Subgroups — #26033211

Order 11392.

348 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152307	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152305	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152007	C₂	1	1	yes	maximal, characteristic, complemented
4	#19152005	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151998	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151900	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151900	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151898	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151896	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151878	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151876	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151726	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151710	C₂	1	1	yes	maximal, characteristic
14	D₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
15	#442×C₈₉	C₂	1	1	yes	maximal, characteristic
16	#15668012	C₂²	1	1	yes	characteristic
17	#15668011	C₂²	1	1	yes	characteristic
18	#15668004	C₂²	1	1	yes	characteristic
19	#15668003	C₂²	1	1	yes	characteristic
20	#15668003	C₂²	1	1	yes	characteristic
21	#15668003	C₂²	1	1	yes	characteristic
22	#15667991	—	2	1	no	—
23	#15667991	—	2	1	no	—
24	#15667989	—	2	1	no	—
25	#15667989	—	2	1	no	—
26	#15667989	—	2	1	no	—
27	#15667989	—	2	1	no	—
28	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
29	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
30	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
32	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊Dic₈₉	—	2	1	no	—
34	C₂×#13083553	—	2	1	no	—
35	#15667946	C₂²	1	1	yes	characteristic
36	#15667946	C₂²	1	1	yes	characteristic
37	#15667945	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667945	C₂²	1	1	yes	characteristic
40	#15667944	C₂²	1	1	yes	characteristic
41	#15667944	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	C₄⋊#10623494	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667941	C₂²	1	1	yes	characteristic
46	#15667933	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic
48	#15667933	C₂²	1	1	yes	characteristic
49	#15667933	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic
56	#102×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#97×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	C₂²×Dic₈₉	—	2	1	no	—
63	C₂²×Dic₈₉	—	2	1	no	—
64	C₂²×Dic₈₉	—	2	1	no	—
65	C₂²×Dic₈₉	—	2	1	no	—
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	—	2	1	no	—
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
73	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₈₉	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
90	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
91	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	—	2	1	no	—
94	C₂²×C₄×C₈₉	—	2	1	no	—
95	D₈×C₈₉	Q₈	1	4	yes	complemented
96	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₂²⋊C₄×C₈₉	—	2	1	no	—
101	C₂²⋊C₄×C₈₉	—	2	1	no	—
102	C₄²×C₈₉	C₂³	1	1	yes	characteristic
