Subgroups — C₂×C₁₃×#2091 #15710061

Order 3328.

1102 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	C₂×C₁₃×#430	C₂	1	3	yes	maximal, complemented
2	C₂×C₁₃×#407	C₂	1	3	yes	maximal
3	C₁₃×#2091	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	3	yes	—
7	C₂²×C₁₃×M₁₆	C₂²	1	3	yes	—
8	C₁₃×#430	—	2	6	no	—
9	C₁₃×#430	C₂²	1	12	yes	complemented
10	C₂×C₁₃×#108	—	2	6	no	—
11	C₁₃×#407	—	2	6	no	—
12	C₁₃×#407	C₂²	1	12	yes	—
13	C₂×C₁₃×C₂²⋊C₈	—	2	6	no	—
14	C₁₃×#338	C₂²	1	4	yes	complemented
15	C₂⁵×C₁₃	Q₈	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	—	2	3	no	—
17	C₂³×C₄×C₁₃	—	2	3	no	—
18	C₂³×C₄×C₁₃	—	2	3	no	—
19	C₂³×C₄×C₁₃	—	2	3	no	—
20	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	—	2	6	no	—
25	C₂×C₁₃×M₁₆	—	2	6	no	—
26	C₂×C₁₃×M₁₆	—	2	3	no	—
27	C₂×C₁₃×M₁₆	—	2	6	no	—
28	C₂×C₁₃×M₁₆	—	2	6	no	—
29	C₂×C₁₃×M₁₆	D₄	1	6	yes	—
30	C₂×C₁₃×M₁₆	C₂³	1	6	yes	—
31	C₂×C₁₃×M₁₆	C₂³	1	6	yes	—
32	C₂²×C₈×C₁₃	—	2	3	no	—
33	C₂²×C₈×C₁₃	D₄	1	6	yes	—
34	C₁₃×#108	—	2	24	no	—
35	C₁₃×C₂²⋊C₈	—	2	24	no	—
36	C₂×#2091	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂⁴×C₁₃	—	2	3	no	—
38	C₂⁴×C₁₃	—	2	3	no	—
39	C₂⁴×C₁₃	—	4	4	no	—
40	C₂⁴×C₁₃	C₂×Q₈	1	2	yes	—
41	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	3	no	—
43	C₂²×C₄×C₁₃	—	2	3	no	—
44	C₂²×C₄×C₁₃	—	2	6	no	—
45	C₂²×C₄×C₁₃	—	2	6	no	—
46	C₂²×C₄×C₁₃	—	2	6	no	—
47	C₂²×C₄×C₁₃	—	2	6	no	—
48	C₂²×C₄×C₁₃	—	4	3	no	—
49	C₂²×C₄×C₁₃	—	4	3	no	—
50	C₂²×C₄×C₁₃	—	4	6	no	—
51	C₂²×C₄×C₁₃	—	4	3	no	—
52	C₂²×C₄×C₁₃	—	4	1	no	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	#41	1	3	yes	—
58	C₂²×C₄×C₁₃	#41	1	3	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₁₃×M₁₆	—	2	12	no	—
61	C₁₃×M₁₆	—	2	12	no	—
62	C₁₃×M₁₆	—	2	12	no	—
63	C₁₃×M₁₆	—	4	6	no	—
64	C₂×C₈×C₁₃	—	2	6	no	—
65	C₂×C₈×C₁₃	—	2	6	no	—
66	C₂×C₈×C₁₃	—	2	6	no	—
67	C₂×C₈×C₁₃	—	4	3	no	—
68	C₂×C₈×C₁₃	—	4	3	no	—
69	C₂×C₈×C₁₃	C₂×D₄	1	12	yes	—
70	C₂×#430	C₂×C₁₃	1	3	yes	complemented
71	C₂×#407	C₂×C₁₃	1	3	yes	—
72	#2091	C₂×C₁₃	1	8	yes	direct product factor
73	C₂×#338	C₂×C₁₃	1	1	yes	characteristic, complemented
74	C₂³×C₁₃	—	2	12	no	—
75	C₂³×C₁₃	—	2	6	no	—
76	C₂³×C₁₃	—	4	2	no	—
77	C₂³×C₁₃	—	4	4	no	—
78	C₂³×C₁₃	—	4	6	no	—
79	C₂³×C₁₃	—	4	12	no	—
80	C₂³×C₁₃	—	4	3	no	—
81	C₂³×C₁₃	—	4	1	no	—
82	C₂³×C₁₃	C₂²⋊Q₈	1	3	yes	—
83	C₂³×C₁₃	C₂²⋊Q₈	1	3	yes	—
84	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	—	2	6	no	—
86	C₂×C₄×C₁₃	—	4	3	no	—
87	C₂×C₄×C₁₃	—	4	1	no	—
88	C₂×C₄×C₁₃	—	4	6	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	4	6	no	—
91	C₂×C₄×C₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	C₂²⋊Q₈	1	3	yes	—
