Subgroups — C₁₃×#33950 #15720289

Order 3328.

448 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3383	C₂	1	2	yes	maximal
2	C₁₃×#2253	C₂	1	2	yes	maximal
3	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2229	C₂	1	4	yes	maximal
5	C₁₃×#2225	C₂	1	2	yes	maximal
6	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
8	C₄×C₁₃×Q₃₂	C₂	1	2	yes	maximal
9	C₁₃×#557	—	2	2	no	—
10	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
11	C₁₃×#500	C₂²	1	2	yes	—
12	C₁₃×#499	C₂²	1	4	yes	—
13	C₁₃×#479	—	2	4	no	—
14	C₁₃×#475	—	2	2	no	—
15	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
17	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	—
18	C₁₃×#367	C₂²	1	4	yes	—
19	C₁₃×#366	C₂²	1	2	yes	—
20	C₁₃×#366	C₂²	1	2	yes	—
21	C₁₃×#366	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₁₃×#358	C₂²	1	4	yes	—
25	C₁₃×#358	C₂²	1	2	yes	—
26	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
28	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
29	C₁₃×C₄⋊Q₈	—	2	2	no	—
30	C₁₃×C₄⋊Q₈	—	4	2	no	—
31	C₁₃×#125	—	2	4	no	—
32	C₁₃×#125	—	4	4	no	—
33	C₄×Q₈×C₁₃	—	2	2	no	—
34	C₄×Q₈×C₁₃	—	2	4	no	—
35	C₄×Q₈×C₁₃	—	4	2	no	—
36	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
37	C₁₃×Q₃₂	Q₈	1	4	yes	complemented
38	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
39	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
40	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
41	C₁₃×#107	—	2	2	no	—
42	C₁₃×#107	C₂³	1	4	yes	—
43	C₁₃×#107	C₂³	1	2	yes	—
44	C₁₃×#106	—	2	4	no	—
45	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
46	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
47	C₁₃×#103	—	2	4	no	—
48	C₁₃×#103	—	2	2	no	—
49	C₄×C₈×C₁₃	C₂³	1	2	yes	—
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	#33950	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×Q₈×C₁₃	—	2	2	no	—
53	C₂×Q₈×C₁₃	—	4	2	no	—
54	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
55	C₁₃×Q₁₆	—	2	2	no	—
56	C₁₃×Q₁₆	C₂×Q₈	1	4	yes	—
57	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	—	2	4	no	—
62	C₁₃×C₄⋊C₄	—	4	2	no	—
63	C₁₃×C₄⋊C₄	—	4	4	no	—
64	C₁₃×C₄⋊C₄	—	4	4	no	—
65	C₁₃×C₄⋊C₄	—	4	4	no	—
66	C₁₃×C₄⋊C₄	—	4	2	no	—
67	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
69	C₄²×C₁₃	—	4	4	no	—
70	C₄²×C₁₃	—	4	2	no	—
71	C₄²×C₁₃	C₂×D₄	1	2	yes	—
72	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₁₃×C₁₆	—	2	1	no	—
74	C₁₃×C₁₆	—	2	2	no	—
75	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
76	#3383	C₂×C₁₃	1	2	yes	—
77	#2253	C₂×C₁₃	1	2	yes	—
78	#2252	C₂×C₁₃	1	1	yes	characteristic
79	#2229	C₂×C₁₃	1	4	yes	—
80	#2225	C₂×C₁₃	1	2	yes	—
81	Q₈×C₁₆	C₂×C₁₃	1	1	yes	characteristic
82	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic
83	C₄×Q₃₂	C₂×C₁₃	1	2	yes	—
84	Q₈×C₁₃	—	2	2	no	—
85	Q₈×C₁₃	—	2	4	no	—
86	Q₈×C₁₃	—	4	2	no	—
87	Q₈×C₁₃	—	8	4	no	—
88	C₂×C₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	4	4	no	—
91	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₁₃	—	2	2	no	—
97	C₈×C₁₃	—	2	1	no	—
98	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
99	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
100	#557	—	2	2	no	—
101	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
102	#500	C₂²×C₁₃	1	2	yes	—
103	#499	C₂²×C₁₃	1	4	yes	—
104	#479	—	2	4	no	—
105	#475	—	2	2	no	—
106	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
107	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
108	C₄×Q₁₆	C₂²×C₁₃	1	4	yes	—
109	#367	C₂²×C₁₃	1	4	yes	—
110	#366	C₂²×C₁₃	1	2	yes	—
