	2	no	—
276	C₂²⋊C₄	—	89	2	no	—
277	C₂²⋊C₄	—	89	2	no	—
278	C₂²⋊C₄	—	89	2	no	—
279	C₄²	—	89	2	no	—
280	C₄²	C₈×D₈₉	1	1	yes	characteristic
281	C₄²	C₈×D₈₉	1	1	yes	characteristic
282	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
283	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	178	2	no	—
289	C₁₆	—	178	2	no	—
290	C₂³	—	89	2	no	—
291	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
292	Q₈	—	178	2	no	—
293	D₄	—	178	4	no	—
294	D₄	—	178	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	89	2	no	—
300	C₂×C₄	—	89	2	no	—
301	C₂×C₄	—	89	2	no	—
302	C₂×C₄	—	178	2	no	—
303	C₂×C₄	—	178	2	no	—
304	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
305	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	178	2	no	—
314	C₈	—	178	2	no	—
315	C₈	#15668058	1	1	yes	characteristic
316	C₈	#15668059	1	1	yes	characteristic
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	178	2	no	—
320	C₂²	—	178	2	no	—
321	C₂²	—	178	2	no	—
322	C₂²	#19152014	1	2	yes	—
323	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	178	2	no	—
329	C₄	—	178	2	no	—
330	C₄	—	178	2	no	—
331	C₄	#19152016	1	2	yes	—
332	C₄	#19152078	1	1	yes	characteristic
333	C₄	#19152364	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	356	2	no	—
336	C₂	#26029360	1	1	yes	characteristic
337	C₂	C₂×#19152014	1	1	yes	characteristic
338	C₂	C₂×#19152016	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2162	C₂
prime radical	#15667986	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2162	D₈₉
hypocenter	C₈₉	#33508
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #26022160	C₂	semidirect
3. C₁₆×Dic₁₇₈	C₂	semidirect
4. #26022085	C₂	semidirect
5. #26022074	C₂	semidirect
6. C₁₆×D₃₅₆	C₂	semidirect
7. #26022069	C₂	semidirect
8. #26022058	C₂	semidirect
9. #26022009	C₂	semidirect
10. C₂×C₄⋊#13083549	C₂	semidirect
11. C₈₉×#2162	C₂	semidirect
178. #2162	D₈₉	semidirect
181. C₈₉	#33508	semidirect

Maximal subgroups

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

subgroup	index
#26029227	2
#26022160	2
C₁₆×Dic₁₇₈	2
#26022085	2
#26022074	2
C₁₆×D₃₅₆	2
#26022069	2
#26022058	2
#26022009	2
C₂×C₄⋊#13083549	2
C₈₉×#2162	2
#33508	89