93	C₂×C₄×C₁₃	C₂²⋊Q₈	1	3	yes	—
94	C₂×C₄×C₁₃	#123	1	3	yes	—
95	C₂×C₄×C₁₃	#123	1	3	yes	—
96	C₂×C₄×C₁₃	#123	1	6	yes	—
97	C₂×C₄×C₁₃	#127	1	3	yes	—
98	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
100	C₂×C₄×C₁₃	C₂×#41	1	3	yes	—
101	C₈×C₁₃	—	2	12	no	—
102	C₈×C₁₃	—	4	6	no	—
103	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
104	C₂²×M₁₆	C₂²×C₁₃	1	3	yes	—
105	C₂²×M₁₆	C₂²×C₁₃	1	3	yes	—
106	#430	—	2	6	no	—
107	#430	C₂²×C₁₃	1	12	yes	complemented
108	C₂×#108	—	2	6	no	—
109	#407	—	2	6	no	—
110	#407	C₂²×C₁₃	1	12	yes	—
111	C₂×C₂²⋊C₈	—	2	6	no	—
112	#338	C₂²×C₁₃	1	4	yes	complemented
113	C₂²×C₁₃	—	2	6	no	—
114	C₂²×C₁₃	—	2	12	no	—
115	C₂²×C₁₃	—	4	3	no	—
116	C₂²×C₁₃	—	4	1	no	—
117	C₂²×C₁₃	—	4	6	no	—
118	C₂²×C₁₃	—	4	12	no	—
119	C₂²×C₁₃	—	4	4	no	—
120	C₂²×C₁₃	—	4	2	no	—
121	C₂²×C₁₃	#399	1	3	yes	—
122	C₂²×C₁₃	#399	1	1	yes	characteristic
123	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	3	yes	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	#399	1	3	yes	—
126	C₄×C₁₃	#399	1	1	yes	characteristic
127	C₄×C₁₃	C₂×#123	1	3	yes	—
128	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
129	C₂⁵	Q₈×C₁₃	1	1	yes	characteristic
130	C₂³×C₄	—	2	3	no	—
131	C₂³×C₄	—	2	3	no	—
132	C₂³×C₄	—	2	3	no	—
133	C₂³×C₄	—	2	3	no	—
134	C₂³×C₄	Q₈×C₁₃	1	1	yes	characteristic
135	C₂³×C₄	Q₈×C₁₃	1	2	yes	—
136	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
137	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
138	C₂×M₁₆	—	2	6	no	—
139	C₂×M₁₆	—	2	6	no	—
140	C₂×M₁₆	—	2	3	no	—
141	C₂×M₁₆	—	2	6	no	—
142	C₂×M₁₆	—	2	6	no	—
143	C₂×M₁₆	D₄×C₁₃	1	6	yes	—
144	C₂×M₁₆	C₂³×C₁₃	1	6	yes	—
145	C₂×M₁₆	C₂³×C₁₃	1	6	yes	—
146	C₂²×C₈	—	2	3	no	—
147	C₂²×C₈	D₄×C₁₃	1	6	yes	—
148	#108	—	2	24	no	—
149	C₂²⋊C₈	—	2	24	no	—
150	C₂×C₁₃	—	2	3	no	—
151	C₂×C₁₃	—	2	3	no	—
152	C₂×C₁₃	—	4	4	no	—
153	C₂×C₁₃	#2091	1	2	yes	direct product factor
154	C₂×C₁₃	C₂×#399	1	1	yes	characteristic
155	C₂⁴	—	2	3	no	—
156	C₂⁴	—	2	3	no	—
157	C₂⁴	—	4	4	no	—
158	C₂⁴	C₂×Q₈×C₁₃	1	2	yes	—
159	C₂⁴	C₂×Q₈×C₁₃	1	1	yes	characteristic
160	C₂²×C₄	—	2	3	no	—
161	C₂²×C₄	—	2	3	no	—
162	C₂²×C₄	—	2	6	no	—
163	C₂²×C₄	—	2	6	no	—
164	C₂²×C₄	—	2	6	no	—
165	C₂²×C₄	—	2	6	no	—
166	C₂²×C₄	—	4	3	no	—
167	C₂²×C₄	—	4	3	no	—
168	C₂²×C₄	—	4	6	no	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	—	4	3	no	—
171	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
172	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
173	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
174	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
175	C₂²×C₄	C₁₃×#41	1	3	yes	—
