Subgroups — C₁₃×#3372 #13112430

Order 1664.

340 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	C₁₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#554	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#491	C₂	1	2	yes	maximal, complemented
4	C₁₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#479	C₂	1	2	yes	maximal, complemented
6	C₁₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#461	C₂	1	2	yes	maximal
8	C₁₃×#461	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#440	C₂	1	2	yes	maximal
11	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×#125	—	2	1	no	—
18	C₁₃×#125	—	2	2	no	—
19	C₁₃×#125	C₂²	1	2	yes	—
20	C₁₃×C₂²⋊Q₈	—	2	1	no	—
21	C₁₃×C₂²⋊Q₈	—	2	2	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	1	yes	characteristic
26	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₃	C₂²	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₁₃×#107	C₂²	1	2	yes	—
30	C₁₃×#106	C₂²	1	1	yes	characteristic
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×#103	C₂²	1	2	yes	—
38	C₁₃×#102	C₂²	1	1	yes	characteristic
39	C₁₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	—	2	1	no	—
42	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
45	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	2	no	—
48	C₁₃×SD₁₆	—	2	2	no	—
49	C₁₃×SD₁₆	—	2	4	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	2	no	—
54	C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₁₃×C₄⋊C₄	—	2	1	no	—
57	C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	—	2	1	no	—
61	C₁₃×C₄⋊C₄	—	2	2	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×C₄⋊C₄	C₂³	1	2	yes	—
67	C₁₃×C₂²⋊C₄	—	2	1	no	—
68	C₁₃×C₂²⋊C₄	—	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	#3372	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	1	no	—
79	Q₈×C₁₃	—	2	2	no	—
80	Q₈×C₁₃	—	8	1	no	—
81	D₄×C₁₃	—	2	1	no	—
82	D₄×C₁₃	#41	1	2	yes	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	—	8	2	no	—
90	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
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	#557	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#554	C₂×C₁₃	1	1	yes	characteristic
100	#491	C₂×C₁₃	1	2	yes	complemented
101	#487	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#479	C₂×C₁₃	1	2	yes	complemented
103	#477	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#461	C₂×C₁₃	1	2	yes	—
105	#461	C₂×C₁₃	1	1	yes	characteristic
106	#446	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#440	C₂×C₁₃	1	2	yes	—
108	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic
109	C₂²×C₁₃	—	2	2	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	1	no	—
116	C₄×C₁₃	—	4	2	no	—
117	C₄×C₁₃	—	4	1	no	—
118	C₄×C₁₃	—	4	1	no	—
119	C₄×C₁₃	—	4	1	no	—
120	C₄×C₁₃	—	4	2	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×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
126	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
127	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
128	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
129	#125	—	2	1	no	—
130	#125	—	2	2	no	—
131	#125	C₂²×C₁₃	1	2	yes	—
132	C₂²⋊Q₈	—	2	1	no	—
133	C₂²⋊Q₈	—	2	2	no	—
134	C₄×Q₈	—	2	1	no	—
135	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
136	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
137	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
138	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
139	C₄×D₄	C₂²×C₁₃	1	2	yes	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	#107	C₂²×C₁₃	1	2	yes	—
142	#106	C₂²×C₁₃	1	1	yes	characteristic
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	#103	C₂²×C₁₃	1	2	yes	—
150	#102	C₂²×C₁₃	1	1	yes	characteristic
151	#97	C₂²×C₁₃	1	2	yes	—
152	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×C₁₃	—	4	2	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	1	no	—
158	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
159	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
160	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
161	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	2	no	—
164	SD₁₆	—	2	2	no	—
165	SD₁₆	—	2	4	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	2	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
179	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
180	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
182	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
183	C₂²⋊C₄	—	2	1	no	—
184	C₂²⋊C₄	—	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	#3372	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	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	8	1	no	—
197	D₄	—	2	1	no	—
198	D₄	C₁₃×#41	1	2	yes	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
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	—	2	2	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	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	2	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	—	4	2	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. C₁₃×#557	C₂	semidirect
3. C₁₃×#491	C₂	semidirect
4. C₁₃×#487	C₂	semidirect
5. C₁₃×#479	C₂	semidirect
6. C₁₃×#477	C₂	semidirect
9. C₁₃×#446	C₂	semidirect
73. #3372	C₁₃	direct (N × H)
98. #557	C₂×C₁₃	semidirect
100. #491	C₂×C₁₃	semidirect
101. #487	C₂×C₁₃	semidirect
102. #479	C₂×C₁₃	semidirect
103. #477	C₂×C₁₃	semidirect
106. #446	C₂×C₁₃	semidirect
189. C₁₃	#3372	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×#557	2
C₁₃×#554	2
C₁₃×#491	2
C₁₃×#487	2
C₁₃×#479	2
C₁₃×#477	2
C₁₃×#461	2
C₁₃×#446	2
C₁₃×#440	2
C₄×C₁₃×SD₁₆	2
#3372	13

Subgroups — C13×#3372 #13112430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3372 #13112430