Subgroups — C₂×C₁₃×#2657 #15720961

Order 3328.

5142 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#127	C₂	1	3	yes	maximal, complemented
2	C₂²×C₁₃×#124	C₂	1	3	yes	maximal, complemented
3	C₁₃×#2657	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₃×#392	C₂	1	8	yes	maximal, complemented
5	C₂×C₁₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×D₄×C₁₃	C₂²	1	3	yes	—
8	C₂³×D₄×C₁₃	C₂²	1	12	yes	—
9	C₂×C₁₃×#127	—	2	24	no	—
10	C₂×C₁₃×#127	C₂²	1	24	yes	complemented
11	C₂×C₁₃×#124	—	2	24	no	—
12	C₂×C₁₃×#124	C₂²	1	24	yes	complemented
13	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	12	yes	—
14	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
15	C₁₃×#392	C₂²	1	64	yes	complemented
16	C₁₃×#376	C₂²	1	8	yes	complemented
17	C₂×C₁₃×#95	C₂²	1	4	yes	—
18	C₂⁵×C₁₃	C₂³	1	6	yes	—
19	C₂²×Q₈×C₁₃	—	2	12	no	—
20	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
21	C₂²×D₄×C₁₃	—	2	12	no	—
22	C₂²×D₄×C₁₃	—	2	48	no	—
23	C₂²×D₄×C₁₃	—	2	24	no	—
24	C₂²×D₄×C₁₃	—	4	48	no	—
25	C₂²×D₄×C₁₃	C₂³	1	12	yes	—
26	C₂²×D₄×C₁₃	C₂³	1	48	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₁₃×#127	—	2	96	no	—
31	C₁₃×#124	—	2	96	no	—
32	C₂×C₁₃×C₂²⋊C₄	—	2	48	no	—
33	C₂×C₁₃×C₂²⋊C₄	C₂³	1	48	yes	—
34	C₂×C₄²×C₁₃	D₄	1	24	yes	complemented
35	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
36	C₁₃×#95	C₂³	1	16	yes	—
37	C₂×#2657	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂⁴×C₁₃	—	2	12	no	—
39	C₂⁴×C₁₃	—	2	24	no	—
40	C₂⁴×C₁₃	—	2	48	no	—
41	C₂⁴×C₁₃	C₂⁴	1	12	yes	—
42	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
43	C₂×Q₈×C₁₃	—	2	24	no	—
44	C₂×Q₈×C₁₃	—	4	16	no	—
45	C₂×D₄×C₁₃	—	2	24	no	—
46	C₂×D₄×C₁₃	—	2	96	no	—
47	C₂×D₄×C₁₃	—	2	48	no	—
48	C₂×D₄×C₁₃	—	4	96	no	—
49	C₂×D₄×C₁₃	—	4	192	no	—
50	C₂×D₄×C₁₃	—	4	48	no	—
51	C₂²×C₄×C₁₃	—	2	24	no	—
52	C₂²×C₄×C₁₃	—	2	12	no	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
55	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
57	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
58	C₁₃×C₂²⋊C₄	—	2	96	no	—
59	C₄²×C₁₃	C₂×D₄	1	48	yes	complemented
60	C₂²×#127	C₂×C₁₃	1	3	yes	complemented
61	C₂²×#124	C₂×C₁₃	1	3	yes	complemented
62	#2657	C₂×C₁₃	1	16	yes	direct product factor
63	C₂×#392	C₂×C₁₃	1	8	yes	complemented
64	C₂×#376	C₂×C₁₃	1	1	yes	characteristic, complemented
65	C₂³×C₁₃	—	2	12	no	—
66	C₂³×C₁₃	—	2	24	no	—
67	C₂³×C₁₃	—	2	48	no	—
68	C₂³×C₁₃	—	4	24	no	—
