Subgroups — #26031578

Order 11392.

398 conjugacy classes of subgroups.

The subgroup lattice has 284 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	#19151895	C₂	1	2	yes	maximal, complemented
3	#19151893	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151888	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151832	C₂	1	2	yes	maximal, complemented
6	#19151829	C₂	1	2	yes	maximal, complemented
7	#19151810	C₂	1	2	yes	maximal, complemented
8	#19151798	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151793	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic, complemented
11	#529×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	C₂²×#10623494	C₂²	1	2	yes	—
16	#15668012	C₂²	1	1	yes	characteristic
17	#15668009	C₂²	1	1	yes	characteristic
18	#15668000	—	2	1	no	—
19	#15667986	—	2	1	no	—
20	#15667986	C₂²	1	2	yes	—
21	C₄×D₃₅₆	—	2	1	no	—
22	C₄×D₃₅₆	C₂²	1	2	yes	—
23	#15667945	C₂²	1	2	yes	—
24	#15667943	C₂²	1	2	yes	—
25	C₄⋊#10623494	C₂²	1	1	yes	characteristic
26	C₄⋊#10623494	C₂²	1	1	yes	characteristic
27	C₄⋊#10623494	C₂²	1	2	yes	—
28	#15667936	C₂²	1	2	yes	—
29	#15667933	C₂²	1	2	yes	—
30	#15667932	C₂²	1	2	yes	—
31	#15667932	C₂²	1	2	yes	—
32	#126×C₈₉	—	2	1	no	—
33	#125×C₈₉	C₂²	1	1	yes	characteristic
34	#124×C₈₉	C₂²	1	1	yes	characteristic
35	#123×C₈₉	—	2	1	no	—
36	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	#121×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₈₉	—	2	1	no	—
40	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
41	#117×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	—	4	2	no	—
51	#13083559	—	4	2	no	—
52	C₂×#10623494	—	4	2	no	—
53	C₂×#10623494	C₂³	1	2	yes	—
54	C₂×#10623494	C₂³	1	2	yes	—
55	C₄⋊Dic₈₉	—	2	1	no	—
56	C₄⋊Dic₈₉	C₂³	1	2	yes	—
57	#13083551	—	2	2	no	—
58	#13083551	—	2	2	no	—
59	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₈₉	—	2	1	no	—
61	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	—	2	1	no	—
63	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₈₉	—	2	1	no	—
67	C₄⋊C₄×C₈₉	—	2	1	no	—
68	C₄⋊C₄×C₈₉	—	2	1	no	—
69	C₄⋊C₄×C₈₉	—	2	1	no	—
70	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₈₉	—	2	1	no	—
73	C₂²⋊C₄×C₈₉	—	2	1	no	—
74	C₂²⋊C₄×C₈₉	—	2	1	no	—
75	C₂²⋊C₄×C₈₉	—	2	1	no	—
76	C₂²⋊C₄×C₈₉	—	2	1	no	—
77	C₄²×C₈₉	—	2	1	no	—
78	C₄²×C₈₉	D₄	1	1	yes	characteristic
79	C₄²×C₈₉	D₄	1	1	yes	characteristic
80	C₄²×C₈₉	C₂³	1	1	yes	characteristic
81	C₄²×C₈₉	C₂³	1	1	yes	characteristic
82	C₂³×C₈₉	—	2	1	no	—
83	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₈₉	—	2	2	no	—
85	Q₈×C₈₉	—	2	1	no	—
86	Q₈×C₈₉	—	2	1	no	—
87	D₄×C₈₉	—	2	1	no	—
88	D₄×C₈₉	—	2	1	no	—
89	D₄×C₈₉	—	4	2	no	—
90	C₂×C₄×C₈₉	—	2	2	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	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	—	4	2	no	—
98	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₈₉	—	4	4	no	—
105	C₂×Dic₈₉	—	2	2	no	—
106	D₃₅₆	—	2	2	no	—
107	D₃₅₆	—	2	2	no	—
108	D₃₅₆	—	4	2	no	—
109	D₃₅₆	—	4	2	no	—
110	C₄×D₈₉	—	4	4	no	—
111	Dic₁₇₈	—	4	2	no	—
112	#10623494	—	2	2	no	—
113	#10623494	—	2	2	no	—
114	C₂²×C₈₉	—	2	1	no	—
115	C₂²×C₈₉	—	2	1	no	—
116	C₂²×C₈₉	—	4	1	no	—
117	C₂²×C₈₉	—	4	1	no	—
118	C₂²×C₈₉	—	4	1	no	—
119	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
120	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₈₉	—	4	2	no	—
122	C₂×D₈₉	—	4	2	no	—
123	C₂×D₈₉	—	4	2	no	—
124	C₄×C₈₉	—	2	1	no	—
125	C₄×C₈₉	—	2	1	no	—
126	C₄×C₈₉	—	2	1	no	—
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	4	1	no	—
129	C₄×C₈₉	—	4	1	no	—
130	C₄×C₈₉	—	4	1	no	—
131	C₄×C₈₉	#135	1	2	yes	—
132	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
133	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
134	Dic₈₉	—	4	2	no	—
135	C₂×C₈₉	—	2	2	no	—
136	C₂×C₈₉	—	8	1	no	—
137	C₂×C₈₉	#537	1	1	yes	characteristic
138	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
139	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
140	D₈₉	—	8	2	no	—
