Subgroups — C₁₃×#2712 #13111770

Order 1664.

556 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#124	C₂	1	2	yes	maximal, complemented
2	C₁₃×#397	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#394	C₂	1	2	yes	maximal, complemented
4	C₁₃×#393	C₂	1	2	yes	maximal, complemented
5	C₁₃×#392	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#391	C₂	1	2	yes	maximal, complemented
7	C₁₃×#388	C₂	1	2	yes	maximal, complemented
8	C₁₃×#386	C₂	1	2	yes	maximal, complemented
9	C₁₃×#385	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
12	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#124	—	2	8	no	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
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	1	yes	characteristic
22	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
26	C₁₃×#95	C₂²	1	2	yes	—
27	C₁₃×#95	C₂²	1	2	yes	—
28	C₁₃×#95	C₂²	1	2	yes	—
29	C₁₃×#95	C₂²	1	1	yes	characteristic
30	C₁₃×#95	C₂²	1	2	yes	—
31	C₁₃×#95	C₂²	1	1	yes	characteristic
32	C₂⁴×C₁₃	C₂³	1	2	yes	—
33	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×Q₈×C₁₃	—	2	4	no	—
35	C₂×Q₈×C₁₃	—	2	4	no	—
36	C₂×Q₈×C₁₃	—	4	2	no	—
37	C₂×D₄×C₁₃	—	2	4	no	—
38	C₂×D₄×C₁₃	—	2	4	no	—
39	C₂×D₄×C₁₃	—	4	2	no	—
40	C₂×D₄×C₁₃	—	4	1	no	—
41	C₂×D₄×C₁₃	—	4	2	no	—
42	C₂²×C₄×C₁₃	—	4	1	no	—
43	C₂²×C₄×C₁₃	—	4	2	no	—
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
51	C₁₃×C₄⋊C₄	—	4	2	no	—
52	C₁₃×C₄⋊C₄	—	4	1	no	—
53	C₁₃×C₂²⋊C₄	—	2	4	no	—
54	C₁₃×C₂²⋊C₄	—	2	4	no	—
55	C₁₃×C₂²⋊C₄	—	2	4	no	—
56	C₁₃×C₂²⋊C₄	—	2	4	no	—
57	C₁₃×C₂²⋊C₄	—	4	2	no	—
58	C₁₃×C₂²⋊C₄	—	4	1	no	—
59	C₁₃×C₂²⋊C₄	—	4	2	no	—
60	C₁₃×C₂²⋊C₄	D₄	1	4	yes	complemented
61	C₄²×C₁₃	—	2	4	no	—
62	#2712	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₁₃	—	2	2	no	—
64	C₂³×C₁₃	—	2	1	no	—
65	C₂³×C₁₃	—	2	2	no	—
66	C₂³×C₁₃	—	2	1	no	—
67	C₂³×C₁₃	—	2	2	no	—
68	C₂³×C₁₃	—	2	2	no	—
69	C₂³×C₁₃	—	2	2	no	—
70	C₂³×C₁₃	—	2	2	no	—
71	C₂³×C₁₃	—	2	2	no	—
72	C₂³×C₁₃	—	2	2	no	—
73	C₂³×C₁₃	—	2	2	no	—
74	C₂³×C₁₃	C₂×D₄	1	2	yes	—
75	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
76	Q₈×C₁₃	—	4	8	no	—
77	D₄×C₁₃	—	4	8	no	—
78	D₄×C₁₃	—	4	4	no	—
79	C₂×C₄×C₁₃	—	2	2	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	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	4	4	no	—
97	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
98	C₂×C₄×C₁₃	#41	1	4	yes	—
99	C₂×#124	C₂×C₁₃	1	2	yes	complemented
100	#397	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#394	C₂×C₁₃	1	2	yes	complemented
102	#393	C₂×C₁₃	1	2	yes	complemented
103	#392	C₂×C₁₃	1	1	yes	characteristic
104	#391	C₂×C₁₃	1	2	yes	complemented
105	#388	C₂×C₁₃	1	2	yes	complemented
106	#386	C₂×C₁₃	1	2	yes	complemented
107	#385	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂²×C₁₃	—	2	2	no	—
109	C₂²×C₁₃	—	2	4	no	—
110	C₂²×C₁₃	—	4	2	no	—
111	C₂²×C₁₃	—	4	1	no	—
112	C₂²×C₁₃	—	4	1	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	—	4	2	no	—
116	C₂²×C₁₃	—	4	2	no	—
117	C₂²×C₁₃	—	4	2	no	—
118	C₂²×C₁₃	—	4	2	no	—
119	C₂²×C₁₃	—	4	2	no	—
120	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₂²×C₁₃	C₂×#41	1	2	yes	—
122	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
124	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
125	C₄×C₁₃	—	2	4	no	—
126	C₄×C₁₃	—	2	4	no	—
127	C₄×C₁₃	—	4	2	no	—
128	C₄×C₁₃	—	4	2	no	—
