Subgroups — D₁₁₃⋊#423 #28407210

Order 14464.

576 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₁₃⋊M₁₆	C₂	1	2	yes	maximal
2	#20299603	C₂	1	2	yes	maximal, complemented
3	#20299602	C₂	1	2	yes	maximal, complemented
4	#20299601	C₂	1	4	yes	maximal, complemented
5	#20299599	C₂	1	4	yes	maximal, complemented
6	C₂×C₄²×D₁₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14268657	C₂²	1	4	yes	—
8	D₁₁₃⋊M₁₆	—	2	8	no	—
9	C₂×#14268656	C₂²	1	4	yes	—
10	C₂×#14268655	C₂²	1	4	yes	—
11	#16813689	C₂²	1	8	yes	complemented
12	C₄⋊#11744753	C₂²	1	8	yes	complemented
13	C₂²×C₄×D₁₁₃	C₄	1	2	yes	—
14	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₁₁₃	C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
17	C₄²×D₁₁₃	C₄	1	1	yes	characteristic
18	C₄²×D₁₁₃	C₄	1	2	yes	—
19	C₄²×D₁₁₃	C₄	1	1	yes	characteristic
20	C₄²×D₁₁₃	C₂²	1	2	yes	—
21	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
22	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₁₃	C₄	1	1	yes	characteristic
24	#14268657	—	2	8	no	—
25	#14268656	—	2	8	no	—
26	#14268655	—	2	8	no	—
27	C₂×#11744753	C₂³	1	8	yes	—
28	C₂³×D₁₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₁₁₃	C₂×C₄	1	2	yes	—
30	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×D₁₁₃	—	2	8	no	—
32	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₁₃	D₄	1	4	yes	—
39	C₂×C₄×D₁₁₃	Q₈	1	4	yes	—
40	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
42	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
47	C₄×Dic₁₁₃	C₂³	1	2	yes	—
48	C₄×Dic₁₁₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
50	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
54	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×D₁₁₃	—	2	4	no	—
56	C₂²×D₁₁₃	C₄⋊C₄	1	4	yes	—
57	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₁₃	—	2	4	no	—
60	C₂×C₄×C₁₁₃	C₄⋊C₄	1	4	yes	—
61	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×Dic₁₁₃	—	2	4	no	—
66	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
67	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
69	C₂×Dic₁₁₃	C₂×D₄	1	2	yes	—
70	C₂×Dic₁₁₃	C₂×Q₈	1	2	yes	—
71	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
72	C₄×D₁₁₃	—	2	4	no	—
73	C₄×D₁₁₃	—	2	4	no	—
74	C₄×D₁₁₃	—	2	4	no	—
75	C₄×D₁₁₃	C₄⋊C₄	1	4	yes	—
76	C₄×D₁₁₃	C₄⋊C₄	1	4	yes	—
77	C₄×D₁₁₃	M₁₆	1	8	yes	complemented
78	C₄×D₁₁₃	C₂×D₄	1	2	yes	—
79	C₄×D₁₁₃	C₂×D₄	1	2	yes	—
80	C₄×D₁₁₃	C₂×Q₈	1	2	yes	—
81	C₄×D₁₁₃	C₂×Q₈	1	2	yes	—
82	#11744753	—	2	8	no	—
83	C₂²×C₁₁₃	—	2	2	no	—
84	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
86	C₂×D₁₁₃	—	2	2	no	—
87	C₂×D₁₁₃	—	2	8	no	—
88	C₂×D₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×D₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×D₁₁₃	C₂×M₁₆	1	4	yes	—
91	C₄×C₁₁₃	—	2	2	no	—
92	C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₄×C₁₁₃	C₂×M₁₆	1	4	yes	complemented
95	Dic₁₁₃	—	2	2	no	—
96	Dic₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
97	Dic₁₁₃	C₂×M₁₆	1	4	yes	complemented
98	Dic₁₁₃	C₂²×D₄	1	1	yes	characteristic
99	Dic₁₁₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₁₁₃	—	2	2	no	—
101	C₂×C₁₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₁₁₃	C₂²×M₁₆	1	2	yes	—
103	D₁₁₃	—	2	2	no	—
104	D₁₁₃	#423	1	4	yes	complemented
105	C₂×#423	—	113	1	no	maximal, Sylow
106	C₁₁₃	C₂×#423	1	1	yes	characteristic, complemented, Sylow
107	C₂²×M₁₆	—	113	2	no	—
108	C₂²×C₄²	—	113	1	no	—
109	#423	—	113	4	no	—
110	#423	—	113	2	no	—
111	#423	—	113	2	no	—
112	C₂×C₄⋊C₈	—	113	4	no	—
113	C₂³×C₄	—	113	1	no	—
