Subgroups — C₁₃×#9691 #15696030

Order 3328.

570 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2868	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#333	C₂	1	4	yes	maximal
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	4	yes	—
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
8	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	—	2	1	no	—
11	C₁₃×#384	—	2	1	no	—
12	C₁₃×#384	—	2	1	no	—
13	C₁₃×#384	—	2	2	no	—
14	C₄³×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#336	—	2	1	no	—
16	C₁₃×#333	—	2	8	no	—
17	C₂²×Q₈×C₁₃	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₁₃×C₄⋊Q₈	—	2	4	no	—
21	C₁₃×C₄⋊Q₈	—	2	4	no	—
22	C₁₃×C₄⋊Q₈	—	4	8	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×C₄⋊C₈	—	2	4	no	—
33	C₁₃×C₄⋊C₈	—	2	2	no	—
34	C₁₃×C₄⋊C₈	—	4	4	no	—
35	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
36	C₄×C₈×C₁₃	—	4	2	no	—
37	#9691	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂×Q₈×C₁₃	—	2	8	no	—
39	C₂×Q₈×C₁₃	—	4	8	no	—
40	C₂²×C₄×C₁₃	—	2	4	no	—
41	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	—	2	4	no	—
47	C₂×C₈×C₁₃	—	4	2	no	—
48	C₂×C₈×C₁₃	—	4	2	no	—
49	C₂×C₈×C₁₃	—	4	2	no	—
50	C₂×C₈×C₁₃	—	4	4	no	—
51	C₁₃×C₄⋊C₄	—	4	8	no	—
52	C₁₃×C₄⋊C₄	—	4	8	no	—
53	C₁₃×C₄⋊C₄	—	4	4	no	—
54	C₁₃×C₄⋊C₄	—	4	2	no	—
55	C₁₃×C₄⋊C₄	—	4	2	no	—
56	C₄²×C₁₃	—	2	4	no	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	SD₁₆	1	2	yes	—
59	C₄²×C₁₃	SD₁₆	1	4	yes	—
60	C₄²×C₁₃	Q₁₆	1	4	yes	—
61	C₄²×C₁₃	Q₁₆	1	2	yes	—
62	C₄²×C₁₃	C₂×D₄	1	2	yes	—
63	C₄²×C₁₃	#41	1	2	yes	—
64	#2868	C₂×C₁₃	1	1	yes	characteristic
65	#1842	C₂×C₁₃	1	1	yes	characteristic
66	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
67	C₂×#333	C₂×C₁₃	1	4	yes	—
68	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	Q₈×C₁₃	—	8	8	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	4	4	no	—
74	C₂×C₄×C₁₃	—	4	8	no	—
75	C₂×C₄×C₁₃	#100	1	4	yes	—
76	C₂×C₄×C₁₃	#103	1	4	yes	—
77	C₂×C₄×C₁₃	#103	1	4	yes	—
78	C₂×C₄×C₁₃	#103	1	4	yes	—
79	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₁₃	#121	1	2	yes	—
81	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₁₃	#137	1	2	yes	—
88	C₈×C₁₃	—	4	4	no	—
89	C₈×C₁₃	—	8	2	no	—
90	C₂×C₄⋊Q₈	C₄×C₁₃	1	4	yes	—
91	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
92	C₂×C₄⋊C₈	—	2	2	no	—
93	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
94	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
95	C₂×C₄×C₈	—	2	1	no	—
96	#384	—	2	1	no	—
97	#384	—	2	1	no	—
98	#384	—	2	2	no	—
99	C₄³	C₂²×C₁₃	1	1	yes	characteristic
100	#336	—	2	1	no	—
101	#333	—	2	8	no	—
102	C₂²×C₁₃	#390	1	1	yes	characteristic
103	C₂²×C₁₃	C₂×#100	1	2	yes	—
104	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
105	C₂²×C₁₃	C₂×#103	1	2	yes	—
106	C₂²×C₁₃	#419	1	1	yes	characteristic
107	C₄×C₁₃	—	2	4	no	—
108	C₄×C₁₃	—	2	2	no	—
109	C₄×C₁₃	—	8	4	no	—
110	C₄×C₁₃	#333	1	8	yes	complemented
111	C₄×C₁₃	#461	1	2	yes	—
112	C₄×C₁₃	#462	1	2	yes	—
113	C₄×C₁₃	#487	1	2	yes	—
