Subgroups — D₄×C₃₂×C₁₂₇ #44577317

Order 32512.

214 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	C₂²×C₃₂×C₁₂₇	C₂	1	2	yes	maximal, complemented
2	D₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×C₂²⋊C₃₂	C₂	1	2	yes	maximal, complemented
5	C₄×C₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
7	C₈×D₄×C₁₂₇	C₄	1	1	yes	characteristic
8	C₂×C₃₂×C₁₂₇	—	2	4	no	—
9	C₂×C₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₃₂×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₁₂₇	C₄	1	1	yes	characteristic
13	C₂²⋊C₁₆×C₁₂₇	C₄	1	2	yes	—
14	C₄×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
16	C₄×D₄×C₁₂₇	C₈	1	1	yes	characteristic
17	C₂×C₁₆×C₁₂₇	—	2	2	no	—
18	C₂×C₁₆×C₁₂₇	—	2	2	no	—
19	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₈×C₁₂₇	C₈	1	1	yes	characteristic
23	C₂²⋊C₈×C₁₂₇	C₈	1	2	yes	—
24	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₃₂×C₁₂₇	—	2	2	no	—
26	C₃₂×C₁₂₇	—	2	1	no	—
27	C₃₂×C₁₂₇	D₄	1	2	yes	direct product factor
28	C₂×D₄×C₁₂₇	C₁₆	1	1	yes	characteristic
29	C₂²×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₁₂₇	—	2	2	no	—
31	C₂×C₈×C₁₂₇	—	2	2	no	—
32	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₁₂₇	C₁₆	1	1	yes	characteristic
36	C₂²⋊C₄×C₁₂₇	C₁₆	1	2	yes	—
37	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
38	C₁₆×C₁₂₇	—	2	2	no	—
39	C₁₆×C₁₂₇	—	2	1	no	—
40	C₁₆×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₁₂₇	#41	1	1	yes	characteristic
42	C₂³×C₁₂₇	C₂×C₁₆	1	2	yes	—
43	D₄×C₁₂₇	C₃₂	1	4	yes	direct product factor
44	C₂×C₄×C₁₂₇	—	2	2	no	—
45	C₂×C₄×C₁₂₇	—	2	2	no	—
46	C₂×C₄×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₁₂₇	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₁₂₇	—	2	2	no	—
51	C₈×C₁₂₇	—	2	1	no	—
52	C₈×C₁₂₇	C₄×D₄	1	1	yes	characteristic
53	C₈×C₁₂₇	#131	1	1	yes	characteristic
54	C₂²×C₁₂₇	—	2	2	no	—
55	C₂²×C₁₂₇	—	2	2	no	—
56	C₂²×C₁₂₇	C₂×C₃₂	1	4	yes	complemented
57	C₂²×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
58	C₄×C₁₂₇	—	2	2	no	—
59	C₄×C₁₂₇	—	2	1	no	—
60	C₄×C₁₂₇	C₂×C₃₂	1	2	yes	complemented
61	C₄×C₁₂₇	C₈×D₄	1	1	yes	characteristic
62	C₄×C₁₂₇	#504	1	1	yes	characteristic
63	D₄×C₃₂	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₁₂₇	—	2	4	no	—
65	C₂×C₁₂₇	D₄×C₁₆	1	1	yes	characteristic
66	C₂×C₁₂₇	C₂²×C₃₂	1	1	yes	characteristic
67	C₂×C₁₂₇	#2258	1	1	yes	characteristic
68	C₂²×C₃₂	C₂×C₁₂₇	1	2	yes	complemented
69	D₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
70	C₄⋊C₃₂	C₂×C₁₂₇	1	1	yes	characteristic, complemented
71	C₂²⋊C₃₂	C₂×C₁₂₇	1	2	yes	complemented
72	C₄×C₃₂	C₂×C₁₂₇	1	1	yes	characteristic, complemented
73	C₁₂₇	D₄×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
75	C₈×D₄	C₄×C₁₂₇	1	1	yes	characteristic
76	C₂×C₃₂	—	2	4	no	—
77	C₂×C₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
78	C₂×C₃₂	C₂²×C₁₂₇	1	2	yes	—
79	C₂×C₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
80	C₄⋊C₁₆	C₄×C₁₂₇	1	1	yes	characteristic
