Subgroups — C₂×C₁₂₇×#1368 #44575671

Order 32512.

254 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#334×C₁₂₇	C₂	1	2	yes	maximal
3	C₁₂₇×#1368	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₈×C₁₂₇	—	2	2	no	—
7	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	#346×C₁₂₇	C₄	1	2	yes	—
9	#346×C₁₂₇	C₂²	1	2	yes	—
10	#334×C₁₂₇	—	2	2	no	—
11	#334×C₁₂₇	C₂²	1	4	yes	—
12	C₂²×C₈×C₁₂₇	—	4	2	no	—
13	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
15	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₁₂₇	—	2	2	no	—
17	C₂×C₁₆×C₁₂₇	C₈	1	4	yes	complemented
18	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₄⋊C₈×C₁₂₇	—	2	4	no	—
21	C₄⋊C₈×C₁₂₇	—	4	2	no	—
22	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	SD₁₆	1	2	yes	—
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	—	4	4	no	—
30	C₂×C₈×C₁₂₇	—	8	4	no	—
31	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
35	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₁₂₇	SD₁₆	1	2	yes	—
39	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₁₂₇	—	2	2	no	—
41	C₁₆×C₁₂₇	C₂×C₈	1	4	yes	complemented
42	C₂³×C₁₂₇	#106	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	—	2	4	no	—
44	C₂×C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
45	C₂×C₄×C₁₂₇	#106	1	2	yes	—
46	C₂×C₄×C₁₂₇	#106	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	#108	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
50	C₈×C₁₂₇	—	2	1	no	—
51	C₈×C₁₂₇	—	2	1	no	—
52	C₈×C₁₂₇	—	8	4	no	—
53	C₈×C₁₂₇	C₄⋊C₈	1	2	yes	—
54	C₈×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
56	C₂²×C₁₂₇	#334	1	2	yes	—
57	C₂²×C₁₂₇	#365	1	4	yes	—
58	C₂²×C₁₂₇	C₂×#106	1	1	yes	characteristic
59	C₄×C₁₂₇	—	2	2	no	—
60	C₄×C₁₂₇	—	2	2	no	—
61	C₄×C₁₂₇	#334	1	2	yes	—
62	C₄×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₄×C₁₂₇	C₂×#108	1	1	yes	characteristic
64	C₂×#1368	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₁₂₇	#1368	1	4	yes	direct product factor
66	C₂×C₁₂₇	C₂×#334	1	1	yes	characteristic
67	C₂×C₁₂₇	C₂×#365	1	2	yes	—
68	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic
69	C₂×#334	C₂×C₁₂₇	1	2	yes	—
70	#1368	C₂×C₁₂₇	1	4	yes	direct product factor
71	C₁₂₇	C₂×#1368	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
73	C₂²×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
74	C₂×C₄⋊C₈	—	2	2	no	—
75	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
76	#346	C₄×C₁₂₇	1	2	yes	—
77	#346	C₂²×C₁₂₇	1	2	yes	—
78	#334	—	2	2	no	—
79	#334	C₂²×C₁₂₇	1	4	yes	—
80	C₂²×C₈	—	4	2	no	—
81	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
82	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
83	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	C₈×C₁₂₇	1	4	yes	complemented
86	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
87	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
88	C₄⋊C₈	—	2	4	no	—
89	C₄⋊C₈	—	4	2	no	—
90	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
91	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
92	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
93	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
94	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
95	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	4	no	—
98	C₂×C₈	—	8	4	no	—
99	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
100	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
101	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
102	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
103	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
104	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
105	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
106	C₄²	SD₁₆×C₁₂₇	1	2	yes	—
107	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
108	C₁₆	—	2	2	no	—
109	C₁₆	C₂×C₈×C₁₂₇	1	4	yes	complemented
110	C₂³	#106×C₁₂₇	1	1	yes	characteristic
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	C₄⋊C₈×C₁₂₇	1	2	yes	—
113	C₂×C₄	#106×C₁₂₇	1	2	yes	—
114	C₂×C₄	#106×C₁₂₇	1	1	yes	characteristic
115	C₂×C₄	#108×C₁₂₇	1	2	yes	—
116	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
117	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
118	C₈	—	2	1	no	—
119	C₈	—	2	1	no	—
120	C₈	—	8	4	no	—
121	C₈	C₄⋊C₈×C₁₂₇	1	2	yes	—
122	C₈	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
123	C₈	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
124	C₂²	#334×C₁₂₇	1	2	yes	—
125	C₂²	#365×C₁₂₇	1	4	yes	—
126	C₂²	C₂×#106×C₁₂₇	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	#334×C₁₂₇	1	2	yes	—
130	C₄	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
131	C₄	C₂×#108×C₁₂₇	1	1	yes	characteristic
132	C₂	C₁₂₇×#1368	1	4	yes	direct product factor
133	C₂	C₂×#334×C₁₂₇	1	1	yes	characteristic
134	C₂	C₂×#365×C₁₂₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	#106
commutator	C₈	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1368	C₂	direct (N × H)
17. C₂×C₁₆×C₁₂₇	C₈	semidirect
41. C₁₆×C₁₂₇	C₂×C₈	semidirect
64. C₂×#1368	C₁₂₇	direct (N × H)
65. C₂×C₁₂₇	#1368	direct (N × H)
70. #1368	C₂×C₁₂₇	direct (N × H)
71. C₁₂₇	C₂×#1368	direct (N × H)
85. C₂×C₁₆	C₈×C₁₂₇	semidirect
109. C₁₆	C₂×C₈×C₁₂₇	semidirect
132. C₂	C₁₂₇×#1368	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×#346×C₁₂₇	2
C₂×#334×C₁₂₇	2
C₁₂₇×#1368	2
C₂×#1368	127

Subgroups — C2×C127×#1368 #44575671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1368 #44575671