Subgroups — C₄×C₁₃×#366 #15699011

Order 3328.

366 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#366	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×#107	C₂	1	2	yes	maximal
4	C₁₃×#1380	C₂	1	2	yes	maximal
5	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#107	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
10	C₁₃×#366	C₄	1	8	yes	direct product factor
11	C₄×C₁₃×C₁₆	C₄	1	4	yes	complemented
12	C₁₃×#340	—	2	2	no	—
13	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
16	C₂×C₄²×C₁₃	—	4	2	no	—
17	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
18	C₂×C₁₃×C₁₆	—	2	2	no	—
19	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
22	C₁₃×#107	—	2	4	no	—
23	C₁₃×#107	C₂×C₄	1	8	yes	—
24	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₃	Q₈	1	2	yes	—
26	C₁₃×#95	—	4	2	no	—
27	C₄×#366	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₄×C₁₃	—	4	4	no	—
29	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃	—	2	2	no	—
33	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
36	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₁₃	#41	1	2	yes	—
38	C₁₃×C₄⋊C₄	—	2	8	no	—
39	C₁₃×C₄⋊C₄	—	4	4	no	—
40	C₄²×C₁₃	—	8	4	no	—
41	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
42	C₄²×C₁₃	Q₁₆	1	2	yes	—
43	C₁₃×C₁₆	—	2	2	no	—
44	C₁₃×C₁₆	C₄²	1	4	yes	complemented
45	C₂×#366	C₂×C₁₃	1	2	yes	—
46	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
47	C₄×#107	C₂×C₁₃	1	2	yes	—
48	#1380	C₂×C₁₃	1	2	yes	—
49	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	4	8	no	—
52	C₂×C₄×C₁₃	—	8	4	no	—
53	C₂×C₄×C₁₃	—	8	4	no	—
54	C₂×C₄×C₁₃	#107	1	2	yes	—
55	C₂×C₄×C₁₃	#107	1	2	yes	—
56	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
57	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
58	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#135	1	2	yes	—
62	C₈×C₁₃	—	2	2	no	—
63	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
64	C₈×C₁₃	#117	1	1	yes	characteristic
65	C₈×C₁₃	C₄×Q₈	1	2	yes	—
66	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
67	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
68	C₂×#107	C₂²×C₁₃	1	4	yes	—
69	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
70	C₄×C₄⋊C₄	—	2	2	no	—
71	#366	C₄×C₁₃	1	8	yes	direct product factor
72	C₄×C₁₆	C₄×C₁₃	1	4	yes	complemented
73	#340	—	2	2	no	—
74	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
75	C₂²×C₁₃	C₄×D₈	1	2	yes	—
76	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
77	C₂²×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
78	C₂²×C₁₃	#508	1	2	yes	—
79	C₄×C₁₃	—	2	2	no	—
80	C₄×C₁₃	—	8	8	no	—
81	C₄×C₁₃	#366	1	4	yes	direct product factor
82	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₄×C₁₃	#427	1	1	yes	characteristic
84	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
85	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
86	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	2	4	no	—
88	C₂×C₄²	—	4	2	no	—
89	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
92	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
93	C₂×C₁₆	C₂×C₄×C₁₃	1	4	yes	—
94	#107	—	2	4	no	—
95	#107	C₂×C₄×C₁₃	1	8	yes	—
96	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
97	C₄×C₈	Q₈×C₁₃	1	2	yes	—
98	#95	—	4	2	no	—
99	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
100	C₂×C₁₃	C₂×#366	1	1	yes	characteristic
101	C₂×C₁₃	#2158	1	1	yes	characteristic
102	C₂×C₁₃	C₄×D₁₆	1	2	yes	—
103	C₂×C₁₃	C₄×Q₃₂	1	2	yes	—
104	C₂²×C₄	—	4	4	no	—
105	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
106	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
107	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
110	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
111	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
112	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
113	C₂×C₈	C₁₃×#41	1	2	yes	—
114	C₄⋊C₄	—	2	8	no	—
115	C₄⋊C₄	—	4	4	no	—
116	C₄²	—	8	4	no	—
117	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
118	C₄²	C₁₃×Q₁₆	1	2	yes	—
119	C₁₆	—	2	2	no	—
120	C₁₆	C₄²×C₁₃	1	4	yes	complemented
121	C₁₃	C₄×#366	1	1	yes	characteristic, direct product factor, Sylow
122	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	8	no	—
125	C₂×C₄	—	8	4	no	—
126	C₂×C₄	—	8	4	no	—
127	C₂×C₄	C₁₃×#107	1	2	yes	—
128	C₂×C₄	C₁₃×#107	1	2	yes	—
129	C₂×C₄	C₁₃×D₁₆	1	2	yes	—
130	C₂×C₄	C₁₃×Q₃₂	1	2	yes	—
131	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
133	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
134	C₂×C₄	C₁₃×#135	1	2	yes	—
135	C₈	—	2	2	no	—
136	C₈	C₂×C₄²×C₁₃	1	1	yes	characteristic
137	C₈	C₁₃×#117	1	1	yes	characteristic
138	C₈	C₄×Q₈×C₁₃	1	2	yes	—
139	C₂²	C₂×C₁₃×#107	1	1	yes	characteristic
140	C₂²	C₄×C₁₃×D₈	1	2	yes	—
141	C₂²	C₂×C₁₃×D₁₆	1	1	yes	characteristic
142	C₂²	C₂×C₁₃×Q₃₂	1	1	yes	characteristic
143	C₂²	C₁₃×#508	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	8	8	no	—
146	C₄	C₁₃×#366	1	4	yes	direct product factor
147	C₄	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
148	C₄	C₁₃×#427	1	1	yes	characteristic
149	C₄	C₄×C₁₃×Q₁₆	1	2	yes	—
150	C₂	C₄×C₁₃×#107	1	1	yes	characteristic
151	C₂	C₂×C₁₃×#366	1	1	yes	characteristic
152	C₂	C₁₃×#2158	1	1	yes	characteristic
153	C₂	C₄×C₁₃×D₁₆	1	2	yes	—
154	C₂	C₄×C₁₃×Q₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	D₈
commutator	C₈	C₂×C₄²×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle, prime radical	C₂³×C₁₃	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₁₃×#366	C₄	direct (N × H)
11. C₄×C₁₃×C₁₆	C₄	semidirect
27. C₄×#366	C₁₃	direct (N × H)
44. C₁₃×C₁₆	C₄²	semidirect
71. #366	C₄×C₁₃	direct (N × H)
72. C₄×C₁₆	C₄×C₁₃	semidirect
81. C₄×C₁₃	#366	direct (N × H)
120. C₁₆	C₄²×C₁₃	semidirect
121. C₁₃	C₄×#366	direct (N × H)
146. C₄	C₁₃×#366	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#366	2
C₂×C₄×C₁₃×C₁₆	2
C₄×C₁₃×#107	2
C₁₃×#1380	2
C₄×#366	13

Subgroups — C4×C13×#366 #15699011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₃×#366 #15699011