Subgroups — C₁₃×#3108 #13112166

Order 1664.

390 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#483	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#480	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#479	C₂	1	2	yes	maximal, complemented
6	C₁₃×#464	C₂	1	2	yes	maximal, complemented
7	C₁₃×#451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#448	C₂	1	1	yes	maximal, characteristic
9	C₄×C₁₃×Q₁₆	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₃×SD₁₆	C₂	1	2	yes	maximal, complemented
11	C₁₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
15	C₁₃×#126	—	2	1	no	—
16	C₁₃×#125	C₂²	1	1	yes	characteristic
17	C₁₃×#124	C₂²	1	1	yes	characteristic
18	C₁₃×#123	—	2	1	no	—
19	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₁₃×#121	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₁₃	—	2	1	no	—
22	C₄×Q₈×C₁₃	—	2	1	no	—
23	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
26	C₄×D₄×C₁₃	—	2	1	no	—
27	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#117	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	—	2	1	no	—
32	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
33	C₁₃×#107	C₂²	1	2	yes	—
34	C₁₃×#106	C₂²	1	2	yes	—
35	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
36	C₁₃×#103	C₂²	1	2	yes	—
37	C₁₃×#103	C₂²	1	2	yes	—
38	C₁₃×#103	C₂²	1	2	yes	—
39	C₁₃×#102	C₂²	1	2	yes	—
40	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂²	1	2	yes	—
43	C₂×Q₈×C₁₃	—	2	4	no	—
44	C₂×Q₈×C₁₃	—	2	1	no	—
45	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
47	C₂×D₄×C₁₃	—	2	1	no	—
48	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
55	C₁₃×Q₁₆	—	2	4	no	—
56	C₁₃×SD₁₆	—	2	4	no	—
57	C₂×C₈×C₁₃	C₂³	1	2	yes	—
58	C₂×C₈×C₁₃	C₂³	1	2	yes	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	—	2	1	no	—
62	C₁₃×C₄⋊C₄	—	2	1	no	—
63	C₁₃×C₄⋊C₄	—	2	1	no	—
64	C₁₃×C₄⋊C₄	—	2	1	no	—
65	C₁₃×C₄⋊C₄	—	2	1	no	—
66	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₁₃×C₂²⋊C₄	—	2	1	no	—
70	C₁₃×C₂²⋊C₄	—	2	1	no	—
71	C₁₃×C₂²⋊C₄	—	2	1	no	—
72	C₁₃×C₂²⋊C₄	—	2	1	no	—
73	C₁₃×C₂²⋊C₄	—	2	1	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	—	2	2	no	—
76	C₄²×C₁₃	—	2	1	no	—
77	C₄²×C₁₃	D₄	1	1	yes	characteristic
78	C₄²×C₁₃	D₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂³	1	1	yes	characteristic
80	C₄²×C₁₃	C₂³	1	1	yes	characteristic
81	#3108	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₁₃	—	2	1	no	—
83	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₁₃	—	2	4	no	—
85	Q₈×C₁₃	—	2	2	no	—
86	Q₈×C₁₃	—	2	2	no	—
87	Q₈×C₁₃	#41	1	4	yes	—
88	D₄×C₁₃	—	2	2	no	—
89	D₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	2	4	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	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	—	2	2	no	—
109	#529	C₂×C₁₃	1	1	yes	characteristic
110	C₂×C₄×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#483	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#480	C₂×C₁₃	1	1	yes	characteristic
113	#479	C₂×C₁₃	1	2	yes	complemented
114	#464	C₂×C₁₃	1	2	yes	complemented
115	#451	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#448	C₂×C₁₃	1	1	yes	characteristic
117	C₄×Q₁₆	C₂×C₁₃	1	2	yes	complemented
118	C₄×SD₁₆	C₂×C₁₃	1	2	yes	complemented
119	#431	C₂×C₁₃	1	1	yes	characteristic, complemented
120	C₂²×C₁₃	—	2	1	no	—
121	C₂²×C₁₃	—	2	1	no	—
122	C₂²×C₁₃	—	4	1	no	—
123	C₂²×C₁₃	—	4	1	no	—
124	C₂²×C₁₃	—	4	1	no	—
125	C₂²×C₁₃	#135	1	2	yes	—
126	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₁₃	—	2	4	no	—
128	C₄×C₁₃	—	2	1	no	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	—	4	1	no	—
133	C₄×C₁₃	—	4	2	no	—
134	C₄×C₁₃	—	4	1	no	—
135	C₄×C₁₃	—	4	1	no	—
