Subgroups — C₁₃×#2377 #13111435

Order 1664.

576 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×#121	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#390	C₂	1	2	yes	maximal, complemented
3	C₁₃×#388	C₂	1	2	yes	maximal, complemented
4	C₁₃×#387	C₂	1	2	yes	maximal, complemented
5	C₁₃×#386	C₂	1	2	yes	maximal, complemented
6	C₁₃×#386	C₂	1	2	yes	maximal, complemented
7	C₁₃×#385	C₂	1	2	yes	maximal, complemented
8	C₁₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
12	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#121	C₄	1	8	yes	complemented
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₂²⋊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×C₄²×C₁₃	C₂²	1	2	yes	—
26	C₁₃×#95	C₂²	1	2	yes	—
27	C₁₃×#95	C₂²	1	1	yes	characteristic
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	2	yes	—
32	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
33	C₂⁴×C₁₃	C₂³	1	2	yes	—
34	C₂×D₄×C₁₃	—	4	2	no	—
35	C₂×D₄×C₁₃	—	4	2	no	—
36	C₂×D₄×C₁₃	—	4	2	no	—
37	C₂×D₄×C₁₃	C₂×C₄	1	4	yes	—
38	C₂×D₄×C₁₃	C₂×C₄	1	8	yes	complemented
39	C₂²×C₄×C₁₃	—	4	2	no	—
40	C₂²×C₄×C₁₃	—	4	2	no	—
41	C₂²×C₄×C₁₃	—	4	2	no	—
42	C₂²×C₄×C₁₃	—	4	2	no	—
43	C₂²×C₄×C₁₃	C₂×C₄	1	4	yes	complemented
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×C₄⋊C₄	—	4	2	no	—
52	C₁₃×C₄⋊C₄	—	4	2	no	—
53	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
54	C₁₃×C₂²⋊C₄	—	4	2	no	—
55	C₁₃×C₂²⋊C₄	—	4	1	no	—
56	C₁₃×C₂²⋊C₄	—	4	2	no	—
57	C₁₃×C₂²⋊C₄	—	4	2	no	—
58	C₁₃×C₂²⋊C₄	—	4	2	no	—
59	C₁₃×C₂²⋊C₄	—	4	1	no	—
60	C₁₃×C₂²⋊C₄	C₂×C₄	1	8	yes	complemented
61	C₄²×C₁₃	—	4	2	no	—
62	#2377	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₁₃	—	2	1	no	—
64	C₂³×C₁₃	—	2	2	no	—
65	C₂³×C₁₃	—	2	2	no	—
66	C₂³×C₁₃	—	2	2	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	1	no	—
72	C₂³×C₁₃	—	2	2	no	—
73	C₂³×C₁₃	C₂²×C₄	1	4	yes	—
74	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
76	D₄×C₁₃	—	4	4	no	—
77	D₄×C₁₃	—	4	8	no	—
78	C₂×C₄×C₁₃	—	2	2	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	2	no	—
89	C₂×C₄×C₁₃	—	2	2	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	—	4	4	no	—
95	C₂×C₄×C₁₃	—	4	8	no	—
96	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
98	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
99	C₂×#121	C₂×C₁₃	1	1	yes	characteristic
100	#390	C₂×C₁₃	1	2	yes	complemented
101	#388	C₂×C₁₃	1	2	yes	complemented
102	#387	C₂×C₁₃	1	2	yes	complemented
103	#386	C₂×C₁₃	1	2	yes	complemented
104	#386	C₂×C₁₃	1	2	yes	complemented
105	#385	C₂×C₁₃	1	2	yes	complemented
106	#381	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#380	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂²×C₁₃	—	2	2	no	—
109	C₂²×C₁₃	—	2	2	no	—
110	C₂²×C₁₃	—	2	4	no	—
111	C₂²×C₁₃	—	2	4	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	2	4	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	C₂³×C₄	1	1	yes	characteristic
121	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
122	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
123	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	—	2	4	no	—
128	C₄×C₁₃	—	4	2	no	—
129	C₄×C₁₃	—	4	2	no	—
130	C₄×C₁₃	—	4	2	no	—
131	C₄×C₁₃	—	4	2	no	—
132	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
133	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
134	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
135	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
136	#121	C₄×C₁₃	1	8	yes	complemented
137	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
138	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
142	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
