Subgroups — C₁₃×#17314 #15703653

Order 3328.

914 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#431	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1450	C₂	1	4	yes	maximal, complemented
4	C₈×C₁₃×M₁₆	C₂	1	8	yes	maximal, complemented
5	C₂²×C₁₃×M₁₆	C₄	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×#431	C₄	1	4	yes	—
8	C₁₃×#431	C₄	1	4	yes	—
9	C₁₃×#431	C₂²	1	4	yes	—
10	C₁₃×#431	C₂²	1	4	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₄	1	4	yes	—
12	C₂×C₁₃×C₂²⋊C₈	C₄	1	4	yes	—
13	C₄×C₁₃×M₁₆	C₄	1	4	yes	—
14	C₄×C₁₃×M₁₆	C₂²	1	4	yes	—
15	C₂×C₁₃×#97	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
18	C₁₃×#322	C₂²	1	8	yes	complemented
19	C₈²×C₁₃	C₂²	1	8	yes	complemented
20	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₃×M₁₆	C₈	1	16	yes	complemented
23	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
34	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
35	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
36	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
37	C₁₃×#97	C₂×C₄	1	8	yes	—
38	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
40	C₄×C₈×C₁₃	C₂³	1	4	yes	—
41	C₄×C₈×C₁₃	C₂³	1	8	yes	—
42	#17314	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂⁴×C₁₃	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	—	4	4	no	—
45	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
46	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
47	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
48	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
49	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₁₃×M₁₆	C₂×C₈	1	32	yes	complemented
57	C₂×C₈×C₁₃	—	4	8	no	—
58	C₂×C₈×C₁₃	C₂×C₈	1	16	yes	complemented
59	C₂×C₈×C₁₃	C₂²×C₄	1	16	yes	—
60	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
61	C₄²×C₁₃	C₄²	1	1	yes	characteristic
62	C₄²×C₁₃	C₄²	1	1	yes	characteristic
63	C₄²×C₁₃	C₄²	1	2	yes	—
64	C₄²×C₁₃	C₄²	1	2	yes	—
65	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
67	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₂×#431	C₂×C₁₃	1	2	yes	—
72	C₂×C₄×M₁₆	C₂×C₁₃	1	1	yes	characteristic
73	#1450	C₂×C₁₃	1	4	yes	complemented
74	C₈×M₁₆	C₂×C₁₃	1	8	yes	complemented
75	C₂³×C₁₃	—	4	2	no	—
76	C₂³×C₁₃	C₄×C₈	1	4	yes	—
77	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
78	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
79	C₂×C₄×C₁₃	—	2	4	no	—
80	C₂×C₄×C₁₃	—	2	4	no	—
81	C₂×C₄×C₁₃	—	2	4	no	—
82	C₂×C₄×C₁₃	—	4	2	no	—
83	C₂×C₄×C₁₃	—	4	2	no	—
84	C₂×C₄×C₁₃	—	4	2	no	—
85	C₂×C₄×C₁₃	C₄×C₈	1	8	yes	—
86	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
87	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
88	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
94	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
96	C₈×C₁₃	—	2	16	no	—
97	C₈×C₁₃	C₂²×C₈	1	16	yes	—
98	C₂²×M₁₆	C₄×C₁₃	1	2	yes	—
99	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
100	#431	C₄×C₁₃	1	4	yes	—
101	#431	C₄×C₁₃	1	4	yes	—
102	#431	C₂²×C₁₃	1	4	yes	—
103	#431	C₂²×C₁₃	1	4	yes	—
104	C₂×C₄⋊C₈	C₄×C₁₃	1	4	yes	—
105	C₂×C₂²⋊C₈	C₄×C₁₃	1	4	yes	—
106	C₄×M₁₆	C₄×C₁₃	1	4	yes	—
107	C₄×M₁₆	C₂²×C₁₃	1	4	yes	—
108	C₂×#97	C₄×C₁₃	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
110	C₂×C₄×C₈	C₂²×C₁₃	1	4	yes	—
111	#322	C₂²×C₁₃	1	8	yes	complemented
112	C₈²	C₂²×C₁₃	1	8	yes	complemented
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	—	2	2	no	—
116	C₂²×C₁₃	—	4	4	no	—
117	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
118	C₂²×C₁₃	C₂×C₄×C₈	1	4	yes	—
119	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
124	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
125	C₄×C₁₃	C₂³×C₈	1	2	yes	—
126	C₄×C₁₃	#568	1	4	yes	—
127	C₂³×C₄	C₂×C₄×C₁₃	1	2	yes	—
128	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
129	C₂×M₁₆	C₈×C₁₃	1	16	yes	complemented
130	C₂×M₁₆	C₂×C₄×C₁₃	1	8	yes	—
131	C₂²×C₈	C₂×C₄×C₁₃	1	8	yes	—
132	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
133	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
134	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
135	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
136	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