69	C₂³×C₁₃	—	4	96	no	—
70	C₂³×C₁₃	—	4	48	no	—
71	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
72	C₂³×C₁₃	H(5,𝔽₂)	1	8	yes	—
73	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
74	Q₈×C₁₃	—	4	16	no	—
75	D₄×C₁₃	—	4	192	no	—
76	D₄×C₁₃	—	4	48	no	—
77	C₂×C₄×C₁₃	—	2	48	no	—
78	C₂×C₄×C₁₃	—	2	24	no	—
79	C₂×C₄×C₁₃	—	2	12	no	—
80	C₂×C₄×C₁₃	—	4	8	no	—
81	C₂×C₄×C₁₃	#127	1	16	yes	—
82	C₂×C₄×C₁₃	C₂²×D₄	1	24	yes	—
83	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
84	C₂³×Q₈	C₂²×C₁₃	1	1	yes	characteristic
85	C₂³×D₄	C₂²×C₁₃	1	3	yes	—
86	C₂³×D₄	C₂²×C₁₃	1	12	yes	—
87	C₂×#127	—	2	24	no	—
88	C₂×#127	C₂²×C₁₃	1	24	yes	complemented
89	C₂×#124	—	2	24	no	—
90	C₂×#124	C₂²×C₁₃	1	24	yes	complemented
91	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	12	yes	—
92	C₂²×C₄²	C₂²×C₁₃	1	3	yes	—
93	#392	C₂²×C₁₃	1	64	yes	complemented
94	#376	C₂²×C₁₃	1	8	yes	complemented
95	C₂×#95	C₂²×C₁₃	1	4	yes	—
96	C₂²×C₁₃	—	4	12	no	—
97	C₂²×C₁₃	—	4	48	no	—
98	C₂²×C₁₃	—	4	24	no	—
99	C₂²×C₁₃	—	8	48	no	—
100	C₂²×C₁₃	C₂×#127	1	4	yes	—
101	C₂²×C₁₃	#535	1	24	yes	—
102	C₂²×C₁₃	C₂³×D₄	1	3	yes	—
103	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	4	yes	—
104	C₄×C₁₃	—	2	24	no	—
105	C₄×C₁₃	—	4	4	no	—
106	C₄×C₁₃	C₂×#127	1	8	yes	—
107	C₂⁵	C₂³×C₁₃	1	6	yes	—
108	C₂²×Q₈	—	2	12	no	—
109	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
110	C₂²×D₄	—	2	12	no	—
111	C₂²×D₄	—	2	48	no	—
112	C₂²×D₄	—	2	24	no	—
113	C₂²×D₄	—	4	48	no	—
114	C₂²×D₄	C₂³×C₁₃	1	12	yes	—
115	C₂²×D₄	C₂³×C₁₃	1	48	yes	—
116	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
117	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
118	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
119	#127	—	2	96	no	—
120	#124	—	2	96	no	—
121	C₂×C₂²⋊C₄	—	2	48	no	—
122	C₂×C₂²⋊C₄	C₂³×C₁₃	1	48	yes	—
123	C₂×C₄²	D₄×C₁₃	1	24	yes	complemented
124	C₂×C₄²	C₂³×C₁₃	1	12	yes	—
125	#95	C₂³×C₁₃	1	16	yes	—
126	C₂×C₁₃	—	8	12	no	—
127	C₂×C₁₃	#2657	1	8	yes	direct product factor
128	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
129	C₂×C₁₃	C₂×#535	1	6	yes	—
130	C₂⁴	—	2	12	no	—
131	C₂⁴	—	2	24	no	—
132	C₂⁴	—	2	48	no	—
133	C₂⁴	C₂⁴×C₁₃	1	12	yes	—
134	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
135	C₂×Q₈	—	2	24	no	—
136	C₂×Q₈	—	4	16	no	—
137	C₂×D₄	—	2	24	no	—
138	C₂×D₄	—	2	96	no	—
139	C₂×D₄	—	2	48	no	—
140	C₂×D₄	—	4	96	no	—
141	C₂×D₄	—	4	192	no	—
142	C₂×D₄	—	4	48	no	—
