Subgroups — #36709174

Order 22784.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029170	C₂	1	1	yes	maximal, characteristic
2	#26022425	C₂	1	4	yes	maximal, complemented
3	#26022420	C₂	1	4	yes	maximal, complemented
4	#26022024	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151420	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022018	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021987	C₂	1	1	yes	maximal, characteristic
8	C₂×#19151407	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#3163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668034	C₂²	1	1	yes	characteristic
11	#19152364	—	2	1	no	—
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	#19151721	—	2	2	no	—
15	#19151716	C₂²	1	4	yes	—
16	C₂×#15667948	C₂²	1	1	yes	characteristic
17	#19151616	—	2	1	no	—
18	#19151420	C₂²	1	2	yes	complemented
19	#19151420	C₂²	1	4	yes	complemented
20	#19151420	C₂²	1	2	yes	complemented
21	#19151419	C₂²	1	4	yes	complemented
22	#19151407	C₂²	1	2	yes	—
23	#19151407	C₂²	1	4	yes	—
24	#19151407	C₂²	1	2	yes	—
25	#544×C₈₉	—	2	1	no	—
26	#501×C₈₉	—	2	2	no	—
27	#499×C₈₉	C₂²	1	2	yes	complemented
28	#498×C₈₉	C₂²	1	2	yes	complemented
29	#428×C₈₉	—	2	1	no	—
30	#427×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
32	#405×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#13083574	—	2	1	no	—
34	#15668034	—	2	1	no	—
35	#15668034	—	2	1	no	—
36	#15668034	Q₈	1	2	yes	complemented
37	#15668034	C₂³	1	2	yes	—
38	C₂×Dic₃₅₆	Q₈	1	1	yes	characteristic, complemented
39	C₂×#13083563	—	2	1	no	—
40	C₂×D₇₁₂	Q₈	1	1	yes	characteristic, complemented
41	#15668020	—	4	2	no	—
42	#15668018	—	2	4	no	—
43	C₂×#13083558	—	2	1	no	—
44	M₁₆×D₈₉	—	4	2	no	—
45	#15667952	—	2	1	no	—
46	#15667952	—	2	1	no	—
47	#15667952	C₂³	1	2	yes	—
48	#15667952	C₂³	1	2	yes	—
49	C₂×#13083549	—	2	1	no	—
50	C₂×#13083549	—	2	1	no	—
51	#15667948	—	2	1	no	—
52	#15667948	C₂³	1	2	yes	—
53	#15667940	—	2	4	no	—
54	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₈₉	—	2	2	no	—
57	C₄⋊Q₈×C₈₉	—	4	1	no	—
58	#125×C₈₉	—	2	2	no	—
59	#125×C₈₉	—	4	1	no	—
60	C₂²⋊Q₈×C₈₉	—	4	2	no	—
61	#117×C₈₉	—	2	1	no	—
62	#117×C₈₉	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
64	#107×C₈₉	—	2	1	no	—
65	#107×C₈₉	—	2	2	no	—
66	#106×C₈₉	—	2	1	no	—
67	#106×C₈₉	—	2	2	no	—
68	#106×C₈₉	C₂³	1	2	yes	—
69	#106×C₈₉	C₂³	1	1	yes	characteristic
70	#106×C₈₉	C₂³	1	1	yes	characteristic
71	#97×C₈₉	D₄	1	2	yes	—
72	C₄×C₈×C₈₉	C₂³	1	2	yes	—
73	#13083574	—	2	2	no	—
74	#13083574	—	2	2	no	—
75	#13083574	—	4	1	no	—
76	#13083574	—	4	1	no	—
77	C₂×Dic₁₇₈	—	2	1	no	—
78	C₂×D₃₅₆	—	2	1	no	—
79	C₂×C₂²⋊D₈₉	—	4	1	no	—
80	C₂×C₄×D₈₉	—	4	1	no	—
81	Dic₃₅₆	—	2	1	no	—
82	Dic₃₅₆	C₂×Q₈	1	2	yes	—
83	#13083563	—	2	2	no	—
84	#13083563	—	2	2	no	—
85	D₇₁₂	—	2	1	no	—
86	D₇₁₂	C₂×Q₈	1	2	yes	—
87	#13083558	—	2	2	no	—
88	#13083558	—	4	1	no	—
89	#13083557	—	4	4	no	—
90	C₂×#10623494	—	4	1	no	—
91	C₈×D₈₉	—	4	4	no	—
92	#13083549	—	2	2	no	—
93	#13083549	—	2	2	no	—
94	C₂×Q₈×C₈₉	—	4	2	no	—
95	C₂²×C₄×C₈₉	—	4	1	no	—
96	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
97	M₁₆×C₈₉	SD₁₆	1	2	yes	—
98	M₁₆×C₈₉	SD₁₆	1	2	yes	—
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
104	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₈₉	—	2	1	no	—
107	C₄⋊C₄×C₈₉	—	2	2	no	—
108	C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₄⋊C₄×C₈₉	—	2	1	no	—
110	C₄⋊C₄×C₈₉	—	4	2	no	—
111	C₄⋊C₄×C₈₉	—	4	2	no	—