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	2	yes	—
147	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
148	#95	C₂²×C₁₃	1	2	yes	—
149	#95	C₂²×C₁₃	1	1	yes	characteristic
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	2	yes	—
154	C₂×C₁₃	—	4	2	no	—
155	C₂×C₁₃	—	4	4	no	—
156	C₂×C₁₃	#518	1	1	yes	characteristic
157	C₂×C₁₃	#518	1	1	yes	characteristic
158	C₂×C₁₃	#520	1	1	yes	characteristic
159	C₂×C₁₃	#560	1	2	yes	—
160	C₂×C₁₃	#562	1	2	yes	—
161	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
162	C₂⁴	C₂³×C₁₃	1	2	yes	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂×D₄	C₂×C₄×C₁₃	1	4	yes	—
167	C₂×D₄	C₂×C₄×C₁₃	1	8	yes	complemented
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	C₂×C₄×C₁₃	1	4	yes	complemented
173	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
174	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
175	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
176	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
177	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
179	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	complemented
183	C₂²⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	4	1	no	—
185	C₂²⋊C₄	—	4	2	no	—
186	C₂²⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	4	1	no	—
189	C₂²⋊C₄	C₂×C₄×C₁₃	1	8	yes	complemented
190	C₄²	—	4	2	no	—
191	C₁₃	#2377	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³	—	2	1	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	2	no	—
202	C₂³	C₂²×C₄×C₁₃	1	4	yes	—
203	C₂³	C₂²×C₄×C₁₃	1	2	yes	—
204	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
205	D₄	—	4	4	no	—
206	D₄	—	4	8	no	—
207	C₂×C₄	—	2	2	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	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	C₂²×C₄×C₁₃	1	2	yes	—
226	C₂×C₄	C₂²×C₄×C₁₃	1	4	yes	—
227	C₂×C₄	C₂²×C₄×C₁₃	1	2	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	4	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	C₂³×C₄×C₁₃	1	1	yes	characteristic
241	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
242	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
243	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
244	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	2	no	—
252	C₂	—	4	2	no	—
253	C₂	—	4	4	no	—
254	C₂	C₁₃×#518	1	1	yes	characteristic
255	C₂	C₁₃×#518	1	1	yes	characteristic
256	C₂	C₁₃×#520	1	1	yes	characteristic
257	C₂	C₁₃×#560	1	2	yes	—
258	C₂	C₁₃×#562	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₃
Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#390	C₂	semidirect
3. C₁₃×#388	C₂	semidirect
4. C₁₃×#387	C₂	semidirect
5. C₁₃×#386	C₂	semidirect
6. C₁₃×#386	C₂	semidirect
7. C₁₃×#385	C₂	semidirect
8. C₁₃×#381	C₂	semidirect
9. C₁₃×#380	C₂	semidirect
14. C₁₃×#121	C₄	semidirect
38. C₂×D₄×C₁₃	C₂×C₄	semidirect
43. C₂²×C₄×C₁₃	C₂×C₄	semidirect
53. C₁₃×C₄⋊C₄	C₂×C₄	semidirect
60. C₁₃×C₂²⋊C₄	C₂×C₄	semidirect
62. #2377	C₁₃	direct (N × H)
100. #390	C₂×C₁₃	semidirect
101. #388	C₂×C₁₃	semidirect
102. #387	C₂×C₁₃	semidirect
103. #386	C₂×C₁₃	semidirect
104. #386	C₂×C₁₃	semidirect
105. #385	C₂×C₁₃	semidirect
106. #381	C₂×C₁₃	semidirect
107. #380	C₂×C₁₃	semidirect
136. #121	C₄×C₁₃	semidirect
167. C₂×D₄	C₂×C₄×C₁₃	semidirect
172. C₂²×C₄	C₂×C₄×C₁₃	semidirect
182. C₄⋊C₄	C₂×C₄×C₁₃	semidirect
189. C₂²⋊C₄	C₂×C₄×C₁₃	semidirect
191. C₁₃	#2377	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
C₁₃×#390	2
C₁₃×#388	2
C₁₃×#387	2
C₁₃×#386	2
C₁₃×#385	2
C₁₃×#381	2
C₁₃×#380	2
#2377	13

Subgroups — C13×#2377 #13111435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2377 #13111435