Subgroups — C₂³×C₁₃×#135 #15749419

Order 3328.

11422 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×#41	C₂	1	2	yes	maximal, complemented
2	C₂²×C₁₃×#135	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₁₃×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₂³×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×#41	—	2	28	no	—
8	C₂²×C₁₃×#41	C₂²	1	56	yes	complemented
9	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
10	C₂³×D₄×C₁₃	—	2	2	no	—
11	C₂³×D₄×C₁₃	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₁₃	—	2	2	no	—
13	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#135	C₂²	1	448	yes	direct product factor
15	C₂²×C₁₃×Q₁₆	C₂²	1	28	yes	complemented
16	C₂²×C₁₃×SD₁₆	C₂²	1	56	yes	complemented
17	C₂²×C₁₃×D₈	C₂²	1	28	yes	complemented
18	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₁₃	C₂²	1	28	yes	complemented
21	C₂⁵×C₁₃	—	2	2	no	—
22	C₂⁵×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₁₃×#41	—	2	336	no	—
24	C₂×C₁₃×#41	C₂³	1	224	yes	complemented
25	C₂²×Q₈×C₁₃	—	2	14	no	—
26	C₂²×Q₈×C₁₃	C₂³	1	28	yes	—
27	C₂²×D₄×C₁₃	—	2	56	no	—
28	C₂²×D₄×C₁₃	—	2	14	no	—
29	C₂²×D₄×C₁₃	C₂³	1	28	yes	—
30	C₂³×C₄×C₁₃	—	2	2	no	—
31	C₂³×C₄×C₁₃	—	2	56	no	—
32	C₂³×C₄×C₁₃	D₄	1	14	yes	—
33	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
36	C₁₃×#135	C₂³	1	512	yes	direct product factor
37	C₂×C₁₃×Q₁₆	C₂³	1	112	yes	complemented
38	C₂×C₁₃×SD₁₆	C₂³	1	224	yes	complemented
39	C₂×C₁₃×D₈	C₂³	1	112	yes	complemented
40	C₂²×C₈×C₁₃	C₂³	1	14	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	14	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	112	yes	complemented
43	C₂³×#135	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂⁴×C₁₃	—	2	8	no	—
45	C₂⁴×C₁₃	—	2	28	no	—
46	C₂⁴×C₁₃	—	4	16	no	—
47	C₂⁴×C₁₃	C₂×D₄	1	14	yes	—
48	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₁₃×#41	—	2	448	no	—
50	C₁₃×#41	C₂⁴	1	128	yes	complemented
51	C₂×Q₈×C₁₃	—	2	84	no	—
52	C₂×Q₈×C₁₃	C₂⁴	1	56	yes	—
53	C₂×D₄×C₁₃	—	2	224	no	—
54	C₂×D₄×C₁₃	—	2	84	no	—
55	C₂×D₄×C₁₃	C₂⁴	1	56	yes	—
56	C₂²×C₄×C₁₃	—	2	28	no	—
57	C₂²×C₄×C₁₃	—	2	224	no	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	14	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	7	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	84	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	7	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
