Subgroups — C₂×C₁₃×#3415 #15748971

Order 3328.

2126 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×#576	C₂	1	2	yes	maximal, complemented
2	C₁₃×#3415	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#509	C₂	1	6	yes	maximal, complemented
4	C₂×C₁₃×#508	C₂	1	3	yes	maximal, complemented
5	C₂²×C₁₃×D₁₆	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×H(5,𝔽₂)	—	2	2	no	—
8	C₁₃×#576	—	2	8	no	—
9	C₁₃×#576	C₂²	1	16	yes	complemented
10	C₂×C₁₃×AGL(1,ℤ/8)	—	2	6	no	—
11	C₂×C₁₃×#135	C₂²	1	6	yes	complemented
12	C₂²×C₁₃×D₈	C₂²	1	6	yes	complemented
13	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
14	C₁₃×#509	C₂²	1	48	yes	complemented
15	C₁₃×#508	C₂²	1	24	yes	complemented
16	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₁₃×SD₃₂	C₂²	1	6	yes	complemented
18	C₂×C₁₃×D₁₆	C₂²	1	3	yes	complemented
19	C₂×C₁₃×D₁₆	C₂²	1	6	yes	complemented
20	C₂×C₁₃×D₁₆	C₂²	1	24	yes	complemented
21	C₁₃×#504	C₂²	1	8	yes	complemented
22	C₂×C₁₃×M₃₂	C₂²	1	3	yes	complemented
23	C₂²×C₁₃×C₁₆	C₂²	1	3	yes	complemented
24	C₁₃×H(5,𝔽₂)	—	2	16	no	—
25	C₁₃×H(5,𝔽₂)	—	4	8	no	—
26	C₂×C₁₃×#41	—	2	6	no	—
27	C₂×C₁₃×#41	—	4	2	no	—
28	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
29	C₂²×D₄×C₁₃	—	2	6	no	—
30	C₂²×D₄×C₁₃	—	4	6	no	—
31	C₁₃×AGL(1,ℤ/8)	—	2	48	no	—
32	C₁₃×#135	—	2	12	no	—
33	C₁₃×#135	C₂³	1	24	yes	complemented
34	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
35	C₂×C₁₃×SD₁₆	—	2	6	no	—
36	C₂×C₁₃×D₈	—	2	6	no	—
37	C₂×C₁₃×D₈	—	2	12	no	—
38	C₂×C₁₃×D₈	C₂³	1	6	yes	—
39	C₂×C₁₃×D₈	C₂³	1	24	yes	complemented
40	C₁₃×#131	D₄	1	4	yes	—
41	C₁₃×#131	C₂³	1	4	yes	—
42	C₂×C₁₃×M₁₆	D₄	1	3	yes	—
43	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
44	C₁₃×Q₃₂	C₂³	1	4	yes	complemented
45	C₁₃×SD₃₂	C₂³	1	24	yes	complemented
46	C₁₃×D₁₆	C₂³	1	12	yes	complemented
47	C₁₃×D₁₆	C₂³	1	24	yes	complemented
48	C₁₃×M₃₂	C₂³	1	12	yes	complemented
49	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
50	C₂×C₁₃×C₁₆	C₂³	1	3	yes	—
51	C₂×C₁₃×C₁₆	C₂³	1	12	yes	complemented
52	C₂×#3415	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	—	4	6	no	—
54	C₁₃×#41	—	2	24	no	—
55	C₁₃×#41	—	4	16	no	—
56	C₁₃×#41	—	4	12	no	—
57	C₁₃×#41	D₈	1	4	yes	—
58	C₁₃×#41	C₂×D₄	1	4	yes	—
59	C₂×Q₈×C₁₃	—	2	2	no	—
60	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
61	C₂×D₄×C₁₃	—	2	6	no	—
62	C₂×D₄×C₁₃	—	2	24	no	—
63	C₂×D₄×C₁₃	—	4	6	no	—
64	C₂×D₄×C₁₃	—	4	48	no	—
65	C₂×D₄×C₁₃	—	4	6	no	—
66	C₂×D₄×C₁₃	—	4	12	no	—
67	C₂×D₄×C₁₃	D₈	1	3	yes	—
