Subgroups — #36469693

Order 22784.

428 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151716	C₂	1	1	yes	maximal, characteristic
2	#26022002	C₂	1	1	yes	maximal, characteristic
3	#26021989	C₂	1	1	yes	maximal, characteristic
4	#26020131	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018367	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	C₂²	1	1	yes	characteristic
9	C₂×#15668018	—	2	1	no	—
10	#19151716	—	2	1	no	—
11	#19151716	—	2	1	no	—
12	#19151716	C₄	1	2	yes	complemented
13	#19151716	C₄	1	2	yes	complemented
14	C₂×#15667948	C₂²	1	1	yes	characteristic
15	C₂×#15667948	C₂²	1	1	yes	characteristic
16	#19151636	C₂²	1	1	yes	characteristic
17	C₂×#15667940	—	2	1	no	—
18	#19151553	—	2	1	no	—
19	#19151415	C₄	1	2	yes	complemented
20	#19151412	C₄	1	2	yes	complemented
21	#19151409	C₄	1	4	yes	complemented
22	#19151358	—	2	1	no	—
23	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	#346×C₈₉	—	2	1	no	—
27	C₄×C₁₆×C₈₉	—	2	1	no	—
28	C₂×#13083574	—	2	1	no	—
29	#15668034	—	2	1	no	—
30	#15668034	—	2	1	no	—
31	#15668034	C₂×C₄	1	2	yes	—
32	#15668034	C₂×C₄	1	2	yes	—
33	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
34	C₂×#13083563	—	2	1	no	—
35	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
36	#15668018	—	2	2	no	—
37	#15668018	—	2	2	no	—
38	#15668018	—	4	1	no	—
39	#15668018	—	4	1	no	—
40	C₂×#13083558	—	2	1	no	—
41	C₂×#13083558	—	2	1	no	—
42	C₂×#13083557	—	4	1	no	—
43	C₂×C₈×D₈₉	—	4	1	no	—
44	#15667971	—	2	1	no	—
45	#15667948	—	2	1	no	—
46	#15667948	—	2	1	no	—
47	#15667948	C₂×C₄	1	2	yes	—
48	#15667948	C₂×C₄	1	2	yes	—
49	#15667946	—	2	1	no	—
50	#15667944	C₂×C₄	1	2	yes	—
51	#15667940	—	2	2	no	—
52	#15667940	—	2	2	no	—
53	#15667922	—	4	1	no	—
54	C₈×Dic₈₉	—	4	1	no	—
55	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
58	M₃₂×C₈₉	—	2	1	no	—
59	M₃₂×C₈₉	—	2	1	no	—
60	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
61	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
62	C₂×C₁₆×C₈₉	—	2	1	no	—
63	C₂×C₁₆×C₈₉	—	2	1	no	—
64	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
67	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
68	#13083574	—	2	2	no	—
69	#13083574	—	2	2	no	—
70	#13083574	—	4	1	no	—
71	#13083574	—	4	1	no	—
72	C₂×Dic₁₇₈	—	2	1	no	—
73	C₂×D₃₅₆	—	2	1	no	—
74	C₂×C₂²⋊D₈₉	—	4	1	no	—
75	C₂×C₄×D₈₉	—	4	1	no	—
76	Dic₃₅₆	—	2	1	no	—
77	Dic₃₅₆	C₄²	1	2	yes	—
78	#13083563	—	2	2	no	—
79	#13083563	—	2	2	no	—
80	D₇₁₂	—	2	1	no	—
81	D₇₁₂	C₄²	1	2	yes	—
82	#13083558	—	2	2	no	—
83	#13083558	—	2	2	no	—
84	#13083558	—	4	1	no	—
85	#13083558	—	4	1	no	—
86	#13083557	—	4	2	no	—
87	#13083557	—	4	2	no	—
88	C₂×#10623494	—	4	1	no	—
89	C₂×#10623494	—	4	1	no	—
90	C₈×D₈₉	—	4	2	no	—
91	C₈×D₈₉	—	4	2	no	—
92	C₄⋊Dic₈₉	—	2	2	no	—
93	C₂²⋊Dic₈₉	—	4	1	no	—
94	C₄×Dic₈₉	—	4	1	no	—
95	C₂²×C₄×C₈₉	—	2	1	no	—
96	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
100	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
101	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
102	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
108	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
109	C₄²×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
112	C₁₆×C₈₉	—	2	2	no	—
113	C₁₆×C₈₉	—	2	2	no	—
114	C₂³×C₈₉	#135	1	1	yes	characteristic
115	C₂²×D₈₉	—	4	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
125	C₂²⋊D₈₉	—	4	2	no	—
126	C₂²⋊D₈₉	—	4	2	no	—
127	C₂×Dic₈₉	—	4	1	no	—
