Subgroups — C₁₃×#3357 #13112415

Order 1664.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#555	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#485	C₂	1	2	yes	maximal, complemented
5	C₁₃×#480	C₂	1	2	yes	maximal, complemented
6	C₁₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#468	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#428	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×#125	—	2	1	no	—
17	C₁₃×#124	—	2	2	no	—
18	C₁₃×C₂²⋊Q₈	—	2	2	no	—
19	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₁₃×#121	—	2	1	no	—
21	C₁₃×#121	C₂²	1	2	yes	—
22	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₃	—	2	1	no	—
24	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#117	—	2	1	no	—
27	C₁₃×#117	C₂²	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₁₃×#107	C₂²	1	2	yes	—
30	C₁₃×#107	C₂²	1	2	yes	—
31	C₁₃×#107	C₂²	1	2	yes	—
32	C₁₃×#106	C₂²	1	1	yes	characteristic
33	C₁₃×#106	C₂²	1	2	yes	—
34	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₁₃×#103	C₂²	1	1	yes	characteristic
36	C₁₃×#103	C₂²	1	2	yes	—
37	C₁₃×#102	C₂²	1	1	yes	characteristic
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	—	2	1	no	—
42	C₂×Q₈×C₁₃	—	2	1	no	—
43	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₃	—	2	2	no	—
45	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
46	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	2	no	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₁₃×SD₁₆	—	2	2	no	—
51	C₁₃×M₁₆	—	2	4	no	—
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂³	1	2	yes	—
54	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₁₃×C₄⋊C₄	—	2	1	no	—
57	C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₁₃×C₄⋊C₄	—	2	1	no	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
63	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
65	C₁₃×C₂²⋊C₄	—	2	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	2	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	—	2	1	no	—
69	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
70	C₄²×C₁₃	—	2	2	no	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	C₂³	1	1	yes	characteristic
73	#3357	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₁₃	—	2	1	no	—
75	C₂³×C₁₃	C₂×D₄	1	2	yes	—
76	Q₈×C₁₃	—	2	1	no	—
77	Q₈×C₁₃	—	2	1	no	—
78	Q₈×C₁₃	—	8	1	no	—
79	Q₈×C₁₃	—	8	1	no	—
80	D₄×C₁₃	—	2	2	no	—
81	D₄×C₁₃	—	2	1	no	—
82	D₄×C₁₃	—	2	1	no	—
83	D₄×C₁₃	—	8	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	—	8	2	no	—
93	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₈×C₁₃	—	2	2	no	—
98	C₈×C₁₃	—	2	1	no	—
99	C₈×C₁₃	#41	1	2	yes	—
100	#555	C₂×C₁₃	1	1	yes	characteristic
101	#554	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#500	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#485	C₂×C₁₃	1	2	yes	complemented
104	#480	C₂×C₁₃	1	2	yes	complemented
105	#477	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#476	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#468	C₂×C₁₃	1	2	yes	complemented
108	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#436	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#428	C₂×C₁₃	1	2	yes	complemented
111	C₂²×C₁₃	—	2	2	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	—	4	1	no	—
115	C₂²×C₁₃	—	4	1	no	—
116	C₂²×C₁₃	—	4	1	no	—
117	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₄×C₁₃	—	2	2	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	4	1	no	—
121	C₄×C₁₃	—	4	1	no	—
122	C₄×C₁₃	—	4	1	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	#137	1	2	yes	—
126	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
127	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
129	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
130	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
131	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
132	#125	—	2	1	no	—
133	#124	—	2	2	no	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
136	#121	—	2	1	no	—
137	#121	C₂²×C₁₃	1	2	yes	—
138	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
139	C₄×D₄	—	2	1	no	—
140	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
141	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
142	#117	—	2	1	no	—
143	#117	C₂²×C₁₃	1	2	yes	—
144	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
145	#107	C₂²×C₁₃	1	2	yes	—
146	#107	C₂²×C₁₃	1	2	yes	—
147	#107	C₂²×C₁₃	1	2	yes	—
148	#106	C₂²×C₁₃	1	1	yes	characteristic
149	#106	C₂²×C₁₃	1	2	yes	—
150	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
151	#103	C₂²×C₁₃	1	1	yes	characteristic
152	#103	C₂²×C₁₃	1	2	yes	—
153	#102	C₂²×C₁₃	1	1	yes	characteristic
154	#102	C₂²×C₁₃	1	2	yes	—
155	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
156	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₁₃	—	4	2	no	—
158	C₂×C₁₃	—	8	1	no	—
159	C₂×C₁₃	#547	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
161	C₂×C₁₃	#576	1	1	yes	characteristic
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
167	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
171	SD₁₆	—	2	2	no	—
172	M₁₆	—	2	4	no	—
173	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
174	C₂×C₈	C₂³×C₁₃	1	2	yes	—
175	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
182	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
184	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
185	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	1	no	—
193	C₄²	C₂³×C₁₃	1	1	yes	characteristic
194	C₁₃	#3357	1	1	yes	characteristic, direct product factor, Sylow
195	C₂³	—	2	1	no	—
196	C₂³	C₂×D₄×C₁₃	1	2	yes	—
197	Q₈	—	2	1	no	—
198	Q₈	—	2	1	no	—
199	Q₈	—	8	1	no	—
200	Q₈	—	8	1	no	—
201	D₄	—	2	2	no	—
202	D₄	—	2	1	no	—
203	D₄	—	2	1	no	—
204	D₄	—	8	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
215	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
216	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
217	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	1	no	—
220	C₈	C₁₃×#41	1	2	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	C₁₃×#137	1	2	yes	—
236	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
237	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
238	C₂	—	4	2	no	—
239	C₂	—	8	1	no	—
240	C₂	C₁₃×#547	1	1	yes	characteristic
241	C₂	C₂×C₁₃×#137	1	1	yes	characteristic
242	C₂	C₁₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#555	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#554	C₂	semidirect
3. C₁₃×#500	C₂	semidirect
4. C₁₃×#485	C₂	semidirect
5. C₁₃×#480	C₂	semidirect
6. C₁₃×#477	C₂	semidirect
7. C₁₃×#476	C₂	semidirect
8. C₁₃×#468	C₂	semidirect
9. C₄×C₁₃×SD₁₆	C₂	semidirect
10. C₁₃×#436	C₂	semidirect
11. C₁₃×#428	C₂	semidirect
73. #3357	C₁₃	direct (N × H)
101. #554	C₂×C₁₃	semidirect
102. #500	C₂×C₁₃	semidirect
103. #485	C₂×C₁₃	semidirect
104. #480	C₂×C₁₃	semidirect
105. #477	C₂×C₁₃	semidirect
106. #476	C₂×C₁₃	semidirect
107. #468	C₂×C₁₃	semidirect
108. C₄×SD₁₆	C₂×C₁₃	semidirect
109. #436	C₂×C₁₃	semidirect
110. #428	C₂×C₁₃	semidirect
194. C₁₃	#3357	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#555	2
C₁₃×#554	2
C₁₃×#500	2
C₁₃×#485	2
C₁₃×#480	2
C₁₃×#477	2
C₁₃×#476	2
C₁₃×#468	2
C₄×C₁₃×SD₁₆	2
C₁₃×#436	2
C₁₃×#428	2
#3357	13

Subgroups — C13×#3357 #13112415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3357 #13112415