Subgroups — C₁₃×#33947 #15720286

Order 3328.

476 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3380	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2253	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2228	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2226	C₂	1	2	yes	maximal, complemented
6	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic
8	C₄×C₁₃×D₁₆	C₂	1	2	yes	maximal
9	C₁₃×#554	—	2	2	no	—
10	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
11	C₁₃×#500	C₂²	1	2	yes	—
12	C₁₃×#499	C₂²	1	4	yes	—
13	C₁₃×#478	—	2	4	no	—
14	C₁₃×#476	—	2	2	no	—
15	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×D₈	C₂²	1	2	yes	—
17	C₄×C₁₃×D₈	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×#357	C₂²	1	4	yes	—
25	C₁₃×#357	C₂²	1	2	yes	—
26	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
28	C₂×C₁₃×D₈	C₂³	1	2	yes	—
29	C₁₃×C₄⋊Q₈	—	2	2	no	—
30	C₁₃×#125	—	2	4	no	—
31	C₁₃×C₂²⋊Q₈	—	4	2	no	—
32	C₁₃×C₂²⋊Q₈	—	4	4	no	—
33	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
34	C₄×D₄×C₁₃	—	2	2	no	—
35	C₄×D₄×C₁₃	—	2	4	no	—
36	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
37	C₁₃×D₁₆	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×#102	—	2	4	no	—
48	C₁₃×#102	—	2	2	no	—
49	C₄×C₈×C₁₃	C₂³	1	2	yes	—
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	#33947	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
54	C₂×D₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	4	2	no	—
56	C₂²×C₄×C₁₃	—	4	4	no	—
57	C₁₃×D₈	—	2	2	no	—
58	C₁₃×D₈	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	—	2	4	no	—
64	C₁₃×C₄⋊C₄	—	4	4	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₁₃×C₄⋊C₄	—	4	4	no	—
67	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
69	C₁₃×C₂²⋊C₄	—	4	2	no	—
70	C₁₃×C₂²⋊C₄	—	4	4	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	#3380	C₂×C₁₃	1	2	yes	complemented
77	#2253	C₂×C₁₃	1	2	yes	complemented
78	#2252	C₂×C₁₃	1	1	yes	characteristic, complemented
79	#2228	C₂×C₁₃	1	4	yes	complemented
80	#2226	C₂×C₁₃	1	2	yes	complemented
81	Q₈×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
82	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic
83	C₄×D₁₆	C₂×C₁₃	1	2	yes	—
84	C₂³×C₁₃	—	4	2	no	—
85	Q₈×C₁₃	—	2	2	no	—
86	Q₈×C₁₃	—	16	2	no	—
87	D₄×C₁₃	—	2	4	no	—
88	D₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	4	4	no	—
91	C₂×C₄×C₁₃	—	8	4	no	—
92	C₂×C₄×C₁₃	—	16	4	no	—
93	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
98	C₈×C₁₃	—	2	2	no	—
99	C₈×C₁₃	—	2	1	no	—
100	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
101	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
102	#554	—	2	2	no	—
103	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
104	#500	C₂²×C₁₃	1	2	yes	—
105	#499	C₂²×C₁₃	1	4	yes	—
106	#478	—	2	4	no	—
107	#476	—	2	2	no	—
108	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
109	C₄×D₈	C₂²×C₁₃	1	2	yes	—
110	C₄×D₈	C₂²×C₁₃	1	4	yes	—
111	#367	C₂²×C₁₃	1	4	yes	—
112	#366	C₂²×C₁₃	1	2	yes	—
113	#366	C₂²×C₁₃	1	1	yes	characteristic
114	#366	C₂²×C₁₃	1	2	yes	—
115	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
116	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
117	#357	C₂²×C₁₃	1	4	yes	—
118	#357	C₂²×C₁₃	1	2	yes	—
119	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
120	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
121	C₂²×C₁₃	—	4	4	no	—
122	C₂²×C₁₃	—	8	2	no	—
123	C₂²×C₁₃	—	8	2	no	—
124	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	—	2	1	no	—
128	C₄×C₁₃	—	8	2	no	—
129	C₄×C₁₃	—	8	4	no	—
130	C₄×C₁₃	#508	1	2	yes	—
