Subgroups — C₄×C₄⋊Dic₂₁ #13056569

Order 1344.

388 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₁	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₂₁	C₂	1	2	yes	maximal
4	#10620670	C₂	1	2	yes	maximal
5	C₄×C₄⋊Dic₇	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₁	C₂²	1	4	yes	—
10	C₃×C₄²×C₇	C₄	1	4	yes	complemented
11	C₄⋊Dic₂₁	C₄	1	8	yes	direct product factor
12	C₄×Dic₂₁	—	2	4	no	—
13	C₂×C₄²×C₇	D₃	1	1	yes	characteristic
14	C₂×C₄⋊Dic₇	—	3	2	no	—
15	C₂×C₄×Dic₇	—	3	2	no	—
16	C₇⋊#95	—	3	2	no	—
17	C₄×C₄⋊Dic₃	—	7	1	no	maximal, Hall
18	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
19	C₂×C₃×C₄×C₇	—	2	2	no	—
20	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
25	C₂×C₃×C₄×C₇	Q₈	1	2	yes	—
26	C₂×Dic₂₁	—	2	4	no	—
27	C₂×Dic₂₁	—	2	4	no	—
28	C₂×Dic₂₁	C₂×C₄	1	8	yes	—
29	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
30	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
31	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
32	C₂²×Dic₇	—	3	4	no	—
33	C₄²×C₇	Dic₃	1	4	yes	complemented
34	C₄⋊Dic₇	—	3	8	no	—
35	C₄×Dic₇	—	6	4	no	—
36	C₂×C₃×C₄²	D₇	1	1	yes	characteristic
37	C₂×C₄⋊Dic₃	—	7	2	no	—
38	C₂×C₄×Dic₃	—	7	2	no	—
39	C₃⋊#95	—	7	2	no	—
40	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₃×C₇	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₃×C₇	#41	1	2	yes	—
45	C₃×C₄×C₇	—	2	2	no	—
46	C₃×C₄×C₇	C₄²	1	4	yes	complemented
47	C₃×C₄×C₇	C₄⋊C₄	1	4	yes	complemented
48	Dic₂₁	—	2	8	no	—
49	C₄×C₄⋊C₄	—	21	1	no	Sylow
50	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
51	C₂×C₄×C₇	—	2	2	no	—
52	C₂×C₄×C₇	Dic₆	1	2	yes	—
53	C₂×C₄×C₇	C₄×D₃	1	4	yes	—
54	C₂×C₄×C₇	D₁₂	1	2	yes	—
55	C₂×C₄×C₇	C₂×Dic₃	1	2	yes	—
56	C₂×C₄×C₇	C₂×Dic₃	1	2	yes	—
57	C₂×C₄×C₇	C₂×Dic₃	1	2	yes	—
58	C₂×Dic₇	—	3	8	no	—
59	C₂×Dic₇	—	6	4	no	—
60	C₂×Dic₇	—	6	4	no	—
61	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
64	C₂²×Dic₃	—	7	4	no	—
65	C₃×C₄²	Dic₇	1	4	yes	complemented
66	C₄⋊Dic₃	—	7	8	no	—
67	C₄×Dic₃	—	14	4	no	—
68	C₂×C₃×C₇	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₇	#117	1	1	yes	characteristic
71	C₂×C₃×C₇	C₄×D₄	1	2	yes	—
72	C₂×C₃×C₇	C₄×Q₈	1	2	yes	—
73	C₂×C₄⋊C₄	—	21	2	no	—
74	C₂×C₄²	—	21	2	no	—
75	C₂×C₄²	D₂₁	1	1	yes	characteristic
76	#95	—	21	2	no	—
77	C₂²×C₇	C₂×Dic₆	1	1	yes	characteristic
78	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
79	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
80	C₂²×C₇	#235	1	2	yes	—
81	C₂²×C₇	C₂²×Dic₃	1	1	yes	characteristic
82	C₄×C₇	—	2	2	no	—
83	C₄×C₇	C₄×Dic₃	1	4	yes	complemented
84	C₄×C₇	C₄⋊Dic₃	1	4	yes	complemented
85	Dic₇	—	6	8	no	—
86	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
87	C₂×C₃×C₄	—	2	2	no	—
88	C₂×C₃×C₄	Dic₁₄	1	2	yes	—
89	C₂×C₃×C₄	C₄×D₇	1	4	yes	—
90	C₂×C₃×C₄	D₂₈	1	2	yes	—
91	C₂×C₃×C₄	C₂×Dic₇	1	2	yes	—
92	C₂×C₃×C₄	C₂×Dic₇	1	2	yes	—
93	C₂×C₃×C₄	C₂×Dic₇	1	2	yes	—
94	C₂×Dic₃	—	7	8	no	—
95	C₂×Dic₃	—	14	4	no	—
96	C₂×Dic₃	—	14	4	no	—
