Subgroups — C₁₂₇×#21971 #44585644

Order 32512.

662 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2921	C₂	1	2	yes	maximal
2	C₂×#424×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1950	C₂	1	8	yes	maximal, complemented
4	C₁₂₇×#1811	C₂	1	2	yes	maximal
5	C₁₂₇×#1810	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
9	#433×C₁₂₇	—	2	8	no	—
10	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
11	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
12	#424×C₁₂₇	C₄	1	8	yes	—
13	#423×C₁₂₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
15	#407×C₁₂₇	C₂²	1	4	yes	—
16	C₄×M₁₆×C₁₂₇	C₂²	1	4	yes	—
17	#341×C₁₂₇	C₂²	1	4	yes	—
18	#341×C₁₂₇	C₂²	1	4	yes	—
19	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₂₇	—	4	2	no	—
21	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
22	C₂×M₁₆×C₁₂₇	C₂³	1	8	yes	—
23	C₂²×C₈×C₁₂₇	—	4	2	no	—
24	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
26	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
27	#117×C₁₂₇	C₂×C₄	1	4	yes	—
28	#117×C₁₂₇	D₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₁₂₇	Q₈	1	2	yes	—
30	C₂×C₂²⋊C₄×C₁₂₇	Q₈	1	2	yes	—
31	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
32	#108×C₁₂₇	—	4	4	no	—
33	#108×C₁₂₇	—	4	4	no	—
34	C₄⋊C₈×C₁₂₇	—	2	8	no	—
35	C₄⋊C₈×C₁₂₇	C₂×C₄	1	16	yes	—
36	C₂²⋊C₈×C₁₂₇	—	2	8	no	—
37	#97×C₁₂₇	—	2	4	no	—
38	C₄×C₈×C₁₂₇	—	2	4	no	—
39	C₂⁴×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	—	2	2	no	—
41	C₂²×C₄×C₁₂₇	—	2	2	no	—
42	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
46	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
47	M₁₆×C₁₂₇	—	2	8	no	—
48	M₁₆×C₁₂₇	—	2	8	no	—
49	M₁₆×C₁₂₇	—	4	4	no	—
50	C₂×C₈×C₁₂₇	—	2	4	no	—
51	C₂×C₈×C₁₂₇	—	2	4	no	—
52	C₂×C₈×C₁₂₇	—	2	8	no	—
53	C₂×C₈×C₁₂₇	—	4	4	no	—
54	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
56	C₄⋊C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
57	C₂²⋊C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
58	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
59	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
60	C₂³×C₁₂₇	—	2	2	no	—
61	C₂³×C₁₂₇	—	4	2	no	—
62	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
63	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	—	2	2	no	—
65	C₂×C₄×C₁₂₇	—	2	2	no	—
66	C₂×C₄×C₁₂₇	—	2	2	no	—
67	C₂×C₄×C₁₂₇	—	2	4	no	—
68	C₂×C₄×C₁₂₇	—	4	4	no	—
69	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	8	yes	—
70	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
74	C₂×C₄×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
75	C₈×C₁₂₇	—	2	4	no	—
76	C₈×C₁₂₇	—	2	4	no	—
77	C₈×C₁₂₇	—	4	8	no	—
78	C₂²×C₁₂₇	—	2	2	no	—
79	C₂²×C₁₂₇	—	2	2	no	—
80	C₂²×C₁₂₇	—	2	4	no	—
81	C₂²×C₁₂₇	—	4	4	no	—
82	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₁₂₇	#544	1	2	yes	—
84	C₄×C₁₂₇	—	2	2	no	—
85	C₄×C₁₂₇	—	2	8	no	—
86	C₄×C₁₂₇	#518	1	1	yes	characteristic
87	C₄×C₁₂₇	#519	1	1	yes	characteristic
88	C₄×C₁₂₇	#537	1	2	yes	—
89	#21971	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×C₁₂₇	—	2	2	no	—
91	C₂×C₁₂₇	—	4	2	no	—
92	C₂×C₁₂₇	#2316	1	1	yes	characteristic
93	C₂×C₁₂₇	#2916	1	2	yes	—
94	#2921	C₂×C₁₂₇	1	2	yes	—
95	C₂×#424	C₂×C₁₂₇	1	1	yes	characteristic
96	#1950	C₂×C₁₂₇	1	8	yes	complemented
97	#1811	C₂×C₁₂₇	1	2	yes	—
98	#1810	C₂×C₁₂₇	1	2	yes	—
99	C₁₂₇	#21971	1	1	yes	characteristic, direct product factor, Sylow
100	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
