Subgroups — C₁₃×#22738 #15709077

Order 3328.

822 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₃×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₁₃×#1948	C₂	1	6	yes	maximal
4	C₁₃×#1834	C₂	1	3	yes	maximal
5	C₄×C₁₃×#106	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₁₃	—	2	2	no	—
7	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₁₃	C₄	1	8	yes	complemented
9	C₂×C₁₃×#106	C₂²	1	6	yes	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
12	C₁₃×#389	—	2	6	no	—
13	C₁₃×#384	C₂²	1	6	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
15	C₁₃×#340	C₂²	1	1	yes	characteristic
16	C₁₃×#340	C₂²	1	3	yes	—
17	C₁₃×#340	C₂²	1	6	yes	—
18	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
21	C₄×Q₈×C₁₃	—	4	8	no	—
22	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
23	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
24	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
26	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
27	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
28	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
29	C₂×C₄²×C₁₃	—	2	6	no	—
30	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
31	C₁₃×#106	—	2	12	no	—
32	C₁₃×C₄⋊C₈	C₂×C₄	1	12	yes	—
33	C₄×C₈×C₁₃	C₂×C₄	1	12	yes	—
34	C₁₃×#95	—	2	6	no	—
35	#22738	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
37	C₂×Q₈×C₁₃	SD₁₆	1	8	yes	—
38	C₂²×C₄×C₁₃	—	2	2	no	—
39	C₂²×C₄×C₁₃	—	2	6	no	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
43	C₂×C₈×C₁₃	—	2	6	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
47	C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₁₃×C₄⋊C₄	—	2	6	no	—
49	C₁₃×C₄⋊C₄	—	2	6	no	—
50	C₁₃×C₄⋊C₄	—	4	12	no	—
51	C₁₃×C₄⋊C₄	—	8	6	no	—
52	C₁₃×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
53	C₁₃×C₄⋊C₄	#41	1	4	yes	—
54	C₄²×C₁₃	—	4	12	no	—
55	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
56	C₄²×C₁₃	C₂×Q₈	1	6	yes	—
57	C₂×C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic
58	Q₈×C₄⋊C₄	C₂×C₁₃	1	2	yes	—
59	#1948	C₂×C₁₃	1	6	yes	—
60	#1834	C₂×C₁₃	1	3	yes	—
61	C₄×#106	C₂×C₁₃	1	3	yes	—
62	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
63	Q₈×C₁₃	#106	1	16	yes	complemented
64	C₂×C₄×C₁₃	—	2	3	no	—
65	C₂×C₄×C₁₃	—	2	3	no	—
66	C₂×C₄×C₁₃	—	2	4	no	—
67	C₂×C₄×C₁₃	—	4	2	no	—
68	C₂×C₄×C₁₃	—	4	2	no	—
69	C₂×C₄×C₁₃	—	4	6	no	—
70	C₂×C₄×C₁₃	—	4	6	no	—
71	C₂×C₄×C₁₃	—	4	6	no	—
72	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
73	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
74	C₂×C₄×C₁₃	C₂×SD₁₆	1	12	yes	—
75	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
78	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
79	C₈×C₁₃	—	2	6	no	—
80	C₈×C₁₃	—	2	2	no	—
81	C₂×C₄×Q₈	—	2	2	no	—
82	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
83	C₈×Q₈	C₄×C₁₃	1	8	yes	complemented
84	C₂×#106	C₂²×C₁₃	1	6	yes	—
85	C₂×C₄⋊C₈	C₂²×C₁₃	1	3	yes	—
86	C₂×C₄×C₈	C₂²×C₁₃	1	3	yes	—
87	#389	—	2	6	no	—
88	#384	C₂²×C₁₃	1	6	yes	—
89	C₄×C₄⋊C₄	C₂²×C₁₃	1	6	yes	—
90	#340	C₂²×C₁₃	1	3	yes	—
91	#340	C₂²×C₁₃	1	6	yes	—
92	#340	C₂²×C₁₃	1	1	yes	characteristic
93	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₁₃	#548	1	2	yes	—
95	C₂²×C₁₃	C₂²×SD₁₆	1	2	yes	—
96	C₂²×C₁₃	#576	1	1	yes	characteristic
97	C₂²×C₁₃	#578	1	1	yes	characteristic
98	C₄×C₁₃	—	2	6	no	—
99	C₄×C₁₃	—	4	4	no	—
100	C₄×C₁₃	—	8	6	no	—
101	C₄×C₁₃	C₂×#106	1	12	yes	—
