Subgroups — #36709175

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	#26030780	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029169	C₂	1	1	yes	maximal, characteristic
3	#26022426	C₂	1	4	yes	maximal, complemented
4	#26022421	C₂	1	4	yes	maximal, complemented
5	#26022025	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151420	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022017	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021988	C₂	1	1	yes	maximal, characteristic
9	C₂×#360×C₈₉	C₂	1	1	yes	maximal, characteristic
10	#19152365	—	2	1	no	—
11	#19152229	—	2	1	no	—
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	#19152005	—	2	2	no	—
14	#19151998	C₂²	1	2	yes	complemented
15	#19151995	C₂²	1	2	yes	complemented
16	#19151720	—	2	2	no	—
17	#19151715	C₂²	1	4	yes	—
18	C₂×#15667948	C₂²	1	1	yes	characteristic
19	#19151638	—	2	1	no	—
20	#19151636	C₂²	1	1	yes	characteristic
21	#19151615	—	2	1	no	—
22	#19151553	C₂²	1	1	yes	characteristic
23	#19151421	C₂²	1	2	yes	complemented
24	#19151421	C₂²	1	2	yes	complemented
25	#19151420	C₂²	1	4	yes	complemented
26	#19151418	C₂²	1	4	yes	complemented
27	#19151408	C₂²	1	2	yes	—
28	#19151408	C₂²	1	2	yes	—
29	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
32	#360×C₈₉	C₂²	1	4	yes	—
33	#15668019	—	2	4	no	—
34	#15668019	—	4	2	no	—
35	C₂×#13083558	—	2	1	no	—
36	C₂×#13083558	—	4	2	no	—
37	C₂×#13083558	D₄	1	1	yes	characteristic
38	#15668011	—	2	2	no	—
39	#15668011	—	4	1	no	—
40	#15668003	—	2	2	no	—
41	#15668003	—	4	1	no	—
42	#15667991	—	4	2	no	—
43	#15667971	—	2	1	no	—
44	#15667971	D₄	1	1	yes	characteristic
45	#15667952	—	2	1	no	—
46	#15667952	C₂³	1	2	yes	—
47	C₂×#13083549	—	2	1	no	—
48	#15667948	—	2	1	no	—
49	#15667948	C₂³	1	2	yes	—
50	#15667946	—	2	1	no	—
51	#15667945	—	2	2	no	—
52	#15667944	C₂³	1	1	yes	characteristic
53	#15667944	C₂³	1	1	yes	characteristic
54	#15667943	—	2	2	no	—
55	#15667939	—	2	4	no	—
56	#15667922	D₄	1	2	yes	—
57	C₈×Dic₈₉	C₂³	1	2	yes	—
58	C₂×#41×C₈₉	—	2	1	no	—
59	#135×C₈₉	—	2	1	no	—
60	#135×C₈₉	—	2	1	no	—
61	#135×C₈₉	Q₈	1	2	yes	complemented
62	#135×C₈₉	C₂³	1	2	yes	—
63	C₂×Q₁₆×C₈₉	Q₈	1	1	yes	characteristic, complemented
64	C₂×SD₁₆×C₈₉	—	2	1	no	—
65	C₂×D₈×C₈₉	Q₈	1	1	yes	characteristic, complemented
66	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
68	M₃₂×C₈₉	—	2	1	no	—
69	M₃₂×C₈₉	C₂³	1	2	yes	—
70	C₂×C₁₆×C₈₉	—	2	1	no	—
71	#106×C₈₉	—	2	1	no	—
72	#106×C₈₉	C₂³	1	2	yes	—
73	C₂×Dic₁₇₈	—	4	2	no	—
74	#13083558	—	2	2	no	—
75	#13083558	—	4	1	no	—
76	#13083558	—	4	4	no	—
77	#13083558	SD₁₆	1	2	yes	—
78	#13083558	SD₁₆	1	2	yes	—
79	C₂×#10623494	—	4	1	no	—
80	C₂×#10623494	—	4	4	no	—
81	C₂×#10623494	C₂×D₄	1	2	yes	—
82	C₄⋊Dic₈₉	—	2	1	no	—
83	C₄⋊Dic₈₉	—	2	1	no	—
84	C₄⋊Dic₈₉	—	2	1	no	—
85	C₄⋊Dic₈₉	—	4	2	no	—
86	#13083553	—	4	2	no	—
87	#13083553	—	4	2	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₂²⋊Dic₈₉	—	4	1	no	—
90	C₄×Dic₈₉	—	4	1	no	—
91	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
92	#13083549	—	2	2	no	—
93	#41×C₈₉	—	2	2	no	—
94	#41×C₈₉	—	2	2	no	—
95	#41×C₈₉	—	4	1	no	—
96	#41×C₈₉	—	4	1	no	—
97	C₂×Q₈×C₈₉	—	2	1	no	—
98	C₂×D₄×C₈₉	—	2	1	no	—
99	C₂×D₄×C₈₉	—	4	1	no	—
100	C₂²×C₄×C₈₉	—	4	1	no	—
101	C₂²×C₄×C₈₉	—	4	1	no	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₈₉	—	2	1	no	—
