Subgroups — C₁₁₃×#13206 #43281226

Order 28928.

590 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#357×C₁₁₃	C₂	1	4	yes	maximal, complemented
3	C₂×#346×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₁₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₁₁₃	—	2	1	no	—
6	C₂²×D₈×C₁₁₃	C₂²	1	2	yes	—
7	C₂×#127×C₁₁₃	—	2	1	no	—
8	C₂²×C₁₆×C₁₁₃	C₂²	1	2	yes	—
9	#493×C₁₁₃	—	2	2	no	—
10	#493×C₁₁₃	C₄	1	4	yes	complemented
11	C₂×#107×C₁₁₃	C₂²	1	2	yes	—
12	C₂×#106×C₁₁₃	—	2	1	no	—
13	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
14	#382×C₁₁₃	—	2	1	no	—
15	#357×C₁₁₃	—	2	8	no	—
16	#346×C₁₁₃	—	2	2	no	—
17	C₂²×D₄×C₁₁₃	—	2	2	no	—
18	C₂²×D₄×C₁₁₃	—	4	2	no	—
19	C₂×D₈×C₁₁₃	—	2	4	no	—
20	C₂×D₈×C₁₁₃	—	2	4	no	—
21	C₂×D₈×C₁₁₃	—	2	2	no	—
22	C₂×D₈×C₁₁₃	—	2	4	no	—
23	C₂×D₈×C₁₁₃	—	4	2	no	—
24	C₂×D₈×C₁₁₃	—	4	2	no	—
25	C₂×D₈×C₁₁₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
28	#127×C₁₁₃	—	2	4	no	—
29	#127×C₁₁₃	—	4	2	no	—
30	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₁₃	—	2	2	no	—
33	C₂×C₁₆×C₁₁₃	—	2	2	no	—
34	C₂×C₁₆×C₁₁₃	D₄	1	4	yes	complemented
35	#107×C₁₁₃	—	2	4	no	—
36	#106×C₁₁₃	—	4	2	no	—
37	C₄×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₁₃	D₄	1	2	yes	—
39	#95×C₁₁₃	—	4	2	no	—
40	C₂⁴×C₁₁₃	—	4	2	no	—
41	C₂×D₄×C₁₁₃	—	2	4	no	—
42	C₂×D₄×C₁₁₃	—	4	8	no	—
43	C₂×D₄×C₁₁₃	—	4	2	no	—
44	C₂×D₄×C₁₁₃	—	4	2	no	—
45	C₂×D₄×C₁₁₃	—	4	2	no	—
46	C₂×D₄×C₁₁₃	—	4	2	no	—
47	C₂×D₄×C₁₁₃	—	4	2	no	—
48	C₂×D₄×C₁₁₃	—	8	8	no	—
49	C₂²×C₄×C₁₁₃	—	4	2	no	—
50	C₂²×C₄×C₁₁₃	D₈	1	2	yes	—
51	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
52	D₈×C₁₁₃	—	2	8	no	—
53	D₈×C₁₁₃	—	4	8	no	—
54	D₈×C₁₁₃	—	4	4	no	—
55	C₂×C₈×C₁₁₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₁₃	D₈	1	2	yes	—
57	C₂×C₈×C₁₁₃	SD₁₆	1	2	yes	—
58	C₂×C₈×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₁₃	#41	1	2	yes	—
62	C₄⋊C₄×C₁₁₃	—	4	2	no	—
63	C₄⋊C₄×C₁₁₃	—	4	2	no	—
64	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₁₁₃	SD₁₆	1	2	yes	—
66	C₁₆×C₁₁₃	—	2	4	no	—
67	C₂³×C₁₁₃	—	4	4	no	—
68	C₂³×C₁₁₃	—	8	2	no	—
69	C₂³×C₁₁₃	—	8	2	no	—
70	C₂³×C₁₁₃	—	8	8	no	—
71	C₂³×C₁₁₃	C₂×D₈	1	1	yes	characteristic
72	D₄×C₁₁₃	—	4	4	no	—
73	D₄×C₁₁₃	—	4	4	no	—
74	D₄×C₁₁₃	—	8	16	no	—
75	D₄×C₁₁₃	—	8	2	no	—
76	D₄×C₁₁₃	—	8	2	no	—
77	C₂×C₄×C₁₁₃	—	2	4	no	—
78	C₂×C₄×C₁₁₃	—	8	2	no	—
79	C₂×C₄×C₁₁₃	—	8	4	no	—
80	C₂×C₄×C₁₁₃	#102	1	4	yes	—
81	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₁₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₁₁₃	#127	1	1	yes	characteristic
