Subgroups — C₁₃×#3094 #13112152

Order 1664.

462 conjugacy classes of subgroups.

The subgroup lattice has 252 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	C₂×C₁₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#482	C₂	1	4	yes	maximal, complemented
4	C₁₃×#481	C₂	1	2	yes	maximal, complemented
5	C₁₃×#480	C₂	1	2	yes	maximal, complemented
6	C₁₃×#428	C₂	1	2	yes	maximal, complemented
7	C₁₃×#418	C₂	1	2	yes	maximal, complemented
8	C₁₃×#407	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₁₃	—	2	1	no	—
10	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
13	C₁₃×#121	—	2	1	no	—
14	C₁₃×#121	—	2	1	no	—
15	C₁₃×#121	C₂²	1	2	yes	—
16	C₁₃×#121	C₂²	1	2	yes	—
17	C₁₃×#117	—	2	1	no	—
18	C₁₃×#117	—	2	1	no	—
19	C₁₃×#117	C₂²	1	2	yes	—
20	C₁₃×#117	C₂²	1	2	yes	complemented
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
25	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
26	C₂×C₄²×C₁₃	—	2	1	no	—
27	C₁₃×#107	C₂²	1	4	yes	complemented
28	C₁₃×#106	C₂²	1	4	yes	complemented
29	C₁₃×#102	C₂²	1	4	yes	complemented
30	C₁₃×#102	C₂²	1	4	yes	complemented
31	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
32	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
33	C₂⁴×C₁₃	—	2	1	no	—
34	C₂⁴×C₁₃	D₄	1	1	yes	characteristic
35	C₂×D₄×C₁₃	—	2	2	no	—
36	C₂×D₄×C₁₃	—	2	2	no	—
37	C₂×D₄×C₁₃	—	2	1	no	—
38	C₂×D₄×C₁₃	—	2	4	no	—
39	C₂×D₄×C₁₃	—	4	4	no	—
40	C₂×D₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	—	2	4	no	—
42	C₂²×C₄×C₁₃	—	2	1	no	—
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×M₁₆	—	2	4	no	—
52	C₂×C₈×C₁₃	C₂³	1	4	yes	—
53	C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₁₃×C₄⋊C₄	—	2	1	no	—
56	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
57	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
58	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
59	C₁₃×C₂²⋊C₄	—	2	2	no	—
60	C₁₃×C₂²⋊C₄	—	2	2	no	—
61	C₁₃×C₂²⋊C₄	—	2	2	no	—
62	C₁₃×C₂²⋊C₄	—	2	2	no	—
63	C₄²×C₁₃	—	2	2	no	—
64	C₄²×C₁₃	—	2	2	no	—
65	#3094	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₁₃	—	2	2	no	—
67	C₂³×C₁₃	—	2	2	no	—
68	C₂³×C₁₃	—	2	2	no	—
69	C₂³×C₁₃	—	4	1	no	—
70	C₂³×C₁₃	—	4	2	no	—
71	C₂³×C₁₃	—	4	2	no	—
72	C₂³×C₁₃	—	4	2	no	—
73	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
76	D₄×C₁₃	—	2	4	no	—
77	D₄×C₁₃	—	4	4	no	—
78	D₄×C₁₃	—	4	4	no	—
79	D₄×C₁₃	—	4	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	—	4	2	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	4	2	no	—
91	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#41	1	4	yes	—
95	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₈×C₁₃	—	2	4	no	—
97	C₂×#121	C₂×C₁₃	1	1	yes	characteristic
98	C₂×#117	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#482	C₂×C₁₃	1	4	yes	complemented
100	#481	C₂×C₁₃	1	2	yes	complemented
101	#480	C₂×C₁₃	1	2	yes	complemented
102	#428	C₂×C₁₃	1	2	yes	complemented
103	#418	C₂×C₁₃	1	2	yes	complemented
104	#407	C₂×C₁₃	1	1	yes	characteristic, complemented
105	C₂²×C₁₃	—	2	1	no	—
106	C₂²×C₁₃	—	2	1	no	—
107	C₂²×C₁₃	—	2	1	no	—
108	C₂²×C₁₃	—	2	1	no	—
109	C₂²×C₁₃	—	2	1	no	—
110	C₂²×C₁₃	—	2	1	no	—
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	—	4	4	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	—	4	2	no	—
116	C₂²×C₁₃	—	8	4	no	—
117	C₂²×C₁₃	#123	1	2	yes	—
118	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
119	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	4	2	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	#123	1	2	yes	—
126	C₄×C₁₃	C₂×#41	1	2	yes	—
127	C₂²×D₄	—	2	1	no	—
128	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
129	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
131	#121	—	2	1	no	—
132	#121	—	2	1	no	—
133	#121	C₂²×C₁₃	1	2	yes	—
134	#121	C₂²×C₁₃	1	2	yes	—
135	#117	—	2	1	no	—
136	#117	—	2	1	no	—
137	#117	C₂²×C₁₃	1	2	yes	—
138	#117	C₂²×C₁₃	1	2	yes	complemented
139	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic, complemented
