Subgroups — C₂×C₁₃×#2242 #15720035

Order 3328.

610 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#495	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2242	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#358	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#357	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×SD₁₆	—	2	1	no	—
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#125	—	2	1	no	—
11	C₂×C₁₃×#124	—	2	1	no	—
12	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
13	C₁₃×#499	—	2	2	no	—
14	C₁₃×#499	C₂²	1	4	yes	complemented
15	C₁₃×#495	—	2	2	no	—
16	C₁₃×#495	C₂²	1	4	yes	—
17	C₂×C₁₃×#107	C₂²	1	2	yes	—
18	C₂×C₁₃×#106	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#358	C₂²	1	8	yes	complemented
21	C₁₃×#357	C₂²	1	8	yes	—
22	C₄×C₁₃×C₁₆	C₂²	1	4	yes	complemented
23	C₂²×Q₈×C₁₃	—	2	1	no	—
24	C₂²×D₄×C₁₃	—	2	1	no	—
25	C₂×C₁₃×Q₁₆	—	2	1	no	—
26	C₂×C₁₃×Q₁₆	—	2	4	no	—
27	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
28	C₂×C₁₃×SD₁₆	—	2	4	no	—
29	C₂×C₁₃×SD₁₆	—	4	4	no	—
30	C₂×C₁₃×D₈	—	2	1	no	—
31	C₂×C₁₃×D₈	—	2	4	no	—
32	C₂×C₁₃×D₈	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×#125	—	2	4	no	—
36	C₁₃×#125	—	4	2	no	—
37	C₁₃×#124	—	2	4	no	—
38	C₁₃×#124	—	4	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
41	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
42	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
43	C₂×C₁₃×C₁₆	—	2	4	no	—
44	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
45	C₁₃×#107	—	2	2	no	—
46	C₁₃×#107	C₂³	1	4	yes	—
47	C₁₃×#106	—	2	4	no	—
48	C₄×C₈×C₁₃	D₄	1	2	yes	—
49	C₄×C₈×C₁₃	C₂³	1	2	yes	—
50	C₂×#2242	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂⁴×C₁₃	—	4	1	no	—
52	C₂×Q₈×C₁₃	—	2	2	no	—
53	C₂×Q₈×C₁₃	—	4	1	no	—
54	C₂×Q₈×C₁₃	—	4	2	no	—
55	C₂×Q₈×C₁₃	—	4	2	no	—
56	C₂×D₄×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	4	2	no	—
58	C₂×D₄×C₁₃	—	4	1	no	—
59	C₂×D₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	D₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₁₃×Q₁₆	—	2	4	no	—
65	C₁₃×Q₁₆	—	4	2	no	—
66	C₁₃×SD₁₆	—	4	8	no	—
67	C₁₃×D₈	—	2	4	no	—
68	C₁₃×D₈	—	4	2	no	—
69	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃	#41	1	4	yes	—
73	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	2	4	no	—
75	C₁₃×C₄⋊C₄	—	4	2	no	—
76	C₁₃×C₄⋊C₄	—	4	8	no	—
77	C₁₃×C₂²⋊C₄	—	4	8	no	—
78	C₄²×C₁₃	D₈	1	2	yes	—
79	C₄²×C₁₃	C₂×D₄	1	2	yes	—
80	C₁₃×C₁₆	—	2	4	no	—
81	C₂×#499	C₂×C₁₃	1	1	yes	characteristic, complemented
82	C₂×#495	C₂×C₁₃	1	1	yes	characteristic
83	#2242	C₂×C₁₃	1	8	yes	direct product factor
84	C₂×#358	C₂×C₁₃	1	2	yes	complemented
85	C₂×#357	C₂×C₁₃	1	2	yes	—
86	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
87	C₂³×C₁₃	—	4	2	no	—
88	C₂³×C₁₃	—	8	2	no	—
89	C₂³×C₁₃	—	8	4	no	—
90	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₁₃	—	4	2	no	—
92	Q₈×C₁₃	—	4	2	no	—
93	Q₈×C₁₃	—	8	2	no	—
94	D₄×C₁₃	—	4	2	no	—
95	D₄×C₁₃	—	4	2	no	—
96	D₄×C₁₃	—	8	2	no	—
97	C₂×C₄×C₁₃	—	2	4	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	4	no	—
100	C₂×C₄×C₁₃	—	8	2	no	—
101	C₂×C₄×C₁₃	—	8	4	no	—
102	C₂×C₄×C₁₃	#124	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
104	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₁₃	#124	1	2	yes	—
108	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
109	C₈×C₁₃	C₂×#41	1	2	yes	—
110	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
111	C₂²×SD₁₆	—	2	1	no	—
112	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
113	C₂×#125	—	2	1	no	—
114	C₂×#124	—	2	1	no	—
115	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
116	#499	—	2	2	no	—
117	#499	C₂²×C₁₃	1	4	yes	complemented
118	#495	—	2	2	no	—
119	#495	C₂²×C₁₃	1	4	yes	—
120	C₂×#107	C₂²×C₁₃	1	2	yes	—
121	C₂×#106	—	2	1	no	—
122	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
123	#358	C₂²×C₁₃	1	8	yes	complemented
124	#357	C₂²×C₁₃	1	8	yes	—
125	C₄×C₁₆	C₂²×C₁₃	1	4	yes	complemented
