Subgroups — C₂×C₁₃×#3458 #15749023

Order 3328.

7022 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	C₁₃×#3458	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₁₃×#544	C₂	1	12	yes	maximal, complemented
3	C₂×C₁₃×#543	C₂	1	4	yes	maximal
4	C₂×C₁₃×#533	C₂	1	12	yes	maximal, complemented
5	C₂²×C₁₃×#117	C₂	1	3	yes	maximal
6	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×#544	C₂²	1	192	yes	complemented
8	C₁₃×#543	C₂²	1	64	yes	—
9	C₁₃×#533	C₂²	1	192	yes	complemented
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	24	yes	complemented
11	C₂×C₁₃×#125	C₂²	1	24	yes	complemented
12	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	48	yes	—
13	C₂×C₄×Q₈×C₁₃	C₂²	1	16	yes	complemented
14	C₂×C₁₃×#117	C₂²	1	12	yes	—
15	C₂×C₁₃×#117	C₂²	1	12	yes	—
16	C₂×C₁₃×#117	C₂²	1	48	yes	—
17	C₂²×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
18	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	6	yes	—
19	C₂²×C₄²×C₁₃	C₂²	1	6	yes	—
20	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
21	C₂²×Q₈×C₁₃	C₂³	1	16	yes	—
22	C₂³×C₄×C₁₃	Q₈	1	16	yes	complemented
23	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	8	yes	—
27	C₁₃×C₄⋊Q₈	C₂³	1	192	yes	complemented
28	C₁₃×#125	C₂³	1	192	yes	complemented
29	C₁₃×C₂²⋊Q₈	C₂³	1	384	yes	—
30	C₄×Q₈×C₁₃	C₂³	1	128	yes	complemented
31	C₁₃×#117	C₂³	1	96	yes	—
32	C₁₃×#117	C₂³	1	96	yes	—
33	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
36	C₂×C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
37	C₂×C₁₃×C₂²⋊C₄	C₂³	1	48	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	48	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
41	C₂×#3458	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂⁴×C₁₃	—	4	4	no	—
43	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
44	C₂⁴×C₁₃	C₂⁴	1	3	yes	—
45	C₂⁴×C₁₃	C₂⁴	1	4	yes	—
46	C₂×Q₈×C₁₃	—	4	32	no	—
47	C₂×Q₈×C₁₃	C₂⁴	1	64	yes	—
48	C₂²×C₄×C₁₃	—	4	24	no	—
49	C₂²×C₄×C₁₃	C₂×Q₈	1	24	yes	—
50	C₂²×C₄×C₁₃	C₂×Q₈	1	24	yes	—
51	C₂²×C₄×C₁₃	C₂×Q₈	1	64	yes	complemented
52	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
53	C₂²×C₄×C₁₃	C₂⁴	1	48	yes	—
54	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
55	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
56	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
57	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₁₃×C₄⋊C₄	C₂⁴	1	96	yes	—
59	C₁₃×C₄⋊C₄	C₂⁴	1	192	yes	—
60	C₁₃×C₂²⋊C₄	C₂⁴	1	96	yes	—
61	C₄²×C₁₃	C₂⁴	1	48	yes	—
62	C₄²×C₁₃	C₂⁴	1	48	yes	—
63	#3458	C₂×C₁₃	1	32	yes	direct product factor
64	C₂×#544	C₂×C₁₃	1	12	yes	complemented
65	C₂×#543	C₂×C₁₃	1	4	yes	—
66	C₂×#533	C₂×C₁₃	1	12	yes	complemented
67	C₂²×#117	C₂×C₁₃	1	3	yes	—
