Subgroups — C₁₃×#33713 #15720052

Order 3328.

834 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#508	C₂	1	4	yes	maximal, complemented
2	C₁₃×#3147	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2249	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2248	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	—	2	2	no	—
9	C₂×C₁₃×#135	C₂²	1	4	yes	—
10	C₁₃×#532	—	2	2	no	—
11	C₁₃×#508	—	2	16	no	—
12	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	—
13	C₂×C₁₃×SD₃₂	C₂²	1	8	yes	—
14	C₂×C₁₃×D₁₆	C₂²	1	4	yes	—
15	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
16	C₁₃×#495	—	2	2	no	—
17	C₁₃×#495	C₂²	1	4	yes	—
18	C₁₃×#494	C₂²	1	2	yes	—
19	C₁₃×#493	C₂²	1	2	yes	—
20	C₁₃×#492	—	2	2	no	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
23	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
24	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
25	C₂×C₁₃×#41	—	2	4	no	—
26	C₁₃×#135	—	2	16	no	—
27	C₁₃×#135	—	4	8	no	—
28	C₂×C₁₃×Q₁₆	—	2	2	no	—
29	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
30	C₂×C₁₃×SD₁₆	—	2	4	no	—
31	C₂×C₁₃×SD₁₆	—	2	4	no	—
32	C₂×C₁₃×D₈	—	2	2	no	—
33	C₂×C₁₃×D₈	C₂³	1	4	yes	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×C₄⋊Q₈	—	2	2	no	—
37	C₁₃×#127	—	2	2	no	—
38	C₁₃×#124	—	2	4	no	—
39	C₁₃×#121	—	4	4	no	—
40	C₄×D₄×C₁₃	—	4	4	no	—
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₁₃×Q₃₂	—	2	8	no	—
43	C₁₃×SD₃₂	—	2	16	no	—
44	C₁₃×D₁₆	—	2	8	no	—
45	C₂×C₁₃×C₁₆	D₄	1	8	yes	complemented
46	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
47	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
48	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	#33713	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₁₃×#41	—	4	8	no	—
54	C₁₃×#41	—	4	8	no	—
55	C₂×Q₈×C₁₃	—	2	4	no	—
56	C₂×D₄×C₁₃	—	2	4	no	—
57	C₂×D₄×C₁₃	—	4	4	no	—
58	C₂×D₄×C₁₃	—	4	4	no	—
59	C₂²×C₄×C₁₃	—	4	4	no	—
60	C₂²×C₄×C₁₃	D₈	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₁₃×Q₁₆	—	2	8	no	—
63	C₁₃×Q₁₆	—	4	4	no	—
64	C₁₃×SD₁₆	—	4	8	no	—
65	C₁₃×SD₁₆	—	4	8	no	—
66	C₁₃×D₈	—	2	8	no	—
67	C₁₃×D₈	—	4	4	no	—
68	C₂×C₈×C₁₃	D₈	1	4	yes	—
69	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	4	4	no	—
75	C₁₃×C₂²⋊C₄	—	4	8	no	—
76	C₄²×C₁₃	D₈	1	2	yes	—
77	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₁₃×C₁₆	C₂×D₄	1	4	yes	—
80	C₁₃×C₁₆	C₂×D₄	1	4	yes	—
81	C₂×#508	C₂×C₁₃	1	4	yes	complemented
82	#3147	C₂×C₁₃	1	2	yes	complemented
83	#2249	C₂×C₁₃	1	4	yes	complemented
84	#2248	C₂×C₁₃	1	2	yes	complemented
85	#2247	C₂×C₁₃	1	1	yes	characteristic, complemented
86	#2246	C₂×C₁₃	1	1	yes	characteristic, complemented
87	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
88	C₂³×C₁₃	—	4	4	no	—
89	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
90	Q₈×C₁₃	—	4	4	no	—
91	Q₈×C₁₃	—	4	4	no	—
92	D₄×C₁₃	—	4	4	no	—
93	D₄×C₁₃	—	4	4	no	—
94	D₄×C₁₃	—	8	8	no	—
95	D₄×C₁₃	—	8	8	no	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	4	4	no	—
98	C₂×C₄×C₁₃	—	8	8	no	—
99	C₂×C₄×C₁₃	#127	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×D₈	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	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₁₃	#127	1	2	yes	—
106	C₈×C₁₃	C₂×D₈	1	2	yes	—
107	C₈×C₁₃	C₂×D₈	1	2	yes	—
108	C₈×C₁₃	C₂²×D₄	1	2	yes	—
109	C₂×#135	—	2	2	no	—
110	C₂×#135	C₂²×C₁₃	1	4	yes	—
111	#532	—	2	2	no	—
112	#508	—	2	16	no	—
113	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	—
114	C₂×SD₃₂	C₂²×C₁₃	1	8	yes	—
115	C₂×D₁₆	C₂²×C₁₃	1	4	yes	—
116	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
117	#495	—	2	2	no	—
118	#495	C₂²×C₁₃	1	4	yes	—
119	#494	C₂²×C₁₃	1	2	yes	—
120	#493	C₂²×C₁₃	1	2	yes	—
121	#492	—	2	2	no	—
122	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
123	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
124	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
125	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	8	4	no	—
128	C₂²×C₁₃	—	8	8	no	—
129	C₂²×C₁₃	#493	1	2	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	#493	1	2	yes	—
134	C₄×C₁₃	#508	1	4	yes	complemented
135	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
