Subgroups — C₁₂₇×#12063 #44575736

Order 32512.

242 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2169	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₂₇×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₂₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₃₂×C₁₂₇	—	2	1	no	—
9	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
10	C₄×D₈×C₁₂₇	C₄	1	1	yes	characteristic
11	C₈×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₈×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²⋊C₁₆×C₁₂₇	—	2	1	no	—
16	C₂²⋊C₁₆×C₁₂₇	—	2	1	no	—
17	C₄×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
19	#335×C₁₂₇	C₄	1	1	yes	characteristic
20	D₄⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
21	D₄⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
22	C₈²×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×D₈×C₁₂₇	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₁₂₇	—	2	1	no	—
25	C₂²×C₈×C₁₂₇	—	2	1	no	—
26	C₄×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₁₂₇	—	4	2	no	—
29	C₂×C₁₆×C₁₂₇	—	2	1	no	—
30	C₂×C₁₆×C₁₂₇	—	2	1	no	—
31	C₂×C₁₆×C₁₂₇	—	2	1	no	—
32	C₂×C₁₆×C₁₂₇	—	4	2	no	—
33	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
35	#107×C₁₂₇	C₈	1	1	yes	characteristic
36	C₄⋊C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
38	#102×C₁₂₇	C₈	1	1	yes	characteristic
39	#102×C₁₂₇	C₈	1	1	yes	characteristic
40	C₂²⋊C₈×C₁₂₇	—	2	1	no	—
41	C₂²⋊C₈×C₁₂₇	—	2	1	no	—
42	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
46	C₂×D₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₂₇	—	2	1	no	—
48	C₂²×C₄×C₁₂₇	—	2	1	no	—
49	D₈×C₁₂₇	—	2	2	no	—
50	C₂×C₈×C₁₂₇	—	2	1	no	—
51	C₂×C₈×C₁₂₇	—	2	1	no	—
52	C₂×C₈×C₁₂₇	—	2	1	no	—
53	C₂×C₈×C₁₂₇	—	4	1	no	—
54	C₂×C₈×C₁₂₇	—	4	1	no	—
55	C₂×C₈×C₁₂₇	—	4	1	no	—
56	C₂×C₈×C₁₂₇	—	4	1	no	—
57	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
58	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
62	C₄⋊C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
63	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
64	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
65	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
66	C₁₆×C₁₂₇	—	2	1	no	—
67	C₁₆×C₁₂₇	—	2	1	no	—
68	C₁₆×C₁₂₇	—	4	1	no	—
69	C₁₆×C₁₂₇	—	4	1	no	—
70	C₁₆×C₁₂₇	—	4	1	no	—
71	C₂³×C₁₂₇	—	2	1	no	—
72	C₂³×C₁₂₇	—	2	1	no	—
73	D₄×C₁₂₇	—	2	1	no	—
74	D₄×C₁₂₇	—	2	1	no	—
75	D₄×C₁₂₇	—	2	1	no	—
76	D₄×C₁₂₇	M₃₂	1	2	yes	complemented
77	C₂×C₄×C₁₂₇	—	2	1	no	—
78	C₂×C₄×C₁₂₇	—	2	1	no	—
79	C₂×C₄×C₁₂₇	—	4	1	no	—
80	C₂×C₄×C₁₂₇	—	4	1	no	—
81	C₂×C₄×C₁₂₇	—	4	1	no	—
82	C₂×C₄×C₁₂₇	—	4	1	no	—
83	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	#131	1	1	yes	characteristic
86	C₈×C₁₂₇	—	2	2	no	—
87	C₈×C₁₂₇	—	2	1	no	—
88	C₈×C₁₂₇	—	2	1	no	—
89	C₈×C₁₂₇	—	2	1	no	—
90	C₈×C₁₂₇	—	4	1	no	—
91	C₈×C₁₂₇	—	4	1	no	—
92	C₈×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₈×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂²×C₁₂₇	—	2	2	no	—
95	C₂²×C₁₂₇	—	4	1	no	—
96	C₂²×C₁₂₇	—	4	1	no	—
97	C₂²×C₁₂₇	—	4	1	no	—
98	C₂²×C₁₂₇	—	4	1	no	—
99	C₂²×C₁₂₇	—	4	1	no	—
100	C₂²×C₁₂₇	C₈×D₄	1	1	yes	characteristic
101	C₄×C₁₂₇	—	2	1	no	—
102	C₄×C₁₂₇	—	4	1	no	—
103	C₄×C₁₂₇	—	4	1	no	—
104	C₄×C₁₂₇	#442	1	1	yes	characteristic
105	C₄×C₁₂₇	#444	1	1	yes	characteristic
106	C₄×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
107	C₄×C₁₂₇	#504	1	1	yes	characteristic
108	#12063	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₁₂₇	—	4	2	no	—
110	C₂×C₁₂₇	—	8	1	no	—
111	C₂×C₁₂₇	#1580	1	1	yes	characteristic
112	C₂×C₁₂₇	#2168	1	1	yes	characteristic
113	C₂×C₁₂₇	#2170	1	1	yes	characteristic
114	#2169	C₂×C₁₂₇	1	1	yes	characteristic, complemented
115	D₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
116	C₈×D₈	C₂×C₁₂₇	1	1	yes	characteristic
117	#1367	C₂×C₁₂₇	1	1	yes	characteristic, complemented
118	D₄⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
119	D₄⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
120	#1312	C₂×C₁₂₇	1	1	yes	characteristic, complemented
121	C₁₂₇	#12063	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×M₃₂	—	2	1	no	—
