Subgroups — C₁₃×#33494 #15719833

Order 3328.

556 conjugacy classes of subgroups.

The subgroup lattice has 200 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×#2168	C₂	1	8	yes	maximal, complemented
3	C₁₃×#2163	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2113	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#131	C₂²	1	2	yes	—
6	C₁₃×#532	C₄	1	1	yes	characteristic
7	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
8	C₂²×C₁₃×C₁₆	C₂²	1	4	yes	—
9	C₁₃×#436	C₄	1	4	yes	—
10	C₈×D₄×C₁₃	C₂²	1	4	yes	—
11	C₁₃×#423	C₄	1	1	yes	characteristic
12	C₁₃×#408	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	—
15	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
16	C₁₃×C₂²⋊C₁₆	C₂²	1	8	yes	—
17	C₁₃×#346	C₂²	1	4	yes	—
18	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
19	C₁₃×#131	—	4	2	no	—
20	C₁₃×#131	—	4	2	no	—
21	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
26	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
27	C₁₃×#127	C₈	1	1	yes	characteristic
28	C₁₃×#124	C₈	1	2	yes	—
29	C₁₃×#121	C₈	1	4	yes	—
30	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₁₃×M₃₂	—	4	4	no	—
33	C₂×C₁₃×C₁₆	—	4	4	no	—
34	C₂×C₁₃×C₁₆	C₂³	1	8	yes	—
35	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
36	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₃	C₂³	1	2	yes	—
39	#33494	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₁₃×#41	—	4	2	no	—
41	C₁₃×#41	—	4	2	no	—
42	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
43	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
44	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
45	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₁₃×M₁₆	—	4	2	no	—
50	C₁₃×M₁₆	—	4	4	no	—
51	C₂×C₈×C₁₃	—	2	4	no	—
52	C₂×C₈×C₁₃	—	2	4	no	—
53	C₂×C₈×C₁₃	—	2	2	no	—
54	C₂×C₈×C₁₃	—	4	1	no	—
55	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
59	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
60	C₁₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
61	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
64	C₁₃×C₁₆	—	2	8	no	—
65	#2931	C₂×C₁₃	1	1	yes	characteristic
66	#2168	C₂×C₁₃	1	8	yes	complemented
67	#2163	C₂×C₁₃	1	2	yes	complemented
68	#2113	C₂×C₁₃	1	4	yes	complemented
69	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
70	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
71	Q₈×C₁₃	—	4	2	no	—
72	D₄×C₁₃	—	4	4	no	—
73	D₄×C₁₃	—	4	2	no	—
74	D₄×C₁₃	—	4	4	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
85	C₈×C₁₃	—	2	4	no	—
86	C₈×C₁₃	—	2	1	no	—
87	C₈×C₁₃	—	2	2	no	—
88	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
89	C₂×#131	C₂²×C₁₃	1	2	yes	—
90	#532	C₄×C₁₃	1	1	yes	characteristic
91	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
92	C₂²×C₁₆	C₂²×C₁₃	1	4	yes	—
93	#436	C₄×C₁₃	1	4	yes	—
94	C₈×D₄	C₂²×C₁₃	1	4	yes	—
95	#423	C₄×C₁₃	1	1	yes	characteristic
96	#408	C₄×C₁₃	1	2	yes	—
97	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
98	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	—
99	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
100	C₂²⋊C₁₆	C₂²×C₁₃	1	8	yes	—
101	#346	C₂²×C₁₃	1	4	yes	—
102	C₂²×C₁₃	—	2	2	no	—
103	C₂²×C₁₃	—	2	8	no	—
104	C₂²×C₁₃	—	2	1	no	—
105	C₂²×C₁₃	—	2	4	no	—
106	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	—	2	2	no	—
109	C₄×C₁₃	—	2	1	no	—
110	C₄×C₁₃	—	2	4	no	—
111	C₄×C₁₃	#518	1	1	yes	characteristic
112	C₄×C₁₃	#568	1	1	yes	characteristic
113	C₂×#41	C₂×C₄×C₁₃	1	2	yes	—
114	#131	—	4	2	no	—
115	#131	—	4	2	no	—
116	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
117	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
118	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
119	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
120	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
121	C₄⋊Q₈	C₈×C₁₃	1	1	yes	characteristic
122	#127	C₈×C₁₃	1	1	yes	characteristic
123	#124	C₈×C₁₃	1	2	yes	—
124	#121	C₈×C₁₃	1	4	yes	—
125	C₄×D₄	C₂×C₄×C₁₃	1	4	yes	—
126	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
127	M₃₂	—	4	4	no	—
128	C₂×C₁₆	—	4	4	no	—
129	C₂×C₁₆	C₂³×C₁₃	1	8	yes	—
130	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
131	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
132	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
133	C₄×C₈	C₂³×C₁₃	1	2	yes	—
134	C₂×C₁₃	—	4	1	no	—
135	C₂×C₁₃	—	4	4	no	—
136	C₂×C₁₃	#2972	1	1	yes	characteristic
137	C₂×C₁₃	#3407	1	2	yes	—
138	#41	—	4	2	no	—
139	#41	—	4	2	no	—
140	C₂×Q₈	C₂×C₈×C₁₃	1	2	yes	—
141	C₂×D₄	C₂×C₈×C₁₃	1	4	yes	—
142	C₂×D₄	C₂×C₈×C₁₃	1	4	yes	—
143	C₂×D₄	C₂×C₈×C₁₃	1	2	yes	—
144	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
145	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
146	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
147	M₁₆	—	4	2	no	—
148	M₁₆	—	4	4	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
154	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
155	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
156	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
157	C₄⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
158	C₂²⋊C₄	C₂×C₈×C₁₃	1	8	yes	—
159	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
160	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
161	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
162	C₁₆	—	2	8	no	—
163	C₁₃	#33494	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³	C₂²×C₈×C₁₃	1	4	yes	—
165	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
166	Q₈	—	4	2	no	—
167	D₄	—	4	4	no	—
168	D₄	—	4	2	no	—
169	D₄	—	4	4	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
177	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
178	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
179	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	2	1	no	—
182	C₈	—	2	2	no	—
183	C₈	C₁₃×H(5,𝔽₂)	1	2	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	8	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	4	no	—
188	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	4	no	—
193	C₄	C₁₃×#518	1	1	yes	characteristic
194	C₄	C₁₃×#568	1	1	yes	characteristic
195	C₂	—	4	1	no	—
196	C₂	—	4	4	no	—
197	C₂	C₁₃×#2972	1	1	yes	characteristic
198	C₂	C₁₃×#3407	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×#2168	C₂	semidirect
3. C₁₃×#2163	C₂	semidirect
4. C₁₃×#2113	C₂	semidirect
39. #33494	C₁₃	direct (N × H)
66. #2168	C₂×C₁₃	semidirect
67. #2163	C₂×C₁₃	semidirect
68. #2113	C₂×C₁₃	semidirect
163. C₁₃	#33494	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×#2168	2
C₁₃×#2163	2
C₁₃×#2113	2
#33494	13

Subgroups — C13×#33494 #15719833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33494 #15719833