Subgroups — C₂×C₁₃×#2219 #15719941

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#467	C₂	1	2	yes	maximal
4	C₁₃×#2219	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#358	C₂	1	2	yes	maximal, complemented
7	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₂²⋊Q₈	—	2	2	no	—
10	C₂×C₁₃×Q₃₂	—	2	4	no	—
11	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	complemented
12	C₂×C₁₃×M₃₂	—	2	4	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#467	—	2	4	no	—
17	C₁₃×#467	C₂²	1	8	yes	—
18	C₂×C₁₃×#107	C₂²	1	2	yes	—
19	C₂×C₁₃×#103	—	2	2	no	—
20	C₁₃×#367	C₂²	1	4	yes	complemented
21	C₁₃×#358	C₂²	1	8	yes	complemented
22	C₂²×Q₈×C₁₃	—	2	2	no	—
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₁₃×Q₁₆	—	2	4	no	—
25	C₂×C₁₃×Q₁₆	—	2	2	no	—
26	C₂×C₁₃×Q₁₆	—	2	4	no	—
27	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×C₂²⋊Q₈	—	2	8	no	—
35	C₁₃×C₂²⋊Q₈	—	4	4	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
38	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
39	C₁₃×Q₃₂	—	2	8	no	—
40	C₁₃×M₃₂	—	2	8	no	—
41	C₂×C₁₃×C₁₆	—	2	2	no	—
42	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
43	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
44	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
45	C₁₃×#107	—	2	2	no	—
46	C₁₃×#107	C₂³	1	4	yes	—
47	C₁₃×#103	—	2	8	no	—
48	C₂×#2219	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
50	C₂×Q₈×C₁₃	—	2	4	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂×Q₈×C₁₃	—	4	4	no	—
53	C₂×Q₈×C₁₃	—	4	4	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	4	2	no	—
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	D₈	1	2	yes	—
59	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₁₃×Q₁₆	—	2	4	no	—
63	C₁₃×Q₁₆	—	2	4	no	—
64	C₁₃×Q₁₆	—	4	4	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	4	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃	#41	1	2	yes	—
73	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	2	4	no	—
75	C₁₃×C₄⋊C₄	—	4	2	no	—
76	C₁₃×C₄⋊C₄	—	4	8	no	—
77	C₁₃×C₂²⋊C₄	—	4	8	no	—
78	C₁₃×C₁₆	—	2	2	no	—
79	C₁₃×C₁₆	C₂×D₄	1	4	yes	—
80	C₂²×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
81	C₂²×M₃₂	C₂×C₁₃	1	1	yes	characteristic
82	C₂×#467	C₂×C₁₃	1	2	yes	—
83	#2219	C₂×C₁₃	1	8	yes	direct product factor
84	C₂×#367	C₂×C₁₃	1	1	yes	characteristic, complemented
85	C₂×#358	C₂×C₁₃	1	2	yes	complemented
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	2	4	no	—
88	C₂³×C₁₃	C₂×D₈	1	2	yes	—
89	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
90	Q₈×C₁₃	—	4	4	no	—
91	Q₈×C₁₃	—	4	4	no	—
92	Q₈×C₁₃	—	8	4	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	4	4	no	—
98	C₂×C₄×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	8	4	no	—
100	C₂×C₄×C₁₃	—	8	4	no	—
101	C₂×C₄×C₁₃	#121	1	2	yes	—
102	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#135	1	2	yes	—
105	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₁₃	—	2	2	no	—
107	C₈×C₁₃	#121	1	2	yes	—
108	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
111	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
112	C₂×C₂²⋊Q₈	—	2	2	no	—
113	C₂×Q₃₂	—	2	4	no	—
114	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	complemented
115	C₂×M₃₂	—	2	4	no	—
116	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
117	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
118	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
119	#467	—	2	4	no	—
120	#467	C₂²×C₁₃	1	8	yes	—
121	C₂×#107	C₂²×C₁₃	1	2	yes	—
122	C₂×#103	—	2	2	no	—
123	#367	C₂²×C₁₃	1	4	yes	complemented
124	#358	C₂²×C₁₃	1	8	yes	complemented
125	C₂²×C₁₃	—	2	4	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	4	4	no	—
128	C₂²×C₁₃	#466	1	2	yes	—
129	C₂²×C₁₃	#510	1	4	yes	—
130	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
131	C₄×C₁₃	—	2	2	no	—
