Subgroups — C₁₃×#13182 #15699521

Order 3328.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2161	C₂	1	2	yes	maximal
2	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1389	C₂	1	2	yes	maximal
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₁₃×#108	—	2	1	no	—
8	C₂×C₁₃×#108	C₂²	1	2	yes	—
9	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
10	C₁₃×#427	C₂²	1	1	yes	characteristic
11	C₁₃×#423	—	2	1	no	—
12	C₁₃×#405	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#368	—	2	4	no	—
15	C₁₃×#364	C₄	1	2	yes	complemented
16	C₁₃×#364	C₄	1	2	yes	complemented
17	C₁₃×#339	—	2	1	no	—
18	C₂×C₁₃×M₁₆	—	2	1	no	—
19	C₂×C₁₃×M₁₆	—	2	2	no	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₁₃×#117	—	2	1	no	—
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₁₃×M₃₂	—	2	2	no	—
25	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₁₃×C₁₆	—	2	2	no	—
27	C₁₃×#108	—	2	2	no	—
28	C₁₃×#108	—	2	1	no	—
29	C₁₃×#108	—	2	2	no	—
30	C₁₃×#108	—	4	1	no	—
31	C₁₃×#108	—	4	1	no	—
32	C₁₃×#108	D₄	1	2	yes	—
33	C₁₃×#107	—	2	1	no	—
34	C₁₃×#106	Q₈	1	2	yes	—
35	C₁₃×C₄⋊C₈	—	4	2	no	—
36	C₁₃×#97	—	4	1	no	—
37	C₄×C₈×C₁₃	—	4	1	no	—
38	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
42	#13182	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₁₃×M₁₆	—	2	2	no	—
46	C₁₃×M₁₆	—	4	1	no	—
47	C₁₃×M₁₆	—	4	2	no	—
48	C₁₃×M₁₆	—	4	2	no	—
49	C₂×C₈×C₁₃	—	4	1	no	—
50	C₂×C₈×C₁₃	—	4	2	no	—
51	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
54	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
55	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
56	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
62	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₁₃×C₂²⋊C₄	—	4	1	no	—
65	C₄²×C₁₃	—	2	1	no	—
66	C₄²×C₁₃	—	4	1	no	—
67	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
69	C₁₃×C₁₆	—	2	4	no	—
70	#2161	C₂×C₁₃	1	2	yes	—
71	C₂×#364	C₂×C₁₃	1	1	yes	characteristic
72	#1848	C₂×C₁₃	1	1	yes	characteristic
73	#1772	C₂×C₁₃	1	1	yes	characteristic
74	#1389	C₂×C₁₃	1	2	yes	—
75	C₂³×C₁₃	#137	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₈×C₁₃	—	4	2	no	—
87	C₈×C₁₃	—	8	2	no	—
88	C₈×C₁₃	#107	1	1	yes	characteristic
89	C₈×C₁₃	#107	1	1	yes	characteristic
90	C₈×C₁₃	#107	1	1	yes	characteristic
91	C₈×C₁₃	#107	1	1	yes	characteristic
92	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
93	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
94	C₈×C₁₃	#125	1	1	yes	characteristic
95	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
97	C₂×#108	—	2	1	no	—
98	C₂×#108	C₂²×C₁₃	1	2	yes	—
99	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
100	#427	C₂²×C₁₃	1	1	yes	characteristic
101	#423	—	2	1	no	—
102	#405	—	2	1	no	—
103	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
104	#368	—	2	4	no	—
105	#364	C₄×C₁₃	1	2	yes	complemented
106	#364	C₄×C₁₃	1	2	yes	complemented
107	#339	—	2	1	no	—
108	C₂²×C₁₃	—	4	1	no	—
109	C₂²×C₁₃	#428	1	1	yes	characteristic
110	C₂²×C₁₃	#462	1	1	yes	characteristic
111	C₂²×C₁₃	#474	1	1	yes	characteristic
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	4	1	no	—
114	C₄×C₁₃	—	8	2	no	—
115	C₄×C₁₃	#389	1	1	yes	characteristic
116	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
117	C₄×C₁₃	#459	1	1	yes	characteristic
118	C₄×C₁₃	#477	1	1	yes	characteristic
119	C₂×M₁₆	—	2	1	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
122	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
123	#117	—	2	1	no	—
124	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	C₂×C₄×C₁₃	1	4	yes	—
127	C₂×C₁₆	—	2	2	no	—
128	#108	—	2	2	no	—
129	#108	—	2	1	no	—
130	#108	—	2	2	no	—
131	#108	—	4	1	no	—
132	#108	—	4	1	no	—
133	#108	D₄×C₁₃	1	2	yes	—
134	#107	—	2	1	no	—
135	#106	Q₈×C₁₃	1	2	yes	—
136	C₄⋊C₈	—	4	2	no	—
137	#97	—	4	1	no	—
138	C₄×C₈	—	4	1	no	—
139	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
140	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
141	C₄×C₈	Q₈×C₁₃	1	1	yes	characteristic
142	C₄×C₈	Q₈×C₁₃	1	1	yes	characteristic
143	C₂×C₁₃	—	2	1	no	—
144	C₂×C₁₃	—	2	1	no	—
145	C₂×C₁₃	—	2	1	no	—
146	C₂×C₁₃	#1947	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	4	1	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
158	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
159	C₂×C₈	C₁₃×Q₁₆	1	1	yes	characteristic
160	C₂×C₈	C₁₃×Q₁₆	1	1	yes	characteristic
161	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
162	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
163	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
164	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
165	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
166	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
167	C₄⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	4	1	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	4	1	no	—
171	C₄²	C₁₃×C₄⋊C₄	1	1	yes	characteristic
172	C₄²	C₁₃×C₄⋊C₄	1	1	yes	characteristic
173	C₁₆	—	2	4	no	—
174	C₁₃	#13182	1	1	yes	characteristic, direct product factor, Sylow
175	C₂³	C₁₃×#137	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
182	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
183	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
184	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
185	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₈	—	4	2	no	—
187	C₈	—	8	2	no	—
188	C₈	C₁₃×#107	1	1	yes	characteristic
189	C₈	C₁₃×#107	1	1	yes	characteristic
190	C₈	C₁₃×#107	1	1	yes	characteristic
191	C₈	C₁₃×#107	1	1	yes	characteristic
192	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
193	C₈	C₄×Q₈×C₁₃	1	1	yes	characteristic
194	C₈	C₁₃×#125	1	1	yes	characteristic
195	C₈	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	C₁₃×#428	1	1	yes	characteristic
198	C₂²	C₁₃×#462	1	1	yes	characteristic
199	C₂²	C₁₃×#474	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	8	2	no	—
203	C₄	C₁₃×#389	1	1	yes	characteristic
204	C₄	C₂×C₁₃×#107	1	1	yes	characteristic
205	C₄	C₁₃×#459	1	1	yes	characteristic
206	C₄	C₁₃×#477	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	C₁₃×#1947	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	#459
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1947
prime radical	C₂³×C₁₃	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₁₃×#364	C₄	semidirect
16. C₁₃×#364	C₄	semidirect
42. #13182	C₁₃	direct (N × H)
105. #364	C₄×C₁₃	semidirect
106. #364	C₄×C₁₃	semidirect
174. C₁₃	#13182	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×#2161	2
C₂×C₁₃×#364	2
C₁₃×#1848	2
C₁₃×#1772	2
C₁₃×#1389	2
#13182	13

Subgroups — C13×#13182 #15699521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13182 #15699521