84	C₂×C₄×C₁₁₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₁₁₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₁₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₈×C₁₁₃	#102	1	4	yes	—
88	C₈×C₁₁₃	C₄×D₄	1	2	yes	—
89	C₈×C₁₁₃	#121	1	2	yes	—
90	C₂²×C₁₁₃	—	8	8	no	—
91	C₂²×C₁₁₃	—	8	4	no	—
92	C₂²×C₁₁₃	—	16	4	no	—
93	C₂²×C₁₁₃	—	16	4	no	—
94	C₂²×C₁₁₃	C₂×#102	1	1	yes	characteristic
95	C₂²×C₁₁₃	#486	1	1	yes	characteristic
96	C₂²×C₁₁₃	#493	1	1	yes	characteristic
97	C₂²×C₁₁₃	#509	1	4	yes	—
98	C₄×C₁₁₃	—	2	4	no	—
99	C₄×C₁₁₃	—	16	2	no	—
100	C₄×C₁₁₃	#390	1	1	yes	characteristic
101	C₄×C₁₁₃	C₂×#102	1	1	yes	characteristic
102	C₄×C₁₁₃	#486	1	1	yes	characteristic
103	C₄×C₁₁₃	#492	1	1	yes	characteristic
104	#13206	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂×C₁₁₃	—	16	4	no	—
106	C₂×C₁₁₃	#1957	1	1	yes	characteristic
107	C₂×C₁₁₃	#2140	1	2	yes	—
108	C₂×C₁₁₃	#2244	1	2	yes	—
109	C₂×C₁₁₃	#2250	1	2	yes	—
110	C₂×#493	C₂×C₁₁₃	1	1	yes	characteristic
111	C₂×#357	C₂×C₁₁₃	1	4	yes	complemented
112	C₂×#346	C₂×C₁₁₃	1	1	yes	characteristic, complemented
113	#1846	C₂×C₁₁₃	1	1	yes	characteristic, complemented
114	C₁₁₃	#13206	1	1	yes	characteristic, direct product factor, Sylow
115	C₂²×D₈	—	2	1	no	—
116	C₂²×D₈	C₂²×C₁₁₃	1	2	yes	—
117	C₂×#127	—	2	1	no	—
118	C₂²×C₁₆	C₂²×C₁₁₃	1	2	yes	—
119	#493	—	2	2	no	—
120	#493	C₄×C₁₁₃	1	4	yes	complemented
121	C₂×#107	C₂²×C₁₁₃	1	2	yes	—
122	C₂×#106	—	2	1	no	—
123	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
124	#382	—	2	1	no	—
125	#357	—	2	8	no	—
126	#346	—	2	2	no	—
127	C₂²×D₄	—	2	2	no	—
128	C₂²×D₄	—	4	2	no	—
129	C₂×D₈	—	2	4	no	—
130	C₂×D₈	—	2	4	no	—
131	C₂×D₈	—	2	2	no	—
132	C₂×D₈	—	2	4	no	—
133	C₂×D₈	—	4	2	no	—
134	C₂×D₈	—	4	2	no	—
135	C₂×D₈	C₂×C₄×C₁₁₃	1	4	yes	—
136	C₂²×C₈	D₄×C₁₁₃	1	1	yes	characteristic
137	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
138	#127	—	2	4	no	—
139	#127	—	4	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	D₄×C₁₁₃	1	4	yes	complemented
145	#107	—	2	4	no	—
146	#106	—	4	2	no	—
147	C₄×C₈	C₂×C₄×C₁₁₃	1	2	yes	—
148	C₄×C₈	D₄×C₁₁₃	1	2	yes	—
149	#95	—	4	2	no	—
150	C₂⁴	—	4	2	no	—
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	4	8	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂×D₄	—	4	2	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂×D₄	—	4	2	no	—
157	C₂×D₄	—	4	2	no	—
158	C₂×D₄	—	8	8	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	D₈×C₁₁₃	1	2	yes	—
161	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
