Subgroups — C₁₂₇×#12015 #44575688

Order 32512.

246 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2162	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1370	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1367	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
7	#431×C₁₂₇	C₄	1	1	yes	characteristic
8	C₄×M₁₆×C₁₂₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
11	C₄⋊C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₄⋊C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²⋊C₁₆×C₁₂₇	—	2	1	no	—
15	C₄×C₁₆×C₁₂₇	—	2	1	no	—
16	#322×C₁₂₇	C₄	1	2	yes	—
17	C₈²×C₁₂₇	C₂²	1	2	yes	—
18	C₂×M₁₆×C₁₂₇	C₈	1	1	yes	characteristic
19	C₂×M₁₆×C₁₂₇	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₂₇	—	2	2	no	—
24	C₂×C₁₆×C₁₂₇	—	2	1	no	—
25	C₂×C₁₆×C₁₂₇	—	2	1	no	—
26	C₂×C₁₆×C₁₂₇	—	4	2	no	—
27	C₄⋊C₈×C₁₂₇	—	2	1	no	—
28	C₂²⋊C₈×C₁₂₇	—	2	1	no	—
29	#97×C₁₂₇	C₈	1	2	yes	—
30	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
39	M₁₆×C₁₂₇	—	2	2	no	—
40	M₁₆×C₁₂₇	C₁₆	1	4	yes	complemented
41	C₂×C₈×C₁₂₇	—	2	1	no	—
42	C₂×C₈×C₁₂₇	—	2	1	no	—
43	C₂×C₈×C₁₂₇	—	2	2	no	—
44	C₂×C₈×C₁₂₇	—	4	1	no	—
45	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
54	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
55	C₁₆×C₁₂₇	—	4	1	no	—
56	C₁₆×C₁₂₇	—	4	2	no	—
57	C₁₆×C₁₂₇	—	4	1	no	—
58	C₂³×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	—	2	1	no	—
60	C₂×C₄×C₁₂₇	—	2	1	no	—
61	C₂×C₄×C₁₂₇	—	4	1	no	—
62	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₁₂₇	M₃₂	1	2	yes	—
67	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₁₂₇	—	2	1	no	—
71	C₈×C₁₂₇	—	2	1	no	—
72	C₈×C₁₂₇	—	2	1	no	—
73	C₈×C₁₂₇	—	2	2	no	—
74	C₈×C₁₂₇	—	4	2	no	—
75	C₈×C₁₂₇	C₂×C₁₆	1	4	yes	complemented
76	C₈×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₈×C₁₂₇	#137	1	1	yes	characteristic
78	C₂²×C₁₂₇	—	2	1	no	—
79	C₂²×C₁₂₇	—	2	1	no	—
80	C₂²×C₁₂₇	C₄⋊C₁₆	1	2	yes	—
81	C₂²×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₄×C₁₂₇	—	2	1	no	—
83	C₄×C₁₂₇	—	2	1	no	—
84	C₄×C₁₂₇	—	2	1	no	—
85	C₄×C₁₂₇	C₄⋊C₁₆	1	2	yes	—
86	C₄×C₁₂₇	#428	1	1	yes	characteristic
87	C₄×C₁₂₇	#430	1	1	yes	characteristic
88	C₄×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
89	C₄×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
90	#12015	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₁₂₇	—	2	2	no	—
92	C₂×C₁₂₇	#1565	1	1	yes	characteristic
93	C₂×C₁₂₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
94	C₂×C₁₂₇	#2153	1	1	yes	characteristic
95	#2162	C₂×C₁₂₇	1	1	yes	characteristic
96	C₂×C₄⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
97	C₈×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
98	#1370	C₂×C₁₂₇	1	2	yes	complemented
99	#1367	C₂×C₁₂₇	1	2	yes	complemented
100	C₁₂₇	#12015	1	1	yes	characteristic, direct product factor, Sylow
101	C₂²×C₁₆	—	2	1	no	—
102	#431	C₄×C₁₂₇	1	1	yes	characteristic
103	C₄×M₁₆	C₄×C₁₂₇	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
105	C₄⋊C₁₆	—	2	1	no	—
106	C₄⋊C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
107	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
108	C₄⋊C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
109	C₂²⋊C₁₆	—	2	1	no	—
110	C₄×C₁₆	—	2	1	no	—
111	#322	C₄×C₁₂₇	1	2	yes	—
112	C₈²	C₂²×C₁₂₇	1	2	yes	—
113	C₂×M₁₆	C₈×C₁₂₇	1	1	yes	characteristic
114	C₂×M₁₆	C₈×C₁₂₇	1	1	yes	characteristic
115	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
116	C₂²×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
117	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	4	2	no	—
122	C₄⋊C₈	—	2	1	no	—
123	C₂²⋊C₈	—	2	1	no	—
124	#97	C₈×C₁₂₇	1	2	yes	—
125	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
126	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
127	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
128	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
129	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
130	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
131	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
133	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	C₁₆×C₁₂₇	1	4	yes	complemented
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
141	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
142	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
143	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
144	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
145	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
146	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
147	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
148	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
149	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
150	C₁₆	—	4	1	no	—
151	C₁₆	—	4	2	no	—
152	C₁₆	—	4	1	no	—
153	C₂³	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
158	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
160	C₂×C₄	C₂×C₁₆×C₁₂₇	1	2	yes	—
161	C₂×C₄	M₃₂×C₁₂₇	1	2	yes	—
162	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
163	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
164	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	4	2	no	—
170	C₈	C₂×C₁₆×C₁₂₇	1	4	yes	complemented
171	C₈	AGL(1,ℤ/8)×C₁₂₇	1	1	yes	characteristic
172	C₈	#137×C₁₂₇	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊C₁₆×C₁₂₇	1	2	yes	—
176	C₂²	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₄⋊C₁₆×C₁₂₇	1	2	yes	—
181	C₄	#428×C₁₂₇	1	1	yes	characteristic
182	C₄	#430×C₁₂₇	1	1	yes	characteristic
183	C₄	C₂²×C₁₆×C₁₂₇	1	1	yes	characteristic
184	C₄	C₂×M₃₂×C₁₂₇	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₁₂₇×#1565	1	1	yes	characteristic
187	C₂	C₂×C₄⋊C₁₆×C₁₂₇	1	1	yes	characteristic
188	C₂	C₁₂₇×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂×D₄
commutator	C₄	C₂²×C₁₆×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₂₇	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1370	C₂	semidirect
5. C₁₂₇×#1367	C₂	semidirect
40. M₁₆×C₁₂₇	C₁₆	semidirect
75. C₈×C₁₂₇	C₂×C₁₆	semidirect
90. #12015	C₁₂₇	direct (N × H)
98. #1370	C₂×C₁₂₇	semidirect
99. #1367	C₂×C₁₂₇	semidirect
100. C₁₂₇	#12015	direct (N × H)
135. M₁₆	C₁₆×C₁₂₇	semidirect
170. C₈	C₂×C₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2162	2
C₂×C₄⋊C₁₆×C₁₂₇	2
C₈×M₁₆×C₁₂₇	2
C₁₂₇×#1370	2
C₁₂₇×#1367	2
#12015	127

Subgroups — C127×#12015 #44575688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12015 #44575688