132	C₄×C₁₃	—	8	4	no	—
133	C₄×C₁₃	—	8	4	no	—
134	C₄×C₁₃	#466	1	2	yes	—
135	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
136	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
137	C₂²×Q₈	—	2	2	no	—
138	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
139	C₂×Q₁₆	—	2	4	no	—
140	C₂×Q₁₆	—	2	2	no	—
141	C₂×Q₁₆	—	2	4	no	—
142	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
146	C₂²×C₈	D₄×C₁₃	1	2	yes	—
147	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
148	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
149	C₂²⋊Q₈	—	2	8	no	—
150	C₂²⋊Q₈	—	4	4	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	4	2	no	—
153	C₂×C₂²⋊C₄	—	4	2	no	—
154	Q₃₂	—	2	8	no	—
155	M₃₂	—	2	8	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	D₄×C₁₃	1	4	yes	—
158	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
159	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
160	#107	—	2	2	no	—
161	#107	C₂³×C₁₃	1	4	yes	—
162	#103	—	2	8	no	—
163	C₂×C₁₃	—	4	2	no	—
164	C₂×C₁₃	#2219	1	4	yes	direct product factor
165	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
166	C₂×C₁₃	C₂×#510	1	2	yes	—
167	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
177	C₂²×C₄	C₁₃×D₈	1	2	yes	—
178	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
179	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
180	Q₁₆	—	2	4	no	—
181	Q₁₆	—	2	4	no	—
182	Q₁₆	—	4	4	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
188	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
189	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
190	C₂×C₈	C₁₃×#41	1	2	yes	—
191	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	8	no	—
195	C₂²⋊C₄	—	4	8	no	—
196	C₁₆	—	2	2	no	—
197	C₁₆	C₂×D₄×C₁₃	1	4	yes	—
198	C₁₃	C₂×#2219	1	1	yes	characteristic, direct product factor, Sylow
199	C₂³	—	2	2	no	—
200	C₂³	—	2	4	no	—
201	C₂³	C₂×C₁₃×D₈	1	2	yes	—
202	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
203	Q₈	—	4	4	no	—
204	Q₈	—	4	4	no	—
205	Q₈	—	8	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	—	8	4	no	—
214	C₂×C₄	C₁₃×#121	1	2	yes	—
215	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
216	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
217	C₂×C₄	C₁₃×#135	1	2	yes	—
218	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	C₁₃×#121	1	2	yes	—
221	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
222	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
223	C₂²	—	2	2	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	4	4	no	—
226	C₂²	C₁₃×#466	1	2	yes	—
227	C₂²	C₁₃×#510	1	4	yes	—
228	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	8	4	no	—
231	C₄	—	8	4	no	—
232	C₄	C₁₃×#466	1	2	yes	—
233	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
234	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
235	C₂	—	4	2	no	—
236	C₂	C₁₃×#2219	1	4	yes	direct product factor
237	C₂	C₂×C₁₃×#466	1	1	yes	characteristic
238	C₂	C₂×C₁₃×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×Q₃₂	C₂	semidirect
4. C₁₃×#2219	C₂	direct (N × H)
5. C₂×C₁₃×#367	C₂	semidirect
6. C₂×C₁₃×#358	C₂	semidirect
11. C₂×C₁₃×Q₃₂	C₂²	semidirect
20. C₁₃×#367	C₂²	semidirect
21. C₁₃×#358	C₂²	semidirect
48. C₂×#2219	C₁₃	direct (N × H)
80. C₂²×Q₃₂	C₂×C₁₃	semidirect
83. #2219	C₂×C₁₃	direct (N × H)
84. C₂×#367	C₂×C₁₃	semidirect
85. C₂×#358	C₂×C₁₃	semidirect
114. C₂×Q₃₂	C₂²×C₁₃	semidirect
123. #367	C₂²×C₁₃	semidirect
124. #358	C₂²×C₁₃	semidirect
164. C₂×C₁₃	#2219	direct (N × H)
198. C₁₃	C₂×#2219	direct (N × H)
236. C₂	C₁₃×#2219	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×Q₃₂	2
C₂²×C₁₃×M₃₂	2
C₂×C₁₃×#467	2
C₁₃×#2219	2
C₂×C₁₃×#367	2
C₂×C₁₃×#358	2
C₂×#2219	13

Subgroups — C2×C13×#2219 #15719941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2219 #15719941