Subgroups — C₁₃×#3389 #13112447

Order 1664.

348 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#554	C₂	1	2	yes	maximal, complemented
2	C₁₃×#501	C₂	1	2	yes	maximal, complemented
3	C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#478	C₂	1	2	yes	maximal, complemented
5	C₁₃×#478	C₂	1	2	yes	maximal, complemented
6	C₁₃×#474	C₂	1	2	yes	maximal, complemented
7	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#442	C₂	1	2	yes	maximal
9	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
11	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₁₃×#125	C₂²	1	2	yes	—
13	C₁₃×#125	C₂²	1	2	yes	—
14	C₁₃×C₂²⋊Q₈	—	2	2	no	—
15	C₁₃×C₂²⋊Q₈	—	2	2	no	—
16	C₁₃×C₂²⋊Q₈	—	2	2	no	—
17	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₃	C₂²	1	2	yes	—
19	C₄×D₄×C₁₃	C₂²	1	2	yes	—
20	C₄×D₄×C₁₃	C₂²	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₁₃×#107	C₂²	1	1	yes	characteristic
23	C₁₃×#107	C₂²	1	1	yes	characteristic
24	C₁₃×#107	C₂²	1	2	yes	—
25	C₁₃×#107	C₂²	1	1	yes	characteristic
26	C₁₃×#106	C₂²	1	2	yes	—
27	C₁₃×#106	C₂²	1	2	yes	—
28	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₁₃×#102	C₂²	1	2	yes	—
31	C₁₃×#102	C₂²	1	2	yes	—
32	C₁₃×#102	C₂²	1	2	yes	—
33	C₁₃×#97	C₂²	1	2	yes	—
34	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₁₃	—	2	2	no	—
36	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂×D₄×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃	—	2	2	no	—
39	C₂²×C₄×C₁₃	—	2	2	no	—
40	C₂²×C₄×C₁₃	—	2	2	no	—
41	C₁₃×D₈	Q₈	1	4	yes	complemented
42	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂³	1	2	yes	—
44	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
51	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
54	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₁₃×C₂²⋊C₄	—	2	2	no	—
56	C₁₃×C₂²⋊C₄	—	2	2	no	—
57	C₁₃×C₂²⋊C₄	—	2	2	no	—
58	C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₄²×C₁₃	C₂³	1	2	yes	—
60	#3389	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₁₃	—	2	2	no	—
62	Q₈×C₁₃	—	4	1	no	—
63	Q₈×C₁₃	—	8	2	no	—
64	D₄×C₁₃	—	2	2	no	—
65	D₄×C₁₃	C₂×Q₈	1	4	yes	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	4	4	no	—
70	C₂×C₄×C₁₃	—	8	2	no	—
71	C₂×C₄×C₁₃	—	8	2	no	—
72	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₈×C₁₃	—	2	1	no	—
78	C₈×C₁₃	—	2	2	no	—
79	C₈×C₁₃	C₂×Q₈	1	2	yes	—
80	#554	C₂×C₁₃	1	2	yes	complemented
81	#501	C₂×C₁₃	1	2	yes	complemented
82	#499	C₂×C₁₃	1	1	yes	characteristic, complemented
83	#478	C₂×C₁₃	1	2	yes	complemented
84	#478	C₂×C₁₃	1	2	yes	complemented
85	#474	C₂×C₁₃	1	2	yes	complemented
86	#446	C₂×C₁₃	1	1	yes	characteristic, complemented
87	#442	C₂×C₁₃	1	2	yes	—
88	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
89	C₂²×C₁₃	—	2	4	no	—
90	C₂²×C₁₃	—	4	2	no	—
91	C₂²×C₁₃	—	4	2	no	—
92	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
93	C₄×C₁₃	—	2	2	no	—
94	C₄×C₁₃	—	2	1	no	—
95	C₄×C₁₃	—	2	1	no	—
96	C₄×C₁₃	—	2	2	no	—
97	C₄×C₁₃	—	4	2	no	—
98	C₄×C₁₃	—	4	2	no	—
99	C₄×C₁₃	—	4	2	no	—
100	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
101	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
103	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
104	#125	C₂²×C₁₃	1	2	yes	—
