Subgroups — C₁₂₇×#12906 #44576579

Order 32512.

374 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×#367×C₁₂₇	C₂	1	4	yes	maximal
2	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1845	C₂	1	2	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	#367×C₁₂₇	—	2	8	no	—
10	C₄×C₁₆×C₁₂₇	C₄	1	4	yes	complemented
11	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
14	C₂×C₄⋊C₄×C₁₂₇	—	4	4	no	—
15	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
17	C₂×C₁₆×C₁₂₇	D₄	1	4	yes	—
18	C₂×C₁₆×C₁₂₇	Q₈	1	4	yes	—
19	#107×C₁₂₇	—	2	8	no	—
20	#107×C₁₂₇	—	4	4	no	—
21	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₁₂₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₂₇	—	4	4	no	—
24	C₂²×C₄×C₁₂₇	D₈	1	2	yes	—
25	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
28	C₂×C₈×C₁₂₇	Q₁₆	1	2	yes	—
29	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
31	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
32	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
33	C₄⋊C₄×C₁₂₇	—	4	4	no	—
34	C₄⋊C₄×C₁₂₇	—	4	4	no	—
35	C₄⋊C₄×C₁₂₇	—	8	8	no	—
36	C₄²×C₁₂₇	C₄⋊C₄	1	2	yes	—
37	C₄²×C₁₂₇	Q₁₆	1	2	yes	—
38	C₁₆×C₁₂₇	C₄⋊C₄	1	8	yes	complemented
39	C₂³×C₁₂₇	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	—	8	4	no	—
41	C₂×C₄×C₁₂₇	—	8	8	no	—
42	C₂×C₄×C₁₂₇	#107	1	4	yes	—
43	C₂×C₄×C₁₂₇	SD₃₂	1	8	yes	—
44	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
50	C₈×C₁₂₇	#107	1	4	yes	—
51	C₈×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
52	C₈×C₁₂₇	C₄⋊Q₈	1	2	yes	—
53	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
54	C₂²×C₁₂₇	#493	1	1	yes	characteristic
55	C₂²×C₁₂₇	#500	1	1	yes	characteristic
56	C₂²×C₁₂₇	C₂×SD₃₂	1	4	yes	—
57	C₄×C₁₂₇	—	16	4	no	—
58	C₄×C₁₂₇	#367	1	8	yes	complemented
59	C₄×C₁₂₇	#384	1	1	yes	characteristic
60	C₄×C₁₂₇	C₂×#107	1	1	yes	characteristic
61	C₄×C₁₂₇	#494	1	1	yes	characteristic
62	C₄×C₁₂₇	#500	1	1	yes	characteristic
63	#12906	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₁₂₇	#1845	1	1	yes	characteristic
65	C₂×C₁₂₇	C₂×#367	1	2	yes	—
66	C₂×C₁₂₇	#2248	1	2	yes	—
67	C₂×C₁₂₇	#2254	1	2	yes	—
68	C₂×#367	C₂×C₁₂₇	1	4	yes	—
69	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
70	#1845	C₂×C₁₂₇	1	2	yes	—
71	C₁₂₇	#12906	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
73	C₂×#107	—	2	2	no	—
74	C₂×#107	C₂²×C₁₂₇	1	4	yes	—
75	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
76	#384	—	2	2	no	—
77	#367	—	2	8	no	—
78	C₄×C₁₆	C₄×C₁₂₇	1	4	yes	complemented
79	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
80	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
81	C₂×C₄⋊C₄	—	2	4	no	—
82	C₂×C₄⋊C₄	—	4	4	no	—
83	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
84	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
85	C₂×C₁₆	D₄×C₁₂₇	1	4	yes	—
86	C₂×C₁₆	Q₈×C₁₂₇	1	4	yes	—
87	#107	—	2	8	no	—
88	#107	—	4	4	no	—
89	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
90	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
91	C₂²×C₄	—	4	4	no	—
92	C₂²×C₄	D₈×C₁₂₇	1	2	yes	—
93	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
94	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
95	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
96	C₂×C₈	Q₁₆×C₁₂₇	1	2	yes	—
97	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
98	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
99	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
100	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	4	4	no	—
103	C₄⋊C₄	—	8	8	no	—
104	C₄²	C₄⋊C₄×C₁₂₇	1	2	yes	—
105	C₄²	Q₁₆×C₁₂₇	1	2	yes	—
106	C₁₆	C₄⋊C₄×C₁₂₇	1	8	yes	complemented
107	C₂³	C₂×D₈×C₁₂₇	1	1	yes	characteristic
108	C₂×C₄	—	8	8	no	—
109	C₂×C₄	—	8	4	no	—
110	C₂×C₄	#107×C₁₂₇	1	4	yes	—
111	C₂×C₄	SD₃₂×C₁₂₇	1	8	yes	—
112	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
113	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
114	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
115	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
116	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
117	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
118	C₈	#107×C₁₂₇	1	4	yes	—
119	C₈	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
120	C₈	C₄⋊Q₈×C₁₂₇	1	2	yes	—
121	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
122	C₂²	#493×C₁₂₇	1	1	yes	characteristic
123	C₂²	#500×C₁₂₇	1	1	yes	characteristic
124	C₂²	C₂×SD₃₂×C₁₂₇	1	4	yes	—
125	C₄	—	16	4	no	—
126	C₄	#367×C₁₂₇	1	8	yes	complemented
127	C₄	#384×C₁₂₇	1	1	yes	characteristic
128	C₄	C₂×#107×C₁₂₇	1	1	yes	characteristic
129	C₄	#494×C₁₂₇	1	1	yes	characteristic
130	C₄	#500×C₁₂₇	1	1	yes	characteristic
131	C₂	C₁₂₇×#1845	1	1	yes	characteristic
132	C₂	C₂×#367×C₁₂₇	1	2	yes	—
133	C₂	C₁₂₇×#2248	1	2	yes	—
134	C₂	C₁₂₇×#2254	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
10. C₄×C₁₆×C₁₂₇	C₄	semidirect
38. C₁₆×C₁₂₇	C₄⋊C₄	semidirect
58. C₄×C₁₂₇	#367	semidirect
63. #12906	C₁₂₇	direct (N × H)
71. C₁₂₇	#12906	direct (N × H)
78. C₄×C₁₆	C₄×C₁₂₇	semidirect
106. C₁₆	C₄⋊C₄×C₁₂₇	semidirect
126. C₄	#367×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#367×C₁₂₇	2
C₂×C₄×C₁₆×C₁₂₇	2
C₁₂₇×#1845	2
#12906	127

Subgroups — C127×#12906 #44576579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12906 #44576579