Subgroups — C₁₃×#7167 #15693506

Order 3328.

286 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃×C₁₆	—	2	1	no	—
5	C₂²×C₁₃×C₁₆	—	2	1	no	—
6	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	—	2	1	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₁₃×C₄⋊C₁₆	—	2	1	no	—
11	C₁₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
12	C₄×C₁₃×C₁₆	—	2	1	no	—
13	C₄×C₁₃×C₁₆	C₄	1	2	yes	complemented
14	C₁₃×#336	—	2	1	no	—
15	C₂²×C₈×C₁₃	—	2	1	no	—
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
19	C₂×C₄²×C₁₃	—	2	1	no	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	—	2	2	no	—
22	C₂×C₁₃×C₁₆	—	2	2	no	—
23	C₂×C₁₃×C₁₆	—	2	2	no	—
24	C₂×C₁₃×C₁₆	—	2	2	no	—
25	C₂×C₁₃×C₁₆	—	4	1	no	—
26	C₂×C₁₃×C₁₆	—	4	1	no	—
27	C₁₃×C₄⋊C₈	—	2	1	no	—
28	C₁₃×C₄⋊C₈	C₈	1	2	yes	—
29	C₄×C₈×C₁₃	—	4	1	no	—
30	C₄×C₈×C₁₃	—	4	1	no	—
31	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
35	C₁₃×#95	—	2	1	no	—
36	#7167	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₄×C₁₃	—	2	2	no	—
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	—	4	1	no	—
45	C₂×C₈×C₁₃	—	4	1	no	—
46	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
51	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	C₁₆	1	4	yes	complemented
56	C₄²×C₁₃	—	4	1	no	—
57	C₄²×C₁₃	—	4	1	no	—
58	C₄²×C₁₃	C₄²	1	1	yes	characteristic
59	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
62	C₁₃×C₁₆	—	2	4	no	—
63	C₁₃×C₁₆	—	4	2	no	—
64	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic
65	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
66	C₈×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
67	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	4	no	—
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	—	4	2	no	—
72	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
83	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
84	C₈×C₁₃	—	2	1	no	—
85	C₈×C₁₃	—	2	1	no	—
86	C₈×C₁₃	—	4	2	no	—
87	C₈×C₁₃	#102	1	1	yes	characteristic
88	C₈×C₁₃	#103	1	1	yes	characteristic
89	C₈×C₁₃	#106	1	1	yes	characteristic
90	C₈×C₁₃	#107	1	1	yes	characteristic
91	C₂²×C₁₆	—	2	1	no	—
92	C₂²×C₁₆	—	2	1	no	—
93	C₂×C₄⋊C₈	C₄×C₁₃	1	1	yes	characteristic
94	C₂×C₄×C₈	—	2	1	no	—
95	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
96	C₄×C₄⋊C₄	C₄×C₁₃	1	1	yes	characteristic
97	C₄⋊C₁₆	—	2	1	no	—
98	C₄⋊C₁₆	C₄×C₁₃	1	2	yes	complemented
99	C₄×C₁₆	—	2	1	no	—
100	C₄×C₁₆	C₄×C₁₃	1	2	yes	complemented
101	#336	—	2	1	no	—
102	C₂²×C₁₃	D₄⋊C₈	1	1	yes	characteristic
103	C₂²×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
104	C₂²×C₁₃	#336	1	1	yes	characteristic
105	C₂²×C₁₃	C₂²⋊C₁₆	1	2	yes	—
106	C₂²×C₁₃	#350	1	2	yes	—
107	C₄×C₁₃	—	2	1	no	—
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	—	4	4	no	—
110	C₄×C₁₃	D₄⋊C₈	1	1	yes	characteristic
111	C₄×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
112	C₄×C₁₃	#339	1	1	yes	characteristic
113	C₄×C₁₃	#340	1	1	yes	characteristic
114	C₄×C₁₃	C₄×C₁₆	1	1	yes	characteristic
115	C₄×C₁₃	#346	1	1	yes	characteristic
116	C₄×C₁₃	C₄⋊C₁₆	1	2	yes	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
119	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₈×C₁₃	1	2	yes	—
121	C₂×C₄²	—	2	1	no	—
122	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	4	1	no	—
128	C₂×C₁₆	—	4	1	no	—
129	C₄⋊C₈	—	2	1	no	—
130	C₄⋊C₈	C₈×C₁₃	1	2	yes	—
131	C₄×C₈	—	4	1	no	—
