Subgroups — #36707996

Order 22784.

552 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	C₈×#13083573	C₂	1	1	yes	maximal, characteristic
2	#26022160	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022157	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022077	C₂	1	2	yes	maximal, complemented
5	#26022074	C₂	1	2	yes	maximal, complemented
6	D₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022054	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022052	C₂	1	2	yes	maximal, complemented
9	C₄×#15667951	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151373	C₂	1	2	yes	maximal, complemented
11	D₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13083573	C₄	1	1	yes	characteristic
13	C₂×#15667951	C₂²	1	1	yes	characteristic
14	C₂×#15667951	C₂²	1	1	yes	characteristic
15	C₂²×#13083549	C₂²	1	2	yes	—
16	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
17	#19151722	C₄	1	1	yes	characteristic
18	D₄×#10623494	C₄	1	1	yes	characteristic
19	C₈×C₂²⋊D₈₉	C₂²	1	2	yes	—
20	#19151661	C₄	1	2	yes	—
21	#19151650	C₄	1	1	yes	characteristic
22	#19151648	C₄	1	2	yes	—
23	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
24	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
25	C₄⋊#13083549	C₂²	1	1	yes	characteristic
26	#19151416	C₂²	1	2	yes	—
27	#19151416	C₂²	1	2	yes	—
28	C₂²⋊#13083549	C₂²	1	2	yes	—
29	#19151377	C₂²	1	2	yes	—
30	#19151373	C₂²	1	2	yes	—
31	#19151373	C₂²	1	2	yes	—
32	#19151373	C₂²	1	1	yes	characteristic
33	#19151373	C₂²	1	2	yes	—
34	#19151373	C₂²	1	1	yes	characteristic
35	C₄×#13083549	C₂²	1	1	yes	characteristic
36	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
37	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083573	C₈	1	1	yes	characteristic
42	C₂²×#10623494	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
45	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
46	#15667977	C₈	1	1	yes	characteristic
47	D₄×Dic₈₉	C₈	1	1	yes	characteristic
48	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
49	#15667972	C₈	1	2	yes	—
50	#15667970	C₈	1	1	yes	characteristic
51	#15667968	C₈	1	2	yes	—
52	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
53	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
54	#15667951	—	2	2	no	—
55	#15667951	—	2	2	no	—
56	C₂×#13083549	—	2	4	no	—
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	2	yes	—
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
61	#15667941	C₂×C₄	1	2	yes	—
62	#15667941	C₂×C₄	1	2	yes	—
63	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
64	#15667924	C₂×C₄	1	2	yes	—
65	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
68	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
69	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
70	C₈×Dic₈₉	C₂³	1	2	yes	—
71	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
72	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₂×C₁₆×C₈₉	—	2	4	no	—
74	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	#13083573	—	2	4	no	—
81	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
82	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
83	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
84	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×#10623494	—	2	4	no	—
89	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
90	C₂×#10623494	C₂²×C₄	1	2	yes	—
91	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
92	C₈×D₈₉	—	2	1	no	—
93	C₈×D₈₉	—	2	1	no	—
94	C₈×D₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
96	#13083553	C₂×C₈	1	2	yes	—
97	#13083553	C₂×C₈	1	2	yes	—
98	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
99	#13083551	C₂×C₈	1	2	yes	—
100	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
102	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
103	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
105	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