137	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
138	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
139	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
140	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
141	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
142	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
143	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
144	#97	C₂×C₄×C₁₃	1	8	yes	—
145	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
146	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	—
147	C₄×C₈	C₂³×C₁₃	1	4	yes	—
148	C₄×C₈	C₂³×C₁₃	1	8	yes	—
149	C₂×C₁₃	—	2	4	no	—
150	C₂×C₁₃	—	2	2	no	—
151	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
152	C₂×C₁₃	#2873	1	2	yes	—
153	C₂⁴	C₄²×C₁₃	1	1	yes	characteristic
154	C₂²×C₄	—	4	4	no	—
155	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
156	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
157	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
158	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
159	C₂²×C₄	C₄²×C₁₃	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₈×C₁₃	1	8	yes	—
161	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
162	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
163	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
164	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
166	M₁₆	C₂×C₈×C₁₃	1	32	yes	complemented
167	C₂×C₈	—	4	8	no	—
168	C₂×C₈	C₂×C₈×C₁₃	1	16	yes	complemented
169	C₂×C₈	C₂²×C₄×C₁₃	1	16	yes	—
170	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
171	C₄²	C₄²×C₁₃	1	1	yes	characteristic
172	C₄²	C₄²×C₁₃	1	1	yes	characteristic
173	C₄²	C₄²×C₁₃	1	2	yes	—
174	C₄²	C₄²×C₁₃	1	2	yes	—
175	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
176	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
177	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
178	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
179	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
180	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
181	C₁₃	#17314	1	1	yes	characteristic, direct product factor, Sylow
182	C₂³	—	4	2	no	—
183	C₂³	C₄×C₈×C₁₃	1	4	yes	—
184	C₂³	C₂×C₄²×C₁₃	1	1	yes	characteristic
185	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	C₄×C₈×C₁₃	1	8	yes	—
193	C₂×C₄	C₄×C₈×C₁₃	1	4	yes	—
194	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
195	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
196	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
197	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
198	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
200	C₂×C₄	C₂²×C₈×C₁₃	1	8	yes	—
201	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
202	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
203	C₈	—	2	16	no	—
204	C₈	C₂²×C₈×C₁₃	1	16	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	4	no	—
209	C₂²	C₂×C₄×C₈×C₁₃	1	2	yes	—
210	C₂²	C₂×C₄×C₈×C₁₃	1	4	yes	—
211	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	C₂×C₄×C₈×C₁₃	1	2	yes	—
216	C₄	C₂×C₄×C₈×C₁₃	1	4	yes	—
217	C₄	C₂³×C₈×C₁₃	1	2	yes	—
218	C₄	C₁₃×#568	1	4	yes	—
219	C₂	—	2	4	no	—
220	C₂	—	2	2	no	—
221	C₂	C₂²×C₄×C₈×C₁₃	1	1	yes	characteristic
222	C₂	C₁₃×#2873	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₃	C₂⁴
commutator	C₂	C₂²×C₄×C₈×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×C₄²
prime radical	C₂⁴×C₁₃	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1450	C₂	semidirect
4. C₈×C₁₃×M₁₆	C₂	semidirect
18. C₁₃×#322	C₂²	semidirect
19. C₈²×C₁₃	C₂²	semidirect
22. C₂×C₁₃×M₁₆	C₈	semidirect
42. #17314	C₁₃	direct (N × H)
56. C₁₃×M₁₆	C₂×C₈	semidirect
58. C₂×C₈×C₁₃	C₂×C₈	semidirect
73. #1450	C₂×C₁₃	semidirect
74. C₈×M₁₆	C₂×C₁₃	semidirect
111. #322	C₂²×C₁₃	semidirect
112. C₈²	C₂²×C₁₃	semidirect
129. C₂×M₁₆	C₈×C₁₃	semidirect
166. M₁₆	C₂×C₈×C₁₃	semidirect
168. C₂×C₈	C₂×C₈×C₁₃	semidirect
181. C₁₃	#17314	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×#431	2
C₂×C₄×C₁₃×M₁₆	2
C₁₃×#1450	2
C₈×C₁₃×M₁₆	2
#17314	13

Subgroups — C13×#17314 #15703653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17314 #15703653