103	C₂³×C₈₉	—	2	1	no	—
104	C₂³×C₈₉	—	2	1	no	—
105	D₄×C₈₉	—	2	1	no	—
106	D₄×C₈₉	—	2	1	no	—
107	D₄×C₈₉	C₂×Q₈	1	2	yes	—
108	D₄×C₈₉	C₂×Q₈	1	2	yes	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	4	2	no	—
112	C₂×C₄×C₈₉	—	8	1	no	—
113	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₈₉	—	2	1	no	—
117	C₂×Dic₈₉	—	2	1	no	—
118	C₂×Dic₈₉	—	2	1	no	—
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	—	4	2	no	—
121	C₂×Dic₈₉	—	8	1	no	—
122	C₂×Dic₈₉	—	8	1	no	—
123	C₂×Dic₈₉	—	8	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	Dic₁₇₈	—	4	1	no	—
129	Dic₁₇₈	—	8	1	no	—
130	Dic₁₇₈	—	8	1	no	—
131	C₈×C₈₉	—	2	1	no	—
132	C₈×C₈₉	C₂×Q₈	1	2	yes	—
133	#10623494	—	2	1	no	—
134	#10623494	—	2	1	no	—
135	C₂²×C₈₉	—	2	2	no	—
136	C₂²×C₈₉	—	2	2	no	—
137	C₂²×C₈₉	—	4	1	no	—
138	C₂²×C₈₉	—	4	1	no	—
139	C₂²×C₈₉	—	4	1	no	—
140	C₂²×C₈₉	—	4	1	no	—
141	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	4	1	no	—
145	C₄×C₈₉	—	4	1	no	—
146	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
147	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₈₉	—	2	1	no	—
149	Dic₈₉	—	2	1	no	—
150	Dic₈₉	—	2	1	no	—
151	Dic₈₉	—	2	1	no	—
152	Dic₈₉	—	4	1	no	—
153	Dic₈₉	—	4	1	no	—
154	Dic₈₉	—	4	1	no	—
155	Dic₈₉	—	4	1	no	—
156	C₂×C₈₉	—	4	2	no	—
157	C₂×C₈₉	—	4	2	no	—
158	C₂×C₈₉	D₄×Q₈	1	1	yes	characteristic
159	C₂×C₈₉	#575	1	1	yes	characteristic
160	C₂×C₈₉	#576	1	1	yes	characteristic
161	#3389	—	89	1	no	maximal, Sylow
162	C₈₉	#3389	1	1	yes	characteristic, complemented, Sylow
163	#554	—	89	1	no	—
164	#554	—	89	1	no	—
165	#501	—	89	1	no	—
166	#501	—	89	1	no	—
167	#499	—	89	1	no	—
168	#478	—	89	1	no	—
169	#478	—	89	1	no	—
170	#478	—	89	1	no	—
171	#478	—	89	1	no	—
172	#474	—	89	1	no	—
173	#474	—	89	1	no	—
174	#446	—	89	1	no	—
175	#442	—	89	1	no	—
176	#442	D₈₉	1	1	yes	characteristic
177	C₄×D₈	—	89	1	no	—
178	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
179	C₄⋊Q₈	—	89	1	no	—
180	C₄⋊Q₈	—	89	1	no	—
181	#125	—	89	1	no	—
182	#125	—	89	1	no	—
183	#125	—	89	1	no	—
184	#125	—	89	1	no	—
185	C₂²⋊Q₈	—	178	1	no	—
186	C₂²⋊Q₈	—	178	1	no	—
187	C₂²⋊Q₈	—	178	1	no	—
188	C₂²⋊Q₈	—	178	1	no	—
189	C₂²⋊Q₈	—	178	1	no	—
190	C₂²⋊Q₈	—	178	1	no	—
191	C₄×Q₈	—	89	1	no	—
192	C₄×D₄	—	89	1	no	—
193	C₄×D₄	—	89	1	no	—
194	C₄×D₄	—	89	1	no	—
195	C₄×D₄	—	89	1	no	—
196	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
197	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
198	C₂×C₄⋊C₄	—	178	1	no	—
199	C₂×C₄⋊C₄	—	178	1	no	—
200	#107	—	89	1	no	—
201	#107	—	89	1	no	—
202	#107	—	89	1	no	—
203	#107	—	89	1	no	—
204	#107	—	89	1	no	—
205	#106	—	89	1	no	—
206	#106	—	89	1	no	—
207	#106	—	89	1	no	—
208	#106	C₂×D₈₉	1	1	yes	characteristic
209	C₄⋊C₈	—	89	1	no	—
210	C₄⋊C₈	—	89	1	no	—
211	C₄⋊C₈	—	89	1	no	—
212	#102	—	89	1	no	—
213	#102	—	89	1	no	—
214	#102	—	89	1	no	—
215	#102	—	89	1	no	—
216	#102	C₂×D₈₉	1	1	yes	characteristic
217	#102	C₂×D₈₉	1	1	yes	characteristic
218	#97	—	89	1	no	—
219	#97	C₂×D₈₉	1	1	yes	characteristic
220	C₄×C₈	—	89	1	no	—
221	C₂×Q₈	—	89	1	no	—
222	C₂×Q₈	—	178	1	no	—
223	C₂×Q₈	—	178	1	no	—