63	C₁₃×Q₁₆	C₂⁴	1	64	yes	complemented
64	C₁₃×SD₁₆	C₂⁴	1	128	yes	complemented
65	C₁₃×D₈	C₂⁴	1	64	yes	complemented
66	C₂×C₈×C₁₃	C₂⁴	1	28	yes	—
67	C₂×C₈×C₁₃	C₂⁴	1	28	yes	—
68	C₂×C₈×C₁₃	C₂⁴	1	64	yes	complemented
69	C₂³×#41	C₂×C₁₃	1	2	yes	complemented
70	C₂²×#135	C₂×C₁₃	1	56	yes	direct product factor
71	C₂³×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
72	C₂³×SD₁₆	C₂×C₁₃	1	2	yes	complemented
73	C₂³×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
74	C₂⁴×C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
75	C₂³×C₁₃	—	2	56	no	—
76	C₂³×C₁₃	—	2	56	no	—
77	C₂³×C₁₃	—	4	112	no	—
78	C₂³×C₁₃	#135	1	8	yes	direct product factor
79	C₂³×C₁₃	C₂²×D₄	1	28	yes	—
80	C₂³×C₁₃	C₂²×D₄	1	7	yes	—
81	Q₈×C₁₃	—	2	56	no	—
82	Q₈×C₁₃	C₂⁵	1	16	yes	—
83	D₄×C₁₃	—	2	128	no	—
84	D₄×C₁₃	—	2	56	no	—
85	D₄×C₁₃	C₂⁵	1	16	yes	—
86	C₂×C₄×C₁₃	—	2	56	no	—
87	C₂×C₄×C₁₃	—	2	128	no	—
88	C₂×C₄×C₁₃	C₂²×D₄	1	28	yes	—
89	C₂×C₄×C₁₃	C₂²×D₄	1	21	yes	—
90	C₂×C₄×C₁₃	C₂²×D₄	1	56	yes	—
91	C₂×C₄×C₁₃	C₂⁵	1	7	yes	—
92	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
93	C₈×C₁₃	C₂⁵	1	8	yes	—
94	C₈×C₁₃	C₂⁵	1	8	yes	—
95	C₂²×#41	—	2	28	no	—
96	C₂²×#41	C₂²×C₁₃	1	56	yes	complemented
97	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
98	C₂³×D₄	—	2	2	no	—
99	C₂³×D₄	C₂²×C₁₃	1	2	yes	—
100	C₂⁴×C₄	—	2	2	no	—
101	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
102	C₂×#135	C₂²×C₁₃	1	448	yes	direct product factor
103	C₂²×Q₁₆	C₂²×C₁₃	1	28	yes	complemented
104	C₂²×SD₁₆	C₂²×C₁₃	1	56	yes	complemented
105	C₂²×D₈	C₂²×C₁₃	1	28	yes	complemented
106	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
107	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
108	C₂³×C₈	C₂²×C₁₃	1	28	yes	complemented
109	C₂²×C₁₃	—	2	56	no	—
110	C₂²×C₁₃	—	2	16	no	—
111	C₂²×C₁₃	—	4	112	no	—
112	C₂²×C₁₃	C₂×#135	1	28	yes	direct product factor
113	C₂²×C₁₃	C₂³×D₄	1	8	yes	—
114	C₂²×C₁₃	C₂³×D₄	1	7	yes	—
115	C₄×C₁₃	—	2	16	no	—
116	C₄×C₁₃	C₂³×D₄	1	8	yes	—
117	C₄×C₁₃	C₂³×D₄	1	7	yes	—
118	C₄×C₁₃	C₂⁶	1	1	yes	characteristic
119	C₂⁵	—	2	2	no	—
120	C₂⁵	D₄×C₁₃	1	1	yes	characteristic
121	C₂×#41	—	2	336	no	—
122	C₂×#41	C₂³×C₁₃	1	224	yes	complemented
123	C₂²×Q₈	—	2	14	no	—
124	C₂²×Q₈	C₂³×C₁₃	1	28	yes	—
125	C₂²×D₄	—	2	56	no	—
126	C₂²×D₄	—	2	14	no	—
