Subgroups — C₂×C₁₃×#3407 #15748962

Order 3328.

2074 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3407	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#504	C₂	1	8	yes	maximal, complemented
4	C₂²×C₁₃×M₃₂	C₂	1	6	yes	maximal, complemented
5	C₂²×C₁₃×#41	C₄	1	1	yes	characteristic
6	C₂×C₁₃×#131	C₄	1	4	yes	—
7	C₂×C₁₃×#131	C₄	1	8	yes	—
8	C₂×C₁₃×#131	C₂²	1	4	yes	—
9	C₂×C₁₃×#131	C₂²	1	8	yes	—
10	C₂²×C₁₃×M₁₆	C₄	1	3	yes	—
11	C₂³×C₈×C₁₃	C₂²	1	3	yes	—
12	C₁₃×#504	C₂²	1	64	yes	complemented
13	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
14	C₂×C₁₃×M₃₂	C₂²	1	12	yes	complemented
15	C₂×C₁₃×M₃₂	C₂²	1	48	yes	complemented
16	C₂²×C₁₃×C₁₆	C₂²	1	12	yes	complemented
17	C₂×C₁₃×#41	C₈	1	4	yes	—
18	C₂×C₁₃×#41	C₈	1	8	yes	—
19	C₂×C₁₃×#41	C₂×C₄	1	4	yes	—
20	C₂×C₁₃×#41	C₂×C₄	1	8	yes	—
21	C₂²×Q₈×C₁₃	C₈	1	1	yes	characteristic
22	C₂²×D₄×C₁₃	C₈	1	3	yes	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
24	C₁₃×#131	C₂×C₄	1	16	yes	—
25	C₁₃×#131	C₂×C₄	1	32	yes	—
26	C₁₃×#131	C₂³	1	16	yes	—
27	C₂×C₁₃×M₁₆	C₂×C₄	1	12	yes	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	24	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	6	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	12	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×M₃₂	C₂³	1	16	yes	—
36	C₁₃×M₃₂	C₂³	1	48	yes	complemented
37	C₂×C₁₃×C₁₆	C₂³	1	8	yes	—
38	C₂×C₁₃×C₁₆	C₂³	1	48	yes	complemented
39	C₂×#3407	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂⁴×C₁₃	C₂×C₈	1	3	yes	—
41	C₁₃×#41	C₂×C₈	1	16	yes	—
42	C₁₃×#41	C₂×C₈	1	32	yes	—
43	C₁₃×#41	C₂²×C₄	1	16	yes	—
44	C₂×Q₈×C₁₃	C₂×C₈	1	8	yes	—
45	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
46	C₂×D₄×C₁₃	C₂×C₈	1	12	yes	—
47	C₂×D₄×C₁₃	C₂×C₈	1	24	yes	—
48	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
49	C₂²×C₄×C₁₃	C₂×C₈	1	6	yes	—
50	C₂²×C₄×C₁₃	C₂×C₈	1	12	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
54	C₁₃×M₁₆	C₂²×C₄	1	48	yes	—
55	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
58	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
59	C₂×C₈×C₁₃	C₂²×C₄	1	24	yes	—
60	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
61	C₂×C₈×C₁₃	C₂⁴	1	12	yes	—
62	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
63	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₁₃×C₁₆	C₂⁴	1	16	yes	—
65	C₂²×#131	C₂×C₁₃	1	1	yes	characteristic
66	#3407	C₂×C₁₃	1	16	yes	direct product factor
67	C₂×#504	C₂×C₁₃	1	8	yes	complemented
68	C₂²×M₃₂	C₂×C₁₃	1	6	yes	complemented
69	C₂³×C₁₃	—	4	6	no	—
70	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
71	C₂³×C₁₃	C₂²×C₈	1	6	yes	—
72	C₂³×C₁₃	C₂²×C₈	1	12	yes	—
73	Q₈×C₁₃	C₂²×C₈	1	16	yes	—
74	D₄×C₁₃	C₂²×C₈	1	48	yes	—
75	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
