Subgroups — #26029326

Order 11392.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152145	C₂	1	1	yes	maximal, characteristic
2	#19151676	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151675	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151672	C₂	1	2	yes	maximal, complemented
5	#19151669	C₂	1	2	yes	maximal, complemented
6	C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151654	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151649	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151564	C₂	1	2	yes	maximal, complemented
10	#19151563	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13083574	C₂²	1	2	yes	—
13	C₂×#13083558	C₂²	1	2	yes	—
14	C₂×#13083557	C₂²	1	2	yes	—
15	C₂×C₈×D₈₉	C₂²	1	2	yes	—
16	#15668011	C₄	1	1	yes	characteristic
17	#15668008	C₄	1	1	yes	characteristic
18	#15667998	C₄	1	2	yes	—
19	#15667987	C₄	1	1	yes	characteristic
20	#15667985	C₄	1	2	yes	—
21	#15667983	C₄	1	1	yes	characteristic
22	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₈₉	C₂²	1	2	yes	—
24	C₄²×D₈₉	C₂²	1	1	yes	characteristic
25	C₄⋊#10623494	C₂²	1	1	yes	characteristic
26	#15667941	C₂²	1	1	yes	characteristic
27	#15667941	C₂²	1	1	yes	characteristic
28	C₂²⋊#10623494	C₂²	1	2	yes	—
29	#15667924	C₂²	1	1	yes	characteristic
30	#15667924	C₂²	1	1	yes	characteristic
31	#15667924	C₂²	1	2	yes	—
32	#15667924	C₂²	1	2	yes	—
33	#15667924	C₂²	1	2	yes	—
34	#15667922	C₂²	1	2	yes	—
35	C₄×#10623494	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
37	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
40	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
41	#13083574	—	4	2	no	—
42	#13083574	—	4	2	no	—
43	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
44	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
45	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₈₉	C₂³	1	2	yes	—
52	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
53	#13083558	—	4	2	no	—
54	#13083557	—	4	2	no	—
55	C₂×#10623494	C₂³	1	2	yes	—
56	C₂×#10623494	C₂³	1	1	yes	characteristic
57	C₂×#10623494	C₂³	1	1	yes	characteristic
58	C₈×D₈₉	—	4	2	no	—
59	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
60	#13083553	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₈₉	C₂×C₄	1	2	yes	—
62	#13083551	C₂×C₄	1	2	yes	—
63	#13083551	C₂×C₄	1	2	yes	—
64	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
69	C₂×C₈×C₈₉	—	4	2	no	—
70	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₈₉	C₂³	1	2	yes	—
72	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
75	C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	C₂³×C₈₉	C₂²×C₄	1	2	yes	—
77	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
78	C₂²×D₈₉	C₂²×C₄	1	1	yes	characteristic
79	D₄×C₈₉	—	2	2	no	—
80	C₂×C₄×C₈₉	—	2	2	no	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₈₉	—	4	2	no	—
87	C₂²⋊D₈₉	—	4	2	no	—
88	C₂²⋊D₈₉	—	4	2	no	—
89	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
93	D₃₅₆	—	2	2	no	—
94	D₃₅₆	—	4	2	no	—
95	C₄×D₈₉	—	2	2	no	—
96	C₄×D₈₉	—	2	2	no	—
97	C₄×D₈₉	—	2	2	no	—
98	C₄×D₈₉	—	2	1	no	—
99	C₄×D₈₉	—	2	1	no	—
100	C₄×D₈₉	—	4	1	no	—
101	Dic₁₇₈	—	4	2	no	—
102	C₈×C₈₉	—	2	1	no	—
103	C₈×C₈₉	—	2	2	no	—
104	C₈×C₈₉	—	2	1	no	—
105	#10623494	—	2	2	no	—
106	#10623494	—	2	1	no	—
107	#10623494	—	2	1	no	—
108	C₂²×C₈₉	—	2	2	no	—
109	C₂²×C₈₉	—	2	2	no	—
110	C₂²×C₈₉	—	2	2	no	—
111	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₈₉	—	2	1	no	—
113	C₂×D₈₉	—	2	2	no	—
114	C₂×D₈₉	—	2	1	no	—
115	C₂×D₈₉	—	2	2	no	—
116	C₂×D₈₉	—	2	1	no	—
117	C₂×D₈₉	—	2	2	no	—
118	C₄×C₈₉	—	2	2	no	—
119	C₄×C₈₉	—	2	1	no	—
120	C₄×C₈₉	#131	1	2	yes	—
121	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
123	Dic₈₉	—	2	1	no	—
124	Dic₈₉	—	2	2	no	—
125	Dic₈₉	—	2	1	no	—
126	Dic₈₉	—	2	1	no	—
127	C₂×C₈₉	—	4	2	no	—
128	C₂×C₈₉	#518	1	1	yes	characteristic
129	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
130	C₂×C₈₉	#568	1	1	yes	characteristic
131	D₈₉	—	4	1	no	—
132	D₈₉	—	4	2	no	—
133	#2981	—	89	1	no	maximal, Sylow
