Subgroups — C₁₃×#7180 #15693519

Order 3328.

240 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×M₃₂	—	2	1	no	—
5	C₁₃×#431	—	2	1	no	—
6	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
7	C₁₃×#427	C₄	1	1	yes	characteristic
8	C₁₃×#405	—	2	1	no	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#364	—	2	1	no	—
11	C₁₃×#364	C₄	1	2	yes	complemented
12	C₁₃×#339	—	2	1	no	—
13	C₁₃×#322	—	2	1	no	—
14	C₈²×C₁₃	C₄	1	2	yes	complemented
15	C₂×C₁₃×M₁₆	—	2	1	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₁₃×#117	—	2	1	no	—
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₁₃×M₃₂	—	2	2	no	—
21	C₁₃×M₃₂	—	4	1	no	—
22	C₂×C₁₃×C₁₆	—	4	1	no	—
23	C₁₃×#108	—	2	1	no	—
24	C₁₃×#108	C₈	1	2	yes	complemented
25	C₁₃×#107	—	2	1	no	—
26	C₁₃×#106	C₈	1	2	yes	complemented
27	C₁₃×C₄⋊C₈	—	2	2	no	—
28	C₁₃×C₂²⋊C₈	—	2	2	no	—
29	C₁₃×#97	—	4	1	no	—
30	C₄×C₈×C₁₃	—	2	2	no	—
31	C₄×C₈×C₁₃	—	4	1	no	—
32	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
36	#7180	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₄×C₁₃	—	2	1	no	—
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₁₃×M₁₆	—	2	2	no	—
40	C₁₃×M₁₆	—	4	1	no	—
41	C₂×C₈×C₁₃	—	2	4	no	—
42	C₂×C₈×C₁₃	—	4	1	no	—
43	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₁₃×C₂²⋊C₄	—	4	1	no	—
57	C₄²×C₁₃	—	2	1	no	—
58	C₄²×C₁₃	—	4	1	no	—
59	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
61	C₁₃×C₁₆	—	4	2	no	—
62	C₂×#364	C₂×C₁₃	1	1	yes	characteristic
63	#1772	C₂×C₁₃	1	1	yes	characteristic
64	#1450	C₂×C₁₃	1	1	yes	characteristic
65	C₂³×C₁₃	#104	1	1	yes	characteristic
66	C₂×C₄×C₁₃	—	2	1	no	—
67	C₂×C₄×C₁₃	—	2	1	no	—
68	C₂×C₄×C₁₃	—	4	2	no	—
69	C₂×C₄×C₁₃	—	4	2	no	—
70	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
76	C₈×C₁₃	—	4	4	no	—
77	C₈×C₁₃	—	4	2	no	—
78	C₈×C₁₃	C₄×C₈	1	1	yes	characteristic
79	C₈×C₁₃	#97	1	1	yes	characteristic
80	C₈×C₁₃	#102	1	1	yes	characteristic
81	C₈×C₁₃	#103	1	1	yes	characteristic
82	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
83	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
84	C₈×C₁₃	#106	1	1	yes	characteristic
85	C₈×C₁₃	#107	1	1	yes	characteristic
86	C₂×M₃₂	—	2	1	no	—
87	#431	—	2	1	no	—
88	C₂×#108	C₄×C₁₃	1	1	yes	characteristic
89	#427	C₄×C₁₃	1	1	yes	characteristic
90	#405	—	2	1	no	—
91	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
92	#364	—	2	1	no	—
93	#364	C₄×C₁₃	1	2	yes	complemented
94	#339	—	2	1	no	—
95	#322	—	2	1	no	—
96	C₈²	C₄×C₁₃	1	2	yes	complemented
97	C₂²×C₁₃	—	4	1	no	—
98	C₂²×C₁₃	D₄⋊C₈	1	1	yes	characteristic
99	C₂²×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
100	C₂²×C₁₃	#339	1	1	yes	characteristic
101	C₄×C₁₃	—	2	2	no	—
102	C₄×C₁₃	—	4	1	no	—
103	C₄×C₁₃	—	8	2	no	—
104	C₄×C₁₃	D₄⋊C₈	1	1	yes	characteristic
105	C₄×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
106	C₄×C₁₃	#336	1	1	yes	characteristic
107	C₄×C₁₃	#340	1	1	yes	characteristic
108	C₂×M₁₆	—	2	1	no	—
109	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
110	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
111	#117	—	2	1	no	—
112	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
113	M₃₂	—	2	2	no	—
114	M₃₂	—	4	1	no	—
115	C₂×C₁₆	—	4	1	no	—
116	#108	—	2	1	no	—
117	#108	C₈×C₁₃	1	2	yes	complemented
