Subgroups — C₁₃×#3254 #13112312

Order 1664.

386 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×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#477	C₂	1	2	yes	maximal, complemented
4	C₁₃×#474	C₂	1	2	yes	maximal, complemented
5	C₁₃×#468	C₂	1	4	yes	maximal, complemented
6	C₁₃×#461	C₂	1	2	yes	maximal, complemented
7	C₁₃×#460	C₂	1	2	yes	maximal, complemented
8	C₁₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#41	—	2	1	no	—
10	C₂×C₁₃×SD₁₆	C₂²	1	4	yes	—
11	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
12	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×#127	C₂²	1	1	yes	characteristic
16	C₁₃×#125	—	2	1	no	—
17	C₁₃×#124	—	2	1	no	—
18	C₁₃×C₂²⋊Q₈	—	2	1	no	—
19	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₁₃×#121	—	2	1	no	—
21	C₁₃×#121	C₂²	1	2	yes	—
22	C₄×Q₈×C₁₃	—	2	1	no	—
23	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
24	C₄×D₄×C₁₃	—	2	1	no	—
25	C₄×D₄×C₁₃	C₂²	1	2	yes	—
26	C₁₃×#117	—	2	1	no	—
27	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#107	C₂²	1	4	yes	—
29	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₁₃×#103	C₂²	1	4	yes	—
32	C₁₃×#102	C₂²	1	4	yes	—
33	C₁₃×#41	—	4	4	no	—
34	C₂×Q₈×C₁₃	—	2	1	no	—
35	C₂×Q₈×C₁₃	—	2	1	no	—
36	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
37	C₂×D₄×C₁₃	—	2	2	no	—
38	C₂×D₄×C₁₃	—	2	1	no	—
39	C₂×D₄×C₁₃	—	2	1	no	—
40	C₂×D₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×SD₁₆	—	4	4	no	—
46	C₁₃×M₁₆	—	4	2	no	—
47	C₂×C₈×C₁₃	C₂³	1	4	yes	—
48	C₁₃×C₄⋊C₄	—	2	1	no	—
49	C₁₃×C₄⋊C₄	—	2	1	no	—
50	C₁₃×C₄⋊C₄	—	2	1	no	—
51	C₁₃×C₄⋊C₄	—	2	1	no	—
52	C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₁₃×C₂²⋊C₄	—	2	2	no	—
57	C₁₃×C₂²⋊C₄	—	2	2	no	—
58	C₁₃×C₂²⋊C₄	—	2	2	no	—
59	C₄²×C₁₃	—	2	2	no	—
60	C₄²×C₁₃	D₄	1	1	yes	characteristic
61	C₄²×C₁₃	D₄	1	1	yes	characteristic
62	C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₄²×C₁₃	C₂³	1	1	yes	characteristic
64	#3254	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₁₃	—	2	2	no	—
66	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₁₃	—	2	2	no	—
68	Q₈×C₁₃	—	2	2	no	—
69	Q₈×C₁₃	—	4	2	no	—
70	Q₈×C₁₃	—	4	2	no	—
71	D₄×C₁₃	—	2	2	no	—
72	D₄×C₁₃	—	2	2	no	—
73	D₄×C₁₃	—	4	2	no	—
74	D₄×C₁₃	—	4	4	no	—
75	D₄×C₁₃	—	4	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	4	4	no	—
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	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₈×C₁₃	—	2	4	no	—
91	#533	C₂×C₁₃	1	1	yes	characteristic, complemented
92	#532	C₂×C₁₃	1	1	yes	characteristic
93	#477	C₂×C₁₃	1	2	yes	complemented
94	#474	C₂×C₁₃	1	2	yes	complemented
95	#468	C₂×C₁₃	1	4	yes	complemented
96	#461	C₂×C₁₃	1	2	yes	complemented
97	#460	C₂×C₁₃	1	2	yes	complemented
98	#423	C₂×C₁₃	1	1	yes	characteristic, complemented
99	C₂²×C₁₃	—	2	1	no	—
100	C₂²×C₁₃	—	2	1	no	—
101	C₂²×C₁₃	—	2	1	no	—
102	C₂²×C₁₃	—	4	2	no	—
103	C₂²×C₁₃	—	4	2	no	—
104	C₂²×C₁₃	—	4	2	no	—
105	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₁₃	—	2	1	no	—
107	C₄×C₁₃	—	2	1	no	—
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	—	4	2	no	—
110	C₄×C₁₃	—	4	2	no	—
111	C₄×C₁₃	—	4	2	no	—
112	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
113	C₄×C₁₃	#137	1	2	yes	—
114	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×#41	—	2	1	no	—
117	C₂×SD₁₆	C₂²×C₁₃	1	4	yes	—
