Subgroups — C₁₃×#13180 #15699519

Order 3328.

294 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2159	C₂	1	2	yes	maximal
2	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1387	C₂	1	2	yes	maximal
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
8	C₁₃×#427	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#107	C₂²	1	2	yes	—
10	C₂×C₁₃×#106	—	2	1	no	—
11	C₁₃×#405	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×#382	—	2	1	no	—
14	C₁₃×#367	—	2	2	no	—
15	C₁₃×#366	—	2	2	no	—
16	C₁₃×#364	C₄	1	4	yes	complemented
17	C₁₃×#339	—	2	1	no	—
18	C₂×C₁₃×M₁₆	—	2	1	no	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₁₃×#117	—	2	1	no	—
22	C₂×C₁₃×C₄⋊C₄	—	2	2	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	1	no	—
28	C₁₃×#108	Q₈	1	2	yes	—
29	C₁₃×#107	—	2	2	no	—
30	C₁₃×#107	—	2	2	no	—
31	C₁₃×#107	D₄	1	2	yes	—
32	C₁₃×#106	—	2	1	no	—
33	C₁₃×#106	—	4	1	no	—
34	C₁₃×#106	—	4	1	no	—
35	C₁₃×#97	—	4	1	no	—
36	C₄×C₈×C₁₃	—	4	1	no	—
37	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₃	Q₈	1	2	yes	—
39	C₁₃×#95	—	4	2	no	—
40	#13180	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂²×C₄×C₁₃	—	2	1	no	—
42	C₂²×C₄×C₁₃	—	4	2	no	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₁₃×M₁₆	—	2	2	no	—
45	C₁₃×M₁₆	—	4	1	no	—
46	C₂×C₈×C₁₃	—	4	1	no	—
47	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
49	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
50	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
51	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₁₃×C₄⋊C₄	—	4	2	no	—
59	C₁₃×C₄⋊C₄	—	4	2	no	—
60	C₁₃×C₂²⋊C₄	—	4	1	no	—
61	C₄²×C₁₃	—	2	1	no	—
62	C₄²×C₁₃	—	4	1	no	—
63	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
64	C₁₃×C₁₆	—	2	4	no	—
65	#2159	C₂×C₁₃	1	2	yes	—
66	C₂×#364	C₂×C₁₃	1	1	yes	characteristic
67	#1846	C₂×C₁₃	1	1	yes	characteristic
68	#1772	C₂×C₁₃	1	1	yes	characteristic
69	#1387	C₂×C₁₃	1	2	yes	—
70	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	4	2	no	—
73	C₂×C₄×C₁₃	—	4	2	no	—
74	C₂×C₄×C₁₃	—	8	4	no	—
75	C₂×C₄×C₁₃	—	8	2	no	—
76	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
82	C₈×C₁₃	—	4	2	no	—
83	C₈×C₁₃	#107	1	2	yes	—
84	C₈×C₁₃	#107	1	2	yes	—
85	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
86	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
87	C₈×C₁₃	#125	1	1	yes	characteristic
88	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
89	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
90	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
91	#427	C₂²×C₁₃	1	1	yes	characteristic
92	C₂×#107	C₂²×C₁₃	1	2	yes	—
93	C₂×#106	—	2	1	no	—
94	#405	—	2	1	no	—
95	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
96	#382	—	2	1	no	—
97	#367	—	2	2	no	—
98	#366	—	2	2	no	—
99	#364	C₄×C₁₃	1	4	yes	complemented
100	#339	—	2	1	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	#428	1	1	yes	characteristic
103	C₂²×C₁₃	#459	1	1	yes	characteristic
104	C₂²×C₁₃	#477	1	1	yes	characteristic
105	C₄×C₁₃	—	4	2	no	—
106	C₄×C₁₃	—	8	2	no	—
107	C₄×C₁₃	—	16	2	no	—
108	C₄×C₁₃	#389	1	1	yes	characteristic
109	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
110	C₄×C₁₃	#462	1	1	yes	characteristic
111	C₄×C₁₃	#474	1	1	yes	characteristic
112	C₂×M₁₆	—	2	1	no	—
113	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
115	#117	—	2	1	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
118	M₃₂	—	2	2	no	—
119	M₃₂	C₂×C₄×C₁₃	1	4	yes	—
120	C₂×C₁₆	—	2	2	no	—
121	#108	—	2	1	no	—
122	#108	Q₈×C₁₃	1	2	yes	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	D₄×C₁₃	1	2	yes	—
126	#106	—	2	1	no	—
127	#106	—	4	1	no	—
128	#106	—	4	1	no	—
129	#97	—	4	1	no	—
130	C₄×C₈	—	4	1	no	—
131	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
132	C₄×C₈	Q₈×C₁₃	1	2	yes	—
133	#95	—	4	2	no	—
134	C₂×C₁₃	—	2	1	no	—
135	C₂×C₁₃	—	2	1	no	—
136	C₂×C₁₃	#1947	1	1	yes	characteristic
137	C₂×C₁₃	#2212	1	2	yes	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
141	M₁₆	—	2	2	no	—
142	M₁₆	—	4	1	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
145	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
146	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
147	C₂×C₈	C₁₃×Q₁₆	1	2	yes	—
148	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
149	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
150	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
152	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
153	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₂²⋊C₄	—	4	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	4	1	no	—
160	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
161	C₁₆	—	2	4	no	—
162	C₁₃	#13180	1	1	yes	characteristic, direct product factor, Sylow
163	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	8	4	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
171	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
173	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
174	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
175	C₈	—	4	2	no	—
176	C₈	C₁₃×#107	1	2	yes	—
177	C₈	C₁₃×#107	1	2	yes	—
178	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
179	C₈	C₄×Q₈×C₁₃	1	1	yes	characteristic
180	C₈	C₁₃×#125	1	1	yes	characteristic
181	C₈	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	C₁₃×#428	1	1	yes	characteristic
184	C₂²	C₁₃×#459	1	1	yes	characteristic
185	C₂²	C₁₃×#477	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	8	2	no	—
188	C₄	—	16	2	no	—
189	C₄	C₁₃×#389	1	1	yes	characteristic
190	C₄	C₂×C₁₃×#107	1	1	yes	characteristic
191	C₄	C₁₃×#462	1	1	yes	characteristic
192	C₄	C₁₃×#474	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₁₃×#1947	1	1	yes	characteristic
196	C₂	C₁₃×#2212	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#459
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂³×C₁₃	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
16. C₁₃×#364	C₄	semidirect
40. #13180	C₁₃	direct (N × H)
99. #364	C₄×C₁₃	semidirect
162. C₁₃	#13180	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×#2159	2
C₂×C₁₃×#364	2
C₁₃×#1846	2
C₁₃×#1772	2
C₁₃×#1387	2
#13180	13

Subgroups — C13×#13180 #15699519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13180 #15699519