Subgroups — C₁₃×#3498 #13112556

Order 1664.

1748 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	C₁₃×#550	C₂	1	4	yes	maximal, complemented
3	D₄²×C₁₃	C₂	1	12	yes	maximal, complemented
4	C₁₃×#535	C₂	1	6	yes	maximal, complemented
5	C₁₃×#532	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₃×#127	C₂	1	3	yes	maximal, complemented
7	C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×H(5,𝔽₂)	—	2	16	no	—
9	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#41	C₂²	1	6	yes	complemented
11	C₂×C₁₃×#41	C₂²	1	4	yes	complemented
12	C₂²×D₄×C₁₃	C₂²	1	6	yes	complemented
13	C₂²×D₄×C₁₃	C₂²	1	12	yes	complemented
14	C₂²×D₄×C₁₃	C₂²	1	12	yes	complemented
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₁₃×#127	C₂²	1	3	yes	complemented
17	C₁₃×#127	C₂²	1	6	yes	complemented
18	C₁₃×#127	C₂²	1	12	yes	complemented
19	C₁₃×#124	C₂²	1	6	yes	complemented
20	C₁₃×#121	C₂²	1	12	yes	complemented
21	C₁₃×#121	C₂²	1	24	yes	complemented
22	C₁₃×#120	C₂²	1	24	yes	complemented
23	C₄×Q₈×C₁₃	C₂²	1	2	yes	complemented
24	C₄×D₄×C₁₃	C₂²	1	6	yes	complemented
25	C₄×D₄×C₁₃	C₂²	1	12	yes	complemented
26	C₁₃×#117	C₂²	1	3	yes	complemented
27	C₂×C₄²×C₁₃	C₂²	1	3	yes	complemented
28	C₂⁴×C₁₃	C₂³	1	12	yes	—
29	C₁₃×#41	—	2	16	no	—
30	C₁₃×#41	—	2	24	no	—
31	C₁₃×#41	D₄	1	8	yes	complemented
32	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
33	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×D₄×C₁₃	—	2	48	no	—
35	C₂×D₄×C₁₃	—	2	24	no	—
36	C₂×D₄×C₁₃	—	4	24	no	—
37	C₂×D₄×C₁₃	C₂³	1	12	yes	—
38	C₂×D₄×C₁₃	C₂³	1	12	yes	—
39	C₂×D₄×C₁₃	C₂³	1	12	yes	—
40	C₂×D₄×C₁₃	C₂³	1	6	yes	—
41	C₂×D₄×C₁₃	C₂³	1	24	yes	—
42	C₂×D₄×C₁₃	C₂³	1	3	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	12	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
46	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
47	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
48	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
49	C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
50	C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₄²×C₁₃	C₂³	1	3	yes	—
52	C₄²×C₁₃	C₂³	1	6	yes	—
53	#3498	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₁₃	—	2	24	no	—
55	C₂³×C₁₃	—	4	24	no	—
56	C₂³×C₁₃	C₂⁴	1	12	yes	—
57	C₂³×C₁₃	C₂⁴	1	3	yes	—
58	Q₈×C₁₃	—	2	4	no	—
59	Q₈×C₁₃	C₂×D₄	1	4	yes	complemented
60	D₄×C₁₃	—	2	24	no	—
61	D₄×C₁₃	—	2	24	no	—
62	D₄×C₁₃	—	2	12	no	—
63	D₄×C₁₃	—	2	24	no	—
64	D₄×C₁₃	—	4	24	no	—
65	D₄×C₁₃	C₂×D₄	1	12	yes	complemented
66	C₂×C₄×C₁₃	—	2	24	no	—
67	C₂×C₄×C₁₃	—	2	12	no	—
68	C₂×C₄×C₁₃	C₂×D₄	1	12	yes	complemented
69	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
70	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
71	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
73	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
