Subgroups — C₂×C₁₃×#2169 #15719770

Order 3328.

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

	subgroup	G/H
center	C₂²×C₈×C₁₃	C₂³
commutator	C₂²	C₂³×C₈×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×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
1. C₂²×C₁₃×M₃₂	C₂	semidirect
3. C₁₃×#2169	C₂	direct (N × H)
4. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
5. C₂×C₁₃×C₂²⋊C₁₆	C₂	semidirect
6. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
10. C₂×C₁₃×M₃₂	C₂²	semidirect
19. C₁₃×C₄⋊C₁₆	C₂²	semidirect
20. C₁₃×C₂²⋊C₁₆	C₂²	semidirect
21. C₄×C₁₃×C₁₆	C₂²	semidirect
39. C₂×C₁₃×C₁₆	D₄	semidirect
47. C₂×#2169	C₁₃	direct (N × H)
76. C₁₃×C₁₆	C₂×D₄	semidirect
77. C₂²×M₃₂	C₂×C₁₃	semidirect
79. #2169	C₂×C₁₃	direct (N × H)
80. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
81. C₂×C₂²⋊C₁₆	C₂×C₁₃	semidirect
82. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
94. C₂×C₄×C₁₃	M₃₂	semidirect
110. C₂×M₃₂	C₂²×C₁₃	semidirect
119. C₄⋊C₁₆	C₂²×C₁₃	semidirect
120. C₂²⋊C₁₆	C₂²×C₁₃	semidirect
121. C₄×C₁₆	C₂²×C₁₃	semidirect
133. C₄×C₁₃	C₂×M₃₂	semidirect
153. C₂×C₁₆	D₄×C₁₃	semidirect
162. C₂×C₁₃	#2169	direct (N × H)
194. C₁₆	C₂×D₄×C₁₃	semidirect
195. C₁₃	C₂×#2169	direct (N × H)
207. C₂×C₄	C₁₃×M₃₂	semidirect
231. C₄	C₂×C₁₃×M₃₂	semidirect
235. C₂	C₁₃×#2169	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×M₃₂	2
C₂×C₈×D₄×C₁₃	2
C₁₃×#2169	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₂×C₄×C₁₃×C₁₆	2
C₂×#2169	13

Subgroups — C2×C13×#2169 #15719770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2169 #15719770