Subgroups — C₁₃×#3341 #13112399

Order 1664.

364 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#555	C₂	1	2	yes	maximal, complemented
2	C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#482	C₂	1	4	yes	maximal, complemented
4	C₁₃×#476	C₂	1	2	yes	maximal, complemented
5	C₁₃×#466	C₂	1	2	yes	maximal, complemented
6	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#427	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
11	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₁₃×#124	—	2	4	no	—
13	C₁₃×#121	—	2	2	no	—
14	C₁₃×#121	C₂²	1	4	yes	—
15	C₄×D₄×C₁₃	C₂²	1	2	yes	—
16	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#117	—	2	2	no	—
18	C₁₃×#117	C₂²	1	4	yes	—
19	C₁₃×#107	C₂²	1	1	yes	characteristic
20	C₁₃×#107	C₂²	1	2	yes	—
21	C₁₃×#107	C₂²	1	2	yes	—
22	C₁₃×#106	C₂²	1	4	yes	—
23	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₁₃×#102	C₂²	1	2	yes	—
25	C₁₃×#102	C₂²	1	4	yes	—
26	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₁₃	—	2	2	no	—
29	C₂×D₄×C₁₃	—	2	4	no	—
30	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
31	C₂×D₄×C₁₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₃	—	2	2	no	—
33	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
34	C₁₃×D₈	—	2	2	no	—
35	C₂×C₈×C₁₃	—	2	4	no	—
36	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂³	1	2	yes	—
38	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
42	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
43	C₁₃×C₂²⋊C₄	—	2	4	no	—
44	C₁₃×C₂²⋊C₄	—	2	4	no	—
45	C₁₃×C₂²⋊C₄	—	2	2	no	—
46	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
47	C₄²×C₁₃	—	2	4	no	—
48	C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	#3341	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₁₃	—	2	2	no	—
51	C₂³×C₁₃	C₂×D₄	1	2	yes	—
52	Q₈×C₁₃	—	8	2	no	—
53	D₄×C₁₃	—	2	2	no	—
54	D₄×C₁₃	—	2	2	no	—
55	D₄×C₁₃	—	2	2	no	—
56	D₄×C₁₃	—	8	4	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	4	no	—
61	C₂×C₄×C₁₃	—	4	4	no	—
62	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₈×C₁₃	—	2	2	no	—
67	C₈×C₁₃	—	2	1	no	—
68	C₈×C₁₃	#41	1	2	yes	—
69	#555	C₂×C₁₃	1	2	yes	complemented
70	#500	C₂×C₁₃	1	1	yes	characteristic, complemented
71	#482	C₂×C₁₃	1	4	yes	complemented
72	#476	C₂×C₁₃	1	2	yes	complemented
73	#466	C₂×C₁₃	1	2	yes	complemented
74	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
75	C₈×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
76	#427	C₂×C₁₃	1	2	yes	complemented
77	C₂²×C₁₃	—	2	2	no	—
78	C₂²×C₁₃	—	2	2	no	—
79	C₂²×C₁₃	—	2	2	no	—
80	C₂²×C₁₃	—	4	2	no	—
81	C₂²×C₁₃	—	4	2	no	—
82	C₂²×C₁₃	—	4	2	no	—
83	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₁₃	—	2	2	no	—
85	C₄×C₁₃	—	2	1	no	—
86	C₄×C₁₃	—	4	2	no	—
87	C₄×C₁₃	—	4	4	no	—
88	C₄×C₁₃	#135	1	2	yes	—
89	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
90	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
91	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
92	C₂²×C₈	C₂²×C₁₃	1	2	yes	—
93	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
94	#124	—	2	4	no	—
95	#121	—	2	2	no	—
96	#121	C₂²×C₁₃	1	4	yes	—
97	C₄×D₄	C₂²×C₁₃	1	2	yes	—
98	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
99	#117	—	2	2	no	—
100	#117	C₂²×C₁₃	1	4	yes	—
101	#107	C₂²×C₁₃	1	1	yes	characteristic
102	#107	C₂²×C₁₃	1	2	yes	—
103	#107	C₂²×C₁₃	1	2	yes	—
104	#106	C₂²×C₁₃	1	4	yes	—
105	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
106	#102	C₂²×C₁₃	1	2	yes	—
107	#102	C₂²×C₁₃	1	4	yes	—
108	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
109	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
110	C₂×C₁₃	—	4	2	no	—
111	C₂×C₁₃	—	8	2	no	—
112	C₂×C₁₃	#547	1	1	yes	characteristic
113	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
114	C₂×C₁₃	#576	1	1	yes	characteristic
115	C₂×Q₈	—	2	2	no	—
116	C₂×D₄	—	2	4	no	—
117	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
118	C₂×D₄	C₂³×C₁₃	1	2	yes	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
121	D₈	—	2	2	no	—
122	C₂×C₈	—	2	4	no	—
123	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
124	C₂×C₈	C₂³×C₁₃	1	2	yes	—
125	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
128	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
129	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	2	no	—
133	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
134	C₄²	—	2	4	no	—
135	C₄²	C₂³×C₁₃	1	1	yes	characteristic
136	C₁₃	#3341	1	1	yes	characteristic, direct product factor, Sylow
137	C₂³	—	2	2	no	—
138	C₂³	C₂×D₄×C₁₃	1	2	yes	—
139	Q₈	—	8	2	no	—
140	D₄	—	2	2	no	—
141	D₄	—	2	2	no	—
142	D₄	—	2	2	no	—
143	D₄	—	8	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
150	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
151	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
152	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	2	1	no	—
155	C₈	C₁₃×#41	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	2	no	—
161	C₂²	—	4	2	no	—
162	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	1	no	—
165	C₄	—	4	2	no	—
166	C₄	—	4	4	no	—
167	C₄	C₁₃×#135	1	2	yes	—
168	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
169	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
170	C₂	—	4	2	no	—
171	C₂	—	8	2	no	—
172	C₂	C₁₃×#547	1	1	yes	characteristic
173	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
174	C₂	C₁₃×#576	1	1	yes	characteristic

	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 ◁ C₄×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#555	C₂	semidirect
2. C₁₃×#500	C₂	semidirect
3. C₁₃×#482	C₂	semidirect
4. C₁₃×#476	C₂	semidirect
5. C₁₃×#466	C₂	semidirect
6. C₄×C₁₃×D₈	C₂	semidirect
7. C₈×D₄×C₁₃	C₂	semidirect
8. C₁₃×#427	C₂	semidirect
49. #3341	C₁₃	direct (N × H)
69. #555	C₂×C₁₃	semidirect
70. #500	C₂×C₁₃	semidirect
71. #482	C₂×C₁₃	semidirect
72. #476	C₂×C₁₃	semidirect
73. #466	C₂×C₁₃	semidirect
74. C₄×D₈	C₂×C₁₃	semidirect
75. C₈×D₄	C₂×C₁₃	semidirect
76. #427	C₂×C₁₃	semidirect
136. C₁₃	#3341	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×#555	2
C₁₃×#500	2
C₁₃×#482	2
C₁₃×#476	2
C₁₃×#466	2
C₄×C₁₃×D₈	2
C₈×D₄×C₁₃	2
C₁₃×#427	2
#3341	13

Subgroups — C13×#3341 #13112399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3341 #13112399