Subgroups — C₁₃×#13235 #15699574

Order 3328.

510 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#357	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1970	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
7	C₂×C₁₃×D₁₆	—	2	2	no	—
8	C₂×C₁₃×D₁₆	—	2	2	no	—
9	C₂×C₁₃×D₁₆	C₄	1	4	yes	complemented
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#102	—	2	2	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#390	—	2	2	no	—
16	C₁₃×#366	—	2	2	no	—
17	C₁₃×#357	—	2	4	no	—
18	C₄×C₁₃×C₁₆	—	2	2	no	—
19	C₂²×D₄×C₁₃	—	2	2	no	—
20	C₂×C₁₃×D₈	—	2	2	no	—
21	C₂×C₁₃×D₈	—	2	2	no	—
22	C₂×C₁₃×D₈	—	2	2	no	—
23	C₂×C₁₃×D₈	—	2	2	no	—
24	C₂×C₁₃×D₈	—	2	2	no	—
25	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₁₃×C₂²⋊C₄	—	4	4	no	—
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₁₃×D₁₆	—	2	8	no	—
32	C₂×C₁₃×C₁₆	—	2	1	no	—
33	C₂×C₁₃×C₁₆	—	2	1	no	—
34	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
36	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
37	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
38	C₁₃×#107	—	2	2	no	—
39	C₁₃×#107	—	2	2	no	—
40	C₁₃×#102	—	4	4	no	—
41	C₄×C₈×C₁₃	—	2	1	no	—
42	C₄×C₈×C₁₃	—	2	1	no	—
43	#13235	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂⁴×C₁₃	—	4	2	no	—
45	C₂×D₄×C₁₃	—	2	4	no	—
46	C₂×D₄×C₁₃	—	4	2	no	—
47	C₂×D₄×C₁₃	—	4	2	no	—
48	C₂×D₄×C₁₃	—	4	2	no	—
49	C₂×D₄×C₁₃	—	4	2	no	—
50	C₂×D₄×C₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	—	4	2	no	—
52	C₂²×C₄×C₁₃	D₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₁₃×D₈	—	2	4	no	—
55	C₁₃×D₈	—	2	4	no	—
56	C₁₃×D₈	—	4	2	no	—
57	C₁₃×D₈	—	4	2	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	D₈	1	2	yes	—
61	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
66	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	—	4	2	no	—
68	C₁₃×C₄⋊C₄	—	4	2	no	—
69	C₁₃×C₂²⋊C₄	—	8	8	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₁₃×C₁₆	—	2	2	no	—
73	C₁₃×C₁₆	C₂²⋊C₄	1	4	yes	complemented
74	C₂²×D₁₆	C₂×C₁₃	1	1	yes	characteristic
75	C₂×#366	C₂×C₁₃	1	1	yes	characteristic, complemented
76	C₂×#357	C₂×C₁₃	1	2	yes	complemented
77	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
78	#1970	C₂×C₁₃	1	2	yes	complemented
79	C₂³×C₁₃	—	4	4	no	—
80	C₂³×C₁₃	—	8	2	no	—
81	C₂³×C₁₃	—	8	2	no	—
82	C₂³×C₁₃	—	8	4	no	—
83	C₂³×C₁₃	—	8	4	no	—
84	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
85	D₄×C₁₃	—	4	4	no	—
86	D₄×C₁₃	—	4	4	no	—
87	D₄×C₁₃	—	8	2	no	—
88	D₄×C₁₃	—	8	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	8	2	no	—
92	C₂×C₄×C₁₃	—	8	4	no	—
93	C₂×C₄×C₁₃	D₁₆	1	4	yes	complemented
94	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
100	C₈×C₁₃	—	2	2	no	—
101	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
103	C₈×C₁₃	#121	1	1	yes	characteristic
104	C₈×C₁₃	#124	1	1	yes	characteristic
105	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
106	C₂×D₁₆	—	2	2	no	—
107	C₂×D₁₆	—	2	2	no	—
108	C₂×D₁₆	C₄×C₁₃	1	4	yes	complemented
109	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
110	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
111	C₂×#107	C₂²×C₁₃	1	2	yes	—
112	C₂×#102	—	2	2	no	—
113	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
114	#390	—	2	2	no	—
115	#366	—	2	2	no	—
116	#357	—	2	4	no	—
117	C₄×C₁₆	—	2	2	no	—
118	C₂²×C₁₃	—	8	8	no	—
119	C₂²×C₁₃	—	8	4	no	—
120	C₂²×C₁₃	—	16	4	no	—
121	C₂²×C₁₃	—	16	2	no	—
122	C₂²×C₁₃	—	16	2	no	—
123	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
124	C₂²×C₁₃	#466	1	1	yes	characteristic
125	C₂²×C₁₃	#493	1	1	yes	characteristic
126	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
127	C₂²×C₁₃	#508	1	2	yes	—
128	C₄×C₁₃	—	2	4	no	—
129	C₄×C₁₃	—	16	2	no	—
130	C₄×C₁₃	#390	1	1	yes	characteristic
131	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
132	C₄×C₁₃	#466	1	1	yes	characteristic
133	C₄×C₁₃	#495	1	1	yes	characteristic
134	C₂²×D₄	—	2	2	no	—
135	C₂×D₈	—	2	2	no	—