68	C₂³×C₁₃	—	2	12	no	—
69	C₂³×C₁₃	—	4	16	no	—
70	C₂³×C₁₃	—	4	8	no	—
71	C₂³×C₁₃	C₂²×Q₈	1	12	yes	—
72	C₂³×C₁₃	C₂²×Q₈	1	16	yes	—
73	C₂³×C₁₃	C₂⁵	1	6	yes	—
74	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
75	Q₈×C₁₃	—	4	64	no	—
76	C₂×C₄×C₁₃	—	2	48	no	—
77	C₂×C₄×C₁₃	—	4	48	no	—
78	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
79	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
80	C₂×C₄×C₁₃	C₂²×Q₈	1	48	yes	—
81	C₂×C₄×C₁₃	C₂²×Q₈	1	48	yes	—
82	C₂×C₄×C₁₃	C₂⁵	1	48	yes	—
83	C₂×C₄×C₁₃	C₂⁵	1	2	yes	—
84	C₂×C₄×C₁₃	C₂⁵	1	6	yes	—
85	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
86	#544	C₂²×C₁₃	1	192	yes	complemented
87	#543	C₂²×C₁₃	1	64	yes	—
88	#533	C₂²×C₁₃	1	192	yes	complemented
89	C₂×C₄⋊Q₈	C₂²×C₁₃	1	24	yes	complemented
90	C₂×#125	C₂²×C₁₃	1	24	yes	complemented
91	C₂×C₂²⋊Q₈	C₂²×C₁₃	1	48	yes	—
92	C₂×C₄×Q₈	C₂²×C₁₃	1	16	yes	complemented
93	C₂×#117	C₂²×C₁₃	1	12	yes	—
94	C₂×#117	C₂²×C₁₃	1	12	yes	—
95	C₂×#117	C₂²×C₁₃	1	48	yes	—
96	C₂²×C₄⋊C₄	C₂²×C₁₃	1	6	yes	—
97	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	6	yes	—
98	C₂²×C₄²	C₂²×C₁₃	1	6	yes	—
99	C₂²×C₁₃	—	2	12	no	—
100	C₂²×C₁₃	—	2	24	no	—
101	C₂²×C₁₃	—	4	16	no	—
102	C₂²×C₁₃	C₂³×Q₈	1	12	yes	—
103	C₂²×C₁₃	C₂³×Q₈	1	2	yes	—
104	C₂²×C₁₃	#585	1	4	yes	—
105	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
106	C₄×C₁₃	—	2	48	no	—
107	C₄×C₁₃	C₂³×Q₈	1	12	yes	—
108	C₄×C₁₃	C₂³×Q₈	1	4	yes	—
109	C₂⁵	C₂³×C₁₃	1	1	yes	characteristic
110	C₂²×Q₈	C₂³×C₁₃	1	16	yes	—
111	C₂³×C₄	Q₈×C₁₃	1	16	yes	complemented
112	C₂³×C₄	C₂³×C₁₃	1	12	yes	—
113	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
114	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
115	C₂³×C₄	C₂³×C₁₃	1	8	yes	—
116	C₄⋊Q₈	C₂³×C₁₃	1	192	yes	complemented
117	#125	C₂³×C₁₃	1	192	yes	complemented
118	C₂²⋊Q₈	C₂³×C₁₃	1	384	yes	—
119	C₄×Q₈	C₂³×C₁₃	1	128	yes	complemented
120	#117	C₂³×C₁₃	1	96	yes	—
121	#117	C₂³×C₁₃	1	96	yes	—
122	C₂×C₄⋊C₄	C₂³×C₁₃	1	24	yes	—
123	C₂×C₄⋊C₄	C₂³×C₁₃	1	48	yes	—
124	C₂×C₄⋊C₄	C₂³×C₁₃	1	48	yes	—
125	C₂×C₂²⋊C₄	C₂³×C₁₃	1	24	yes	—
126	C₂×C₂²⋊C₄	C₂³×C₁₃	1	48	yes	—
127	C₂×C₄²	C₂³×C₁₃	1	12	yes	—
128	C₂×C₄²	C₂³×C₁₃	1	12	yes	—
129	C₂×C₄²	C₂³×C₁₃	1	48	yes	—
130	C₂×C₁₃	—	2	12	no	—
131	C₂×C₁₃	#3458	1	4	yes	direct product factor
132	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
133	C₂×C₁₃	C₂×#585	1	2	yes	—
134	C₂⁴	—	4	4	no	—
135	C₂⁴	C₂×Q₈×C₁₃	1	8	yes	—
136	C₂⁴	C₂⁴×C₁₃	1	3	yes	—
137	C₂⁴	C₂⁴×C₁₃	1	4	yes	—
138	C₂×Q₈	—	4	32	no	—
139	C₂×Q₈	C₂⁴×C₁₃	1	64	yes	—
140	C₂²×C₄	—	4	24	no	—
141	C₂²×C₄	C₂×Q₈×C₁₃	1	24	yes	—
142	C₂²×C₄	C₂×Q₈×C₁₃	1	24	yes	—
143	C₂²×C₄	C₂×Q₈×C₁₃	1	64	yes	complemented
144	C₂²×C₄	C₂⁴×C₁₃	1	24	yes	—
