Subgroups — C₄×#368×C₁₂₇ #44576348

Order 32512.

314 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×#368×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#108×C₁₂₇	C₂	1	2	yes	maximal
4	C₁₂₇×#1381	C₂	1	2	yes	maximal
5	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
8	C₄×M₁₆×C₁₂₇	—	2	2	no	—
9	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	#368×C₁₂₇	C₄	1	8	yes	direct product factor
11	C₄×C₁₆×C₁₂₇	C₄	1	4	yes	—
12	#341×C₁₂₇	—	2	2	no	—
13	C₂×M₁₆×C₁₂₇	—	2	4	no	—
14	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₁₂₇	—	2	2	no	—
18	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
21	#108×C₁₂₇	—	2	4	no	—
22	#108×C₁₂₇	C₂×C₄	1	8	yes	—
23	#97×C₁₂₇	—	4	2	no	—
24	C₄×C₈×C₁₂₇	—	4	2	no	—
25	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₁₂₇	Q₈	1	2	yes	—
27	C₂²×C₄×C₁₂₇	Q₁₆	1	2	yes	—
28	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
29	M₁₆×C₁₂₇	—	2	8	no	—
30	M₁₆×C₁₂₇	—	4	4	no	—
31	C₂×C₈×C₁₂₇	—	2	2	no	—
32	C₂×C₈×C₁₂₇	—	4	4	no	—
33	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
36	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	#41	1	2	yes	—
38	C₄²×C₁₂₇	C₄⋊C₄	1	2	yes	—
39	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
40	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
41	C₁₆×C₁₂₇	—	2	2	no	—
42	C₁₆×C₁₂₇	C₄²	1	4	yes	—
43	C₂³×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	—	2	4	no	—
45	C₂×C₄×C₁₂₇	#107	1	2	yes	—
46	C₂×C₄×C₁₂₇	#107	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	#135	1	2	yes	—
51	C₈×C₁₂₇	—	2	2	no	—
52	C₈×C₁₂₇	—	4	8	no	—
53	C₈×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
54	C₈×C₁₂₇	#117	1	1	yes	characteristic
55	C₈×C₁₂₇	C₄×Q₈	1	2	yes	—
56	C₂²×C₁₂₇	—	2	2	no	—
57	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
58	C₂²×C₁₂₇	C₄×Q₁₆	1	2	yes	—
59	C₄×C₁₂₇	—	2	2	no	—
60	C₄×C₁₂₇	#368	1	4	yes	direct product factor
61	C₄×C₁₂₇	C₄×C₄⋊C₄	1	1	yes	characteristic
62	C₄×C₁₂₇	#427	1	1	yes	characteristic
63	C₄×C₁₂₇	C₄×D₈	1	2	yes	—
64	C₄×#368	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₁₂₇	—	2	2	no	—
66	C₂×C₁₂₇	C₄×#107	1	1	yes	characteristic
67	C₂×C₁₂₇	C₂×#368	1	2	yes	—
68	C₂×#368	C₂×C₁₂₇	1	2	yes	—
69	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
70	C₄×#108	C₂×C₁₂₇	1	2	yes	—
71	#1381	C₂×C₁₂₇	1	2	yes	—
72	C₁₂₇	C₄×#368	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
74	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
75	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
76	C₄×M₁₆	—	2	2	no	—
77	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
78	#368	C₄×C₁₂₇	1	8	yes	direct product factor
79	C₄×C₁₆	C₄×C₁₂₇	1	4	yes	—
80	#341	—	2	2	no	—
81	C₂×M₁₆	—	2	4	no	—
82	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
83	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
84	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
87	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
88	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
89	#108	—	2	4	no	—
90	#108	C₂×C₄×C₁₂₇	1	8	yes	—
91	#97	—	4	2	no	—
92	C₄×C₈	—	4	2	no	—
93	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
94	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
95	C₂²×C₄	Q₁₆×C₁₂₇	1	2	yes	—
96	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
97	M₁₆	—	2	8	no	—
98	M₁₆	—	4	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	4	no	—
101	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
102	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
103	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
104	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
105	C₂×C₈	#41×C₁₂₇	1	2	yes	—
106	C₄²	C₄⋊C₄×C₁₂₇	1	2	yes	—
107	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
108	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
109	C₁₆	—	2	2	no	—
110	C₁₆	C₄²×C₁₂₇	1	4	yes	—
111	C₂³	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	#107×C₁₂₇	1	2	yes	—
114	C₂×C₄	#107×C₁₂₇	1	2	yes	—
115	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
116	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
117	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
118	C₂×C₄	#135×C₁₂₇	1	2	yes	—
119	C₈	—	2	2	no	—
120	C₈	—	4	8	no	—
121	C₈	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
122	C₈	#117×C₁₂₇	1	1	yes	characteristic
123	C₈	C₄×Q₈×C₁₂₇	1	2	yes	—
124	C₂²	—	2	2	no	—
125	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
126	C₂²	C₄×Q₁₆×C₁₂₇	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	#368×C₁₂₇	1	4	yes	direct product factor
129	C₄	C₄×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
130	C₄	#427×C₁₂₇	1	1	yes	characteristic
131	C₄	C₄×D₈×C₁₂₇	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₄×#107×C₁₂₇	1	1	yes	characteristic
134	C₂	C₂×#368×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₂₇	D₈
commutator	C₈	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#107
prime radical	C₂³×C₁₂₇	C₂×Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×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. #368×C₁₂₇	C₄	direct (N × H)
60. C₄×C₁₂₇	#368	direct (N × H)
64. C₄×#368	C₁₂₇	direct (N × H)
72. C₁₂₇	C₄×#368	direct (N × H)
78. #368	C₄×C₁₂₇	direct (N × H)
128. C₄	#368×C₁₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#368×C₁₂₇	2
C₂×C₄×C₁₆×C₁₂₇	2
C₄×#108×C₁₂₇	2
C₁₂₇×#1381	2
C₄×#368	127

Subgroups — C4×#368×C127 #44576348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#368×C₁₂₇ #44576348