Subgroups — C₁₃×#3176 #13112234

Order 1664.

386 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#540	C₂	1	2	yes	maximal, complemented
2	C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#490	C₂	1	2	yes	maximal, complemented
5	C₁₃×#469	C₂	1	2	yes	maximal, complemented
6	C₁₃×#466	C₂	1	2	yes	maximal, complemented
7	C₁₃×#442	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×D₈	C₂	1	2	yes	maximal, complemented
9	C₁₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×D₈	C₂²	1	2	yes	—
11	C₂×C₁₃×D₈	C₂²	1	2	yes	—
12	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#127	C₂²	1	2	yes	—
15	C₁₃×#125	C₂²	1	2	yes	—
16	C₁₃×#123	—	2	4	no	—
17	C₁₃×#121	—	2	4	no	—
18	C₁₃×#121	C₂²	1	4	yes	—
19	C₄×D₄×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×#102	C₂²	1	4	yes	—
24	C₁₃×#102	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×D₄×C₁₃	—	2	4	no	—
30	C₂×D₄×C₁₃	—	2	4	no	—
31	C₂×D₄×C₁₃	C₂³	1	4	yes	—
32	C₂²×C₄×C₁₃	—	2	4	no	—
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×D₈	—	2	4	no	—
35	C₁₃×D₈	—	2	4	no	—
36	C₁₃×M₁₆	—	2	2	no	—
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
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	—	2	4	no	—
47	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
48	C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	#3176	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₁₃	—	2	4	no	—
52	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
53	D₄×C₁₃	—	2	4	no	—
54	D₄×C₁₃	—	2	4	no	—
55	D₄×C₁₃	—	8	4	no	—
56	D₄×C₁₃	—	8	4	no	—
57	C₂×C₄×C₁₃	—	2	1	no	—
58	C₂×C₄×C₁₃	—	2	1	no	—
59	C₂×C₄×C₁₃	—	2	4	no	—
60	C₂×C₄×C₁₃	—	8	4	no	—
61	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₈×C₁₃	—	2	1	no	—
66	C₈×C₁₃	—	2	1	no	—
67	C₈×C₁₃	#41	1	2	yes	—
68	C₈×C₁₃	#41	1	2	yes	—
69	#540	C₂×C₁₃	1	2	yes	complemented
70	#499	C₂×C₁₃	1	1	yes	characteristic, complemented
71	#493	C₂×C₁₃	1	1	yes	characteristic, complemented
72	#490	C₂×C₁₃	1	2	yes	complemented
73	#469	C₂×C₁₃	1	2	yes	complemented
74	#466	C₂×C₁₃	1	2	yes	complemented
75	#442	C₂×C₁₃	1	2	yes	complemented
76	C₄×D₈	C₂×C₁₃	1	2	yes	complemented
77	#405	C₂×C₁₃	1	1	yes	characteristic, complemented
78	C₂²×C₁₃	—	2	1	no	—
79	C₂²×C₁₃	—	2	2	no	—
80	C₂²×C₁₃	—	4	4	no	—
81	C₂²×C₁₃	—	4	4	no	—
82	C₂²×C₁₃	—	4	4	no	—
83	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₁₃	—	2	1	no	—
85	C₄×C₁₃	—	2	2	no	—
86	C₄×C₁₃	—	2	2	no	—
87	C₄×C₁₃	—	4	4	no	—
88	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
89	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
90	C₂×D₈	C₂²×C₁₃	1	2	yes	—
91	C₂×D₈	C₂²×C₁₃	1	2	yes	—
92	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
94	#127	C₂²×C₁₃	1	2	yes	—
95	#125	C₂²×C₁₃	1	2	yes	—
96	#123	—	2	4	no	—
97	#121	—	2	4	no	—
98	#121	C₂²×C₁₃	1	4	yes	—
99	C₄×D₄	C₂²×C₁₃	1	4	yes	—
100	#117	C₂²×C₁₃	1	1	yes	characteristic
101	#107	C₂²×C₁₃	1	2	yes	—
102	#106	C₂²×C₁₃	1	2	yes	—
103	#102	C₂²×C₁₃	1	4	yes	—
104	#102	C₂²×C₁₃	1	4	yes	—
105	#97	C₂²×C₁₃	1	1	yes	characteristic
106	#97	C₂²×C₁₃	1	1	yes	characteristic
107	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
108	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
109	C₂×C₁₃	—	4	1	no	—
110	C₂×C₁₃	—	8	4	no	—
111	C₂×C₁₃	#532	1	1	yes	characteristic
112	C₂×C₁₃	#576	1	2	yes	—
113	C₂×D₄	—	2	4	no	—
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	C₂³×C₁₃	1	4	yes	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
118	D₈	—	2	4	no	—
119	D₈	—	2	4	no	—
120	M₁₆	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
123	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
124	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
125	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
128	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
129	C₂²⋊C₄	—	2	4	no	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
132	C₄²	C₂³×C₁₃	1	1	yes	characteristic
133	C₄²	C₂³×C₁₃	1	1	yes	characteristic
134	C₁₃	#3176	1	1	yes	characteristic, direct product factor, Sylow
135	C₂³	—	2	4	no	—
136	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
137	D₄	—	2	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	8	4	no	—
140	D₄	—	8	4	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	8	4	no	—
145	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
146	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
147	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
148	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	C₁₃×#41	1	2	yes	—
152	C₈	C₁₃×#41	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	4	4	no	—
158	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	4	4	no	—
163	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
164	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
165	C₂	—	4	1	no	—
166	C₂	—	8	4	no	—
167	C₂	C₁₃×#532	1	1	yes	characteristic
168	C₂	C₁₃×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
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
1. C₁₃×#540	C₂	semidirect
2. C₁₃×#499	C₂	semidirect
3. C₁₃×#493	C₂	semidirect
4. C₁₃×#490	C₂	semidirect
5. C₁₃×#469	C₂	semidirect
6. C₁₃×#466	C₂	semidirect
7. C₁₃×#442	C₂	semidirect
8. C₄×C₁₃×D₈	C₂	semidirect
9. C₁₃×#405	C₂	semidirect
50. #3176	C₁₃	direct (N × H)
69. #540	C₂×C₁₃	semidirect
70. #499	C₂×C₁₃	semidirect
71. #493	C₂×C₁₃	semidirect
72. #490	C₂×C₁₃	semidirect
73. #469	C₂×C₁₃	semidirect
74. #466	C₂×C₁₃	semidirect
75. #442	C₂×C₁₃	semidirect
76. C₄×D₈	C₂×C₁₃	semidirect
77. #405	C₂×C₁₃	semidirect
134. C₁₃	#3176	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×#540	2
C₁₃×#499	2
C₁₃×#493	2
C₁₃×#490	2
C₁₃×#469	2
C₁₃×#466	2
C₁₃×#442	2
C₄×C₁₃×D₈	2
C₁₃×#405	2
#3176	13

Subgroups — C13×#3176 #13112234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3176 #13112234