Subgroups — C₁₃×#3205 #13112263

Order 1664.

532 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#493	C₂	1	2	yes	maximal, complemented
4	C₁₃×#486	C₂	1	2	yes	maximal, complemented
5	C₁₃×#459	C₂	1	4	yes	maximal, complemented
6	C₁₃×#447	C₂	1	4	yes	maximal, complemented
7	C₁₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#41	—	2	1	no	—
9	C₂²×D₄×C₁₃	—	2	1	no	—
10	C₂²×D₄×C₁₃	—	2	1	no	—
11	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
12	C₂×C₁₃×D₈	C₂²	1	8	yes	complemented
13	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#127	—	2	1	no	—
15	C₁₃×#127	—	2	1	no	—
16	C₁₃×#127	C₂²	1	2	yes	—
17	C₁₃×#127	C₂²	1	2	yes	—
18	C₁₃×#127	C₂²	1	1	yes	characteristic, complemented
19	C₁₃×#124	—	2	1	no	—
20	C₁₃×#121	—	2	1	no	—
21	C₁₃×#121	C₂²	1	2	yes	—
22	C₄×D₄×C₁₃	—	2	1	no	—
23	C₄×D₄×C₁₃	C₂²	1	2	yes	complemented
24	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×C₄⋊C₈	C₂²	1	2	yes	complemented
26	C₁₃×#102	C₂²	1	8	yes	complemented
27	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈×C₁₃	C₂²	1	2	yes	complemented
29	C₂⁴×C₁₃	—	2	2	no	—
30	C₁₃×#41	—	4	4	no	—
31	C₂×Q₈×C₁₃	—	2	1	no	—
32	C₂×D₄×C₁₃	—	2	2	no	—
33	C₂×D₄×C₁₃	—	2	1	no	—
34	C₂×D₄×C₁₃	—	2	1	no	—
35	C₂×D₄×C₁₃	—	2	2	no	—
36	C₂×D₄×C₁₃	—	2	2	no	—
37	C₂×D₄×C₁₃	—	2	2	no	—
38	C₂×D₄×C₁₃	—	2	2	no	—
39	C₂×D₄×C₁₃	—	2	2	no	—
40	C₂×D₄×C₁₃	—	4	4	no	—
41	C₂×D₄×C₁₃	—	4	4	no	—
42	C₂×D₄×C₁₃	—	4	4	no	—
43	C₂×D₄×C₁₃	C₂³	1	4	yes	—
44	C₂×D₄×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×D₈	—	4	8	no	—
50	C₂×C₈×C₁₃	C₂³	1	4	yes	—
51	C₁₃×C₄⋊C₄	—	2	1	no	—
52	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₁₃×C₂²⋊C₄	—	2	2	no	—
54	C₁₃×C₂²⋊C₄	—	2	2	no	—
55	C₄²×C₁₃	D₄	1	1	yes	characteristic
56	C₄²×C₁₃	D₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	#3205	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₁₃	—	2	2	no	—
61	C₂³×C₁₃	—	2	4	no	—
62	C₂³×C₁₃	—	4	4	no	—
63	C₂³×C₁₃	—	4	4	no	—
64	C₂³×C₁₃	—	4	4	no	—
65	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₁₃	—	4	2	no	—
67	D₄×C₁₃	—	2	4	no	—
68	D₄×C₁₃	—	2	8	no	—
69	D₄×C₁₃	—	4	4	no	—
70	D₄×C₁₃	—	4	4	no	—
71	D₄×C₁₃	—	4	4	no	—
72	D₄×C₁₃	—	4	4	no	—
73	D₄×C₁₃	—	4	4	no	—
74	D₄×C₁₃	—	4	2	no	—
75	D₄×C₁₃	—	4	2	no	—
76	D₄×C₁₃	—	4	4	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	4	4	no	—
81	C₂×C₄×C₁₃	D₈	1	4	yes	complemented
82	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₈×C₁₃	—	2	4	no	—
89	#532	C₂×C₁₃	1	1	yes	characteristic, complemented
90	C₂×#127	C₂×C₁₃	1	1	yes	characteristic, complemented
91	#493	C₂×C₁₃	1	2	yes	complemented
92	#486	C₂×C₁₃	1	2	yes	complemented
93	#459	C₂×C₁₃	1	4	yes	complemented
94	#447	C₂×C₁₃	1	4	yes	complemented
95	#431	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	—	2	1	no	—
98	C₂²×C₁₃	—	4	4	no	—
99	C₂²×C₁₃	—	4	2	no	—
100	C₂²×C₁₃	—	4	2	no	—
101	C₂²×C₁₃	—	4	4	no	—
102	C₂²×C₁₃	—	4	4	no	—
103	C₂²×C₁₃	—	4	2	no	—
104	C₂²×C₁₃	—	8	8	no	—
105	C₂²×C₁₃	C₂×D₈	1	2	yes	—
106	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₁₃	—	2	1	no	—
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	—	4	2	no	—
110	C₄×C₁₃	C₂×D₈	1	2	yes	—
111	C₄×C₁₃	C₂×D₈	1	2	yes	—
112	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
113	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
114	C₂×#41	—	2	1	no	—
115	C₂²×D₄	—	2	1	no	—
116	C₂²×D₄	—	2	1	no	—
117	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
118	C₂×D₈	C₂²×C₁₃	1	8	yes	complemented
119	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
120	#127	—	2	1	no	—
121	#127	—	2	1	no	—
122	#127	C₂²×C₁₃	1	2	yes	—
123	#127	C₂²×C₁₃	1	2	yes	—
124	#127	C₂²×C₁₃	1	1	yes	characteristic, complemented
125	#124	—	2	1	no	—
126	#121	—	2	1	no	—
127	#121	C₂²×C₁₃	1	2	yes	—
