Subgroups — #26033201

Order 11392.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 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	#19152307	C₂	1	1	yes	maximal, characteristic
3	#19152006	C₂	1	1	yes	maximal, characteristic, complemented
4	#19152005	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151997	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151906	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151904	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151896	C₂	1	1	yes	maximal, characteristic
9	#19151886	C₂	1	1	yes	maximal, characteristic
10	#19151882	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151725	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151695	C₂	1	1	yes	maximal, characteristic
13	#19151695	C₂	1	1	yes	maximal, characteristic
14	C₄×#13083562	C₂	1	1	yes	maximal, characteristic
15	#478×C₈₉	C₂	1	1	yes	maximal, characteristic
16	C₂×#13083562	C₂²	1	1	yes	characteristic
17	#15668011	—	2	1	no	—
18	#15668011	C₂²	1	1	yes	characteristic
19	#15668011	C₂²	1	1	yes	characteristic
20	#15668004	C₂²	1	1	yes	characteristic
21	#15668003	C₂²	1	1	yes	characteristic
22	#15668003	C₂²	1	1	yes	characteristic
23	#15668002	—	2	1	no	—
24	#15668002	—	2	1	no	—
25	#15667991	—	2	1	no	—
26	#15667989	—	2	1	no	—
27	#15667989	—	2	1	no	—
28	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
30	#15667977	—	2	1	no	—
31	#15667977	C₂²	1	1	yes	characteristic
32	#15667977	C₂²	1	1	yes	characteristic
33	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
34	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
35	C₂×#13083553	—	2	1	no	—
36	#15667946	C₂²	1	1	yes	characteristic
37	#15667945	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667944	C₂²	1	1	yes	characteristic
40	#15667943	C₂²	1	1	yes	characteristic
41	#15667943	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	#15667941	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667937	C₂²	1	1	yes	characteristic
46	#15667937	C₂²	1	1	yes	characteristic
47	#15667935	C₂²	1	1	yes	characteristic
48	#15667933	C₂²	1	1	yes	characteristic
49	#15667922	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₄×#10623494	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	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂²×Dic₈₉	—	2	1	no	—
64	C₂²×Dic₈₉	—	2	1	no	—
65	#13083562	Q₈	1	4	yes	complemented
66	C₂×#10623494	C₂³	1	1	yes	characteristic
67	C₂×#10623494	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₈₉	—	2	1	no	—
69	C₄⋊Dic₈₉	—	2	1	no	—
70	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
71	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
74	#13083553	—	2	1	no	—
75	#13083553	—	2	1	no	—
76	#13083553	—	2	1	no	—
77	#13083553	—	2	1	no	—
78	#13083553	—	2	1	no	—
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	D₄	1	1	yes	characteristic
84	#13083553	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	C₄×Dic₈₉	—	2	1	no	—
