Subgroups — C₁₂₇×#12559 #44576232

Order 32512.

554 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₁₂₇	C₂	1	2	yes	maximal
3	C₁₂₇×#1381	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₁₂₇	—	2	2	no	—
5	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂×M₃₂×C₁₂₇	—	2	4	no	—
7	C₂×M₃₂×C₁₂₇	C₄	1	4	yes	—
8	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
9	C₂×#108×C₁₂₇	—	2	2	no	—
10	C₂×#108×C₁₂₇	—	2	4	no	—
11	C₂×#108×C₁₂₇	C₄	1	8	yes	—
12	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
13	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
14	C₂×M₁₆×C₁₂₇	—	2	8	no	—
15	C₂×M₁₆×C₁₂₇	—	2	4	no	—
16	C₂×M₁₆×C₁₂₇	—	4	2	no	—
17	C₂×M₁₆×C₁₂₇	—	4	4	no	—
18	C₂²×C₈×C₁₂₇	—	4	2	no	—
19	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₂₇	D₄	1	4	yes	—
23	C₂²×C₈×C₁₂₇	Q₈	1	2	yes	—
24	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
25	M₃₂×C₁₂₇	—	2	8	no	—
26	C₂×C₁₆×C₁₂₇	—	2	4	no	—
27	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
28	#108×C₁₂₇	—	2	8	no	—
29	#108×C₁₂₇	—	2	4	no	—
30	#108×C₁₂₇	C₂×C₄	1	8	yes	—
31	C₂⁴×C₁₂₇	Q₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
33	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₁₂₇	D₈	1	2	yes	—
35	C₂²×C₄×C₁₂₇	SD₁₆	1	4	yes	—
36	C₂²×C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₁₂₇	—	2	8	no	—
39	M₁₆×C₁₂₇	—	4	8	no	—
40	M₁₆×C₁₂₇	—	4	4	no	—
41	C₂×C₈×C₁₂₇	—	4	8	no	—
42	C₂×C₈×C₁₂₇	—	4	4	no	—
43	C₂×C₈×C₁₂₇	C₄²	1	4	yes	—
44	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	4	yes	—
46	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₂₇	C₄⋊C₄	1	8	yes	—
48	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
52	C₁₆×C₁₂₇	—	2	4	no	—
53	C₂³×C₁₂₇	—	4	2	no	—
54	C₂³×C₁₂₇	#103	1	4	yes	—
55	C₂³×C₁₂₇	#107	1	2	yes	—
56	C₂³×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	#95	1	4	yes	—
58	C₂×C₄×C₁₂₇	#102	1	4	yes	—
59	C₂×C₄×C₁₂₇	#102	1	4	yes	—
60	C₂×C₄×C₁₂₇	#103	1	4	yes	—
61	C₂×C₄×C₁₂₇	#106	1	4	yes	—
62	C₂×C₄×C₁₂₇	#107	1	2	yes	—
63	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
67	C₈×C₁₂₇	—	4	8	no	—
68	C₈×C₁₂₇	#95	1	4	yes	—
69	C₈×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
70	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₈×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂²×C₁₂₇	—	2	4	no	—
73	C₂²×C₁₂₇	—	2	2	no	—
74	C₂²×C₁₂₇	—	4	4	no	—
75	C₂²×C₁₂₇	#340	1	4	yes	—
76	C₂²×C₁₂₇	C₂×#103	1	2	yes	—
77	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
78	C₄×C₁₂₇	#340	1	4	yes	—
79	C₄×C₁₂₇	C₂×#95	1	1	yes	characteristic
80	C₄×C₁₂₇	C₂×#102	1	2	yes	—
81	C₄×C₁₂₇	C₂×#106	1	1	yes	characteristic
82	#12559	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₁₂₇	—	2	4	no	—
84	C₂×C₁₂₇	—	2	2	no	—
85	C₂×C₁₂₇	C₂×#340	1	1	yes	characteristic
86	C₂×C₁₂₇	#2161	1	2	yes	—
87	C₂²×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
88	C₂²×#108	C₂×C₁₂₇	1	2	yes	—
89	#1381	C₂×C₁₂₇	1	4	yes	complemented
90	C₁₂₇	#12559	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²×M₁₆	—	2	2	no	—
92	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
93	C₂×M₃₂	—	2	4	no	—
94	C₂×M₃₂	C₄×C₁₂₇	1	4	yes	—
95	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
96	C₂×#108	—	2	2	no	—
97	C₂×#108	—	2	4	no	—
98	C₂×#108	C₄×C₁₂₇	1	8	yes	—
99	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
