Subgroups — C₁₂₇×#12000 #44575673

Order 32512.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1368	C₂	1	4	yes	maximal, complemented
5	C₂×M₃₂×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂×M₃₂×C₁₂₇	C₄	1	1	yes	characteristic
7	#431×C₁₂₇	—	2	1	no	—
8	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
9	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	#346×C₁₂₇	C₂²	1	2	yes	—
11	C₄×C₁₆×C₁₂₇	C₄	1	2	yes	—
12	#335×C₁₂₇	—	2	1	no	—
13	#334×C₁₂₇	—	2	1	no	—
14	#334×C₁₂₇	C₂²	1	2	yes	—
15	#334×C₁₂₇	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₂₇	—	4	1	no	—
17	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
20	M₃₂×C₁₂₇	—	2	2	no	—
21	M₃₂×C₁₂₇	C₈	1	4	yes	complemented
22	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₄⋊C₈×C₁₂₇	—	2	2	no	—
25	C₄⋊C₈×C₁₂₇	—	2	2	no	—
26	C₄⋊C₈×C₁₂₇	—	4	1	no	—
27	C₂²⋊C₈×C₁₂₇	—	4	1	no	—
28	C₄×C₈×C₁₂₇	—	4	1	no	—
29	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₁₂₇	—	2	2	no	—
37	C₂×C₈×C₁₂₇	—	4	2	no	—
38	C₂×C₈×C₁₂₇	—	4	2	no	—
39	C₂×C₈×C₁₂₇	—	8	2	no	—
40	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
44	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
48	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
49	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₁₆×C₁₂₇	—	2	2	no	—
51	C₁₆×C₁₂₇	C₂×C₈	1	4	yes	complemented
52	C₂³×C₁₂₇	#106	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	—	2	2	no	—
54	C₂×C₄×C₁₂₇	—	4	1	no	—
55	C₂×C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₁₂₇	#106	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	#106	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	#106	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	#108	1	2	yes	—
60	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₁₂₇	—	2	1	no	—
64	C₈×C₁₂₇	—	2	1	no	—
65	C₈×C₁₂₇	—	8	2	no	—
66	C₈×C₁₂₇	—	8	2	no	—
67	C₈×C₁₂₇	C₄⋊C₈	1	2	yes	—
68	C₈×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
69	C₈×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂²×C₁₂₇	—	2	1	no	—
71	C₂²×C₁₂₇	—	2	1	no	—
72	C₂²×C₁₂₇	#334	1	2	yes	—
73	C₂²×C₁₂₇	C₂×#106	1	1	yes	characteristic
74	C₄×C₁₂₇	—	2	1	no	—
75	C₄×C₁₂₇	—	2	1	no	—
76	C₄×C₁₂₇	—	2	1	no	—
77	C₄×C₁₂₇	#334	1	2	yes	—
78	C₄×C₁₂₇	#365	1	2	yes	—
79	C₄×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₄×C₁₂₇	C₂×#108	1	1	yes	characteristic
81	#12000	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₁₂₇	—	2	2	no	—
83	C₂×C₁₂₇	C₂×#334	1	1	yes	characteristic
84	C₂×C₁₂₇	C₂×#365	1	1	yes	characteristic
85	C₂×C₁₂₇	#2155	1	1	yes	characteristic
86	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
87	#1564	C₂×C₁₂₇	1	1	yes	characteristic
88	C₂×#334	C₂×C₁₂₇	1	1	yes	characteristic
89	#1368	C₂×C₁₂₇	1	4	yes	complemented
90	C₁₂₇	#12000	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×M₃₂	C₄×C₁₂₇	1	1	yes	characteristic
92	C₂×M₃₂	C₄×C₁₂₇	1	1	yes	characteristic
93	#431	—	2	1	no	—
94	C₂×C₄⋊C₈	—	2	1	no	—
95	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
96	#346	C₂²×C₁₂₇	1	2	yes	—
97	C₄×C₁₆	C₄×C₁₂₇	1	2	yes	—
98	#335	—	2	1	no	—
99	#334	—	2	1	no	—
100	#334	C₂²×C₁₂₇	1	2	yes	—
101	#334	C₂²×C₁₂₇	1	2	yes	—
102	C₂²×C₈	—	4	1	no	—
103	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
104	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
105	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	C₈×C₁₂₇	1	4	yes	complemented
108	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
109	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	4	1	no	—
113	C₂²⋊C₈	—	4	1	no	—
114	C₄×C₈	—	4	1	no	—
115	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
116	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
117	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
118	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
119	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
120	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
121	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	8	2	no	—
126	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
127	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
128	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
129	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
130	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
131	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
132	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
133	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
134	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
135	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
136	C₁₆	—	2	2	no	—
137	C₁₆	C₂×C₈×C₁₂₇	1	4	yes	complemented
138	C₂³	#106×C₁₂₇	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	C₄⋊C₈×C₁₂₇	1	2	yes	—
142	C₂×C₄	#106×C₁₂₇	1	1	yes	characteristic
143	C₂×C₄	#106×C₁₂₇	1	1	yes	characteristic
144	C₂×C₄	#106×C₁₂₇	1	1	yes	characteristic
145	C₂×C₄	#108×C₁₂₇	1	2	yes	—
146	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
147	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
148	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	8	2	no	—
153	C₈	C₄⋊C₈×C₁₂₇	1	2	yes	—
154	C₈	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
155	C₈	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#334×C₁₂₇	1	2	yes	—
159	C₂²	C₂×#106×C₁₂₇	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	#334×C₁₂₇	1	2	yes	—
164	C₄	#365×C₁₂₇	1	2	yes	—
165	C₄	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
166	C₄	C₂×#108×C₁₂₇	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	C₂×#334×C₁₂₇	1	1	yes	characteristic
169	C₂	C₂×#365×C₁₂₇	1	1	yes	characteristic
170	C₂	C₁₂₇×#2155	1	1	yes	characteristic

	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
4. C₁₂₇×#1368	C₂	semidirect
21. M₃₂×C₁₂₇	C₈	semidirect
51. C₁₆×C₁₂₇	C₂×C₈	semidirect
81. #12000	C₁₂₇	direct (N × H)
89. #1368	C₂×C₁₂₇	semidirect
90. C₁₂₇	#12000	direct (N × H)
107. M₃₂	C₈×C₁₂₇	semidirect
137. C₁₆	C₂×C₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1564	2
C₂×#334×C₁₂₇	2
C₁₂₇×#1368	2
#12000	127

Subgroups — C127×#12000 #44575673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12000 #44575673