Subgroups — C₁₃×#3381 #13112439

Order 1664.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#554	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#499	C₂	1	2	yes	maximal, complemented
4	C₁₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#479	C₂	1	2	yes	maximal, complemented
6	C₁₃×#478	C₂	1	2	yes	maximal
7	C₁₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#474	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic
11	C₄×C₁₃×SD₁₆	C₂	1	2	yes	maximal
12	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
13	C₁₃×C₄⋊Q₈	—	2	1	no	—
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×#125	—	2	2	no	—
17	C₁₃×#125	C₂²	1	2	yes	—
18	C₁₃×#125	C₂²	1	2	yes	—
19	C₁₃×C₂²⋊Q₈	—	2	1	no	—
20	C₁₃×C₂²⋊Q₈	—	2	2	no	—
21	C₄×Q₈×C₁₃	—	2	1	no	—
22	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₃	C₂²	1	2	yes	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₁₃×#107	C₂²	1	2	yes	—
29	C₁₃×#107	C₂²	1	2	yes	—
30	C₁₃×#106	C₂²	1	2	yes	—
31	C₁₃×#106	C₂²	1	1	yes	characteristic
32	C₁₃×#106	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
35	C₁₃×#103	C₂²	1	2	yes	—
36	C₁₃×#103	C₂²	1	1	yes	characteristic
37	C₁₃×#102	C₂²	1	2	yes	—
38	C₁₃×#102	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₃	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₃	—	2	1	no	—
42	C₂×Q₈×C₁₃	—	2	1	no	—
43	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	2	no	—
48	C₁₃×SD₁₆	Q₈	1	4	yes	complemented
49	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂³	1	2	yes	—
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	—	2	1	no	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	2	2	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	D₄	1	2	yes	—
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×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₁₃×C₂²⋊C₄	—	2	1	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₄²×C₁₃	—	2	2	no	—
69	C₄²×C₁₃	—	2	1	no	—
70	C₄²×C₁₃	C₂³	1	2	yes	—
71	C₄²×C₁₃	C₂³	1	1	yes	characteristic
72	#3381	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₁₃	—	2	1	no	—
74	Q₈×C₁₃	—	2	2	no	—
75	Q₈×C₁₃	—	2	1	no	—
76	Q₈×C₁₃	—	4	2	no	—
77	Q₈×C₁₃	—	8	1	no	—
78	Q₈×C₁₃	C₂×Q₈	1	2	yes	—
79	D₄×C₁₃	—	2	1	no	—
80	D₄×C₁₃	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	4	2	no	—
87	C₂×C₄×C₁₃	—	8	2	no	—
88	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₈×C₁₃	—	2	1	no	—
94	C₈×C₁₃	—	2	2	no	—
95	C₈×C₁₃	C₂×Q₈	1	2	yes	—
96	#557	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#554	C₂×C₁₃	1	1	yes	characteristic
98	#499	C₂×C₁₃	1	2	yes	complemented
99	#498	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#479	C₂×C₁₃	1	2	yes	complemented
101	#478	C₂×C₁₃	1	2	yes	—
102	#477	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#474	C₂×C₁₃	1	1	yes	characteristic
104	C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
105	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic
106	C₄×SD₁₆	C₂×C₁₃	1	2	yes	—
107	C₂²×C₁₃	—	2	2	no	—
108	C₂²×C₁₃	—	4	1	no	—
109	C₂²×C₁₃	—	4	1	no	—
110	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₁₃	—	2	2	no	—
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	—	4	1	no	—
116	C₄×C₁₃	—	4	2	no	—
117	C₄×C₁₃	—	4	1	no	—
118	C₄×C₁₃	—	4	2	no	—
119	C₄×C₁₃	#135	1	2	yes	—
120	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
121	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
123	C₄⋊Q₈	—	2	1	no	—
124	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
125	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
126	#125	—	2	2	no	—
127	#125	C₂²×C₁₃	1	2	yes	—
128	#125	C₂²×C₁₃	1	2	yes	—
129	C₂²⋊Q₈	—	2	1	no	—
130	C₂²⋊Q₈	—	2	2	no	—
131	C₄×Q₈	—	2	1	no	—
132	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
133	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
134	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
135	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
136	C₄×D₄	C₂²×C₁₃	1	2	yes	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	#107	C₂²×C₁₃	1	2	yes	—
139	#107	C₂²×C₁₃	1	2	yes	—
140	#106	C₂²×C₁₃	1	2	yes	—
141	#106	C₂²×C₁₃	1	1	yes	characteristic
142	#106	C₂²×C₁₃	1	2	yes	—
143	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
144	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
145	#103	C₂²×C₁₃	1	2	yes	—
146	#103	C₂²×C₁₃	1	1	yes	characteristic
147	#102	C₂²×C₁₃	1	2	yes	—
148	#102	C₂²×C₁₃	1	1	yes	characteristic
149	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₄×C₈	C₂²×C₁₃	1	2	yes	—
151	C₂×C₁₃	—	4	2	no	—
152	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
153	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
154	C₂×C₁₃	#577	1	1	yes	characteristic
155	C₂×Q₈	—	2	1	no	—
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
158	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
159	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	2	no	—
162	SD₁₆	Q₈×C₁₃	1	4	yes	complemented
163	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
164	C₂×C₈	C₂³×C₁₃	1	2	yes	—
165	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
175	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
176	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
177	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
178	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
179	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
180	C₂²⋊C₄	—	2	1	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	1	no	—
184	C₄²	C₂³×C₁₃	1	2	yes	—
185	C₄²	C₂³×C₁₃	1	1	yes	characteristic
186	C₁₃	#3381	1	1	yes	characteristic, direct product factor, Sylow
187	C₂³	—	2	1	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	2	1	no	—
190	Q₈	—	4	2	no	—
191	Q₈	—	8	1	no	—
192	Q₈	C₂×Q₈×C₁₃	1	2	yes	—
193	D₄	—	2	1	no	—
194	D₄	C₂×Q₈×C₁₃	1	2	yes	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
203	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
204	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
206	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	2	no	—
209	C₈	C₂×Q₈×C₁₃	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	1	no	—
218	C₄	—	4	1	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	2	no	—
222	C₄	C₁₃×#135	1	2	yes	—
223	C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
224	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
225	C₂	—	4	2	no	—
226	C₂	D₄×Q₈×C₁₃	1	1	yes	characteristic
227	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
228	C₂	C₁₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₂×D₄×C₁₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#557	C₂	semidirect
3. C₁₃×#499	C₂	semidirect
4. C₁₃×#498	C₂	semidirect
5. C₁₃×#479	C₂	semidirect
7. C₁₃×#477	C₂	semidirect
9. C₈×Q₈×C₁₃	C₂	semidirect
48. C₁₃×SD₁₆	Q₈	semidirect
72. #3381	C₁₃	direct (N × H)
96. #557	C₂×C₁₃	semidirect
98. #499	C₂×C₁₃	semidirect
99. #498	C₂×C₁₃	semidirect
100. #479	C₂×C₁₃	semidirect
102. #477	C₂×C₁₃	semidirect
104. C₈×Q₈	C₂×C₁₃	semidirect
162. SD₁₆	Q₈×C₁₃	semidirect
186. C₁₃	#3381	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×#557	2
C₁₃×#554	2
C₁₃×#499	2
C₁₃×#498	2
C₁₃×#479	2
C₁₃×#478	2
C₁₃×#477	2
C₁₃×#474	2
C₈×Q₈×C₁₃	2
C₄×C₁₃×SD₁₆	2
#3381	13

Subgroups — C13×#3381 #13112439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3381 #13112439