100	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
101	C₂×M₁₆	—	2	8	no	—
102	C₂×M₁₆	—	2	4	no	—
103	C₂×M₁₆	—	4	2	no	—
104	C₂×M₁₆	—	4	4	no	—
105	C₂²×C₈	—	4	2	no	—
106	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
107	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
108	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
109	C₂²×C₈	D₄×C₁₂₇	1	4	yes	—
110	C₂²×C₈	Q₈×C₁₂₇	1	2	yes	—
111	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
112	M₃₂	—	2	8	no	—
113	C₂×C₁₆	—	2	4	no	—
114	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
115	#108	—	2	8	no	—
116	#108	—	2	4	no	—
117	#108	C₂×C₄×C₁₂₇	1	8	yes	—
118	C₂⁴	Q₁₆×C₁₂₇	1	1	yes	characteristic
119	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
120	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	2	yes	—
121	C₂²×C₄	D₈×C₁₂₇	1	2	yes	—
122	C₂²×C₄	SD₁₆×C₁₂₇	1	4	yes	—
123	C₂²×C₄	Q₁₆×C₁₂₇	1	1	yes	characteristic
124	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
125	M₁₆	—	2	8	no	—
126	M₁₆	—	4	8	no	—
127	M₁₆	—	4	4	no	—
128	C₂×C₈	—	4	8	no	—
129	C₂×C₈	—	4	4	no	—
130	C₂×C₈	C₄²×C₁₂₇	1	4	yes	—
131	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	4	yes	—
132	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	4	yes	—
133	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
134	C₂×C₈	C₄⋊C₄×C₁₂₇	1	8	yes	—
135	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
136	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
137	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
138	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
139	C₁₆	—	2	4	no	—
140	C₂³	—	4	2	no	—
141	C₂³	#103×C₁₂₇	1	4	yes	—
142	C₂³	#107×C₁₂₇	1	2	yes	—
143	C₂³	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
144	C₂×C₄	#95×C₁₂₇	1	4	yes	—
145	C₂×C₄	#102×C₁₂₇	1	4	yes	—
146	C₂×C₄	#102×C₁₂₇	1	4	yes	—
147	C₂×C₄	#103×C₁₂₇	1	4	yes	—
148	C₂×C₄	#106×C₁₂₇	1	4	yes	—
149	C₂×C₄	#107×C₁₂₇	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
152	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
153	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
154	C₈	—	4	8	no	—
155	C₈	#95×C₁₂₇	1	4	yes	—
156	C₈	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
157	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
158	C₈	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	4	no	—
162	C₂²	#340×C₁₂₇	1	4	yes	—
163	C₂²	C₂×#103×C₁₂₇	1	2	yes	—
164	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
165	C₄	#340×C₁₂₇	1	4	yes	—
166	C₄	C₂×#95×C₁₂₇	1	1	yes	characteristic
167	C₄	C₂×#102×C₁₂₇	1	2	yes	—
168	C₄	C₂×#106×C₁₂₇	1	1	yes	characteristic
169	C₂	—	2	4	no	—
170	C₂	—	2	2	no	—
171	C₂	C₂×#340×C₁₂₇	1	1	yes	characteristic
172	C₂	C₁₂₇×#2161	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂×D₈
commutator	C₈	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#107
prime radical	C₂⁴×C₁₂₇	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ C₂³×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1381	C₂	semidirect
82. #12559	C₁₂₇	direct (N × H)
89. #1381	C₂×C₁₂₇	semidirect
90. C₁₂₇	#12559	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₁₂₇	2
C₂²×#108×C₁₂₇	2
C₁₂₇×#1381	2
#12559	127

Subgroups — C127×#12559 #44576232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12559 #44576232