Subgroups — C₁₂₇×#13191 #44576864

Order 32512.

450 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2931	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#350×C₁₂₇	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#131×C₁₂₇	C₂²	1	2	yes	—
6	#532×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
8	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
9	#436×C₁₂₇	—	2	2	no	—
10	C₈×D₄×C₁₂₇	—	2	2	no	—
11	#423×C₁₂₇	C₄	1	1	yes	characteristic
12	#408×C₁₂₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
15	#350×C₁₂₇	—	2	8	no	—
16	C₄×C₁₆×C₁₂₇	—	2	2	no	—
17	C₂×#41×C₁₂₇	C₂×C₄	1	2	yes	—
18	#131×C₁₂₇	—	2	4	no	—
19	#131×C₁₂₇	—	2	4	no	—
20	C₂×M₁₆×C₁₂₇	—	2	2	no	—
21	C₂×M₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₂₇	—	2	2	no	—
23	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₄⋊Q₈×C₁₂₇	C₈	1	1	yes	characteristic
26	#127×C₁₂₇	C₈	1	1	yes	characteristic
27	#124×C₁₂₇	C₈	1	2	yes	—
28	#121×C₁₂₇	—	2	2	no	—
29	C₄×D₄×C₁₂₇	—	2	2	no	—
30	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	M₃₂×C₁₂₇	—	2	4	no	—
32	C₂×C₁₆×C₁₂₇	—	2	2	no	—
33	C₂×C₁₆×C₁₂₇	—	2	2	no	—
34	C₂×C₁₆×C₁₂₇	—	2	2	no	—
35	C₂×C₁₆×C₁₂₇	D₄	1	4	yes	complemented
36	C₄⋊C₈×C₁₂₇	—	2	2	no	—
37	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
38	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
39	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
40	#41×C₁₂₇	—	2	4	no	—
41	#41×C₁₂₇	—	2	4	no	—
42	C₂×Q₈×C₁₂₇	C₂×C₈	1	2	yes	—
43	C₂×D₄×C₁₂₇	—	2	2	no	—
44	C₂×D₄×C₁₂₇	—	2	2	no	—
45	C₂×D₄×C₁₂₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₂₇	—	2	2	no	—
47	C₂²×C₄×C₁₂₇	M₁₆	1	2	yes	—
48	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₁₂₇	—	2	4	no	—
50	M₁₆×C₁₂₇	—	4	4	no	—
51	C₂×C₈×C₁₂₇	—	2	2	no	—
52	C₂×C₈×C₁₂₇	—	2	2	no	—
53	C₂×C₈×C₁₂₇	—	4	2	no	—
54	C₂×C₈×C₁₂₇	—	4	2	no	—
55	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	#41	1	2	yes	—
60	C₄⋊C₄×C₁₂₇	—	2	2	no	—
61	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
62	C₄²×C₁₂₇	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
65	C₁₆×C₁₂₇	—	2	4	no	—
66	C₁₆×C₁₂₇	—	4	2	no	—
67	C₂³×C₁₂₇	—	2	2	no	—
68	C₂³×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
69	Q₈×C₁₂₇	—	2	4	no	—
70	D₄×C₁₂₇	—	2	4	no	—
71	D₄×C₁₂₇	—	4	4	no	—
72	D₄×C₁₂₇	—	4	4	no	—
73	C₂×C₄×C₁₂₇	—	2	4	no	—
74	C₂×C₄×C₁₂₇	—	2	2	no	—
75	C₂×C₄×C₁₂₇	—	4	2	no	—
76	C₂×C₄×C₁₂₇	—	4	2	no	—
77	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	#131	1	2	yes	—
83	C₈×C₁₂₇	—	2	2	no	—
84	C₈×C₁₂₇	—	4	2	no	—
85	C₈×C₁₂₇	#121	1	2	yes	—
86	C₈×C₁₂₇	#124	1	1	yes	characteristic
87	C₈×C₁₂₇	#127	1	1	yes	characteristic
88	C₂²×C₁₂₇	—	2	2	no	—
89	C₂²×C₁₂₇	—	4	4	no	—
90	C₂²×C₁₂₇	—	4	2	no	—
91	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
92	C₂²×C₁₂₇	#436	1	2	yes	—
93	C₄×C₁₂₇	—	2	2	no	—
94	C₄×C₁₂₇	—	4	2	no	—
95	C₄×C₁₂₇	#350	1	4	yes	complemented
96	C₄×C₁₂₇	#390	1	1	yes	characteristic
97	C₄×C₁₂₇	#408	1	1	yes	characteristic
98	C₄×C₁₂₇	C₈×D₄	1	2	yes	—
99	#13191	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₁₂₇	—	2	2	no	—
101	C₂×C₁₂₇	—	8	2	no	—
102	C₂×C₁₂₇	#1954	1	1	yes	characteristic
103	C₂×C₁₂₇	C₂×#350	1	2	yes	—
104	#2931	C₂×C₁₂₇	1	1	yes	characteristic
105	#2150	C₂×C₁₂₇	1	1	yes	characteristic, complemented
106	C₂×#350	C₂×C₁₂₇	1	4	yes	complemented
107	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
108	C₁₂₇	#13191	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×#131	C₂²×C₁₂₇	1	2	yes	—
110	#532	C₄×C₁₂₇	1	1	yes	characteristic
111	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
112	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