128	C₂×Dic₈₉	—	4	2	no	—
129	D₃₅₆	—	2	2	no	—
130	D₃₅₆	—	4	1	no	—
131	C₄×D₈₉	—	4	2	no	—
132	C₄×D₈₉	—	4	2	no	—
133	Dic₁₇₈	—	2	2	no	—
134	Dic₁₇₈	—	4	1	no	—
135	C₈×C₈₉	#102	1	1	yes	characteristic
136	C₈×C₈₉	#102	1	1	yes	characteristic
137	C₈×C₈₉	#102	1	1	yes	characteristic
138	C₈×C₈₉	#102	1	1	yes	characteristic
139	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
140	C₈×C₈₉	#117	1	1	yes	characteristic
141	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
142	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
143	#10623494	—	4	2	no	—
144	#10623494	—	4	2	no	—
145	C₂²×C₈₉	—	4	1	no	—
146	C₂²×C₈₉	#416	1	1	yes	characteristic
147	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
148	C₂²×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
149	C₂×D₈₉	—	4	2	no	—
150	C₂×D₈₉	—	8	2	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	4	1	no	—
153	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
155	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
156	C₄×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
157	Dic₈₉	—	4	2	no	—
158	Dic₈₉	—	8	2	no	—
159	#12646	—	89	1	no	maximal, Sylow
160	C₂×C₈₉	—	2	1	no	—
161	C₂×C₈₉	—	2	1	no	—
162	C₂×C₈₉	—	2	1	no	—
163	C₂×C₈₉	C₄×#102	1	1	yes	characteristic
164	D₈₉	—	8	2	no	—
165	C₂×#443	—	89	1	no	—
166	#2148	—	89	1	no	—
167	#2145	—	89	1	no	—
168	C₄×M₃₂	D₈₉	1	1	yes	characteristic, complemented
169	#1772	—	89	1	no	—
170	#1389	—	89	1	no	—
171	#1388	—	89	1	no	—
172	C₈₉	#12646	1	1	yes	characteristic, complemented, Sylow
173	C₂×#135	—	89	1	no	—
174	C₂×#131	—	178	1	no	—
175	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
176	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
177	#443	—	89	2	no	—
178	#443	—	89	2	no	—
179	#443	—	178	1	no	—
180	#443	—	178	1	no	—
181	C₂×#108	—	89	1	no	—
182	C₂×#108	—	89	1	no	—
183	#427	—	89	1	no	—
184	C₂×#104	—	178	1	no	—
185	#405	—	178	1	no	—
186	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
187	#362	—	89	2	no	—
188	#361	—	89	2	no	—
189	#360	—	89	4	no	—
190	#346	—	2	1	no	—
191	C₄×C₁₆	—	2	1	no	—
192	#339	—	178	1	no	—
193	C₂×#41	—	178	1	no	—
194	#135	—	89	2	no	—
195	#135	—	89	2	no	—
196	#135	—	178	1	no	—
197	#135	—	178	1	no	—
198	C₂×Q₁₆	—	89	1	no	—
199	C₂×SD₁₆	—	178	1	no	—
200	C₂×D₈	—	89	1	no	—
201	#131	—	356	1	no	—
202	#131	—	356	1	no	—
203	#131	—	178	2	no	—
204	#131	—	178	2	no	—
205	C₂×M₁₆	—	356	1	no	—
206	C₂×M₁₆	—	178	1	no	—
207	C₂×M₁₆	—	178	1	no	—
208	C₂²×C₈	—	356	1	no	—
209	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
211	#117	—	178	1	no	—
212	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
213	M₃₂	—	2	1	no	—
214	M₃₂	—	2	1	no	—
215	M₃₂	C₄×D₈₉	1	2	yes	—
216	M₃₂	C₄×D₈₉	1	2	yes	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	2	1	no	—
219	#108	—	89	2	no	—
220	#108	—	89	2	no	—
221	#108	—	178	1	no	—
222	#108	—	178	1	no	—
223	#107	—	178	1	no	—
224	#106	—	89	2	no	—
225	#104	—	178	2	no	—
226	#104	—	178	2	no	—
227	#97	—	356	1	no	—
228	C₄×C₈	—	356	1	no	—
229	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
230	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
231	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
232	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
233	#41	—	356	1	no	—
234	#41	—	356	1	no	—
235	#41	—	178	2	no	—
236	#41	—	178	2	no	—
237	C₂×Q₈	—	178	1	no	—
238	C₂×D₄	—	356	1	no	—
239	C₂×D₄	—	178	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	356	1	no	—