102	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
103	C₄×C₁₃	#519	1	1	yes	characteristic
104	C₄×C₁₃	#557	1	2	yes	—
105	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
106	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
107	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
108	C₄×Q₈	—	4	8	no	—
109	C₄×Q₈	C₂×C₄×C₁₃	1	4	yes	—
110	C₄×Q₈	Q₈×C₁₃	1	4	yes	—
111	C₂×C₄⋊C₄	—	2	6	no	—
112	C₂×C₄⋊C₄	—	2	6	no	—
113	C₂×C₄⋊C₄	D₄×C₁₃	1	3	yes	—
114	C₂×C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
115	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
116	C₂×C₄²	—	2	6	no	—
117	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
118	#106	—	2	12	no	—
119	C₄⋊C₈	C₂×C₄×C₁₃	1	12	yes	—
120	C₄×C₈	C₂×C₄×C₁₃	1	12	yes	—
121	#95	—	2	6	no	—
122	C₂×C₁₃	#2351	1	1	yes	characteristic
123	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
124	C₂×C₁₃	#2913	1	1	yes	characteristic
125	C₂×C₁₃	#3359	1	2	yes	—
126	C₂×C₁₃	#3362	1	2	yes	—
127	C₂×Q₈	C₁₃×C₄⋊C₄	1	4	yes	—
128	C₂×Q₈	C₁₃×SD₁₆	1	8	yes	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	6	no	—
131	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
132	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
133	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
134	C₂×C₈	—	2	6	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
137	C₂×C₈	C₂²×C₄×C₁₃	1	6	yes	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	6	no	—
140	C₄⋊C₄	—	2	6	no	—
141	C₄⋊C₄	—	4	12	no	—
142	C₄⋊C₄	—	8	6	no	—
143	C₄⋊C₄	C₁₃×C₄⋊C₄	1	12	yes	—
144	C₄⋊C₄	C₁₃×#41	1	4	yes	—
145	C₄²	—	4	12	no	—
146	C₄²	C₂²×C₄×C₁₃	1	6	yes	—
147	C₄²	C₂×Q₈×C₁₃	1	6	yes	—
148	C₁₃	#22738	1	1	yes	characteristic, direct product factor, Sylow
149	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
150	Q₈	C₁₃×#106	1	16	yes	complemented
151	C₂×C₄	—	2	3	no	—
152	C₂×C₄	—	2	3	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	6	no	—
157	C₂×C₄	—	4	6	no	—
158	C₂×C₄	—	4	6	no	—
159	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	6	yes	—
160	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	6	yes	—
161	C₂×C₄	C₂×C₁₃×SD₁₆	1	12	yes	—
162	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
163	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
164	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
165	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
166	C₈	—	2	6	no	—
167	C₈	—	2	2	no	—
168	C₂²	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
169	C₂²	C₁₃×#548	1	2	yes	—
170	C₂²	C₂²×C₁₃×SD₁₆	1	2	yes	—
171	C₂²	C₁₃×#576	1	1	yes	characteristic
172	C₂²	C₁₃×#578	1	1	yes	characteristic
173	C₄	—	2	6	no	—
174	C₄	—	4	4	no	—
175	C₄	—	8	6	no	—
176	C₄	C₂×C₁₃×#106	1	12	yes	—
177	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
178	C₄	C₁₃×#519	1	1	yes	characteristic
179	C₄	C₁₃×#557	1	2	yes	—
180	C₂	C₁₃×#2351	1	1	yes	characteristic
181	C₂	C₂²×C₁₃×#106	1	1	yes	characteristic
182	C₂	C₁₃×#2913	1	1	yes	characteristic
183	C₂	C₁₃×#3359	1	2	yes	—
184	C₂	C₁₃×#3362	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₁₃	C₄	semidirect
35. #22738	C₁₃	direct (N × H)
63. Q₈×C₁₃	#106	semidirect
83. C₈×Q₈	C₄×C₁₃	semidirect
148. C₁₃	#22738	direct (N × H)
150. Q₈	C₁₃×#106	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₁₃	2
Q₈×C₁₃×C₄⋊C₄	2
C₁₃×#1948	2
C₁₃×#1834	2
C₄×C₁₃×#106	2
#22738	13

Subgroups — C13×#22738 #15709077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#22738 #15709077