Subgroups — C₁₃×#2639 #13111697

Order 1664.

576 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#121	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#400	C₂	1	2	yes	maximal, complemented
4	C₁₃×#397	C₂	1	2	yes	maximal, complemented
5	C₁₃×#394	C₂	1	2	yes	maximal, complemented
6	C₁₃×#393	C₂	1	2	yes	maximal, complemented
7	C₁₃×#386	C₂	1	4	yes	maximal, complemented
8	C₂×C₁₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
11	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×C₂²⋊Q₈	—	2	4	no	—
15	C₁₃×#121	—	2	4	no	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
22	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₁₃×#95	C₂²	1	4	yes	—
24	C₁₃×#95	C₂²	1	4	yes	—
25	C₁₃×#95	C₂²	1	2	yes	—
26	C₁₃×#95	C₂²	1	2	yes	—
27	C₂⁴×C₁₃	C₂³	1	2	yes	—
28	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₁₃	—	2	2	no	—
30	C₂×Q₈×C₁₃	—	4	2	no	—
31	C₂×D₄×C₁₃	—	2	4	no	—
32	C₂×D₄×C₁₃	—	2	8	no	—
33	C₂×D₄×C₁₃	—	2	2	no	—
34	C₂×D₄×C₁₃	—	4	2	no	—
35	C₂²×C₄×C₁₃	—	2	4	no	—
36	C₂²×C₄×C₁₃	—	4	2	no	—
37	C₂²×C₄×C₁₃	—	4	2	no	—
38	C₂²×C₄×C₁₃	—	4	2	no	—
39	C₂²×C₄×C₁₃	D₄	1	4	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₁₃×C₄⋊C₄	—	2	4	no	—
49	C₁₃×C₂²⋊C₄	—	2	4	no	—
50	C₁₃×C₂²⋊C₄	—	2	4	no	—
51	C₁₃×C₂²⋊C₄	—	4	4	no	—
52	C₁₃×C₂²⋊C₄	—	4	4	no	—
53	#2639	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₁₃	—	2	2	no	—
55	C₂³×C₁₃	—	2	1	no	—
56	C₂³×C₁₃	—	2	1	no	—
57	C₂³×C₁₃	—	2	2	no	—
58	C₂³×C₁₃	—	2	4	no	—
59	C₂³×C₁₃	—	2	2	no	—
60	C₂³×C₁₃	—	2	4	no	—
61	C₂³×C₁₃	—	2	2	no	—
62	C₂³×C₁₃	C₂×D₄	1	2	yes	—
63	C₂³×C₁₃	#41	1	4	yes	—
64	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
65	Q₈×C₁₃	—	4	4	no	—
66	D₄×C₁₃	—	4	4	no	—
67	D₄×C₁₃	—	4	8	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	4	no	—
74	C₂×C₄×C₁₃	—	2	4	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	—	2	4	no	—
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	4	4	no	—
81	C₂×C₄×C₁₃	—	4	8	no	—
82	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₂²⋊Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
85	C₂×#121	C₂×C₁₃	1	1	yes	characteristic
86	#400	C₂×C₁₃	1	2	yes	complemented
87	#397	C₂×C₁₃	1	2	yes	complemented
88	#394	C₂×C₁₃	1	2	yes	complemented
89	#393	C₂×C₁₃	1	2	yes	complemented
90	#386	C₂×C₁₃	1	4	yes	complemented
91	C₂×#95	C₂×C₁₃	1	1	yes	characteristic, complemented
92	C₂²×C₁₃	—	2	2	no	—
93	C₂²×C₁₃	—	2	2	no	—
94	C₂²×C₁₃	—	2	4	no	—
95	C₂²×C₁₃	—	2	4	no	—
96	C₂²×C₁₃	—	4	2	no	—
97	C₂²×C₁₃	—	4	2	no	—
98	C₂²×C₁₃	—	4	2	no	—
99	C₂²×C₁₃	—	4	4	no	—
100	C₂²×C₁₃	—	4	4	no	—
101	C₂²×C₁₃	#124	1	4	yes	—
102	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
103	C₂²×C₁₃	C₂×#41	1	2	yes	—
104	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
105	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	—	2	2	no	—
109	C₄×C₁₃	—	4	2	no	—
110	C₄×C₁₃	—	4	2	no	—
111	C₄×C₁₃	—	4	2	no	—
112	C₄×C₁₃	—	4	4	no	—
113	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
114	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
115	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
116	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
117	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