106	#13083549	—	2	2	no	—
107	#13083549	—	2	1	no	—
108	#13083549	#41	1	2	yes	complemented
109	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
117	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
118	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₁₆×C₈₉	—	2	2	no	—
120	C₁₆×C₈₉	—	2	1	no	—
121	C₁₆×C₈₉	#41	1	2	yes	complemented
122	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
123	C₂²×D₈₉	C₂²×C₈	1	1	yes	characteristic
124	D₄×C₈₉	M₃₂	1	4	yes	complemented
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	—	2	2	no	—
127	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
131	C₂²⋊D₈₉	—	2	4	no	—
132	C₂×Dic₈₉	—	2	4	no	—
133	C₂×Dic₈₉	—	4	2	no	—
134	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
135	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
136	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
137	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
138	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
139	C₄×D₈₉	—	2	1	no	—
140	C₄×D₈₉	—	2	1	no	—
141	C₄×D₈₉	—	2	2	no	—
142	C₄×D₈₉	—	4	1	no	—
143	Dic₁₇₈	—	2	2	no	—
144	C₈×C₈₉	—	2	2	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
147	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
148	#10623494	—	2	1	no	—
149	#10623494	—	2	2	no	—
150	#10623494	#131	1	2	yes	—
151	C₂²×C₈₉	—	2	2	no	—
152	C₂²×C₈₉	—	2	2	no	—
153	C₂²×C₈₉	C₂×M₃₂	1	4	yes	—
154	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	2	1	no	—
157	C₂×D₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	2	no	—
160	C₄×C₈₉	C₂×M₃₂	1	2	yes	—
161	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
162	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
163	Dic₈₉	—	2	2	no	—
164	Dic₈₉	—	2	1	no	—
165	Dic₈₉	—	2	1	no	—
166	Dic₈₉	—	2	2	no	—
167	Dic₈₉	—	2	1	no	—
168	#33484	—	89	1	no	maximal, Sylow
169	C₂×C₈₉	—	2	4	no	—
170	C₂×C₈₉	C₈×#41	1	1	yes	characteristic
171	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
172	C₂×C₈₉	#3407	1	1	yes	characteristic
173	D₈₉	—	4	1	no	—
174	C₈×#41	—	89	1	no	—
175	#2183	—	89	1	no	—
176	#2183	—	89	1	no	—
177	#2168	—	89	2	no	—
178	#2168	—	89	2	no	—
179	D₄×C₁₆	—	89	1	no	—
180	D₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
181	#2163	—	89	1	no	—
182	#2163	—	89	2	no	—
183	C₄×M₃₂	—	89	1	no	—
184	C₂×#346	—	89	2	no	—
185	C₈₉	#33484	1	1	yes	characteristic, complemented, Sylow
186	C₄×#41	—	89	1	no	—
187	C₂×M₃₂	—	89	1	no	—
188	C₂×M₃₂	—	89	1	no	—
189	C₂²×C₁₆	—	89	2	no	—
190	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
191	C₈×Q₈	—	89	1	no	—
192	C₈×Q₈	—	89	1	no	—
193	C₈×D₄	—	89	2	no	—
194	C₈×D₄	—	89	1	no	—
195	C₈×D₄	—	89	2	no	—
196	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
197	#431	—	89	1	no	—
198	#431	—	89	2	no	—
199	C₂×C₄×C₈	—	89	2	no	—
200	C₂×C₄×C₈	—	89	1	no	—
201	C₄⋊C₁₆	—	89	2	no	—
202	C₄⋊C₁₆	—	89	2	no	—
203	C₄⋊C₁₆	—	89	1	no	—
204	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
205	C₂²⋊C₁₆	—	89	2	no	—
206	C₂²⋊C₁₆	—	89	2	no	—
207	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
208	#346	—	89	1	no	—
209	#346	—	89	2	no	—
210	#346	—	89	2	no	—
211	#346	—	89	2	no	—
212	#346	—	89	1	no	—
213	C₄×C₁₆	—	89	1	no	—
214	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
215	C₂×#41	—	89	1	no	—
216	C₂²×C₈	—	89	2	no	—
217	C₂²×C₈	—	89	1	no	—
218	C₂²×C₈	—	89	1	no	—
219	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
220	C₄×Q₈	—	89	1	no	—
221	C₄×Q₈	—	89	1	no	—
222	C₄×D₄	—	89	1	no	—
223	C₄×D₄	—	89	2	no	—
224	C₄×D₄	—	89	2	no	—
225	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
226	#117	—	89	1	no	—
227	#117	—	89	2	no	—
228	C₂×C₄²	—	89	2	no	—
229	C₂×C₄²	—	89	1	no	—
230	M₃₂	—	178	2	no	—
231	M₃₂	—	178	2	no	—
232	C₂×C₁₆	—	2	4	no	—
233	C₂×C₁₆	—	89	1	no	—
234	C₂×C₁₆	—	89	1	no	—
235	C₂×C₁₆	—	89	2	no	—
236	C₂×C₁₆	—	178	4	no	—
237	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
238	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
