Subgroups — #26029237

Order 11392.

362 conjugacy classes of subgroups.

The subgroup lattice has 224 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	#19151726	C₂	1	2	yes	maximal, complemented
3	#19151674	C₂	1	2	yes	maximal, complemented
4	#19151671	C₂	1	2	yes	maximal, complemented
5	#19151667	C₂	1	2	yes	maximal, complemented
6	#19151657	C₂	1	2	yes	maximal, complemented
7	#19151655	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151552	C₂	1	2	yes	maximal, complemented
9	C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13083557	C₂²	1	2	yes	—
11	C₂×C₈×D₈₉	C₂²	1	2	yes	—
12	#15668006	C₄	1	2	yes	—
13	#15668004	C₄	1	1	yes	characteristic
14	#15668000	C₄	1	1	yes	characteristic
15	#15667995	C₄	1	2	yes	—
16	#15667991	C₄	1	1	yes	characteristic
17	#15667986	C₄	1	1	yes	characteristic
18	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
19	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₈₉	C₂²	1	2	yes	—
21	#15667969	C₂²	1	2	yes	—
22	C₄⋊#10623494	C₂²	1	2	yes	—
23	#15667941	C₂²	1	2	yes	—
24	#15667941	C₂²	1	2	yes	—
25	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
26	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
27	#15667924	C₂²	1	2	yes	—
28	#15667924	C₂²	1	2	yes	—
29	#15667923	C₂²	1	2	yes	—
30	#15667922	C₂²	1	2	yes	—
31	C₈×Dic₈₉	C₂²	1	2	yes	—
32	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
33	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
34	#97×C₈₉	C₂²	1	2	yes	—
35	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
38	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₈₉	C₂³	1	2	yes	—
41	#13083557	—	2	4	no	—
42	C₂×#10623494	C₂³	1	2	yes	—
43	C₂×#10623494	C₂³	1	2	yes	—
44	C₈×D₈₉	—	2	4	no	—
45	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
47	#13083553	C₂×C₄	1	2	yes	—
48	#13083553	C₂×C₄	1	2	yes	—
49	#13083553	C₂×C₄	1	2	yes	—
50	#13083553	C₂×C₄	1	2	yes	—
51	C₂²⋊Dic₈₉	C₂×C₄	1	2	yes	—
52	#13083551	C₂×C₄	1	2	yes	—
53	#13083551	C₂×C₄	1	2	yes	—
54	#13083551	C₂×C₄	1	2	yes	—
55	#13083551	C₂×C₄	1	2	yes	—
56	C₄×Dic₈₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
60	M₁₆×C₈₉	—	2	4	no	—
61	C₂×C₈×C₈₉	C₂³	1	2	yes	—
62	C₂×C₈×C₈₉	C₂³	1	2	yes	—
63	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₈₉	C₂³	1	1	yes	characteristic
66	C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₈₉	—	2	2	no	—
70	C₂×C₄×C₈₉	—	4	1	no	—
71	C₂×C₄×C₈₉	—	4	1	no	—
72	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₂²⋊D₈₉	—	4	2	no	—
79	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
80	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
81	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
82	D₃₅₆	—	4	1	no	—
83	C₄×D₈₉	—	2	2	no	—
84	C₄×D₈₉	—	2	2	no	—
85	Dic₁₇₈	—	4	1	no	—
86	C₈×C₈₉	—	2	2	no	—
87	C₈×C₈₉	#41	1	4	yes	—
88	#10623494	—	2	2	no	—
89	#10623494	—	2	2	no	—
90	C₂²×C₈₉	—	2	1	no	—
91	C₂²×C₈₉	—	2	1	no	—
92	C₂²×C₈₉	—	2	1	no	—
93	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₈₉	—	2	2	no	—
95	C₂×D₈₉	—	2	2	no	—
96	C₂×D₈₉	—	2	2	no	—
97	C₄×C₈₉	—	2	1	no	—
98	C₄×C₈₉	—	2	1	no	—
99	C₄×C₈₉	—	2	1	no	—
100	C₄×C₈₉	—	2	1	no	—
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	C₂×#41	1	2	yes	—
103	Dic₈₉	—	2	2	no	—
104	Dic₈₉	—	2	2	no	—
105	Dic₈₉	—	2	2	no	—
106	C₂×C₈₉	—	4	1	no	—
107	C₂×C₈₉	C₄×#41	1	1	yes	characteristic
