Subgroups — C₁₂₇×#12011 #44575684

Order 32512.

220 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×#364×C₁₂₇	C₂	1	2	yes	maximal
2	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1373	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₂₇	—	2	2	no	—
5	C₂×M₃₂×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂×M₃₂×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	#364×C₁₂₇	—	2	2	no	—
9	#364×C₁₂₇	C₂²	1	4	yes	—
10	#346×C₁₂₇	C₄	1	2	yes	—
11	C₄×C₁₆×C₁₂₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₂₇	Q₈	1	1	yes	characteristic
15	M₃₂×C₁₂₇	—	2	2	no	—
16	M₃₂×C₁₂₇	—	2	4	no	—
17	M₃₂×C₁₂₇	—	4	2	no	—
18	M₃₂×C₁₂₇	C₈	1	4	yes	—
19	C₂×C₁₆×C₁₂₇	—	4	2	no	—
20	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
27	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
32	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
33	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
34	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₁₂₇	—	2	1	no	—
37	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
39	C₁₆×C₁₂₇	—	2	2	no	—
40	C₁₆×C₁₂₇	—	4	4	no	—
41	C₁₆×C₁₂₇	C₂×C₈	1	4	yes	—
42	C₂³×C₁₂₇	#108	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	—	2	1	no	—
44	C₂×C₄×C₁₂₇	—	2	1	no	—
45	C₂×C₄×C₁₂₇	—	4	2	no	—
46	C₂×C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
47	C₂×C₄×C₁₂₇	#107	1	2	yes	—
48	C₂×C₄×C₁₂₇	#108	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₁₂₇	C₄⋊C₈	1	2	yes	—
51	C₈×C₁₂₇	#107	1	2	yes	—
52	C₈×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
53	C₈×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
54	C₈×C₁₂₇	C₂×D₈	1	1	yes	characteristic
55	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
56	C₂²×C₁₂₇	—	4	1	no	—
57	C₂²×C₁₂₇	#335	1	2	yes	—
58	C₂²×C₁₂₇	C₂×#108	1	1	yes	characteristic
59	C₄×C₁₂₇	—	2	2	no	—
60	C₄×C₁₂₇	—	4	1	no	—
61	C₄×C₁₂₇	#335	1	2	yes	—
62	C₄×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₄×C₁₂₇	C₂×#107	1	1	yes	characteristic
64	#12011	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₁₂₇	—	2	1	no	—
66	C₂×C₁₂₇	—	2	2	no	—
67	C₂×C₁₂₇	C₂×#335	1	1	yes	characteristic
68	C₂×#364	C₂×C₁₂₇	1	2	yes	—
69	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
70	#1373	C₂×C₁₂₇	1	4	yes	complemented
71	C₁₂₇	#12011	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×M₃₂	—	2	2	no	—
73	C₂×M₃₂	C₄×C₁₂₇	1	1	yes	characteristic
74	C₂×M₃₂	C₄×C₁₂₇	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
76	#364	—	2	2	no	—
77	#364	C₂²×C₁₂₇	1	4	yes	—
78	#346	C₄×C₁₂₇	1	2	yes	—
79	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
80	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
81	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
82	C₂×C₄²	Q₈×C₁₂₇	1	1	yes	characteristic
83	M₃₂	—	2	2	no	—
84	M₃₂	—	2	4	no	—
85	M₃₂	—	4	2	no	—
86	M₃₂	C₈×C₁₂₇	1	4	yes	—
87	C₂×C₁₆	—	4	2	no	—
88	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
89	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
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	—	2	1	no	—
95	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
96	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
98	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
99	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
100	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
101	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
102	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
103	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
106	C₄²	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
107	C₁₆	—	2	2	no	—
108	C₁₆	—	4	4	no	—
109	C₁₆	C₂×C₈×C₁₂₇	1	4	yes	—
110	C₂³	#108×C₁₂₇	1	1	yes	characteristic
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	C₄⋊C₈×C₁₂₇	1	2	yes	—
115	C₂×C₄	#107×C₁₂₇	1	2	yes	—
116	C₂×C₄	#108×C₁₂₇	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
118	C₈	C₄⋊C₈×C₁₂₇	1	2	yes	—
119	C₈	#107×C₁₂₇	1	2	yes	—
120	C₈	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
121	C₈	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
122	C₈	C₂×D₈×C₁₂₇	1	1	yes	characteristic
123	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
124	C₂²	—	4	1	no	—
125	C₂²	#335×C₁₂₇	1	2	yes	—
126	C₂²	C₂×#108×C₁₂₇	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	4	1	no	—
129	C₄	#335×C₁₂₇	1	2	yes	—
130	C₄	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
131	C₄	C₂×#107×C₁₂₇	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	2	no	—
134	C₂	C₂×#335×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₂₇	C₂×D₈
commutator	C₈	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	C₂×#335
prime radical	C₂³×C₁₂₇	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1373	C₂	semidirect
64. #12011	C₁₂₇	direct (N × H)
70. #1373	C₂×C₁₂₇	semidirect
71. C₁₂₇	#12011	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×#364×C₁₂₇	2
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1373	2
#12011	127

Subgroups — C127×#12011 #44575684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12011 #44575684