74	C₂×H(5,𝔽₂)	C₂×C₁₃	1	2	yes	complemented
75	#550	C₂×C₁₃	1	4	yes	complemented
76	D₄²	C₂×C₁₃	1	12	yes	complemented
77	#535	C₂×C₁₃	1	6	yes	complemented
78	#532	C₂×C₁₃	1	3	yes	complemented
79	C₂×#127	C₂×C₁₃	1	3	yes	complemented
80	C₄×#41	C₂×C₁₃	1	1	yes	characteristic, complemented
81	C₂²×C₁₃	—	2	6	no	—
82	C₂²×C₁₃	—	2	12	no	—
83	C₂²×C₁₃	—	2	24	no	—
84	C₂²×C₁₃	—	8	24	no	—
85	C₂²×C₁₃	C₂²×D₄	1	6	yes	—
86	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
87	C₄×C₁₃	—	2	4	no	—
88	C₄×C₁₃	—	2	6	no	—
89	C₄×C₁₃	C₂²×D₄	1	6	yes	—
90	C₄×C₁₃	C₂²×D₄	1	2	yes	—
91	C₄×C₁₃	H(5,𝔽₂)	1	4	yes	complemented
92	H(5,𝔽₂)	—	2	16	no	—
93	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
94	C₂×#41	C₂²×C₁₃	1	6	yes	complemented
95	C₂×#41	C₂²×C₁₃	1	4	yes	complemented
96	C₂²×D₄	C₂²×C₁₃	1	6	yes	complemented
97	C₂²×D₄	C₂²×C₁₃	1	12	yes	complemented
98	C₂²×D₄	C₂²×C₁₃	1	12	yes	complemented
99	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
100	#127	C₂²×C₁₃	1	3	yes	complemented
101	#127	C₂²×C₁₃	1	6	yes	complemented
102	#127	C₂²×C₁₃	1	12	yes	complemented
103	#124	C₂²×C₁₃	1	6	yes	complemented
104	#121	C₂²×C₁₃	1	12	yes	complemented
105	#121	C₂²×C₁₃	1	24	yes	complemented
106	#120	C₂²×C₁₃	1	24	yes	complemented
107	C₄×Q₈	C₂²×C₁₃	1	2	yes	complemented
108	C₄×D₄	C₂²×C₁₃	1	6	yes	complemented
109	C₄×D₄	C₂²×C₁₃	1	12	yes	complemented
110	#117	C₂²×C₁₃	1	3	yes	complemented
111	C₂×C₄²	C₂²×C₁₃	1	3	yes	complemented
112	C₂×C₁₃	—	2	6	no	—
113	C₂×C₁₃	—	4	12	no	—
114	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
115	C₂×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
116	C₂⁴	C₂³×C₁₃	1	12	yes	—
117	#41	—	2	16	no	—
118	#41	—	2	24	no	—
119	#41	D₄×C₁₃	1	8	yes	complemented
120	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
121	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
122	C₂×D₄	—	2	48	no	—
123	C₂×D₄	—	2	24	no	—
124	C₂×D₄	—	4	24	no	—
125	C₂×D₄	C₂³×C₁₃	1	12	yes	—
126	C₂×D₄	C₂³×C₁₃	1	12	yes	—
127	C₂×D₄	C₂³×C₁₃	1	12	yes	—
128	C₂×D₄	C₂³×C₁₃	1	6	yes	—
129	C₂×D₄	C₂³×C₁₃	1	24	yes	—
130	C₂×D₄	C₂³×C₁₃	1	3	yes	—
131	C₂²×C₄	C₂³×C₁₃	1	6	yes	—
132	C₂²×C₄	C₂³×C₁₃	1	12	yes	—
133	C₂²×C₄	C₂³×C₁₃	1	3	yes	—
134	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
135	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
136	C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
137	C₂²⋊C₄	C₂³×C₁₃	1	24	yes	—
138	C₄²	C₂³×C₁₃	1	1	yes	characteristic
139	C₄²	C₂³×C₁₃	1	3	yes	—
140	C₄²	C₂³×C₁₃	1	6	yes	—
141	C₁₃	#3498	1	1	yes	characteristic, direct product factor, Sylow
142	C₂³	—	2	24	no	—
143	C₂³	—	4	24	no	—
144	C₂³	C₂⁴×C₁₃	1	12	yes	—
145	C₂³	C₂⁴×C₁₃	1	3	yes	—
146	Q₈	—	2	4	no	—
147	Q₈	C₂×D₄×C₁₃	1	4	yes	complemented
148	D₄	—	2	24	no	—
149	D₄	—	2	24	no	—
150	D₄	—	2	12	no	—
151	D₄	—	2	24	no	—
152	D₄	—	4	24	no	—
153	D₄	C₂×D₄×C₁₃	1	12	yes	complemented