88	C₄×Dic₈₉	—	2	1	no	—
89	C₄×Dic₈₉	—	2	1	no	—
90	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
91	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	—	2	1	no	—
94	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	—	2	1	no	—
97	C₄⋊C₄×C₈₉	—	2	1	no	—
98	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂²⋊C₄×C₈₉	—	2	1	no	—
102	C₄²×C₈₉	C₂³	1	1	yes	characteristic
103	C₂³×C₈₉	—	2	1	no	—
104	D₄×C₈₉	—	2	1	no	—
105	D₄×C₈₉	C₂×Q₈	1	2	yes	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	8	1	no	—
110	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₂×Dic₈₉	—	2	1	no	—
114	C₂×Dic₈₉	—	2	1	no	—
115	C₂×Dic₈₉	—	2	1	no	—
116	C₂×Dic₈₉	—	2	1	no	—
117	C₂×Dic₈₉	—	4	2	no	—
118	C₂×Dic₈₉	—	8	1	no	—
119	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
123	Dic₁₇₈	—	2	1	no	—
124	Dic₁₇₈	—	4	1	no	—
125	Dic₁₇₈	—	4	2	no	—
126	Dic₁₇₈	—	8	1	no	—
127	Dic₁₇₈	C₂×Q₈	1	2	yes	—
128	C₈×C₈₉	—	2	1	no	—
129	C₈×C₈₉	—	2	1	no	—
130	#10623494	—	2	1	no	—
131	#10623494	C₂×Q₈	1	2	yes	—
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	4	1	no	—
134	C₂²×C₈₉	—	4	1	no	—
135	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	—	4	1	no	—
139	C₄×C₈₉	—	4	1	no	—
140	C₄×C₈₉	—	4	1	no	—
141	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
142	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
143	Dic₈₉	—	2	2	no	—
144	Dic₈₉	—	2	1	no	—
145	Dic₈₉	—	2	1	no	—
146	Dic₈₉	—	2	1	no	—
147	Dic₈₉	—	2	1	no	—
148	Dic₈₉	—	4	1	no	—
149	Dic₈₉	—	4	1	no	—
150	Dic₈₉	—	4	1	no	—
151	C₂×C₈₉	—	4	2	no	—
152	C₂×C₈₉	D₄×Q₈	1	1	yes	characteristic
153	C₂×C₈₉	#575	1	1	yes	characteristic
154	C₂×C₈₉	#577	1	1	yes	characteristic
155	#3388	—	89	1	no	maximal, Sylow
156	C₈₉	#3388	1	1	yes	characteristic, complemented, Sylow
157	#557	—	89	1	no	—
158	#554	—	89	1	no	—
159	#501	—	89	1	no	—
160	#501	—	89	1	no	—
161	#499	—	89	1	no	—
162	#479	—	89	1	no	—
163	#479	—	89	1	no	—
164	#478	—	89	1	no	—
165	#478	D₈₉	1	1	yes	characteristic
166	#476	—	89	1	no	—
167	#475	—	89	1	no	—
168	#446	—	89	1	no	—
169	#440	—	89	1	no	—
170	#440	—	89	1	no	—
171	C₄×SD₁₆	—	89	1	no	—
172	C₂×SD₁₆	—	89	1	no	—
173	C₄⋊Q₈	—	89	1	no	—
174	C₄⋊Q₈	—	89	1	no	—
175	C₄⋊Q₈	—	178	1	no	—
176	#125	—	89	1	no	—
177	#125	—	89	1	no	—
178	#125	—	89	1	no	—
179	#125	—	178	1	no	—
180	#125	—	178	1	no	—
181	#125	C₂×D₈₉	1	1	yes	characteristic
182	C₂²⋊Q₈	—	178	1	no	—
183	C₂²⋊Q₈	—	178	1	no	—
184	C₂²⋊Q₈	—	178	1	no	—
185	C₄×Q₈	—	89	1	no	—
186	C₄×Q₈	—	89	1	no	—
187	C₄×Q₈	—	89	1	no	—
188	C₄×Q₈	—	89	1	no	—
189	C₄×Q₈	—	178	1	no	—
190	C₄×D₄	—	89	1	no	—
191	C₄×D₄	—	89	1	no	—
192	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
193	C₂×C₄⋊C₄	—	178	1	no	—
194	#107	—	89	1	no	—
195	#107	—	89	1	no	—
196	#107	—	89	1	no	—
197	#107	C₂×D₈₉	1	1	yes	characteristic
198	#106	—	89	1	no	—
199	#106	—	89	1	no	—
200	#106	—	89	1	no	—
201	#106	—	89	1	no	—
202	#106	C₂×D₈₉	1	1	yes	characteristic
203	C₄⋊C₈	—	89	1	no	—
204	C₄⋊C₈	—	89	1	no	—
205	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
