Subgroups — #26029256

Order 11392.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152132	C₂	1	1	yes	maximal, characteristic
2	#19151728	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151677	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151674	C₂	1	2	yes	maximal, complemented
6	#19151667	C₂	1	2	yes	maximal, complemented
7	C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151660	C₂	1	2	yes	maximal, complemented
9	#19151651	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151555	C₂	1	2	yes	maximal, complemented
11	C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13083557	C₂²	1	2	yes	—
13	C₂×C₈×D₈₉	C₂²	1	2	yes	—
14	#15668006	C₄	1	2	yes	—
15	#15668004	C₄	1	1	yes	characteristic
16	#15668000	C₄	1	1	yes	characteristic
17	#15667995	C₄	1	2	yes	—
18	#15667991	C₄	1	1	yes	characteristic
19	#15667986	C₄	1	1	yes	characteristic
20	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
21	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₈₉	C₂²	1	2	yes	—
23	#15667969	C₂²	1	2	yes	—
24	C₄⋊#10623494	C₂²	1	1	yes	characteristic
25	C₄⋊#10623494	C₂²	1	1	yes	characteristic
26	#15667941	C₂²	1	2	yes	—
27	#15667941	C₂²	1	2	yes	—
28	C₂²⋊#10623494	C₂²	1	2	yes	—
29	#15667924	C₂²	1	2	yes	—
30	#15667924	C₂²	1	2	yes	—
31	#15667922	C₂²	1	2	yes	—
32	C₄×#10623494	C₂²	1	2	yes	—
33	C₈×Dic₈₉	C₂²	1	2	yes	—
34	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
37	#97×C₈₉	C₂²	1	1	yes	characteristic
38	#97×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
42	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₈₉	C₂³	1	2	yes	—
45	#13083557	—	2	4	no	—
46	C₂×#10623494	C₂³	1	2	yes	—
47	C₂×#10623494	C₂³	1	2	yes	—
48	C₈×D₈₉	—	2	4	no	—
49	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
51	#13083553	C₂×C₄	1	2	yes	—
52	#13083553	C₂×C₄	1	2	yes	—
53	#13083553	C₂×C₄	1	2	yes	—
54	#13083553	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₈₉	C₂×C₄	1	2	yes	—
56	#13083551	C₂×C₄	1	2	yes	—
57	#13083551	C₂×C₄	1	2	yes	—
58	#13083551	C₂×C₄	1	2	yes	—
59	#13083551	C₂×C₄	1	2	yes	—
60	C₄×Dic₈₉	C₂³	1	2	yes	—
61	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
64	M₁₆×C₈₉	—	2	2	no	—
65	M₁₆×C₈₉	—	2	2	no	—
66	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₄²×C₈₉	C₂³	1	1	yes	characteristic
73	C₄²×C₈₉	C₂³	1	1	yes	characteristic
74	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	—	2	1	no	—
78	C₂×C₄×C₈₉	—	2	1	no	—
79	C₂×C₄×C₈₉	—	4	1	no	—
80	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
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	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
88	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
89	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
90	D₃₅₆	—	2	2	no	—
91	C₄×D₈₉	—	2	2	no	—
92	C₄×D₈₉	—	2	2	no	—
93	Dic₁₇₈	—	2	2	no	—
94	C₈×C₈₉	—	2	1	no	—
95	C₈×C₈₉	—	2	1	no	—
96	C₈×C₈₉	#41	1	2	yes	—
97	C₈×C₈₉	#41	1	2	yes	—
98	#10623494	—	2	2	no	—
99	#10623494	—	2	2	no	—
100	C₂²×C₈₉	—	2	1	no	—
101	C₂²×C₈₉	—	2	1	no	—
102	C₂²×C₈₉	—	2	1	no	—
103	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
104	C₂×D₈₉	—	2	2	no	—
105	C₂×D₈₉	—	2	2	no	—
106	C₂×D₈₉	—	2	2	no	—
107	C₄×C₈₉	—	2	1	no	—
108	C₄×C₈₉	—	2	1	no	—
109	C₄×C₈₉	—	2	1	no	—
110	C₄×C₈₉	—	2	1	no	—
111	C₄×C₈₉	#131	1	2	yes	—
112	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
113	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
114	Dic₈₉	—	2	2	no	—
115	Dic₈₉	—	2	2	no	—
116	Dic₈₉	—	2	2	no	—
117	C₂×C₈₉	—	4	1	no	—
118	C₂×C₈₉	C₄×#41	1	1	yes	characteristic
119	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
120	C₂×C₈₉	#568	1	1	yes	characteristic
121	D₈₉	—	4	2	no	—