242	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
243	Q₁₆	—	89	2	no	—
244	Q₁₆	—	178	1	no	—
245	SD₁₆	—	178	2	no	—
246	SD₁₆	—	178	2	no	—
247	D₈	—	89	2	no	—
248	D₈	—	178	1	no	—
249	M₁₆	—	356	1	no	—
250	M₁₆	—	356	1	no	—
251	M₁₆	—	356	2	no	—
252	M₁₆	—	356	2	no	—
253	M₁₆	—	178	2	no	—
254	M₁₆	—	178	2	no	—
255	C₂×C₈	—	356	1	no	—
256	C₂×C₈	—	356	1	no	—
257	C₂×C₈	—	356	2	no	—
258	C₂×C₈	—	356	2	no	—
259	C₂×C₈	#13083551	1	1	yes	characteristic
260	C₂×C₈	#13083551	1	1	yes	characteristic
261	C₂×C₈	D₇₁₂	1	1	yes	characteristic
262	C₂×C₈	D₇₁₂	1	1	yes	characteristic
263	C₂×C₈	#13083563	1	1	yes	characteristic
264	C₂×C₈	#13083563	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	C₂×C₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
269	C₂×C₈	#13083574	1	1	yes	characteristic
270	C₂×C₈	#13083574	1	1	yes	characteristic
271	C₄⋊C₄	—	178	2	no	—
272	C₂²⋊C₄	—	356	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	356	1	no	—
275	C₄²	#13083551	1	1	yes	characteristic
276	C₄²	#13083551	1	1	yes	characteristic
277	C₁₆	—	2	2	no	—
278	C₁₆	—	2	2	no	—
279	C₂³	—	356	1	no	—
280	C₂³	#15668034	1	1	yes	characteristic
281	Q₈	—	356	1	no	—
282	Q₈	—	178	2	no	—
283	D₄	—	356	2	no	—
284	D₄	—	356	2	no	—
285	D₄	—	356	1	no	—
286	D₄	—	178	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	356	1	no	—
291	C₂×C₄	—	356	2	no	—
292	C₂×C₄	—	356	2	no	—
293	C₂×C₄	—	356	2	no	—
294	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
295	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
296	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
297	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
298	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
299	C₂×C₄	#15668034	1	1	yes	characteristic
300	C₈	—	356	2	no	—
301	C₈	—	356	2	no	—
302	C₈	#15667934	1	1	yes	characteristic
303	C₈	#15667934	1	1	yes	characteristic
304	C₈	#15667934	1	1	yes	characteristic
305	C₈	#15667934	1	1	yes	characteristic
306	C₈	C₄²×D₈₉	1	1	yes	characteristic
307	C₈	#15667969	1	1	yes	characteristic
308	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
309	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
310	C₂²	—	4	1	no	—
311	C₂²	—	356	2	no	—
312	C₂²	—	712	2	no	—
313	C₂²	#19151595	1	1	yes	characteristic
314	C₂²	C₄×D₇₁₂	1	1	yes	characteristic
315	C₂²	C₄×#13083563	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	356	2	no	—
319	C₄	—	712	2	no	—
320	C₄	C₄×#13083551	1	1	yes	characteristic
321	C₄	C₂×#15667934	1	1	yes	characteristic
322	C₄	C₄×D₇₁₂	1	1	yes	characteristic
323	C₄	C₄×#13083563	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	712	2	no	—
328	C₂	C₄×#15667934	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#15667934
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×M₃₂×C₈₉	C₂
socle	C₂×C₈₉	C₄×#102
prime radical	C₂×#15668034	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#12646
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
4. #26020131	C₂	semidirect
5. #26018367	C₂	semidirect
6. #26018364	C₂	semidirect
7. C₄×M₃₂×C₈₉	C₂	semidirect
12. #19151716	C₄	semidirect
13. #19151716	C₄	semidirect
19. #19151415	C₄	semidirect
20. #19151412	C₄	semidirect
21. #19151409	C₄	semidirect
168. C₄×M₃₂	D₈₉	semidirect
172. C₈₉	#12646	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151716	2
#26022002	2
#26021989	2
#26020131	2
#26018367	2
#26018364	2
C₄×M₃₂×C₈₉	2
#12646	89