68	C₂²×C₄×C₁₃	—	4	6	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
70	C₁₃×Q₁₆	—	2	2	no	—
71	C₁₃×Q₁₆	C₂⁴	1	4	yes	—
72	C₁₃×SD₁₆	—	2	24	no	—
73	C₁₃×D₈	—	2	6	no	—
74	C₁₃×D₈	—	2	24	no	—
75	C₁₃×D₈	C₂⁴	1	12	yes	—
76	C₁₃×M₁₆	C₂×D₄	1	12	yes	—
77	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
78	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
79	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
81	C₁₃×C₁₆	C₂⁴	1	2	yes	—
82	C₁₃×C₁₆	C₂⁴	1	6	yes	—
83	C₂×#576	C₂×C₁₃	1	2	yes	complemented
84	#3415	C₂×C₁₃	1	16	yes	direct product factor
85	C₂×#509	C₂×C₁₃	1	6	yes	complemented
86	C₂×#508	C₂×C₁₃	1	3	yes	complemented
87	C₂²×D₁₆	C₂×C₁₃	1	3	yes	complemented
88	C₂×#504	C₂×C₁₃	1	1	yes	characteristic, complemented
89	C₂³×C₁₃	—	4	6	no	—
90	C₂³×C₁₃	—	4	24	no	—
91	C₂³×C₁₃	—	16	12	no	—
92	C₂³×C₁₃	C₂×D₈	1	3	yes	—
93	Q₈×C₁₃	—	2	4	no	—
94	Q₈×C₁₃	—	4	2	no	—
95	Q₈×C₁₃	C₂×D₈	1	4	yes	—
96	D₄×C₁₃	—	2	12	no	—
97	D₄×C₁₃	—	4	24	no	—
98	D₄×C₁₃	—	4	24	no	—
99	D₄×C₁₃	—	4	6	no	—
100	D₄×C₁₃	C₂×D₈	1	12	yes	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	—	4	24	no	—
103	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
104	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
105	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
106	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₁₃	C₂²×D₄	1	6	yes	—
109	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
110	C₂×H(5,𝔽₂)	—	2	2	no	—
111	#576	—	2	8	no	—
112	#576	C₂²×C₁₃	1	16	yes	complemented
113	C₂×AGL(1,ℤ/8)	—	2	6	no	—
114	C₂×#135	C₂²×C₁₃	1	6	yes	complemented
115	C₂²×D₈	C₂²×C₁₃	1	6	yes	complemented
116	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
117	#509	C₂²×C₁₃	1	48	yes	complemented
118	#508	C₂²×C₁₃	1	24	yes	complemented
119	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic, complemented
120	C₂×SD₃₂	C₂²×C₁₃	1	6	yes	complemented
121	C₂×D₁₆	C₂²×C₁₃	1	3	yes	complemented
122	C₂×D₁₆	C₂²×C₁₃	1	6	yes	complemented
123	C₂×D₁₆	C₂²×C₁₃	1	24	yes	complemented
124	#504	C₂²×C₁₃	1	8	yes	complemented
125	C₂×M₃₂	C₂²×C₁₃	1	3	yes	complemented
126	C₂²×C₁₆	C₂²×C₁₃	1	3	yes	complemented
127	C₂²×C₁₃	—	2	6	no	—
128	C₂²×C₁₃	—	4	12	no	—
129	C₂²×C₁₃	—	8	12	no	—
130	C₂²×C₁₃	—	16	24	no	—
131	C₂²×C₁₃	C₂²×D₈	1	6	yes	—
132	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
133	C₄×C₁₃	—	4	4	no	—
134	C₄×C₁₃	C₂²×D₈	1	6	yes	—
135	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
137	H(5,𝔽₂)	—	2	16	no	—
138	H(5,𝔽₂)	—	4	8	no	—
139	C₂×#41	—	2	6	no	—
140	C₂×#41	—	4	2	no	—
141	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