101	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
102	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
103	#433	—	2	8	no	—
104	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
105	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
106	#424	C₄×C₁₂₇	1	8	yes	—
107	#423	C₂²×C₁₂₇	1	4	yes	—
108	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	—
109	#407	C₂²×C₁₂₇	1	4	yes	—
110	C₄×M₁₆	C₂²×C₁₂₇	1	4	yes	—
111	#341	C₂²×C₁₂₇	1	4	yes	—
112	#341	C₂²×C₁₂₇	1	4	yes	—
113	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
114	C₂×M₁₆	—	4	2	no	—
115	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
116	C₂×M₁₆	C₂³×C₁₂₇	1	8	yes	—
117	C₂²×C₈	—	4	2	no	—
118	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
119	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
120	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
121	#117	C₂×C₄×C₁₂₇	1	4	yes	—
122	#117	D₄×C₁₂₇	1	4	yes	—
123	C₂×C₄⋊C₄	Q₈×C₁₂₇	1	2	yes	—
124	C₂×C₂²⋊C₄	Q₈×C₁₂₇	1	2	yes	—
125	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
126	#108	—	4	4	no	—
127	#108	—	4	4	no	—
128	C₄⋊C₈	—	2	8	no	—
129	C₄⋊C₈	C₂×C₄×C₁₂₇	1	16	yes	—
130	C₂²⋊C₈	—	2	8	no	—
131	#97	—	2	4	no	—
132	C₄×C₈	—	2	4	no	—
133	C₂⁴	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
137	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
138	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
139	C₂²×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
140	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
141	M₁₆	—	2	8	no	—
142	M₁₆	—	2	8	no	—
143	M₁₆	—	4	4	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	2	8	no	—
147	C₂×C₈	—	4	4	no	—
148	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
149	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
150	C₄⋊C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
151	C₂²⋊C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
152	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
153	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
154	C₂³	—	2	2	no	—
155	C₂³	—	4	2	no	—
156	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
157	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	8	yes	—
164	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
165	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
166	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
167	C₂×C₄	H(5,𝔽₂)×C₁₂₇	1	2	yes	—
168	C₂×C₄	2₋¹⁺⁴×C₁₂₇	1	2	yes	—
169	C₈	—	2	4	no	—
170	C₈	—	2	4	no	—
171	C₈	—	4	8	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	4	4	no	—
176	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
177	C₂²	#544×C₁₂₇	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	2	8	no	—
180	C₄	#518×C₁₂₇	1	1	yes	characteristic
181	C₄	#519×C₁₂₇	1	1	yes	characteristic
182	C₄	#537×C₁₂₇	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	4	2	no	—
185	C₂	C₁₂₇×#2316	1	1	yes	characteristic
186	C₂	C₁₂₇×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₂₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×#117×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1950	C₂	semidirect
89. #21971	C₁₂₇	direct (N × H)
96. #1950	C₂×C₁₂₇	semidirect
99. C₁₂₇	#21971	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#2921	2
C₂×#424×C₁₂₇	2
C₁₂₇×#1950	2
C₁₂₇×#1811	2
C₁₂₇×#1810	2
#21971	127

Subgroups — C127×#21971 #44585644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21971 #44585644