154	C₂×C₄	—	2	24	no	—
155	C₂×C₄	—	2	12	no	—
156	C₂×C₄	C₂×D₄×C₁₃	1	12	yes	complemented
157	C₂×C₄	C₂⁴×C₁₃	1	6	yes	—
158	C₂×C₄	C₂⁴×C₁₃	1	4	yes	—
159	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
160	C₂×C₄	C₂⁴×C₁₃	1	3	yes	—
161	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
162	C₂²	—	2	6	no	—
163	C₂²	—	2	12	no	—
164	C₂²	—	2	24	no	—
165	C₂²	—	8	24	no	—
166	C₂²	C₂²×D₄×C₁₃	1	6	yes	—
167	C₂²	C₂⁵×C₁₃	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	2	6	no	—
170	C₄	C₂²×D₄×C₁₃	1	6	yes	—
171	C₄	C₂²×D₄×C₁₃	1	2	yes	—
172	C₄	C₁₃×H(5,𝔽₂)	1	4	yes	complemented
173	C₂	—	2	6	no	—
174	C₂	—	4	12	no	—
175	C₂	C₂³×D₄×C₁₃	1	1	yes	characteristic
176	C₂	C₂×C₁₃×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×H(5,𝔽₂)	C₂	semidirect
2. C₁₃×#550	C₂	semidirect
3. D₄²×C₁₃	C₂	semidirect
4. C₁₃×#535	C₂	semidirect
5. C₁₃×#532	C₂	semidirect
6. C₂×C₁₃×#127	C₂	semidirect
7. C₄×C₁₃×#41	C₂	semidirect
10. C₂×C₁₃×#41	C₂²	semidirect
11. C₂×C₁₃×#41	C₂²	semidirect
12. C₂²×D₄×C₁₃	C₂²	semidirect
13. C₂²×D₄×C₁₃	C₂²	semidirect
14. C₂²×D₄×C₁₃	C₂²	semidirect
15. C₁₃×C₄⋊Q₈	C₂²	semidirect
16. C₁₃×#127	C₂²	semidirect
17. C₁₃×#127	C₂²	semidirect
18. C₁₃×#127	C₂²	semidirect
19. C₁₃×#124	C₂²	semidirect
20. C₁₃×#121	C₂²	semidirect
21. C₁₃×#121	C₂²	semidirect
22. C₁₃×#120	C₂²	semidirect
23. C₄×Q₈×C₁₃	C₂²	semidirect
24. C₄×D₄×C₁₃	C₂²	semidirect
25. C₄×D₄×C₁₃	C₂²	semidirect
26. C₁₃×#117	C₂²	semidirect
27. C₂×C₄²×C₁₃	C₂²	semidirect
31. C₁₃×#41	D₄	semidirect
53. #3498	C₁₃	direct (N × H)
59. Q₈×C₁₃	C₂×D₄	semidirect
65. D₄×C₁₃	C₂×D₄	semidirect
68. C₂×C₄×C₁₃	C₂×D₄	semidirect
74. C₂×H(5,𝔽₂)	C₂×C₁₃	semidirect
75. #550	C₂×C₁₃	semidirect
76. D₄²	C₂×C₁₃	semidirect
77. #535	C₂×C₁₃	semidirect
78. #532	C₂×C₁₃	semidirect
79. C₂×#127	C₂×C₁₃	semidirect
80. C₄×#41	C₂×C₁₃	semidirect
91. C₄×C₁₃	H(5,𝔽₂)	semidirect
94. C₂×#41	C₂²×C₁₃	semidirect
95. C₂×#41	C₂²×C₁₃	semidirect
96. C₂²×D₄	C₂²×C₁₃	semidirect
97. C₂²×D₄	C₂²×C₁₃	semidirect
98. C₂²×D₄	C₂²×C₁₃	semidirect
99. C₄⋊Q₈	C₂²×C₁₃	semidirect
100. #127	C₂²×C₁₃	semidirect
101. #127	C₂²×C₁₃	semidirect
102. #127	C₂²×C₁₃	semidirect
103. #124	C₂²×C₁₃	semidirect
104. #121	C₂²×C₁₃	semidirect
105. #121	C₂²×C₁₃	semidirect
106. #120	C₂²×C₁₃	semidirect
107. C₄×Q₈	C₂²×C₁₃	semidirect
108. C₄×D₄	C₂²×C₁₃	semidirect
109. C₄×D₄	C₂²×C₁₃	semidirect
110. #117	C₂²×C₁₃	semidirect
111. C₂×C₄²	C₂²×C₁₃	semidirect
119. #41	D₄×C₁₃	semidirect
141. C₁₃	#3498	direct (N × H)
147. Q₈	C₂×D₄×C₁₃	semidirect
153. D₄	C₂×D₄×C₁₃	semidirect
156. C₂×C₄	C₂×D₄×C₁₃	semidirect
172. C₄	C₁₃×H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×H(5,𝔽₂)	2
C₁₃×#550	2
D₄²×C₁₃	2
C₁₃×#535	2
C₁₃×#532	2
C₂×C₁₃×#127	2
C₄×C₁₃×#41	2
#3498	13

Subgroups — C13×#3498 #13112556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3498 #13112556