114	C₂³×C₄	—	113	2	no	—
115	C₂×M₁₆	—	113	4	no	—
116	C₂×M₁₆	—	113	4	no	—
117	C₂×M₁₆	—	226	8	no	—
118	C₂²×C₈	—	113	4	no	—
119	C₂×C₄²	—	113	1	no	—
120	C₂×C₄²	—	113	2	no	—
121	C₂×C₄²	—	113	1	no	—
122	C₂×C₄²	—	113	1	no	—
123	C₂×C₄²	—	113	2	no	—
124	C₂×C₄²	—	113	2	no	—
125	C₂×C₄²	—	113	1	no	—
126	C₂×C₄²	—	113	1	no	—
127	C₂×C₄²	#90673	1	1	yes	characteristic
128	C₄⋊C₈	—	113	8	no	—
129	C₄⋊C₈	—	113	8	no	—
130	C₂⁴	—	113	1	no	—
131	C₂²×C₄	—	113	1	no	—
132	C₂²×C₄	—	113	4	no	—
133	C₂²×C₄	—	113	4	no	—
134	C₂²×C₄	—	113	1	no	—
135	C₂²×C₄	—	113	1	no	—
136	C₂²×C₄	—	113	2	no	—
137	C₂²×C₄	—	113	1	no	—
138	C₂²×C₄	—	113	1	no	—
139	C₂²×C₄	—	113	2	no	—
140	C₂²×C₄	—	113	2	no	—
141	C₂²×C₄	—	113	2	no	—
142	C₂²×C₄	—	113	2	no	—
143	C₂²×C₄	—	226	8	no	—
144	C₂²×C₄	C₂×#90673	1	2	yes	—
145	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
146	M₁₆	—	226	8	no	—
147	M₁₆	—	226	8	no	—
148	C₂×C₈	—	113	8	no	—
149	C₂×C₈	—	226	8	no	—
150	C₄²	—	113	2	no	—
151	C₄²	—	113	2	no	—
152	C₄²	—	113	1	no	—
153	C₄²	—	113	2	no	—
154	C₄²	—	113	2	no	—
155	C₄²	—	113	1	no	—
156	C₄²	—	113	2	no	—
157	C₄²	C₂×#90673	1	1	yes	characteristic
158	C₄²	C₂×#90673	1	1	yes	characteristic
159	C₄²	C₂×#90673	1	2	yes	—
160	C₂³	—	113	1	no	—
161	C₂³	—	113	4	no	—
162	C₂³	—	113	1	no	—
163	C₂³	—	226	4	no	—
164	C₂³	C₂²×#90673	1	1	yes	characteristic
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	113	2	no	—
167	C₂×C₄	—	113	2	no	—
168	C₂×C₄	—	113	1	no	—
169	C₂×C₄	—	113	1	no	—
170	C₂×C₄	—	113	2	no	—
171	C₂×C₄	—	113	2	no	—
172	C₂×C₄	—	113	2	no	—
173	C₂×C₄	—	113	4	no	—
174	C₂×C₄	—	113	4	no	—
175	C₂×C₄	—	113	2	no	—
176	C₂×C₄	—	113	2	no	—
177	C₂×C₄	—	113	8	no	—
178	C₂×C₄	—	113	2	no	—
179	C₂×C₄	—	226	4	no	—
180	C₂×C₄	—	226	4	no	—
181	C₂×C₄	—	226	4	no	—
182	C₂×C₄	—	226	4	no	—
183	C₂×C₄	C₄⋊#90673	1	4	yes	—
184	C₂×C₄	C₂²×#90673	1	2	yes	—
185	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
186	C₂×C₄	C₂²×#90673	1	2	yes	—
187	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
188	C₈	—	226	8	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	113	2	no	—
191	C₂²	—	113	4	no	—
192	C₂²	—	113	2	no	—
193	C₂²	—	226	8	no	—
194	C₂²	—	226	2	no	—
195	C₂²	C₂×C₄⋊#90673	1	2	yes	—
196	C₂²	C₂³×#90673	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	113	1	no	—
199	C₄	—	113	2	no	—
200	C₄	—	113	1	no	—
201	C₄	—	113	4	no	—
202	C₄	—	226	2	no	—
203	C₄	C₂×C₄⋊#90673	1	2	yes	—
204	C₄	C₂×C₄⋊#90673	1	2	yes	—
205	C₄	D₁₁₃⋊M₁₆	1	4	yes	complemented
206	C₂	—	2	2	no	—
207	C₂	—	113	4	no	—
208	C₂	—	226	2	no	—
209	C₂	C₂²×C₄⋊#90673	1	1	yes	characteristic
210	C₂	C₂×D₁₁₃⋊M₁₆	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×#90673
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Frattini	C₂²	C₂³×#90673
Fitting	C₂×C₄²×C₁₁₃	C₄
prime radical	C₂³×D₁₁₃	C₂×C₄
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	C₂×C₄²	#90673
hypocenter	C₁₁₃	C₂×#423
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #20299603	C₂	semidirect
3. #20299602	C₂	semidirect
4. #20299601	C₂	semidirect
5. #20299599	C₂	semidirect
11. #16813689	C₂²	semidirect
12. C₄⋊#11744753	C₂²	semidirect
77. C₄×D₁₁₃	M₁₆	semidirect
94. C₄×C₁₁₃	C₂×M₁₆	semidirect
97. Dic₁₁₃	C₂×M₁₆	semidirect
104. D₁₁₃	#423	semidirect
106. C₁₁₃	C₂×#423	semidirect
205. C₄	D₁₁₃⋊M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₁₃⋊M₁₆	2
#20299603	2
#20299602	2
#20299601	2
#20299599	2
C₂×C₄²×D₁₁₃	2
C₂×#423	113

Subgroups — D113⋊#423 #28407210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₁₃⋊#423 #28407210