Subgroups — C₂⁷×#106 #72314601

Order 4096.

46838779 conjugacy classes of subgroups.

The subgroup lattice has 125 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁷×C₄⋊C₄	C₂	1	2	yes	maximal
2	C₂⁶×#106	C₂	1	508	yes	maximal, direct product factor
3	C₂⁵×#106	C₂²	1	42672	yes	direct product factor
4	C₂⁶×C₄⋊C₄	—	2	254	no	—
5	C₂⁶×C₄⋊C₄	C₂²	1	508	yes	—
6	C₂⁸×C₈	C₂	1	1	yes	maximal, characteristic
7	C₂⁸×C₄	—	2	2	no	—
8	C₂⁸×C₄	C₂²	1	1	yes	characteristic
9	C₂⁷×C₈	C₄	1	256	yes	complemented
10	C₂⁷×C₈	C₂²	1	254	yes	—
11	C₂⁹	D₄	1	1	yes	characteristic
12	C₂⁷×C₄	—	2	508	no	—
13	C₂⁷×C₄	—	4	256	no	—
14	C₂⁷×C₄	C₂×C₄	1	128	yes	—
15	C₂⁷×C₄	D₄	1	127	yes	—
16	C₂⁷×C₄	Q₈	1	128	yes	—
17	C₂⁷×C₄	C₂³	1	127	yes	—
18	C₂⁶×C₈	C₂×C₄	1	32512	yes	complemented
19	C₂⁶×C₈	C₂³	1	10668	yes	—
20	C₂⁵×C₄⋊C₄	—	2	32004	no	—
21	C₂⁵×C₄⋊C₄	C₂³	1	21336	yes	—
22	C₂⁴×#106	C₂³	1	755904	yes	direct product factor
23	C₂⁸	C₄⋊C₄	1	128	yes	—
24	C₂⁸	SD₁₆	1	256	yes	—
25	C₂⁸	C₂×D₄	1	127	yes	—
26	C₂⁶×C₄	—	2	21336	no	—
27	C₂⁶×C₄	—	4	32512	no	—
28	C₂⁶×C₄	C₄⋊C₄	1	16256	yes	—
29	C₂⁶×C₄	C₂²×C₄	1	8128	yes	—
30	C₂⁶×C₄	C₂×D₄	1	8001	yes	—
31	C₂⁶×C₄	C₂×Q₈	1	8128	yes	—
32	C₂⁶×C₄	C₂⁴	1	2667	yes	—
33	C₂⁵×C₈	C₂²×C₄	1	682752	yes	complemented
34	C₂⁵×C₈	C₂⁴	1	94488	yes	—
35	C₂⁴×C₄⋊C₄	—	2	661416	no	—
36	C₂⁴×C₄⋊C₄	C₂⁴	1	188976	yes	—
37	C₂³×#106	C₂⁴	1	3023616	yes	direct product factor
38	C₂⁷	#106	1	16384	yes	direct product factor
39	C₂⁷	C₂×C₄⋊C₄	1	8128	yes	—
40	C₂⁷	C₂×SD₁₆	1	16256	yes	—
41	C₂⁷	C₂²×D₄	1	2667	yes	—
42	C₂⁵×C₄	—	2	188976	no	—
43	C₂⁵×C₄	—	4	682752	no	—
44	C₂⁵×C₄	C₂×C₄⋊C₄	1	512064	yes	—
45	C₂⁵×C₄	C₂³×C₄	1	85344	yes	—
46	C₂⁵×C₄	C₂²×D₄	1	82677	yes	—
47	C₂⁵×C₄	C₂²×Q₈	1	85344	yes	—
48	C₂⁵×C₄	C₂⁵	1	11811	yes	—
49	C₂⁴×C₈	C₂³×C₄	1	3023616	yes	complemented
50	C₂⁴×C₈	C₂⁵	1	188976	yes	—
51	C₂³×C₄⋊C₄	—	2	2834640	no	—
52	C₂³×C₄⋊C₄	C₂⁵	1	377952	yes	—
53	C₂²×#106	C₂⁵	1	2731008	yes	direct product factor
54	C₂⁶	C₂×#106	1	520192	yes	direct product factor
55	C₂⁶	C₂²×C₄⋊C₄	1	85344	yes	—
56	C₂⁶	C₂²×SD₁₆	1	170688	yes	—
57	C₂⁶	C₂³×D₄	1	11811	yes	—
58	C₂⁴×C₄	—	2	377952	no	—
59	C₂⁴×C₄	—	4	3023616	no	—
60	C₂⁴×C₄	C₂²×C₄⋊C₄	1	2645664	yes	—
61	C₂⁴×C₄	C₂⁴×C₄	1	188976	yes	—
62	C₂⁴×C₄	C₂³×D₄	1	177165	yes	—
63	C₂⁴×C₄	C₂³×Q₈	1	188976	yes	—
64	C₂⁴×C₄	C₂⁶	1	11811	yes	—
65	C₂³×C₈	C₂⁴×C₄	1	3023616	yes	complemented
66	C₂³×C₈	C₂⁶	1	85344	yes	—
67	C₂²×C₄⋊C₄	—	2	2645664	no	—
68	C₂²×C₄⋊C₄	C₂⁶	1	170688	yes	—
69	C₂×#106	C₂⁶	1	520192	yes	direct product factor
70	C₂⁵	C₂²×#106	1	2731008	yes	direct product factor
71	C₂⁵	C₂³×C₄⋊C₄	1	188976	yes	—
72	C₂⁵	C₂³×SD₁₆	1	377952	yes	—