128	C₄×D₄	—	2	1	no	—
129	C₄×D₄	C₂²×C₁₃	1	2	yes	complemented
130	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
131	C₄⋊C₈	C₂²×C₁₃	1	2	yes	complemented
132	#102	C₂²×C₁₃	1	8	yes	complemented
133	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
134	C₄×C₈	C₂²×C₁₃	1	2	yes	complemented
135	C₂×C₁₃	—	2	2	no	—
136	C₂×C₁₃	—	8	2	no	—
137	C₂×C₁₃	—	8	4	no	—
138	C₂×C₁₃	#535	1	1	yes	characteristic
139	C₂×C₁₃	C₂²×D₈	1	1	yes	characteristic
140	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂⁴	—	2	2	no	—
142	#41	—	4	4	no	—
143	C₂×Q₈	—	2	1	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	—	4	4	no	—
153	C₂×D₄	—	4	4	no	—
154	C₂×D₄	—	4	4	no	—
155	C₂×D₄	C₂³×C₁₃	1	4	yes	—
156	C₂×D₄	C₂³×C₁₃	1	2	yes	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
159	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
161	D₈	—	4	8	no	—
162	C₂×C₈	C₂³×C₁₃	1	4	yes	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₄²	D₄×C₁₃	1	1	yes	characteristic
168	C₄²	D₄×C₁₃	1	1	yes	characteristic
169	C₄²	C₂³×C₁₃	1	1	yes	characteristic
170	C₄²	C₂³×C₁₃	1	1	yes	characteristic
171	C₁₃	#3205	1	1	yes	characteristic, direct product factor, Sylow
172	C₂³	—	2	2	no	—
173	C₂³	—	2	4	no	—
174	C₂³	—	4	4	no	—
175	C₂³	—	4	4	no	—
176	C₂³	—	4	4	no	—
177	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	D₄	—	2	4	no	—
180	D₄	—	2	8	no	—
181	D₄	—	4	4	no	—
182	D₄	—	4	4	no	—
183	D₄	—	4	4	no	—
184	D₄	—	4	4	no	—
185	D₄	—	4	4	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	4	no	—
188	D₄	—	4	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₁₃×D₈	1	4	yes	complemented
194	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
195	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
196	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
197	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
198	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
199	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
200	C₈	—	2	4	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	4	no	—
207	C₂²	—	4	4	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	8	8	no	—
210	C₂²	C₂×C₁₃×D₈	1	2	yes	—
211	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	C₂×C₁₃×D₈	1	2	yes	—
216	C₄	C₂×C₁₃×D₈	1	2	yes	—
217	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
218	C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
219	C₂	—	2	2	no	—
220	C₂	—	8	4	no	—
221	C₂	—	8	2	no	—
222	C₂	C₁₃×#535	1	1	yes	characteristic
223	C₂	C₂²×C₁₃×D₈	1	1	yes	characteristic
224	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#532	C₂	semidirect
2. C₂×C₁₃×#127	C₂	semidirect
3. C₁₃×#493	C₂	semidirect
4. C₁₃×#486	C₂	semidirect
5. C₁₃×#459	C₂	semidirect
6. C₁₃×#447	C₂	semidirect
7. C₁₃×#431	C₂	semidirect
12. C₂×C₁₃×D₈	C₂²	semidirect
13. C₁₃×C₄⋊Q₈	C₂²	semidirect
18. C₁₃×#127	C₂²	semidirect
23. C₄×D₄×C₁₃	C₂²	semidirect
25. C₁₃×C₄⋊C₈	C₂²	semidirect
26. C₁₃×#102	C₂²	semidirect
28. C₄×C₈×C₁₃	C₂²	semidirect
59. #3205	C₁₃	direct (N × H)
81. C₂×C₄×C₁₃	D₈	semidirect
89. #532	C₂×C₁₃	semidirect
90. C₂×#127	C₂×C₁₃	semidirect
91. #493	C₂×C₁₃	semidirect
92. #486	C₂×C₁₃	semidirect
93. #459	C₂×C₁₃	semidirect
94. #447	C₂×C₁₃	semidirect
95. #431	C₂×C₁₃	semidirect
118. C₂×D₈	C₂²×C₁₃	semidirect
119. C₄⋊Q₈	C₂²×C₁₃	semidirect
124. #127	C₂²×C₁₃	semidirect
129. C₄×D₄	C₂²×C₁₃	semidirect
131. C₄⋊C₈	C₂²×C₁₃	semidirect
132. #102	C₂²×C₁₃	semidirect
134. C₄×C₈	C₂²×C₁₃	semidirect
171. C₁₃	#3205	direct (N × H)
193. C₂×C₄	C₁₃×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#532	2
C₂×C₁₃×#127	2
C₁₃×#493	2
C₁₃×#486	2
C₁₃×#459	2
C₁₃×#447	2
C₁₃×#431	2
#3205	13

Subgroups — C13×#3205 #13112263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3205 #13112263