Subgroups — C₁₃×#33901 #15720240

Order 3328.

548 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	C₁₃×#3334	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2232	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2224	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2215	C₂	1	2	yes	maximal, complemented
6	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#366	C₂	1	2	yes	maximal, complemented
9	C₁₃×#547	—	2	2	no	—
10	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
12	C₁₃×#500	C₂²	1	2	yes	—
13	C₁₃×#480	—	2	4	no	—
14	C₁₃×#474	—	2	2	no	—
15	C₁₃×#466	C₂²	1	4	yes	—
16	C₄×C₁₃×D₈	C₂²	1	2	yes	—
17	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#107	C₂²	1	4	yes	—
19	C₁₃×#366	C₂²	1	2	yes	—
20	C₁₃×#366	C₂²	1	2	yes	—
21	C₁₃×#366	C₂²	1	1	yes	characteristic
22	C₁₃×#366	C₂²	1	4	yes	—
23	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₁₃×#357	C₂²	1	2	yes	—
25	C₁₃×#357	C₂²	1	4	yes	—
26	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
28	C₂×C₁₃×#41	—	4	2	no	—
29	C₂×C₁₃×D₈	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
31	C₁₃×C₄⋊Q₈	—	2	2	no	—
32	C₁₃×#123	—	4	4	no	—
33	C₁₃×C₂²⋊Q₈	—	4	2	no	—
34	C₁₃×#121	—	2	4	no	—
35	C₄×D₄×C₁₃	—	2	2	no	—
36	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
38	C₁₃×D₁₆	—	2	2	no	—
39	C₂×C₁₃×C₁₆	—	2	4	no	—
40	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
43	C₁₃×#107	—	2	4	no	—
44	C₁₃×#107	—	2	2	no	—
45	C₁₃×#107	C₂³	1	2	yes	—
46	C₁₃×#107	C₂³	1	4	yes	—
47	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
48	C₁₃×#102	—	2	2	no	—
49	C₁₃×#102	—	2	4	no	—
50	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	#33901	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₁₃×#41	—	8	8	no	—
54	C₂×Q₈×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	—	2	2	no	—
56	C₂×D₄×C₁₃	—	4	4	no	—
57	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	4	4	no	—
59	C₂²×C₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₁₃×D₈	—	2	2	no	—
62	C₁₃×D₈	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	—	2	2	no	—
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₄⋊C₄	—	4	4	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
73	C₁₃×C₂²⋊C₄	—	4	2	no	—
74	C₁₃×C₂²⋊C₄	—	4	4	no	—
75	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
76	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₁₃×C₁₆	—	2	2	no	—
78	C₁₃×C₁₆	—	2	1	no	—
79	C₁₃×C₁₆	#41	1	2	yes	complemented
80	#3334	C₂×C₁₃	1	2	yes	complemented
81	#2252	C₂×C₁₃	1	1	yes	characteristic, complemented
82	#2232	C₂×C₁₃	1	4	yes	complemented
83	#2224	C₂×C₁₃	1	2	yes	complemented
84	#2215	C₂×C₁₃	1	2	yes	complemented
85	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
86	D₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
87	C₂×#366	C₂×C₁₃	1	2	yes	complemented
88	C₂³×C₁₃	—	4	2	no	—
89	C₂³×C₁₃	C₂×D₈	1	2	yes	—
90	Q₈×C₁₃	—	8	4	no	—
91	D₄×C₁₃	—	4	2	no	—
92	D₄×C₁₃	—	4	2	no	—
93	D₄×C₁₃	—	8	8	no	—
94	D₄×C₁₃	D₁₆	1	4	yes	complemented
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	8	8	no	—
100	C₂×C₄×C₁₃	—	8	4	no	—
101	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
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	—	2	2	no	—
106	C₈×C₁₃	—	2	1	no	—
107	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
108	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
109	#547	—	2	2	no	—
110	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
111	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
112	#500	C₂²×C₁₃	1	2	yes	—
113	#480	—	2	4	no	—
114	#474	—	2	2	no	—
115	#466	C₂²×C₁₃	1	4	yes	—
116	C₄×D₈	C₂²×C₁₃	1	2	yes	—
117	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
118	C₂×#107	C₂²×C₁₃	1	4	yes	—
119	#366	C₂²×C₁₃	1	1	yes	characteristic
120	#366	C₂²×C₁₃	1	2	yes	—
121	#366	C₂²×C₁₃	1	4	yes	—
122	#366	C₂²×C₁₃	1	2	yes	—
123	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
124	#357	C₂²×C₁₃	1	2	yes	—
125	#357	C₂²×C₁₃	1	4	yes	—
126	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
127	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	8	2	no	—
131	C₂²×C₁₃	—	8	2	no	—
132	C₂²×C₁₃	—	8	2	no	—
133	C₂²×C₁₃	C₂×D₁₆	1	4	yes	—
134	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	8	2	no	—
138	C₄×C₁₃	—	8	4	no	—
139	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
140	C₄×C₁₃	#547	1	1	yes	characteristic
141	C₄×C₁₃	#578	1	1	yes	characteristic
142	C₂×#41	—	4	2	no	—
143	C₂×D₈	C₂³×C₁₃	1	2	yes	—
144	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
145	C₄⋊Q₈	—	2	2	no	—
146	#123	—	4	4	no	—
147	C₂²⋊Q₈	—	4	2	no	—
148	#121	—	2	4	no	—
149	C₄×D₄	—	2	2	no	—
150	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
151	C₂×C₄⋊C₄	—	2	4	no	—
152	D₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	4	no	—
154	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
155	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
156	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
157	#107	—	2	4	no	—
158	#107	—	2	2	no	—
159	#107	C₂³×C₁₃	1	2	yes	—
160	#107	C₂³×C₁₃	1	4	yes	—
161	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
162	#102	—	2	2	no	—
163	#102	—	2	4	no	—
164	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
165	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
166	C₂×C₁₃	—	2	4	no	—
167	C₂×C₁₃	—	16	2	no	—
168	C₂×C₁₃	#3334	1	1	yes	characteristic
169	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
170	C₂×C₁₃	#3417	1	1	yes	characteristic
171	#41	—	8	8	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
176	C₂²×C₄	—	4	4	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
179	D₈	—	2	2	no	—
180	D₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
184	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
185	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
191	C₂²⋊C₄	—	4	2	no	—
192	C₂²⋊C₄	—	4	4	no	—
193	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
194	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
195	C₁₆	—	2	2	no	—
196	C₁₆	—	2	1	no	—
197	C₁₆	C₁₃×#41	1	2	yes	complemented
198	C₁₃	#33901	1	1	yes	characteristic, direct product factor, Sylow
199	C₂³	—	4	2	no	—
200	C₂³	C₂×C₁₃×D₈	1	2	yes	—
201	Q₈	—	8	4	no	—
202	D₄	—	4	2	no	—
203	D₄	—	4	2	no	—
204	D₄	—	8	8	no	—
205	D₄	C₁₃×D₁₆	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	8	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
213	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
214	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
219	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	8	2	no	—
223	C₂²	—	8	2	no	—
224	C₂²	—	8	2	no	—
225	C₂²	C₂×C₁₃×D₁₆	1	4	yes	—
226	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	8	2	no	—
230	C₄	—	8	4	no	—
231	C₄	C₂×C₁₃×D₁₆	1	2	yes	—
232	C₄	C₁₃×#547	1	1	yes	characteristic
233	C₄	C₁₃×#578	1	1	yes	characteristic
234	C₂	—	2	4	no	—
235	C₂	—	16	2	no	—
236	C₂	C₁₃×#3334	1	1	yes	characteristic
237	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic
238	C₂	C₁₃×#3417	1	1	yes	characteristic

