Subgroups — C₁₃×#2555 #13111613

Order 1664.

696 conjugacy classes of subgroups.

The subgroup lattice has 286 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	C₂×C₁₃×#121	C₂	1	2	yes	maximal, complemented
3	C₁₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#394	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#392	C₂	1	2	yes	maximal, complemented
6	C₁₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#388	C₂	1	2	yes	maximal, complemented
8	C₁₃×#386	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
12	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#124	—	2	8	no	—
17	C₁₃×#121	—	2	8	no	—
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	2	yes	—
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	2	yes	—
25	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#95	C₂²	1	2	yes	—
30	C₁₃×#95	C₂²	1	2	yes	—
31	C₁₃×#95	C₂²	1	1	yes	characteristic
32	C₁₃×#95	C₂²	1	1	yes	characteristic
33	C₂⁴×C₁₃	C₂³	1	2	yes	—
34	C₂⁴×C₁₃	C₂³	1	2	yes	—
35	C₂×Q₈×C₁₃	—	2	4	no	—
36	C₂×Q₈×C₁₃	—	4	1	no	—
37	C₂×D₄×C₁₃	—	2	4	no	—
38	C₂×D₄×C₁₃	—	2	4	no	—
39	C₂×D₄×C₁₃	—	2	4	no	—
40	C₂×D₄×C₁₃	—	2	4	no	—
41	C₂×D₄×C₁₃	—	2	2	no	—
42	C₂×D₄×C₁₃	—	2	4	no	—
43	C₂×D₄×C₁₃	—	4	1	no	—
44	C₂×D₄×C₁₃	—	4	1	no	—
45	C₂×D₄×C₁₃	—	4	2	no	—
46	C₂×D₄×C₁₃	—	4	2	no	—
47	C₂×D₄×C₁₃	—	4	1	no	—
48	C₂²×C₄×C₁₃	—	2	4	no	—
49	C₂²×C₄×C₁₃	—	2	4	no	—
50	C₂²×C₄×C₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×C₄⋊C₄	—	2	4	no	—
59	C₁₃×C₄⋊C₄	—	4	1	no	—
60	C₁₃×C₂²⋊C₄	—	2	4	no	—
61	C₁₃×C₂²⋊C₄	—	2	4	no	—
62	C₁₃×C₂²⋊C₄	—	2	4	no	—
63	C₁₃×C₂²⋊C₄	—	2	4	no	—
64	C₁₃×C₂²⋊C₄	—	2	4	no	—
65	C₁₃×C₂²⋊C₄	—	2	4	no	—
66	C₁₃×C₂²⋊C₄	D₄	1	8	yes	complemented
67	C₄²×C₁₃	—	2	4	no	—
68	C₄²×C₁₃	—	4	1	no	—
69	#2555	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	—	2	2	no	—
75	C₂³×C₁₃	—	2	2	no	—
76	C₂³×C₁₃	—	2	2	no	—
77	C₂³×C₁₃	—	2	2	no	—
78	C₂³×C₁₃	—	2	2	no	—
79	C₂³×C₁₃	—	2	2	no	—
80	C₂³×C₁₃	—	2	2	no	—
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	—	2	2	no	—
83	C₂³×C₁₃	C₂×D₄	1	4	yes	—
84	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
85	Q₈×C₁₃	—	4	4	no	—
86	D₄×C₁₃	—	4	8	no	—
87	D₄×C₁₃	—	4	8	no	—
88	D₄×C₁₃	—	4	4	no	—
89	D₄×C₁₃	—	4	4	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	4	8	no	—
106	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
107	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
108	C₂×C₄×C₁₃	#41	1	2	yes	—
109	C₂×C₄×C₁₃	#41	1	4	yes	—
110	C₂×#124	C₂×C₁₃	1	2	yes	complemented
111	C₂×#121	C₂×C₁₃	1	2	yes	complemented
112	#395	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#394	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#392	C₂×C₁₃	1	2	yes	complemented
115	#390	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#388	C₂×C₁₃	1	2	yes	complemented
117	#386	C₂×C₁₃	1	2	yes	complemented
118	C₄×C₂²⋊C₄	C₂×C₁₃	1	2	yes	complemented
119	C₂²×C₁₃	—	2	4	no	—
120	C₂²×C₁₃	—	2	4	no	—
121	C₂²×C₁₃	—	2	4	no	—
122	C₂²×C₁₃	—	4	2	no	—
123	C₂²×C₁₃	—	4	2	no	—
124	C₂²×C₁₃	—	4	2	no	—
125	C₂²×C₁₃	—	4	2	no	—
126	C₂²×C₁₃	—	4	2	no	—
127	C₂²×C₁₃	—	4	2	no	—
128	C₂²×C₁₃	—	4	2	no	—
129	C₂²×C₁₃	—	4	2	no	—
130	C₂²×C₁₃	—	4	2	no	—
131	C₂²×C₁₃	—	4	2	no	—
132	C₂²×C₁₃	—	4	2	no	—
133	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
134	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×#41	1	2	yes	—
136	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	4	no	—
140	C₄×C₁₃	—	2	4	no	—
141	C₄×C₁₃	—	2	4	no	—
142	C₄×C₁₃	—	4	2	no	—
143	C₄×C₁₃	—	4	1	no	—
144	C₄×C₁₃	—	4	1	no	—
145	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
146	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
147	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
148	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
149	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
150	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
151	#124	—	2	8	no	—
152	#121	—	2	8	no	—
153	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
154	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
156	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
161	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
162	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
163	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
164	#95	C₂²×C₁₃	1	2	yes	—
165	#95	C₂²×C₁₃	1	2	yes	—
166	#95	C₂²×C₁₃	1	1	yes	characteristic
167	#95	C₂²×C₁₃	1	1	yes	characteristic
168	C₂×C₁₃	—	4	4	no	—
169	C₂×C₁₃	—	8	2	no	—
170	C₂×C₁₃	#532	1	2	yes	—
171	C₂×C₁₃	#538	1	1	yes	characteristic
172	C₂×C₁₃	D₄²	1	1	yes	characteristic
173	C₂×C₁₃	#546	1	2	yes	—
174	C₂×C₁₃	#555	1	1	yes	characteristic
175	C₂⁴	C₂³×C₁₃	1	2	yes	—
176	C₂⁴	C₂³×C₁₃	1	2	yes	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
194	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
197	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
198	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
199	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	2	4	no	—
208	C₂²⋊C₄	D₄×C₁₃	1	8	yes	complemented
209	C₄²	—	2	4	no	—
210	C₄²	—	4	1	no	—
211	C₁₃	#2555	1	1	yes	characteristic, direct product factor, Sylow
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	C₂×D₄×C₁₃	1	4	yes	—
226	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
227	Q₈	—	4	4	no	—
228	D₄	—	4	8	no	—
229	D₄	—	4	8	no	—
230	D₄	—	4	4	no	—
231	D₄	—	4	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
249	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
250	C₂×C₄	C₁₃×#41	1	2	yes	—
251	C₂×C₄	C₁₃×#41	1	4	yes	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₂²×D₄×C₁₃	1	2	yes	—
267	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
268	C₂²	C₂×C₁₃×#41	1	2	yes	—
269	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
270	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	2	4	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₂	—	4	4	no	—
279	C₂	—	8	2	no	—
280	C₂	C₁₃×#532	1	2	yes	—
281	C₂	C₁₃×#538	1	1	yes	characteristic
282	C₂	D₄²×C₁₃	1	1	yes	characteristic
283	C₂	C₁₃×#546	1	2	yes	—
284	C₂	C₁₃×#555	1	1	yes	characteristic