162	D₈	—	2	8	no	—
163	D₈	—	4	8	no	—
164	D₈	—	4	4	no	—
165	C₂×C₈	C₂²⋊C₄×C₁₁₃	1	2	yes	—
166	C₂×C₈	D₈×C₁₁₃	1	2	yes	—
167	C₂×C₈	SD₁₆×C₁₁₃	1	2	yes	—
168	C₂×C₈	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
169	C₂×C₈	C₂×D₄×C₁₁₃	1	2	yes	—
170	C₂×C₈	C₂×D₄×C₁₁₃	1	1	yes	characteristic
171	C₂×C₈	#41×C₁₁₃	1	2	yes	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
175	C₄²	SD₁₆×C₁₁₃	1	2	yes	—
176	C₁₆	—	2	4	no	—
177	C₂³	—	4	4	no	—
178	C₂³	—	8	2	no	—
179	C₂³	—	8	2	no	—
180	C₂³	—	8	8	no	—
181	C₂³	C₂×D₈×C₁₁₃	1	1	yes	characteristic
182	D₄	—	4	4	no	—
183	D₄	—	4	4	no	—
184	D₄	—	8	16	no	—
185	D₄	—	8	2	no	—
186	D₄	—	8	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	—	8	4	no	—
190	C₂×C₄	#102×C₁₁₃	1	4	yes	—
191	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
192	C₂×C₄	#124×C₁₁₃	1	1	yes	characteristic
193	C₂×C₄	#127×C₁₁₃	1	1	yes	characteristic
194	C₂×C₄	C₂×D₈×C₁₁₃	1	1	yes	characteristic
195	C₂×C₄	C₂×SD₁₆×C₁₁₃	1	1	yes	characteristic
196	C₂×C₄	C₂×SD₁₆×C₁₁₃	1	1	yes	characteristic
197	C₈	#102×C₁₁₃	1	4	yes	—
198	C₈	C₄×D₄×C₁₁₃	1	2	yes	—
199	C₈	#121×C₁₁₃	1	2	yes	—
200	C₂²	—	8	8	no	—
201	C₂²	—	8	4	no	—
202	C₂²	—	16	4	no	—
203	C₂²	—	16	4	no	—
204	C₂²	C₂×#102×C₁₁₃	1	1	yes	characteristic
205	C₂²	#486×C₁₁₃	1	1	yes	characteristic
206	C₂²	#493×C₁₁₃	1	1	yes	characteristic
207	C₂²	#509×C₁₁₃	1	4	yes	—
208	C₄	—	2	4	no	—
209	C₄	—	16	2	no	—
210	C₄	#390×C₁₁₃	1	1	yes	characteristic
211	C₄	C₂×#102×C₁₁₃	1	1	yes	characteristic
212	C₄	#486×C₁₁₃	1	1	yes	characteristic
213	C₄	#492×C₁₁₃	1	1	yes	characteristic
214	C₂	—	16	4	no	—
215	C₂	C₁₁₃×#1957	1	1	yes	characteristic
216	C₂	C₁₁₃×#2140	1	2	yes	—
217	C₂	C₁₁₃×#2244	1	2	yes	—
218	C₂	C₁₁₃×#2250	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₁₃
prime radical	C₂×#493×C₁₁₃	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁₃ ◁ C₂×C₄²×C₁₁₃ ◁ C₂×C₄×C₈×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#357×C₁₁₃	C₂	semidirect
3. C₂×#346×C₁₁₃	C₂	semidirect
4. C₁₁₃×#1846	C₂	semidirect
10. #493×C₁₁₃	C₄	semidirect
34. C₂×C₁₆×C₁₁₃	D₄	semidirect
104. #13206	C₁₁₃	direct (N × H)
111. C₂×#357	C₂×C₁₁₃	semidirect
112. C₂×#346	C₂×C₁₁₃	semidirect
113. #1846	C₂×C₁₁₃	semidirect
114. C₁₁₃	#13206	direct (N × H)
120. #493	C₄×C₁₁₃	semidirect
144. C₂×C₁₆	D₄×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₁₁₃	2
C₂×#357×C₁₁₃	2
C₂×#346×C₁₁₃	2
C₁₁₃×#1846	2
#13206	113

Subgroups — C113×#13206 #43281226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#13206 #43281226