104	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	—
105	SD₁₆×C₈₉	—	2	2	no	—
106	SD₁₆×C₈₉	—	2	2	no	—
107	D₈×C₈₉	—	2	1	no	—
108	D₈×C₈₉	C₂×Q₈	1	2	yes	—
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
112	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
113	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₈₉	—	2	2	no	—
116	C₄⋊C₄×C₈₉	—	4	1	no	—
117	C₁₆×C₈₉	—	2	2	no	—
118	C₂³×C₈₉	—	4	1	no	—
119	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	Q₈×C₈₉	—	2	2	no	—
121	Q₈×C₈₉	—	4	1	no	—
122	D₄×C₈₉	—	2	2	no	—
123	D₄×C₈₉	—	4	2	no	—
124	D₄×C₈₉	—	4	2	no	—
125	D₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	16	1	no	—
131	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₈₉	—	2	2	no	—
138	C₂×Dic₈₉	—	4	1	no	—
139	C₂×Dic₈₉	—	4	1	no	—
140	Dic₁₇₈	—	16	2	no	—
141	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
142	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
144	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
145	#10623494	—	4	2	no	—
146	#10623494	C₂×SD₁₆	1	4	yes	—
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	—	4	2	no	—
149	C₂²×C₈₉	—	16	1	no	—
150	C₂²×C₈₉	#476	1	1	yes	characteristic
151	C₂²×C₈₉	#476	1	1	yes	characteristic
152	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₄×C₈₉	—	4	2	no	—
154	C₄×C₈₉	—	8	2	no	—
155	C₄×C₈₉	#476	1	1	yes	characteristic
156	C₄×C₈₉	#476	1	1	yes	characteristic
157	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
159	Dic₈₉	—	4	2	no	—
160	Dic₈₉	—	8	1	no	—
161	Dic₈₉	—	8	1	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	—	8	2	no	—
167	C₂×C₈₉	C₂×#476	1	1	yes	characteristic
168	#3163	—	89	1	no	—
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	D₈₉	1	1	yes	characteristic
177	C₈₉	#33638	1	1	yes	characteristic, complemented, Sylow
178	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
179	#568	—	178	1	no	—
180	#544	—	178	1	no	—
181	C₂×M₃₂	—	89	1	no	—
182	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
183	#501	—	178	2	no	—
184	#499	—	89	2	no	—
185	#498	—	89	2	no	—
186	#444	—	178	2	no	—
187	#443	—	89	4	no	—
188	C₂×#108	—	89	1	no	—
189	#428	—	178	1	no	—
190	#427	—	89	1	no	—
191	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
192	#421	—	178	1	no	—
193	#405	—	89	1	no	—
194	#365	—	89	2	no	—
195	#365	—	89	2	no	—
196	#365	—	89	4	no	—
197	#364	—	89	4	no	—
198	#360	—	89	2	no	—
199	#360	—	89	2	no	—
200	#360	C₂×D₈₉	1	4	yes	—
201	C₂×#41	—	2	1	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	Dic₁₇₈	1	2	yes	complemented
205	#135	C₂²×D₈₉	1	2	yes	—
206	C₂×Q₁₆	Dic₁₇₈	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	Dic₁₇₈	1	1	yes	characteristic, complemented
209	#131	—	356	2	no	—
210	#131	—	178	4	no	—
211	C₂×M₁₆	—	89	1	no	—
212	C₂×M₁₆	—	356	2	no	—
213	C₂×M₁₆	—	178	1	no	—
214	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
215	C₄⋊Q₈	—	356	1	no	—
216	C₄⋊Q₈	—	178	2	no	—
217	#125	—	356	1	no	—
218	#125	—	178	2	no	—
219	C₂²⋊Q₈	—	356	2	no	—
220	#117	—	89	1	no	—
221	#117	—	178	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	89	2	no	—
225	M₃₂	—	178	1	no	—
226	M₃₂	C₂²×D₈₉	1	2	yes	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	178	1	no	—
229	#108	—	89	2	no	—
230	#108	—	178	1	no	—
231	#107	—	178	2	no	—
232	#107	—	178	1	no	—
