Subgroups — C₁₂₇×#12888 #44576561

Order 32512.

342 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1845	C₂	1	2	yes	maximal
3	C₁₂₇×#1380	C₂	1	4	yes	maximal
4	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
5	C₂×#107×C₁₂₇	—	2	2	no	—
6	C₂×#107×C₁₂₇	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	#384×C₁₂₇	—	2	2	no	—
9	C₄×C₁₆×C₁₂₇	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
11	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₂₇	—	4	2	no	—
14	C₂×C₄⋊C₄×C₁₂₇	—	4	2	no	—
15	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₁₂₇	—	2	4	no	—
17	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
18	#107×C₁₂₇	—	2	8	no	—
19	#107×C₁₂₇	—	4	4	no	—
20	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₂₇	Q₈	1	2	yes	—
22	C₂²×C₄×C₁₂₇	—	4	4	no	—
23	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
25	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₁₂₇	SD₁₆	1	4	yes	—
28	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₁₂₇	#41	1	4	yes	—
31	C₄⋊C₄×C₁₂₇	—	4	4	no	—
32	C₄⋊C₄×C₁₂₇	—	4	4	no	—
33	C₄⋊C₄×C₁₂₇	—	8	4	no	—
34	C₄⋊C₄×C₁₂₇	—	8	4	no	—
35	C₄²×C₁₂₇	C₄⋊C₄	1	2	yes	—
36	C₄²×C₁₂₇	Q₁₆	1	2	yes	—
37	C₁₆×C₁₂₇	—	2	4	no	—
38	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	—	8	4	no	—
40	C₂×C₄×C₁₂₇	—	8	4	no	—
41	C₂×C₄×C₁₂₇	—	8	4	no	—
42	C₂×C₄×C₁₂₇	#107	1	2	yes	—
43	C₂×C₄×C₁₂₇	#107	1	2	yes	—
44	C₂×C₄×C₁₂₇	D₁₆	1	2	yes	—
45	C₂×C₄×C₁₂₇	SD₃₂	1	4	yes	—
46	C₂×C₄×C₁₂₇	Q₃₂	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	#124	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
51	C₈×C₁₂₇	#106	1	4	yes	—
52	C₈×C₁₂₇	#117	1	2	yes	—
53	C₈×C₁₂₇	#125	1	2	yes	—
54	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
55	C₂²×C₁₂₇	#486	1	2	yes	—
56	C₂²×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
57	C₂²×C₁₂₇	C₂×SD₃₂	1	2	yes	—
58	C₂²×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
59	C₄×C₁₂₇	—	16	4	no	—
60	C₄×C₁₂₇	#366	1	4	yes	—
61	C₄×C₁₂₇	#367	1	4	yes	—
62	C₄×C₁₂₇	#382	1	1	yes	characteristic
63	C₄×C₁₂₇	C₂×#106	1	1	yes	characteristic
64	C₄×C₁₂₇	#487	1	2	yes	—
65	#12888	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₁₂₇	#1834	1	1	yes	characteristic
67	C₂×C₁₂₇	C₂×#366	1	1	yes	characteristic
68	C₂×C₁₂₇	C₂×#367	1	1	yes	characteristic
69	C₂×C₁₂₇	#2240	1	2	yes	—
70	C₂×C₁₂₇	#2241	1	2	yes	—
71	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
72	#1845	C₂×C₁₂₇	1	2	yes	—
73	#1380	C₂×C₁₂₇	1	4	yes	—
74	C₁₂₇	#12888	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
76	C₂×#107	—	2	2	no	—
77	C₂×#107	C₂²×C₁₂₇	1	4	yes	—
78	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
79	#384	—	2	2	no	—
80	C₄×C₁₆	C₄×C₁₂₇	1	4	yes	complemented
81	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
82	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
83	C₂×C₄⋊C₄	—	2	4	no	—
84	C₂×C₄⋊C₄	—	4	2	no	—
85	C₂×C₄⋊C₄	—	4	2	no	—
86	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
89	#107	—	2	8	no	—
90	#107	—	4	4	no	—
91	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
92	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
93	C₂²×C₄	—	4	4	no	—
94	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
95	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
96	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
98	C₂×C₈	SD₁₆×C₁₂₇	1	4	yes	—
99	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
100	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
101	C₂×C₈	#41×C₁₂₇	1	4	yes	—
102	C₄⋊C₄	—	4	4	no	—
103	C₄⋊C₄	—	4	4	no	—
104	C₄⋊C₄	—	8	4	no	—
105	C₄⋊C₄	—	8	4	no	—
106	C₄²	C₄⋊C₄×C₁₂₇	1	2	yes	—
107	C₄²	Q₁₆×C₁₂₇	1	2	yes	—
108	C₁₆	—	2	4	no	—
109	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
110	C₂×C₄	—	8	4	no	—
111	C₂×C₄	—	8	4	no	—
112	C₂×C₄	—	8	4	no	—
113	C₂×C₄	#107×C₁₂₇	1	2	yes	—
114	C₂×C₄	#107×C₁₂₇	1	2	yes	—
115	C₂×C₄	D₁₆×C₁₂₇	1	2	yes	—
116	C₂×C₄	SD₃₂×C₁₂₇	1	4	yes	—
117	C₂×C₄	Q₃₂×C₁₂₇	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
119	C₂×C₄	#124×C₁₂₇	1	2	yes	—
120	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
121	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
122	C₈	#106×C₁₂₇	1	4	yes	—
123	C₈	#117×C₁₂₇	1	2	yes	—
124	C₈	#125×C₁₂₇	1	2	yes	—
125	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
126	C₂²	#486×C₁₂₇	1	2	yes	—
127	C₂²	C₂×D₁₆×C₁₂₇	1	1	yes	characteristic
128	C₂²	C₂×SD₃₂×C₁₂₇	1	2	yes	—
129	C₂²	C₂×Q₃₂×C₁₂₇	1	1	yes	characteristic
130	C₄	—	16	4	no	—
131	C₄	#366×C₁₂₇	1	4	yes	—
132	C₄	#367×C₁₂₇	1	4	yes	—
133	C₄	#382×C₁₂₇	1	1	yes	characteristic
134	C₄	C₂×#106×C₁₂₇	1	1	yes	characteristic
135	C₄	#487×C₁₂₇	1	2	yes	—
136	C₂	C₁₂₇×#1834	1	1	yes	characteristic
137	C₂	C₂×#366×C₁₂₇	1	1	yes	characteristic
138	C₂	C₂×#367×C₁₂₇	1	1	yes	characteristic
139	C₂	C₁₂₇×#2240	1	2	yes	—
140	C₂	C₁₂₇×#2241	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
LCS	G ▷ C₂×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
9. C₄×C₁₆×C₁₂₇	C₄	semidirect
65. #12888	C₁₂₇	direct (N × H)
74. C₁₂₇	#12888	direct (N × H)
80. C₄×C₁₆	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆×C₁₂₇	2
C₁₂₇×#1845	2
C₁₂₇×#1380	2
#12888	127

Subgroups — C127×#12888 #44576561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12888 #44576561