Subgroups — C₁₃×#21960 #15708299

Order 3328.

818 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	C₂²×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2283	C₂	1	2	yes	maximal
3	C₁₃×#1949	C₂	1	8	yes	maximal, complemented
4	C₁₃×#1808	C₂	1	2	yes	maximal
5	C₁₃×#1807	C₂	1	2	yes	maximal
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#107	C₂²	1	4	yes	—
10	C₂×C₁₃×#106	C₂²	1	4	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₄	1	8	yes	complemented
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
13	C₁₃×#383	—	2	2	no	—
14	C₁₃×#382	C₂²	1	4	yes	—
15	C₁₃×#381	—	2	2	no	—
16	C₁₃×#380	C₂²	1	4	yes	—
17	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
18	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
19	C₁₃×#376	—	2	2	no	—
20	C₁₃×#340	C₂²	1	4	yes	—
21	C₁₃×#340	C₂²	1	4	yes	—
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	—	4	2	no	—
26	C₂²×C₈×C₁₃	—	4	2	no	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
31	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
32	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
33	C₂×C₄²×C₁₃	—	2	4	no	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₃	D₄	1	2	yes	—
36	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
38	C₁₃×#107	—	4	4	no	—
39	C₁₃×#106	—	4	4	no	—
40	C₁₃×C₄⋊C₈	C₂×C₄	1	16	yes	—
41	C₁₃×#95	—	2	4	no	—
42	C₁₃×#95	—	2	4	no	—
43	C₁₃×#95	—	2	4	no	—
44	C₁₃×#95	—	2	4	no	—
45	#21960	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	4	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	—	2	8	no	—
51	C₂²×C₄×C₁₃	—	4	2	no	—
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₁₃	—	2	4	no	—
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	—	4	4	no	—
63	C₂×C₈×C₁₃	—	4	4	no	—
64	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
66	C₁₃×C₄⋊C₄	—	2	8	no	—
67	C₁₃×C₄⋊C₄	—	2	8	no	—
68	C₁₃×C₂²⋊C₄	—	4	8	no	—
69	C₄²×C₁₃	—	4	8	no	—
70	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
71	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
72	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
73	C₂²×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
74	#2283	C₂×C₁₃	1	2	yes	—
75	#1949	C₂×C₁₃	1	8	yes	complemented
76	#1808	C₂×C₁₃	1	2	yes	—
77	#1807	C₂×C₁₃	1	2	yes	—
78	C₂³×C₁₃	—	2	2	no	—
79	C₂³×C₁₃	—	2	2	no	—
80	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₁₃	#135	1	4	yes	—
82	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	4	no	—
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	—	2	8	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	4	no	—
90	C₂×C₄×C₁₃	—	4	4	no	—
91	C₂×C₄×C₁₃	—	4	8	no	—
92	C₂×C₄×C₁₃	—	4	8	no	—
93	C₂×C₄×C₁₃	—	4	8	no	—
94	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
100	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
101	C₈×C₁₃	—	2	4	no	—
102	C₈×C₁₃	—	2	4	no	—
103	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
104	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
105	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
106	C₂×#107	C₂²×C₁₃	1	4	yes	—
107	C₂×#106	C₂²×C₁₃	1	4	yes	—
108	C₂×C₄⋊C₈	C₄×C₁₃	1	8	yes	complemented
109	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
110	#383	—	2	2	no	—
111	#382	C₂²×C₁₃	1	4	yes	—
112	#381	—	2	2	no	—
113	#380	C₂²×C₁₃	1	4	yes	—
114	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
115	C₄×C₂²⋊C₄	—	2	2	no	—
116	#376	—	2	2	no	—
117	#340	C₂²×C₁₃	1	4	yes	—
118	#340	C₂²×C₁₃	1	4	yes	—
119	C₂²×C₁₃	—	2	2	no	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	—	2	8	no	—
122	C₂²×C₁₃	#427	1	4	yes	—
123	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₁₃	#537	1	2	yes	—
125	C₂²×C₁₃	C₂×#135	1	2	yes	—
126	C₂²×C₁₃	#575	1	2	yes	—
127	C₄×C₁₃	—	2	2	no	—
128	C₄×C₁₃	—	2	4	no	—
129	C₄×C₁₃	—	2	4	no	—
130	C₄×C₁₃	—	8	8	no	—
131	C₄×C₁₃	#518	1	1	yes	characteristic
132	C₄×C₁₃	#519	1	1	yes	characteristic
133	C₄×C₁₃	#544	1	2	yes	—
134	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
135	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
136	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
137	C₂²×C₈	—	4	2	no	—
138	C₂²×C₈	—	4	2	no	—
139	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
140	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
141	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
142	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
143	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
144	C₂×C₂²⋊C₄	—	2	4	no	—
145	C₂×C₄²	—	2	4	no	—
146	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
147	C₂×C₄²	D₄×C₁₃	1	2	yes	—
148	C₂×C₄²	Q₈×C₁₃	1	4	yes	—
149	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
150	#107	—	4	4	no	—
151	#106	—	4	4	no	—
152	C₄⋊C₈	C₂×C₄×C₁₃	1	16	yes	—
153	#95	—	2	4	no	—
154	#95	—	2	4	no	—
155	#95	—	2	4	no	—
156	#95	—	2	4	no	—
157	C₂×C₁₃	—	2	4	no	—
158	C₂×C₁₃	#2316	1	1	yes	characteristic
159	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
160	C₂×C₁₃	#2911	1	1	yes	characteristic
161	C₂×C₁₃	#3250	1	4	yes	—
162	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
169	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
170	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
171	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
173	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
174	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
175	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
181	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
182	C₄⋊C₄	—	2	8	no	—
183	C₄⋊C₄	—	2	8	no	—
184	C₂²⋊C₄	—	4	8	no	—
185	C₄²	—	4	8	no	—
186	C₄²	C₁₃×C₄⋊C₄	1	8	yes	—
187	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
188	C₄²	C₂×Q₈×C₁₃	1	4	yes	—
189	C₁₃	#21960	1	1	yes	characteristic, direct product factor, Sylow
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
193	C₂³	C₁₃×#135	1	4	yes	—
194	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
207	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
208	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
209	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
211	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
212	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
213	C₈	—	2	4	no	—
214	C₈	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	8	no	—
218	C₂²	C₁₃×#427	1	4	yes	—
219	C₂²	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
220	C₂²	C₁₃×#537	1	2	yes	—
221	C₂²	C₂×C₁₃×#135	1	2	yes	—
222	C₂²	C₁₃×#575	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	4	no	—
226	C₄	—	8	8	no	—
227	C₄	C₁₃×#518	1	1	yes	characteristic
228	C₄	C₁₃×#519	1	1	yes	characteristic
229	C₄	C₁₃×#544	1	2	yes	—
230	C₂	—	2	4	no	—
231	C₂	C₁₃×#2316	1	1	yes	characteristic
232	C₂	C₂×C₁₃×#427	1	1	yes	characteristic
233	C₂	C₁₃×#2911	1	1	yes	characteristic
234	C₂	C₁₃×#3250	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1949	C₂	semidirect
11. C₂×C₁₃×C₄⋊C₈	C₄	semidirect
45. #21960	C₁₃	direct (N × H)
75. #1949	C₂×C₁₃	semidirect
108. C₂×C₄⋊C₈	C₄×C₁₃	semidirect
189. C₁₃	#21960	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×C₄⋊C₈	2
C₁₃×#2283	2
C₁₃×#1949	2
C₁₃×#1808	2
C₁₃×#1807	2
#21960	13

Subgroups — C13×#21960 #15708299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#21960 #15708299