Subgroups — C₁₃×#33740 #15720079

Order 3328.

532 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3176	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2244	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2218	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2215	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2177	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×D₁₆	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#540	—	2	2	no	—
11	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
12	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₁₃×#499	C₂²	1	2	yes	—
16	C₁₃×#493	C₂²	1	2	yes	—
17	C₁₃×#490	—	2	2	no	—
18	C₁₃×#469	—	2	2	no	—
19	C₁₃×#466	C₂²	1	4	yes	—
20	C₁₃×#442	—	2	2	no	—
21	C₄×C₁₃×D₈	C₂²	1	4	yes	—
22	C₁₃×#405	C₂²	1	1	yes	characteristic
23	C₁₃×#367	C₂²	1	2	yes	—
24	C₁₃×#366	C₂²	1	2	yes	—
25	C₁₃×#357	C₂²	1	4	yes	—
26	C₁₃×#357	C₂²	1	4	yes	—
27	C₁₃×#346	C₂²	1	1	yes	characteristic
28	C₁₃×#346	C₂²	1	1	yes	characteristic
29	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
31	C₂×C₁₃×D₈	—	2	2	no	—
32	C₂×C₁₃×D₈	C₂³	1	4	yes	—
33	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×#127	—	2	2	no	—
36	C₁₃×#125	—	2	2	no	—
37	C₁₃×#123	—	4	4	no	—
38	C₁₃×#121	—	2	4	no	—
39	C₁₃×#121	—	4	4	no	—
40	C₄×D₄×C₁₃	—	2	4	no	—
41	C₁₃×#117	D₄	1	1	yes	characteristic
42	C₁₃×D₁₆	—	2	4	no	—
43	C₁₃×D₁₆	—	2	4	no	—
44	C₁₃×M₃₂	—	2	2	no	—
45	C₂×C₁₃×C₁₆	—	2	2	no	—
46	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
48	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
50	C₁₃×#107	C₂³	1	4	yes	—
51	C₁₃×#106	—	2	2	no	—
52	C₁₃×#102	—	2	4	no	—
53	C₁₃×#102	—	2	4	no	—
54	C₁₃×#97	D₄	1	1	yes	characteristic
55	C₁₃×#97	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	#33740	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×D₄×C₁₃	—	2	4	no	—
60	C₂×D₄×C₁₃	—	4	4	no	—
61	C₂×D₄×C₁₃	—	4	4	no	—
62	C₂²×C₄×C₁₃	—	4	4	no	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₁₃×D₈	—	2	4	no	—
65	C₁₃×D₈	—	2	4	no	—
66	C₁₃×D₈	—	4	4	no	—
67	C₁₃×M₁₆	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
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	—	2	4	no	—
75	C₁₃×C₄⋊C₄	—	4	4	no	—
76	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
77	C₁₃×C₂²⋊C₄	—	4	4	no	—
78	C₁₃×C₂²⋊C₄	—	4	4	no	—
79	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
80	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×C₁₆	—	2	1	no	—
83	C₁₃×C₁₆	—	2	1	no	—
84	C₁₃×C₁₆	#41	1	2	yes	—
85	C₁₃×C₁₆	#41	1	2	yes	—
86	#3176	C₂×C₁₃	1	2	yes	complemented
87	#2253	C₂×C₁₃	1	1	yes	characteristic, complemented
88	#2246	C₂×C₁₃	1	1	yes	characteristic, complemented
89	#2244	C₂×C₁₃	1	2	yes	complemented
90	#2218	C₂×C₁₃	1	2	yes	complemented
91	#2215	C₂×C₁₃	1	2	yes	complemented
92	#2177	C₂×C₁₃	1	2	yes	complemented
93	C₄×D₁₆	C₂×C₁₃	1	2	yes	complemented
94	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
95	C₂³×C₁₃	—	4	4	no	—
96	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
97	D₄×C₁₃	—	4	4	no	—
98	D₄×C₁₃	—	4	4	no	—
99	D₄×C₁₃	—	16	4	no	—
100	D₄×C₁₃	—	16	4	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	16	4	no	—
105	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₁₃	—	2	1	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	#135	1	2	yes	—
112	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
113	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
114	#540	—	2	2	no	—
115	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
116	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
117	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
118	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
119	#499	C₂²×C₁₃	1	2	yes	—
120	#493	C₂²×C₁₃	1	2	yes	—
121	#490	—	2	2	no	—
122	#469	—	2	2	no	—
123	#466	C₂²×C₁₃	1	4	yes	—
124	#442	—	2	2	no	—
125	C₄×D₈	C₂²×C₁₃	1	4	yes	—
126	#405	C₂²×C₁₃	1	1	yes	characteristic
127	#367	C₂²×C₁₃	1	2	yes	—
128	#366	C₂²×C₁₃	1	2	yes	—
129	#357	C₂²×C₁₃	1	4	yes	—
130	#357	C₂²×C₁₃	1	4	yes	—
131	#346	C₂²×C₁₃	1	1	yes	characteristic
132	#346	C₂²×C₁₃	1	1	yes	characteristic