132	C₄×C₈	—	4	1	no	—
133	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
134	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
135	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
136	C₄×C₈	Q₈×C₁₃	1	1	yes	characteristic
137	#95	—	2	1	no	—
138	C₂×C₁₃	#1279	1	1	yes	characteristic
139	C₂×C₁₃	D₄⋊C₁₆	1	2	yes	—
140	C₂×C₁₃	Q₈⋊C₁₆	1	2	yes	—
141	C₂×C₁₃	#1374	1	1	yes	characteristic
142	C₂×C₁₃	#1379	1	1	yes	characteristic
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
146	C₂²×C₄	C₁₃×M₁₆	1	1	yes	characteristic
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
157	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
158	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
159	C₂×C₈	C₁₃×Q₁₆	1	1	yes	characteristic
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	C₁₃×C₁₆	1	4	yes	complemented
162	C₄²	—	4	1	no	—
163	C₄²	—	4	1	no	—
164	C₄²	C₄²×C₁₃	1	1	yes	characteristic
165	C₄²	C₁₃×C₄⋊C₄	1	1	yes	characteristic
166	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
167	C₄²	C₁₃×M₁₆	1	1	yes	characteristic
168	C₁₆	—	2	4	no	—
169	C₁₆	—	4	2	no	—
170	C₁₃	#7167	1	1	yes	characteristic, direct product factor, Sylow
171	C₂³	C₁₃×C₂²⋊C₈	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	C₁₃×#95	1	1	yes	characteristic
177	C₂×C₄	C₄×C₈×C₁₃	1	1	yes	characteristic
178	C₂×C₄	C₁₃×#97	1	1	yes	characteristic
179	C₂×C₄	C₁₃×C₂²⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
181	C₂×C₄	C₁₃×#103	1	1	yes	characteristic
182	C₂×C₄	C₁₃×#104	1	1	yes	characteristic
183	C₂×C₄	C₁₃×#104	1	1	yes	characteristic
184	C₂×C₄	C₁₃×C₄⋊C₈	1	1	yes	characteristic
185	C₂×C₄	C₁₃×C₄⋊C₈	1	1	yes	characteristic
186	C₂×C₄	C₂×C₁₃×C₁₆	1	2	yes	—
187	C₂×C₄	C₁₃×M₃₂	1	2	yes	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	4	2	no	—
191	C₈	C₁₃×#102	1	1	yes	characteristic
192	C₈	C₁₃×#103	1	1	yes	characteristic
193	C₈	C₁₃×#106	1	1	yes	characteristic
194	C₈	C₁₃×#107	1	1	yes	characteristic
195	C₂²	C₁₃×D₄⋊C₈	1	1	yes	characteristic
196	C₂²	C₁₃×Q₈⋊C₈	1	1	yes	characteristic
197	C₂²	C₁₃×#336	1	1	yes	characteristic
198	C₂²	C₁₃×C₂²⋊C₁₆	1	2	yes	—
199	C₂²	C₁₃×#350	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	4	no	—
203	C₄	C₁₃×D₄⋊C₈	1	1	yes	characteristic
204	C₄	C₁₃×Q₈⋊C₈	1	1	yes	characteristic
205	C₄	C₁₃×#339	1	1	yes	characteristic
206	C₄	C₁₃×#340	1	1	yes	characteristic
207	C₄	C₄×C₁₃×C₁₆	1	1	yes	characteristic
208	C₄	C₁₃×#346	1	1	yes	characteristic
209	C₄	C₁₃×C₄⋊C₁₆	1	2	yes	—
210	C₂	C₁₃×#1279	1	1	yes	characteristic
211	C₂	C₁₃×D₄⋊C₁₆	1	2	yes	—
212	C₂	C₁₃×Q₈⋊C₁₆	1	2	yes	—
213	C₂	C₁₃×#1374	1	1	yes	characteristic
214	C₂	C₁₃×#1379	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₁₃	D₄
commutator	C₄	C₄×C₁₃×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₃
socle, prime radical	C₂³×C₁₃	C₂²⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ 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×C₄⋊C₁₆	C₄	semidirect
13. C₄×C₁₃×C₁₆	C₄	semidirect
36. #7167	C₁₃	direct (N × H)
55. C₁₃×C₄⋊C₄	C₁₆	semidirect
98. C₄⋊C₁₆	C₄×C₁₃	semidirect
100. C₄×C₁₆	C₄×C₁₃	semidirect
161. C₄⋊C₄	C₁₃×C₁₆	semidirect
170. C₁₃	#7167	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×C₄⋊C₁₆	2
C₂×C₄×C₁₃×C₁₆	2
C₈×C₁₃×C₄⋊C₄	2
#7167	13

Subgroups — C13×#7167 #15693506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#7167 #15693506