118	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
119	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
120	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
121	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
122	#127	C₂²×C₁₃	1	1	yes	characteristic
123	#125	—	2	1	no	—
124	#124	—	2	1	no	—
125	C₂²⋊Q₈	—	2	1	no	—
126	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
127	#121	—	2	1	no	—
128	#121	C₂²×C₁₃	1	2	yes	—
129	C₄×Q₈	—	2	1	no	—
130	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
131	C₄×D₄	—	2	1	no	—
132	C₄×D₄	C₂²×C₁₃	1	2	yes	—
133	#117	—	2	1	no	—
134	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
135	#107	C₂²×C₁₃	1	4	yes	—
136	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
137	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
138	#103	C₂²×C₁₃	1	4	yes	—
139	#102	C₂²×C₁₃	1	4	yes	—
140	C₂×C₁₃	—	4	1	no	—
141	C₂×C₁₃	—	8	2	no	—
142	C₂×C₁₃	#537	1	1	yes	characteristic
143	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
145	#41	—	4	4	no	—
146	C₂×Q₈	—	2	1	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	—	2	1	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	C₂³×C₁₃	1	2	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
155	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
157	SD₁₆	—	4	4	no	—
158	M₁₆	—	4	2	no	—
159	C₂×C₈	C₂³×C₁₃	1	4	yes	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
167	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	D₄×C₁₃	1	1	yes	characteristic
173	C₄²	D₄×C₁₃	1	1	yes	characteristic
174	C₄²	C₂³×C₁₃	1	1	yes	characteristic
175	C₄²	C₂³×C₁₃	1	1	yes	characteristic
176	C₁₃	#3254	1	1	yes	characteristic, direct product factor, Sylow
177	C₂³	—	2	2	no	—
178	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
179	Q₈	—	2	2	no	—
180	Q₈	—	2	2	no	—
181	Q₈	—	4	2	no	—
182	Q₈	—	4	2	no	—
183	D₄	—	2	2	no	—
184	D₄	—	2	2	no	—
185	D₄	—	4	2	no	—
186	D₄	—	4	4	no	—
187	D₄	—	4	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
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	C₂×D₄×C₁₃	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
201	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
202	C₈	—	2	4	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
217	C₄	C₁₃×#137	1	2	yes	—
218	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
219	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
220	C₂	—	4	1	no	—
221	C₂	—	8	2	no	—
222	C₂	C₁₃×#537	1	1	yes	characteristic
223	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
224	C₂	C₂×C₁₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#532	C₂
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×#533	C₂	semidirect
3. C₁₃×#477	C₂	semidirect
4. C₁₃×#474	C₂	semidirect
5. C₁₃×#468	C₂	semidirect
6. C₁₃×#461	C₂	semidirect
7. C₁₃×#460	C₂	semidirect
8. C₁₃×#423	C₂	semidirect
64. #3254	C₁₃	direct (N × H)
91. #533	C₂×C₁₃	semidirect
93. #477	C₂×C₁₃	semidirect
94. #474	C₂×C₁₃	semidirect
95. #468	C₂×C₁₃	semidirect
96. #461	C₂×C₁₃	semidirect
97. #460	C₂×C₁₃	semidirect
98. #423	C₂×C₁₃	semidirect
176. C₁₃	#3254	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×#533	2
C₁₃×#532	2
C₁₃×#477	2
C₁₃×#474	2
C₁₃×#468	2
C₁₃×#461	2
C₁₃×#460	2
C₁₃×#423	2
#3254	13

Subgroups — C13×#3254 #13112312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3254 #13112312