Subgroups — C₁₃×#12654 #15698993

Order 3328.

382 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×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1726	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1374	C₂	1	2	yes	maximal
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
8	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
9	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
10	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#376	C₄	1	1	yes	characteristic
15	C₁₃×C₄⋊C₁₆	C₄	1	8	yes	complemented
16	C₁₃×#346	—	2	2	no	—
17	C₄×C₁₃×C₁₆	—	2	2	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃×C₁₆	—	2	2	no	—
26	C₂×C₁₃×C₁₆	—	2	2	no	—
27	C₂×C₁₃×C₁₆	—	2	2	no	—
28	C₂×C₁₃×C₁₆	—	2	2	no	—
29	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
31	C₁₃×#97	—	2	2	no	—
32	C₁₃×#97	—	2	2	no	—
33	C₁₃×#97	C₈	1	4	yes	—
34	C₄×C₈×C₁₃	—	2	2	no	—
35	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₁₃	D₄	1	2	yes	—
37	C₄×C₈×C₁₃	Q₈	1	2	yes	—
38	C₁₃×#95	C₈	1	4	yes	—
39	#12654	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	—	2	1	no	—
45	C₂×C₈×C₁₃	—	2	1	no	—
46	C₂×C₈×C₁₃	—	2	1	no	—
47	C₂×C₈×C₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	C₄²	1	2	yes	—
49	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
53	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₁₃	#41	1	2	yes	—
57	C₄²×C₁₃	—	2	1	no	—
58	C₄²×C₁₃	—	2	1	no	—
59	C₄²×C₁₃	C₄²	1	2	yes	—
60	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
61	C₄²×C₁₃	C₂×C₈	1	2	yes	—
62	C₄²×C₁₃	M₁₆	1	2	yes	—
63	C₁₃×C₁₆	—	2	4	no	—
64	C₁₃×C₁₆	—	2	4	no	—
65	C₂×C₄⋊C₁₆	C₂×C₁₃	1	2	yes	—
66	C₂×#346	C₂×C₁₃	1	1	yes	characteristic
67	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
68	#1726	C₂×C₁₃	1	1	yes	characteristic
69	#1374	C₂×C₁₃	1	2	yes	—
70	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	2	4	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	4	no	—
75	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
76	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
77	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
83	C₈×C₁₃	—	2	2	no	—
84	C₈×C₁₃	—	2	2	no	—
85	C₈×C₁₃	C₄⋊C₈	1	4	yes	—
86	C₈×C₁₃	C₄×D₄	1	2	yes	—
87	C₈×C₁₃	C₄×Q₈	1	2	yes	—
88	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
89	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
90	C₂×#97	C₄×C₁₃	1	1	yes	characteristic
91	C₂×#97	C₄×C₁₃	1	1	yes	characteristic
92	C₂×#97	C₄×C₁₃	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
96	#376	C₄×C₁₃	1	1	yes	characteristic
97	C₄⋊C₁₆	C₄×C₁₃	1	8	yes	complemented
98	#346	—	2	2	no	—
99	C₄×C₁₆	—	2	2	no	—
100	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
101	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₁₃	#431	1	1	yes	characteristic
103	C₂²×C₁₃	#504	1	4	yes	—
104	C₄×C₁₃	—	2	4	no	—
105	C₄×C₁₃	—	2	4	no	—
106	C₄×C₁₃	—	4	2	no	—
107	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
109	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₄×C₁₃	#431	1	1	yes	characteristic
111	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
112	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
113	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
115	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
116	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
117	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	C₂×C₄×C₁₃	1	4	yes	—
123	C₂×C₁₆	C₂×C₄×C₁₃	1	4	yes	—
124	#97	—	2	2	no	—
125	#97	—	2	2	no	—
126	#97	C₈×C₁₃	1	4	yes	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
129	C₄×C₈	D₄×C₁₃	1	2	yes	—
130	C₄×C₈	Q₈×C₁₃	1	2	yes	—
131	#95	C₈×C₁₃	1	4	yes	—
132	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
133	C₂×C₁₃	#2108	1	2	yes	—
134	C₂×C₁₃	#2151	1	2	yes	—
135	C₂×C₁₃	#2164	1	2	yes	—
136	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
137	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
138	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
139	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	C₄²×C₁₃	1	2	yes	—
145	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
146	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
147	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
148	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
149	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
150	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
152	C₂×C₈	C₁₃×#41	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	C₄²×C₁₃	1	2	yes	—
156	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
157	C₄²	C₂×C₈×C₁₃	1	2	yes	—
158	C₄²	C₁₃×M₁₆	1	2	yes	—
159	C₁₆	—	2	4	no	—
160	C₁₆	—	2	4	no	—
161	C₁₃	#12654	1	1	yes	characteristic, direct product factor, Sylow
162	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	C₄×C₈×C₁₃	1	4	yes	—
168	C₂×C₄	C₁₃×C₄⋊C₈	1	4	yes	—
169	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
170	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
172	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
174	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	C₁₃×C₄⋊C₈	1	4	yes	—
178	C₈	C₄×D₄×C₁₃	1	2	yes	—
179	C₈	C₄×Q₈×C₁₃	1	2	yes	—
180	C₂²	C₂×C₄×C₈×C₁₃	1	1	yes	characteristic
181	C₂²	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
182	C₂²	C₁₃×#431	1	1	yes	characteristic
183	C₂²	C₁₃×#504	1	4	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
188	C₄	C₄×C₁₃×M₁₆	1	1	yes	characteristic
189	C₄	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
190	C₄	C₁₃×#431	1	1	yes	characteristic
191	C₂	C₄×C₁₃×C₄⋊C₈	1	1	yes	characteristic
192	C₂	C₁₃×#2108	1	2	yes	—
193	C₂	C₁₃×#2151	1	2	yes	—
194	C₂	C₁₃×#2164	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
15. C₁₃×C₄⋊C₁₆	C₄	semidirect
39. #12654	C₁₃	direct (N × H)
97. C₄⋊C₁₆	C₄×C₁₃	semidirect
161. C₁₃	#12654	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×C₁₃×#346	2
C₂×C₄×C₁₃×C₁₆	2
C₁₃×#1726	2
C₁₃×#1374	2
#12654	13

Subgroups — C13×#12654 #15698993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12654 #15698993