239	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
240	C₄⋊C₈	—	89	2	no	—
241	C₄⋊C₈	—	89	2	no	—
242	C₄⋊C₈	—	89	1	no	—
243	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
244	C₂²⋊C₈	—	89	2	no	—
245	C₂²⋊C₈	—	89	2	no	—
246	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
247	C₄×C₈	—	89	2	no	—
248	C₄×C₈	—	89	1	no	—
249	C₄×C₈	—	89	2	no	—
250	C₄×C₈	—	89	1	no	—
251	C₄×C₈	—	89	2	no	—
252	C₄×C₈	—	89	1	no	—
253	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
254	#41	—	178	4	no	—
255	C₂×Q₈	—	89	1	no	—
256	C₂×D₄	—	89	2	no	—
257	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
258	C₂²×C₄	—	89	2	no	—
259	C₂²×C₄	—	89	2	no	—
260	C₂²×C₄	—	89	1	no	—
261	C₂²×C₄	—	89	1	no	—
262	C₂²×C₄	—	89	1	no	—
263	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	89	1	no	—
267	C₂×C₈	—	89	1	no	—
268	C₂×C₈	—	89	2	no	—
269	C₂×C₈	—	178	2	no	—
270	C₂×C₈	—	178	4	no	—
271	C₂×C₈	—	178	1	no	—
272	C₂×C₈	—	178	1	no	—
273	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
274	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
276	C₄⋊C₄	—	89	2	no	—
277	C₄⋊C₄	—	89	2	no	—
278	C₄⋊C₄	—	89	1	no	—
279	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
280	C₂²⋊C₄	—	89	2	no	—
281	C₂²⋊C₄	—	89	2	no	—
282	C₂²⋊C₄	C₈×D₈₉	1	2	yes	—
283	C₄²	—	89	2	no	—
284	C₄²	—	89	1	no	—
285	C₄²	—	89	1	no	—
286	C₄²	—	89	2	no	—
287	C₄²	—	89	1	no	—
288	C₄²	—	89	2	no	—
289	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
290	C₁₆	—	2	1	no	—
291	C₁₆	—	2	2	no	—
292	C₁₆	—	89	2	no	—
293	C₁₆	—	178	2	no	—
294	C₁₆	—	178	1	no	—
295	C₁₆	#13083573	1	2	yes	complemented
296	C₂³	—	89	1	no	—
297	C₂³	C₂×C₈×D₈₉	1	2	yes	—
298	Q₈	—	178	2	no	—
299	D₄	—	178	4	no	—
300	D₄	#15667951	1	4	yes	complemented
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	89	1	no	—
304	C₂×C₄	—	89	1	no	—
305	C₂×C₄	—	89	1	no	—
306	C₂×C₄	—	89	2	no	—
307	C₂×C₄	—	89	2	no	—
308	C₂×C₄	—	356	1	no	—
309	C₂×C₄	—	356	2	no	—
310	C₂×C₄	—	178	4	no	—
311	C₂×C₄	—	178	2	no	—
312	C₂×C₄	—	178	1	no	—
313	C₂×C₄	—	178	1	no	—
314	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
315	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
316	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	2	1	no	—
320	C₈	—	89	2	no	—
321	C₈	—	178	1	no	—
322	C₈	—	178	2	no	—
323	C₈	#41×D₈₉	1	1	yes	characteristic
324	C₈	C₂×#13083573	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	178	1	no	—
328	C₂²	—	178	1	no	—
329	C₂²	—	178	1	no	—
330	C₂²	C₂×#15667951	1	4	yes	—
331	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	178	1	no	—
335	C₄	—	178	1	no	—
336	C₄	—	178	2	no	—
337	C₄	—	178	1	no	—
338	C₄	—	178	2	no	—
339	C₄	C₂×#15667951	1	2	yes	—
340	C₄	C₄×#13083573	1	1	yes	characteristic
341	C₄	#131×D₈₉	1	1	yes	characteristic
342	C₂	—	2	4	no	—
343	C₂	—	356	1	no	—
344	C₂	C₈×#13083573	1	1	yes	characteristic
345	C₂	C₂²×#15667951	1	1	yes	characteristic
346	C₂	#26033353	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	D₄×C₁₆×C₈₉	C₂
prime radical	C₂×#13083573	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	D₄×C₁₆	D₈₉
hypocenter	C₈₉	#33484
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #26022160	C₂	semidirect
3. #26022157	C₂	semidirect
4. #26022077	C₂	semidirect
5. #26022074	C₂	semidirect
6. D₄×#13083549	C₂	semidirect
7. #26022054	C₂	semidirect
8. #26022052	C₂	semidirect
9. C₄×#15667951	C₂	semidirect
10. C₂×#19151373	C₂	semidirect
11. D₄×C₁₆×C₈₉	C₂	semidirect
108. #13083549	#41	semidirect
121. C₁₆×C₈₉	#41	semidirect
124. D₄×C₈₉	M₃₂	semidirect
180. D₄×C₁₆	D₈₉	semidirect
185. C₈₉	#33484	semidirect
295. C₁₆	#13083573	semidirect
300. D₄	#15667951	semidirect

Maximal subgroups

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

subgroup	index
C₈×#13083573	2
#26022160	2
#26022157	2
#26022077	2
#26022074	2
D₄×#13083549	2
#26022054	2
#26022052	2
C₄×#15667951	2
C₂×#19151373	2
D₄×C₁₆×C₈₉	2
#33484	89