Subgroups — C₂×C₁₃×#2216 #15719937

Order 3328.

702 conjugacy classes of subgroups.

The subgroup lattice has 248 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	C₂³×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#467	C₂	1	2	yes	maximal
4	C₁₃×#2216	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#366	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	C₂²×C₁₃×C₁₆	—	2	4	no	—
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	4	yes	—
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×#366	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	C₂×C₁₃×C₁₆	—	2	2	no	—
41	C₂×C₁₃×C₁₆	—	2	8	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×#2216	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	C₂³×C₁₆	C₂×C₁₃	1	1	yes	characteristic
82	C₂×#467	C₂×C₁₃	1	2	yes	—
83	#2216	C₂×C₁₃	1	8	yes	direct product factor
84	C₂×#366	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	2	no	—
88	C₂³×C₁₃	Q₃₂	1	4	yes	complemented
89	C₂³×C₁₃	C₂×D₈	1	2	yes	—
90	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₁₃	—	4	4	no	—
92	Q₈×C₁₃	—	4	4	no	—
93	Q₈×C₁₃	—	8	4	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	4	no	—
98	C₂×C₄×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	4	4	no	—
100	C₂×C₄×C₁₃	—	8	4	no	—
101	C₂×C₄×C₁₃	—	8	4	no	—
102	C₂×C₄×C₁₃	#121	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#135	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	#121	1	2	yes	—
109	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
111	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
112	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
113	C₂×C₂²⋊Q₈	—	2	2	no	—
114	C₂×Q₃₂	—	2	4	no	—
115	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	complemented
116	C₂²×C₁₆	—	2	4	no	—
117	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
118	C₂²×C₁₆	C₂²×C₁₃	1	4	yes	—
119	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
120	#467	—	2	4	no	—
121	#467	C₂²×C₁₃	1	8	yes	—
122	C₂×#107	C₂²×C₁₃	1	2	yes	—
123	C₂×#103	—	2	2	no	—
124	#366	C₂²×C₁₃	1	4	yes	complemented
125	#358	C₂²×C₁₃	1	8	yes	complemented
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	8	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	#466	1	2	yes	—
130	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
131	C₂²×C₁₃	C₂×Q₃₂	1	4	yes	complemented
132	C₂²×C₁₃	#508	1	2	yes	—
133	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	8	4	no	—
136	C₄×C₁₃	—	8	4	no	—
137	C₄×C₁₃	#466	1	2	yes	—
138	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
139	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
140	C₂²×Q₈	—	2	2	no	—
141	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
142	C₂×Q₁₆	—	2	4	no	—
143	C₂×Q₁₆	—	2	2	no	—
144	C₂×Q₁₆	—	2	4	no	—
145	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
149	C₂²×C₈	D₄×C₁₃	1	2	yes	—
150	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
151	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
152	C₂²⋊Q₈	—	2	8	no	—
153	C₂²⋊Q₈	—	4	4	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	4	2	no	—
156	C₂×C₂²⋊C₄	—	4	2	no	—
157	Q₃₂	—	2	8	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	8	no	—
160	C₂×C₁₆	D₄×C₁₃	1	4	yes	—
161	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
162	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
163	#107	—	2	2	no	—
164	#107	C₂³×C₁₃	1	4	yes	—
165	#103	—	2	8	no	—
166	C₂×C₁₃	—	2	4	no	—
167	C₂×C₁₃	#2216	1	4	yes	direct product factor
168	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
169	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
170	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
171	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
181	C₂²×C₄	C₁₃×D₈	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
183	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
184	Q₁₆	—	2	4	no	—
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	4	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
192	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
193	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
194	C₂×C₈	C₁₃×#41	1	2	yes	—
195	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	8	no	—
199	C₂²⋊C₄	—	4	8	no	—
200	C₁₆	—	2	2	no	—
201	C₁₆	C₂×D₄×C₁₃	1	4	yes	—
202	C₁₃	C₂×#2216	1	1	yes	characteristic, direct product factor, Sylow
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	C₁₃×Q₃₂	1	4	yes	complemented
206	C₂³	C₂×C₁₃×D₈	1	2	yes	—
207	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
208	Q₈	—	4	4	no	—
209	Q₈	—	4	4	no	—
210	Q₈	—	8	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	C₁₃×#121	1	2	yes	—
220	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
221	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
222	C₂×C₄	C₁₃×#135	1	2	yes	—
223	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	C₁₃×#121	1	2	yes	—
226	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
227	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	8	no	—
231	C₂²	C₁₃×#466	1	2	yes	—
232	C₂²	C₂×C₁₃×Q₃₂	1	2	yes	—
233	C₂²	C₂×C₁₃×Q₃₂	1	4	yes	complemented
234	C₂²	C₁₃×#508	1	2	yes	—
235	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	8	4	no	—
238	C₄	—	8	4	no	—
239	C₄	C₁₃×#466	1	2	yes	—
240	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
241	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
242	C₂	—	2	4	no	—
243	C₂	C₁₃×#2216	1	4	yes	direct product factor
244	C₂	C₂×C₁₃×#466	1	1	yes	characteristic
245	C₂	C₂²×C₁₃×Q₃₂	1	1	yes	characteristic
246	C₂	C₂×C₁₃×#508	1	1	yes	characteristic

	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×#2216	C₂	direct (N × H)
5. C₂×C₁₃×#366	C₂	semidirect
6. C₂×C₁₃×#358	C₂	semidirect
11. C₂×C₁₃×Q₃₂	C₂²	semidirect
20. C₁₃×#366	C₂²	semidirect
21. C₁₃×#358	C₂²	semidirect
48. C₂×#2216	C₁₃	direct (N × H)
80. C₂²×Q₃₂	C₂×C₁₃	semidirect
83. #2216	C₂×C₁₃	direct (N × H)
84. C₂×#366	C₂×C₁₃	semidirect
85. C₂×#358	C₂×C₁₃	semidirect
88. C₂³×C₁₃	Q₃₂	semidirect
115. C₂×Q₃₂	C₂²×C₁₃	semidirect
124. #366	C₂²×C₁₃	semidirect
125. #358	C₂²×C₁₃	semidirect
131. C₂²×C₁₃	C₂×Q₃₂	semidirect
167. C₂×C₁₃	#2216	direct (N × H)
202. C₁₃	C₂×#2216	direct (N × H)
205. C₂³	C₁₃×Q₃₂	semidirect
233. C₂²	C₂×C₁₃×Q₃₂	semidirect
243. C₂	C₁₃×#2216	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
C₂³×C₁₃×C₁₆	2
C₂×C₁₃×#467	2
C₁₃×#2216	2
C₂×C₁₃×#366	2
C₂×C₁₃×#358	2
C₂×#2216	13

Subgroups — C2×C13×#2216 #15719937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2216 #15719937