Subgroups — #26030403

Order 11392.

430 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152148	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151922	C₂	1	2	yes	maximal, complemented
3	#19151883	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151878	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151798	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151793	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151743	C₂	1	2	yes	maximal, complemented
8	#19151708	C₂	1	2	yes	maximal, complemented
9	#19151696	C₂	1	2	yes	maximal, complemented
10	#19151655	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13083574	—	2	1	no	—
13	C₂×#13083561	C₂²	1	2	yes	—
14	C₂×#13083559	C₂²	1	2	yes	—
15	#15668012	C₂²	1	1	yes	characteristic
16	#15668009	C₂²	1	1	yes	characteristic
17	#15668000	—	2	1	no	—
18	#15667986	—	2	1	no	—
19	#15667986	C₂²	1	2	yes	—
20	C₄×D₃₅₆	—	2	1	no	—
21	C₄×D₃₅₆	C₂²	1	2	yes	—
22	#15667945	C₂²	1	2	yes	—
23	#15667943	C₂²	1	2	yes	—
24	C₄⋊#10623494	C₂²	1	1	yes	characteristic
25	C₄⋊#10623494	C₂²	1	1	yes	characteristic
26	#15667936	C₂²	1	2	yes	—
27	#15667933	C₂²	1	2	yes	—
28	#15667932	C₂²	1	2	yes	—
29	#15667932	C₂²	1	2	yes	—
30	C₂²⋊#10623494	C₂²	1	2	yes	—
31	#15667923	C₂²	1	2	yes	—
32	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₈₉	—	2	1	no	—
36	C₄×D₄×C₈₉	—	2	1	no	—
37	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
39	#117×C₈₉	—	2	1	no	—
40	#117×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	—	2	1	no	—
42	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
43	#13083574	—	4	4	no	—
44	C₂×Dic₁₇₈	—	2	1	no	—
45	C₂×D₃₅₆	—	2	1	no	—
46	C₂×D₃₅₆	—	2	1	no	—
47	C₂×D₃₅₆	C₂³	1	2	yes	—
48	C₂×C₂²⋊D₈₉	—	2	2	no	—
49	C₂×C₄×D₈₉	—	2	2	no	—
50	#13083561	—	2	4	no	—
51	#13083559	—	2	4	no	—
52	C₂×#10623494	C₂³	1	2	yes	—
53	C₂×#10623494	C₂³	1	2	yes	—
54	C₄⋊Dic₈₉	—	2	1	no	—
55	C₄⋊Dic₈₉	C₂³	1	2	yes	—
56	#13083551	—	2	2	no	—
57	#13083551	—	2	2	no	—
58	#41×C₈₉	—	2	2	no	—
59	#41×C₈₉	—	2	2	no	—
60	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₈₉	—	2	1	no	—
62	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	—	2	1	no	—
64	C₂²×C₄×C₈₉	—	2	1	no	—
65	C₂²×C₄×C₈₉	—	2	1	no	—
66	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₈₉	—	2	1	no	—
70	C₄⋊C₄×C₈₉	—	2	1	no	—
71	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
73	C₂²⋊C₄×C₈₉	—	2	1	no	—
74	C₂²⋊C₄×C₈₉	—	2	1	no	—
75	C₂²⋊C₄×C₈₉	—	2	1	no	—
76	C₄²×C₈₉	—	2	1	no	—
77	C₄²×C₈₉	—	2	1	no	—
78	C₄²×C₈₉	—	2	1	no	—
79	C₄²×C₈₉	D₄	1	1	yes	characteristic
80	C₄²×C₈₉	D₄	1	1	yes	characteristic
81	C₄²×C₈₉	C₂³	1	1	yes	characteristic
82	C₄²×C₈₉	C₂³	1	1	yes	characteristic
83	C₂³×C₈₉	—	2	1	no	—
84	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₈₉	—	2	2	no	—
86	Q₈×C₈₉	—	2	1	no	—
87	Q₈×C₈₉	#41	1	2	yes	complemented
88	D₄×C₈₉	—	2	4	no	—
89	D₄×C₈₉	—	2	1	no	—
90	D₄×C₈₉	#41	1	2	yes	complemented
91	C₂×C₄×C₈₉	—	2	4	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	4	1	no	—
99	C₂×C₄×C₈₉	—	4	2	no	—
100	C₂×C₄×C₈₉	—	4	2	no	—
101	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
107	C₂²⋊D₈₉	—	4	4	no	—
108	C₂×Dic₈₉	—	2	2	no	—
109	D₃₅₆	—	2	2	no	—
110	D₃₅₆	—	2	2	no	—
111	D₃₅₆	—	4	2	no	—
112	D₃₅₆	—	4	2	no	—
113	C₄×D₈₉	—	4	4	no	—
114	Dic₁₇₈	—	4	2	no	—
115	#10623494	—	2	2	no	—
116	#10623494	—	2	2	no	—
117	C₂²×C₈₉	—	2	1	no	—
118	C₂²×C₈₉	—	2	2	no	—
119	C₂²×C₈₉	—	2	1	no	—
120	C₂²×C₈₉	—	2	1	no	—
121	C₂²×C₈₉	—	4	1	no	—
122	C₂²×C₈₉	—	4	1	no	—
123	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×D₈₉	—	4	2	no	—
125	C₂×D₈₉	—	4	2	no	—
126	C₂×D₈₉	—	4	2	no	—
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	2	2	no	—
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	—	4	1	no	—
133	C₄×C₈₉	—	4	1	no	—
134	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
135	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
136	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
137	Dic₈₉	—	4	2	no	—
138	C₂×C₈₉	—	4	1	no	—
139	C₂×C₈₉	—	4	2	no	—
140	C₂×C₈₉	#525	1	1	yes	characteristic
141	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₈₉	#575	1	1	yes	characteristic
