Subgroups — D₁₁₃⋊#568 #28407566

Order 14464.

1460 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299678	C₂	1	2	yes	maximal, complemented
2	#20299677	C₂	1	8	yes	maximal, complemented
3	#20299676	C₂	1	6	yes	maximal, complemented
4	D₁₁₃⋊#131	C₂	1	8	yes	maximal, complemented
5	C₂×D₁₁₃⋊M₁₆	C₂	1	6	yes	maximal, complemented
6	C₂×#41×D₁₁₃	C₂	1	1	yes	maximal, characteristic
7	#16813710	C₂²	1	16	yes	complemented
8	#16813709	C₂²	1	48	yes	complemented
9	C₂×#14268657	C₂²	1	12	yes	complemented
10	D₁₁₃⋊M₁₆	C₂²	1	4	yes	—
11	D₁₁₃⋊M₁₆	C₂²	1	12	yes	complemented
12	D₁₁₃⋊M₁₆	C₂²	1	12	yes	complemented
13	D₁₁₃⋊M₁₆	C₂²	1	12	yes	complemented
14	C₂×#14268656	C₂²	1	12	yes	complemented
15	C₂×#14268655	C₂²	1	12	yes	complemented
16	C₂×#14268648	C₂²	1	1	yes	characteristic
17	C₂×#14268647	C₄	1	3	yes	—
18	C₂×#14268646	C₂²	1	3	yes	—
19	#41×D₁₁₃	C₄	1	4	yes	—
20	#41×D₁₁₃	C₄	1	4	yes	—
21	#41×D₁₁₃	C₂²	1	4	yes	—
22	#41×D₁₁₃	C₂²	1	4	yes	—
23	C₂×Q₈×D₁₁₃	C₄	1	1	yes	characteristic
24	C₂×D₄×D₁₁₃	C₄	1	3	yes	—
25	C₂²×C₄×D₁₁₃	C₂²	1	3	yes	—
26	C₂×#41×C₁₁₃	C₄	1	1	yes	characteristic
27	#14268657	C₂³	1	8	yes	—
28	#14268657	C₂³	1	24	yes	complemented
29	#14268656	C₂³	1	8	yes	—
30	#14268656	C₂³	1	24	yes	complemented
31	#14268655	C₂³	1	8	yes	—
32	#14268655	C₂³	1	24	yes	complemented
33	C₂×#11744753	C₂³	1	24	yes	complemented
34	C₂³×D₁₁₃	C₂×C₄	1	3	yes	—
35	#14268648	C₂×C₄	1	4	yes	—
36	#14268648	C₂³	1	4	yes	—
37	#14268647	C₂×C₄	1	12	yes	—
38	#14268647	C₂×C₄	1	12	yes	—
39	#14268646	C₂×C₄	1	12	yes	—
40	#14268646	C₂³	1	12	yes	—
41	C₂×Dic₂₂₆	C₂×C₄	1	3	yes	—
42	Q₈×D₁₁₃	C₂×C₄	1	4	yes	—
43	Q₈×D₁₁₃	C₂×C₄	1	4	yes	—
44	C₂×D₄₅₂	C₂×C₄	1	3	yes	—
45	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	6	yes	—
46	D₄×D₁₁₃	C₂×C₄	1	12	yes	—
47	D₄×D₁₁₃	C₂×C₄	1	12	yes	—
48	C₂²×Dic₁₁₃	C₂³	1	3	yes	—
49	C₂×C₄×D₁₁₃	C₂×C₄	1	3	yes	—
50	C₂×C₄×D₁₁₃	C₂×C₄	1	6	yes	—
51	C₂×C₄×D₁₁₃	C₂×C₄	1	6	yes	—
52	C₂×C₄×D₁₁₃	C₂×C₄	1	3	yes	—
53	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₁₃	C₂³	1	6	yes	—
55	C₂×C₄×D₁₁₃	C₂³	1	6	yes	—
56	C₂×C₄×D₁₁₃	C₂³	1	3	yes	—
57	#41×C₁₁₃	C₂×C₄	1	4	yes	—
58	#41×C₁₁₃	C₂×C₄	1	4	yes	—
59	C₂×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂×D₄×C₁₁₃	C₂×C₄	1	3	yes	—
61	C₂²×C₄×C₁₁₃	C₂×C₄	1	3	yes	—
62	C₂³×C₁₁₃	C₂²×C₄	1	3	yes	—
63	C₂²×D₁₁₃	—	4	6	no	—
64	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₁₁₃	C₂²×C₄	1	6	yes	—
66	C₂²×D₁₁₃	C₂²×C₄	1	6	yes	—
67	C₂²×D₁₁₃	C₂²×C₄	1	3	yes	—
68	Q₈×C₁₁₃	C₂²×C₄	1	4	yes	—
69	D₄×C₁₁₃	C₂²×C₄	1	12	yes	—
70	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₁₃	C₂²×C₄	1	3	yes	—
72	C₂×C₄×C₁₁₃	C₂²×C₄	1	6	yes	—
73	C₂×C₄×C₁₁₃	C₂²×C₄	1	6	yes	—
74	C₂²⋊D₁₁₃	C₂²×C₄	1	24	yes	—
75	C₂×Dic₁₁₃	C₂²×C₄	1	3	yes	—