	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 ◁ C₄×D₄×C₁₃ ◁ C₈×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3334	C₂	semidirect
2. C₁₃×#2252	C₂	semidirect
3. C₁₃×#2232	C₂	semidirect
4. C₁₃×#2224	C₂	semidirect
5. C₁₃×#2215	C₂	semidirect
6. C₄×C₁₃×D₁₆	C₂	semidirect
7. D₄×C₁₃×C₁₆	C₂	semidirect
8. C₂×C₁₃×#366	C₂	semidirect
52. #33901	C₁₃	direct (N × H)
79. C₁₃×C₁₆	#41	semidirect
80. #3334	C₂×C₁₃	semidirect
81. #2252	C₂×C₁₃	semidirect
82. #2232	C₂×C₁₃	semidirect
83. #2224	C₂×C₁₃	semidirect
84. #2215	C₂×C₁₃	semidirect
85. C₄×D₁₆	C₂×C₁₃	semidirect
86. D₄×C₁₆	C₂×C₁₃	semidirect
87. C₂×#366	C₂×C₁₃	semidirect
94. D₄×C₁₃	D₁₆	semidirect
197. C₁₆	C₁₃×#41	semidirect
198. C₁₃	#33901	direct (N × H)
205. D₄	C₁₃×D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#3334	2
C₁₃×#2252	2
C₁₃×#2232	2
C₁₃×#2224	2
C₁₃×#2215	2
C₄×C₁₃×D₁₆	2
D₄×C₁₃×C₁₆	2
C₂×C₁₃×#366	2
#33901	13

Subgroups — C13×#33901 #15720240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33901 #15720240