143	D₈₉	—	8	2	no	—
144	#3115	—	89	1	no	maximal, Sylow
145	C₈₉	#3115	1	1	yes	characteristic, complemented, Sylow
146	#532	—	89	1	no	—
147	C₄×#41	D₈₉	1	1	yes	characteristic, complemented
148	#482	—	89	2	no	—
149	#475	—	89	1	no	—
150	#474	—	89	1	no	—
151	#460	—	89	1	no	—
152	#459	—	89	1	no	—
153	#449	—	89	2	no	—
154	#442	—	89	2	no	—
155	#440	—	89	2	no	—
156	#432	—	89	1	no	—
157	C₂×#41	—	178	1	no	—
158	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
159	C₂×SD₁₆	—	89	2	no	—
160	C₂×D₈	—	89	2	no	—
161	C₄⋊Q₈	—	89	1	no	—
162	#127	—	89	1	no	—
163	#124	—	178	1	no	—
164	#121	—	89	2	no	—
165	#121	—	178	1	no	—
166	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
167	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	89	2	no	—
171	C₄×D₄	—	178	1	no	—
172	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
173	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
174	#117	—	2	1	no	—
175	#117	C₂×D₈₉	1	1	yes	characteristic
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
178	#107	—	89	2	no	—
179	#106	—	89	2	no	—
180	C₄⋊C₈	—	89	1	no	—
181	C₄⋊C₈	—	89	1	no	—
182	#103	—	89	2	no	—
183	#102	—	89	2	no	—
184	#102	—	89	2	no	—
185	#102	—	89	2	no	—
186	C₂²⋊C₈	—	89	2	no	—
187	#97	—	89	2	no	—
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	#41	—	356	4	no	—
191	C₂×Q₈	—	178	1	no	—
192	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	89	2	no	—
195	C₂×D₄	—	178	2	no	—
196	C₂×D₄	—	178	1	no	—
197	C₂×D₄	—	178	1	no	—
198	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	178	2	no	—
203	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
206	SD₁₆	—	178	4	no	—
207	D₈	—	178	4	no	—
208	C₂×C₈	—	89	2	no	—
209	C₂×C₈	—	89	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	89	2	no	—
213	C₄⋊C₄	—	178	1	no	—
214	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	178	2	no	—
220	C₂²⋊C₄	—	178	2	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
225	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
226	C₄²	C₂²×D₈₉	1	1	yes	characteristic
227	C₄²	C₂²×D₈₉	1	1	yes	characteristic
228	C₂³	—	2	1	no	—
229	C₂³	—	178	2	no	—
230	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
231	Q₈	—	2	1	no	—
232	Q₈	—	356	2	no	—
233	Q₈	#13083574	1	2	yes	complemented
234	D₄	—	2	4	no	—
235	D₄	—	2	1	no	—
236	D₄	—	356	2	no	—
237	D₄	—	356	4	no	—
238	D₄	—	356	2	no	—
239	D₄	—	178	2	no	—
240	D₄	—	178	2	no	—
241	D₄	#13083574	1	2	yes	complemented
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	356	4	no	—
253	C₂×C₄	—	178	2	no	—
254	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
259	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
260	C₈	—	178	2	no	—
261	C₈	—	178	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	356	2	no	—
269	C₂²	—	356	2	no	—
270	C₂²	—	356	2	no	—
271	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	356	2	no	—
280	C₄	#15668037	1	2	yes	—
281	C₄	C₂×#13083574	1	1	yes	characteristic
282	C₄	C₂×#13083574	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	4	2	no	—
285	C₂	—	712	2	no	—
286	C₂	#19152109	1	1	yes	characteristic
287	C₂	C₂×#15668037	1	1	yes	characteristic
288	C₂	#19152396	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₄×#41×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	C₄×#41	D₈₉
hypocenter	C₈₉	#3115
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #19152148	C₂	semidirect
2. #19151922	C₂	semidirect
3. #19151883	C₂	semidirect
4. #19151878	C₂	semidirect
5. #19151798	C₂	semidirect
6. #19151793	C₂	semidirect
7. #19151743	C₂	semidirect
8. #19151708	C₂	semidirect
9. #19151696	C₂	semidirect
10. #19151655	C₂	semidirect
11. C₄×#41×C₈₉	C₂	semidirect
87. Q₈×C₈₉	#41	semidirect
90. D₄×C₈₉	#41	semidirect
145. C₈₉	#3115	semidirect
147. C₄×#41	D₈₉	semidirect
233. Q₈	#13083574	semidirect
241. D₄	#13083574	semidirect

Maximal subgroups

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

subgroup	index
#19152148	2
#19151922	2
#19151883	2
#19151878	2
#19151798	2
#19151793	2
#19151743	2
#19151708	2
#19151696	2
#19151655	2
C₄×#41×C₈₉	2
#3115	89