Subgroups — #26032971

Order 11392.

372 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	#19152320	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151959	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151958	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151947	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151893	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151892	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151877	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151815	C₂	1	1	yes	maximal, characteristic
10	#19151813	C₂	1	1	yes	maximal, characteristic
11	#19151801	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151726	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151695	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151678	C₂	1	1	yes	maximal, characteristic
15	#440×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₇₈	—	2	1	no	—
17	C₂×#13083562	C₂²	1	1	yes	characteristic
18	C₂×#13083559	C₂²	1	1	yes	characteristic
19	#15668012	C₂²	1	1	yes	characteristic
20	#15668011	C₂²	1	1	yes	characteristic
21	#15668011	C₂²	1	1	yes	characteristic
22	#15668010	C₂²	1	1	yes	characteristic
23	#15668007	—	2	1	no	—
24	#15668006	—	2	1	no	—
25	#15668001	C₂²	1	1	yes	characteristic
26	#15668000	C₂²	1	1	yes	characteristic
27	#15667991	—	2	1	no	—
28	#15667990	—	2	1	no	—
29	#15667989	—	2	1	no	—
30	#15667989	—	2	1	no	—
31	C₄×Dic₁₇₈	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	#15667974	C₂²	1	1	yes	characteristic
36	#15667969	—	2	1	no	—
37	#15667945	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	C₄⋊#10623494	C₂²	1	1	yes	characteristic
40	#15667941	C₂²	1	1	yes	characteristic
41	#15667941	C₂²	1	1	yes	characteristic
42	#15667937	C₂²	1	1	yes	characteristic
43	#15667937	C₂²	1	1	yes	characteristic
44	#15667936	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667934	C₂²	1	1	yes	characteristic
48	#15667934	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#103×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	—	2	1	no	—
62	C₂×Dic₁₇₈	—	4	4	no	—
63	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	#13083562	—	2	2	no	—
70	#13083559	—	2	2	no	—
71	C₂×#10623494	C₂³	1	1	yes	characteristic
72	C₂×#10623494	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₉	—	2	1	no	—
74	C₄⋊Dic₈₉	D₄	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	D₄	1	1	yes	characteristic
83	#13083553	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₈₉	—	2	1	no	—
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
93	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	—	2	1	no	—
97	SD₁₆×C₈₉	—	2	2	no	—
98	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₈₉	—	2	1	no	—
104	C₄²×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	Q₈×C₈₉	—	2	1	no	—
109	Q₈×C₈₉	—	2	1	no	—
110	D₄×C₈₉	—	2	1	no	—
111	D₄×C₈₉	#41	1	2	yes	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	4	2	no	—
116	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	—	4	2	no	—
123	C₂×Dic₈₉	—	4	2	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	C₄×D₈₉	—	4	2	no	—
131	Dic₁₇₈	—	2	2	no	—
132	Dic₁₇₈	—	2	1	no	—
133	Dic₁₇₈	—	2	1	no	—
134	Dic₁₇₈	—	4	2	no	—
135	Dic₁₇₈	—	4	2	no	—
136	Dic₁₇₈	—	4	2	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	—	2	1	no	—
139	#10623494	—	2	1	no	—
140	#10623494	—	2	1	no	—
141	C₂²×C₈₉	—	2	2	no	—
142	C₂²×C₈₉	—	4	1	no	—
143	C₂²×C₈₉	—	4	1	no	—
144	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₈₉	—	4	1	no	—
146	C₂×D₈₉	—	4	1	no	—
147	C₂×D₈₉	—	4	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	4	1	no	—
151	C₄×C₈₉	—	4	1	no	—
152	C₄×C₈₉	—	4	1	no	—
153	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
154	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
155	Dic₈₉	—	2	1	no	—
156	Dic₈₉	—	2	1	no	—
157	Dic₈₉	—	2	1	no	—
158	Dic₈₉	—	4	1	no	—
159	Dic₈₉	—	4	1	no	—
160	Dic₈₉	—	4	1	no	—
161	Dic₈₉	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₈₉	—	4	2	no	—
163	C₂×C₈₉	#548	1	1	yes	characteristic
164	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×C₈₉	#578	1	1	yes	characteristic
166	D₈₉	—	8	1	no	—
167	#3367	—	89	1	no	maximal, Sylow
168	C₈₉	#3367	1	1	yes	characteristic, complemented, Sylow
169	#556	—	89	1	no	—
170	D₄×Q₈	—	89	1	no	—
171	#489	—	89	1	no	—
172	#489	—	89	1	no	—
173	#487	—	89	1	no	—
174	#477	—	89	1	no	—
175	#477	—	89	1	no	—
176	#474	—	89	1	no	—
177	#463	—	89	1	no	—
178	#463	—	89	1	no	—
179	#461	—	89	1	no	—