76	C₂×Dic₁₁₃	C₂²×C₄	1	6	yes	—
77	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₁₁₃	C₂⁴	1	6	yes	—
79	D₄₅₂	C₂²×C₄	1	12	yes	—
80	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₁₁₃	C₂²×C₄	1	6	yes	—
83	C₄×D₁₁₃	C₂²×C₄	1	6	yes	—
84	C₄×D₁₁₃	C₂²×C₄	1	6	yes	—
85	C₄×D₁₁₃	C₂²×C₄	1	6	yes	—
86	C₄×D₁₁₃	C₂²×C₄	1	6	yes	—
87	C₄×D₁₁₃	C₂⁴	1	1	yes	characteristic
88	C₄×D₁₁₃	C₂⁴	1	1	yes	characteristic
89	C₄×D₁₁₃	C₂⁴	1	6	yes	—
90	Dic₂₂₆	C₂²×C₄	1	12	yes	—
91	#11744753	C₂⁴	1	16	yes	—
92	C₂²×C₁₁₃	—	4	3	no	—
93	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
94	C₂²×C₁₁₃	C₂³×C₄	1	6	yes	—
95	C₂×D₁₁₃	—	2	2	no	—
96	C₂×D₁₁₃	—	2	12	no	—
97	C₂×D₁₁₃	—	4	6	no	—
98	C₂×D₁₁₃	—	4	3	no	—
99	C₂×D₁₁₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×D₁₁₃	C₂³×C₄	1	6	yes	—
101	C₂×D₁₁₃	C₂³×C₄	1	1	yes	characteristic
102	C₄×C₁₁₃	C₂³×C₄	1	6	yes	—
103	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
104	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
105	Dic₁₁₃	C₂³×C₄	1	6	yes	—
106	Dic₁₁₃	C₂³×C₄	1	1	yes	characteristic
107	Dic₁₁₃	C₂⁵	1	1	yes	characteristic
108	C₂×C₁₁₃	—	2	1	no	—
109	C₂×C₁₁₃	—	2	6	no	—
110	C₂×C₁₁₃	C₂⁴×C₄	1	1	yes	characteristic
111	D₁₁₃	—	2	1	no	—
112	D₁₁₃	—	2	6	no	—
113	D₁₁₃	#568	1	2	yes	complemented
114	C₂×#568	—	113	1	no	maximal, Sylow
115	C₁₁₃	C₂×#568	1	1	yes	characteristic, complemented, Sylow
116	C₂²×#41	—	113	1	no	—
117	#568	—	113	6	no	—
118	#568	—	113	8	no	—
119	#568	—	113	2	no	—
120	C₂×#131	—	113	8	no	—
121	C₂²×M₁₆	—	113	6	no	—
122	C₂×#41	—	113	4	no	—
123	C₂×#41	—	113	4	no	—
124	C₂×#41	—	113	4	no	—
125	C₂×#41	—	113	4	no	—
126	C₂×#41	—	113	1	no	—
127	C₂×#41	—	113	3	no	—
128	C₂×#41	—	113	3	no	—
129	C₂×#41	#90673	1	1	yes	characteristic
130	C₂²×Q₈	—	113	1	no	—
131	C₂²×D₄	—	113	3	no	—
132	C₂³×C₄	—	113	3	no	—
133	#131	—	113	48	no	—
134	#131	—	113	16	no	—
135	C₂×M₁₆	—	113	4	no	—
136	C₂×M₁₆	—	113	12	no	—
137	C₂×M₁₆	—	113	12	no	—
138	C₂×M₁₆	—	113	12	no	—
139	C₂×M₁₆	—	113	12	no	—
140	C₂×M₁₆	—	113	12	no	—
141	C₂²×C₈	—	113	12	no	—
142	C₂⁴	—	113	3	no	—
143	#41	—	113	4	no	—
144	#41	—	113	4	no	—
145	#41	—	113	12	no	—
146	#41	—	113	12	no	—
147	#41	—	113	12	no	—
148	#41	—	113	12	no	—
149	#41	C₂×#90673	1	4	yes	—
150	#41	C₂×#90673	1	4	yes	—
151	C₂×Q₈	—	113	4	no	—
152	C₂×Q₈	—	113	4	no	—
153	C₂×Q₈	—	113	3	no	—
154	C₂×Q₈	C₂×#90673	1	1	yes	characteristic
155	C₂×D₄	—	113	6	no	—
156	C₂×D₄	—	113	12	no	—
157	C₂×D₄	—	113	12	no	—
158	C₂×D₄	—	113	3	no	—
159	C₂×D₄	C₂×#90673	1	3	yes	—
160	C₂²×C₄	—	113	3	no	—
161	C₂²×C₄	—	113	6	no	—
162	C₂²×C₄	—	113	1	no	—
163	C₂²×C₄	—	113	6	no	—
164	C₂²×C₄	—	113	3	no	—
165	C₂²×C₄	—	113	3	no	—
166	C₂²×C₄	—	113	3	no	—
167	C₂²×C₄	—	113	6	no	—
168	C₂²×C₄	—	113	6	no	—