133	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
134	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
135	C₂²×C₁₃	—	2	2	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	—	8	4	no	—
138	C₂²×C₁₃	—	8	4	no	—
139	C₂²×C₁₃	—	8	4	no	—
140	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	8	4	no	—
145	C₄×C₁₃	#532	1	1	yes	characteristic
146	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
147	C₂×D₈	—	2	2	no	—
148	C₂×D₈	C₂³×C₁₃	1	4	yes	—
149	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
151	#127	—	2	2	no	—
152	#125	—	2	2	no	—
153	#123	—	4	4	no	—
154	#121	—	2	4	no	—
155	#121	—	4	4	no	—
156	C₄×D₄	—	2	4	no	—
157	#117	D₄×C₁₃	1	1	yes	characteristic
158	D₁₆	—	2	4	no	—
159	D₁₆	—	2	4	no	—
160	M₃₂	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
163	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
164	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
165	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
166	#107	C₂³×C₁₃	1	4	yes	—
167	#106	—	2	2	no	—
168	#102	—	2	4	no	—
169	#102	—	2	4	no	—
170	#97	D₄×C₁₃	1	1	yes	characteristic
171	#97	D₄×C₁₃	1	1	yes	characteristic
172	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
173	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
174	C₂×C₁₃	—	4	1	no	—
175	C₂×C₁₃	—	16	4	no	—
176	C₂×C₁₃	#3169	1	1	yes	characteristic
177	C₂×C₁₃	#3415	1	2	yes	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	4	4	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
183	D₈	—	2	4	no	—
184	D₈	—	2	4	no	—
185	D₈	—	4	4	no	—
186	M₁₆	—	2	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
190	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
191	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
192	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
196	C₂²⋊C₄	—	4	4	no	—
197	C₂²⋊C₄	—	4	4	no	—
198	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
199	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
200	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
201	C₁₆	—	2	1	no	—
202	C₁₆	—	2	1	no	—
203	C₁₆	C₁₃×#41	1	2	yes	—
204	C₁₆	C₁₃×#41	1	2	yes	—
205	C₁₃	#33740	1	1	yes	characteristic, direct product factor, Sylow
206	C₂³	—	4	4	no	—
207	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
208	D₄	—	4	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	16	4	no	—
211	D₄	—	16	4	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	16	4	no	—
216	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
217	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
218	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
219	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	C₁₃×#135	1	2	yes	—
223	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
224	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
225	C₂²	—	2	2	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	8	4	no	—
228	C₂²	—	8	4	no	—
229	C₂²	—	8	4	no	—
230	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	—	8	4	no	—
235	C₄	C₁₃×#532	1	1	yes	characteristic
236	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
237	C₂	—	4	1	no	—
238	C₂	—	16	4	no	—
239	C₂	C₁₃×#3169	1	1	yes	characteristic
240	C₂	C₁₃×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3176	C₂	semidirect
2. C₁₃×#2253	C₂	semidirect
3. C₁₃×#2246	C₂	semidirect
4. C₁₃×#2244	C₂	semidirect
5. C₁₃×#2218	C₂	semidirect
6. C₁₃×#2215	C₂	semidirect
7. C₁₃×#2177	C₂	semidirect
8. C₄×C₁₃×D₁₆	C₂	semidirect
9. C₁₃×#2108	C₂	semidirect
58. #33740	C₁₃	direct (N × H)
86. #3176	C₂×C₁₃	semidirect
87. #2253	C₂×C₁₃	semidirect
88. #2246	C₂×C₁₃	semidirect
89. #2244	C₂×C₁₃	semidirect
90. #2218	C₂×C₁₃	semidirect
91. #2215	C₂×C₁₃	semidirect
92. #2177	C₂×C₁₃	semidirect
93. C₄×D₁₆	C₂×C₁₃	semidirect
94. #2108	C₂×C₁₃	semidirect
205. C₁₃	#33740	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#3176	2
C₁₃×#2253	2
C₁₃×#2246	2
C₁₃×#2244	2
C₁₃×#2218	2
C₁₃×#2215	2
C₁₃×#2177	2
C₄×C₁₃×D₁₆	2
C₁₃×#2108	2
#33740	13

Subgroups — C13×#33740 #15720079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33740 #15720079