Subgroups — #26032984

Order 11392.

382 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	#19152325	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152269	C₂	1	1	yes	maximal, characteristic
3	#19151964	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151961	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151943	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151897	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151877	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151799	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151797	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151797	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151727	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151698	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151697	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151685	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	—	2	1	no	—
17	C₂×#13083563	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	#15668011	—	2	1	no	—
21	#15668011	C₂²	1	1	yes	characteristic
22	#15668009	C₂²	1	1	yes	characteristic
23	#15668008	C₂²	1	1	yes	characteristic
24	#15668008	C₂²	1	1	yes	characteristic
25	#15668003	C₂²	1	1	yes	characteristic
26	#15668002	—	2	1	no	—
27	#15668002	—	2	1	no	—
28	#15667986	—	2	1	no	—
29	#15667985	—	2	1	no	—
30	#15667985	—	2	1	no	—
31	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
33	#15667977	—	2	1	no	—
34	#15667977	C₂²	1	1	yes	characteristic
35	#15667977	C₂²	1	1	yes	characteristic
36	#15667974	C₂²	1	1	yes	characteristic
37	#15667974	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667941	C₂²	1	1	yes	characteristic
40	#15667941	C₂²	1	1	yes	characteristic
41	#15667935	C₂²	1	1	yes	characteristic
42	#15667934	C₂²	1	1	yes	characteristic
43	#15667934	C₂²	1	1	yes	characteristic
44	#15667933	C₂²	1	1	yes	characteristic
45	#15667932	C₂²	1	1	yes	characteristic
46	#15667932	C₂²	1	1	yes	characteristic
47	#15667932	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	#15667923	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	4	4	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	—	2	1	no	—
65	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
66	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂×C₂²⋊D₈₉	—	2	1	no	—
70	C₂×C₂²⋊D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	—	2	1	no	—
73	#13083563	—	2	2	no	—
74	#13083563	—	2	2	no	—
75	#13083562	—	2	2	no	—
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₂×#10623494	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	—	2	1	no	—
79	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
80	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	D₄	1	1	yes	characteristic
87	#13083553	D₄	1	1	yes	characteristic
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
94	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	—	2	1	no	—
97	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₄⋊C₄×C₈₉	—	2	1	no	—
101	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₉	—	2	1	no	—
105	C₄²×C₈₉	C₂³	1	1	yes	characteristic
106	C₂³×C₈₉	—	2	1	no	—
107	C₂²×D₈₉	—	2	1	no	—