113	#436	—	2	2	no	—
114	C₈×D₄	—	2	2	no	—
115	#423	C₄×C₁₂₇	1	1	yes	characteristic
116	#408	C₄×C₁₂₇	1	2	yes	—
117	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
118	C₄⋊C₁₆	—	2	2	no	—
119	#350	—	2	8	no	—
120	C₄×C₁₆	—	2	2	no	—
121	C₂×#41	C₂×C₄×C₁₂₇	1	2	yes	—
122	#131	—	2	4	no	—
123	#131	—	2	4	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂×M₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
128	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
129	C₄⋊Q₈	C₈×C₁₂₇	1	1	yes	characteristic
130	#127	C₈×C₁₂₇	1	1	yes	characteristic
131	#124	C₈×C₁₂₇	1	2	yes	—
132	#121	—	2	2	no	—
133	C₄×D₄	—	2	2	no	—
134	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
135	M₃₂	—	2	4	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	D₄×C₁₂₇	1	4	yes	complemented
140	C₄⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	4	no	—
142	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
143	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
144	#41	—	2	4	no	—
145	#41	—	2	4	no	—
146	C₂×Q₈	C₂×C₈×C₁₂₇	1	2	yes	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	C₂×C₈×C₁₂₇	1	2	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	M₁₆×C₁₂₇	1	2	yes	—
152	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
153	M₁₆	—	2	4	no	—
154	M₁₆	—	4	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
160	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
161	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
162	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
163	C₂×C₈	#41×C₁₂₇	1	2	yes	—
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	4	no	—
166	C₄²	C₂²⋊C₄×C₁₂₇	1	2	yes	—
167	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
168	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
169	C₁₆	—	2	4	no	—
170	C₁₆	—	4	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
173	Q₈	—	2	4	no	—
174	D₄	—	2	4	no	—
175	D₄	—	4	4	no	—
176	D₄	—	4	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
182	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
184	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
185	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
186	C₂×C₄	#131×C₁₂₇	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	4	2	no	—
189	C₈	#121×C₁₂₇	1	2	yes	—
190	C₈	#124×C₁₂₇	1	1	yes	characteristic
191	C₈	#127×C₁₂₇	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	4	4	no	—
194	C₂²	—	4	2	no	—
195	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
196	C₂²	#436×C₁₂₇	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	#350×C₁₂₇	1	4	yes	complemented
200	C₄	#390×C₁₂₇	1	1	yes	characteristic
201	C₄	#408×C₁₂₇	1	1	yes	characteristic
202	C₄	C₈×D₄×C₁₂₇	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	8	2	no	—
205	C₂	C₁₂₇×#1954	1	1	yes	characteristic
206	C₂	C₂×#350×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₈
prime radical	#532×C₁₂₇	C₄
LCS	G ▷ C₂×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
2. C₁₂₇×#2150	C₂	semidirect
3. C₂×#350×C₁₂₇	C₂	semidirect
4. C₂×C₄×C₁₆×C₁₂₇	C₂	semidirect
35. C₂×C₁₆×C₁₂₇	D₄	semidirect
95. C₄×C₁₂₇	#350	semidirect
99. #13191	C₁₂₇	direct (N × H)
105. #2150	C₂×C₁₂₇	semidirect
106. C₂×#350	C₂×C₁₂₇	semidirect
107. C₂×C₄×C₁₆	C₂×C₁₂₇	semidirect
108. C₁₂₇	#13191	direct (N × H)
139. C₂×C₁₆	D₄×C₁₂₇	semidirect
199. C₄	#350×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2931	2
C₁₂₇×#2150	2
C₂×#350×C₁₂₇	2
C₂×C₄×C₁₆×C₁₂₇	2
#13191	127

Subgroups — C127×#13191 #44576864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13191 #44576864