Subgroups — C₁₃×#3177 #13112235

Order 1664.

338 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#541	C₂	1	2	yes	maximal
2	C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#491	C₂	1	2	yes	maximal, complemented
5	C₁₃×#470	C₂	1	2	yes	maximal
6	C₁₃×#467	C₂	1	2	yes	maximal
7	C₁₃×#441	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×Q₁₆	C₂	1	2	yes	maximal, complemented
9	C₁₃×#405	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
12	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₁₃×#126	—	2	4	no	—
16	C₁₃×#125	—	2	4	no	—
17	C₁₃×#125	C₂²	1	2	yes	—
18	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
19	C₄×Q₈×C₁₃	C₂²	1	4	yes	—
20	C₁₃×#117	C₂²	1	1	yes	characteristic
21	C₁₃×#107	C₂²	1	2	yes	—
22	C₁₃×#106	C₂²	1	2	yes	—
23	C₁₃×#103	C₂²	1	4	yes	—
24	C₁₃×#103	C₂²	1	4	yes	—
25	C₁₃×#97	C₂²	1	1	yes	characteristic
26	C₁₃×#97	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₁₃×Q₁₆	—	2	4	no	—
32	C₁₃×Q₁₆	—	2	4	no	—
33	C₁₃×M₁₆	—	2	2	no	—
34	C₂×C₈×C₁₃	—	2	2	no	—
35	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊C₄	—	2	4	no	—
40	C₁₃×C₄⋊C₄	—	2	4	no	—
41	C₁₃×C₄⋊C₄	—	2	4	no	—
42	C₁₃×C₄⋊C₄	—	2	4	no	—
43	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
44	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
45	C₁₃×C₂²⋊C₄	—	2	4	no	—
46	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
47	C₄²×C₁₃	—	2	4	no	—
48	C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	#3177	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
52	Q₈×C₁₃	—	2	4	no	—
53	Q₈×C₁₃	—	2	4	no	—
54	C₂×C₄×C₁₃	—	2	1	no	—
55	C₂×C₄×C₁₃	—	2	1	no	—
56	C₂×C₄×C₁₃	—	2	4	no	—
57	C₂×C₄×C₁₃	—	2	4	no	—
58	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₈×C₁₃	—	2	1	no	—
63	C₈×C₁₃	—	2	1	no	—
64	C₈×C₁₃	#41	1	2	yes	—
65	C₈×C₁₃	#41	1	2	yes	—
66	#541	C₂×C₁₃	1	2	yes	—
67	#499	C₂×C₁₃	1	1	yes	characteristic, complemented
68	#494	C₂×C₁₃	1	1	yes	characteristic, complemented
69	#491	C₂×C₁₃	1	2	yes	complemented
70	#470	C₂×C₁₃	1	2	yes	—
71	#467	C₂×C₁₃	1	2	yes	—
72	#441	C₂×C₁₃	1	2	yes	complemented
73	C₄×Q₁₆	C₂×C₁₃	1	2	yes	complemented
74	#405	C₂×C₁₃	1	1	yes	characteristic
75	C₂²×C₁₃	—	2	1	no	—
76	C₂²×C₁₃	—	2	2	no	—
77	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₁₃	—	2	1	no	—
79	C₄×C₁₃	—	2	2	no	—
80	C₄×C₁₃	—	2	2	no	—
81	C₄×C₁₃	—	4	4	no	—
82	C₄×C₁₃	—	4	4	no	—
83	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
84	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
85	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
86	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
87	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
89	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
90	#126	—	2	4	no	—
91	#125	—	2	4	no	—
92	#125	C₂²×C₁₃	1	2	yes	—
93	C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
94	C₄×Q₈	C₂²×C₁₃	1	4	yes	—
95	#117	C₂²×C₁₃	1	1	yes	characteristic
96	#107	C₂²×C₁₃	1	2	yes	—
97	#106	C₂²×C₁₃	1	2	yes	—
98	#103	C₂²×C₁₃	1	4	yes	—
99	#103	C₂²×C₁₃	1	4	yes	—
100	#97	C₂²×C₁₃	1	1	yes	characteristic
101	#97	C₂²×C₁₃	1	1	yes	characteristic
102	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
103	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
104	C₂×C₁₃	—	4	1	no	—
105	C₂×C₁₃	#532	1	1	yes	characteristic
106	C₂×C₁₃	#578	1	2	yes	—
107	C₂×Q₈	C₂³×C₁₃	1	4	yes	—
108	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
109	Q₁₆	—	2	4	no	—
110	Q₁₆	—	2	4	no	—
111	M₁₆	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
114	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
115	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
116	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
122	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
123	C₂²⋊C₄	—	2	4	no	—
124	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
125	C₄²	—	2	4	no	—
126	C₄²	C₂³×C₁₃	1	1	yes	characteristic
127	C₄²	C₂³×C₁₃	1	1	yes	characteristic
128	C₁₃	#3177	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
130	Q₈	—	2	4	no	—
131	Q₈	—	2	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
137	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
138	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
139	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	C₁₃×#41	1	2	yes	—
143	C₈	C₁₃×#41	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	4	4	no	—
151	C₄	—	4	4	no	—
152	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
153	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
154	C₂	—	4	1	no	—
155	C₂	C₁₃×#532	1	1	yes	characteristic
156	C₂	C₁₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₂³×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#499	C₂	semidirect
3. C₁₃×#494	C₂	semidirect
4. C₁₃×#491	C₂	semidirect
7. C₁₃×#441	C₂	semidirect
8. C₄×C₁₃×Q₁₆	C₂	semidirect
50. #3177	C₁₃	direct (N × H)
67. #499	C₂×C₁₃	semidirect
68. #494	C₂×C₁₃	semidirect
69. #491	C₂×C₁₃	semidirect
72. #441	C₂×C₁₃	semidirect
73. C₄×Q₁₆	C₂×C₁₃	semidirect
128. C₁₃	#3177	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×#541	2
C₁₃×#499	2
C₁₃×#494	2
C₁₃×#491	2
C₁₃×#470	2
C₁₃×#467	2
C₁₃×#441	2
C₄×C₁₃×Q₁₆	2
C₁₃×#405	2
#3177	13

Subgroups — C13×#3177 #13112235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3177 #13112235