224	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
225	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	178	1	no	—
229	C₂²×C₄	—	178	1	no	—
230	C₂²×C₄	—	178	1	no	—
231	C₂²×C₄	—	178	1	no	—
232	D₈	Dic₁₇₈	1	4	yes	complemented
233	C₂×C₈	—	89	1	no	—
234	C₂×C₈	—	89	1	no	—
235	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₄⋊C₄	—	89	1	no	—
238	C₄⋊C₄	—	89	1	no	—
239	C₄⋊C₄	—	89	1	no	—
240	C₄⋊C₄	—	89	1	no	—
241	C₄⋊C₄	—	89	1	no	—
242	C₄⋊C₄	—	89	1	no	—
243	C₄⋊C₄	—	89	1	no	—
244	C₄⋊C₄	—	178	1	no	—
245	C₄⋊C₄	—	178	1	no	—
246	C₄⋊C₄	—	178	1	no	—
247	C₄⋊C₄	—	178	1	no	—
248	C₄⋊C₄	—	178	1	no	—
249	C₄⋊C₄	—	178	1	no	—
250	C₄⋊C₄	—	178	1	no	—
251	C₄⋊C₄	—	178	1	no	—
252	C₄⋊C₄	—	178	1	no	—
253	C₄⋊C₄	—	178	1	no	—
254	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	178	1	no	—
259	C₂²⋊C₄	—	178	1	no	—
260	C₂²⋊C₄	—	178	1	no	—
261	C₂²⋊C₄	—	178	1	no	—
262	C₄²	—	89	1	no	—
263	C₄²	—	89	1	no	—
264	C₄²	C₂²×D₈₉	1	1	yes	characteristic
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	Q₈	—	356	1	no	—
268	Q₈	—	712	1	no	—
269	Q₈	—	712	1	no	—
270	D₄	—	2	1	no	—
271	D₄	—	2	1	no	—
272	D₄	C₂×Dic₁₇₈	1	2	yes	—
273	D₄	C₂×Dic₁₇₈	1	2	yes	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	89	1	no	—
279	C₂×C₄	—	89	1	no	—
280	C₂×C₄	—	89	1	no	—
281	C₂×C₄	—	89	1	no	—
282	C₂×C₄	—	356	2	no	—
283	C₂×C₄	—	178	1	no	—
284	C₂×C₄	—	178	1	no	—
285	C₂×C₄	—	178	1	no	—
286	C₂×C₄	—	178	1	no	—
287	C₂×C₄	—	712	1	no	—
288	C₂×C₄	—	712	1	no	—
289	C₂×C₄	—	712	1	no	—
290	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
291	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	178	1	no	—
295	C₈	—	178	1	no	—
296	C₈	C₂×Dic₁₇₈	1	2	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	356	1	no	—
309	C₄	—	356	1	no	—
310	C₄	—	356	1	no	—
311	C₄	—	356	1	no	—
312	C₄	—	178	1	no	—
313	C₄	—	178	1	no	—
314	C₄	—	178	1	no	—
315	C₄	—	178	1	no	—
316	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
317	C₄	#15668059	1	1	yes	characteristic
318	C₂	—	4	2	no	—
319	C₂	—	4	2	no	—
320	C₂	D₄×Dic₁₇₈	1	1	yes	characteristic
321	C₂	#19152394	1	1	yes	characteristic
322	C₂	#19152402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#442×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₈×C₈₉	C₂²
hypercenter	#442	D₈₉
hypocenter	C₈₉	#3389
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #442

Complements

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

N	H ≅ G/N	type
1. #19152307	C₂	semidirect
2. #19152305	C₂	semidirect
3. #19152007	C₂	semidirect
4. #19152005	C₂	semidirect
5. #19151998	C₂	semidirect
6. #19151900	C₂	semidirect
7. #19151900	C₂	semidirect
8. #19151898	C₂	semidirect
9. #19151896	C₂	semidirect
10. #19151878	C₂	semidirect
11. #19151876	C₂	semidirect
12. #19151726	C₂	semidirect
95. D₈×C₈₉	Q₈	semidirect
162. C₈₉	#3389	semidirect
232. D₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#19152307	2
#19152305	2
#19152007	2
#19152005	2
#19151998	2
#19151900	2
#19151898	2
#19151896	2
#19151878	2
#19151876	2
#19151726	2
#19151710	2
D₈×Dic₈₉	2
#442×C₈₉	2
#3389	89