112	C₄⋊C₄×C₈₉	—	4	2	no	—
113	C₄⋊C₄×C₈₉	—	4	1	no	—
114	C₂²⋊C₄×C₈₉	—	2	1	no	—
115	C₂²⋊C₄×C₈₉	—	4	1	no	—
116	C₄²×C₈₉	—	4	1	no	—
117	C₄²×C₈₉	C₂×D₄	1	2	yes	—
118	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂²×D₈₉	—	4	1	no	—
120	Q₈×C₈₉	—	16	2	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	—	4	1	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	—	16	1	no	—
126	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₈₉	—	4	2	no	—
133	C₂²⋊D₈₉	—	4	2	no	—
134	C₂×Dic₈₉	—	4	1	no	—
135	D₃₅₆	—	2	2	no	—
136	D₃₅₆	—	4	1	no	—
137	C₄×D₈₉	—	4	2	no	—
138	C₄×D₈₉	—	4	2	no	—
139	Dic₁₇₈	—	2	2	no	—
140	Dic₁₇₈	—	4	1	no	—
141	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
142	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
144	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
145	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
146	#10623494	—	4	2	no	—
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	#476	1	1	yes	characteristic
149	C₂²×C₈₉	#476	1	1	yes	characteristic
150	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×D₈₉	—	4	2	no	—
152	C₂×D₈₉	—	16	1	no	—
153	C₄×C₈₉	—	4	2	no	—
154	C₄×C₈₉	—	8	1	no	—
155	C₄×C₈₉	—	8	2	no	—
156	C₄×C₈₉	—	8	1	no	—
157	C₄×C₈₉	#476	1	1	yes	characteristic
158	C₄×C₈₉	#476	1	1	yes	characteristic
159	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
160	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
161	Dic₈₉	—	4	2	no	—
162	#33638	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	—	2	1	no	—
164	C₂×C₈₉	—	2	1	no	—
165	C₂×C₈₉	—	2	1	no	—
166	C₂×C₈₉	C₂×#476	1	1	yes	characteristic
167	D₈₉	—	8	2	no	—
168	#3163	D₈₉	1	1	yes	characteristic, complemented
169	#2957	—	89	1	no	—
170	#2231	—	89	4	no	—
171	#2230	—	89	4	no	—
172	#2155	—	89	1	no	—
173	C₂×#365	—	89	1	no	—
174	#2153	—	89	1	no	—
175	#2145	—	89	1	no	—
176	C₂×#360	—	89	1	no	—
177	C₈₉	#33638	1	1	yes	characteristic, complemented, Sylow
178	C₂×#135	—	89	1	no	—
179	#568	—	178	1	no	—
180	#544	—	2	1	no	—
181	C₂×M₃₂	—	89	1	no	—
182	C₂×M₃₂	—	89	1	no	—
183	#501	—	2	2	no	—
184	#499	C₂×D₈₉	1	2	yes	complemented
185	#498	C₂×D₈₉	1	2	yes	complemented
186	#444	—	178	2	no	—
187	#443	—	89	4	no	—
188	C₂×#108	—	89	1	no	—
189	#428	—	2	1	no	—
190	#427	C₂×D₈₉	1	1	yes	characteristic
191	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
192	#421	—	178	1	no	—
193	#405	C₂×D₈₉	1	1	yes	characteristic
194	#365	—	89	4	no	—
195	#365	—	89	2	no	—
196	#365	—	89	2	no	—
197	#364	—	89	4	no	—
198	#360	—	89	4	no	—
199	#360	—	89	2	no	—
200	#360	—	89	2	no	—
201	C₂×#41	—	178	1	no	—
202	#135	—	89	2	no	—
203	#135	—	89	2	no	—
204	#135	—	178	1	no	—
205	#135	—	178	1	no	—
206	C₂×Q₁₆	—	89	1	no	—
207	C₂×SD₁₆	—	178	1	no	—
208	C₂×D₈	—	89	1	no	—
209	#131	—	356	2	no	—
210	#131	—	178	4	no	—
211	C₂×M₁₆	—	356	2	no	—
212	C₂×M₁₆	—	178	1	no	—
213	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	—	4	1	no	—
217	#125	—	2	2	no	—
218	#125	—	4	1	no	—
219	C₂²⋊Q₈	—	4	2	no	—
220	#117	—	2	1	no	—
221	#117	C₂²⋊D₈₉	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	2	1	no	—
223	M₃₂	—	89	2	no	—
224	M₃₂	—	89	2	no	—
225	M₃₂	—	178	1	no	—
226	M₃₂	—	178	1	no	—
227	C₂×C₁₆	—	178	1	no	—
228	C₂×C₁₆	—	178	1	no	—
229	#108	—	89	2	no	—
230	#108	—	178	1	no	—
231	#107	—	2	1	no	—
232	#107	—	2	2	no	—
233	#106	—	2	1	no	—
234	#106	—	2	2	no	—
235	#106	C₂²×D₈₉	1	2	yes	—
236	#106	C₂²×D₈₉	1	1	yes	characteristic
237	#106	C₂²×D₈₉	1	1	yes	characteristic
