Subgroups — C₁₂₇×#11999 #44575672

Order 32512.

234 conjugacy classes of subgroups.

The subgroup lattice has 134 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	C₁₂₇×#1564	C₂	1	2	yes	maximal
3	C₁₂₇×#1368	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
6	#431×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	#335×C₁₂₇	—	2	2	no	—
11	#334×C₁₂₇	C₂²	1	4	yes	—
12	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₁₂₇	—	2	2	no	—
16	C₂×C₁₆×C₁₂₇	C₈	1	4	yes	complemented
17	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₄⋊C₈×C₁₂₇	—	2	4	no	—
20	C₂²⋊C₈×C₁₂₇	—	4	2	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	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
34	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₁₂₇	SD₁₆	1	2	yes	—
38	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₁₆×C₁₂₇	—	2	2	no	—
40	C₁₆×C₁₂₇	C₂×C₈	1	4	yes	—
41	C₂³×C₁₂₇	#106	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	—	4	2	no	—
43	C₂×C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₁₂₇	#106	1	2	yes	—
45	C₂×C₄×C₁₂₇	#106	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	#108	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
49	C₈×C₁₂₇	—	2	1	no	—
50	C₈×C₁₂₇	—	2	1	no	—
51	C₈×C₁₂₇	—	8	4	no	—
52	C₈×C₁₂₇	C₄⋊C₈	1	2	yes	—
53	C₈×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
55	C₂²×C₁₂₇	—	2	2	no	—
56	C₂²×C₁₂₇	#334	1	2	yes	—
57	C₂²×C₁₂₇	C₂×#106	1	1	yes	characteristic
58	C₄×C₁₂₇	—	2	2	no	—
59	C₄×C₁₂₇	—	2	2	no	—
60	C₄×C₁₂₇	#334	1	2	yes	—
61	C₄×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
62	C₄×C₁₂₇	C₂×#108	1	1	yes	characteristic
63	#11999	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₁₂₇	—	2	2	no	—
65	C₂×C₁₂₇	C₂×#334	1	1	yes	characteristic
66	C₂×C₁₂₇	#2155	1	2	yes	—
67	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic
68	#1564	C₂×C₁₂₇	1	2	yes	—
69	#1368	C₂×C₁₂₇	1	4	yes	complemented
70	C₁₂₇	#11999	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
72	C₂²×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
73	#431	—	2	2	no	—
74	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
75	#346	C₄×C₁₂₇	1	2	yes	—
76	#346	C₂²×C₁₂₇	1	2	yes	—
77	#335	—	2	2	no	—
78	#334	C₂²×C₁₂₇	1	4	yes	—
79	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
80	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
81	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₈×C₁₂₇	1	4	yes	complemented
84	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
85	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
86	C₄⋊C₈	—	2	4	no	—
87	C₂²⋊C₈	—	4	2	no	—
88	C₄×C₈	—	4	2	no	—
89	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
90	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
91	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
92	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
93	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
94	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	4	no	—
97	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
98	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
99	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
100	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
101	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
102	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
103	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
104	C₄²	SD₁₆×C₁₂₇	1	2	yes	—
105	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
106	C₁₆	—	2	2	no	—
107	C₁₆	C₂×C₈×C₁₂₇	1	4	yes	—
108	C₂³	#106×C₁₂₇	1	1	yes	characteristic
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	C₄⋊C₈×C₁₂₇	1	2	yes	—
111	C₂×C₄	#106×C₁₂₇	1	2	yes	—
112	C₂×C₄	#106×C₁₂₇	1	1	yes	characteristic
113	C₂×C₄	#108×C₁₂₇	1	2	yes	—
114	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
115	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
116	C₈	—	2	1	no	—
117	C₈	—	2	1	no	—
118	C₈	—	8	4	no	—
119	C₈	C₄⋊C₈×C₁₂₇	1	2	yes	—
120	C₈	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
121	C₈	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	#334×C₁₂₇	1	2	yes	—
124	C₂²	C₂×#106×C₁₂₇	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	2	no	—
127	C₄	#334×C₁₂₇	1	2	yes	—
128	C₄	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
129	C₄	C₂×#108×C₁₂₇	1	1	yes	characteristic
130	C₂	—	2	2	no	—
131	C₂	C₂×#334×C₁₂₇	1	1	yes	characteristic
132	C₂	C₁₂₇×#2155	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#106
commutator	C₈	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#106
prime radical	C₂³×C₁₂₇	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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	semidirect
16. C₂×C₁₆×C₁₂₇	C₈	semidirect
63. #11999	C₁₂₇	direct (N × H)
69. #1368	C₂×C₁₂₇	semidirect
70. C₁₂₇	#11999	direct (N × H)
83. C₂×C₁₆	C₈×C₁₂₇	semidirect

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
C₁₂₇×#1564	2
C₁₂₇×#1368	2
#11999	127

Subgroups — C127×#11999 #44575672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11999 #44575672