134	C₈₉	#2981	1	1	yes	characteristic, complemented, Sylow
135	#532	—	89	1	no	—
136	#436	—	89	1	no	—
137	#436	—	89	1	no	—
138	#435	—	89	2	no	—
139	#435	—	89	2	no	—
140	C₈×D₄	—	89	1	no	—
141	C₈×D₄	D₈₉	1	1	yes	characteristic, complemented
142	#432	—	89	1	no	—
143	#431	—	89	1	no	—
144	#408	—	89	2	no	—
145	#408	—	89	2	no	—
146	C₂×#41	—	89	2	no	—
147	C₂×M₁₆	—	89	2	no	—
148	C₂×M₁₆	—	89	2	no	—
149	C₂²×C₈	—	89	2	no	—
150	C₂²×C₈	C₂×D₈₉	1	2	yes	—
151	C₄⋊Q₈	—	89	1	no	—
152	#127	—	89	1	no	—
153	#124	—	89	2	no	—
154	#121	—	89	1	no	—
155	#121	—	89	1	no	—
156	#121	—	89	2	no	—
157	C₄×D₄	—	89	2	no	—
158	C₄×D₄	—	89	1	no	—
159	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
160	C₂×C₄²	—	89	1	no	—
161	C₄⋊C₈	—	89	1	no	—
162	C₄⋊C₈	—	89	1	no	—
163	C₄⋊C₈	—	89	1	no	—
164	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
165	C₂²⋊C₈	—	89	2	no	—
166	C₂²⋊C₈	—	89	1	no	—
167	C₂²⋊C₈	—	89	1	no	—
168	C₂²⋊C₈	—	89	2	no	—
169	C₂²⋊C₈	—	89	2	no	—
170	C₂²⋊C₈	—	89	2	no	—
171	C₂²⋊C₈	C₂×D₈₉	1	2	yes	—
172	#97	—	89	2	no	—
173	C₄×C₈	—	89	1	no	—
174	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
175	#41	—	356	2	no	—
176	#41	—	356	2	no	—
177	C₂×Q₈	—	89	2	no	—
178	C₂×D₄	—	89	2	no	—
179	C₂×D₄	—	89	2	no	—
180	C₂×D₄	—	89	2	no	—
181	C₂×D₄	—	89	2	no	—
182	C₂×D₄	—	89	1	no	—
183	C₂×D₄	C₄×D₈₉	1	1	yes	characteristic
184	C₂²×C₄	—	89	1	no	—
185	C₂²×C₄	—	89	1	no	—
186	C₂²×C₄	—	89	1	no	—
187	C₂²×C₄	—	89	2	no	—
188	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
189	M₁₆	—	356	2	no	—
190	M₁₆	—	356	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	89	1	no	—
193	C₂×C₈	—	89	1	no	—
194	C₂×C₈	—	89	2	no	—
195	C₂×C₈	—	356	2	no	—
196	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
197	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₈₉	1	2	yes	—
199	C₄⋊C₄	—	89	2	no	—
200	C₄⋊C₄	—	89	1	no	—
201	C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
202	C₂²⋊C₄	—	89	2	no	—
203	C₂²⋊C₄	—	89	2	no	—
204	C₂²⋊C₄	—	89	2	no	—
205	C₂²⋊C₄	C₄×D₈₉	1	2	yes	—
206	C₄²	—	89	1	no	—
207	C₄²	—	89	1	no	—
208	C₄²	—	89	1	no	—
209	C₄²	C₂²×D₈₉	1	1	yes	characteristic
210	C₂³	—	89	2	no	—
211	C₂³	—	89	1	no	—
212	C₂³	C₂×C₄×D₈₉	1	2	yes	—
213	Q₈	—	356	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	356	2	no	—
216	D₄	—	356	2	no	—
217	D₄	—	356	2	no	—
218	D₄	—	356	2	no	—
219	D₄	—	178	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	89	1	no	—
223	C₂×C₄	—	89	1	no	—
224	C₂×C₄	—	89	1	no	—
225	C₂×C₄	—	89	2	no	—
226	C₂×C₄	—	356	1	no	—
227	C₂×C₄	—	178	2	no	—
228	C₂×C₄	—	178	2	no	—
229	C₂×C₄	—	178	1	no	—
230	C₂×C₄	—	178	2	no	—
231	C₂×C₄	—	178	1	no	—
232	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	178	1	no	—
240	C₈	—	178	2	no	—
241	C₈	—	178	1	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	178	1	no	—
246	C₂²	—	178	2	no	—
247	C₂²	—	178	1	no	—
248	C₂²	—	178	2	no	—
249	C₂²	—	178	1	no	—
250	C₂²	—	178	2	no	—
251	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	178	1	no	—
255	C₄	—	178	2	no	—
256	C₄	—	178	1	no	—
257	C₄	—	178	1	no	—
258	C₄	#15668018	1	2	yes	—
259	C₄	#15668057	1	1	yes	characteristic
260	C₄	#15668058	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	356	1	no	—
263	C₂	—	356	2	no	—
264	C₂	#19152070	1	1	yes	characteristic
265	C₂	C₂×#15668018	1	1	yes	characteristic
266	C₂	#19152363	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	C₈×D₄×C₈₉	C₂
prime radical	#19152145	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₈×D₄	D₈₉
hypocenter	C₈₉	#2981
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #19151676	C₂	semidirect
3. #19151675	C₂	semidirect
4. #19151672	C₂	semidirect
5. #19151669	C₂	semidirect
6. C₈×D₃₅₆	C₂	semidirect
7. #19151654	C₂	semidirect
8. #19151649	C₂	semidirect
9. #19151564	C₂	semidirect
10. #19151563	C₂	semidirect
11. C₈×D₄×C₈₉	C₂	semidirect
134. C₈₉	#2981	semidirect
141. C₈×D₄	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152145	2
#19151676	2
#19151675	2
#19151672	2
#19151669	2
C₈×D₃₅₆	2
#19151654	2
#19151649	2
#19151564	2
#19151563	2
C₈×D₄×C₈₉	2
#2981	89