108	C₂²×D₈₉	—	2	1	no	—
109	D₄×C₈₉	—	2	1	no	—
110	D₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	4	2	no	—
115	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₈₉	—	4	2	no	—
119	C₂²⋊D₈₉	—	4	2	no	—
120	C₂²⋊D₈₉	—	8	1	no	—
121	C₂²⋊D₈₉	—	8	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
128	D₃₅₆	—	2	1	no	—
129	D₃₅₆	—	2	1	no	—
130	D₃₅₆	—	2	1	no	—
131	D₃₅₆	—	2	1	no	—
132	D₃₅₆	—	4	2	no	—
133	D₃₅₆	—	8	1	no	—
134	C₄×D₈₉	—	4	2	no	—
135	C₄×D₈₉	—	4	2	no	—
136	Dic₁₇₈	—	2	1	no	—
137	Dic₁₇₈	—	2	2	no	—
138	Dic₁₇₈	—	4	2	no	—
139	Dic₁₇₈	#41	1	2	yes	complemented
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	#10623494	—	2	1	no	—
143	#10623494	—	2	1	no	—
144	C₂²×C₈₉	—	4	1	no	—
145	C₂²×C₈₉	—	4	1	no	—
146	C₂²×C₈₉	—	4	1	no	—
147	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₈₉	—	4	1	no	—
149	C₂×D₈₉	—	4	1	no	—
150	C₂×D₈₉	—	4	1	no	—
151	C₂×D₈₉	—	4	1	no	—
152	C₂×D₈₉	—	4	1	no	—
153	C₂×D₈₉	—	4	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	4	1	no	—
157	C₄×C₈₉	—	4	1	no	—
158	C₄×C₈₉	—	4	1	no	—
159	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
160	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₈₉	—	2	2	no	—
162	Dic₈₉	—	2	1	no	—
163	Dic₈₉	—	2	1	no	—
164	Dic₈₉	—	2	1	no	—
165	Dic₈₉	—	4	1	no	—
166	Dic₈₉	AGL(1,ℤ/8)	1	2	yes	—
167	C₂×C₈₉	—	8	1	no	—
168	C₂×C₈₉	#548	1	1	yes	characteristic
169	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₈₉	#577	1	1	yes	characteristic
171	D₈₉	—	8	1	no	—
172	D₈₉	—	8	1	no	—
173	#3368	—	89	1	no	maximal, Sylow
174	C₈₉	#3368	1	1	yes	characteristic, complemented, Sylow
175	#557	—	89	1	no	—
176	#550	—	89	1	no	—
177	#490	—	89	1	no	—
178	#490	—	89	1	no	—
179	#486	—	89	1	no	—
180	#478	—	89	1	no	—
181	#478	D₈₉	1	1	yes	characteristic, complemented
182	#474	—	89	1	no	—
183	#460	—	89	1	no	—
184	#460	—	89	1	no	—
185	#460	—	89	1	no	—
186	#446	—	89	1	no	—
187	#440	—	89	1	no	—
188	#440	—	89	1	no	—
189	C₄×SD₁₆	—	89	1	no	—
190	C₂×#41	—	178	1	no	—
191	C₂×SD₁₆	—	89	1	no	—
192	C₂×SD₁₆	—	89	1	no	—
193	C₂×SD₁₆	—	89	1	no	—
194	C₄⋊Q₈	—	89	1	no	—
195	C₄⋊Q₈	—	178	1	no	—
196	#127	—	89	1	no	—
197	#127	—	89	1	no	—
198	#127	—	89	1	no	—
199	#125	—	89	1	no	—
200	#125	—	178	1	no	—
201	#125	—	178	1	no	—
202	#125	C₂×D₈₉	1	1	yes	characteristic
203	#121	—	178	1	no	—
204	#121	—	178	1	no	—
205	#121	—	178	1	no	—
206	C₄×Q₈	—	89	1	no	—
207	C₄×Q₈	—	89	1	no	—
208	C₄×Q₈	—	89	1	no	—
209	C₄×Q₈	—	89	1	no	—
210	C₄×Q₈	—	178	1	no	—
211	C₄×D₄	—	89	1	no	—
212	C₄×D₄	—	89	1	no	—
213	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
214	#107	—	89	1	no	—
215	#107	C₂×D₈₉	1	1	yes	characteristic
216	#106	C₂×D₈₉	1	1	yes	characteristic
217	C₄⋊C₈	—	89	1	no	—
218	C₄⋊C₈	—	89	1	no	—
219	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
220	#103	—	89	1	no	—
221	#102	—	89	1	no	—
222	#102	—	89	1	no	—
223	#102	—	89	1	no	—
224	#102	—	89	1	no	—
225	#102	—	89	1	no	—
226	#102	—	89	1	no	—
227	#102	—	89	1	no	—
228	#102	C₂×D₈₉	1	1	yes	characteristic
229	#102	C₂×D₈₉	1	1	yes	characteristic
230	#97	—	89	1	no	—
231	#97	—	89	1	no	—
232	C₄×C₈	—	89	1	no	—
233	#41	—	356	4	no	—
234	C₂×Q₈	—	89	1	no	—
235	C₂×Q₈	—	89	1	no	—
236	C₂×Q₈	—	178	1	no	—