131	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
132	C₄×C₁₃	#578	1	1	yes	characteristic
133	C₂×D₈	C₂³×C₁₃	1	2	yes	—
134	C₄⋊Q₈	—	2	2	no	—
135	#125	—	2	4	no	—
136	C₂²⋊Q₈	—	4	2	no	—
137	C₂²⋊Q₈	—	4	4	no	—
138	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
139	C₄×D₄	—	2	2	no	—
140	C₄×D₄	—	2	4	no	—
141	C₂×C₄⋊C₄	—	4	2	no	—
142	D₁₆	Q₈×C₁₃	1	4	yes	complemented
143	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
144	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
145	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
146	#107	—	2	2	no	—
147	#107	C₂³×C₁₃	1	4	yes	—
148	#107	C₂³×C₁₃	1	2	yes	—
149	#106	—	2	4	no	—
150	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
151	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
152	#102	—	2	4	no	—
153	#102	—	2	2	no	—
154	C₄×C₈	C₂³×C₁₃	1	2	yes	—
155	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
156	C₂×C₁₃	—	8	4	no	—
157	C₂×C₁₃	Q₈×D₈	1	1	yes	characteristic
158	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
159	C₂×C₁₃	#3415	1	1	yes	characteristic
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
162	C₂×D₄	—	2	2	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	4	no	—
165	D₈	—	2	2	no	—
166	D₈	C₂×Q₈×C₁₃	1	4	yes	—
167	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
168	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
169	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
176	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
177	C₂²⋊C₄	—	4	2	no	—
178	C₂²⋊C₄	—	4	4	no	—
179	C₄²	C₂×D₄×C₁₃	1	2	yes	—
180	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
181	C₁₆	—	2	1	no	—
182	C₁₆	—	2	2	no	—
183	C₁₆	C₂×Q₈×C₁₃	1	2	yes	—
184	C₁₃	#33947	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³	—	4	2	no	—
186	Q₈	—	2	2	no	—
187	Q₈	—	16	2	no	—
188	D₄	—	2	4	no	—
189	D₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	—	16	4	no	—
194	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
195	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
196	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
197	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
198	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	1	no	—
201	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
202	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
203	C₂²	—	4	4	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	8	2	no	—
206	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	1	no	—
210	C₄	—	8	2	no	—
211	C₄	—	8	4	no	—
212	C₄	C₁₃×#508	1	2	yes	—
213	C₄	D₄×Q₈×C₁₃	1	1	yes	characteristic
214	C₄	C₁₃×#578	1	1	yes	characteristic
215	C₂	—	8	4	no	—
216	C₂	Q₈×C₁₃×D₈	1	1	yes	characteristic
217	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
218	C₂	C₁₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×D₁₆	C₂²
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
1. C₁₃×#3380	C₂	semidirect
2. C₁₃×#2253	C₂	semidirect
3. C₁₃×#2252	C₂	semidirect
4. C₁₃×#2228	C₂	semidirect
5. C₁₃×#2226	C₂	semidirect
6. Q₈×C₁₃×C₁₆	C₂	semidirect
37. C₁₃×D₁₆	Q₈	semidirect
51. #33947	C₁₃	direct (N × H)
76. #3380	C₂×C₁₃	semidirect
77. #2253	C₂×C₁₃	semidirect
78. #2252	C₂×C₁₃	semidirect
79. #2228	C₂×C₁₃	semidirect
80. #2226	C₂×C₁₃	semidirect
81. Q₈×C₁₆	C₂×C₁₃	semidirect
142. D₁₆	Q₈×C₁₃	semidirect
184. C₁₃	#33947	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×#3380	2
C₁₃×#2253	2
C₁₃×#2252	2
C₁₃×#2228	2
C₁₃×#2226	2
Q₈×C₁₃×C₁₆	2
C₄×C₁₃×D₁₆	2
#33947	13

Subgroups — C13×#33947 #15720286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33947 #15720286