78	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
79	C₂×C₄×C₁₃	C₂²×C₈	1	24	yes	—
80	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
81	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
82	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
84	C₈×C₁₃	C₂³×C₄	1	2	yes	—
85	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
86	C₈×C₁₃	C₂³×C₄	1	12	yes	—
87	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
88	C₂²×#41	C₄×C₁₃	1	1	yes	characteristic
89	C₂×#131	C₄×C₁₃	1	4	yes	—
90	C₂×#131	C₄×C₁₃	1	8	yes	—
91	C₂×#131	C₂²×C₁₃	1	4	yes	—
92	C₂×#131	C₂²×C₁₃	1	8	yes	—
93	C₂²×M₁₆	C₄×C₁₃	1	3	yes	—
94	C₂³×C₈	C₂²×C₁₃	1	3	yes	—
95	#504	C₂²×C₁₃	1	64	yes	complemented
96	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
97	C₂×M₃₂	C₂²×C₁₃	1	12	yes	complemented
98	C₂×M₃₂	C₂²×C₁₃	1	48	yes	complemented
99	C₂²×C₁₆	C₂²×C₁₃	1	12	yes	complemented
100	C₂²×C₁₃	—	2	2	no	—
101	C₂²×C₁₃	—	2	12	no	—
102	C₂²×C₁₃	—	4	12	no	—
103	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
104	C₂²×C₁₃	C₂³×C₈	1	12	yes	—
105	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
106	C₄×C₁₃	C₂³×C₈	1	2	yes	—
107	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
108	C₄×C₁₃	C₂³×C₈	1	12	yes	—
109	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
110	C₂×#41	C₈×C₁₃	1	4	yes	—
111	C₂×#41	C₈×C₁₃	1	8	yes	—
112	C₂×#41	C₂×C₄×C₁₃	1	4	yes	—
113	C₂×#41	C₂×C₄×C₁₃	1	8	yes	—
114	C₂²×Q₈	C₈×C₁₃	1	1	yes	characteristic
115	C₂²×D₄	C₈×C₁₃	1	3	yes	—
116	C₂³×C₄	C₂×C₄×C₁₃	1	3	yes	—
117	#131	C₂×C₄×C₁₃	1	16	yes	—
118	#131	C₂×C₄×C₁₃	1	32	yes	—
119	#131	C₂³×C₁₃	1	16	yes	—
120	C₂×M₁₆	C₂×C₄×C₁₃	1	12	yes	—
121	C₂×M₁₆	C₂×C₄×C₁₃	1	24	yes	—
122	C₂²×C₈	C₂×C₄×C₁₃	1	3	yes	—
123	C₂²×C₈	C₂×C₄×C₁₃	1	6	yes	—
124	C₂²×C₈	C₂×C₄×C₁₃	1	12	yes	—
125	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₁₃	1	6	yes	—
127	C₂²×C₈	C₂³×C₁₃	1	12	yes	—
128	M₃₂	C₂³×C₁₃	1	16	yes	—
129	M₃₂	C₂³×C₁₃	1	48	yes	complemented
130	C₂×C₁₆	C₂³×C₁₃	1	8	yes	—
131	C₂×C₁₆	C₂³×C₁₃	1	48	yes	complemented
132	C₂×C₁₃	—	2	2	no	—
133	C₂×C₁₃	—	2	12	no	—
134	C₂×C₁₃	#3407	1	2	yes	direct product factor
135	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
136	C₂⁴	C₂×C₈×C₁₃	1	3	yes	—
137	#41	C₂×C₈×C₁₃	1	16	yes	—
138	#41	C₂×C₈×C₁₃	1	32	yes	—
139	#41	C₂²×C₄×C₁₃	1	16	yes	—
140	C₂×Q₈	C₂×C₈×C₁₃	1	8	yes	—
141	C₂×Q₈	C₂×C₈×C₁₃	1	4	yes	—
142	C₂×D₄	C₂×C₈×C₁₃	1	12	yes	—
143	C₂×D₄	C₂×C₈×C₁₃	1	24	yes	—
144	C₂²×C₄	C₂×C₈×C₁₃	1	3	yes	—
145	C₂²×C₄	C₂×C₈×C₁₃	1	6	yes	—
146	C₂²×C₄	C₂×C₈×C₁₃	1	12	yes	—
147	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
148	C₂²×C₄	C₂²×C₄×C₁₃	1	6	yes	—
149	C₂²×C₄	C₂²×C₄×C₁₃	1	12	yes	—
150	M₁₆	C₂²×C₄×C₁₃	1	48	yes	—
151	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
