Subgroups — C₄²×C₁₃×Q₁₆ #15706838

Order 3328.

1110 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×Q₁₆	C₂	1	3	yes	maximal
2	C₄²×Q₈×C₁₃	C₂	1	2	yes	maximal
3	C₄×C₁₃×#107	C₂	1	3	yes	maximal
4	C₄×C₁₃×#103	C₂	1	6	yes	maximal
5	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	C₂²	1	6	yes	—
8	C₄×C₁₃×Q₁₆	C₄	1	24	yes	direct product factor
9	C₂×C₁₃×#107	C₂²	1	3	yes	—
10	C₂×C₁₃×#103	C₂²	1	6	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
13	C₄×C₁₃×C₄⋊C₄	—	2	6	no	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
15	C₄³×C₁₃	—	2	2	no	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#340	C₂²	1	6	yes	—
18	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
19	C₂×C₁₃×Q₁₆	C₂×C₄	1	12	yes	—
20	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₁₃	—	2	12	no	—
23	C₄×Q₈×C₁₃	C₂×C₄	1	24	yes	—
24	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
25	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
26	C₂×C₄²×C₁₃	—	2	6	no	—
27	C₂×C₄²×C₁₃	—	2	6	no	—
28	C₂×C₄²×C₁₃	D₄	1	3	yes	—
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
31	C₁₃×#107	C₂×C₄	1	12	yes	—
32	C₁₃×#103	C₂×C₄	1	24	yes	—
33	C₄×C₈×C₁₃	—	2	6	no	—
34	C₄×C₈×C₁₃	C₂×C₄	1	12	yes	—
35	C₁₃×#95	—	2	6	no	—
36	C₄²×Q₁₆	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×Q₈×C₁₃	—	2	6	no	—
38	C₂×Q₈×C₁₃	C₂²×C₄	1	12	yes	—
39	C₂²×C₄×C₁₃	—	2	2	no	—
40	C₂²×C₄×C₁₃	—	2	6	no	—
41	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
43	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
44	C₁₃×Q₁₆	C₄²	1	16	yes	direct product factor
45	C₂×C₈×C₁₃	—	2	6	no	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
48	C₁₃×C₄⋊C₄	—	2	24	no	—
49	C₁₃×C₄⋊C₄	—	2	6	no	—
50	C₁₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
51	C₄²×C₁₃	—	2	24	no	—
52	C₄²×C₁₃	—	2	6	no	—
53	C₄²×C₁₃	—	4	12	no	—
54	C₄²×C₁₃	Q₁₆	1	4	yes	direct product factor
55	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
56	C₄²×C₁₃	#41	1	6	yes	—
57	C₂×C₄×Q₁₆	C₂×C₁₃	1	3	yes	—
58	C₄²×Q₈	C₂×C₁₃	1	2	yes	—
59	C₄×#107	C₂×C₁₃	1	3	yes	—
60	C₄×#103	C₂×C₁₃	1	6	yes	—
61	C₄²×C₈	C₂×C₁₃	1	1	yes	characteristic
62	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
63	Q₈×C₁₃	—	2	12	no	—
64	Q₈×C₁₃	C₂×C₄²	1	8	yes	—
65	C₂×C₄×C₁₃	—	2	12	no	—
66	C₂×C₄×C₁₃	—	2	6	no	—
67	C₂×C₄×C₁₃	—	2	12	no	—
68	C₂×C₄×C₁₃	—	4	12	no	—
69	C₂×C₄×C₁₃	C₄×D₄	1	6	yes	—
70	C₂×C₄×C₁₃	C₄×D₄	1	6	yes	—
71	C₂×C₄×C₁₃	C₂×Q₁₆	1	6	yes	—
72	C₂×C₄×C₁₃	#135	1	6	yes	—
73	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
74	C₂×C₄×C₁₃	C₂×#41	1	3	yes	—
75	C₈×C₁₃	—	2	6	no	—
76	C₈×C₁₃	C₂×C₄²	1	4	yes	—
77	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
78	C₂×C₄×Q₈	C₂²×C₁₃	1	6	yes	—
79	C₄×Q₁₆	C₄×C₁₃	1	24	yes	direct product factor
80	C₂×#107	C₂²×C₁₃	1	3	yes	—
81	C₂×#103	C₂²×C₁₃	1	6	yes	—
82	C₂×C₄×C₈	C₂²×C₁₃	1	3	yes	—
83	C₂×C₄×C₈	C₂²×C₁₃	1	3	yes	—
84	C₄×C₄⋊C₄	—	2	6	no	—
85	C₄×C₄⋊C₄	C₂²×C₁₃	1	6	yes	—
86	C₄³	—	2	2	no	—
87	C₄³	C₂²×C₁₃	1	1	yes	characteristic
88	#340	C₂²×C₁₃	1	6	yes	—
89	C₂²×C₁₃	C₂×C₄×D₄	1	3	yes	—
90	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
91	C₂²×C₁₃	C₂×#135	1	3	yes	—
92	C₄×C₁₃	—	2	6	no	—
93	C₄×C₁₃	—	2	8	no	—
94	C₄×C₁₃	—	4	12	no	—
95	C₄×C₁₃	C₄×Q₁₆	1	12	yes	direct product factor