118	#107	—	2	1	no	—
119	#106	C₈×C₁₃	1	2	yes	complemented
120	C₄⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	2	2	no	—
122	#97	—	4	1	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	4	1	no	—
125	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
126	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
127	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
128	C₄×C₈	Q₈×C₁₃	1	1	yes	characteristic
129	C₂×C₁₃	—	2	1	no	—
130	C₂×C₁₃	—	2	1	no	—
131	C₂×C₁₃	—	2	1	no	—
132	C₂×C₁₃	#1279	1	1	yes	characteristic
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
135	M₁₆	—	2	2	no	—
136	M₁₆	—	4	1	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	C₄²×C₁₃	1	1	yes	characteristic
140	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
144	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
145	C₂×C₈	C₁₃×M₁₆	1	1	yes	characteristic
146	C₂×C₈	C₁₃×M₁₆	1	1	yes	characteristic
147	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
148	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
149	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
150	C₂×C₈	C₁₃×Q₁₆	1	1	yes	characteristic
151	C₄⋊C₄	—	2	2	no	—
152	C₂²⋊C₄	—	4	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	4	1	no	—
155	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
156	C₄²	C₁₃×C₄⋊C₄	1	1	yes	characteristic
157	C₁₆	—	4	2	no	—
158	C₁₃	#7180	1	1	yes	characteristic, direct product factor, Sylow
159	C₂³	C₁₃×#104	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	C₁₃×#95	1	1	yes	characteristic
165	C₂×C₄	C₁₃×C₂²⋊C₈	1	1	yes	characteristic
166	C₂×C₄	C₁₃×C₂²⋊C₈	1	1	yes	characteristic
167	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
168	C₂×C₄	C₁₃×#103	1	1	yes	characteristic
169	C₂×C₄	C₁₃×#104	1	1	yes	characteristic
170	C₈	—	4	4	no	—
171	C₈	—	4	2	no	—
172	C₈	C₄×C₈×C₁₃	1	1	yes	characteristic
173	C₈	C₁₃×#97	1	1	yes	characteristic
174	C₈	C₁₃×#102	1	1	yes	characteristic
175	C₈	C₁₃×#103	1	1	yes	characteristic
176	C₈	C₁₃×C₄⋊C₈	1	1	yes	characteristic
177	C₈	C₁₃×C₄⋊C₈	1	1	yes	characteristic
178	C₈	C₁₃×#106	1	1	yes	characteristic
179	C₈	C₁₃×#107	1	1	yes	characteristic
180	C₂²	—	4	1	no	—
181	C₂²	C₁₃×D₄⋊C₈	1	1	yes	characteristic
182	C₂²	C₁₃×Q₈⋊C₈	1	1	yes	characteristic
183	C₂²	C₁₃×#339	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	8	2	no	—
187	C₄	C₁₃×D₄⋊C₈	1	1	yes	characteristic
188	C₄	C₁₃×Q₈⋊C₈	1	1	yes	characteristic
189	C₄	C₁₃×#336	1	1	yes	characteristic
190	C₄	C₁₃×#340	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	C₁₃×#1279	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃	#106
commutator	C₈	C₄×C₈×C₁₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₃
socle	C₂×C₁₃	#1279
prime radical	C₂³×C₁₃	#104
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₁₃×#364	C₄	semidirect
14. C₈²×C₁₃	C₄	semidirect
24. C₁₃×#108	C₈	semidirect
26. C₁₃×#106	C₈	semidirect
36. #7180	C₁₃	direct (N × H)
93. #364	C₄×C₁₃	semidirect
96. C₈²	C₄×C₁₃	semidirect
117. #108	C₈×C₁₃	semidirect
119. #106	C₈×C₁₃	semidirect
158. C₁₃	#7180	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#364	2
C₁₃×#1772	2
C₁₃×#1450	2
#7180	13

Subgroups — C13×#7180 #15693519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#7180 #15693519