Subgroups — C₁₃×#33493 #15719832

Order 3328.

572 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2169	C₂	1	4	yes	maximal, complemented
3	D₄×C₁₃×C₁₆	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2162	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2113	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₃×#131	C₂²	1	2	yes	—
7	C₁₃×#532	C₄	1	1	yes	characteristic
8	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
9	C₂²×C₁₃×C₁₆	C₂²	1	4	yes	—
10	C₁₃×#436	C₄	1	4	yes	—
11	C₈×D₄×C₁₃	C₂²	1	4	yes	—
12	C₁₃×#423	C₄	1	1	yes	characteristic
13	C₁₃×#408	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
18	C₁₃×C₂²⋊C₁₆	C₂²	1	8	yes	—
19	C₄×C₁₃×C₁₆	C₂²	1	4	yes	—
20	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
21	C₁₃×#131	—	4	2	no	—
22	C₁₃×#131	—	4	2	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
28	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
29	C₁₃×#127	C₈	1	1	yes	characteristic
30	C₁₃×#124	C₈	1	2	yes	—
31	C₁₃×#121	C₈	1	4	yes	—
32	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₁₃×M₃₂	—	4	4	no	—
35	C₂×C₁₃×C₁₆	—	4	4	no	—
36	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
37	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
38	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
39	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
40	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₃	C₂³	1	2	yes	—
42	#33493	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₁₃×#41	—	4	2	no	—
44	C₁₃×#41	—	4	2	no	—
45	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
46	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
47	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
48	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₁₃×M₁₆	—	4	2	no	—
53	C₁₃×M₁₆	—	4	4	no	—
54	C₂×C₈×C₁₃	—	2	4	no	—
55	C₂×C₈×C₁₃	—	2	4	no	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	4	1	no	—
59	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
64	C₁₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
65	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
68	C₁₃×C₁₆	—	2	4	no	—
69	C₁₃×C₁₆	—	2	4	no	—
70	#2931	C₂×C₁₃	1	1	yes	characteristic
71	#2169	C₂×C₁₃	1	4	yes	complemented
72	D₄×C₁₆	C₂×C₁₃	1	4	yes	complemented
73	#2162	C₂×C₁₃	1	2	yes	complemented
74	#2113	C₂×C₁₃	1	4	yes	complemented
75	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
76	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	Q₈×C₁₃	—	4	2	no	—
78	D₄×C₁₃	—	2	8	no	—
79	D₄×C₁₃	—	4	2	no	—
80	D₄×C₁₃	—	4	4	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	4	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	4	no	—
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
89	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
91	C₈×C₁₃	—	2	4	no	—
92	C₈×C₁₃	—	2	1	no	—
93	C₈×C₁₃	—	2	2	no	—
94	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
95	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
96	C₂×#131	C₂²×C₁₃	1	2	yes	—
97	#532	C₄×C₁₃	1	1	yes	characteristic
98	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
99	C₂²×C₁₆	C₂²×C₁₃	1	4	yes	—
100	#436	C₄×C₁₃	1	4	yes	—
101	C₈×D₄	C₂²×C₁₃	1	4	yes	—
102	#423	C₄×C₁₃	1	1	yes	characteristic
103	#408	C₄×C₁₃	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
105	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
106	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
107	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
108	C₂²⋊C₁₆	C₂²×C₁₃	1	8	yes	—
109	C₄×C₁₆	C₂²×C₁₃	1	4	yes	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	—	2	8	no	—
112	C₂²×C₁₃	—	2	1	no	—
113	C₂²×C₁₃	—	2	4	no	—
114	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	2	4	no	—
118	C₄×C₁₃	#504	1	4	yes	—
119	C₄×C₁₃	#518	1	1	yes	characteristic
120	C₄×C₁₃	#568	1	1	yes	characteristic
121	C₂×#41	C₂×C₄×C₁₃	1	2	yes	—
122	#131	—	4	2	no	—
123	#131	—	4	2	no	—
124	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
125	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
126	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
127	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
128	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
129	C₄⋊Q₈	C₈×C₁₃	1	1	yes	characteristic
130	#127	C₈×C₁₃	1	1	yes	characteristic
131	#124	C₈×C₁₃	1	2	yes	—
132	#121	C₈×C₁₃	1	4	yes	—
133	C₄×D₄	C₂×C₄×C₁₃	1	4	yes	—
134	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
135	M₃₂	—	4	4	no	—
136	C₂×C₁₆	—	4	4	no	—
137	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
138	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
139	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
140	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
141	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
142	C₄×C₈	C₂³×C₁₃	1	2	yes	—
143	C₂×C₁₃	—	4	1	no	—
144	C₂×C₁₃	—	4	4	no	—
145	C₂×C₁₃	#2972	1	1	yes	characteristic
146	C₂×C₁₃	C₂×#504	1	2	yes	—
147	#41	—	4	2	no	—
148	#41	—	4	2	no	—
149	C₂×Q₈	C₂×C₈×C₁₃	1	2	yes	—
150	C₂×D₄	C₂×C₈×C₁₃	1	4	yes	—
151	C₂×D₄	C₂×C₈×C₁₃	1	4	yes	—
152	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	—
153	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
154	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
155	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
156	M₁₆	—	4	2	no	—
157	M₁₆	—	4	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
164	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
165	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
166	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
167	C₄⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
168	C₂²⋊C₄	C₂×C₈×C₁₃	1	8	yes	—
169	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
170	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
171	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
172	C₁₆	—	2	4	no	—
173	C₁₆	—	2	4	no	—
174	C₁₃	#33493	1	1	yes	characteristic, direct product factor, Sylow
175	C₂³	C₂²×C₈×C₁₃	1	4	yes	—
176	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
177	Q₈	—	4	2	no	—
178	D₄	—	2	8	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
188	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
189	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
190	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
191	C₈	—	2	4	no	—
192	C₈	—	2	1	no	—
193	C₈	—	2	2	no	—
194	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
195	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	4	no	—
200	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	4	no	—
204	C₄	C₁₃×#504	1	4	yes	—
205	C₄	C₁₃×#518	1	1	yes	characteristic
206	C₄	C₁₃×#568	1	1	yes	characteristic
207	C₂	—	4	1	no	—
208	C₂	—	4	4	no	—
209	C₂	C₁₃×#2972	1	1	yes	characteristic
210	C₂	C₂×C₁₃×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₁₃×#532	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2169	C₂	semidirect
3. D₄×C₁₃×C₁₆	C₂	semidirect
4. C₁₃×#2162	C₂	semidirect
5. C₁₃×#2113	C₂	semidirect
42. #33493	C₁₃	direct (N × H)
71. #2169	C₂×C₁₃	semidirect
72. D₄×C₁₆	C₂×C₁₃	semidirect
73. #2162	C₂×C₁₃	semidirect
74. #2113	C₂×C₁₃	semidirect
174. C₁₃	#33493	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×#2931	2
C₁₃×#2169	2
D₄×C₁₃×C₁₆	2
C₁₃×#2162	2
C₁₃×#2113	2
#33493	13

Subgroups — C13×#33493 #15719832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33493 #15719832