143	C₂²×C₄	—	2	24	no	—
144	C₂²×C₄	—	2	12	no	—
145	C₂²×C₄	C₂×D₄×C₁₃	1	24	yes	—
146	C₂²×C₄	C₂×D₄×C₁₃	1	12	yes	—
147	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
148	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
149	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
150	C₂²⋊C₄	—	2	96	no	—
151	C₄²	C₂×D₄×C₁₃	1	48	yes	complemented
152	C₁₃	C₂×#2657	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³	—	2	12	no	—
154	C₂³	—	2	24	no	—
155	C₂³	—	2	48	no	—
156	C₂³	—	4	24	no	—
157	C₂³	—	4	96	no	—
158	C₂³	—	4	48	no	—
159	C₂³	C₂²×D₄×C₁₃	1	6	yes	—
160	C₂³	C₁₃×H(5,𝔽₂)	1	8	yes	—
161	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
162	Q₈	—	4	16	no	—
163	D₄	—	4	192	no	—
164	D₄	—	4	48	no	—
165	C₂×C₄	—	2	48	no	—
166	C₂×C₄	—	2	24	no	—
167	C₂×C₄	—	2	12	no	—
168	C₂×C₄	—	4	8	no	—
169	C₂×C₄	C₁₃×#127	1	16	yes	—
170	C₂×C₄	C₂²×D₄×C₁₃	1	24	yes	—
171	C₂×C₄	C₂²×D₄×C₁₃	1	12	yes	—
172	C₂²	—	4	12	no	—
173	C₂²	—	4	48	no	—
174	C₂²	—	4	24	no	—
175	C₂²	—	8	48	no	—
176	C₂²	C₂×C₁₃×#127	1	4	yes	—
177	C₂²	C₁₃×#535	1	24	yes	—
178	C₂²	C₂³×D₄×C₁₃	1	3	yes	—
179	C₂²	C₂×C₁₃×H(5,𝔽₂)	1	4	yes	—
180	C₄	—	2	24	no	—
181	C₄	—	4	4	no	—
182	C₄	C₂×C₁₃×#127	1	8	yes	—
183	C₂	—	8	12	no	—
184	C₂	C₁₃×#2657	1	8	yes	direct product factor
185	C₂	C₂²×C₁₃×#127	1	1	yes	characteristic
186	C₂	C₂×C₁₃×#535	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×#127	C₂	semidirect
2. C₂²×C₁₃×#124	C₂	semidirect
3. C₁₃×#2657	C₂	direct (N × H)
4. C₂×C₁₃×#392	C₂	semidirect
5. C₂×C₁₃×#376	C₂	semidirect
10. C₂×C₁₃×#127	C₂²	semidirect
12. C₂×C₁₃×#124	C₂²	semidirect
15. C₁₃×#392	C₂²	semidirect
16. C₁₃×#376	C₂²	semidirect
34. C₂×C₄²×C₁₃	D₄	semidirect
37. C₂×#2657	C₁₃	direct (N × H)
59. C₄²×C₁₃	C₂×D₄	semidirect
60. C₂²×#127	C₂×C₁₃	semidirect
61. C₂²×#124	C₂×C₁₃	semidirect
62. #2657	C₂×C₁₃	direct (N × H)
63. C₂×#392	C₂×C₁₃	semidirect
64. C₂×#376	C₂×C₁₃	semidirect
88. C₂×#127	C₂²×C₁₃	semidirect
90. C₂×#124	C₂²×C₁₃	semidirect
93. #392	C₂²×C₁₃	semidirect
94. #376	C₂²×C₁₃	semidirect
123. C₂×C₄²	D₄×C₁₃	semidirect
127. C₂×C₁₃	#2657	direct (N × H)
151. C₄²	C₂×D₄×C₁₃	semidirect
152. C₁₃	C₂×#2657	direct (N × H)
184. C₂	C₁₃×#2657	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×#127	2
C₂²×C₁₃×#124	2
C₁₃×#2657	2
C₂×C₁₃×#392	2
C₂×C₁₃×#376	2
C₂×#2657	13

Subgroups — C2×C13×#2657 #15720961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2657 #15720961