118	C₂²⋊Q₈	—	2	4	no	—
119	#121	—	2	4	no	—
120	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
123	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
124	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
125	C₂×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
126	C₂×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
127	#95	C₂²×C₁₃	1	2	yes	—
128	#95	C₂²×C₁₃	1	2	yes	—
129	#95	C₂²×C₁₃	1	4	yes	—
130	#95	C₂²×C₁₃	1	4	yes	—
131	C₂×C₁₃	—	2	4	no	—
132	C₂×C₁₃	—	8	2	no	—
133	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
134	C₂×C₁₃	#534	1	1	yes	characteristic
135	C₂×C₁₃	#537	1	1	yes	characteristic
136	C₂×C₁₃	#538	1	2	yes	—
137	C₂×C₁₃	#539	1	2	yes	—
138	C₂⁴	C₂³×C₁₃	1	2	yes	—
139	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×D₄	—	2	4	no	—
143	C₂×D₄	—	2	8	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	4	2	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	D₄×C₁₃	1	4	yes	—
151	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
152	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
153	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
154	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
155	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₂²⋊C₄	—	2	4	no	—
162	C₂²⋊C₄	—	4	4	no	—
163	C₂²⋊C₄	—	4	4	no	—
164	C₁₃	#2639	1	1	yes	characteristic, direct product factor, Sylow
165	C₂³	—	2	2	no	—
166	C₂³	—	2	1	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	4	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	4	no	—
172	C₂³	—	2	2	no	—
173	C₂³	C₂×D₄×C₁₃	1	2	yes	—
174	C₂³	C₁₃×#41	1	4	yes	—
175	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
176	Q₈	—	4	4	no	—
177	D₄	—	4	4	no	—
178	D₄	—	4	8	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	4	8	no	—
193	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
194	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	4	no	—
203	C₂²	—	4	4	no	—
204	C₂²	C₁₃×#124	1	4	yes	—
205	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
206	C₂²	C₂×C₁₃×#41	1	2	yes	—
207	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
208	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
209	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	4	no	—
216	C₂	—	2	4	no	—
217	C₂	—	8	2	no	—
218	C₂	C₂×C₁₃×#124	1	1	yes	characteristic
219	C₂	C₁₃×#534	1	1	yes	characteristic
220	C₂	C₁₃×#537	1	1	yes	characteristic
221	C₂	C₁₃×#538	1	2	yes	—
222	C₂	C₁₃×#539	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×C₂²⋊Q₈	C₂	semidirect
3. C₁₃×#400	C₂	semidirect
4. C₁₃×#397	C₂	semidirect
5. C₁₃×#394	C₂	semidirect
6. C₁₃×#393	C₂	semidirect
7. C₁₃×#386	C₂	semidirect
8. C₂×C₁₃×#95	C₂	semidirect
53. #2639	C₁₃	direct (N × H)
84. C₂×C₂²⋊Q₈	C₂×C₁₃	semidirect
86. #400	C₂×C₁₃	semidirect
87. #397	C₂×C₁₃	semidirect
88. #394	C₂×C₁₃	semidirect
89. #393	C₂×C₁₃	semidirect
90. #386	C₂×C₁₃	semidirect
91. C₂×#95	C₂×C₁₃	semidirect
164. C₁₃	#2639	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×C₂²⋊Q₈	2
C₂×C₁₃×#121	2
C₁₃×#400	2
C₁₃×#397	2
C₁₃×#394	2
C₁₃×#393	2
C₁₃×#386	2
C₂×C₁₃×#95	2
#2639	13

Subgroups — C13×#2639 #13111697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2639 #13111697