Finite Group Databasebeta

← C₂⁷×SD₁₆ #72314423

Subgroups — C₂⁷×SD₁₆ #72314423

Order 2048.

20276840 conjugacy classes of subgroups; only the 9059432 normal subgroups are shown.

normal subgroups only

This group has 20276840 conjugacy classes of subgroups — too many to draw. Only its 9059432 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁷×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁶×SD₁₆	C₂	1	508	yes	maximal, direct product factor
3	C₂⁷×D₄	C₂	1	1	yes	maximal, characteristic
4	C₂⁷×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁹	—	2	1	no	—
6	C₂⁶×Q₈	—	2	127	no	—
7	C₂⁶×Q₈	C₂²	1	254	yes	complemented
8	C₂⁶×D₄	—	2	127	no	—
9	C₂⁶×D₄	C₂²	1	254	yes	—
10	C₂⁷×C₄	—	2	1	no	—
11	C₂⁷×C₄	C₂²	1	1	yes	characteristic
12	C₂⁵×SD₁₆	C₂²	1	42672	yes	direct product factor
13	C₂⁶×C₈	C₂²	1	254	yes	complemented
14	C₂⁸	—	2	254	no	—
15	C₂⁸	—	4	128	no	—
16	C₂⁸	D₄	1	1	yes	characteristic
17	C₂⁵×Q₈	—	2	16002	no	—
18	C₂⁵×Q₈	C₂³	1	10668	yes	complemented
19	C₂⁵×D₄	—	2	16002	no	—
20	C₂⁵×D₄	C₂³	1	10668	yes	—
21	C₂⁶×C₄	—	2	254	no	—
22	C₂⁶×C₄	D₄	1	127	yes	—
23	C₂⁶×C₄	C₂³	1	127	yes	—
24	C₂⁴×SD₁₆	C₂³	1	755904	yes	direct product factor
25	C₂⁵×C₈	C₂³	1	10668	yes	complemented
26	C₂⁷	—	2	10668	no	—
27	C₂⁷	—	4	16256	no	—
28	C₂⁷	SD₁₆	1	128	yes	direct product factor
29	C₂⁷	C₂×D₄	1	127	yes	—
30	C₂⁴×Q₈	—	2	330708	no	—
31	C₂⁴×Q₈	C₂⁴	1	94488	yes	complemented
32	C₂⁴×D₄	—	2	330708	no	—
33	C₂⁴×D₄	C₂⁴	1	94488	yes	—
34	C₂⁵×C₄	—	2	10668	no	—
35	C₂⁵×C₄	C₂×D₄	1	8001	yes	—
36	C₂⁵×C₄	C₂⁴	1	2667	yes	—
37	C₂³×SD₁₆	C₂⁴	1	3023616	yes	direct product factor
38	C₂⁴×C₈	C₂⁴	1	94488	yes	complemented
39	C₂⁶	—	2	94488	no	—
40	C₂⁶	—	4	341376	no	—
41	C₂⁶	C₂×SD₁₆	1	8128	yes	direct product factor
42	C₂⁶	C₂²×D₄	1	2667	yes	—
43	C₂³×Q₈	—	2	1417320	no	—
44	C₂³×Q₈	C₂⁵	1	188976	yes	complemented
45	C₂³×D₄	—	2	1417320	no	—
46	C₂³×D₄	C₂⁵	1	188976	yes	—
47	C₂⁴×C₄	—	2	94488	no	—
48	C₂⁴×C₄	C₂²×D₄	1	82677	yes	—
49	C₂⁴×C₄	C₂⁵	1	11811	yes	—
50	C₂²×SD₁₆	C₂⁵	1	2731008	yes	direct product factor
51	C₂³×C₈	C₂⁵	1	188976	yes	complemented
52	C₂⁵	—	2	188976	no	—
53	C₂⁵	—	4	1511808	no	—
54	C₂⁵	C₂²×SD₁₆	1	85344	yes	direct product factor
55	C₂⁵	C₂³×D₄	1	11811	yes	—
56	C₂²×Q₈	—	2	1322832	no	—
57	C₂²×Q₈	C₂⁶	1	85344	yes	complemented
58	C₂²×D₄	—	2	1322832	no	—
59	C₂²×D₄	C₂⁶	1	85344	yes	—
60	C₂³×C₄	—	2	188976	no	—
61	C₂³×C₄	C₂³×D₄	1	177165	yes	—
62	C₂³×C₄	C₂⁶	1	11811	yes	—
63	C₂×SD₁₆	C₂⁶	1	520192	yes	direct product factor
64	C₂²×C₈	C₂⁶	1	85344	yes	complemented
65	C₂⁴	—	2	85344	no	—
66	C₂⁴	—	4	1511808	no	—
67	C₂⁴	C₂³×SD₁₆	1	188976	yes	direct product factor
68	C₂⁴	C₂⁴×D₄	1	11811	yes	—
69	C₂×Q₈	—	2	256032	no	—
70	C₂×Q₈	C₂⁷	1	8128	yes	complemented
71	C₂×D₄	—	2	256032	no	—
72	C₂×D₄	C₂⁷	1	8128	yes	—
73	C₂²×C₄	—	2	85344	no	—
74	C₂²×C₄	C₂⁴×D₄	1	82677	yes	—
75	C₂²×C₄	C₂⁷	1	2667	yes	—
76	SD₁₆	C₂⁷	1	16384	yes	direct product factor
77	C₂×C₈	C₂⁷	1	8128	yes	complemented
78	C₂³	—	2	8128	no	—
79	C₂³	—	4	341376	no	—
80	C₂³	C₂⁴×SD₁₆	1	94488	yes	direct product factor
81	C₂³	C₂⁵×D₄	1	2667	yes	—
82	Q₈	—	2	8128	no	—
83	Q₈	C₂⁸	1	128	yes	complemented
84	D₄	—	2	8128	no	—
85	D₄	C₂⁸	1	128	yes	—
86	C₂×C₄	—	2	8128	no	—
87	C₂×C₄	C₂⁵×D₄	1	8001	yes	—
88	C₂×C₄	C₂⁸	1	127	yes	—
89	C₈	C₂⁸	1	128	yes	complemented
90	C₂²	—	2	128	no	—
91	C₂²	—	4	16256	no	—
92	C₂²	C₂⁵×SD₁₆	1	10668	yes	direct product factor
93	C₂²	C₂⁶×D₄	1	127	yes	—
94	C₄	—	2	128	no	—
95	C₄	C₂⁶×D₄	1	127	yes	—
96	C₄	C₂⁹	1	1	yes	characteristic
97	C₂	—	4	128	no	—
98	C₂	C₂⁷×D₄	1	1	yes	characteristic
99	C₂	C₂⁶×SD₁₆	1	254	yes	direct product factor