142	C₂²×D₄	—	2	6	no	—
143	C₂²×D₄	—	4	6	no	—
144	AGL(1,ℤ/8)	—	2	48	no	—
145	#135	—	2	12	no	—
146	#135	C₂³×C₁₃	1	24	yes	complemented
147	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
148	C₂×SD₁₆	—	2	6	no	—
149	C₂×D₈	—	2	6	no	—
150	C₂×D₈	—	2	12	no	—
151	C₂×D₈	C₂³×C₁₃	1	6	yes	—
152	C₂×D₈	C₂³×C₁₃	1	24	yes	complemented
153	#131	D₄×C₁₃	1	4	yes	—
154	#131	C₂³×C₁₃	1	4	yes	—
155	C₂×M₁₆	D₄×C₁₃	1	3	yes	—
156	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
157	Q₃₂	C₂³×C₁₃	1	4	yes	complemented
158	SD₃₂	C₂³×C₁₃	1	24	yes	complemented
159	D₁₆	C₂³×C₁₃	1	12	yes	complemented
160	D₁₆	C₂³×C₁₃	1	24	yes	complemented
161	M₃₂	C₂³×C₁₃	1	12	yes	complemented
162	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
163	C₂×C₁₆	C₂³×C₁₃	1	3	yes	—
164	C₂×C₁₆	C₂³×C₁₃	1	12	yes	complemented
165	C₂×C₁₃	—	2	6	no	—
166	C₂×C₁₃	—	8	12	no	—
167	C₂×C₁₃	#3415	1	2	yes	direct product factor
168	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
169	C₂⁴	—	4	6	no	—
170	#41	—	2	24	no	—
171	#41	—	4	16	no	—
172	#41	—	4	12	no	—
173	#41	C₁₃×D₈	1	4	yes	—
174	#41	C₂×D₄×C₁₃	1	4	yes	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
177	C₂×D₄	—	2	6	no	—
178	C₂×D₄	—	2	24	no	—
179	C₂×D₄	—	4	6	no	—
180	C₂×D₄	—	4	48	no	—
181	C₂×D₄	—	4	6	no	—
182	C₂×D₄	—	4	12	no	—
183	C₂×D₄	C₁₃×D₈	1	3	yes	—
184	C₂²×C₄	—	4	6	no	—
185	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
186	Q₁₆	—	2	2	no	—
187	Q₁₆	C₂⁴×C₁₃	1	4	yes	—
188	SD₁₆	—	2	24	no	—
189	D₈	—	2	6	no	—
190	D₈	—	2	24	no	—
191	D₈	C₂⁴×C₁₃	1	12	yes	—
192	M₁₆	C₂×D₄×C₁₃	1	12	yes	—
193	C₂×C₈	C₂×D₄×C₁₃	1	3	yes	—
194	C₂×C₈	C₂×D₄×C₁₃	1	6	yes	—
195	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
196	C₂×C₈	C₂⁴×C₁₃	1	6	yes	—
197	C₁₆	C₂⁴×C₁₃	1	2	yes	—
198	C₁₆	C₂⁴×C₁₃	1	6	yes	—
199	C₁₃	C₂×#3415	1	1	yes	characteristic, direct product factor, Sylow
200	C₂³	—	4	6	no	—
201	C₂³	—	4	24	no	—
202	C₂³	—	16	12	no	—
203	C₂³	C₂×C₁₃×D₈	1	3	yes	—
204	Q₈	—	2	4	no	—
205	Q₈	—	4	2	no	—
206	Q₈	C₂×C₁₃×D₈	1	4	yes	—
207	D₄	—	2	12	no	—
208	D₄	—	4	24	no	—
209	D₄	—	4	24	no	—
210	D₄	—	4	6	no	—
211	D₄	C₂×C₁₃×D₈	1	12	yes	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	24	no	—
214	C₂×C₄	C₂×C₁₃×D₈	1	3	yes	—
215	C₂×C₄	C₂×C₁₃×D₈	1	6	yes	—
216	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
218	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
219	C₈	C₂²×D₄×C₁₃	1	6	yes	—
220	C₈	C₂⁵×C₁₃	1	1	yes	characteristic
221	C₂²	—	2	6	no	—
222	C₂²	—	4	12	no	—
223	C₂²	—	8	12	no	—
224	C₂²	—	16	24	no	—
