Subgroups — C₁₃×#3394 #13112452

Order 1664.

372 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	C₁₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#495	C₂	1	2	yes	maximal, complemented
4	C₁₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#464	C₂	1	2	yes	maximal, complemented
6	C₁₃×#463	C₂	1	2	yes	maximal, complemented
7	C₁₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#459	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, complemented
11	C₄×C₁₃×SD₁₆	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
14	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×D₈	C₂²	1	2	yes	—
17	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₃×#127	C₂²	1	1	yes	characteristic
19	C₁₃×#126	—	2	2	no	—
20	C₁₃×#124	—	2	1	no	—
21	C₁₃×#124	C₂²	1	2	yes	—
22	C₁₃×#124	C₂²	1	2	yes	—
23	C₁₃×#123	—	2	2	no	—
24	C₁₃×C₂²⋊Q₈	—	2	1	no	—
25	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
27	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₁₃	—	2	1	no	—
29	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₁₃	C₂²	1	2	yes	—
31	C₁₃×#117	—	2	1	no	—
32	C₁₃×#106	C₂²	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
44	C₂×D₄×C₁₃	—	2	1	no	—
45	C₂×D₄×C₁₃	—	2	1	no	—
46	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₁₃×Q₁₆	—	4	2	no	—
51	C₁₃×SD₁₆	—	2	4	no	—
52	C₁₃×SD₁₆	—	2	2	no	—
53	C₁₃×SD₁₆	—	4	2	no	—
54	C₁₃×D₈	—	4	2	no	—
55	C₂×C₈×C₁₃	C₂³	1	2	yes	—
56	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×C₄⋊C₄	—	2	1	no	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	—	2	1	no	—
61	C₁₃×C₄⋊C₄	—	2	1	no	—
62	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₁₃×C₂²⋊C₄	—	2	2	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	—	2	2	no	—
69	C₁₃×C₂²⋊C₄	—	2	1	no	—
70	C₁₃×C₂²⋊C₄	—	2	2	no	—
71	C₁₃×C₂²⋊C₄	—	2	2	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	—	2	1	no	—
74	C₄²×C₁₃	C₂³	1	2	yes	—
75	C₄²×C₁₃	C₂³	1	1	yes	characteristic
76	#3394	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₁₃	—	2	1	no	—
78	C₂³×C₁₃	—	2	2	no	—
79	Q₈×C₁₃	—	2	2	no	—
80	Q₈×C₁₃	—	2	1	no	—
81	Q₈×C₁₃	—	2	2	no	—
82	Q₈×C₁₃	—	2	2	no	—
83	Q₈×C₁₃	—	2	1	no	—
84	D₄×C₁₃	—	2	1	no	—
85	D₄×C₁₃	—	2	2	no	—
86	D₄×C₁₃	—	2	1	no	—
87	D₄×C₁₃	—	2	2	no	—
88	D₄×C₁₃	—	4	2	no	—
89	D₄×C₁₃	—	8	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	8	2	no	—
96	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
101	C₈×C₁₃	—	2	2	no	—
102	C₈×C₁₃	—	2	1	no	—
103	C₈×C₁₃	#41	1	2	yes	—
104	#559	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#556	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#495	C₂×C₁₃	1	2	yes	complemented
107	#492	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#464	C₂×C₁₃	1	2	yes	complemented
109	#463	C₂×C₁₃	1	2	yes	complemented
110	#462	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#459	C₂×C₁₃	1	1	yes	characteristic
112	C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₄×SD₁₆	C₂×C₁₃	1	2	yes	complemented
115	C₂²×C₁₃	—	4	1	no	—
116	C₂²×C₁₃	—	4	2	no	—
117	C₂²×C₁₃	—	4	1	no	—
118	C₂²×C₁₃	—	4	2	no	—
119	C₂²×C₁₃	—	4	2	no	—
120	C₂²×C₁₃	—	4	1	no	—
121	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	—	4	1	no	—
126	C₄×C₁₃	—	4	2	no	—
127	C₄×C₁₃	—	4	1	no	—
128	C₄×C₁₃	—	4	1	no	—
129	C₄×C₁₃	#135	1	2	yes	—
130	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
131	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
133	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
134	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
136	C₂×D₈	C₂²×C₁₃	1	2	yes	—
137	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
138	#127	C₂²×C₁₃	1	1	yes	characteristic
139	#126	—	2	2	no	—
140	#124	—	2	1	no	—
141	#124	C₂²×C₁₃	1	2	yes	—
142	#124	C₂²×C₁₃	1	2	yes	—
143	#123	—	2	2	no	—
144	C₂²⋊Q₈	—	2	1	no	—
145	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