145	C₂²×C₄	C₂⁴×C₁₃	1	48	yes	—
146	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
147	C₂²×C₄	C₂⁴×C₁₃	1	3	yes	—
148	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
149	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
150	C₄⋊C₄	C₂⁴×C₁₃	1	96	yes	—
151	C₄⋊C₄	C₂⁴×C₁₃	1	192	yes	—
152	C₂²⋊C₄	C₂⁴×C₁₃	1	96	yes	—
153	C₄²	C₂⁴×C₁₃	1	48	yes	—
154	C₄²	C₂⁴×C₁₃	1	48	yes	—
155	C₁₃	C₂×#3458	1	1	yes	characteristic, direct product factor, Sylow
156	C₂³	—	2	12	no	—
157	C₂³	—	4	16	no	—
158	C₂³	—	4	8	no	—
159	C₂³	C₂²×Q₈×C₁₃	1	12	yes	—
160	C₂³	C₂²×Q₈×C₁₃	1	16	yes	—
161	C₂³	C₂⁵×C₁₃	1	6	yes	—
162	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
163	Q₈	—	4	64	no	—
164	C₂×C₄	—	2	48	no	—
165	C₂×C₄	—	4	48	no	—
166	C₂×C₄	C₂²×Q₈×C₁₃	1	4	yes	—
167	C₂×C₄	C₂²×Q₈×C₁₃	1	12	yes	—
168	C₂×C₄	C₂²×Q₈×C₁₃	1	48	yes	—
169	C₂×C₄	C₂²×Q₈×C₁₃	1	48	yes	—
170	C₂×C₄	C₂⁵×C₁₃	1	48	yes	—
171	C₂×C₄	C₂⁵×C₁₃	1	2	yes	—
172	C₂×C₄	C₂⁵×C₁₃	1	6	yes	—
173	C₂²	—	2	12	no	—
174	C₂²	—	2	24	no	—
175	C₂²	—	4	16	no	—
176	C₂²	C₂³×Q₈×C₁₃	1	12	yes	—
177	C₂²	C₂³×Q₈×C₁₃	1	2	yes	—
178	C₂²	C₁₃×#585	1	4	yes	—
179	C₂²	C₂⁶×C₁₃	1	1	yes	characteristic
180	C₄	—	2	48	no	—
181	C₄	C₂³×Q₈×C₁₃	1	4	yes	—
182	C₄	C₂³×Q₈×C₁₃	1	12	yes	—
183	C₂	—	2	12	no	—
184	C₂	C₁₃×#3458	1	4	yes	direct product factor
185	C₂	C₂⁴×Q₈×C₁₃	1	1	yes	characteristic
186	C₂	C₂×C₁₃×#585	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₃
socle	C₂³×C₁₃	C₂⁵
prime radical	C₂⁵×C₁₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3458	C₂	direct (N × H)
2. C₂×C₁₃×#544	C₂	semidirect
4. C₂×C₁₃×#533	C₂	semidirect
7. C₁₃×#544	C₂²	semidirect
9. C₁₃×#533	C₂²	semidirect
10. C₂×C₁₃×C₄⋊Q₈	C₂²	semidirect
11. C₂×C₁₃×#125	C₂²	semidirect
13. C₂×C₄×Q₈×C₁₃	C₂²	semidirect
22. C₂³×C₄×C₁₃	Q₈	semidirect
27. C₁₃×C₄⋊Q₈	C₂³	semidirect
28. C₁₃×#125	C₂³	semidirect
30. C₄×Q₈×C₁₃	C₂³	semidirect
41. C₂×#3458	C₁₃	direct (N × H)
51. C₂²×C₄×C₁₃	C₂×Q₈	semidirect
63. #3458	C₂×C₁₃	direct (N × H)
64. C₂×#544	C₂×C₁₃	semidirect
66. C₂×#533	C₂×C₁₃	semidirect
86. #544	C₂²×C₁₃	semidirect
88. #533	C₂²×C₁₃	semidirect
89. C₂×C₄⋊Q₈	C₂²×C₁₃	semidirect
90. C₂×#125	C₂²×C₁₃	semidirect
92. C₂×C₄×Q₈	C₂²×C₁₃	semidirect
111. C₂³×C₄	Q₈×C₁₃	semidirect
116. C₄⋊Q₈	C₂³×C₁₃	semidirect
117. #125	C₂³×C₁₃	semidirect
119. C₄×Q₈	C₂³×C₁₃	semidirect
131. C₂×C₁₃	#3458	direct (N × H)
143. C₂²×C₄	C₂×Q₈×C₁₃	semidirect
155. C₁₃	C₂×#3458	direct (N × H)
184. C₂	C₁₃×#3458	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#3458	2
C₂×C₁₃×#544	2
C₂×C₁₃×#543	2
C₂×C₁₃×#533	2
C₂²×C₁₃×#117	2
C₂×#3458	13

Subgroups — C2×C13×#3458 #15749023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#3458 #15749023