Subgroups — C₁₃×#3334 #13112392

Order 1664.

420 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#547	C₂	1	2	yes	maximal, complemented
2	C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#480	C₂	1	4	yes	maximal, complemented
4	C₁₃×#474	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	C₂×C₁₃×#107	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×#41	—	2	2	no	—
10	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
12	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₁₃×#123	—	2	4	no	—
14	C₁₃×C₂²⋊Q₈	—	2	2	no	—
15	C₁₃×#121	C₂²	1	4	yes	—
16	C₄×D₄×C₁₃	C₂²	1	2	yes	—
17	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
19	C₁₃×#107	C₂²	1	1	yes	characteristic
20	C₁₃×#107	C₂²	1	2	yes	—
21	C₁₃×#107	C₂²	1	4	yes	—
22	C₁₃×#107	C₂²	1	2	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	C₁₃×#41	—	4	8	no	—
29	C₂×Q₈×C₁₃	—	2	2	no	—
30	C₂×D₄×C₁₃	—	2	4	no	—
31	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
32	C₂×D₄×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₃	—	2	4	no	—
34	C₂²×C₄×C₁₃	—	2	2	no	—
35	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
36	C₁₃×D₈	—	2	2	no	—
37	C₂×C₈×C₁₃	—	2	4	no	—
38	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊C₄	—	2	4	no	—
42	C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
45	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
46	C₁₃×C₂²⋊C₄	—	2	2	no	—
47	C₁₃×C₂²⋊C₄	—	2	4	no	—
48	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
49	C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	#3334	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₁₃	—	2	2	no	—
52	C₂³×C₁₃	C₂×D₄	1	2	yes	—
53	Q₈×C₁₃	—	4	4	no	—
54	D₄×C₁₃	—	2	2	no	—
55	D₄×C₁₃	—	2	2	no	—
56	D₄×C₁₃	—	4	8	no	—
57	D₄×C₁₃	D₈	1	4	yes	complemented
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	4	no	—
62	C₂×C₄×C₁₃	—	4	8	no	—
63	C₂×C₄×C₁₃	—	4	4	no	—
64	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₈×C₁₃	—	2	2	no	—
69	C₈×C₁₃	—	2	1	no	—
70	C₈×C₁₃	#41	1	2	yes	complemented
71	#547	C₂×C₁₃	1	2	yes	complemented
72	#500	C₂×C₁₃	1	1	yes	characteristic, complemented
73	#480	C₂×C₁₃	1	4	yes	complemented
74	#474	C₂×C₁₃	1	2	yes	complemented
75	#466	C₂×C₁₃	1	2	yes	complemented
76	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
77	C₈×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
78	C₂×#107	C₂×C₁₃	1	2	yes	complemented
79	C₂²×C₁₃	—	2	2	no	—
80	C₂²×C₁₃	—	2	2	no	—
81	C₂²×C₁₃	—	4	2	no	—
82	C₂²×C₁₃	—	4	2	no	—
83	C₂²×C₁₃	—	4	2	no	—
84	C₂²×C₁₃	C₂×D₈	1	4	yes	—
85	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
86	C₄×C₁₃	—	2	2	no	—
87	C₄×C₁₃	—	2	1	no	—
88	C₄×C₁₃	—	4	2	no	—
89	C₄×C₁₃	—	4	4	no	—
90	C₄×C₁₃	C₂×D₈	1	2	yes	—
91	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
92	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
93	C₂×#41	—	2	2	no	—
94	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
95	C₂²×C₈	C₂²×C₁₃	1	2	yes	—
96	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
97	#123	—	2	4	no	—
98	C₂²⋊Q₈	—	2	2	no	—
99	#121	C₂²×C₁₃	1	4	yes	—
100	C₄×D₄	C₂²×C₁₃	1	2	yes	—
101	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
102	C₂×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
103	#107	C₂²×C₁₃	1	1	yes	characteristic
104	#107	C₂²×C₁₃	1	2	yes	—
105	#107	C₂²×C₁₃	1	4	yes	—
106	#107	C₂²×C₁₃	1	2	yes	—
107	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
108	#102	C₂²×C₁₃	1	2	yes	—
109	#102	C₂²×C₁₃	1	4	yes	—
110	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
111	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
112	C₂×C₁₃	—	2	4	no	—
113	C₂×C₁₃	—	8	2	no	—
114	C₂×C₁₃	#547	1	1	yes	characteristic
115	C₂×C₁₃	C₂²×D₈	1	1	yes	characteristic
116	C₂×C₁₃	#578	1	1	yes	characteristic
117	#41	—	4	8	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×D₄	—	2	4	no	—
120	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
121	C₂×D₄	C₂³×C₁₃	1	2	yes	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
125	D₈	—	2	2	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
128	C₂×C₈	C₂³×C₁₃	1	2	yes	—
129	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
133	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
134	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₂²⋊C₄	—	2	4	no	—
137	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
138	C₄²	C₂³×C₁₃	1	1	yes	characteristic
139	C₁₃	#3334	1	1	yes	characteristic, direct product factor, Sylow
140	C₂³	—	2	2	no	—
141	C₂³	C₂×D₄×C₁₃	1	2	yes	—
142	Q₈	—	4	4	no	—
143	D₄	—	2	2	no	—
144	D₄	—	2	2	no	—
145	D₄	—	4	8	no	—
146	D₄	C₁₃×D₈	1	4	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
154	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
155	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
156	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	2	1	no	—
159	C₈	C₁₃×#41	1	2	yes	complemented
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	2	no	—
165	C₂²	C₂×C₁₃×D₈	1	4	yes	—
166	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	4	no	—
171	C₄	C₂×C₁₃×D₈	1	2	yes	—
172	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
173	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
174	C₂	—	2	4	no	—
175	C₂	—	8	2	no	—
176	C₂	C₁₃×#547	1	1	yes	characteristic
177	C₂	C₂²×C₁₃×D₈	1	1	yes	characteristic
178	C₂	C₁₃×#578	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×#547	C₂	semidirect
2. C₁₃×#500	C₂	semidirect
3. C₁₃×#480	C₂	semidirect
4. C₁₃×#474	C₂	semidirect
5. C₁₃×#466	C₂	semidirect
6. C₄×C₁₃×D₈	C₂	semidirect
7. C₈×D₄×C₁₃	C₂	semidirect
8. C₂×C₁₃×#107	C₂	semidirect
50. #3334	C₁₃	direct (N × H)
57. D₄×C₁₃	D₈	semidirect
70. C₈×C₁₃	#41	semidirect
71. #547	C₂×C₁₃	semidirect
72. #500	C₂×C₁₃	semidirect
73. #480	C₂×C₁₃	semidirect
74. #474	C₂×C₁₃	semidirect
75. #466	C₂×C₁₃	semidirect
76. C₄×D₈	C₂×C₁₃	semidirect
77. C₈×D₄	C₂×C₁₃	semidirect
78. C₂×#107	C₂×C₁₃	semidirect
139. C₁₃	#3334	direct (N × H)
146. D₄	C₁₃×D₈	semidirect
159. C₈	C₁₃×#41	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#547	2
C₁₃×#500	2
C₁₃×#480	2
C₁₃×#474	2
C₁₃×#466	2
C₄×C₁₃×D₈	2
C₈×D₄×C₁₃	2
C₂×C₁₃×#107	2
#3334	13

Subgroups — C13×#3334 #13112392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3334 #13112392