Subgroups — C₁₃×#3360 #13112418

Order 1664.

350 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#488	C₂	1	2	yes	maximal, complemented
4	C₁₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#479	C₂	1	2	yes	maximal, complemented
6	C₁₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#463	C₂	1	2	yes	maximal
8	C₁₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₃×SD₁₆	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×#126	—	2	2	no	—
17	C₁₃×#125	C₂²	1	2	yes	—
18	C₁₃×#124	C₂²	1	2	yes	—
19	C₁₃×C₂²⋊Q₈	—	2	1	no	—
20	C₁₃×C₂²⋊Q₈	—	2	1	no	—
21	C₁₃×C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₃	C₂²	1	2	yes	—
27	C₁₃×#117	—	2	1	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₁₃×#107	C₂²	1	1	yes	characteristic
30	C₁₃×#106	C₂²	1	2	yes	—
31	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
33	C₁₃×#103	C₂²	1	2	yes	—
34	C₁₃×#103	C₂²	1	2	yes	—
35	C₁₃×#103	C₂²	1	2	yes	—
36	C₁₃×#102	C₂²	1	1	yes	characteristic
37	C₁₃×#102	C₂²	1	1	yes	characteristic
38	C₁₃×#102	C₂²	1	2	yes	—
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	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
44	C₂×D₄×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	1	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₁₃×SD₁₆	—	2	4	no	—
50	C₁₃×D₈	—	2	2	no	—
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂³	1	2	yes	—
53	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	—	2	1	no	—
56	C₁₃×C₄⋊C₄	—	2	1	no	—
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	1	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	1	yes	characteristic
65	C₁₃×C₂²⋊C₄	—	2	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	1	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	—	2	1	no	—
69	C₁₃×C₂²⋊C₄	—	2	2	no	—
70	C₄²×C₁₃	—	2	2	no	—
71	C₄²×C₁₃	C₂³	1	1	yes	characteristic
72	C₄²×C₁₃	C₂³	1	2	yes	—
73	#3360	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₁₃	—	2	1	no	—
75	C₂³×C₁₃	—	2	1	no	—
76	Q₈×C₁₃	—	2	2	no	—
77	Q₈×C₁₃	—	2	2	no	—
78	Q₈×C₁₃	—	2	2	no	—
79	Q₈×C₁₃	—	8	1	no	—
80	D₄×C₁₃	—	2	1	no	—
81	D₄×C₁₃	—	2	1	no	—
82	D₄×C₁₃	—	2	1	no	—
83	D₄×C₁₃	#41	1	2	yes	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	1	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	—	8	2	no	—
91	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₈×C₁₃	—	2	1	no	—
97	C₈×C₁₃	—	2	2	no	—
98	C₈×C₁₃	—	2	1	no	—
99	#556	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#554	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#488	C₂×C₁₃	1	2	yes	complemented
102	#487	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#479	C₂×C₁₃	1	2	yes	complemented
104	#476	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#463	C₂×C₁₃	1	2	yes	—
106	#461	C₂×C₁₃	1	1	yes	characteristic, complemented
107	C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₄×SD₁₆	C₂×C₁₃	1	2	yes	complemented
109	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	—	4	1	no	—
112	C₂²×C₁₃	—	4	1	no	—
113	C₂²×C₁₃	—	4	1	no	—
114	C₂²×C₁₃	—	4	1	no	—
115	C₂²×C₁₃	—	4	1	no	—
116	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₁₃	—	2	2	no	—
118	C₄×C₁₃	—	2	2	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	4	2	no	—
121	C₄×C₁₃	—	4	1	no	—
122	C₄×C₁₃	—	4	1	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	#135	1	2	yes	—
125	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
127	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
128	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
129	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
130	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
131	#126	—	2	2	no	—
132	#125	C₂²×C₁₃	1	2	yes	—
133	#124	C₂²×C₁₃	1	2	yes	—
134	C₂²⋊Q₈	—	2	1	no	—
135	C₂²⋊Q₈	—	2	1	no	—
136	C₂²⋊Q₈	—	2	2	no	—
137	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
138	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
139	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
140	C₄×D₄	C₂²×C₁₃	1	2	yes	—
141	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
142	#117	—	2	1	no	—
143	C₂×C₄⋊C₄	—	2	1	no	—
144	#107	C₂²×C₁₃	1	1	yes	characteristic
145	#106	C₂²×C₁₃	1	2	yes	—
146	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
148	#103	C₂²×C₁₃	1	2	yes	—
149	#103	C₂²×C₁₃	1	2	yes	—
150	#103	C₂²×C₁₃	1	2	yes	—
151	#102	C₂²×C₁₃	1	1	yes	characteristic
152	#102	C₂²×C₁₃	1	1	yes	characteristic
153	#102	C₂²×C₁₃	1	2	yes	—
154	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₁₃	1	2	yes	—
156	C₂×C₁₃	—	4	2	no	—
157	C₂×C₁₃	—	8	1	no	—
158	C₂×C₁₃	#548	1	1	yes	characteristic
159	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
160	C₂×C₁₃	#577	1	1	yes	characteristic
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
163	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
164	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
165	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	SD₁₆	—	2	4	no	—
170	D₈	—	2	2	no	—
171	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
172	C₂×C₈	C₂³×C₁₃	1	2	yes	—
173	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
181	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
182	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
183	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
184	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	1	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	1	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	C₂³×C₁₃	1	1	yes	characteristic
192	C₄²	C₂³×C₁₃	1	2	yes	—
193	C₁₃	#3360	1	1	yes	characteristic, direct product factor, Sylow
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	Q₈	—	2	2	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	—	8	1	no	—
200	D₄	—	2	1	no	—
201	D₄	—	2	1	no	—
202	D₄	—	2	1	no	—
203	D₄	C₁₃×#41	1	2	yes	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
212	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
213	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
215	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	2	no	—
233	C₄	C₁₃×#135	1	2	yes	—
234	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
235	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
236	C₂	—	4	2	no	—
237	C₂	—	8	1	no	—
238	C₂	C₁₃×#548	1	1	yes	characteristic
239	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
240	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×C₁₃×D₈	C₂²
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×#556	C₂	semidirect
2. C₁₃×#554	C₂	semidirect
3. C₁₃×#488	C₂	semidirect
4. C₁₃×#487	C₂	semidirect
5. C₁₃×#479	C₂	semidirect
6. C₁₃×#476	C₂	semidirect
8. C₁₃×#461	C₂	semidirect
9. C₈×Q₈×C₁₃	C₂	semidirect
10. C₄×C₁₃×SD₁₆	C₂	semidirect
73. #3360	C₁₃	direct (N × H)
99. #556	C₂×C₁₃	semidirect
100. #554	C₂×C₁₃	semidirect
101. #488	C₂×C₁₃	semidirect
102. #487	C₂×C₁₃	semidirect
103. #479	C₂×C₁₃	semidirect
104. #476	C₂×C₁₃	semidirect
106. #461	C₂×C₁₃	semidirect
107. C₈×Q₈	C₂×C₁₃	semidirect
108. C₄×SD₁₆	C₂×C₁₃	semidirect
193. C₁₃	#3360	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×#556	2
C₁₃×#554	2
C₁₃×#488	2
C₁₃×#487	2
C₁₃×#479	2
C₁₃×#476	2
C₁₃×#463	2
C₁₃×#461	2
C₈×Q₈×C₁₃	2
C₄×C₁₃×SD₁₆	2
C₄×C₁₃×D₈	2
#3360	13

Subgroups — C13×#3360 #13112418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3360 #13112418