225	C₂²	C₂²×C₁₃×D₈	1	6	yes	—
226	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
227	C₄	—	4	4	no	—
228	C₄	C₂²×C₁₃×D₈	1	6	yes	—
229	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
230	C₄	C₂³×D₄×C₁₃	1	1	yes	characteristic
231	C₂	—	2	6	no	—
232	C₂	—	8	12	no	—
233	C₂	C₁₃×#3415	1	2	yes	direct product factor
234	C₂	C₂³×C₁₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₁₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₁₃×#41 ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#576	C₂	semidirect
2. C₁₃×#3415	C₂	direct (N × H)
3. C₂×C₁₃×#509	C₂	semidirect
4. C₂×C₁₃×#508	C₂	semidirect
5. C₂²×C₁₃×D₁₆	C₂	semidirect
6. C₂×C₁₃×#504	C₂	semidirect
9. C₁₃×#576	C₂²	semidirect
11. C₂×C₁₃×#135	C₂²	semidirect
12. C₂²×C₁₃×D₈	C₂²	semidirect
14. C₁₃×#509	C₂²	semidirect
15. C₁₃×#508	C₂²	semidirect
16. C₂×C₁₃×Q₃₂	C₂²	semidirect
17. C₂×C₁₃×SD₃₂	C₂²	semidirect
18. C₂×C₁₃×D₁₆	C₂²	semidirect
19. C₂×C₁₃×D₁₆	C₂²	semidirect
20. C₂×C₁₃×D₁₆	C₂²	semidirect
21. C₁₃×#504	C₂²	semidirect
22. C₂×C₁₃×M₃₂	C₂²	semidirect
23. C₂²×C₁₃×C₁₆	C₂²	semidirect
33. C₁₃×#135	C₂³	semidirect
39. C₂×C₁₃×D₈	C₂³	semidirect
44. C₁₃×Q₃₂	C₂³	semidirect
45. C₁₃×SD₃₂	C₂³	semidirect
46. C₁₃×D₁₆	C₂³	semidirect
47. C₁₃×D₁₆	C₂³	semidirect
48. C₁₃×M₃₂	C₂³	semidirect
51. C₂×C₁₃×C₁₆	C₂³	semidirect
52. C₂×#3415	C₁₃	direct (N × H)
83. C₂×#576	C₂×C₁₃	semidirect
84. #3415	C₂×C₁₃	direct (N × H)
85. C₂×#509	C₂×C₁₃	semidirect
86. C₂×#508	C₂×C₁₃	semidirect
87. C₂²×D₁₆	C₂×C₁₃	semidirect
88. C₂×#504	C₂×C₁₃	semidirect
112. #576	C₂²×C₁₃	semidirect
114. C₂×#135	C₂²×C₁₃	semidirect
115. C₂²×D₈	C₂²×C₁₃	semidirect
117. #509	C₂²×C₁₃	semidirect
118. #508	C₂²×C₁₃	semidirect
119. C₂×Q₃₂	C₂²×C₁₃	semidirect
120. C₂×SD₃₂	C₂²×C₁₃	semidirect
121. C₂×D₁₆	C₂²×C₁₃	semidirect
122. C₂×D₁₆	C₂²×C₁₃	semidirect
123. C₂×D₁₆	C₂²×C₁₃	semidirect
124. #504	C₂²×C₁₃	semidirect
125. C₂×M₃₂	C₂²×C₁₃	semidirect
126. C₂²×C₁₆	C₂²×C₁₃	semidirect
146. #135	C₂³×C₁₃	semidirect
152. C₂×D₈	C₂³×C₁₃	semidirect
157. Q₃₂	C₂³×C₁₃	semidirect
158. SD₃₂	C₂³×C₁₃	semidirect
159. D₁₆	C₂³×C₁₃	semidirect
160. D₁₆	C₂³×C₁₃	semidirect
161. M₃₂	C₂³×C₁₃	semidirect
164. C₂×C₁₆	C₂³×C₁₃	semidirect
167. C₂×C₁₃	#3415	direct (N × H)
199. C₁₃	C₂×#3415	direct (N × H)
233. C₂	C₁₃×#3415	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×#576	2
C₁₃×#3415	2
C₂×C₁₃×#509	2
C₂×C₁₃×#508	2
C₂²×C₁₃×D₁₆	2
C₂×C₁₃×#504	2
C₂×#3415	13

Subgroups — C2×C13×#3415 #15748971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#3415 #15748971