Subgroups — C₂×C₁₃×#2240 #15720033

Order 3328.

726 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2240	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#357	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×D₈	—	2	1	no	—
7	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
8	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₁₃×#127	—	2	1	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₁₃×#500	—	2	2	no	—
12	C₁₃×#500	C₂²	1	4	yes	complemented
13	C₁₃×#493	—	2	2	no	—
14	C₁₃×#493	C₂²	1	4	yes	—
15	C₂×C₁₃×#107	—	2	1	no	—
16	C₂×C₁₃×#107	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#357	C₂²	1	16	yes	complemented
19	C₄×C₁₃×C₁₆	C₂²	1	4	yes	complemented
20	C₂²×Q₈×C₁₃	—	4	1	no	—
21	C₂²×D₄×C₁₃	—	2	2	no	—
22	C₂²×D₄×C₁₃	—	4	1	no	—
23	C₂²×D₄×C₁₃	—	4	1	no	—
24	C₂×C₁₃×D₈	—	2	2	no	—
25	C₂×C₁₃×D₈	—	2	4	no	—
26	C₂×C₁₃×D₈	—	2	8	no	—
27	C₂×C₁₃×D₈	—	4	4	no	—
28	C₂×C₁₃×D₈	C₂³	1	4	yes	—
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₁₃×C₄⋊Q₈	—	2	4	no	—
32	C₁₃×C₄⋊Q₈	—	4	2	no	—
33	C₁₃×#127	—	2	4	no	—
34	C₁₃×#127	—	4	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₁₃×C₁₆	—	2	4	no	—
39	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
40	C₁₃×#107	—	2	4	no	—
41	C₁₃×#107	—	2	2	no	—
42	C₁₃×#107	C₂³	1	4	yes	—
43	C₄×C₈×C₁₃	D₄	1	2	yes	—
44	C₄×C₈×C₁₃	C₂³	1	2	yes	—
45	C₂×#2240	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂⁴×C₁₃	—	4	2	no	—
47	C₂×Q₈×C₁₃	—	4	4	no	—
48	C₂×Q₈×C₁₃	—	8	4	no	—
49	C₂×D₄×C₁₃	—	2	4	no	—
50	C₂×D₄×C₁₃	—	4	4	no	—
51	C₂×D₄×C₁₃	—	4	4	no	—
52	C₂×D₄×C₁₃	—	4	4	no	—
53	C₂×D₄×C₁₃	—	4	2	no	—
54	C₂×D₄×C₁₃	—	4	4	no	—
55	C₂×D₄×C₁₃	—	8	4	no	—
56	C₂×D₄×C₁₃	—	8	4	no	—
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₁₃×D₈	—	2	8	no	—
62	C₁₃×D₈	—	4	4	no	—
63	C₁₃×D₈	—	4	8	no	—
64	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
65	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₃	#41	1	4	yes	—
68	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₄⋊C₄	—	4	2	no	—
71	C₁₃×C₄⋊C₄	—	4	4	no	—
72	C₄²×C₁₃	D₈	1	2	yes	—
73	C₄²×C₁₃	C₂×D₄	1	2	yes	—
74	C₁₃×C₁₆	—	2	4	no	—
75	C₂×#500	C₂×C₁₃	1	1	yes	characteristic, complemented
76	C₂×#493	C₂×C₁₃	1	1	yes	characteristic
77	#2240	C₂×C₁₃	1	8	yes	direct product factor
78	C₂×#357	C₂×C₁₃	1	4	yes	complemented
79	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
80	C₂³×C₁₃	—	4	4	no	—
81	C₂³×C₁₃	—	8	4	no	—
82	C₂³×C₁₃	—	8	4	no	—
83	C₂³×C₁₃	—	8	4	no	—
84	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₁₃	—	8	8	no	—
86	D₄×C₁₃	—	4	4	no	—
87	D₄×C₁₃	—	4	4	no	—
88	D₄×C₁₃	—	8	4	no	—
89	D₄×C₁₃	—	8	8	no	—
90	D₄×C₁₃	—	8	8	no	—
91	C₂×C₄×C₁₃	—	4	4	no	—
92	C₂×C₄×C₁₃	—	8	4	no	—
93	C₂×C₄×C₁₃	D₁₆	1	4	yes	—
94	C₂×C₄×C₁₃	SD₃₂	1	4	yes	—
95	C₂×C₄×C₁₃	#124	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
100	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₁₃	#124	1	2	yes	—
102	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
103	C₈×C₁₃	C₂×#41	1	2	yes	—
104	C₂²×D₈	—	2	1	no	—
105	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
106	C₂×C₄⋊Q₈	—	2	1	no	—
107	C₂×#127	—	2	1	no	—
108	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
109	#500	—	2	2	no	—
110	#500	C₂²×C₁₃	1	4	yes	complemented
111	#493	—	2	2	no	—
112	#493	C₂²×C₁₃	1	4	yes	—
113	C₂×#107	—	2	1	no	—
114	C₂×#107	C₂²×C₁₃	1	2	yes	—
115	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
116	#357	C₂²×C₁₃	1	16	yes	complemented
117	C₄×C₁₆	C₂²×C₁₃	1	4	yes	complemented
118	C₂²×C₁₃	—	8	4	no	—
119	C₂²×C₁₃	—	8	4	no	—
120	C₂²×C₁₃	—	8	4	no	—
121	C₂²×C₁₃	—	16	8	no	—
122	C₂²×C₁₃	#486	1	2	yes	—
123	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
124	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
125	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
126	C₄×C₁₃	—	8	4	no	—
127	C₄×C₁₃	#486	1	2	yes	—
128	C₄×C₁₃	C₂×D₁₆	1	4	yes	—
129	C₄×C₁₃	C₂×SD₃₂	1	4	yes	—
130	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
131	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
132	C₂²×Q₈	—	4	1	no	—