126	C₂²×C₁₃	—	8	2	no	—
127	C₂²×C₁₃	—	8	4	no	—
128	C₂²×C₁₃	—	16	4	no	—
129	C₂²×C₁₃	#486	1	2	yes	—
130	C₂²×C₁₃	#508	1	4	yes	—
131	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₁₃	—	2	4	no	—
133	C₄×C₁₃	—	8	2	no	—
134	C₄×C₁₃	—	8	4	no	—
135	C₄×C₁₃	#486	1	2	yes	—
136	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
137	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
138	C₂²×Q₈	—	2	1	no	—
139	C₂²×D₄	—	2	1	no	—
140	C₂×Q₁₆	—	2	1	no	—
141	C₂×Q₁₆	—	2	4	no	—
142	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
143	C₂×SD₁₆	—	2	4	no	—
144	C₂×SD₁₆	—	4	4	no	—
145	C₂×D₈	—	2	1	no	—
146	C₂×D₈	—	2	4	no	—
147	C₂×D₈	C₂³×C₁₃	1	2	yes	—
148	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
150	#125	—	2	4	no	—
151	#125	—	4	2	no	—
152	#124	—	2	4	no	—
153	#124	—	4	2	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	4	2	no	—
156	C₂×C₂²⋊C₄	—	4	2	no	—
157	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
160	#107	—	2	2	no	—
161	#107	C₂³×C₁₃	1	4	yes	—
162	#106	—	2	4	no	—
163	C₄×C₈	D₄×C₁₃	1	2	yes	—
164	C₄×C₈	C₂³×C₁₃	1	2	yes	—
165	C₂×C₁₃	—	16	2	no	—
166	C₂×C₁₃	#2242	1	4	yes	direct product factor
167	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
168	C₂×C₁₃	C₂×#508	1	2	yes	—
169	C₂⁴	—	4	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₁₃×D₈	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
182	Q₁₆	—	2	4	no	—
183	Q₁₆	—	4	2	no	—
184	SD₁₆	—	4	8	no	—
185	D₈	—	2	4	no	—
186	D₈	—	4	2	no	—
187	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
188	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
189	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
190	C₂×C₈	C₁₃×#41	1	4	yes	—
191	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	8	no	—
195	C₂²⋊C₄	—	4	8	no	—
196	C₄²	C₁₃×D₈	1	2	yes	—
197	C₄²	C₂×D₄×C₁₃	1	2	yes	—
198	C₁₆	—	2	4	no	—
199	C₁₃	C₂×#2242	1	1	yes	characteristic, direct product factor, Sylow
200	C₂³	—	4	2	no	—
201	C₂³	—	8	2	no	—
202	C₂³	—	8	4	no	—
203	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
204	Q₈	—	4	2	no	—
205	Q₈	—	4	2	no	—
206	Q₈	—	8	2	no	—
207	D₄	—	4	2	no	—
208	D₄	—	4	2	no	—
209	D₄	—	8	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	C₁₃×#124	1	2	yes	—
216	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
217	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
218	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
219	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
220	C₈	C₁₃×#124	1	2	yes	—
221	C₈	C₂×C₁₃×SD₁₆	1	4	yes	—
222	C₈	C₂×C₁₃×#41	1	2	yes	—
223	C₂²	—	8	2	no	—
224	C₂²	—	8	4	no	—
225	C₂²	—	16	4	no	—
226	C₂²	C₁₃×#486	1	2	yes	—
227	C₂²	C₁₃×#508	1	4	yes	—
228	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	8	2	no	—
231	C₄	—	8	4	no	—
232	C₄	C₁₃×#486	1	2	yes	—
233	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
234	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
235	C₂	—	16	2	no	—
236	C₂	C₁₃×#2242	1	4	yes	direct product factor
237	C₂	C₂×C₁₃×#486	1	1	yes	characteristic
238	C₂	C₂×C₁₃×#508	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#499	C₂	semidirect
3. C₁₃×#2242	C₂	direct (N × H)
4. C₂×C₁₃×#358	C₂	semidirect
6. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
14. C₁₃×#499	C₂²	semidirect
20. C₁₃×#358	C₂²	semidirect
22. C₄×C₁₃×C₁₆	C₂²	semidirect
50. C₂×#2242	C₁₃	direct (N × H)
81. C₂×#499	C₂×C₁₃	semidirect
83. #2242	C₂×C₁₃	direct (N × H)
84. C₂×#358	C₂×C₁₃	semidirect
86. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
117. #499	C₂²×C₁₃	semidirect
123. #358	C₂²×C₁₃	semidirect
125. C₄×C₁₆	C₂²×C₁₃	semidirect
166. C₂×C₁₃	#2242	direct (N × H)
199. C₁₃	C₂×#2242	direct (N × H)
236. C₂	C₁₃×#2242	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#499	2
C₂×C₁₃×#495	2
C₁₃×#2242	2
C₂×C₁₃×#358	2
C₂×C₁₃×#357	2
C₂×C₄×C₁₃×C₁₆	2
C₂×#2242	13

Subgroups — C2×C13×#2242 #15720035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2242 #15720035