Subgroups — C₁₃×#3331 #13112389

Order 1664.

388 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#547	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#483	C₂	1	2	yes	maximal, complemented
5	C₁₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#470	C₂	1	2	yes	maximal, complemented
7	C₁₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#452	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×#417	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×#41	C₂²	1	2	yes	—
13	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₃×#125	—	2	1	no	—
19	C₁₃×#123	—	2	2	no	—
20	C₁₃×C₂²⋊Q₈	—	2	2	no	—
21	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₁₃×#121	—	2	1	no	—
24	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₃	—	2	1	no	—
26	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
30	C₁₃×#106	C₂²	1	1	yes	characteristic
31	C₁₃×#106	C₂²	1	2	yes	—
32	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₁₃×#103	C₂²	1	2	yes	—
34	C₁₃×#103	C₂²	1	2	yes	—
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	C₁₃×#41	—	2	4	no	—
42	C₁₃×#41	—	4	2	no	—
43	C₂×Q₈×C₁₃	—	2	1	no	—
44	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
45	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₃	—	2	2	no	—
47	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
48	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
53	C₁₃×Q₁₆	—	4	2	no	—
54	C₁₃×SD₁₆	—	2	2	no	—
55	C₁₃×M₁₆	—	4	2	no	—
56	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂³	1	2	yes	—
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₄⋊C₄	—	2	1	no	—
63	C₁₃×C₄⋊C₄	—	2	1	no	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	2	1	no	—
66	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
69	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₁₃×C₂²⋊C₄	—	2	2	no	—
71	C₁₃×C₂²⋊C₄	—	2	2	no	—
72	C₁₃×C₂²⋊C₄	—	2	1	no	—
73	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
74	C₄²×C₁₃	—	2	1	no	—
75	C₄²×C₁₃	C₂³	1	1	yes	characteristic
76	#3331	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₁₃	—	2	1	no	—
78	C₂³×C₁₃	C₂×D₄	1	2	yes	—
79	Q₈×C₁₃	—	2	1	no	—
80	Q₈×C₁₃	—	2	2	no	—
81	Q₈×C₁₃	—	2	1	no	—
82	Q₈×C₁₃	—	2	2	no	—
83	Q₈×C₁₃	—	8	1	no	—
84	D₄×C₁₃	—	2	2	no	—
85	D₄×C₁₃	—	2	1	no	—
86	D₄×C₁₃	—	4	4	no	—
87	D₄×C₁₃	#41	1	2	yes	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	4	4	no	—
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	—	8	2	no	—
98	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
102	C₈×C₁₃	—	2	1	no	—
103	C₈×C₁₃	—	2	1	no	—
104	C₈×C₁₃	—	2	2	no	—
105	#554	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#547	C₂×C₁₃	1	1	yes	characteristic
107	#487	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#483	C₂×C₁₃	1	2	yes	complemented
109	#476	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#470	C₂×C₁₃	1	2	yes	complemented
111	#462	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#452	C₂×C₁₃	1	2	yes	complemented
113	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#436	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#417	C₂×C₁₃	1	2	yes	complemented
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	—	2	2	no	—
119	C₂²×C₁₃	—	2	2	no	—
120	C₂²×C₁₃	—	4	1	no	—
121	C₂²×C₁₃	—	4	1	no	—
122	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	—	4	2	no	—
126	C₄×C₁₃	—	4	1	no	—
127	C₄×C₁₃	—	4	1	no	—
128	C₄×C₁₃	—	4	1	no	—
129	C₄×C₁₃	—	4	2	no	—
130	C₄×C₁₃	#137	1	2	yes	—
131	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
132	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×#41	C₂²×C₁₃	1	2	yes	—
134	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
135	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
137	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
138	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
139	#125	—	2	1	no	—
140	#123	—	2	2	no	—
141	C₂²⋊Q₈	—	2	2	no	—
142	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
143	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
144	#121	—	2	1	no	—
145	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
146	C₄×D₄	—	2	1	no	—
147	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
148	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
151	#106	C₂²×C₁₃	1	1	yes	characteristic
152	#106	C₂²×C₁₃	1	2	yes	—
153	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
154	#103	C₂²×C₁₃	1	2	yes	—
155	#103	C₂²×C₁₃	1	2	yes	—
156	#103	C₂²×C₁₃	1	1	yes	characteristic
157	#103	C₂²×C₁₃	1	2	yes	—
158	#102	C₂²×C₁₃	1	1	yes	characteristic
159	#102	C₂²×C₁₃	1	2	yes	—
160	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
161	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×C₁₃	—	4	2	no	—
163	C₂×C₁₃	—	4	2	no	—
164	C₂×C₁₃	#546	1	1	yes	characteristic
165	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
166	C₂×C₁₃	#577	1	1	yes	characteristic
167	#41	—	2	4	no	—
168	#41	—	4	2	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
171	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
174	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
179	Q₁₆	—	4	2	no	—
180	SD₁₆	—	2	2	no	—
181	M₁₆	—	4	2	no	—
182	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
183	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
184	C₂×C₈	C₂³×C₁₃	1	2	yes	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
193	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
194	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
195	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
200	C₄²	—	2	1	no	—
201	C₄²	C₂³×C₁₃	1	1	yes	characteristic
202	C₁₃	#3331	1	1	yes	characteristic, direct product factor, Sylow
203	C₂³	—	2	1	no	—
204	C₂³	C₂×D₄×C₁₃	1	2	yes	—
205	Q₈	—	2	1	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	2	1	no	—
208	Q₈	—	2	2	no	—
209	Q₈	—	8	1	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	1	no	—
212	D₄	—	4	4	no	—
213	D₄	C₁₃×#41	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
225	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
226	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
227	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
228	C₈	—	2	1	no	—
229	C₈	—	2	1	no	—
230	C₈	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	1	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	2	no	—
245	C₄	C₁₃×#137	1	2	yes	—
246	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
247	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
248	C₂	—	4	2	no	—
249	C₂	—	4	2	no	—
250	C₂	C₁₃×#546	1	1	yes	characteristic
251	C₂	C₂×C₁₃×#137	1	1	yes	characteristic
252	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×#547	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
1. C₁₃×#554	C₂	semidirect
3. C₁₃×#487	C₂	semidirect
4. C₁₃×#483	C₂	semidirect
5. C₁₃×#476	C₂	semidirect
6. C₁₃×#470	C₂	semidirect
7. C₁₃×#462	C₂	semidirect
8. C₁₃×#452	C₂	semidirect
9. C₄×C₁₃×SD₁₆	C₂	semidirect
10. C₁₃×#436	C₂	semidirect
11. C₁₃×#417	C₂	semidirect
76. #3331	C₁₃	direct (N × H)
105. #554	C₂×C₁₃	semidirect
107. #487	C₂×C₁₃	semidirect
108. #483	C₂×C₁₃	semidirect
109. #476	C₂×C₁₃	semidirect
110. #470	C₂×C₁₃	semidirect
111. #462	C₂×C₁₃	semidirect
112. #452	C₂×C₁₃	semidirect
113. C₄×SD₁₆	C₂×C₁₃	semidirect
114. #436	C₂×C₁₃	semidirect
115. #417	C₂×C₁₃	semidirect
202. C₁₃	#3331	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×#554	2
C₁₃×#547	2
C₁₃×#487	2
C₁₃×#483	2
C₁₃×#476	2
C₁₃×#470	2
C₁₃×#462	2
C₁₃×#452	2
C₄×C₁₃×SD₁₆	2
C₁₃×#436	2
C₁₃×#417	2
#3331	13

Subgroups — C13×#3331 #13112389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3331 #13112389