233	#106	—	2	1	no	—
234	#106	—	89	1	no	—
235	#106	—	89	1	no	—
236	#106	—	178	2	no	—
237	#106	C₂²×D₈₉	1	2	yes	—
238	#104	—	178	4	no	—
239	#97	—	89	2	no	—
240	C₄×C₈	—	89	2	no	—
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	356	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
252	Q₁₆	—	2	1	no	—
253	Q₁₆	C₂×Dic₁₇₈	1	2	yes	—
254	SD₁₆	—	2	2	no	—
255	SD₁₆	—	2	2	no	—
256	D₈	—	2	1	no	—
257	D₈	C₂×Dic₁₇₈	1	2	yes	—
258	M₁₆	—	89	2	no	—
259	M₁₆	—	89	2	no	—
260	M₁₆	—	356	4	no	—
261	M₁₆	—	356	1	no	—
262	M₁₆	—	178	2	no	—
263	C₂×C₈	—	89	2	no	—
264	C₂×C₈	—	356	4	no	—
265	C₂×C₈	—	356	1	no	—
266	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
268	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
269	C₂×C₈	#13083573	1	1	yes	characteristic
270	C₂×C₈	#13083573	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	356	2	no	—
275	C₄⋊C₄	—	356	2	no	—
276	C₄⋊C₄	—	356	2	no	—
277	C₄⋊C₄	—	178	1	no	—
278	C₄⋊C₄	—	178	1	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₂²⋊C₄	—	356	1	no	—
281	C₂²⋊C₄	—	178	1	no	—
282	C₄²	—	89	2	no	—
283	C₄²	—	356	1	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	178	2	no	—
286	C₂³	—	4	1	no	—
287	C₂³	#15668036	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	1424	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	16	1	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	1	no	—
302	C₂×C₄	—	178	2	no	—
303	C₂×C₄	#15667989	1	1	yes	characteristic
304	C₂×C₄	#15667989	1	1	yes	characteristic
305	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
307	C₂×C₄	#15668036	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
309	C₈	—	89	4	no	—
310	C₈	—	356	2	no	—
311	C₈	#15667989	1	1	yes	characteristic
312	C₈	#15667989	1	1	yes	characteristic
313	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
314	C₈	C₂×#13083573	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	#19151886	1	1	yes	characteristic
319	C₂²	#19151886	1	1	yes	characteristic
320	C₂²	C₂×#15668036	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	356	2	no	—
324	C₄	—	712	1	no	—
325	C₄	—	712	1	no	—
326	C₄	#19151886	1	1	yes	characteristic
327	C₄	#19151886	1	1	yes	characteristic
328	C₄	C₂×#15667989	1	1	yes	characteristic
329	C₄	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×#19151886	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19151886
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#360×C₈₉	C₂
socle	C₂×C₈₉	C₂×#476
prime radical	C₂×#135×C₈₉	C₂²
hypercenter	C₂×#360	D₈₉
hypocenter	C₈₉	#33638
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
1. #26030780	C₂	semidirect
3. #26022426	C₂	semidirect
4. #26022421	C₂	semidirect
5. #26022025	C₂	semidirect
6. C₂×#19151420	C₂	semidirect
7. #26022017	C₂	semidirect
14. #19151998	C₂²	semidirect
15. #19151995	C₂²	semidirect
23. #19151421	C₂²	semidirect
24. #19151421	C₂²	semidirect
25. #19151420	C₂²	semidirect
26. #19151418	C₂²	semidirect
61. #135×C₈₉	Q₈	semidirect
63. C₂×Q₁₆×C₈₉	Q₈	semidirect
65. C₂×D₈×C₈₉	Q₈	semidirect
177. C₈₉	#33638	semidirect
204. #135	Dic₁₇₈	semidirect
206. C₂×Q₁₆	Dic₁₇₈	semidirect
208. C₂×D₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030780	2
#26029169	2
#26022426	2
#26022421	2
#26022025	2
C₂×#19151420	2
#26022017	2
#26021988	2
C₂×#360×C₈₉	2
#33638	89