73	C₂⁵	C₂⁴×D₄	1	11811	yes	—
74	C₂³×C₄	—	2	170688	no	—
75	C₂³×C₄	—	4	3023616	no	—
76	C₂³×C₄	C₂³×C₄⋊C₄	1	2834640	yes	—
77	C₂³×C₄	C₂⁵×C₄	1	94488	yes	—
78	C₂³×C₄	C₂⁴×D₄	1	82677	yes	—
79	C₂³×C₄	C₂⁴×Q₈	1	94488	yes	—
80	C₂³×C₄	C₂⁷	1	2667	yes	—
81	C₂²×C₈	C₂⁵×C₄	1	682752	yes	complemented
82	C₂²×C₈	C₂⁷	1	8128	yes	—
83	C₂×C₄⋊C₄	—	2	512064	no	—
84	C₂×C₄⋊C₄	C₂⁷	1	16256	yes	—
85	#106	C₂⁷	1	16384	yes	direct product factor
86	C₂⁴	C₂³×#106	1	3023616	yes	direct product factor
87	C₂⁴	C₂⁴×C₄⋊C₄	1	94488	yes	—
88	C₂⁴	C₂⁴×SD₁₆	1	188976	yes	—
89	C₂⁴	C₂⁵×D₄	1	2667	yes	—
90	C₂²×C₄	—	2	16256	no	—
91	C₂²×C₄	—	4	682752	no	—
92	C₂²×C₄	C₂⁴×C₄⋊C₄	1	661416	yes	—
93	C₂²×C₄	C₂⁶×C₄	1	10668	yes	—
94	C₂²×C₄	C₂⁵×D₄	1	8001	yes	—
95	C₂²×C₄	C₂⁵×Q₈	1	10668	yes	—
96	C₂²×C₄	C₂⁸	1	127	yes	—
97	C₂×C₈	C₂⁶×C₄	1	32512	yes	complemented
98	C₂×C₈	C₂⁸	1	128	yes	—
99	C₄⋊C₄	—	2	16256	no	—
100	C₄⋊C₄	C₂⁸	1	256	yes	—
101	C₂³	C₂⁴×#106	1	755904	yes	direct product factor
102	C₂³	C₂⁵×C₄⋊C₄	1	10668	yes	—
103	C₂³	C₂⁵×SD₁₆	1	21336	yes	—
104	C₂³	C₂⁶×D₄	1	127	yes	—
105	C₂×C₄	—	2	256	no	—
106	C₂×C₄	—	4	32512	no	—
107	C₂×C₄	C₂⁵×C₄⋊C₄	1	32004	yes	—
108	C₂×C₄	C₂⁷×C₄	1	254	yes	—
109	C₂×C₄	C₂⁶×D₄	1	127	yes	—
110	C₂×C₄	C₂⁶×Q₈	1	254	yes	—
111	C₂×C₄	C₂⁹	1	1	yes	characteristic
112	C₈	C₂⁷×C₄	1	256	yes	complemented
113	C₂²	C₂⁷×D₄	1	1	yes	characteristic
114	C₂²	C₂⁶×C₄⋊C₄	1	254	yes	—
115	C₂²	C₂⁵×#106	1	42672	yes	direct product factor
116	C₂²	C₂⁶×SD₁₆	1	508	yes	—
117	C₄	—	4	256	no	—
118	C₄	C₂⁸×C₄	1	1	yes	characteristic
119	C₄	C₂⁶×C₄⋊C₄	1	254	yes	—
120	C₄	C₂⁷×Q₈	1	1	yes	characteristic
121	C₂	C₂⁶×#106	1	508	yes	direct product factor
122	C₂	C₂⁷×SD₁₆	1	2	yes	—
123	C₂	C₂⁷×C₄⋊C₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂⁶×#106	C₂	direct (N × H)
3. C₂⁵×#106	C₂²	direct (N × H)
9. C₂⁷×C₈	C₄	semidirect
18. C₂⁶×C₈	C₂×C₄	semidirect
22. C₂⁴×#106	C₂³	direct (N × H)
33. C₂⁵×C₈	C₂²×C₄	semidirect
37. C₂³×#106	C₂⁴	direct (N × H)
38. C₂⁷	#106	direct (N × H)
49. C₂⁴×C₈	C₂³×C₄	semidirect
53. C₂²×#106	C₂⁵	direct (N × H)
54. C₂⁶	C₂×#106	direct (N × H)
65. C₂³×C₈	C₂⁴×C₄	semidirect
69. C₂×#106	C₂⁶	direct (N × H)
70. C₂⁵	C₂²×#106	direct (N × H)
81. C₂²×C₈	C₂⁵×C₄	semidirect
85. #106	C₂⁷	direct (N × H)
86. C₂⁴	C₂³×#106	direct (N × H)
97. C₂×C₈	C₂⁶×C₄	semidirect
101. C₂³	C₂⁴×#106	direct (N × H)
112. C₈	C₂⁷×C₄	semidirect
115. C₂²	C₂⁵×#106	direct (N × H)
121. C₂	C₂⁶×#106	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×C₄⋊C₄	2
C₂⁶×#106	2
C₂⁸×C₈	2

Subgroups — C27×#106 #72314601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁷×#106 #72314601