238	#104	—	178	4	no	—
239	#97	C₂²⋊D₈₉	1	2	yes	—
240	C₄×C₈	C₂²×D₈₉	1	2	yes	—
241	#41	—	356	1	no	—
242	#41	—	356	1	no	—
243	#41	—	178	2	no	—
244	#41	—	178	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	178	1	no	—
247	C₂×D₄	—	356	1	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	356	1	no	—
251	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
252	Q₁₆	—	89	2	no	—
253	Q₁₆	—	178	1	no	—
254	SD₁₆	—	178	2	no	—
255	SD₁₆	—	178	2	no	—
256	D₈	—	89	2	no	—
257	D₈	—	178	1	no	—
258	M₁₆	—	356	1	no	—
259	M₁₆	—	356	4	no	—
260	M₁₆	—	178	2	no	—
261	M₁₆	#13083561	1	2	yes	—
262	M₁₆	#13083561	1	2	yes	—
263	C₂×C₈	—	356	1	no	—
264	C₂×C₈	—	356	4	no	—
265	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
267	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
268	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
269	C₂×C₈	#13083575	1	1	yes	characteristic
270	C₂×C₈	#13083575	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
284	C₁₆	—	178	2	no	—
285	C₁₆	—	178	2	no	—
286	C₂³	—	356	1	no	—
287	C₂³	#15668038	1	1	yes	characteristic
288	Q₈	—	16	2	no	—
289	Q₈	—	356	1	no	—
290	Q₈	—	178	2	no	—
291	D₄	—	356	2	no	—
292	D₄	—	356	2	no	—
293	D₄	—	356	1	no	—
294	D₄	—	178	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	16	1	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	356	2	no	—
303	C₂×C₄	#15667988	1	1	yes	characteristic
304	C₂×C₄	#15667988	1	1	yes	characteristic
305	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
306	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
307	C₂×C₄	#15668038	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
309	C₈	—	356	2	no	—
310	C₈	#15667988	1	1	yes	characteristic
311	C₈	#15667988	1	1	yes	characteristic
312	C₈	C₂×#13083561	1	4	yes	—
313	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
314	C₈	C₂×#13083575	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	356	2	no	—
317	C₂²	—	1424	1	no	—
318	C₂²	#19151885	1	1	yes	characteristic
319	C₂²	#19151885	1	1	yes	characteristic
320	C₂²	C₂×#15668038	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	356	2	no	—
326	C₄	#19151885	1	1	yes	characteristic
327	C₄	#19151885	1	1	yes	characteristic
328	C₄	C₂×#15667988	1	1	yes	characteristic
329	C₄	C₂²×#13083561	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	712	2	no	—
334	C₂	C₂×#19151885	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19151885
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3163	C₂
socle	C₂×C₈₉	C₂×#476
prime radical	C₂×#15668034	C₂²
hypercenter	#3163	D₈₉
hypocenter	C₈₉	#33638
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #3163

Complements

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

N	H ≅ G/N	type
2. #26022425	C₂	semidirect
3. #26022420	C₂	semidirect
4. #26022024	C₂	semidirect
5. C₂×#19151420	C₂	semidirect
6. #26022018	C₂	semidirect
9. C₈₉×#3163	C₂	semidirect
18. #19151420	C₂²	semidirect
19. #19151420	C₂²	semidirect
20. #19151420	C₂²	semidirect
21. #19151419	C₂²	semidirect
27. #499×C₈₉	C₂²	semidirect
28. #498×C₈₉	C₂²	semidirect
36. #15668034	Q₈	semidirect
38. C₂×Dic₃₅₆	Q₈	semidirect
40. C₂×D₇₁₂	Q₈	semidirect
168. #3163	D₈₉	semidirect
177. C₈₉	#33638	semidirect
184. #499	C₂×D₈₉	semidirect
185. #498	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029170	2
#26022425	2
#26022420	2
#26022024	2
C₂×#19151420	2
#26022018	2
#26021987	2
C₂×#19151407	2
C₈₉×#3163	2
#33638	89