123	C₂²×C₁₆	—	2	1	no	—
124	C₄×D₈	C₄×C₁₂₇	1	1	yes	characteristic
125	C₈×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
126	C₈×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
127	C₄⋊C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
128	C₄⋊C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
129	C₂²⋊C₁₆	—	2	1	no	—
130	C₂²⋊C₁₆	—	2	1	no	—
131	C₄×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
132	C₄×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
133	#335	C₄×C₁₂₇	1	1	yes	characteristic
134	D₄⋊C₈	C₄×C₁₂₇	1	1	yes	characteristic
135	D₄⋊C₈	C₄×C₁₂₇	1	1	yes	characteristic
136	C₈²	C₂²×C₁₂₇	1	1	yes	characteristic
137	C₂×D₈	C₈×C₁₂₇	1	1	yes	characteristic
138	C₂²×C₈	—	2	1	no	—
139	C₂²×C₈	—	2	1	no	—
140	C₄×D₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
141	C₄×D₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
142	M₃₂	—	4	2	no	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	2	1	no	—
145	C₂×C₁₆	—	2	1	no	—
146	C₂×C₁₆	—	4	2	no	—
147	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
148	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
149	#107	C₈×C₁₂₇	1	1	yes	characteristic
150	C₄⋊C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
151	C₄⋊C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
152	#102	C₈×C₁₂₇	1	1	yes	characteristic
153	#102	C₈×C₁₂₇	1	1	yes	characteristic
154	C₂²⋊C₈	—	2	1	no	—
155	C₂²⋊C₈	—	2	1	no	—
156	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
157	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
158	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
159	C₂×D₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
160	C₂×D₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	D₈	—	2	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
172	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
174	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
175	C₄⋊C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
176	C₄⋊C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
177	C₂²⋊C₄	—	2	1	no	—
178	C₂²⋊C₄	—	2	1	no	—
179	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
180	C₁₆	—	2	1	no	—
181	C₁₆	—	2	1	no	—
182	C₁₆	—	4	1	no	—
183	C₁₆	—	4	1	no	—
184	C₁₆	—	4	1	no	—
185	C₂³	—	2	1	no	—
186	C₂³	—	2	1	no	—
187	D₄	—	2	1	no	—
188	D₄	—	2	1	no	—
189	D₄	—	2	1	no	—
190	D₄	M₃₂×C₁₂₇	1	2	yes	complemented
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
198	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
199	C₂×C₄	#131×C₁₂₇	1	1	yes	characteristic
200	C₈	—	2	2	no	—
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	1	no	—
206	C₈	AGL(1,ℤ/8)×C₁₂₇	1	1	yes	characteristic
207	C₈	AGL(1,ℤ/8)×C₁₂₇	1	1	yes	characteristic
208	C₂²	—	2	2	no	—
209	C₂²	—	4	1	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	C₈×D₄×C₁₂₇	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	#442×C₁₂₇	1	1	yes	characteristic
219	C₄	#444×C₁₂₇	1	1	yes	characteristic
220	C₄	C₂×M₃₂×C₁₂₇	1	1	yes	characteristic
221	C₄	#504×C₁₂₇	1	1	yes	characteristic
222	C₂	—	4	2	no	—
223	C₂	—	8	1	no	—
224	C₂	C₁₂₇×#1580	1	1	yes	characteristic
225	C₂	C₁₂₇×#2168	1	1	yes	characteristic
226	C₂	C₁₂₇×#2170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₈×D₄
prime radical	C₂×D₈×C₁₂₇	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₄×C₁₆×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₂₇×#2169	C₂	semidirect
2. D₄×C₁₆×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1367	C₂	semidirect
5. C₁₂₇×D₄⋊C₁₆	C₂	semidirect
6. C₁₂₇×D₄⋊C₁₆	C₂	semidirect
7. C₁₂₇×#1312	C₂	semidirect
76. D₄×C₁₂₇	M₃₂	semidirect
108. #12063	C₁₂₇	direct (N × H)
114. #2169	C₂×C₁₂₇	semidirect
115. D₄×C₁₆	C₂×C₁₂₇	semidirect
117. #1367	C₂×C₁₂₇	semidirect
118. D₄⋊C₁₆	C₂×C₁₂₇	semidirect
119. D₄⋊C₁₆	C₂×C₁₂₇	semidirect
120. #1312	C₂×C₁₂₇	semidirect
121. C₁₂₇	#12063	direct (N × H)
190. D₄	M₃₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2169	2
D₄×C₁₆×C₁₂₇	2
C₈×D₈×C₁₂₇	2
C₁₂₇×#1367	2
C₁₂₇×D₄⋊C₁₆	2
C₁₂₇×#1312	2
#12063	127

Subgroups — C127×#12063 #44575736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12063 #44575736