81	C₂²⋊C₁₆	C₄×C₁₂₇	1	2	yes	—
82	C₄×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
83	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
84	C₄×D₄	C₈×C₁₂₇	1	1	yes	characteristic
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
88	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
89	C₂×C₁₆	C₂³×C₁₂₇	1	1	yes	characteristic
90	C₄⋊C₈	C₈×C₁₂₇	1	1	yes	characteristic
91	C₂²⋊C₈	C₈×C₁₂₇	1	2	yes	—
92	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
93	C₃₂	—	2	2	no	—
94	C₃₂	—	2	1	no	—
95	C₃₂	D₄×C₁₂₇	1	2	yes	direct product factor
96	C₂×D₄	C₁₆×C₁₂₇	1	1	yes	characteristic
97	C₂²×C₄	C₂×C₈×C₁₂₇	1	2	yes	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
101	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
102	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
103	C₄⋊C₄	C₁₆×C₁₂₇	1	1	yes	characteristic
104	C₂²⋊C₄	C₁₆×C₁₂₇	1	2	yes	—
105	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
106	C₁₆	—	2	2	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	C₂×D₄×C₁₂₇	1	1	yes	characteristic
109	C₁₆	#41×C₁₂₇	1	1	yes	characteristic
110	C₂³	C₂×C₁₆×C₁₂₇	1	2	yes	—
111	D₄	C₃₂×C₁₂₇	1	4	yes	direct product factor
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂×C₁₆×C₁₂₇	1	1	yes	characteristic
115	C₂×C₄	C₂×C₁₆×C₁₂₇	1	2	yes	—
116	C₂×C₄	C₂×C₁₆×C₁₂₇	1	1	yes	characteristic
117	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
118	C₈	—	2	1	no	—
119	C₈	—	2	2	no	—
120	C₈	C₄×D₄×C₁₂₇	1	1	yes	characteristic
121	C₈	#131×C₁₂₇	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂×C₃₂×C₁₂₇	1	4	yes	complemented
125	C₂²	C₂²×C₁₆×C₁₂₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	C₂×C₃₂×C₁₂₇	1	2	yes	complemented
129	C₄	C₈×D₄×C₁₂₇	1	1	yes	characteristic
130	C₄	#504×C₁₂₇	1	1	yes	characteristic
131	C₂	—	2	4	no	—
132	C₂	D₄×C₁₆×C₁₂₇	1	1	yes	characteristic
133	C₂	C₂²×C₃₂×C₁₂₇	1	1	yes	characteristic
134	C₂	C₁₂₇×#2258	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂×C₁₂₇	C₂²
commutator	C₂	C₂²×C₃₂×C₁₂₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂²×C₁₆
prime radical	C₂×D₄×C₁₂₇	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₂×C₁₂₇	C₂	semidirect
3. C₁₂₇×C₄⋊C₃₂	C₂	semidirect
4. C₁₂₇×C₂²⋊C₃₂	C₂	semidirect
5. C₄×C₃₂×C₁₂₇	C₂	semidirect
27. C₃₂×C₁₂₇	D₄	direct (N × H)
43. D₄×C₁₂₇	C₃₂	direct (N × H)
56. C₂²×C₁₂₇	C₂×C₃₂	semidirect
60. C₄×C₁₂₇	C₂×C₃₂	semidirect
63. D₄×C₃₂	C₁₂₇	direct (N × H)
68. C₂²×C₃₂	C₂×C₁₂₇	semidirect
70. C₄⋊C₃₂	C₂×C₁₂₇	semidirect
71. C₂²⋊C₃₂	C₂×C₁₂₇	semidirect
72. C₄×C₃₂	C₂×C₁₂₇	semidirect
73. C₁₂₇	D₄×C₃₂	direct (N × H)
95. C₃₂	D₄×C₁₂₇	direct (N × H)
111. D₄	C₃₂×C₁₂₇	direct (N × H)
124. C₂²	C₂×C₃₂×C₁₂₇	semidirect
128. C₄	C₂×C₃₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₂×C₁₂₇	2
D₄×C₁₆×C₁₂₇	2
C₁₂₇×C₄⋊C₃₂	2
C₁₂₇×C₂²⋊C₃₂	2
C₄×C₃₂×C₁₂₇	2
D₄×C₃₂	127

Subgroups — D4×C32×C127 #44577317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₃₂×C₁₂₇ #44577317