114	C₄×C₁₃	#492	1	1	yes	characteristic
115	C₄×C₁₃	#494	1	1	yes	characteristic
116	C₂²×Q₈	C₂×C₄×C₁₃	1	4	yes	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₄⋊Q₈	—	2	4	no	—
120	C₄⋊Q₈	—	2	4	no	—
121	C₄⋊Q₈	—	4	8	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
127	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
128	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
129	C₂×C₄²	D₄×C₁₃	1	2	yes	—
130	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
131	C₄⋊C₈	—	2	4	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	4	4	no	—
134	C₄⋊C₈	D₄×C₁₃	1	4	yes	—
135	C₄×C₈	—	4	2	no	—
136	C₂×C₁₃	C₂×#333	1	2	yes	—
137	C₂×C₁₃	#1548	1	2	yes	—
138	C₂×C₁₃	#1956	1	1	yes	characteristic
139	C₂×C₁₃	#1965	1	1	yes	characteristic
140	C₂×C₁₃	#1981	1	1	yes	characteristic
141	C₂×Q₈	—	2	8	no	—
142	C₂×Q₈	—	4	8	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
145	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
146	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	4	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	2	2	no	—
161	C₄²	C₁₃×SD₁₆	1	2	yes	—
162	C₄²	C₁₃×SD₁₆	1	4	yes	—
163	C₄²	C₁₃×Q₁₆	1	4	yes	—
164	C₄²	C₁₃×Q₁₆	1	2	yes	—
165	C₄²	C₂×D₄×C₁₃	1	2	yes	—
166	C₄²	C₁₃×#41	1	2	yes	—
167	C₁₃	#9691	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
169	Q₈	—	8	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	8	no	—
175	C₂×C₄	C₁₃×#100	1	4	yes	—
176	C₂×C₄	C₁₃×#103	1	4	yes	—
177	C₂×C₄	C₁₃×#103	1	4	yes	—
178	C₂×C₄	C₁₃×#103	1	4	yes	—
179	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
180	C₂×C₄	C₁₃×#121	1	2	yes	—
181	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
182	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
183	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
184	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
185	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
186	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
187	C₂×C₄	C₁₃×#137	1	2	yes	—
188	C₈	—	4	4	no	—
189	C₈	—	8	2	no	—
190	C₂²	C₁₃×#390	1	1	yes	characteristic
191	C₂²	C₂×C₁₃×#100	1	2	yes	—
192	C₂²	C₂×C₁₃×#103	1	1	yes	characteristic
193	C₂²	C₂×C₁₃×#103	1	2	yes	—
194	C₂²	C₁₃×#419	1	1	yes	characteristic
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	8	4	no	—
198	C₄	C₁₃×#333	1	8	yes	complemented
199	C₄	C₁₃×#461	1	2	yes	—
200	C₄	C₁₃×#462	1	2	yes	—
201	C₄	C₁₃×#487	1	2	yes	—
202	C₄	C₁₃×#492	1	1	yes	characteristic
203	C₄	C₁₃×#494	1	1	yes	characteristic
204	C₂	C₂×C₁₃×#333	1	2	yes	—
205	C₂	C₁₃×#1548	1	2	yes	—
206	C₂	C₁₃×#1956	1	1	yes	characteristic
207	C₂	C₁₃×#1965	1	1	yes	characteristic
208	C₂	C₁₃×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
37. #9691	C₁₃	direct (N × H)
110. C₄×C₁₃	#333	semidirect
167. C₁₃	#9691	direct (N × H)
198. C₄	C₁₃×#333	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#2868	2
C₁₃×#1842	2
C₄×C₁₃×C₄⋊C₈	2
C₂×C₁₃×#333	2
#9691	13

Subgroups — C13×#9691 #15696030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9691 #15696030