Subgroups — C₁₃×#2535 #13111593

Order 1664.

732 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#121	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#397	C₂	1	2	yes	maximal, complemented
5	C₁₃×#392	C₂	1	2	yes	maximal, complemented
6	C₁₃×#386	C₂	1	4	yes	maximal, complemented
7	C₁₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#380	C₂	1	2	yes	maximal, complemented
9	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₁₃	C₂²	1	4	yes	—
11	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
14	C₁₃×#121	—	2	8	no	—
15	C₄×D₄×C₁₃	—	2	4	no	—
16	C₄×D₄×C₁₃	—	2	4	no	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
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	2	yes	—
23	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×#95	C₂²	1	4	yes	—
26	C₁₃×#95	C₂²	1	2	yes	—
27	C₁₃×#95	C₂²	1	2	yes	—
28	C₂⁴×C₁₃	C₂³	1	2	yes	—
29	C₂⁴×C₁₃	C₂³	1	2	yes	—
30	C₂×D₄×C₁₃	—	2	4	no	—
31	C₂×D₄×C₁₃	—	2	8	no	—
32	C₂×D₄×C₁₃	—	2	8	no	—
33	C₂×D₄×C₁₃	—	2	2	no	—
34	C₂×D₄×C₁₃	—	2	2	no	—
35	C₂×D₄×C₁₃	—	2	2	no	—
36	C₂×D₄×C₁₃	—	4	4	no	—
37	C₂×D₄×C₁₃	D₄	1	4	yes	complemented
38	C₂²×C₄×C₁₃	—	2	4	no	—
39	C₂²×C₄×C₁₃	—	2	4	no	—
40	C₂²×C₄×C₁₃	—	2	4	no	—
41	C₂²×C₄×C₁₃	—	2	4	no	—
42	C₂²×C₄×C₁₃	—	4	4	no	—
43	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	—	2	4	no	—
52	C₁₃×C₂²⋊C₄	—	2	8	no	—
53	C₁₃×C₂²⋊C₄	—	2	4	no	—
54	C₁₃×C₂²⋊C₄	—	2	4	no	—
55	C₁₃×C₂²⋊C₄	—	4	4	no	—
56	C₄²×C₁₃	D₄	1	4	yes	complemented
57	#2535	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₁₃	—	2	2	no	—
59	C₂³×C₁₃	—	2	2	no	—
60	C₂³×C₁₃	—	2	2	no	—
61	C₂³×C₁₃	—	2	2	no	—
62	C₂³×C₁₃	—	2	2	no	—
63	C₂³×C₁₃	—	2	2	no	—
64	C₂³×C₁₃	—	2	4	no	—
65	C₂³×C₁₃	—	2	2	no	—
66	C₂³×C₁₃	—	2	4	no	—
67	C₂³×C₁₃	—	2	2	no	—
68	C₂³×C₁₃	C₂×D₄	1	4	yes	—
69	C₂³×C₁₃	#41	1	4	yes	—
70	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
71	D₄×C₁₃	—	2	8	no	—
72	D₄×C₁₃	—	2	8	no	—
73	D₄×C₁₃	—	4	16	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	4	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	4	no	—
82	C₂×C₄×C₁₃	—	2	4	no	—
83	C₂×C₄×C₁₃	—	2	4	no	—
84	C₂×C₄×C₁₃	—	4	8	no	—
85	C₂×C₄×C₁₃	—	4	8	no	—
86	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₁₃	#41	1	2	yes	—
91	C₂×#121	C₂×C₁₃	1	2	yes	complemented
92	C₂×C₄×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
93	C₂×C₄×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
94	#397	C₂×C₁₃	1	2	yes	complemented
95	#392	C₂×C₁₃	1	2	yes	complemented
96	#386	C₂×C₁₃	1	4	yes	complemented
97	#382	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#380	C₂×C₁₃	1	2	yes	complemented
99	C₂²×C₁₃	—	2	4	no	—
100	C₂²×C₁₃	—	2	4	no	—
101	C₂²×C₁₃	—	2	4	no	—
102	C₂²×C₁₃	—	2	8	no	—
103	C₂²×C₁₃	—	2	4	no	—
104	C₂²×C₁₃	—	4	2	no	—
105	C₂²×C₁₃	—	4	2	no	—
106	C₂²×C₁₃	—	4	2	no	—
107	C₂²×C₁₃	—	4	2	no	—
108	C₂²×C₁₃	—	4	2	no	—
109	C₂²×C₁₃	—	4	2	no	—
110	C₂²×C₁₃	—	4	4	no	—
111	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₂²×C₁₃	C₂×#41	1	2	yes	—
115	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
116	C₄×C₁₃	—	2	2	no	—
117	C₄×C₁₃	—	2	2	no	—
118	C₄×C₁₃	—	4	2	no	—
119	C₄×C₁₃	—	4	2	no	—
120	C₄×C₁₃	—	4	4	no	—
121	C₄×C₁₃	#121	1	4	yes	—
122	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
123	C₂²×D₄	C₂²×C₁₃	1	4	yes	—
124	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
125	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
126	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