141	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic, complemented
142	C₂×C₂²⋊C₄	—	2	1	no	—
143	C₂×C₂²⋊C₄	—	2	1	no	—
144	C₂×C₄²	—	2	1	no	—
145	#107	C₂²×C₁₃	1	4	yes	complemented
146	#106	C₂²×C₁₃	1	4	yes	complemented
147	#102	C₂²×C₁₃	1	4	yes	complemented
148	#102	C₂²×C₁₃	1	4	yes	complemented
149	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	complemented
150	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	complemented
151	C₂×C₁₃	—	2	2	no	—
152	C₂×C₁₃	—	2	2	no	—
153	C₂×C₁₃	—	4	1	no	—
154	C₂×C₁₃	—	8	2	no	—
155	C₂×C₁₃	C₂×#123	1	1	yes	characteristic
156	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₁₃	#575	1	1	yes	characteristic
158	C₂⁴	—	2	1	no	—
159	C₂⁴	D₄×C₁₃	1	1	yes	characteristic
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	1	no	—
163	C₂×D₄	—	2	4	no	—
164	C₂×D₄	—	4	4	no	—
165	C₂×D₄	C₂³×C₁₃	1	2	yes	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
171	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
172	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
176	M₁₆	—	2	4	no	—
177	C₂×C₈	C₂³×C₁₃	1	4	yes	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
182	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
183	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₁₃	#3094	1	1	yes	characteristic, direct product factor, Sylow
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	4	1	no	—
195	C₂³	—	4	2	no	—
196	C₂³	—	4	2	no	—
197	C₂³	—	4	2	no	—
198	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
199	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
200	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
201	D₄	—	2	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	4	4	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	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	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
217	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
218	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₄	C₁₃×#41	1	4	yes	—
220	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
221	C₈	—	2	4	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	4	4	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	8	4	no	—
234	C₂²	C₁₃×#123	1	2	yes	—
235	C₂²	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
236	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	2	no	—
242	C₄	C₁₃×#123	1	2	yes	—
243	C₄	C₂×C₁₃×#41	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	2	2	no	—
246	C₂	—	4	1	no	—
247	C₂	—	8	2	no	—
248	C₂	C₂×C₁₃×#123	1	1	yes	characteristic
249	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
250	C₂	C₁₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#117	C₂	semidirect
3. C₁₃×#482	C₂	semidirect
4. C₁₃×#481	C₂	semidirect
5. C₁₃×#480	C₂	semidirect
6. C₁₃×#428	C₂	semidirect
7. C₁₃×#418	C₂	semidirect
8. C₁₃×#407	C₂	semidirect
20. C₁₃×#117	C₂²	semidirect
22. C₂×C₁₃×C₄⋊C₄	C₂²	semidirect
23. C₂×C₁₃×C₄⋊C₄	C₂²	semidirect
27. C₁₃×#107	C₂²	semidirect
28. C₁₃×#106	C₂²	semidirect
29. C₁₃×#102	C₂²	semidirect
30. C₁₃×#102	C₂²	semidirect
31. C₁₃×C₂²⋊C₈	C₂²	semidirect
32. C₁₃×C₂²⋊C₈	C₂²	semidirect
65. #3094	C₁₃	direct (N × H)
98. C₂×#117	C₂×C₁₃	semidirect
99. #482	C₂×C₁₃	semidirect
100. #481	C₂×C₁₃	semidirect
101. #480	C₂×C₁₃	semidirect
102. #428	C₂×C₁₃	semidirect
103. #418	C₂×C₁₃	semidirect
104. #407	C₂×C₁₃	semidirect
138. #117	C₂²×C₁₃	semidirect
140. C₂×C₄⋊C₄	C₂²×C₁₃	semidirect
141. C₂×C₄⋊C₄	C₂²×C₁₃	semidirect
145. #107	C₂²×C₁₃	semidirect
146. #106	C₂²×C₁₃	semidirect
147. #102	C₂²×C₁₃	semidirect
148. #102	C₂²×C₁₃	semidirect
149. C₂²⋊C₈	C₂²×C₁₃	semidirect
150. C₂²⋊C₈	C₂²×C₁₃	semidirect
190. C₁₃	#3094	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×C₁₃×#117	2
C₁₃×#482	2
C₁₃×#481	2
C₁₃×#480	2
C₁₃×#428	2
C₁₃×#418	2
C₁₃×#407	2
#3094	13

Subgroups — C13×#3094 #13112152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3094 #13112152