Subgroups — C₁₃×#3369 #13112427

Order 1664.

338 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#556	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#489	C₂	1	2	yes	maximal
4	C₁₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#476	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#475	C₂	1	2	yes	maximal, complemented
7	C₁₃×#464	C₂	1	2	yes	maximal
8	C₁₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#441	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	—	2	1	no	—
15	C₁₃×C₄⋊Q₈	—	2	2	no	—
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₃×#126	—	2	2	no	—
20	C₁₃×#124	C₂²	1	2	yes	—
21	C₁₃×C₂²⋊Q₈	—	2	1	no	—
22	C₄×Q₈×C₁₃	—	2	1	no	—
23	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
25	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
26	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#117	—	2	1	no	—
29	C₁₃×#106	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	1	yes	characteristic
34	C₁₃×#103	C₂²	1	2	yes	—
35	C₁₃×#103	C₂²	1	2	yes	—
36	C₁₃×#103	C₂²	1	2	yes	—
37	C₁₃×#102	C₂²	1	1	yes	characteristic
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₁₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	—	2	2	no	—
42	C₂×Q₈×C₁₃	—	2	1	no	—
43	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
45	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₁₃×Q₁₆	—	2	4	no	—
49	C₁₃×SD₁₆	—	2	2	no	—
50	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	—	2	1	no	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₁₃×C₄⋊C₄	—	2	1	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	1	yes	characteristic
63	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
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	C₄²×C₁₃	—	2	2	no	—
69	C₄²×C₁₃	—	2	2	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	C₂³	1	1	yes	characteristic
72	C₄²×C₁₃	C₂³	1	2	yes	—
73	#3369	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₁₃	—	2	1	no	—
75	Q₈×C₁₃	—	2	2	no	—
76	Q₈×C₁₃	—	2	1	no	—
77	Q₈×C₁₃	—	2	2	no	—
78	Q₈×C₁₃	—	2	2	no	—
79	Q₈×C₁₃	—	8	2	no	—
80	Q₈×C₁₃	—	8	1	no	—
81	Q₈×C₁₃	#41	1	2	yes	—
82	D₄×C₁₃	—	2	1	no	—
83	D₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₃	—	2	1	no	—
96	C₈×C₁₃	—	2	2	no	—
97	C₈×C₁₃	—	2	1	no	—
98	Q₈²	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#556	C₂×C₁₃	1	1	yes	characteristic
100	#489	C₂×C₁₃	1	2	yes	—
101	#487	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#476	C₂×C₁₃	1	1	yes	characteristic
103	#475	C₂×C₁₃	1	2	yes	complemented
104	#464	C₂×C₁₃	1	2	yes	—
105	#462	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#446	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#441	C₂×C₁₃	1	2	yes	complemented
108	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic
109	C₂²×C₁₃	—	4	1	no	—
110	C₂²×C₁₃	—	4	1	no	—
111	C₂²×C₁₃	—	4	1	no	—
112	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	4	2	no	—
118	C₄×C₁₃	—	4	1	no	—
119	C₄×C₁₃	—	4	2	no	—
120	C₄×C₁₃	—	4	1	no	—
121	C₄×C₁₃	#137	1	2	yes	—
122	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
123	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
125	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
126	C₄⋊Q₈	—	2	1	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
129	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
130	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
131	#126	—	2	2	no	—
132	#124	C₂²×C₁₃	1	2	yes	—
133	C₂²⋊Q₈	—	2	1	no	—
134	C₄×Q₈	—	2	1	no	—
135	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
136	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
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	#117	—	2	1	no	—
141	#106	C₂²×C₁₃	1	1	yes	characteristic
142	#106	C₂²×C₁₃	1	2	yes	—
143	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
144	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
145	#103	C₂²×C₁₃	1	1	yes	characteristic
146	#103	C₂²×C₁₃	1	2	yes	—
147	#103	C₂²×C₁₃	1	2	yes	—
148	#103	C₂²×C₁₃	1	2	yes	—
149	#102	C₂²×C₁₃	1	1	yes	characteristic
150	#102	C₂²×C₁₃	1	2	yes	—
151	#97	C₂²×C₁₃	1	2	yes	—
152	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×C₁₃	—	8	1	no	—
154	C₂×C₁₃	#548	1	1	yes	characteristic
155	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
156	C₂×C₁₃	#577	1	1	yes	characteristic
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
160	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
161	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
162	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	Q₁₆	—	2	4	no	—
165	SD₁₆	—	2	2	no	—
166	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
167	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
168	C₂×C₈	C₂³×C₁₃	1	2	yes	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
177	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
178	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
180	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	1	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	1	no	—
187	C₄²	C₂³×C₁₃	1	1	yes	characteristic
188	C₄²	C₂³×C₁₃	1	2	yes	—
189	C₁₃	#3369	1	1	yes	characteristic, direct product factor, Sylow
190	C₂³	—	2	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	2	1	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	2	2	no	—
195	Q₈	—	8	2	no	—
196	Q₈	—	8	1	no	—
197	Q₈	C₁₃×#41	1	2	yes	—
198	D₄	—	2	1	no	—
199	D₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
208	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	2	no	—
213	C₈	—	2	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	—	4	1	no	—
216	C₂²	—	4	1	no	—
217	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	1	no	—
226	C₄	C₁₃×#137	1	2	yes	—
227	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
228	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
229	C₂	—	8	1	no	—
230	C₂	C₁₃×#548	1	1	yes	characteristic
231	C₂	C₂×C₁₃×#137	1	1	yes	characteristic
232	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×D₄×C₁₃	C₂³
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. Q₈²×C₁₃	C₂	semidirect
4. C₁₃×#487	C₂	semidirect
6. C₁₃×#475	C₂	semidirect
8. C₁₃×#462	C₂	semidirect
9. C₁₃×#446	C₂	semidirect
10. C₁₃×#441	C₂	semidirect
73. #3369	C₁₃	direct (N × H)
98. Q₈²	C₂×C₁₃	semidirect
101. #487	C₂×C₁₃	semidirect
103. #475	C₂×C₁₃	semidirect
105. #462	C₂×C₁₃	semidirect
106. #446	C₂×C₁₃	semidirect
107. #441	C₂×C₁₃	semidirect
189. C₁₃	#3369	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈²×C₁₃	2
C₁₃×#556	2
C₁₃×#489	2
C₁₃×#487	2
C₁₃×#476	2
C₁₃×#475	2
C₁₃×#464	2
C₁₃×#462	2
C₁₃×#446	2
C₁₃×#441	2
C₄×C₁₃×SD₁₆	2
#3369	13

Subgroups — C13×#3369 #13112427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3369 #13112427