127	C₂²×D₄	C₂³×C₁₃	1	28	yes	—
128	C₂³×C₄	—	2	2	no	—
129	C₂³×C₄	—	2	56	no	—
130	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
131	C₂³×C₄	D₄×C₁₃	1	14	yes	—
132	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
133	C₂³×C₄	C₂³×C₁₃	1	14	yes	—
134	#135	C₂³×C₁₃	1	512	yes	direct product factor
135	C₂×Q₁₆	C₂³×C₁₃	1	112	yes	complemented
136	C₂×SD₁₆	C₂³×C₁₃	1	224	yes	complemented
137	C₂×D₈	C₂³×C₁₃	1	112	yes	complemented
138	C₂²×C₈	C₂³×C₁₃	1	14	yes	—
139	C₂²×C₈	C₂³×C₁₃	1	14	yes	—
140	C₂²×C₈	C₂³×C₁₃	1	112	yes	complemented
141	C₂×C₁₃	—	2	8	no	—
142	C₂×C₁₃	—	4	16	no	—
143	C₂×C₁₃	C₂²×#135	1	14	yes	direct product factor
144	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
145	C₂⁴	—	2	8	no	—
146	C₂⁴	—	2	28	no	—
147	C₂⁴	—	4	16	no	—
148	C₂⁴	C₂×D₄×C₁₃	1	14	yes	—
149	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
150	#41	—	2	448	no	—
151	#41	C₂⁴×C₁₃	1	128	yes	complemented
152	C₂×Q₈	—	2	84	no	—
153	C₂×Q₈	C₂⁴×C₁₃	1	56	yes	—
154	C₂×D₄	—	2	224	no	—
155	C₂×D₄	—	2	84	no	—
156	C₂×D₄	C₂⁴×C₁₃	1	56	yes	—
157	C₂²×C₄	—	2	28	no	—
158	C₂²×C₄	—	2	224	no	—
159	C₂²×C₄	C₂×D₄×C₁₃	1	14	yes	—
160	C₂²×C₄	C₂×D₄×C₁₃	1	7	yes	—
161	C₂²×C₄	C₂×D₄×C₁₃	1	84	yes	—
162	C₂²×C₄	C₂⁴×C₁₃	1	7	yes	—
163	C₂²×C₄	C₂⁴×C₁₃	1	28	yes	—
164	Q₁₆	C₂⁴×C₁₃	1	64	yes	complemented
165	SD₁₆	C₂⁴×C₁₃	1	128	yes	complemented
166	D₈	C₂⁴×C₁₃	1	64	yes	complemented
167	C₂×C₈	C₂⁴×C₁₃	1	28	yes	—
168	C₂×C₈	C₂⁴×C₁₃	1	28	yes	—
169	C₂×C₈	C₂⁴×C₁₃	1	64	yes	complemented
170	C₁₃	C₂³×#135	1	1	yes	characteristic, direct product factor, Sylow
171	C₂³	—	2	56	no	—
172	C₂³	—	2	56	no	—
173	C₂³	—	4	112	no	—
174	C₂³	C₁₃×#135	1	8	yes	direct product factor
175	C₂³	C₂²×D₄×C₁₃	1	28	yes	—
176	C₂³	C₂²×D₄×C₁₃	1	7	yes	—
177	Q₈	—	2	56	no	—
178	Q₈	C₂⁵×C₁₃	1	16	yes	—
179	D₄	—	2	128	no	—
180	D₄	—	2	56	no	—
181	D₄	C₂⁵×C₁₃	1	16	yes	—
182	C₂×C₄	—	2	56	no	—
183	C₂×C₄	—	2	128	no	—
184	C₂×C₄	C₂²×D₄×C₁₃	1	28	yes	—
185	C₂×C₄	C₂²×D₄×C₁₃	1	21	yes	—
186	C₂×C₄	C₂²×D₄×C₁₃	1	56	yes	—
187	C₂×C₄	C₂⁵×C₁₃	1	7	yes	—
188	C₂×C₄	C₂⁵×C₁₃	1	8	yes	—
189	C₈	C₂⁵×C₁₃	1	8	yes	—
190	C₈	C₂⁵×C₁₃	1	8	yes	—
191	C₂²	—	2	56	no	—
192	C₂²	—	2	16	no	—
193	C₂²	—	4	112	no	—
194	C₂²	C₂×C₁₃×#135	1	28	yes	direct product factor
195	C₂²	C₂³×D₄×C₁₃	1	8	yes	—
196	C₂²	C₂³×D₄×C₁₃	1	7	yes	—
197	C₄	—	2	16	no	—
198	C₄	C₂³×D₄×C₁₃	1	8	yes	—
199	C₄	C₂³×D₄×C₁₃	1	7	yes	—