206	#103	—	89	1	no	—
207	#103	—	89	1	no	—
208	#103	—	89	1	no	—
209	#102	—	89	1	no	—
210	#102	C₂×D₈₉	1	1	yes	characteristic
211	#102	C₂×D₈₉	1	1	yes	characteristic
212	#97	—	89	1	no	—
213	#97	—	89	1	no	—
214	C₄×C₈	—	89	1	no	—
215	C₂×Q₈	—	89	1	no	—
216	C₂×Q₈	—	89	1	no	—
217	C₂×Q₈	—	178	1	no	—
218	C₂×Q₈	—	178	1	no	—
219	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	178	1	no	—
222	C₂²×C₄	—	178	1	no	—
223	SD₁₆	—	89	4	no	—
224	C₂×C₈	—	89	1	no	—
225	C₂×C₈	—	89	1	no	—
226	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
227	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	89	1	no	—
231	C₄⋊C₄	—	89	1	no	—
232	C₄⋊C₄	—	89	1	no	—
233	C₄⋊C₄	—	89	1	no	—
234	C₄⋊C₄	—	89	1	no	—
235	C₄⋊C₄	—	89	1	no	—
236	C₄⋊C₄	—	178	1	no	—
237	C₄⋊C₄	—	178	1	no	—
238	C₄⋊C₄	—	178	1	no	—
239	C₄⋊C₄	—	178	1	no	—
240	C₄⋊C₄	—	178	1	no	—
241	C₄⋊C₄	—	178	1	no	—
242	C₄⋊C₄	—	178	1	no	—
243	C₄⋊C₄	—	178	1	no	—
244	C₄⋊C₄	—	178	1	no	—
245	C₄⋊C₄	—	178	1	no	—
246	C₄⋊C₄	—	178	1	no	—
247	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	178	1	no	—
252	C₂²⋊C₄	—	178	1	no	—
253	C₄²	—	89	1	no	—
254	C₄²	—	89	1	no	—
255	C₄²	—	178	1	no	—
256	C₄²	—	178	1	no	—
257	C₄²	—	178	1	no	—
258	C₄²	C₂²×D₈₉	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	Q₈	—	89	2	no	—
261	Q₈	—	356	1	no	—
262	Q₈	—	356	2	no	—
263	Q₈	—	178	1	no	—
264	Q₈	—	712	1	no	—
265	D₄	—	2	1	no	—
266	D₄	Q₈×D₈₉	1	2	yes	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	89	1	no	—
272	C₂×C₄	—	89	1	no	—
273	C₂×C₄	—	89	1	no	—
274	C₂×C₄	—	89	1	no	—
275	C₂×C₄	—	356	2	no	—
276	C₂×C₄	—	178	1	no	—
277	C₂×C₄	—	178	1	no	—
278	C₂×C₄	—	178	1	no	—
279	C₂×C₄	—	178	1	no	—
280	C₂×C₄	—	712	1	no	—
281	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
282	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	89	2	no	—
287	C₈	—	178	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	356	1	no	—
298	C₄	—	356	1	no	—
299	C₄	—	356	1	no	—
300	C₄	—	178	1	no	—
301	C₄	—	178	1	no	—
302	C₄	—	178	1	no	—
303	C₄	—	178	1	no	—
304	C₄	—	178	2	no	—
305	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
306	C₄	#15668059	1	1	yes	characteristic
307	C₂	—	4	2	no	—
308	C₂	#19152261	1	1	yes	characteristic
309	C₂	#19152395	1	1	yes	characteristic
310	C₂	#19152408	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	C₂×D₄×C₈₉	C₂³
hypercenter	#478	D₈₉
hypocenter	C₈₉	#3388
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. #19152006	C₂	semidirect
4. #19152005	C₂	semidirect
5. #19151997	C₂	semidirect
6. #19151906	C₂	semidirect
7. #19151904	C₂	semidirect
10. #19151882	C₂	semidirect
11. #19151725	C₂	semidirect
65. #13083562	Q₈	semidirect
156. C₈₉	#3388	semidirect

Maximal subgroups

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

subgroup	index
#19152325	2
#19152307	2
#19152006	2
#19152005	2
#19151997	2
#19151906	2
#19151904	2
#19151896	2
#19151886	2
#19151882	2
#19151725	2
#19151695	2
C₄×#13083562	2
#478×C₈₉	2
#3388	89