Subgroups — C₁₃×#13164 #15699503

Order 3328.

284 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×#2153	C₂	1	2	yes	maximal
2	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1378	C₂	1	2	yes	maximal
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
10	C₁₃×#427	C₄	1	1	yes	characteristic
11	C₁₃×#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×#364	—	2	4	no	—
14	C₁₃×#364	—	2	1	no	—
15	C₁₃×#364	—	2	1	no	—
16	C₁₃×#346	—	2	1	no	—
17	C₄×C₁₃×C₁₆	—	2	1	no	—
18	C₁₃×#339	C₄	1	2	yes	—
19	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₁₃×#117	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₁₃×M₃₂	—	2	1	no	—
25	C₁₃×M₃₂	—	2	1	no	—
26	C₁₃×M₃₂	—	2	2	no	—
27	C₁₃×M₃₂	—	2	2	no	—
28	C₁₃×M₃₂	D₄	1	2	yes	—
29	C₁₃×M₃₂	Q₈	1	2	yes	—
30	C₂×C₁₃×C₁₆	—	2	1	no	—
31	C₂×C₁₃×C₁₆	—	2	1	no	—
32	C₂×C₁₃×C₁₆	—	2	2	no	—
33	C₁₃×#108	C₈	1	2	yes	—
34	C₁₃×#108	C₈	1	2	yes	—
35	C₁₃×#107	—	2	1	no	—
36	C₁₃×#106	—	2	1	no	—
37	C₁₃×#97	—	2	2	no	—
38	C₄×C₈×C₁₃	—	2	2	no	—
39	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
43	#13164	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₁₃×M₁₆	—	2	2	no	—
47	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
60	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₂²⋊C₄	—	2	2	no	—
63	C₄²×C₁₃	—	2	1	no	—
64	C₄²×C₁₃	—	2	2	no	—
65	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
66	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
67	C₁₃×C₁₆	—	2	2	no	—
68	C₁₃×C₁₆	—	2	2	no	—
69	C₁₃×C₁₆	—	4	2	no	—
70	#2153	C₂×C₁₃	1	2	yes	—
71	C₂×#364	C₂×C₁₃	1	1	yes	characteristic
72	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic
73	#1772	C₂×C₁₃	1	1	yes	characteristic
74	#1378	C₂×C₁₃	1	2	yes	—
75	C₂³×C₁₃	#131	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	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
86	C₈×C₁₃	—	2	4	no	—
87	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
88	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
89	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
90	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
91	C₈×C₁₃	#121	1	1	yes	characteristic
92	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
93	C₈×C₁₃	#125	1	1	yes	characteristic
94	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
96	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
97	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
98	C₂×#108	C₄×C₁₃	1	1	yes	characteristic
99	#427	C₄×C₁₃	1	1	yes	characteristic
100	#405	C₂²×C₁₃	1	2	yes	—
101	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
102	#364	—	2	4	no	—
103	#364	—	2	1	no	—
104	#364	—	2	1	no	—
105	#346	—	2	1	no	—
106	C₄×C₁₆	—	2	1	no	—
107	#339	C₄×C₁₃	1	2	yes	—
108	C₂²×C₁₃	—	4	1	no	—
109	C₂²×C₁₃	#424	1	1	yes	characteristic
110	C₂²×C₁₃	#436	1	1	yes	characteristic
111	C₂²×C₁₃	C₈×Q₈	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×C₄⋊C₈	1	1	yes	characteristic
117	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
118	C₄×C₁₃	#446	1	1	yes	characteristic
119	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
120	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
122	#117	C₂×C₄×C₁₃	1	2	yes	—
123	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
124	M₃₂	—	2	1	no	—
125	M₃₂	—	2	1	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	2	2	no	—
128	M₃₂	D₄×C₁₃	1	2	yes	—
129	M₃₂	Q₈×C₁₃	1	2	yes	—
130	C₂×C₁₆	—	2	1	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	—	2	2	no	—
133	#108	C₈×C₁₃	1	2	yes	—
134	#108	C₈×C₁₃	1	2	yes	—
135	#107	—	2	1	no	—
136	#106	—	2	1	no	—
137	#97	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
140	C₄×C₈	D₄×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	#1939	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	C₂×C₈×C₁₃	1	4	yes	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
155	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
156	C₂×C₈	C₁₃×M₁₆	1	1	yes	characteristic
157	C₂×C₈	C₁₃×M₁₆	1	1	yes	characteristic
158	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
159	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
160	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
161	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
162	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
163	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	2	no	—
168	C₄²	C₁₃×C₄⋊C₄	1	1	yes	characteristic
169	C₄²	C₁₃×C₄⋊C₄	1	1	yes	characteristic
170	C₁₆	—	2	2	no	—
171	C₁₆	—	2	2	no	—
172	C₁₆	—	4	2	no	—
173	C₁₃	#13164	1	1	yes	characteristic, direct product factor, Sylow
174	C₂³	C₁₃×#131	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
181	C₂×C₄	C₄×Q₈×C₁₃	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
183	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
184	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	C₁₃×C₄⋊C₈	1	1	yes	characteristic
187	C₈	C₁₃×C₄⋊C₈	1	1	yes	characteristic
188	C₈	C₁₃×C₄⋊C₈	1	1	yes	characteristic
189	C₈	C₁₃×C₄⋊C₈	1	1	yes	characteristic
190	C₈	C₁₃×#121	1	1	yes	characteristic
191	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
192	C₈	C₁₃×#125	1	1	yes	characteristic
193	C₈	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	C₁₃×#424	1	1	yes	characteristic
196	C₂²	C₁₃×#436	1	1	yes	characteristic
197	C₂²	C₈×Q₈×C₁₃	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	8	2	no	—
201	C₄	C₁₃×#389	1	1	yes	characteristic
202	C₄	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
203	C₄	C₈×D₄×C₁₃	1	1	yes	characteristic
204	C₄	C₁₃×#446	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	C₁₃×#1939	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃	#121
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1939
prime radical	C₂³×C₁₃	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
43. #13164	C₁₃	direct (N × H)
173. C₁₃	#13164	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×#2153	2
C₂×C₁₃×#364	2
C₄×C₁₃×M₃₂	2
C₁₃×#1772	2
C₁₃×#1378	2
#13164	13

Subgroups — C13×#13164 #15699503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13164 #15699503