237	C₂×D₄	—	89	1	no	—
238	C₂×D₄	—	89	1	no	—
239	C₂×D₄	—	178	1	no	—
240	C₂×D₄	—	178	1	no	—
241	C₂×D₄	—	178	1	no	—
242	C₂×D₄	—	178	1	no	—
243	C₂×D₄	—	178	1	no	—
244	C₂×D₄	—	178	1	no	—
245	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	178	1	no	—
248	C₂²×C₄	—	178	1	no	—
249	SD₁₆	—	178	2	no	—
250	SD₁₆	—	178	2	no	—
251	SD₁₆	—	178	2	no	—
252	C₂×C₈	—	89	1	no	—
253	C₂×C₈	—	89	1	no	—
254	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	89	1	no	—
259	C₄⋊C₄	—	89	1	no	—
260	C₄⋊C₄	—	89	1	no	—
261	C₄⋊C₄	—	89	1	no	—
262	C₄⋊C₄	—	178	1	no	—
263	C₄⋊C₄	—	178	1	no	—
264	C₄⋊C₄	—	178	1	no	—
265	C₄⋊C₄	—	178	1	no	—
266	C₄⋊C₄	—	178	1	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	178	1	no	—
273	C₂²⋊C₄	—	178	1	no	—
274	C₄²	—	89	1	no	—
275	C₄²	—	89	1	no	—
276	C₄²	—	178	1	no	—
277	C₄²	—	178	1	no	—
278	C₄²	—	178	1	no	—
279	C₄²	C₂²×D₈₉	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	—	178	1	no	—
282	C₂³	—	178	1	no	—
283	Q₈	—	89	2	no	—
284	Q₈	—	356	2	no	—
285	Q₈	—	178	2	no	—
286	Q₈	—	178	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	356	2	no	—
290	D₄	—	356	2	no	—
291	D₄	—	356	2	no	—
292	D₄	—	178	1	no	—
293	D₄	—	178	1	no	—
294	D₄	—	178	1	no	—
295	D₄	—	178	1	no	—
296	D₄	—	712	1	no	—
297	D₄	—	712	1	no	—
298	D₄	—	712	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	89	1	no	—
304	C₂×C₄	—	89	1	no	—
305	C₂×C₄	—	89	1	no	—
306	C₂×C₄	—	89	1	no	—
307	C₂×C₄	—	356	2	no	—
308	C₂×C₄	—	356	2	no	—
309	C₂×C₄	—	178	1	no	—
310	C₂×C₄	—	178	1	no	—
311	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
312	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	178	1	no	—
317	C₈	—	178	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	356	1	no	—
322	C₂²	—	356	1	no	—
323	C₂²	—	356	1	no	—
324	C₂²	—	356	1	no	—
325	C₂²	—	356	1	no	—
326	C₂²	—	356	1	no	—
327	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	89	2	no	—
334	C₄	—	356	1	no	—
335	C₄	—	178	1	no	—
336	C₄	—	178	1	no	—
337	C₄	—	178	2	no	—
338	C₄	—	178	1	no	—
339	C₄	#41×D₈₉	1	1	yes	characteristic
340	C₄	#15668060	1	1	yes	characteristic
341	C₂	—	8	1	no	—
342	C₂	—	712	1	no	—
343	C₂	—	712	1	no	—
344	C₂	#19152258	1	1	yes	characteristic
345	C₂	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
346	C₂	#19152405	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#478×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152269	C₂
hypercenter	#478	D₈₉
hypocenter	C₈₉	#3368
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #19152325	C₂	semidirect
3. #19151964	C₂	semidirect
4. #19151961	C₂	semidirect
5. #19151943	C₂	semidirect
6. #19151897	C₂	semidirect
7. #19151877	C₂	semidirect
8. #19151799	C₂	semidirect
9. #19151797	C₂	semidirect
10. #19151797	C₂	semidirect
11. #19151727	C₂	semidirect
12. #19151698	C₂	semidirect
13. #19151697	C₂	semidirect
14. #19151685	C₂	semidirect
15. #478×C₈₉	C₂	semidirect
139. Dic₁₇₈	#41	semidirect
174. C₈₉	#3368	semidirect
181. #478	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152325	2
#19152269	2
#19151964	2
#19151961	2
#19151943	2
#19151897	2
#19151877	2
#19151799	2
#19151797	2
#19151727	2
#19151698	2
#19151697	2
#19151685	2
#478×C₈₉	2
#3368	89