180	#446	—	89	1	no	—
181	#440	—	89	1	no	—
182	#440	D₈₉	1	1	yes	characteristic, complemented
183	C₄×D₈	—	89	1	no	—
184	C₂²×Q₈	—	178	1	no	—
185	C₂×SD₁₆	—	89	1	no	—
186	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
187	C₂×D₈	—	89	1	no	—
188	C₄⋊Q₈	—	89	1	no	—
189	C₄⋊Q₈	—	89	1	no	—
190	C₄⋊Q₈	—	89	1	no	—
191	C₄⋊Q₈	—	89	1	no	—
192	#126	—	178	1	no	—
193	#126	—	178	1	no	—
194	#124	—	89	1	no	—
195	#124	—	89	1	no	—
196	C₂²⋊Q₈	—	178	1	no	—
197	C₂²⋊Q₈	—	178	1	no	—
198	C₂²⋊Q₈	—	178	1	no	—
199	C₂²⋊Q₈	—	178	1	no	—
200	C₄×Q₈	—	89	1	no	—
201	C₄×Q₈	—	89	1	no	—
202	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
203	C₄×D₄	—	89	1	no	—
204	C₄×D₄	—	89	1	no	—
205	C₄×D₄	—	89	1	no	—
206	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
207	#117	—	178	1	no	—
208	#107	—	89	1	no	—
209	#107	—	89	1	no	—
210	#107	C₂×D₈₉	1	1	yes	characteristic
211	C₄⋊C₈	—	89	1	no	—
212	C₄⋊C₈	—	89	1	no	—
213	C₄⋊C₈	—	89	1	no	—
214	#103	—	89	1	no	—
215	#103	—	89	1	no	—
216	#103	—	89	1	no	—
217	#103	—	89	1	no	—
218	#103	—	89	1	no	—
219	#103	C₂×D₈₉	1	1	yes	characteristic
220	#102	—	89	1	no	—
221	#102	—	89	1	no	—
222	#102	—	89	1	no	—
223	#102	C₂×D₈₉	1	1	yes	characteristic
224	#97	—	89	1	no	—
225	#97	C₂×D₈₉	1	1	yes	characteristic
226	C₄×C₈	—	89	1	no	—
227	C₂×Q₈	—	89	1	no	—
228	C₂×Q₈	—	89	1	no	—
229	C₂×Q₈	—	356	4	no	—
230	C₂×Q₈	—	178	1	no	—
231	C₂×Q₈	—	178	1	no	—
232	C₂×Q₈	—	178	1	no	—
233	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
234	C₂×D₄	—	89	1	no	—
235	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	178	1	no	—
238	C₂²×C₄	—	178	1	no	—
239	C₂²×C₄	—	178	1	no	—
240	SD₁₆	—	2	2	no	—
241	SD₁₆	—	178	2	no	—
242	D₈	—	178	2	no	—
243	C₂×C₈	—	89	1	no	—
244	C₂×C₈	—	89	1	no	—
245	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	89	1	no	—
249	C₄⋊C₄	—	89	1	no	—
250	C₄⋊C₄	—	89	1	no	—
251	C₄⋊C₄	—	89	1	no	—
252	C₄⋊C₄	—	89	1	no	—
253	C₄⋊C₄	—	178	1	no	—
254	C₄⋊C₄	—	178	1	no	—
255	C₄⋊C₄	—	178	1	no	—
256	C₄⋊C₄	—	178	1	no	—
257	C₄⋊C₄	—	178	1	no	—
258	C₄⋊C₄	—	178	1	no	—
259	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	178	1	no	—
263	C₂²⋊C₄	—	178	1	no	—
264	C₂²⋊C₄	—	178	1	no	—
265	C₂²⋊C₄	—	178	1	no	—
266	C₂²⋊C₄	—	178	1	no	—
267	C₂²⋊C₄	—	178	1	no	—
268	C₂²⋊C₄	—	178	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	89	1	no	—
271	C₄²	—	89	1	no	—
272	C₄²	—	178	1	no	—
273	C₄²	C₂²×D₈₉	1	1	yes	characteristic
274	C₂³	—	2	1	no	—
275	C₂³	—	178	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	356	2	no	—
279	Q₈	—	356	2	no	—
280	Q₈	—	356	2	no	—
281	Q₈	—	178	2	no	—
282	Q₈	—	178	1	no	—
283	Q₈	—	178	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	178	1	no	—
286	D₄	—	178	1	no	—
287	D₄	#13083574	1	2	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	89	1	no	—
293	C₂×C₄	—	89	1	no	—
294	C₂×C₄	—	89	1	no	—
295	C₂×C₄	—	89	1	no	—
296	C₂×C₄	—	356	2	no	—
297	C₂×C₄	—	356	2	no	—
298	C₂×C₄	—	356	2	no	—
299	C₂×C₄	—	178	1	no	—
300	C₂×C₄	—	178	1	no	—
301	C₂×C₄	—	178	1	no	—
302	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
303	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	178	1	no	—
308	C₈	—	178	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	356	1	no	—
313	C₂²	—	356	1	no	—
314	C₂²	—	356	1	no	—
315	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	89	2	no	—
322	C₄	—	356	1	no	—
323	C₄	—	356	1	no	—
324	C₄	—	356	1	no	—
325	C₄	—	178	1	no	—
326	C₄	—	178	1	no	—
327	C₄	—	178	1	no	—
328	C₄	C₂×#13083574	1	1	yes	characteristic
329	C₄	#15668059	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	712	1	no	—
332	C₂	#19152257	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
334	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#440×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083559	C₂²
hypercenter	#440	D₈₉
hypocenter	C₈₉	#3367
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #19152320	C₂	semidirect
2. D₄×Dic₁₇₈	C₂	semidirect
3. #19151959	C₂	semidirect
4. #19151958	C₂	semidirect
5. #19151947	C₂	semidirect
6. #19151893	C₂	semidirect
7. #19151892	C₂	semidirect
8. #19151877	C₂	semidirect
11. #19151801	C₂	semidirect
12. #19151726	C₂	semidirect
13. #19151695	C₂	semidirect
15. #440×C₈₉	C₂	semidirect
168. C₈₉	#3367	semidirect
182. #440	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152320	2
D₄×Dic₁₇₈	2
#19151959	2
#19151958	2
#19151947	2
#19151893	2
#19151892	2
#19151877	2
#19151815	2
#19151813	2
#19151801	2
#19151726	2
#19151695	2
#19151678	2
#440×C₈₉	2
#3367	89