200	C₄	C₂⁶×C₁₃	1	1	yes	characteristic
201	C₂	—	2	8	no	—
202	C₂	—	4	16	no	—
203	C₂	C₂²×C₁₃×#135	1	14	yes	direct product factor
204	C₂	C₂⁴×D₄×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₁₃	D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₁₃
socle	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₃ ◁ C₂⁴×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₃×#41	C₂	semidirect
2. C₂²×C₁₃×#135	C₂	direct (N × H)
3. C₂³×C₁₃×Q₁₆	C₂	semidirect
4. C₂³×C₁₃×SD₁₆	C₂	semidirect
5. C₂³×C₁₃×D₈	C₂	semidirect
6. C₂⁴×C₈×C₁₃	C₂	semidirect
8. C₂²×C₁₃×#41	C₂²	semidirect
14. C₂×C₁₃×#135	C₂²	direct (N × H)
15. C₂²×C₁₃×Q₁₆	C₂²	semidirect
16. C₂²×C₁₃×SD₁₆	C₂²	semidirect
17. C₂²×C₁₃×D₈	C₂²	semidirect
20. C₂³×C₈×C₁₃	C₂²	semidirect
24. C₂×C₁₃×#41	C₂³	semidirect
36. C₁₃×#135	C₂³	direct (N × H)
37. C₂×C₁₃×Q₁₆	C₂³	semidirect
38. C₂×C₁₃×SD₁₆	C₂³	semidirect
39. C₂×C₁₃×D₈	C₂³	semidirect
42. C₂²×C₈×C₁₃	C₂³	semidirect
43. C₂³×#135	C₁₃	direct (N × H)
50. C₁₃×#41	C₂⁴	semidirect
63. C₁₃×Q₁₆	C₂⁴	semidirect
64. C₁₃×SD₁₆	C₂⁴	semidirect
65. C₁₃×D₈	C₂⁴	semidirect
68. C₂×C₈×C₁₃	C₂⁴	semidirect
69. C₂³×#41	C₂×C₁₃	semidirect
70. C₂²×#135	C₂×C₁₃	direct (N × H)
71. C₂³×Q₁₆	C₂×C₁₃	semidirect
72. C₂³×SD₁₆	C₂×C₁₃	semidirect
73. C₂³×D₈	C₂×C₁₃	semidirect
74. C₂⁴×C₈	C₂×C₁₃	semidirect
78. C₂³×C₁₃	#135	direct (N × H)
96. C₂²×#41	C₂²×C₁₃	semidirect
102. C₂×#135	C₂²×C₁₃	direct (N × H)
103. C₂²×Q₁₆	C₂²×C₁₃	semidirect
104. C₂²×SD₁₆	C₂²×C₁₃	semidirect
105. C₂²×D₈	C₂²×C₁₃	semidirect
108. C₂³×C₈	C₂²×C₁₃	semidirect
112. C₂²×C₁₃	C₂×#135	direct (N × H)
122. C₂×#41	C₂³×C₁₃	semidirect
134. #135	C₂³×C₁₃	direct (N × H)
135. C₂×Q₁₆	C₂³×C₁₃	semidirect
136. C₂×SD₁₆	C₂³×C₁₃	semidirect
137. C₂×D₈	C₂³×C₁₃	semidirect
140. C₂²×C₈	C₂³×C₁₃	semidirect
143. C₂×C₁₃	C₂²×#135	direct (N × H)
151. #41	C₂⁴×C₁₃	semidirect
164. Q₁₆	C₂⁴×C₁₃	semidirect
165. SD₁₆	C₂⁴×C₁₃	semidirect
166. D₈	C₂⁴×C₁₃	semidirect
169. C₂×C₈	C₂⁴×C₁₃	semidirect
170. C₁₃	C₂³×#135	direct (N × H)
174. C₂³	C₁₃×#135	direct (N × H)
194. C₂²	C₂×C₁₃×#135	direct (N × H)
203. C₂	C₂²×C₁₃×#135	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×#41	2
C₂²×C₁₃×#135	2
C₂³×C₁₃×Q₁₆	2
C₂³×C₁₃×SD₁₆	2
C₂³×C₁₃×D₈	2
C₂⁴×C₈×C₁₃	2
C₂³×#135	13

Subgroups — C23×C13×#135 #15749419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₁₃×#135 #15749419