Subgroups — C₁₃×#13189 #15699528

Order 3328.

534 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
6	C₂×C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₈×D₄×C₁₃	—	2	2	no	—
10	C₈×D₄×C₁₃	—	2	2	no	—
11	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₁₃×C₂²⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊C₁₆	—	2	2	no	—
15	C₁₃×C₂²⋊C₁₆	—	2	8	no	—
16	C₄×C₁₃×C₁₆	—	2	2	no	—
17	C₂²×D₄×C₁₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₃	—	2	4	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	—	2	4	no	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₄×D₄×C₁₃	—	2	2	no	—
27	C₄×D₄×C₁₃	—	2	2	no	—
28	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
29	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×C₁₆	—	2	2	no	—
32	C₂×C₁₃×C₁₆	—	2	2	no	—
33	C₂×C₁₃×C₁₆	—	2	4	no	—
34	C₂×C₁₃×C₁₆	—	2	2	no	—
35	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
36	C₁₃×C₄⋊C₈	—	2	2	no	—
37	C₁₃×C₂²⋊C₈	—	2	4	no	—
38	C₄×C₈×C₁₃	D₄	1	2	yes	—
39	C₄×C₈×C₁₃	D₄	1	2	yes	—
40	#13189	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂⁴×C₁₃	C₂×C₈	1	2	yes	—
42	C₂×D₄×C₁₃	—	2	2	no	—
43	C₂×D₄×C₁₃	—	2	2	no	—
44	C₂×D₄×C₁₃	C₁₆	1	4	yes	complemented
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	—	2	4	no	—
47	C₂²×C₄×C₁₃	—	2	2	no	—
48	C₂²×C₄×C₁₃	—	2	4	no	—
49	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	—	2	2	no	—
54	C₂×C₈×C₁₃	—	2	2	no	—
55	C₂×C₈×C₁₃	—	2	4	no	—
56	C₂×C₈×C₁₃	—	4	4	no	—
57	C₂×C₈×C₁₃	—	4	2	no	—
58	C₂×C₈×C₁₃	—	4	2	no	—
59	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	#41	1	2	yes	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₂²⋊C₄	—	2	4	no	—
66	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₁₃	M₁₆	1	2	yes	—
68	C₁₃×C₁₆	—	2	4	no	—
69	C₁₃×C₁₆	—	4	2	no	—
70	C₂×C₈×D₄	C₂×C₁₃	1	1	yes	characteristic
71	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
72	C₂×C₂²⋊C₁₆	C₂×C₁₃	1	4	yes	complemented
73	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
74	C₂³×C₁₃	—	2	2	no	—
75	C₂³×C₁₃	—	2	4	no	—
76	C₂³×C₁₃	—	2	2	no	—
77	C₂³×C₁₃	—	2	4	no	—
78	C₂³×C₁₃	C₂×C₁₆	1	4	yes	—
79	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	D₄×C₁₃	—	2	8	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	4	no	—
84	C₂×C₄×C₁₃	—	4	2	no	—
85	C₂×C₄×C₁₃	—	4	2	no	—
86	C₂×C₄×C₁₃	—	4	4	no	—
87	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
88	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
90	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#131	1	2	yes	—
95	C₈×C₁₃	—	2	2	no	—
96	C₈×C₁₃	—	4	2	no	—
97	C₈×C₁₃	#121	1	2	yes	—
98	C₈×C₁₃	#124	1	1	yes	characteristic
99	C₈×C₁₃	#127	1	1	yes	characteristic
100	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
101	C₂×C₄×D₄	C₄×C₁₃	1	1	yes	characteristic
102	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
103	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
104	C₈×D₄	—	2	2	no	—
105	C₈×D₄	—	2	2	no	—
106	C₂×C₄⋊C₈	C₄×C₁₃	1	1	yes	characteristic
107	C₂×C₂²⋊C₈	C₄×C₁₃	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
109	C₄⋊C₁₆	—	2	2	no	—
110	C₂²⋊C₁₆	—	2	8	no	—
111	C₄×C₁₆	—	2	2	no	—
112	C₂²×C₁₃	—	2	4	no	—
113	C₂²×C₁₃	—	2	8	no	—
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	—	4	4	no	—
116	C₂²×C₁₃	—	4	2	no	—
117	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₂²×C₁₃	C₈×D₄	1	2	yes	—
119	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
120	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
121	C₂²×C₁₃	#504	1	2	yes	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	C₂²⋊C₁₆	1	4	yes	complemented
125	C₄×C₁₃	#390	1	1	yes	characteristic
126	C₄×C₁₃	#408	1	1	yes	characteristic
127	C₄×C₁₃	#436	1	2	yes	—
128	C₂²×D₄	C₈×C₁₃	1	1	yes	characteristic
129	C₂³×C₄	C₂×C₄×C₁₃	1	2	yes	—
130	C₂²×C₈	—	2	4	no	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	4	no	—
