Subgroups — C₁₃×#12666 #15699005

Order 3328.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#347	C₂	1	2	yes	maximal
3	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1770	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1375	C₂	1	2	yes	maximal
6	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₁₃×#97	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₁₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
14	C₁₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
15	C₁₃×#347	—	2	4	no	—
16	C₁₃×#346	C₄	1	2	yes	complemented
17	C₄×C₁₃×C₁₆	C₄	1	2	yes	complemented
18	C₁₃×#336	C₄	1	2	yes	—
19	C₂²×C₈×C₁₃	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	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₁₃×M₃₂	—	2	2	no	—
26	C₁₃×M₃₂	—	2	1	no	—
27	C₁₃×M₃₂	—	2	1	no	—
28	C₁₃×M₃₂	—	2	2	no	—
29	C₁₃×M₃₂	D₄	1	2	yes	—
30	C₁₃×M₃₂	Q₈	1	2	yes	—
31	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
33	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
34	C₁₃×C₄⋊C₈	—	2	1	no	—
35	C₁₃×C₄⋊C₈	—	2	1	no	—
36	C₁₃×#97	—	2	2	no	—
37	C₁₃×#97	—	2	2	no	—
38	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₁₃×#95	—	2	1	no	—
43	#12666	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₄×C₁₃	—	2	2	no	—
45	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	—	2	1	no	—
49	C₂×C₈×C₁₃	—	2	1	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂×D₄	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	—	4	2	no	—
62	C₄²×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	—	2	2	no	—
64	C₄²×C₁₃	C₄²	1	1	yes	characteristic
65	C₄²×C₁₃	C₄²	1	1	yes	characteristic
66	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
67	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
68	C₁₃×C₁₆	—	2	2	no	—
69	C₁₃×C₁₆	—	2	4	no	—
70	C₁₃×C₁₆	C₄⋊C₄	1	4	yes	complemented
71	#2150	C₂×C₁₃	1	1	yes	characteristic
72	C₂×#347	C₂×C₁₃	1	2	yes	—
73	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic
74	#1770	C₂×C₁₃	1	1	yes	characteristic
75	#1375	C₂×C₁₃	1	2	yes	—
76	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	4	1	no	—
79	C₂×C₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	—	4	2	no	—
81	C₂×C₄×C₁₃	—	4	2	no	—
82	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
92	C₈×C₁₃	—	2	1	no	—
93	C₈×C₁₃	—	2	1	no	—
94	C₈×C₁₃	—	4	2	no	—
95	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₈×C₁₃	#117	1	1	yes	characteristic
97	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
98	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
99	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
100	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
101	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
102	C₂×C₄⋊C₈	C₄×C₁₃	1	1	yes	characteristic
103	C₂×#97	C₂²×C₁₃	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
105	C₄×C₄⋊C₄	C₄×C₁₃	1	1	yes	characteristic
106	C₄⋊C₁₆	C₄×C₁₃	1	2	yes	complemented
107	C₄⋊C₁₆	C₄×C₁₃	1	2	yes	complemented
108	#347	—	2	4	no	—
109	#346	C₄×C₁₃	1	2	yes	complemented
110	C₄×C₁₆	C₄×C₁₃	1	2	yes	complemented
111	#336	C₄×C₁₃	1	2	yes	—
112	C₂²×C₁₃	—	2	1	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
115	C₂²×C₁₃	#436	1	1	yes	characteristic
116	C₂²×C₁₃	#446	1	1	yes	characteristic
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	8	2	no	—
121	C₄×C₁₃	#347	1	2	yes	complemented
122	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₁₃	#405	1	1	yes	characteristic
124	C₄×C₁₃	#436	1	1	yes	characteristic
125	C₄×C₁₃	#446	1	1	yes	characteristic
126	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
127	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
128	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