146	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
148	C₄×D₄	—	2	1	no	—
149	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
150	C₄×D₄	C₂²×C₁₃	1	2	yes	—
151	#117	—	2	1	no	—
152	#106	C₂²×C₁₃	1	1	yes	characteristic
153	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
154	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
155	#103	C₂²×C₁₃	1	2	yes	—
156	#103	C₂²×C₁₃	1	1	yes	characteristic
157	#102	C₂²×C₁₃	1	2	yes	—
158	#102	C₂²×C₁₃	1	1	yes	characteristic
159	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
160	C₄×C₈	C₂²×C₁₃	1	2	yes	—
161	C₂×C₁₃	—	8	1	no	—
162	C₂×C₁₃	—	8	2	no	—
163	C₂×C₁₃	#550	1	1	yes	characteristic
164	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
165	C₂×C₁₃	#577	1	1	yes	characteristic
166	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
167	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
168	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
172	C₂×D₄	C₂³×C₁₃	1	2	yes	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	2	no	—
175	Q₁₆	—	4	2	no	—
176	SD₁₆	—	2	4	no	—
177	SD₁₆	—	2	2	no	—
178	SD₁₆	—	4	2	no	—
179	D₈	—	4	2	no	—
180	C₂×C₈	C₂³×C₁₃	1	2	yes	—
181	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
182	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
188	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
189	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
190	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	1	no	—
199	C₄²	C₂³×C₁₃	1	2	yes	—
200	C₄²	C₂³×C₁₃	1	1	yes	characteristic
201	C₁₃	#3394	1	1	yes	characteristic, direct product factor, Sylow
202	C₂³	—	2	1	no	—
203	C₂³	—	2	2	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	2	1	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	2	2	no	—
208	Q₈	—	2	1	no	—
209	D₄	—	2	1	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	1	no	—
212	D₄	—	2	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	8	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
222	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
223	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
224	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
225	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	2	1	no	—
228	C₈	C₁₃×#41	1	2	yes	—
229	C₂²	—	4	1	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	1	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	1	no	—
235	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	1	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₄	C₁₃×#135	1	2	yes	—
244	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
245	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
246	C₂	—	8	1	no	—
247	C₂	—	8	2	no	—
248	C₂	C₁₃×#550	1	1	yes	characteristic
249	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
250	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	C₁₃×#459	C₂
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×#559	C₂	semidirect
2. C₁₃×#556	C₂	semidirect
3. C₁₃×#495	C₂	semidirect
4. C₁₃×#492	C₂	semidirect
5. C₁₃×#464	C₂	semidirect
6. C₁₃×#463	C₂	semidirect
7. C₁₃×#462	C₂	semidirect
9. C₈×Q₈×C₁₃	C₂	semidirect
10. C₄×C₁₃×SD₁₆	C₂	semidirect
11. C₄×C₁₃×SD₁₆	C₂	semidirect
76. #3394	C₁₃	direct (N × H)
104. #559	C₂×C₁₃	semidirect
105. #556	C₂×C₁₃	semidirect
106. #495	C₂×C₁₃	semidirect
107. #492	C₂×C₁₃	semidirect
108. #464	C₂×C₁₃	semidirect
109. #463	C₂×C₁₃	semidirect
110. #462	C₂×C₁₃	semidirect
112. C₈×Q₈	C₂×C₁₃	semidirect
113. C₄×SD₁₆	C₂×C₁₃	semidirect
114. C₄×SD₁₆	C₂×C₁₃	semidirect
201. C₁₃	#3394	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×#559	2
C₁₃×#556	2
C₁₃×#495	2
C₁₃×#492	2
C₁₃×#464	2
C₁₃×#463	2
C₁₃×#462	2
C₁₃×#459	2
C₈×Q₈×C₁₃	2
C₄×C₁₃×SD₁₆	2
#3394	13

Subgroups — C13×#3394 #13112452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3394 #13112452