133	C₂²×D₄	—	2	2	no	—
134	C₂²×D₄	—	4	1	no	—
135	C₂²×D₄	—	4	1	no	—
136	C₂×D₈	—	2	4	no	—
137	C₂×D₈	—	2	2	no	—
138	C₂×D₈	—	2	8	no	—
139	C₂×D₈	—	4	4	no	—
140	C₂×D₈	C₂³×C₁₃	1	4	yes	—
141	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	—	4	2	no	—
145	#127	—	2	4	no	—
146	#127	—	4	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	4	1	no	—
149	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
150	C₂×C₁₆	—	2	4	no	—
151	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
152	#107	—	2	4	no	—
153	#107	—	2	2	no	—
154	#107	C₂³×C₁₃	1	4	yes	—
155	C₄×C₈	D₄×C₁₃	1	2	yes	—
156	C₄×C₈	C₂³×C₁₃	1	2	yes	—
157	C₂×C₁₃	—	16	4	no	—
158	C₂×C₁₃	#2240	1	4	yes	direct product factor
159	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
160	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
161	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
162	C₂⁴	—	4	2	no	—
163	C₂×Q₈	—	4	4	no	—
164	C₂×Q₈	—	8	4	no	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	4	4	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	—	4	4	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	4	no	—
171	C₂×D₄	—	8	4	no	—
172	C₂×D₄	—	8	4	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
175	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
177	D₈	—	2	8	no	—
178	D₈	—	4	4	no	—
179	D₈	—	4	8	no	—
180	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
181	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
183	C₂×C₈	C₁₃×#41	1	4	yes	—
184	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄²	C₁₃×D₈	1	2	yes	—
189	C₄²	C₂×D₄×C₁₃	1	2	yes	—
190	C₁₆	—	2	4	no	—
191	C₁₃	C₂×#2240	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³	—	4	4	no	—
193	C₂³	—	8	4	no	—
194	C₂³	—	8	4	no	—
195	C₂³	—	8	4	no	—
196	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
197	Q₈	—	8	8	no	—
198	D₄	—	4	4	no	—
199	D₄	—	4	4	no	—
200	D₄	—	8	4	no	—
201	D₄	—	8	8	no	—
202	D₄	—	8	8	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	C₁₃×D₁₆	1	4	yes	—
206	C₂×C₄	C₁₃×SD₃₂	1	4	yes	—
207	C₂×C₄	C₁₃×#124	1	2	yes	—
208	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
209	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
210	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
211	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
212	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
213	C₈	C₁₃×#124	1	2	yes	—
214	C₈	C₂×C₁₃×SD₁₆	1	4	yes	—
215	C₈	C₂×C₁₃×#41	1	2	yes	—
216	C₂²	—	8	4	no	—
217	C₂²	—	8	4	no	—
218	C₂²	—	8	4	no	—
219	C₂²	—	16	8	no	—
220	C₂²	C₁₃×#486	1	2	yes	—
221	C₂²	C₂×C₁₃×D₁₆	1	2	yes	—
222	C₂²	C₂×C₁₃×SD₃₂	1	2	yes	—
223	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
224	C₄	—	8	4	no	—
225	C₄	C₁₃×#486	1	2	yes	—
226	C₄	C₂×C₁₃×D₁₆	1	4	yes	—
227	C₄	C₂×C₁₃×SD₃₂	1	4	yes	—
228	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
229	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
230	C₂	—	16	4	no	—
231	C₂	C₁₃×#2240	1	4	yes	direct product factor
232	C₂	C₂×C₁₃×#486	1	1	yes	characteristic
233	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic
234	C₂	C₂²×C₁₃×SD₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#493	C₂
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×#500	C₂	semidirect
3. C₁₃×#2240	C₂	direct (N × H)
4. C₂×C₁₃×#357	C₂	semidirect
5. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
12. C₁₃×#500	C₂²	semidirect
18. C₁₃×#357	C₂²	semidirect
19. C₄×C₁₃×C₁₆	C₂²	semidirect
45. C₂×#2240	C₁₃	direct (N × H)
75. C₂×#500	C₂×C₁₃	semidirect
77. #2240	C₂×C₁₃	direct (N × H)
78. C₂×#357	C₂×C₁₃	semidirect
79. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
110. #500	C₂²×C₁₃	semidirect
116. #357	C₂²×C₁₃	semidirect
117. C₄×C₁₆	C₂²×C₁₃	semidirect
158. C₂×C₁₃	#2240	direct (N × H)
191. C₁₃	C₂×#2240	direct (N × H)
231. C₂	C₁₃×#2240	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×#500	2
C₂×C₁₃×#493	2
C₁₃×#2240	2
C₂×C₁₃×#357	2
C₂×C₄×C₁₃×C₁₆	2
C₂×#2240	13

Subgroups — C2×C13×#2240 #15720033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2240 #15720033