169	C₂²×C₄	C₂×#90673	1	3	yes	—
170	M₁₆	—	113	8	no	—
171	M₁₆	—	113	8	no	—
172	M₁₆	—	113	24	no	—
173	M₁₆	—	113	24	no	—
174	C₂×C₈	—	113	8	no	—
175	C₂×C₈	—	113	24	no	—
176	C₂×C₈	—	113	24	no	—
177	C₂³	—	113	6	no	—
178	C₂³	—	113	3	no	—
179	C₂³	—	113	1	no	—
180	C₂³	—	113	6	no	—
181	C₂³	—	452	6	no	—
182	C₂³	C₂²×#90673	1	3	yes	—
183	Q₈	—	113	12	no	—
184	Q₈	C₂²×#90673	1	4	yes	—
185	D₄	—	113	24	no	—
186	D₄	—	113	12	no	—
187	D₄	C₂²×#90673	1	12	yes	—
188	C₂×C₄	—	113	6	no	—
189	C₂×C₄	—	113	1	no	—
190	C₂×C₄	—	113	3	no	—
191	C₂×C₄	—	113	6	no	—
192	C₂×C₄	—	113	6	no	—
193	C₂×C₄	—	113	1	no	—
194	C₂×C₄	—	113	1	no	—
195	C₂×C₄	—	113	6	no	—
196	C₂×C₄	—	113	6	no	—
197	C₂×C₄	—	113	6	no	—
198	C₂×C₄	—	113	6	no	—
199	C₂×C₄	—	113	1	no	—
200	C₂×C₄	—	113	1	no	—
201	C₂×C₄	—	113	6	no	—
202	C₂×C₄	C₂²×#90673	1	3	yes	—
203	C₂×C₄	C₂²×#90673	1	6	yes	—
204	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
205	C₂×C₄	C₂²×#90673	1	6	yes	—
206	C₈	—	113	16	no	—
207	C₂²	—	4	3	no	—
208	C₂²	—	113	1	no	—
209	C₂²	—	113	1	no	—
210	C₂²	—	113	6	no	—
211	C₂²	—	452	3	no	—
212	C₂²	—	452	6	no	—
213	C₂²	—	226	2	no	—
214	C₂²	—	226	12	no	—
215	C₂²	C₂³×#90673	1	1	yes	characteristic
216	C₂²	C₂³×#90673	1	6	yes	—
217	C₄	—	113	1	no	—
218	C₄	—	113	1	no	—
219	C₄	—	113	6	no	—
220	C₄	C₂³×#90673	1	1	yes	characteristic
221	C₄	C₂³×#90673	1	6	yes	—
222	C₄	C₂³×#90673	1	1	yes	characteristic
223	C₂	—	2	1	no	—
224	C₂	—	2	6	no	—
225	C₂	—	113	2	no	—
226	C₂	—	226	1	no	—
227	C₂	—	226	6	no	—
228	C₂	C₂⁴×#90673	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂³×#90673
commutator, socle	C₂×C₁₁₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#90673
Fitting	C₂×#41×C₁₁₃	C₄
prime radical	C₂×#41×D₁₁₃	C₂
cosocle	Dic₁₁₃	C₂⁵
hypercenter	C₂×#41	#90673
hypocenter	C₁₁₃	C₂×#568
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #20299678	C₂	semidirect
2. #20299677	C₂	semidirect
3. #20299676	C₂	semidirect
4. D₁₁₃⋊#131	C₂	semidirect
5. C₂×D₁₁₃⋊M₁₆	C₂	semidirect
7. #16813710	C₂²	semidirect
8. #16813709	C₂²	semidirect
9. C₂×#14268657	C₂²	semidirect
11. D₁₁₃⋊M₁₆	C₂²	semidirect
12. D₁₁₃⋊M₁₆	C₂²	semidirect
13. D₁₁₃⋊M₁₆	C₂²	semidirect
14. C₂×#14268656	C₂²	semidirect
15. C₂×#14268655	C₂²	semidirect
28. #14268657	C₂³	semidirect
30. #14268656	C₂³	semidirect
32. #14268655	C₂³	semidirect
33. C₂×#11744753	C₂³	semidirect
113. D₁₁₃	#568	semidirect
115. C₁₁₃	C₂×#568	semidirect

Maximal subgroups

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

subgroup	index
#20299678	2
#20299677	2
#20299676	2
D₁₁₃⋊#131	2
C₂×D₁₁₃⋊M₁₆	2
C₂×#41×D₁₁₃	2
C₂×#568	113

Subgroups — D113⋊#568 #28407566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₁₃⋊#568 #28407566