129	C₄×C₁₃	—	4	1	no	—
130	C₄×C₁₃	—	4	2	no	—
131	C₄×C₁₃	—	4	1	no	—
132	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
133	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
134	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
135	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
136	#124	—	2	8	no	—
137	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
138	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
139	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
140	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
148	#95	C₂²×C₁₃	1	2	yes	—
149	#95	C₂²×C₁₃	1	2	yes	—
150	#95	C₂²×C₁₃	1	2	yes	—
151	#95	C₂²×C₁₃	1	1	yes	characteristic
152	#95	C₂²×C₁₃	1	2	yes	—
153	#95	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×C₁₃	—	4	2	no	—
155	C₂×C₁₃	—	8	2	no	—
156	C₂×C₁₃	#537	1	1	yes	characteristic
157	C₂×C₁₃	#542	1	2	yes	—
158	C₂×C₁₃	#546	1	2	yes	—
159	C₂×C₁₃	#559	1	1	yes	characteristic
160	C₂×C₁₃	H(3,𝔽₄)	1	1	yes	characteristic
161	C₂⁴	C₂³×C₁₃	1	2	yes	—
162	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	—	4	1	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
175	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
176	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
178	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
179	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₂²⋊C₄	—	2	4	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₂²⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	—	4	1	no	—
188	C₂²⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	D₄×C₁₃	1	4	yes	complemented
190	C₄²	—	2	4	no	—
191	C₁₃	#2712	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	C₂×D₄×C₁₃	1	2	yes	—
204	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
205	Q₈	—	4	8	no	—
206	D₄	—	4	8	no	—
207	D₄	—	4	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
227	C₂×C₄	C₁₃×#41	1	4	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	1	no	—
232	C₂²	—	4	1	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
241	C₂²	C₂×C₁₃×#41	1	2	yes	—
242	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
244	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	1	no	—
252	C₂	—	4	2	no	—
253	C₂	—	8	2	no	—
254	C₂	C₁₃×#537	1	1	yes	characteristic
255	C₂	C₁₃×#542	1	2	yes	—
256	C₂	C₁₃×#546	1	2	yes	—
257	C₂	C₁₃×#559	1	1	yes	characteristic
258	C₂	C₁₃×H(3,𝔽₄)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₁₃×#392	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×#124	C₂	semidirect
2. C₁₃×#397	C₂	semidirect
3. C₁₃×#394	C₂	semidirect
4. C₁₃×#393	C₂	semidirect
6. C₁₃×#391	C₂	semidirect
7. C₁₃×#388	C₂	semidirect
8. C₁₃×#386	C₂	semidirect
9. C₁₃×#385	C₂	semidirect
60. C₁₃×C₂²⋊C₄	D₄	semidirect
62. #2712	C₁₃	direct (N × H)
99. C₂×#124	C₂×C₁₃	semidirect
100. #397	C₂×C₁₃	semidirect
101. #394	C₂×C₁₃	semidirect
102. #393	C₂×C₁₃	semidirect
104. #391	C₂×C₁₃	semidirect
105. #388	C₂×C₁₃	semidirect
106. #386	C₂×C₁₃	semidirect
107. #385	C₂×C₁₃	semidirect
189. C₂²⋊C₄	D₄×C₁₃	semidirect
191. C₁₃	#2712	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×#124	2
C₁₃×#397	2
C₁₃×#394	2
C₁₃×#393	2
C₁₃×#392	2
C₁₃×#391	2
C₁₃×#388	2
C₁₃×#386	2
C₁₃×#385	2
#2712	13

Subgroups — C13×#2712 #13111770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2712 #13111770