	subgroup	G/H
center, socle	C₂⁸	D₄
commutator, Frattini, cosocle	C₄	C₂⁹
prime radical	C₂⁷×D₄	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁸ ◁ C₂⁷×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂⁷×Q₈	C₂	semidirect
2. C₂⁶×SD₁₆	C₂	direct (N × H)
4. C₂⁷×C₈	C₂	semidirect
7. C₂⁶×Q₈	C₂²	semidirect
12. C₂⁵×SD₁₆	C₂²	direct (N × H)
13. C₂⁶×C₈	C₂²	semidirect
18. C₂⁵×Q₈	C₂³	semidirect
24. C₂⁴×SD₁₆	C₂³	direct (N × H)
25. C₂⁵×C₈	C₂³	semidirect
28. C₂⁷	SD₁₆	direct (N × H)
31. C₂⁴×Q₈	C₂⁴	semidirect
37. C₂³×SD₁₆	C₂⁴	direct (N × H)
38. C₂⁴×C₈	C₂⁴	semidirect
41. C₂⁶	C₂×SD₁₆	direct (N × H)
44. C₂³×Q₈	C₂⁵	semidirect
50. C₂²×SD₁₆	C₂⁵	direct (N × H)
51. C₂³×C₈	C₂⁵	semidirect
54. C₂⁵	C₂²×SD₁₆	direct (N × H)
57. C₂²×Q₈	C₂⁶	semidirect
63. C₂×SD₁₆	C₂⁶	direct (N × H)
64. C₂²×C₈	C₂⁶	semidirect
67. C₂⁴	C₂³×SD₁₆	direct (N × H)
70. C₂×Q₈	C₂⁷	semidirect
76. SD₁₆	C₂⁷	direct (N × H)
77. C₂×C₈	C₂⁷	semidirect
80. C₂³	C₂⁴×SD₁₆	direct (N × H)
83. Q₈	C₂⁸	semidirect
89. C₈	C₂⁸	semidirect
92. C₂²	C₂⁵×SD₁₆	direct (N × H)
99. C₂	C₂⁶×SD₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁷×Q₈	2
C₂⁶×SD₁₆	2
C₂⁷×D₄	2
C₂⁷×C₈	2