105	#125	C₂²×C₁₃	1	2	yes	—
106	C₂²⋊Q₈	—	2	2	no	—
107	C₂²⋊Q₈	—	2	2	no	—
108	C₂²⋊Q₈	—	2	2	no	—
109	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
110	C₄×D₄	C₂²×C₁₃	1	2	yes	—
111	C₄×D₄	C₂²×C₁₃	1	2	yes	—
112	C₄×D₄	C₂²×C₁₃	1	2	yes	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	#107	C₂²×C₁₃	1	1	yes	characteristic
115	#107	C₂²×C₁₃	1	1	yes	characteristic
116	#107	C₂²×C₁₃	1	2	yes	—
117	#107	C₂²×C₁₃	1	1	yes	characteristic
118	#106	C₂²×C₁₃	1	2	yes	—
119	#106	C₂²×C₁₃	1	2	yes	—
120	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
121	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
122	#102	C₂²×C₁₃	1	2	yes	—
123	#102	C₂²×C₁₃	1	2	yes	—
124	#102	C₂²×C₁₃	1	2	yes	—
125	#97	C₂²×C₁₃	1	2	yes	—
126	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×C₁₃	—	4	4	no	—
128	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
129	C₂×C₁₃	#575	1	1	yes	characteristic
130	C₂×C₁₃	#576	1	1	yes	characteristic
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
133	C₂×D₄	C₂³×C₁₃	1	2	yes	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	D₈	Q₈×C₁₃	1	4	yes	complemented
138	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
139	C₂×C₈	C₂³×C₁₃	1	2	yes	—
140	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
147	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
148	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
149	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
150	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₄²	C₂³×C₁₃	1	1	yes	characteristic
155	C₄²	C₂³×C₁₃	1	2	yes	—
156	C₁₃	#3389	1	1	yes	characteristic, direct product factor, Sylow
157	C₂³	—	2	2	no	—
158	Q₈	—	4	1	no	—
159	Q₈	—	8	2	no	—
160	D₄	—	2	2	no	—
161	D₄	C₂×Q₈×C₁₃	1	4	yes	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
169	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
170	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
171	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
172	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	2	no	—
175	C₈	C₂×Q₈×C₁₃	1	2	yes	—
176	C₂²	—	2	4	no	—
177	C₂²	—	4	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	2	no	—
187	C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
188	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
189	C₂	—	4	4	no	—
190	C₂	D₄×Q₈×C₁₃	1	1	yes	characteristic
191	C₂	C₁₃×#575	1	1	yes	characteristic
192	C₂	C₁₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#554	C₂	semidirect
2. C₁₃×#501	C₂	semidirect
3. C₁₃×#499	C₂	semidirect
4. C₁₃×#478	C₂	semidirect
5. C₁₃×#478	C₂	semidirect
6. C₁₃×#474	C₂	semidirect
7. C₁₃×#446	C₂	semidirect
41. C₁₃×D₈	Q₈	semidirect
60. #3389	C₁₃	direct (N × H)
80. #554	C₂×C₁₃	semidirect
81. #501	C₂×C₁₃	semidirect
82. #499	C₂×C₁₃	semidirect
83. #478	C₂×C₁₃	semidirect
84. #478	C₂×C₁₃	semidirect
85. #474	C₂×C₁₃	semidirect
86. #446	C₂×C₁₃	semidirect
137. D₈	Q₈×C₁₃	semidirect
156. C₁₃	#3389	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×#554	2
C₁₃×#501	2
C₁₃×#499	2
C₁₃×#478	2
C₁₃×#474	2
C₁₃×#446	2
C₁₃×#442	2
C₄×C₁₃×D₈	2
#3389	13

Subgroups — C13×#3389 #13112447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3389 #13112447