	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×#124	C₂	semidirect
2. C₂×C₁₃×#121	C₂	semidirect
3. C₁₃×#395	C₂	semidirect
4. C₁₃×#394	C₂	semidirect
5. C₁₃×#392	C₂	semidirect
6. C₁₃×#390	C₂	semidirect
7. C₁₃×#388	C₂	semidirect
8. C₁₃×#386	C₂	semidirect
9. C₄×C₁₃×C₂²⋊C₄	C₂	semidirect
66. C₁₃×C₂²⋊C₄	D₄	semidirect
69. #2555	C₁₃	direct (N × H)
110. C₂×#124	C₂×C₁₃	semidirect
111. C₂×#121	C₂×C₁₃	semidirect
112. #395	C₂×C₁₃	semidirect
113. #394	C₂×C₁₃	semidirect
114. #392	C₂×C₁₃	semidirect
115. #390	C₂×C₁₃	semidirect
116. #388	C₂×C₁₃	semidirect
117. #386	C₂×C₁₃	semidirect
118. C₄×C₂²⋊C₄	C₂×C₁₃	semidirect
208. C₂²⋊C₄	D₄×C₁₃	semidirect
211. C₁₃	#2555	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
C₂×C₁₃×#121	2
C₁₃×#395	2
C₁₃×#394	2
C₁₃×#392	2
C₁₃×#390	2
C₁₃×#388	2
C₁₃×#386	2
C₄×C₁₃×C₂²⋊C₄	2
#2555	13

Subgroups — C13×#2555 #13111613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2555 #13111613