136	C₂×D₈	—	2	2	no	—
137	C₂×D₈	—	2	2	no	—
138	C₂×D₈	—	2	2	no	—
139	C₂×D₈	—	2	2	no	—
140	C₂×D₈	C₂×C₄×C₁₃	1	4	yes	—
141	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₂²⋊C₄	—	4	4	no	—
145	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
146	D₁₆	—	2	8	no	—
147	C₂×C₁₆	—	2	1	no	—
148	C₂×C₁₆	—	2	1	no	—
149	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
150	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
151	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
152	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
153	#107	—	2	2	no	—
154	#107	—	2	2	no	—
155	#102	—	4	4	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	2	1	no	—
158	C₂×C₁₃	—	16	4	no	—
159	C₂×C₁₃	#1970	1	1	yes	characteristic
160	C₂×C₁₃	C₄×D₁₆	1	2	yes	—
161	C₂×C₁₃	#2215	1	2	yes	—
162	C₂×C₁₃	#2246	1	1	yes	characteristic
163	C₂×C₁₃	#2249	1	1	yes	characteristic
164	C₂⁴	—	4	2	no	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	C₁₃×D₈	1	2	yes	—
173	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
174	D₈	—	2	4	no	—
175	D₈	—	2	4	no	—
176	D₈	—	4	2	no	—
177	D₈	—	4	2	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	C₁₃×D₈	1	2	yes	—
181	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
184	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
185	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
186	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	—	8	8	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₁₆	—	2	2	no	—
193	C₁₆	C₁₃×C₂²⋊C₄	1	4	yes	complemented
194	C₁₃	#13235	1	1	yes	characteristic, direct product factor, Sylow
195	C₂³	—	4	4	no	—
196	C₂³	—	8	2	no	—
197	C₂³	—	8	2	no	—
198	C₂³	—	8	4	no	—
199	C₂³	—	8	4	no	—
200	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
201	D₄	—	4	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	8	2	no	—
204	D₄	—	8	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	C₁₃×D₁₆	1	4	yes	complemented
210	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
212	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
213	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
214	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
215	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
218	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
219	C₈	C₁₃×#121	1	1	yes	characteristic
220	C₈	C₁₃×#124	1	1	yes	characteristic
221	C₂²	—	8	8	no	—
222	C₂²	—	8	4	no	—
223	C₂²	—	16	2	no	—
224	C₂²	—	16	2	no	—
225	C₂²	—	16	4	no	—
226	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
227	C₂²	C₁₃×#466	1	1	yes	characteristic
228	C₂²	C₁₃×#493	1	1	yes	characteristic
229	C₂²	C₂×C₁₃×D₁₆	1	2	yes	—
230	C₂²	C₁₃×#508	1	2	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	16	2	no	—
233	C₄	C₁₃×#390	1	1	yes	characteristic
234	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
235	C₄	C₁₃×#466	1	1	yes	characteristic
236	C₄	C₁₃×#495	1	1	yes	characteristic
237	C₂	—	16	4	no	—
238	C₂	C₁₃×#1970	1	1	yes	characteristic
239	C₂	C₄×C₁₃×D₁₆	1	2	yes	—
240	C₂	C₁₃×#2215	1	2	yes	—
241	C₂	C₁₃×#2246	1	1	yes	characteristic
242	C₂	C₁₃×#2249	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#366	C₂	semidirect
3. C₂×C₁₃×#357	C₂	semidirect
4. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
5. C₁₃×#1970	C₂	semidirect
9. C₂×C₁₃×D₁₆	C₄	semidirect
43. #13235	C₁₃	direct (N × H)
73. C₁₃×C₁₆	C₂²⋊C₄	semidirect
75. C₂×#366	C₂×C₁₃	semidirect
76. C₂×#357	C₂×C₁₃	semidirect
77. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
78. #1970	C₂×C₁₃	semidirect
93. C₂×C₄×C₁₃	D₁₆	semidirect
108. C₂×D₁₆	C₄×C₁₃	semidirect
193. C₁₆	C₁₃×C₂²⋊C₄	semidirect
194. C₁₃	#13235	direct (N × H)
209. C₂×C₄	C₁₃×D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×D₁₆	2
C₂×C₁₃×#366	2
C₂×C₁₃×#357	2
C₂×C₄×C₁₃×C₁₆	2
C₁₃×#1970	2
#13235	13

Subgroups — C13×#13235 #15699574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13235 #15699574