136	C₄×C₁₃	#137	1	2	yes	—
137	C₄×C₁₃	C₂×#41	1	2	yes	—
138	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
139	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
140	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
141	#126	—	2	1	no	—
142	#125	C₂²×C₁₃	1	1	yes	characteristic
143	#124	C₂²×C₁₃	1	1	yes	characteristic
144	#123	—	2	1	no	—
145	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
146	#121	C₂²×C₁₃	1	1	yes	characteristic
147	C₄×Q₈	—	2	1	no	—
148	C₄×Q₈	—	2	1	no	—
149	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
150	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
151	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
152	C₄×D₄	—	2	1	no	—
153	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
154	#117	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
159	#107	C₂²×C₁₃	1	2	yes	—
160	#106	C₂²×C₁₃	1	2	yes	—
161	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
162	#103	C₂²×C₁₃	1	2	yes	—
163	#103	C₂²×C₁₃	1	2	yes	—
164	#103	C₂²×C₁₃	1	2	yes	—
165	#102	C₂²×C₁₃	1	2	yes	—
166	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
167	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
168	C₄×C₈	C₂²×C₁₃	1	2	yes	—
169	C₂×C₁₃	—	2	2	no	—
170	C₂×C₁₃	—	8	1	no	—
171	C₂×C₁₃	#525	1	1	yes	characteristic
172	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
173	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
174	C₂×Q₈	—	2	4	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
177	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
184	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
186	Q₁₆	—	2	4	no	—
187	SD₁₆	—	2	4	no	—
188	C₂×C₈	C₂³×C₁₃	1	2	yes	—
189	C₂×C₈	C₂³×C₁₃	1	2	yes	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
198	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
199	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	1	no	—
208	C₄²	D₄×C₁₃	1	1	yes	characteristic
209	C₄²	D₄×C₁₃	1	1	yes	characteristic
210	C₄²	C₂³×C₁₃	1	1	yes	characteristic
211	C₄²	C₂³×C₁₃	1	1	yes	characteristic
212	C₁₃	#3108	1	1	yes	characteristic, direct product factor, Sylow
213	C₂³	—	2	1	no	—
214	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	2	no	—
217	Q₈	—	2	2	no	—
218	Q₈	C₁₃×#41	1	4	yes	—
219	D₄	—	2	2	no	—
220	D₄	—	4	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
233	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
235	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	C₁₃×#135	1	2	yes	—
246	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	1	no	—
256	C₄	C₁₃×#137	1	2	yes	—
257	C₄	C₂×C₁₃×#41	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	8	1	no	—
260	C₂	C₁₃×#525	1	1	yes	characteristic
261	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
262	C₂	C₂×C₁₃×#137	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₁₃×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₁₃	C₂	semidirect
3. C₁₃×#483	C₂	semidirect
5. C₁₃×#479	C₂	semidirect
6. C₁₃×#464	C₂	semidirect
7. C₁₃×#451	C₂	semidirect
9. C₄×C₁₃×Q₁₆	C₂	semidirect
10. C₄×C₁₃×SD₁₆	C₂	semidirect
11. C₁₃×#431	C₂	semidirect
81. #3108	C₁₃	direct (N × H)
110. C₂×C₄×Q₈	C₂×C₁₃	semidirect
111. #483	C₂×C₁₃	semidirect
113. #479	C₂×C₁₃	semidirect
114. #464	C₂×C₁₃	semidirect
115. #451	C₂×C₁₃	semidirect
117. C₄×Q₁₆	C₂×C₁₃	semidirect
118. C₄×SD₁₆	C₂×C₁₃	semidirect
119. #431	C₂×C₁₃	semidirect
212. C₁₃	#3108	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×#529	2
C₂×C₄×Q₈×C₁₃	2
C₁₃×#483	2
C₁₃×#480	2
C₁₃×#479	2
C₁₃×#464	2
C₁₃×#451	2
C₁₃×#448	2
C₄×C₁₃×Q₁₆	2
C₄×C₁₃×SD₁₆	2
C₁₃×#431	2
#3108	13

Subgroups — C13×#3108 #13112166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3108 #13112166