Subgroups — C₂²×C₁₆×Dic₇ #13506418

Order 1792.

1022 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆×Dic₇	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄×Dic₇	C₄	1	1	yes	characteristic
7	C₂³×C₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂²×C₇×C₁₆	C₄	1	8	yes	complemented
9	C₂²×C₇×C₁₆	C₂²	1	6	yes	—
10	C₂×C₈×Dic₇	C₄	1	6	yes	—
11	C₂×C₈×Dic₇	C₂²	1	6	yes	—
12	C₂²×C₇⋊C₁₆	C₄	1	8	yes	complemented
13	C₂²×C₇⋊C₁₆	C₂²	1	6	yes	—
14	C₁₆×Dic₇	C₂²	1	16	yes	direct product factor
15	C₂²×C₄×C₁₆	—	7	1	no	maximal, Sylow
16	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂³×Dic₇	C₈	1	2	yes	—
18	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
19	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₇×C₈	C₂³	1	3	yes	—
22	C₂×C₄×Dic₇	C₈	1	6	yes	—
23	C₂×C₄×Dic₇	C₂×C₄	1	6	yes	—
24	C₂²×C₇⋊C₈	C₈	1	8	yes	—
25	C₂²×C₇⋊C₈	C₂×C₄	1	6	yes	—
26	C₂×C₇×C₁₆	C₂×C₄	1	24	yes	complemented
27	C₂×C₇×C₁₆	C₂³	1	4	yes	—
28	C₈×Dic₇	C₂×C₄	1	12	yes	—
29	C₈×Dic₇	C₂³	1	4	yes	—
30	C₂×C₇⋊C₁₆	C₂×C₄	1	24	yes	complemented
31	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
32	C₂³×C₁₆	—	7	1	no	—
33	C₂³×C₁₆	D₇	1	1	yes	characteristic
34	C₂²×C₄×C₈	—	7	1	no	—
35	C₂×C₄×C₁₆	—	7	12	no	—
36	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₇	C₄²	1	4	yes	—
38	C₂²×C₄×C₇	C₂×C₈	1	3	yes	—
39	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
40	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
41	C₂²×Dic₇	C₁₆	1	16	yes	direct product factor
42	C₂²×Dic₇	C₂×C₈	1	12	yes	—
43	C₂×C₇×C₈	C₄²	1	12	yes	—
44	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
45	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
46	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
47	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
48	C₄×Dic₇	C₂×C₈	1	12	yes	—
49	C₄×Dic₇	C₂²×C₄	1	4	yes	—
50	C₂×C₇⋊C₈	C₂×C₈	1	24	yes	—
51	C₂×C₇⋊C₈	C₂²×C₄	1	4	yes	—
52	C₇×C₁₆	C₂²×C₄	1	8	yes	complemented
53	C₇⋊C₁₆	C₂²×C₄	1	8	yes	complemented
54	C₂³×C₈	—	7	1	no	—
55	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
56	C₂²×C₄²	—	7	1	no	—
57	C₂²×C₁₆	—	7	8	no	—
58	C₂²×C₁₆	—	7	6	no	—
59	C₂²×C₁₆	Dic₇	1	8	yes	direct product factor
60	C₂²×C₁₆	C₂×D₇	1	6	yes	—
61	C₂×C₄×C₈	—	7	6	no	—
62	C₂×C₄×C₈	—	7	6	no	—
63	C₄×C₁₆	—	7	16	no	—
64	C₂³×C₇	C₄×C₈	1	4	yes	—
65	C₂³×C₇	C₂×C₁₆	1	8	yes	—
66	C₂³×C₇	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₇	C₄×C₈	1	12	yes	—
68	C₂×C₄×C₇	C₂×C₄²	1	6	yes	—
69	C₂×C₄×C₇	C₂²×C₈	1	6	yes	—
70	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×Dic₇	C₂×C₁₆	1	48	yes	direct product factor
73	C₂×Dic₇	C₂²×C₈	1	8	yes	—
74	C₇×C₈	C₂×C₄²	1	6	yes	—
75	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
76	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
77	C₇⋊C₈	C₂²×C₈	1	8	yes	—
78	C₂³×C₄	—	7	2	no	—
79	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
80	C₂²×C₈	—	7	8	no	—
81	C₂²×C₈	—	7	6	no	—
82	C₂²×C₈	C₄×D₇	1	3	yes	—
83	C₂²×C₈	C₄×D₇	1	4	yes	—
84	C₂²×C₈	C₂×Dic₇	1	4	yes	—
85	C₂²×C₈	C₂²×D₇	1	3	yes	—
86	C₂×C₄²	—	7	6	no	—
87	C₂×C₄²	—	7	6	no	—
88	C₂×C₁₆	—	7	24	no	—
89	C₂×C₁₆	—	7	4	no	—