127	#121	—	2	8	no	—
128	C₄×D₄	—	2	4	no	—
129	C₄×D₄	—	2	4	no	—
130	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
135	C₂×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
136	C₂×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
137	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
138	#95	C₂²×C₁₃	1	4	yes	—
139	#95	C₂²×C₁₃	1	2	yes	—
140	#95	C₂²×C₁₃	1	2	yes	—
141	C₂×C₁₃	—	4	4	no	—
142	C₂×C₁₃	—	4	4	no	—
143	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
144	C₂×C₁₃	#525	1	1	yes	characteristic
145	C₂×C₁₃	#535	1	1	yes	characteristic
146	C₂×C₁₃	#546	1	2	yes	—
147	C₂×C₁₃	#553	1	2	yes	—
148	C₂⁴	C₂³×C₁₃	1	2	yes	—
149	C₂⁴	C₂³×C₁₃	1	2	yes	—
150	C₂×D₄	—	2	4	no	—
151	C₂×D₄	—	2	8	no	—
152	C₂×D₄	—	2	8	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	4	4	no	—
157	C₂×D₄	D₄×C₁₃	1	4	yes	complemented
158	C₂²×C₄	—	2	4	no	—
159	C₂²×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	C₂³×C₁₃	1	4	yes	—
164	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
165	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
166	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₂²⋊C₄	—	2	8	no	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	4	4	no	—
176	C₄²	D₄×C₁₃	1	4	yes	complemented
177	C₁₃	#2535	1	1	yes	characteristic, direct product factor, Sylow
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	4	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	4	no	—
187	C₂³	—	2	2	no	—
188	C₂³	C₂×D₄×C₁₃	1	4	yes	—
189	C₂³	C₁₃×#41	1	4	yes	—
190	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
191	D₄	—	2	8	no	—
192	D₄	—	2	8	no	—
193	D₄	—	4	16	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
208	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
209	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
210	C₂×C₄	C₁₃×#41	1	2	yes	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	4	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	4	no	—
223	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
224	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
225	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
226	C₂²	C₂×C₁₃×#41	1	2	yes	—
227	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	4	no	—
233	C₄	C₁₃×#121	1	4	yes	—
234	C₂	—	4	4	no	—
235	C₂	—	4	4	no	—
236	C₂	C₂×C₁₃×#121	1	1	yes	characteristic
237	C₂	C₁₃×#525	1	1	yes	characteristic
238	C₂	C₁₃×#535	1	1	yes	characteristic
239	C₂	C₁₃×#546	1	2	yes	—
240	C₂	C₁₃×#553	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
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×#121	C₂	semidirect
2. C₂×C₄×D₄×C₁₃	C₂	semidirect
3. C₂×C₄×D₄×C₁₃	C₂	semidirect
4. C₁₃×#397	C₂	semidirect
5. C₁₃×#392	C₂	semidirect
6. C₁₃×#386	C₂	semidirect
7. C₁₃×#382	C₂	semidirect
8. C₁₃×#380	C₂	semidirect
37. C₂×D₄×C₁₃	D₄	semidirect
56. C₄²×C₁₃	D₄	semidirect
57. #2535	C₁₃	direct (N × H)
91. C₂×#121	C₂×C₁₃	semidirect
92. C₂×C₄×D₄	C₂×C₁₃	semidirect
93. C₂×C₄×D₄	C₂×C₁₃	semidirect
94. #397	C₂×C₁₃	semidirect
95. #392	C₂×C₁₃	semidirect
96. #386	C₂×C₁₃	semidirect
97. #382	C₂×C₁₃	semidirect
98. #380	C₂×C₁₃	semidirect
157. C₂×D₄	D₄×C₁₃	semidirect
176. C₄²	D₄×C₁₃	semidirect
177. C₁₃	#2535	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×#121	2
C₂×C₄×D₄×C₁₃	2
C₁₃×#397	2
C₁₃×#392	2
C₁₃×#386	2
C₁₃×#382	2
C₁₃×#380	2
#2535	13

Subgroups — C13×#2535 #13111593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2535 #13111593