96	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
97	C₄×C₁₃	C₄×#41	1	3	yes	—
98	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
99	C₂×Q₁₆	C₂×C₄×C₁₃	1	12	yes	—
100	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
101	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
102	C₄×Q₈	—	2	12	no	—
103	C₄×Q₈	C₂×C₄×C₁₃	1	24	yes	—
104	C₂×C₄⋊C₄	—	2	6	no	—
105	C₂×C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
106	C₂×C₄²	—	2	6	no	—
107	C₂×C₄²	—	2	6	no	—
108	C₂×C₄²	D₄×C₁₃	1	3	yes	—
109	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
110	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
111	#107	C₂×C₄×C₁₃	1	12	yes	—
112	#103	C₂×C₄×C₁₃	1	24	yes	—
113	C₄×C₈	—	2	6	no	—
114	C₄×C₈	C₂×C₄×C₁₃	1	12	yes	—
115	#95	—	2	6	no	—
116	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
117	C₂×C₁₃	C₂×C₄×Q₁₆	1	3	yes	—
118	C₂×C₁₃	C₄×#135	1	3	yes	—
119	C₂×Q₈	—	2	6	no	—
120	C₂×Q₈	C₂²×C₄×C₁₃	1	12	yes	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	6	no	—
123	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
124	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
125	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
126	Q₁₆	C₄²×C₁₃	1	16	yes	direct product factor
127	C₂×C₈	—	2	6	no	—
128	C₂×C₈	C₂²×C₄×C₁₃	1	6	yes	—
129	C₂×C₈	C₂²×C₄×C₁₃	1	6	yes	—
130	C₄⋊C₄	—	2	24	no	—
131	C₄⋊C₄	—	2	6	no	—
132	C₄⋊C₄	C₂²×C₄×C₁₃	1	12	yes	—
133	C₄²	—	2	24	no	—
134	C₄²	—	2	6	no	—
135	C₄²	—	4	12	no	—
136	C₄²	C₁₃×Q₁₆	1	4	yes	direct product factor
137	C₄²	C₂²×C₄×C₁₃	1	6	yes	—
138	C₄²	C₁₃×#41	1	6	yes	—
139	C₁₃	C₄²×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
140	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
141	Q₈	—	2	12	no	—
142	Q₈	C₂×C₄²×C₁₃	1	8	yes	—
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	—	2	6	no	—
145	C₂×C₄	—	2	12	no	—
146	C₂×C₄	—	4	12	no	—
147	C₂×C₄	C₄×D₄×C₁₃	1	6	yes	—
148	C₂×C₄	C₄×D₄×C₁₃	1	6	yes	—
149	C₂×C₄	C₂×C₁₃×Q₁₆	1	6	yes	—
150	C₂×C₄	C₁₃×#135	1	6	yes	—
151	C₂×C₄	C₂³×C₄×C₁₃	1	3	yes	—
152	C₂×C₄	C₂×C₁₃×#41	1	3	yes	—
153	C₈	—	2	6	no	—
154	C₈	C₂×C₄²×C₁₃	1	4	yes	—
155	C₂²	C₂×C₄×D₄×C₁₃	1	3	yes	—
156	C₂²	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
157	C₂²	C₂×C₁₃×#135	1	3	yes	—
158	C₄	—	2	8	no	—
159	C₄	—	2	6	no	—
160	C₄	—	4	12	no	—
161	C₄	C₄×C₁₃×Q₁₆	1	12	yes	direct product factor
162	C₄	C₂²×C₄²×C₁₃	1	1	yes	characteristic
163	C₄	C₄×C₁₃×#41	1	3	yes	—
164	C₂	C₄²×D₄×C₁₃	1	1	yes	characteristic
165	C₂	C₂×C₄×C₁₃×Q₁₆	1	3	yes	—
166	C₂	C₄×C₁₃×#135	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₃	D₄
commutator	C₄	C₂²×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle, prime radical	C₂³×C₁₃	C₂²×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₄×C₁₃×Q₁₆	C₄	direct (N × H)
36. C₄²×Q₁₆	C₁₃	direct (N × H)
44. C₁₃×Q₁₆	C₄²	direct (N × H)
54. C₄²×C₁₃	Q₁₆	direct (N × H)
79. C₄×Q₁₆	C₄×C₁₃	direct (N × H)
95. C₄×C₁₃	C₄×Q₁₆	direct (N × H)
126. Q₁₆	C₄²×C₁₃	direct (N × H)
136. C₄²	C₁₃×Q₁₆	direct (N × H)
139. C₁₃	C₄²×Q₁₆	direct (N × H)
161. C₄	C₄×C₁₃×Q₁₆	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×C₄×C₁₃×Q₁₆	2
C₄²×Q₈×C₁₃	2
C₄×C₁₃×#107	2
C₄×C₁₃×#103	2
C₄²×C₈×C₁₃	2
C₄²×Q₁₆	13

Subgroups — C42×C13×Q16 #15706838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₁₃×Q₁₆ #15706838