152	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
153	C₂×C₈	C₂²×C₄×C₁₃	1	6	yes	—
154	C₂×C₈	C₂²×C₄×C₁₃	1	12	yes	—
155	C₂×C₈	C₂²×C₄×C₁₃	1	12	yes	—
156	C₂×C₈	C₂²×C₄×C₁₃	1	24	yes	—
157	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
158	C₂×C₈	C₂⁴×C₁₃	1	12	yes	—
159	C₂×C₈	C₂⁴×C₁₃	1	2	yes	—
160	C₁₆	C₂⁴×C₁₃	1	16	yes	—
161	C₁₃	C₂×#3407	1	1	yes	characteristic, direct product factor, Sylow
162	C₂³	—	4	6	no	—
163	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
164	C₂³	C₂²×C₈×C₁₃	1	6	yes	—
165	C₂³	C₂²×C₈×C₁₃	1	12	yes	—
166	Q₈	C₂²×C₈×C₁₃	1	16	yes	—
167	D₄	C₂²×C₈×C₁₃	1	48	yes	—
168	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
169	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₈×C₁₃	1	12	yes	—
171	C₂×C₄	C₂²×C₈×C₁₃	1	6	yes	—
172	C₂×C₄	C₂²×C₈×C₁₃	1	24	yes	—
173	C₂×C₄	C₂²×C₈×C₁₃	1	12	yes	—
174	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
175	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
176	C₂×C₄	C₂³×C₄×C₁₃	1	12	yes	—
177	C₈	C₂³×C₄×C₁₃	1	2	yes	—
178	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
179	C₈	C₂³×C₄×C₁₃	1	12	yes	—
180	C₈	C₂⁵×C₁₃	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	2	12	no	—
183	C₂²	—	4	12	no	—
184	C₂²	C₂³×C₈×C₁₃	1	2	yes	—
185	C₂²	C₂³×C₈×C₁₃	1	12	yes	—
186	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
187	C₄	C₂³×C₈×C₁₃	1	2	yes	—
188	C₄	C₂³×C₈×C₁₃	1	1	yes	characteristic
189	C₄	C₂³×C₈×C₁₃	1	12	yes	—
190	C₄	C₂⁴×C₄×C₁₃	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	12	no	—
193	C₂	C₁₃×#3407	1	2	yes	direct product factor
194	C₂	C₂⁴×C₈×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₃	C₂⁴
commutator	C₂	C₂⁴×C₈×C₁₃
Frattini, cosocle	C₈	C₂⁵×C₁₃
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂²×C₁₃×#41	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#3407	C₂	direct (N × H)
3. C₂×C₁₃×#504	C₂	semidirect
4. C₂²×C₁₃×M₃₂	C₂	semidirect
12. C₁₃×#504	C₂²	semidirect
14. C₂×C₁₃×M₃₂	C₂²	semidirect
15. C₂×C₁₃×M₃₂	C₂²	semidirect
16. C₂²×C₁₃×C₁₆	C₂²	semidirect
36. C₁₃×M₃₂	C₂³	semidirect
38. C₂×C₁₃×C₁₆	C₂³	semidirect
39. C₂×#3407	C₁₃	direct (N × H)
66. #3407	C₂×C₁₃	direct (N × H)
67. C₂×#504	C₂×C₁₃	semidirect
68. C₂²×M₃₂	C₂×C₁₃	semidirect
95. #504	C₂²×C₁₃	semidirect
97. C₂×M₃₂	C₂²×C₁₃	semidirect
98. C₂×M₃₂	C₂²×C₁₃	semidirect
99. C₂²×C₁₆	C₂²×C₁₃	semidirect
129. M₃₂	C₂³×C₁₃	semidirect
131. C₂×C₁₆	C₂³×C₁₃	semidirect
134. C₂×C₁₃	#3407	direct (N × H)
161. C₁₃	C₂×#3407	direct (N × H)
193. C₂	C₁₃×#3407	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×#131	2
C₁₃×#3407	2
C₂×C₁₃×#504	2
C₂²×C₁₃×M₃₂	2
C₂×#3407	13

Subgroups — C2×C13×#3407 #15748962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#3407 #15748962