97	C₃×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
98	C₂²×C₄	—	21	4	no	—
99	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
100	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
101	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
102	C₄⋊C₄	—	21	8	no	—
103	C₄²	—	42	4	no	—
104	C₄²	Dic₂₁	1	4	yes	complemented
105	C₂×C₇	C₄×Dic₆	1	2	yes	—
106	C₂×C₇	C₄×D₁₂	1	2	yes	—
107	C₂×C₇	C₂×C₄×Dic₃	1	1	yes	characteristic
108	C₂×C₇	C₂×C₄⋊Dic₃	1	1	yes	characteristic
109	C₂×C₇	#926	1	1	yes	characteristic
110	C₂²×C₃	C₂×Dic₁₄	1	1	yes	characteristic
111	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
112	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
113	C₂²×C₃	#1157	1	2	yes	—
114	C₂²×C₃	C₂²×Dic₇	1	1	yes	characteristic
115	C₃×C₄	—	2	2	no	—
116	C₃×C₄	C₄×Dic₇	1	4	yes	complemented
117	C₃×C₄	C₄⋊Dic₇	1	4	yes	complemented
118	Dic₃	—	14	8	no	—
119	C₂³	C₂²×D₂₁	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	21	8	no	—
122	C₂×C₄	—	42	4	no	—
123	C₂×C₄	—	42	4	no	—
124	C₂×C₄	Dic₄₂	1	2	yes	—
125	C₂×C₄	C₄×D₂₁	1	4	yes	—
126	C₂×C₄	D₈₄	1	2	yes	—
127	C₂×C₄	C₂×Dic₂₁	1	2	yes	—
128	C₂×C₄	C₂×Dic₂₁	1	2	yes	—
129	C₂×C₄	C₂×Dic₂₁	1	2	yes	—
130	C₇	C₄×C₄⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₃	C₄×Dic₁₄	1	2	yes	—
132	C₂×C₃	C₄×D₂₈	1	2	yes	—
133	C₂×C₃	C₂×C₄×Dic₇	1	1	yes	characteristic
134	C₂×C₃	C₂×C₄⋊Dic₇	1	1	yes	characteristic
135	C₂×C₃	#6489	1	1	yes	characteristic
136	C₂²	C₂×Dic₄₂	1	1	yes	characteristic
137	C₂²	C₂×C₄×D₂₁	1	2	yes	—
138	C₂²	C₂×D₈₄	1	1	yes	characteristic
139	C₂²	#66728	1	2	yes	—
140	C₂²	C₂²×Dic₂₁	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	42	8	no	—
143	C₄	C₄×Dic₂₁	1	4	yes	complemented
144	C₄	C₄⋊Dic₂₁	1	4	yes	direct product factor
145	C₃	C₄×C₄⋊Dic₇	1	1	yes	characteristic, complemented, Sylow
146	C₂	C₄×Dic₄₂	1	2	yes	—
147	C₂	C₄×D₈₄	1	2	yes	—
148	C₂	C₂×C₄×Dic₂₁	1	1	yes	characteristic
149	C₂	C₂×C₄⋊Dic₂₁	1	1	yes	characteristic
150	C₂	#10621409	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁
commutator	C₂×C₃×C₇	C₂×C₄²
Frattini	C₂³	C₂²×D₂₁
Fitting	C₂×C₃×C₄²×C₇	C₂
socle, prime radical, cosocle	C₂³×C₃×C₇	C₂³
hypercenter	C₂×C₄²	D₂₁
hypocenter	C₃×C₇	C₄×C₄⋊C₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
10. C₃×C₄²×C₇	C₄	semidirect
11. C₄⋊Dic₂₁	C₄	direct (N × H)
33. C₄²×C₇	Dic₃	semidirect
46. C₃×C₄×C₇	C₄²	semidirect
47. C₃×C₄×C₇	C₄⋊C₄	semidirect
65. C₃×C₄²	Dic₇	semidirect
83. C₄×C₇	C₄×Dic₃	semidirect
84. C₄×C₇	C₄⋊Dic₃	semidirect
97. C₃×C₇	C₄×C₄⋊C₄	semidirect
104. C₄²	Dic₂₁	semidirect
116. C₃×C₄	C₄×Dic₇	semidirect
117. C₃×C₄	C₄⋊Dic₇	semidirect
130. C₇	C₄×C₄⋊Dic₃	semidirect
143. C₄	C₄×Dic₂₁	semidirect
144. C₄	C₄⋊Dic₂₁	direct (N × H)
145. C₃	C₄×C₄⋊Dic₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₇	2
C₂×C₄⋊Dic₂₁	2
C₂×C₄×Dic₂₁	2
#10620670	2
C₄×C₄⋊Dic₇	3
C₄×C₄⋊Dic₃	7

Subgroups — C4×C4⋊Dic21 #13056569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄⋊Dic₂₁ #13056569