131	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
132	M₃₂	—	2	2	no	—
133	M₃₂	—	2	1	no	—
134	M₃₂	—	2	1	no	—
135	M₃₂	—	2	2	no	—
136	M₃₂	D₄×C₁₃	1	2	yes	—
137	M₃₂	Q₈×C₁₃	1	2	yes	—
138	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
139	C₂×C₁₆	C₂×C₄×C₁₃	1	4	yes	—
140	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	—	2	1	no	—
143	#97	—	2	2	no	—
144	#97	—	2	2	no	—
145	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
146	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
147	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
148	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
149	#95	—	2	1	no	—
150	C₂×C₁₃	—	2	1	no	—
151	C₂×C₁₃	—	2	1	no	—
152	C₂×C₁₃	#1770	1	1	yes	characteristic
153	C₂×C₁₃	C₂×#347	1	1	yes	characteristic
154	C₂×C₁₃	#2110	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₁₃×M₁₆	1	1	yes	characteristic
157	C₂²×C₄	C₁₃×M₁₆	1	1	yes	characteristic
158	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	C₄²×C₁₃	1	1	yes	characteristic
165	C₂×C₈	C₄²×C₁₃	1	1	yes	characteristic
166	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
167	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
168	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
169	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
170	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
171	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	C₄²×C₁₃	1	1	yes	characteristic
176	C₄²	C₄²×C₁₃	1	1	yes	characteristic
177	C₄²	C₁₃×M₁₆	1	1	yes	characteristic
178	C₄²	C₁₃×M₁₆	1	1	yes	characteristic
179	C₁₆	—	2	2	no	—
180	C₁₆	—	2	4	no	—
181	C₁₆	C₁₃×C₄⋊C₄	1	4	yes	complemented
182	C₁₃	#12666	1	1	yes	characteristic, direct product factor, Sylow
183	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	C₁₃×#97	1	1	yes	characteristic
190	C₂×C₄	C₁₃×#97	1	1	yes	characteristic
191	C₂×C₄	C₁₃×#97	1	1	yes	characteristic
192	C₂×C₄	C₁₃×#97	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
194	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
195	C₂×C₄	C₄×Q₈×C₁₃	1	1	yes	characteristic
196	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
197	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
198	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	4	2	no	—
202	C₈	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
203	C₈	C₁₃×#117	1	1	yes	characteristic
204	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
205	C₈	C₄×Q₈×C₁₃	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×C₁₃×#97	1	1	yes	characteristic
209	C₂²	C₁₃×#436	1	1	yes	characteristic
210	C₂²	C₁₃×#446	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	C₁₃×#347	1	2	yes	complemented
216	C₄	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
217	C₄	C₁₃×#405	1	1	yes	characteristic
218	C₄	C₁₃×#436	1	1	yes	characteristic
219	C₄	C₁₃×#446	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	C₁₃×#1770	1	1	yes	characteristic
223	C₂	C₂×C₁₃×#347	1	1	yes	characteristic
224	C₂	C₁₃×#2110	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂³×C₁₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₁₃×C₄⋊C₁₆	C₄	semidirect
14. C₁₃×C₄⋊C₁₆	C₄	semidirect
16. C₁₃×#346	C₄	semidirect
17. C₄×C₁₃×C₁₆	C₄	semidirect
43. #12666	C₁₃	direct (N × H)
70. C₁₃×C₁₆	C₄⋊C₄	semidirect
106. C₄⋊C₁₆	C₄×C₁₃	semidirect
107. C₄⋊C₁₆	C₄×C₁₃	semidirect
109. #346	C₄×C₁₃	semidirect
110. C₄×C₁₆	C₄×C₁₃	semidirect
121. C₄×C₁₃	#347	semidirect
181. C₁₆	C₁₃×C₄⋊C₄	semidirect
182. C₁₃	#12666	direct (N × H)
215. C₄	C₁₃×#347	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#2150	2
C₂×C₁₃×#347	2
C₄×C₁₃×M₃₂	2
C₁₃×#1770	2
C₁₃×#1375	2
#12666	13

Subgroups — C13×#12666 #15699005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12666 #15699005