Subgroups — C₁₂₇×#20459 #44584132

Order 32512.

994 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₁₂₇	C₂	1	2	yes	maximal
2	C₁₂₇×#2873	C₂	1	2	yes	maximal
3	C₂×#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1778	C₂	1	4	yes	maximal, complemented
5	C₄×#108×C₁₂₇	C₂	1	4	yes	maximal, complemented
6	C₂×#341×C₁₂₇	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₁₂₇	C₂²	1	4	yes	—
8	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
11	#430×C₁₂₇	C₄	1	16	yes	complemented
12	C₂×#108×C₁₂₇	C₂²	1	8	yes	—
13	#405×C₁₂₇	—	2	8	no	—
14	#405×C₁₂₇	C₄	1	8	yes	—
15	C₄×M₁₆×C₁₂₇	C₂²	1	8	yes	—
16	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
18	#341×C₁₂₇	C₂²	1	4	yes	—
19	#341×C₁₂₇	C₂²	1	4	yes	—
20	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₂₇	—	2	4	no	—
22	C₂×M₁₆×C₁₂₇	—	2	8	no	—
23	C₂×M₁₆×C₁₂₇	C₂×C₄	1	16	yes	—
24	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₂×M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
26	C₂×M₁₆×C₁₂₇	C₂³	1	8	yes	—
27	C₂²×C₈×C₁₂₇	—	2	4	no	—
28	C₂²×C₈×C₁₂₇	—	2	4	no	—
29	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
33	#117×C₁₂₇	C₂×C₄	1	4	yes	—
34	#117×C₁₂₇	D₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₂₇	Q₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₁₂₇	Q₈	1	2	yes	—
37	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
38	#108×C₁₂₇	C₂×C₄	1	32	yes	—
39	#97×C₁₂₇	—	2	8	no	—
40	#97×C₁₂₇	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₁₂₇	—	2	8	no	—
42	C₄×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
43	C₂⁴×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
48	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
50	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
51	M₁₆×C₁₂₇	—	2	16	no	—
52	M₁₆×C₁₂₇	—	2	8	no	—
53	M₁₆×C₁₂₇	C₄²	1	16	yes	—
54	M₁₆×C₁₂₇	C₂²×C₄	1	16	yes	—
55	C₂×C₈×C₁₂₇	—	2	8	no	—
56	C₂×C₈×C₁₂₇	—	2	16	no	—
57	C₂×C₈×C₁₂₇	—	2	8	no	—
58	C₂×C₈×C₁₂₇	—	2	2	no	—
59	C₂×C₈×C₁₂₇	—	2	2	no	—
60	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
64	C₄⋊C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
65	C₂²⋊C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
66	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
67	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
68	C₂³×C₁₂₇	—	4	2	no	—
69	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
70	C₂³×C₁₂₇	C₄×Q₈	1	4	yes	—
71	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	—	4	4	no	—
73	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
74	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	8	yes	—
75	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₁₂₇	#117	1	4	yes	—
77	C₂×C₄×C₁₂₇	C₄×D₄	1	8	yes	—
78	C₂×C₄×C₁₂₇	C₄×Q₈	1	4	yes	—
79	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
81	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
83	C₈×C₁₂₇	—	2	2	no	—
84	C₈×C₁₂₇	—	2	2	no	—
85	C₈×C₁₂₇	—	2	16	no	—
86	C₈×C₁₂₇	C₂×C₄²	1	8	yes	—
87	C₂²×C₁₂₇	—	2	2	no	—
88	C₂²×C₁₂₇	—	2	4	no	—
89	C₂²×C₁₂₇	—	4	4	no	—
90	C₂²×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
91	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₁₂₇	C₂×C₄×Q₈	1	2	yes	—
93	C₄×C₁₂₇	—	2	8	no	—
94	C₄×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
95	C₄×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
96	C₄×C₁₂₇	C₂×#117	1	1	yes	characteristic
97	C₄×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
98	#20459	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂×C₁₂₇	—	2	2	no	—
100	C₂×C₁₂₇	—	2	4	no	—
101	C₂×C₁₂₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₁₂₇	#2916	1	2	yes	—
103	C₂×#430	C₂×C₁₂₇	1	2	yes	—
104	#2873	C₂×C₁₂₇	1	2	yes	—
105	C₂×#405	C₂×C₁₂₇	1	1	yes	characteristic
106	#1778	C₂×C₁₂₇	1	4	yes	complemented
107	C₄×#108	C₂×C₁₂₇	1	4	yes	complemented
108	C₂×#341	C₂×C₁₂₇	1	2	yes	—
109	C₁₂₇	#20459	1	1	yes	characteristic, direct product factor, Sylow
110	C₂²×M₁₆	C₂²×C₁₂₇	1	4	yes	—
111	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
112	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