108	C₂×C₈₉	#568	1	1	yes	characteristic
109	C₂×C₈₉	#568	1	1	yes	characteristic
110	D₈₉	—	4	2	no	—
111	#2967	—	89	1	no	maximal, Sylow
112	C₈₉	#2967	1	1	yes	characteristic, complemented, Sylow
113	#529	—	89	1	no	—
114	#446	—	89	2	no	—
115	#436	—	89	2	no	—
116	#435	—	89	2	no	—
117	#435	—	89	2	no	—
118	#433	—	89	2	no	—
119	#432	—	89	1	no	—
120	#405	—	89	2	no	—
121	C₄×M₁₆	D₈₉	1	1	yes	characteristic, complemented
122	C₂×M₁₆	—	89	2	no	—
123	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
124	C₂²×C₈	—	89	2	no	—
125	#126	—	89	2	no	—
126	#125	—	89	1	no	—
127	#124	—	89	1	no	—
128	#123	—	89	2	no	—
129	C₂²⋊Q₈	—	89	1	no	—
130	#121	—	89	1	no	—
131	C₄×Q₈	—	89	1	no	—
132	C₄×D₄	—	89	2	no	—
133	C₄×D₄	—	89	1	no	—
134	#117	—	89	2	no	—
135	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
136	C₄⋊C₈	—	89	2	no	—
137	C₄⋊C₈	—	89	2	no	—
138	C₄⋊C₈	—	89	2	no	—
139	C₂²⋊C₈	—	89	1	no	—
140	C₂²⋊C₈	—	89	1	no	—
141	C₂²⋊C₈	—	89	2	no	—
142	C₂²⋊C₈	—	89	2	no	—
143	#97	—	89	2	no	—
144	#97	—	89	2	no	—
145	#97	C₂×D₈₉	1	2	yes	—
146	C₄×C₈	—	89	2	no	—
147	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
148	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
149	C₂×Q₈	—	89	1	no	—
150	C₂×D₄	—	89	2	no	—
151	C₂×D₄	—	89	1	no	—
152	C₂²×C₄	—	89	2	no	—
153	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
156	M₁₆	—	2	4	no	—
157	M₁₆	—	178	4	no	—
158	C₂×C₈	—	89	2	no	—
159	C₂×C₈	—	89	2	no	—
160	C₂×C₈	—	178	4	no	—
161	C₂×C₈	C₂²×D₈₉	1	2	yes	—
162	C₂×C₈	C₂²×D₈₉	1	2	yes	—
163	C₄⋊C₄	—	89	2	no	—
164	C₄⋊C₄	—	89	2	no	—
165	C₄⋊C₄	—	89	1	no	—
166	C₄⋊C₄	—	89	2	no	—
167	C₄⋊C₄	—	89	1	no	—
168	C₄⋊C₄	—	89	2	no	—
169	C₂²⋊C₄	—	89	2	no	—
170	C₂²⋊C₄	—	89	2	no	—
171	C₂²⋊C₄	—	89	2	no	—
172	C₂²⋊C₄	—	89	2	no	—
173	C₂²⋊C₄	—	89	2	no	—
174	C₄²	—	89	2	no	—
175	C₄²	C₄×D₈₉	1	1	yes	characteristic
176	C₄²	C₄×D₈₉	1	1	yes	characteristic
177	C₄²	C₂²×D₈₉	1	1	yes	characteristic
178	C₄²	C₂²×D₈₉	1	1	yes	characteristic
179	C₂³	—	89	2	no	—
180	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
181	Q₈	—	356	1	no	—
182	D₄	—	356	1	no	—
183	D₄	—	356	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	89	2	no	—
188	C₂×C₄	—	89	2	no	—
189	C₂×C₄	—	89	2	no	—
190	C₂×C₄	—	178	2	no	—
191	C₂×C₄	—	178	2	no	—
192	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	178	2	no	—
200	C₈	—	178	2	no	—
201	C₈	#13083574	1	4	yes	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	178	2	no	—
206	C₂²	—	178	2	no	—
207	C₂²	—	178	2	no	—
208	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	178	2	no	—
215	C₄	—	178	2	no	—
216	C₄	—	178	2	no	—
217	C₄	C₂×#13083574	1	2	yes	—
218	C₂	—	4	1	no	—
219	C₂	—	356	2	no	—
220	C₂	C₄×#13083574	1	1	yes	characteristic
221	C₂	#19152364	1	1	yes	characteristic
222	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	#2967
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. #19151726	C₂	semidirect
3. #19151674	C₂	semidirect
4. #19151671	C₂	semidirect
5. #19151667	C₂	semidirect
6. #19151657	C₂	semidirect
7. #19151655	C₂	semidirect
8. #19151552	C₂	semidirect
9. C₄×M₁₆×C₈₉	C₂	semidirect
112. C₈₉	#2967	semidirect
121. 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
#19151726	2
#19151674	2
#19151671	2
#19151667	2
#19151657	2
#19151655	2
#19151552	2
C₄×M₁₆×C₈₉	2
#2967	89