Subgroups — C₁₂₇×#12023 #44575696

Order 32512.

242 conjugacy classes of subgroups.

The subgroup lattice has 148 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×#1567	C₂	1	2	yes	maximal
3	C₁₂₇×#1372	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₁₂₇	C₄	1	2	yes	—
5	#423×C₁₂₇	—	2	2	no	—
6	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₄×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
8	C₄×C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₁₂₇	C₂²	1	2	yes	—
10	#335×C₁₂₇	—	2	2	no	—
11	#335×C₁₂₇	C₂²	1	4	yes	—
12	C₂×M₁₆×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	4	no	—
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	C₄⋊C₈×C₁₂₇	—	4	2	no	—
21	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
28	M₁₆×C₁₂₇	—	8	4	no	—
29	C₂×C₈×C₁₂₇	—	2	2	no	—
30	C₂×C₈×C₁₂₇	—	2	2	no	—
31	C₂×C₈×C₁₂₇	—	4	4	no	—
32	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
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	D₈	1	1	yes	characteristic
38	C₄²×C₁₂₇	Q₁₆	1	1	yes	characteristic
39	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₁₂₇	—	2	2	no	—
41	C₁₆×C₁₂₇	M₁₆	1	4	yes	complemented
42	C₂³×C₁₂₇	#107	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	—	2	2	no	—
44	C₂×C₄×C₁₂₇	—	2	1	no	—
45	C₂×C₄×C₁₂₇	—	2	1	no	—
46	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	#107	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	SD₃₂	1	4	yes	—
50	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
53	C₈×C₁₂₇	—	2	1	no	—
54	C₈×C₁₂₇	—	2	1	no	—
55	C₈×C₁₂₇	—	8	4	no	—
56	C₈×C₁₂₇	#108	1	2	yes	—
57	C₈×C₁₂₇	C₂×M₁₆	1	2	yes	—
58	C₂²×C₁₂₇	—	2	1	no	—
59	C₂²×C₁₂₇	—	2	1	no	—
60	C₂²×C₁₂₇	#367	1	2	yes	—
61	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
62	C₄×C₁₂₇	—	2	1	no	—
63	C₄×C₁₂₇	—	2	1	no	—
64	C₄×C₁₂₇	#367	1	2	yes	—
65	C₄×C₁₂₇	#368	1	2	yes	—
66	C₄×C₁₂₇	#423	1	1	yes	characteristic
67	C₄×C₁₂₇	C₂×#108	1	1	yes	characteristic
68	C₄×C₁₂₇	C₂×SD₃₂	1	2	yes	—
69	#12023	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×C₁₂₇	—	2	2	no	—
71	C₂×C₁₂₇	#1567	1	1	yes	characteristic
72	C₂×C₁₂₇	C₂×#367	1	1	yes	characteristic
73	C₂×C₁₂₇	C₂×#368	1	1	yes	characteristic
74	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
75	#1567	C₂×C₁₂₇	1	2	yes	—
76	#1372	C₂×C₁₂₇	1	4	yes	complemented
77	C₁₂₇	#12023	1	1	yes	characteristic, direct product factor, Sylow
78	C₂²×C₁₆	C₄×C₁₂₇	1	2	yes	—
79	#423	—	2	2	no	—
80	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
81	C₄×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
82	C₄×C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
83	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
84	#335	—	2	2	no	—
85	#335	C₂²×C₁₂₇	1	4	yes	—
86	C₂×M₁₆	—	4	2	no	—
87	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
88	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
89	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
90	C₂×C₁₆	—	2	4	no	—
91	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
92	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
93	C₄⋊C₈	—	2	4	no	—
94	C₄⋊C₈	—	4	2	no	—
95	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
96	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
97	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
98	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
99	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
100	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
101	C₂²×C₄	Q₁₆×C₁₂₇	1	1	yes	characteristic
102	M₁₆	—	8	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	4	4	no	—
106	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
107	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
108	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
109	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
110	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
111	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
112	C₄²	Q₁₆×C₁₂₇	1	1	yes	characteristic
113	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
114	C₁₆	—	2	2	no	—
115	C₁₆	M₁₆×C₁₂₇	1	4	yes	complemented
116	C₂³	#107×C₁₂₇	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
121	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
122	C₂×C₄	#107×C₁₂₇	1	1	yes	characteristic
123	C₂×C₄	SD₃₂×C₁₂₇	1	4	yes	—
124	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
125	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
126	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
127	C₈	—	2	1	no	—
128	C₈	—	2	1	no	—
129	C₈	—	8	4	no	—
130	C₈	#108×C₁₂₇	1	2	yes	—
131	C₈	C₂×M₁₆×C₁₂₇	1	2	yes	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#367×C₁₂₇	1	2	yes	—
135	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	#367×C₁₂₇	1	2	yes	—
139	C₄	#368×C₁₂₇	1	2	yes	—
140	C₄	#423×C₁₂₇	1	1	yes	characteristic
141	C₄	C₂×#108×C₁₂₇	1	1	yes	characteristic
142	C₄	C₂×SD₃₂×C₁₂₇	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	C₁₂₇×#1567	1	1	yes	characteristic
145	C₂	C₂×#367×C₁₂₇	1	1	yes	characteristic
146	C₂	C₂×#368×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#107
prime radical	C₂³×C₁₂₇	#107
LCS	G ▷ C₂×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×#1372	C₂	semidirect
41. C₁₆×C₁₂₇	M₁₆	semidirect
69. #12023	C₁₂₇	direct (N × H)
76. #1372	C₂×C₁₂₇	semidirect
77. C₁₂₇	#12023	direct (N × H)
115. C₁₆	M₁₆×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×#1567	2
C₁₂₇×#1372	2
#12023	127

Subgroups — C127×#12023 #44575696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12023 #44575696