141	#3247	—	89	1	no	maximal, Sylow
142	C₈₉	#3247	1	1	yes	characteristic, complemented, Sylow
143	#532	—	89	1	no	—
144	#529	D₈₉	1	1	yes	characteristic, complemented
145	#478	—	89	2	no	—
146	#477	—	89	1	no	—
147	#476	—	89	1	no	—
148	#466	—	89	2	no	—
149	#465	—	89	2	no	—
150	#463	—	89	2	no	—
151	#460	—	89	1	no	—
152	#459	—	89	1	no	—
153	C₂×C₄⋊C₈	—	89	1	no	—
154	C₂×#41	—	178	1	no	—
155	C₂×SD₁₆	—	89	2	no	—
156	C₂×D₈	—	89	2	no	—
157	C₂²×C₈	—	89	2	no	—
158	C₄⋊Q₈	—	89	1	no	—
159	#127	—	89	1	no	—
160	#126	—	2	1	no	—
161	#125	C₂×D₈₉	1	1	yes	characteristic
162	#124	—	178	1	no	—
163	#124	C₂×D₈₉	1	1	yes	characteristic
164	#123	—	2	1	no	—
165	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
166	#121	—	89	2	no	—
167	#121	—	178	1	no	—
168	#121	C₂×D₈₉	1	1	yes	characteristic
169	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	—	89	2	no	—
172	C₄×D₄	—	178	1	no	—
173	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
174	#117	—	2	1	no	—
175	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
176	#107	—	89	2	no	—
177	#106	—	89	2	no	—
178	C₄⋊C₈	—	89	1	no	—
179	C₄⋊C₈	—	89	1	no	—
180	C₄⋊C₈	—	89	2	no	—
181	#103	—	89	2	no	—
182	#102	—	89	2	no	—
183	#102	—	89	2	no	—
184	#102	—	89	2	no	—
185	#41	—	356	4	no	—
186	C₂×Q₈	—	178	1	no	—
187	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	89	2	no	—
190	C₂×D₄	—	178	2	no	—
191	C₂×D₄	—	178	1	no	—
192	C₂×D₄	—	178	1	no	—
193	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	178	2	no	—
196	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
199	SD₁₆	—	356	2	no	—
200	D₈	—	356	2	no	—
201	C₂×C₈	—	89	2	no	—
202	C₂×C₈	—	89	2	no	—
203	C₂×C₈	—	356	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	89	2	no	—
209	C₄⋊C₄	—	178	1	no	—
210	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
211	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	178	2	no	—
218	C₂²⋊C₄	—	178	2	no	—
219	C₄²	—	2	1	no	—
220	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
221	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
222	C₄²	C₂²×D₈₉	1	1	yes	characteristic
223	C₄²	C₂²×D₈₉	1	1	yes	characteristic
224	C₂³	—	2	1	no	—
225	C₂³	—	178	2	no	—
226	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
227	Q₈	—	2	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	356	2	no	—
230	D₄	—	2	1	no	—
231	D₄	—	2	1	no	—
232	D₄	—	4	2	no	—
233	D₄	—	356	2	no	—
234	D₄	—	356	4	no	—
235	D₄	—	356	2	no	—
236	D₄	—	178	2	no	—
237	D₄	—	178	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	356	4	no	—
247	C₂×C₄	—	178	2	no	—
248	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
253	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
254	C₈	—	178	2	no	—
255	C₈	—	178	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	356	2	no	—
262	C₂²	—	356	2	no	—
263	C₂²	—	356	2	no	—
264	C₂²	#15668037	1	2	yes	—
265	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	356	2	no	—
274	C₄	#15668032	1	2	yes	—
275	C₄	#15668058	1	1	yes	characteristic
276	C₄	#15668059	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	8	1	no	—
279	C₂	—	712	2	no	—
280	C₂	#19152176	1	1	yes	characteristic
281	C₂	C₂×#15668032	1	1	yes	characteristic
282	C₂	C₂×#15668037	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#529×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#529	D₈₉
hypocenter	C₈₉	#3247
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #19152148	C₂	semidirect
2. #19151895	C₂	semidirect
3. #19151893	C₂	semidirect
4. #19151888	C₂	semidirect
5. #19151832	C₂	semidirect
6. #19151829	C₂	semidirect
7. #19151810	C₂	semidirect
8. #19151798	C₂	semidirect
9. #19151793	C₂	semidirect
10. C₂×C₄⋊#10623494	C₂	semidirect
11. #529×C₈₉	C₂	semidirect
142. C₈₉	#3247	semidirect
144. #529	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152148	2
#19151895	2
#19151893	2
#19151888	2
#19151832	2
#19151829	2
#19151810	2
#19151798	2
#19151793	2
C₂×C₄⋊#10623494	2
#529×C₈₉	2
#3247	89