111	#366	C₂²×C₁₃	1	1	yes	characteristic
112	#366	C₂²×C₁₃	1	2	yes	—
113	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
114	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
115	#358	C₂²×C₁₃	1	4	yes	—
116	#358	C₂²×C₁₃	1	2	yes	—
117	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
118	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
119	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	—	4	4	no	—
124	C₄×C₁₃	—	8	4	no	—
125	C₄×C₁₃	—	8	2	no	—
126	C₄×C₁₃	#508	1	2	yes	—
127	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
128	C₄×C₁₃	#578	1	1	yes	characteristic
129	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	4	2	no	—
132	#125	—	2	4	no	—
133	#125	—	4	4	no	—
134	C₄×Q₈	—	2	2	no	—
135	C₄×Q₈	—	2	4	no	—
136	C₄×Q₈	—	4	2	no	—
137	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
138	Q₃₂	Q₈×C₁₃	1	4	yes	complemented
139	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
140	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
141	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
142	#107	—	2	2	no	—
143	#107	C₂³×C₁₃	1	4	yes	—
144	#107	C₂³×C₁₃	1	2	yes	—
145	#106	—	2	4	no	—
146	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
147	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
148	#103	—	2	4	no	—
149	#103	—	2	2	no	—
150	C₄×C₈	C₂³×C₁₃	1	2	yes	—
151	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
152	C₂×C₁₃	Q₈×D₈	1	1	yes	characteristic
153	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
154	C₂×C₁₃	#3417	1	1	yes	characteristic
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
158	Q₁₆	—	2	2	no	—
159	Q₁₆	C₂×Q₈×C₁₃	1	4	yes	—
160	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
161	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
162	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
171	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
172	C₄²	—	4	4	no	—
173	C₄²	—	4	2	no	—
174	C₄²	C₂×D₄×C₁₃	1	2	yes	—
175	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
176	C₁₆	—	2	1	no	—
177	C₁₆	—	2	2	no	—
178	C₁₆	C₂×Q₈×C₁₃	1	2	yes	—
179	C₁₃	#33950	1	1	yes	characteristic, direct product factor, Sylow
180	Q₈	—	2	2	no	—
181	Q₈	—	2	4	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	8	4	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
188	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
189	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
190	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
191	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	1	no	—
194	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
195	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
196	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	1	no	—
200	C₄	—	4	4	no	—
201	C₄	—	8	4	no	—
202	C₄	—	8	2	no	—
203	C₄	C₁₃×#508	1	2	yes	—
204	C₄	D₄×Q₈×C₁₃	1	1	yes	characteristic
205	C₄	C₁₃×#578	1	1	yes	characteristic
206	C₂	Q₈×C₁₃×D₈	1	1	yes	characteristic
207	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
208	C₂	C₁₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ C₈×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
37. C₁₃×Q₃₂	Q₈	semidirect
51. #33950	C₁₃	direct (N × H)
138. Q₃₂	Q₈×C₁₃	semidirect
179. C₁₃	#33950	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×#3383	2
C₁₃×#2253	2
C₁₃×#2252	2
C₁₃×#2229	2
C₁₃×#2225	2
Q₈×C₁₃×C₁₆	2
C₄×C₁₃×Q₃₂	2
#33950	13

Subgroups — C13×#33950 #15720289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33950 #15720289