122	#2969	—	89	1	no	maximal, Sylow
123	C₈₉	#2969	1	1	yes	characteristic, complemented, Sylow
124	#529	—	89	1	no	—
125	#446	—	89	1	no	—
126	C₈×Q₈	—	89	1	no	—
127	#436	—	89	2	no	—
128	#436	—	89	1	no	—
129	#435	—	89	2	no	—
130	C₈×D₄	—	89	1	no	—
131	#433	—	89	2	no	—
132	#431	—	89	1	no	—
133	#405	—	89	2	no	—
134	C₄×M₁₆	D₈₉	1	1	yes	characteristic, complemented
135	C₂×M₁₆	—	89	2	no	—
136	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
137	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
138	C₂²×C₈	—	89	2	no	—
139	#126	—	89	2	no	—
140	#125	—	89	1	no	—
141	#124	—	89	1	no	—
142	#123	—	89	2	no	—
143	C₂²⋊Q₈	—	89	1	no	—
144	#121	—	89	1	no	—
145	C₄×Q₈	—	89	1	no	—
146	C₄×D₄	—	89	2	no	—
147	C₄×D₄	—	89	1	no	—
148	#117	—	89	2	no	—
149	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
150	C₄⋊C₈	—	89	2	no	—
151	C₄⋊C₈	—	89	2	no	—
152	C₄⋊C₈	—	89	1	no	—
153	C₄⋊C₈	—	89	1	no	—
154	C₂²⋊C₈	—	89	2	no	—
155	C₂²⋊C₈	—	89	2	no	—
156	C₂²⋊C₈	—	89	2	no	—
157	#97	—	89	2	no	—
158	#97	C₂×D₈₉	1	1	yes	characteristic
159	#97	C₂×D₈₉	1	1	yes	characteristic
160	C₄×C₈	—	89	2	no	—
161	C₄×C₈	—	89	2	no	—
162	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
163	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
164	C₂×Q₈	—	89	1	no	—
165	C₂×D₄	—	89	2	no	—
166	C₂×D₄	—	89	1	no	—
167	C₂²×C₄	—	89	2	no	—
168	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
171	M₁₆	—	2	2	no	—
172	M₁₆	—	2	2	no	—
173	M₁₆	—	178	4	no	—
174	C₂×C₈	—	89	2	no	—
175	C₂×C₈	—	89	2	no	—
176	C₂×C₈	—	178	4	no	—
177	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
181	C₄⋊C₄	—	89	2	no	—
182	C₄⋊C₄	—	89	2	no	—
183	C₄⋊C₄	—	89	1	no	—
184	C₄⋊C₄	—	89	2	no	—
185	C₄⋊C₄	—	89	1	no	—
186	C₄⋊C₄	—	89	2	no	—
187	C₂²⋊C₄	—	89	2	no	—
188	C₂²⋊C₄	—	89	2	no	—
189	C₂²⋊C₄	—	89	2	no	—
190	C₂²⋊C₄	—	89	2	no	—
191	C₂²⋊C₄	—	89	2	no	—
192	C₄²	—	89	2	no	—
193	C₄²	C₄×D₈₉	1	1	yes	characteristic
194	C₄²	C₄×D₈₉	1	1	yes	characteristic
195	C₄²	C₂²×D₈₉	1	1	yes	characteristic
196	C₄²	C₂²×D₈₉	1	1	yes	characteristic
197	C₂³	—	89	2	no	—
198	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
199	Q₈	—	178	2	no	—
200	D₄	—	356	2	no	—
201	D₄	—	178	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	89	2	no	—
207	C₂×C₄	—	89	2	no	—
208	C₂×C₄	—	89	2	no	—
209	C₂×C₄	—	178	2	no	—
210	C₂×C₄	—	178	2	no	—
211	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	178	2	no	—
220	C₈	—	178	2	no	—
221	C₈	#13083574	1	2	yes	—
222	C₈	#13083574	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	178	2	no	—
227	C₂²	—	178	2	no	—
228	C₂²	—	178	2	no	—
229	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	178	2	no	—
235	C₄	—	178	2	no	—
236	C₄	—	178	2	no	—
237	C₄	#15668020	1	2	yes	—
238	C₄	C₂×#13083574	1	1	yes	characteristic
239	C₄	C₂×#13083574	1	1	yes	characteristic
240	C₂	—	4	1	no	—
241	C₂	—	356	2	no	—
242	C₂	C₄×#13083574	1	1	yes	characteristic
243	C₂	C₂×#15668020	1	1	yes	characteristic
244	C₂	#19152364	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	C₄×M₁₆×C₈₉	C₂
prime radical	#15667986	C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₄×M₁₆	D₈₉
hypocenter	C₈₉	#2969
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #19151728	C₂	semidirect
3. C₈×Dic₁₇₈	C₂	semidirect
4. #19151677	C₂	semidirect
5. #19151674	C₂	semidirect
6. #19151667	C₂	semidirect
7. C₈×D₃₅₆	C₂	semidirect
8. #19151660	C₂	semidirect
9. #19151651	C₂	semidirect
10. #19151555	C₂	semidirect
11. C₄×M₁₆×C₈₉	C₂	semidirect
123. C₈₉	#2969	semidirect
134. C₄×M₁₆	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152132	2
#19151728	2
C₈×Dic₁₇₈	2
#19151677	2
#19151674	2
#19151667	2
C₈×D₃₅₆	2
#19151660	2
#19151651	2
#19151555	2
C₄×M₁₆×C₈₉	2
#2969	89