136	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
137	C₂×#41	—	2	4	no	—
138	#135	—	2	16	no	—
139	#135	—	4	8	no	—
140	C₂×Q₁₆	—	2	2	no	—
141	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
142	C₂×SD₁₆	—	2	4	no	—
143	C₂×SD₁₆	—	2	4	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	C₂³×C₁₃	1	4	yes	—
146	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
148	C₄⋊Q₈	—	2	2	no	—
149	#127	—	2	2	no	—
150	#124	—	2	4	no	—
151	#121	—	4	4	no	—
152	C₄×D₄	—	4	4	no	—
153	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
154	Q₃₂	—	2	8	no	—
155	SD₃₂	—	2	16	no	—
156	D₁₆	—	2	8	no	—
157	C₂×C₁₆	D₄×C₁₃	1	8	yes	complemented
158	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
159	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
160	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
161	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
162	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
163	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
164	C₂×C₁₃	—	2	2	no	—
165	C₂×C₁₃	—	16	4	no	—
166	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
167	C₂×C₁₃	C₂×#508	1	2	yes	—
168	#41	—	4	8	no	—
169	#41	—	4	8	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂²×C₄	—	4	4	no	—
175	C₂²×C₄	C₁₃×D₈	1	2	yes	—
176	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
177	Q₁₆	—	2	8	no	—
178	Q₁₆	—	4	4	no	—
179	SD₁₆	—	4	8	no	—
180	SD₁₆	—	4	8	no	—
181	D₈	—	2	8	no	—
182	D₈	—	4	4	no	—
183	C₂×C₈	C₁₃×D₈	1	4	yes	—
184	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
185	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
186	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
187	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
188	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
189	C₄⋊C₄	—	4	4	no	—
190	C₂²⋊C₄	—	4	8	no	—
191	C₄²	C₁₃×D₈	1	2	yes	—
192	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
193	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
194	C₁₆	C₂×D₄×C₁₃	1	4	yes	—
195	C₁₆	C₂×D₄×C₁₃	1	4	yes	—
196	C₁₃	#33713	1	1	yes	characteristic, direct product factor, Sylow
197	C₂³	—	4	4	no	—
198	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
199	Q₈	—	4	4	no	—
200	Q₈	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	8	8	no	—
204	D₄	—	8	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	8	no	—
208	C₂×C₄	C₁₃×#127	1	2	yes	—
209	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
210	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
211	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
212	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
213	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
214	C₈	C₁₃×#127	1	2	yes	—
215	C₈	C₂×C₁₃×D₈	1	2	yes	—
216	C₈	C₂×C₁₃×D₈	1	2	yes	—
217	C₈	C₂²×D₄×C₁₃	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	4	no	—
220	C₂²	—	8	8	no	—
221	C₂²	C₁₃×#493	1	2	yes	—
222	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	C₁₃×#493	1	2	yes	—
226	C₄	C₁₃×#508	1	4	yes	complemented
227	C₄	C₂×C₁₃×#127	1	1	yes	characteristic
228	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	16	4	no	—
231	C₂	C₂×C₁₃×#493	1	1	yes	characteristic
232	C₂	C₂×C₁₃×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#508	C₂	semidirect
2. C₁₃×#3147	C₂	semidirect
3. C₁₃×#2249	C₂	semidirect
4. C₁₃×#2248	C₂	semidirect
5. C₁₃×#2247	C₂	semidirect
6. C₁₃×#2246	C₂	semidirect
7. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
45. C₂×C₁₃×C₁₆	D₄	semidirect
52. #33713	C₁₃	direct (N × H)
81. C₂×#508	C₂×C₁₃	semidirect
82. #3147	C₂×C₁₃	semidirect
83. #2249	C₂×C₁₃	semidirect
84. #2248	C₂×C₁₃	semidirect
85. #2247	C₂×C₁₃	semidirect
86. #2246	C₂×C₁₃	semidirect
87. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
134. C₄×C₁₃	#508	semidirect
157. C₂×C₁₆	D₄×C₁₃	semidirect
196. C₁₃	#33713	direct (N × H)
226. C₄	C₁₃×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#508	2
C₁₃×#3147	2
C₁₃×#2249	2
C₁₃×#2248	2
C₁₃×#2247	2
C₁₃×#2246	2
C₂×C₄×C₁₃×C₁₆	2
#33713	13

Subgroups — C13×#33713 #15720052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33713 #15720052