90	C₂×C₁₆	C₂×Dic₇	1	24	yes	direct product factor
91	C₂×C₁₆	C₂²×D₇	1	4	yes	—
92	C₄×C₈	—	7	12	no	—
93	C₄×C₈	—	7	4	no	—
94	C₂²×C₇	C₄×C₁₆	1	16	yes	complemented
95	C₂²×C₇	C₂×C₄×C₈	1	6	yes	—
96	C₂²×C₇	C₂²×C₁₆	1	12	yes	—
97	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
98	C₄×C₇	C₂×C₄×C₈	1	6	yes	—
99	C₄×C₇	C₂²×C₄²	1	1	yes	characteristic
100	C₄×C₇	C₂³×C₈	1	1	yes	characteristic
101	Dic₇	C₂²×C₁₆	1	16	yes	direct product factor
102	C₂⁴	C₈×D₇	1	1	yes	characteristic
103	C₂²×C₄	—	7	12	no	—
104	C₂²×C₄	—	7	16	no	—
105	C₂²×C₄	C₈×D₇	1	4	yes	—
106	C₂²×C₄	C₈×D₇	1	3	yes	—
107	C₂²×C₄	C₄×Dic₇	1	4	yes	—
108	C₂²×C₄	C₂×C₄×D₇	1	3	yes	—
109	C₂×C₈	—	7	24	no	—
110	C₂×C₈	—	7	4	no	—
111	C₂×C₈	C₄×Dic₇	1	12	yes	—
112	C₂×C₈	C₂×C₄×D₇	1	3	yes	—
113	C₂×C₈	C₂×C₄×D₇	1	6	yes	—
114	C₂×C₈	C₂²×Dic₇	1	6	yes	—
115	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
116	C₄²	—	7	12	no	—
117	C₄²	—	7	4	no	—
118	C₁₆	—	7	8	no	—
119	C₁₆	C₂²×Dic₇	1	8	yes	direct product factor
120	C₂×C₇	C₂×C₄×C₁₆	1	12	yes	complemented
121	C₂×C₇	C₂²×C₄×C₈	1	1	yes	characteristic
122	C₂×C₇	C₂³×C₁₆	1	2	yes	—
123	C₂³	D₇×C₁₆	1	8	yes	—
124	C₂³	C₈×Dic₇	1	4	yes	—
125	C₂³	C₂×C₈×D₇	1	3	yes	—
126	C₂×C₄	—	7	8	no	—
127	C₂×C₄	—	7	48	no	—
128	C₂×C₄	C₈×Dic₇	1	12	yes	—
129	C₂×C₄	C₂×C₈×D₇	1	3	yes	—
130	C₂×C₄	C₂×C₈×D₇	1	6	yes	—
131	C₂×C₄	C₂×C₄×Dic₇	1	6	yes	—
132	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
133	C₈	—	7	8	no	—
134	C₈	C₂×C₄×Dic₇	1	6	yes	—
135	C₈	C₂²×C₄×D₇	1	1	yes	characteristic
136	C₈	C₂³×Dic₇	1	1	yes	characteristic
137	C₇	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
138	C₂²	C₁₆×Dic₇	1	16	yes	direct product factor
139	C₂²	C₂×D₇×C₁₆	1	12	yes	—
140	C₂²	C₂×C₈×Dic₇	1	6	yes	—
141	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
142	C₄	—	7	16	no	—
143	C₄	C₂×C₈×Dic₇	1	6	yes	—
144	C₄	C₂²×C₈×D₇	1	1	yes	characteristic
145	C₄	C₂²×C₄×Dic₇	1	1	yes	characteristic
146	C₂	C₂×C₁₆×Dic₇	1	12	yes	direct product factor
147	C₂	C₂²×D₇×C₁₆	1	2	yes	—
148	C₂	C₂²×C₈×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₁₆	D₇
commutator, hypocenter	C₇	C₂²×C₄×C₁₆
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂³×C₇×C₁₆	C₂
socle, prime radical	C₂⁴×C₇	C₂×C₈
cosocle	C₂×C₇×C₈	C₂⁴
LCS	G ▷ C₇
UCS	C₁ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
4. C₂×C₁₆×Dic₇	C₂	direct (N × H)
8. C₂²×C₇×C₁₆	C₄	semidirect
12. C₂²×C₇⋊C₁₆	C₄	semidirect
14. C₁₆×Dic₇	C₂²	direct (N × H)
26. C₂×C₇×C₁₆	C₂×C₄	semidirect
30. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
41. C₂²×Dic₇	C₁₆	direct (N × H)
52. C₇×C₁₆	C₂²×C₄	semidirect
53. C₇⋊C₁₆	C₂²×C₄	semidirect
59. C₂²×C₁₆	Dic₇	direct (N × H)
72. C₂×Dic₇	C₂×C₁₆	direct (N × H)
90. C₂×C₁₆	C₂×Dic₇	direct (N × H)
94. C₂²×C₇	C₄×C₁₆	semidirect
101. Dic₇	C₂²×C₁₆	direct (N × H)
119. C₁₆	C₂²×Dic₇	direct (N × H)
120. C₂×C₇	C₂×C₄×C₁₆	semidirect
137. C₇	C₂²×C₄×C₁₆	semidirect
138. C₂²	C₁₆×Dic₇	direct (N × H)
146. C₂	C₂×C₁₆×Dic₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×C₁₆	2
C₂²×C₈×Dic₇	2
C₂³×C₇⋊C₁₆	2
C₂×C₁₆×Dic₇	2
C₂²×C₄×C₁₆	7

Subgroups — C22×C16×Dic7 #13506418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₆×Dic₇ #13506418