113	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
114	#430	C₄×C₁₂₇	1	16	yes	complemented
115	C₂×#108	C₂²×C₁₂₇	1	8	yes	—
116	#405	—	2	8	no	—
117	#405	C₄×C₁₂₇	1	8	yes	—
118	C₄×M₁₆	C₂²×C₁₂₇	1	8	yes	—
119	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
120	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
121	#341	C₂²×C₁₂₇	1	4	yes	—
122	#341	C₂²×C₁₂₇	1	4	yes	—
123	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
124	C₂×M₁₆	—	2	4	no	—
125	C₂×M₁₆	—	2	8	no	—
126	C₂×M₁₆	C₂×C₄×C₁₂₇	1	16	yes	—
127	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
128	C₂×M₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
129	C₂×M₁₆	C₂³×C₁₂₇	1	8	yes	—
130	C₂²×C₈	—	2	4	no	—
131	C₂²×C₈	—	2	4	no	—
132	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
133	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
136	#117	C₂×C₄×C₁₂₇	1	4	yes	—
137	#117	D₄×C₁₂₇	1	4	yes	—
138	C₂×C₄⋊C₄	Q₈×C₁₂₇	1	2	yes	—
139	C₂×C₂²⋊C₄	Q₈×C₁₂₇	1	2	yes	—
140	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
141	#108	C₂×C₄×C₁₂₇	1	32	yes	—
142	#97	—	2	8	no	—
143	#97	C₂×C₄×C₁₂₇	1	8	yes	—
144	C₄×C₈	—	2	8	no	—
145	C₄×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
146	C₂⁴	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
148	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
149	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
150	C₂²×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
151	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
152	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
153	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
154	M₁₆	—	2	16	no	—
155	M₁₆	—	2	8	no	—
156	M₁₆	C₄²×C₁₂₇	1	16	yes	—
157	M₁₆	C₂²×C₄×C₁₂₇	1	16	yes	—
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	2	16	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
164	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
165	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
166	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
167	C₄⋊C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
168	C₂²⋊C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
169	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
170	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
171	C₂³	—	4	2	no	—
172	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
173	C₂³	C₄×Q₈×C₁₂₇	1	4	yes	—
174	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
177	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	8	yes	—
178	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
179	C₂×C₄	#117×C₁₂₇	1	4	yes	—
180	C₂×C₄	C₄×D₄×C₁₂₇	1	8	yes	—
181	C₂×C₄	C₄×Q₈×C₁₂₇	1	4	yes	—
182	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
183	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
184	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
185	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	2	2	no	—
188	C₈	—	2	16	no	—
189	C₈	C₂×C₄²×C₁₂₇	1	8	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
194	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
195	C₂²	C₂×C₄×Q₈×C₁₂₇	1	2	yes	—
196	C₄	—	2	8	no	—
197	C₄	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
198	C₄	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
199	C₄	C₂×#117×C₁₂₇	1	1	yes	characteristic
200	C₄	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
201	C₂	—	2	2	no	—
202	C₂	—	2	4	no	—
203	C₂	C₂×C₄×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
204	C₂	C₁₂₇×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₂₇	C₂×Q₈
LCS	G ▷ 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
4. C₁₂₇×#1778	C₂	semidirect
5. C₄×#108×C₁₂₇	C₂	semidirect
11. #430×C₁₂₇	C₄	semidirect
98. #20459	C₁₂₇	direct (N × H)
106. #1778	C₂×C₁₂₇	semidirect
107. C₄×#108	C₂×C₁₂₇	semidirect
109. C₁₂₇	#20459	direct (N × H)
114. #430	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#430×C₁₂₇	2
C₁₂₇×#2873	2
C₂×#405×C₁₂₇	2
C₁₂₇×#1778	2
C₄×#108×C₁₂₇	2
C₂×#341×C₁₂₇	2
#20459	127

Subgroups — C127×#20459 #44584132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20459 #44584132