Subgroups — C₁₃×#3335 #13112393

Order 1664.

388 conjugacy classes of subgroups.

The subgroup lattice has 248 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×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#483	C₂	1	2	yes	maximal, complemented
5	C₁₃×#481	C₂	1	2	yes	maximal, complemented
6	C₁₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#465	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₃×#106	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×#41	—	2	1	no	—
13	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×#125	—	2	1	no	—
18	C₁₃×#123	—	2	2	no	—
19	C₁₃×C₂²⋊Q₈	—	2	1	no	—
20	C₁₃×C₂²⋊Q₈	—	2	2	no	—
21	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₁₃×#121	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₃	—	2	1	no	—
25	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₁₃×#106	C₂²	1	1	yes	characteristic
30	C₁₃×#106	C₂²	1	2	yes	—
31	C₁₃×#106	C₂²	1	2	yes	—
32	C₁₃×#106	C₂²	1	2	yes	—
33	C₁₃×#106	C₂²	1	2	yes	—
34	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
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	—	4	4	no	—
42	C₂×Q₈×C₁₃	—	2	1	no	—
43	C₂×Q₈×C₁₃	—	2	1	no	—
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₃	—	2	2	no	—
46	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
52	C₁₃×SD₁₆	—	2	2	no	—
53	C₂×C₈×C₁₃	—	2	4	no	—
54	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂³	1	2	yes	—
56	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₄⋊C₄	—	2	1	no	—
63	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
66	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
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	D₄	1	2	yes	—
72	C₄²×C₁₃	—	2	1	no	—
73	C₄²×C₁₃	C₂³	1	1	yes	characteristic
74	#3335	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₃	—	2	1	no	—
76	C₂³×C₁₃	C₂×D₄	1	2	yes	—
77	Q₈×C₁₃	—	2	1	no	—
78	Q₈×C₁₃	—	2	1	no	—
79	Q₈×C₁₃	—	4	2	no	—
80	Q₈×C₁₃	—	8	1	no	—
81	D₄×C₁₃	—	2	1	no	—
82	D₄×C₁₃	—	2	1	no	—
83	D₄×C₁₃	—	4	4	no	—
84	D₄×C₁₃	SD₁₆	1	4	yes	complemented
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	4	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	—	8	2	no	—
95	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₈×C₁₃	—	2	2	no	—
100	C₈×C₁₃	—	2	1	no	—
101	C₈×C₁₃	#41	1	2	yes	complemented
102	#554	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#547	C₂×C₁₃	1	1	yes	characteristic
104	#498	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#483	C₂×C₁₃	1	2	yes	complemented
106	#481	C₂×C₁₃	1	2	yes	complemented
107	#476	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#475	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#465	C₂×C₁₃	1	2	yes	complemented
110	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₈×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂×#106	C₂×C₁₃	1	2	yes	complemented
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	—	4	1	no	—
116	C₂²×C₁₃	—	4	1	no	—
117	C₂²×C₁₃	—	4	1	no	—
118	C₂²×C₁₃	C₂×SD₁₆	1	4	yes	—
119	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	4	1	no	—
123	C₄×C₁₃	—	4	1	no	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	—	4	1	no	—
126	C₄×C₁₃	—	4	2	no	—
127	C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
128	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
129	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×#41	—	2	1	no	—
131	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
132	C₂²×C₈	C₂²×C₁₃	1	2	yes	—
133	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
134	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
135	#125	—	2	1	no	—
136	#123	—	2	2	no	—
137	C₂²⋊Q₈	—	2	1	no	—
138	C₂²⋊Q₈	—	2	2	no	—
139	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
140	#121	C₂²×C₁₃	1	2	yes	—
141	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
142	C₄×D₄	—	2	1	no	—
143	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
144	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
147	#106	C₂²×C₁₃	1	1	yes	characteristic
148	#106	C₂²×C₁₃	1	2	yes	—
149	#106	C₂²×C₁₃	1	2	yes	—
150	#106	C₂²×C₁₃	1	2	yes	—
151	#106	C₂²×C₁₃	1	2	yes	—
152	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
153	#103	C₂²×C₁₃	1	1	yes	characteristic
154	#103	C₂²×C₁₃	1	2	yes	—
155	#102	C₂²×C₁₃	1	1	yes	characteristic
156	#102	C₂²×C₁₃	1	2	yes	—
157	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
158	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×C₁₃	—	2	4	no	—
160	C₂×C₁₃	—	8	1	no	—
161	C₂×C₁₃	#547	1	1	yes	characteristic
162	C₂×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
163	C₂×C₁₃	#577	1	1	yes	characteristic
164	#41	—	4	4	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
170	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
175	SD₁₆	—	2	2	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₁₃	1	2	yes	—
179	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
188	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
189	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
190	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	1	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
195	C₄²	—	2	1	no	—
196	C₄²	C₂³×C₁₃	1	1	yes	characteristic
197	C₁₃	#3335	1	1	yes	characteristic, direct product factor, Sylow
198	C₂³	—	2	1	no	—
199	C₂³	C₂×D₄×C₁₃	1	2	yes	—
200	Q₈	—	2	1	no	—
201	Q₈	—	2	1	no	—
202	Q₈	—	4	2	no	—
203	Q₈	—	8	1	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	D₄	—	4	4	no	—
207	D₄	C₁₃×SD₁₆	1	4	yes	complemented
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
220	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
221	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	C₁₃×#41	1	2	yes	complemented
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	C₂×C₁₃×SD₁₆	1	4	yes	—
231	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	2	no	—
239	C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
240	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
241	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
242	C₂	—	2	4	no	—
243	C₂	—	8	1	no	—
244	C₂	C₁₃×#547	1	1	yes	characteristic
245	C₂	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
246	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×#498	C₂	semidirect
4. C₁₃×#483	C₂	semidirect
5. C₁₃×#481	C₂	semidirect
6. C₁₃×#476	C₂	semidirect
7. C₁₃×#475	C₂	semidirect
8. C₁₃×#465	C₂	semidirect
9. C₄×C₁₃×SD₁₆	C₂	semidirect
10. C₈×D₄×C₁₃	C₂	semidirect
11. C₂×C₁₃×#106	C₂	semidirect
74. #3335	C₁₃	direct (N × H)
84. D₄×C₁₃	SD₁₆	semidirect
101. C₈×C₁₃	#41	semidirect
102. #554	C₂×C₁₃	semidirect
104. #498	C₂×C₁₃	semidirect
105. #483	C₂×C₁₃	semidirect
106. #481	C₂×C₁₃	semidirect
107. #476	C₂×C₁₃	semidirect
108. #475	C₂×C₁₃	semidirect
109. #465	C₂×C₁₃	semidirect
110. C₄×SD₁₆	C₂×C₁₃	semidirect
111. C₈×D₄	C₂×C₁₃	semidirect
112. C₂×#106	C₂×C₁₃	semidirect
197. C₁₃	#3335	direct (N × H)
207. D₄	C₁₃×SD₁₆	semidirect
224. C₈	C₁₃×#41	semidirect

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×#498	2
C₁₃×#483	2
C₁₃×#481	2
C₁₃×#476	2
C₁₃×#475	2
C₁₃×#465	2
C₄×C₁₃×SD₁₆	2
C₈×D₄×C₁₃	2
C₂×C₁₃×#106	2
#3335	13

Subgroups — C13×#3335 #13112393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3335 #13112393