134	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
135	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
136	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
137	C₄×D₄	—	2	2	no	—
138	C₄×D₄	—	2	2	no	—
139	C₂×C₄⋊C₄	C₈×C₁₃	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	C₈×C₁₃	1	2	yes	—
141	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	4	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	D₄×C₁₃	1	4	yes	complemented
147	C₄⋊C₈	—	2	2	no	—
148	C₂²⋊C₈	—	2	4	no	—
149	C₄×C₈	D₄×C₁₃	1	2	yes	—
150	C₄×C₈	D₄×C₁₃	1	2	yes	—
151	C₂×C₁₃	—	4	4	no	—
152	C₂×C₁₃	#1954	1	1	yes	characteristic
153	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
154	C₂×C₁₃	#2113	1	1	yes	characteristic
155	C₂×C₁₃	D₄×C₁₆	1	2	yes	—
156	C₂×C₁₃	#2169	1	2	yes	—
157	C₂⁴	C₂×C₈×C₁₃	1	2	yes	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	C₁₃×C₁₆	1	4	yes	complemented
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
166	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	4	4	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
176	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
177	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
178	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
179	C₂×C₈	C₁₃×#41	1	2	yes	—
180	C₄⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
183	C₄²	C₁₃×M₁₆	1	2	yes	—
184	C₁₆	—	2	4	no	—
185	C₁₆	—	4	2	no	—
186	C₁₃	#13189	1	1	yes	characteristic, direct product factor, Sylow
187	C₂³	—	2	2	no	—
188	C₂³	—	2	4	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	4	no	—
191	C₂³	C₂×C₁₃×C₁₆	1	4	yes	—
192	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
193	D₄	—	2	8	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
201	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
202	C₂×C₄	C₂×C₁₃×C₁₆	1	2	yes	—
203	C₂×C₄	C₁₃×M₃₂	1	2	yes	—
204	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
206	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
207	C₂×C₄	C₁₃×#131	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	4	2	no	—
210	C₈	C₁₃×#121	1	2	yes	—
211	C₈	C₁₃×#124	1	1	yes	characteristic
212	C₈	C₁₃×#127	1	1	yes	characteristic
213	C₂²	—	2	4	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	4	no	—
217	C₂²	—	4	2	no	—
218	C₂²	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
219	C₂²	C₈×D₄×C₁₃	1	2	yes	—
220	C₂²	C₂²×C₁₃×C₁₆	1	1	yes	characteristic
221	C₂²	C₂×C₁₃×M₃₂	1	1	yes	characteristic
222	C₂²	C₁₃×#504	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	C₁₃×C₂²⋊C₁₆	1	4	yes	complemented
226	C₄	C₁₃×#390	1	1	yes	characteristic
227	C₄	C₁₃×#408	1	1	yes	characteristic
228	C₄	C₁₃×#436	1	2	yes	—
229	C₂	—	4	4	no	—
230	C₂	C₁₃×#1954	1	1	yes	characteristic
231	C₂	C₂×C₁₃×C₂²⋊C₁₆	1	1	yes	characteristic
232	C₂	C₁₃×#2113	1	1	yes	characteristic
233	C₂	D₄×C₁₃×C₁₆	1	2	yes	—
234	C₂	C₁₃×#2169	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₃	C₂³
commutator	C₂²	C₂²×C₁₃×C₁₆
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂²×D₄×C₁₃	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
2. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₁₃×C₂²⋊C₁₆	C₂	semidirect
4. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
35. C₂×C₁₃×C₁₆	D₄	semidirect
40. #13189	C₁₃	direct (N × H)
44. C₂×D₄×C₁₃	C₁₆	semidirect
71. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
72. C₂×C₂²⋊C₁₆	C₂×C₁₃	semidirect
73. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
124. C₄×C₁₃	C₂²⋊C₁₆	semidirect
146. C₂×C₁₆	D₄×C₁₃	semidirect
160. C₂×D₄	C₁₃×C₁₆	semidirect
186. C₁₃	#13189	direct (N × H)
225. C₄	C₁₃×C₂²⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₃	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₂×C₄×C₁₃×C₁₆	2
#13189	13

Subgroups — C13×#13189 #15699528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13189 #15699528