176	C₂²×C₄	C₁₃×#41	1	3	yes	—
177	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
178	M₁₆	—	2	12	no	—
179	M₁₆	—	2	12	no	—
180	M₁₆	—	2	12	no	—
181	M₁₆	—	4	6	no	—
182	C₂×C₈	—	2	6	no	—
183	C₂×C₈	—	2	6	no	—
184	C₂×C₈	—	2	6	no	—
185	C₂×C₈	—	4	3	no	—
186	C₂×C₈	—	4	3	no	—
187	C₂×C₈	C₂×D₄×C₁₃	1	12	yes	—
188	C₁₃	C₂×#2091	1	1	yes	characteristic, direct product factor, Sylow
189	C₂³	—	2	12	no	—
190	C₂³	—	2	6	no	—
191	C₂³	—	4	2	no	—
192	C₂³	—	4	4	no	—
193	C₂³	—	4	6	no	—
194	C₂³	—	4	12	no	—
195	C₂³	—	4	3	no	—
196	C₂³	—	4	1	no	—
197	C₂³	C₁₃×C₂²⋊Q₈	1	3	yes	—
198	C₂³	C₁₃×C₂²⋊Q₈	1	3	yes	—
199	C₂³	C₂²×Q₈×C₁₃	1	1	yes	characteristic
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	4	3	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	6	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	6	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₁₃×C₂²⋊Q₈	1	3	yes	—
208	C₂×C₄	C₁₃×C₂²⋊Q₈	1	3	yes	—
209	C₂×C₄	C₁₃×#123	1	3	yes	—
210	C₂×C₄	C₁₃×#123	1	3	yes	—
211	C₂×C₄	C₁₃×#123	1	6	yes	—
212	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
213	C₂×C₄	C₁₃×#127	1	3	yes	—
214	C₂×C₄	C₂²×D₄×C₁₃	1	3	yes	—
215	C₂×C₄	C₂×C₁₃×#41	1	3	yes	—
216	C₈	—	2	12	no	—
217	C₈	—	4	6	no	—
218	C₂²	—	2	6	no	—
219	C₂²	—	2	12	no	—
220	C₂²	—	4	3	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	6	no	—
223	C₂²	—	4	12	no	—
224	C₂²	—	4	4	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₁₃×#399	1	3	yes	—
227	C₂²	C₁₃×#399	1	1	yes	characteristic
228	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	3	yes	—
229	C₄	—	4	2	no	—
230	C₄	C₁₃×#399	1	3	yes	—
231	C₄	C₁₃×#399	1	1	yes	characteristic
232	C₄	C₂×C₁₃×#123	1	3	yes	—
233	C₄	C₂×C₁₃×#127	1	1	yes	characteristic
234	C₂	—	2	3	no	—
235	C₂	—	2	3	no	—
236	C₂	—	4	4	no	—
237	C₂	C₁₃×#2091	1	2	yes	direct product factor
238	C₂	C₂×C₁₃×#399	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂²×C₁₃	#399
prime radical	C₂⁵×C₁₃	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#430	C₂	semidirect
3. C₁₃×#2091	C₂	direct (N × H)
4. C₂×C₁₃×#338	C₂	semidirect
9. C₁₃×#430	C₂²	semidirect
14. C₁₃×#338	C₂²	semidirect
36. C₂×#2091	C₁₃	direct (N × H)
70. C₂×#430	C₂×C₁₃	semidirect
72. #2091	C₂×C₁₃	direct (N × H)
73. C₂×#338	C₂×C₁₃	semidirect
107. #430	C₂²×C₁₃	semidirect
112. #338	C₂²×C₁₃	semidirect
153. C₂×C₁₃	#2091	direct (N × H)
188. C₁₃	C₂×#2091	direct (N × H)
237. C₂	C₁₃×#2091	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#430	2
C₂×C₁₃×#407	2
C₁₃×#2091	2
C₂×C₁₃×#338	2
C₂×#2091	13

Subgroups — C2×C13×#2091 #15710061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2091 #15710061