Subgroups — C₁₂₇×#7168 #44570841

Order 32512.

254 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2150	C₂	1	1	yes	maximal, characteristic
2	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1770	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₂₇	—	2	1	no	—
5	C₂×M₃₂×C₁₂₇	—	2	1	no	—
6	C₂×C₄⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×#97×C₁₂₇	—	2	1	no	—
8	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₄×C₁₂₇	C₄	1	1	yes	characteristic
10	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
11	C₄⋊C₁₆×C₁₂₇	C₄	1	2	yes	complemented
12	#346×C₁₂₇	—	2	1	no	—
13	C₄×C₁₆×C₁₂₇	C₄	1	2	yes	complemented
14	#336×C₁₂₇	—	2	1	no	—
15	C₂²×C₈×C₁₂₇	—	2	1	no	—
16	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
19	C₂×C₄²×C₁₂₇	—	2	1	no	—
20	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	M₃₂×C₁₂₇	—	2	2	no	—
22	M₃₂×C₁₂₇	—	2	2	no	—
23	M₃₂×C₁₂₇	—	4	1	no	—
24	M₃₂×C₁₂₇	—	4	1	no	—
25	C₂×C₁₆×C₁₂₇	—	2	2	no	—
26	C₂×C₁₆×C₁₂₇	—	2	2	no	—
27	C₄⋊C₈×C₁₂₇	—	2	1	no	—
28	C₄⋊C₈×C₁₂₇	C₈	1	2	yes	—
29	#97×C₁₂₇	—	4	1	no	—
30	#97×C₁₂₇	—	4	1	no	—
31	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
35	#95×C₁₂₇	C₈	1	2	yes	—
36	C₂²×C₄×C₁₂₇	—	2	2	no	—
37	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	—	2	1	no	—
41	C₂×C₈×C₁₂₇	—	2	1	no	—
42	C₂×C₈×C₁₂₇	—	2	2	no	—
43	C₂×C₈×C₁₂₇	—	4	1	no	—
44	C₂×C₈×C₁₂₇	—	4	1	no	—
45	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	D₈	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
53	C₄⋊C₄×C₁₂₇	—	4	2	no	—
54	C₄²×C₁₂₇	—	4	1	no	—
55	C₄²×C₁₂₇	—	4	1	no	—
56	C₄²×C₁₂₇	C₄²	1	1	yes	characteristic
57	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₁₂₇	M₁₆	1	1	yes	characteristic
60	C₁₆×C₁₂₇	—	2	4	no	—
61	C₁₆×C₁₂₇	—	4	2	no	—
62	C₂³×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	—	2	2	no	—
64	C₂×C₄×C₁₂₇	—	4	1	no	—
65	C₂×C₄×C₁₂₇	—	4	2	no	—
66	C₂×C₄×C₁₂₇	—	4	2	no	—
67	C₂×C₄×C₁₂₇	—	4	2	no	—
68	C₂×C₄×C₁₂₇	#95	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	#97	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	#103	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	#104	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	#104	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
78	C₈×C₁₂₇	—	2	1	no	—
79	C₈×C₁₂₇	—	2	1	no	—
80	C₈×C₁₂₇	—	4	2	no	—
81	C₈×C₁₂₇	#102	1	1	yes	characteristic
82	C₈×C₁₂₇	#103	1	1	yes	characteristic
83	C₈×C₁₂₇	#106	1	1	yes	characteristic
84	C₈×C₁₂₇	#107	1	1	yes	characteristic
85	C₂²×C₁₂₇	—	2	1	no	—
86	C₂²×C₁₂₇	—	2	1	no	—
87	C₂²×C₁₂₇	D₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₁₂₇	Q₈⋊C₈	1	1	yes	characteristic
89	C₂²×C₁₂₇	#336	1	1	yes	characteristic
90	C₄×C₁₂₇	—	2	1	no	—
91	C₄×C₁₂₇	—	2	1	no	—
92	C₄×C₁₂₇	—	2	1	no	—
93	C₄×C₁₂₇	—	8	2	no	—
94	C₄×C₁₂₇	D₄⋊C₈	1	1	yes	characteristic
95	C₄×C₁₂₇	Q₈⋊C₈	1	1	yes	characteristic
96	C₄×C₁₂₇	#339	1	1	yes	characteristic
97	C₄×C₁₂₇	#340	1	1	yes	characteristic
98	C₄×C₁₂₇	#347	1	2	yes	—
99	#7168	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₁₂₇	—	2	1	no	—
101	C₂×C₁₂₇	—	2	1	no	—
102	C₂×C₁₂₇	#1279	1	1	yes	characteristic
103	C₂×C₁₂₇	#1375	1	1	yes	characteristic
104	C₂×C₁₂₇	#1378	1	1	yes	characteristic
105	#2150	C₂×C₁₂₇	1	1	yes	characteristic
106	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
107	#1770	C₂×C₁₂₇	1	1	yes	characteristic
108	C₁₂₇	#7168	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×M₃₂	—	2	1	no	—
110	C₂×M₃₂	—	2	1	no	—
111	C₂×C₄⋊C₈	C₄×C₁₂₇	1	1	yes	characteristic
112	C₂×#97	—	2	1	no	—
113	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
114	C₄×C₄⋊C₄	C₄×C₁₂₇	1	1	yes	characteristic
115	C₄⋊C₁₆	—	2	1	no	—
116	C₄⋊C₁₆	C₄×C₁₂₇	1	2	yes	complemented
117	#346	—	2	1	no	—
118	C₄×C₁₆	C₄×C₁₂₇	1	2	yes	complemented
119	#336	—	2	1	no	—
120	C₂²×C₈	—	2	1	no	—
121	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
122	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	2	1	no	—
124	C₂×C₄²	—	2	1	no	—
125	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
126	M₃₂	—	2	2	no	—
127	M₃₂	—	2	2	no	—
128	M₃₂	—	4	1	no	—
129	M₃₂	—	4	1	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₄⋊C₈	—	2	1	no	—
133	C₄⋊C₈	C₈×C₁₂₇	1	2	yes	—
134	#97	—	4	1	no	—
135	#97	—	4	1	no	—
136	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
137	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
138	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
139	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
140	#95	C₈×C₁₂₇	1	2	yes	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
144	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
152	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
153	C₂×C₈	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
154	C₂×C₈	D₈×C₁₂₇	1	1	yes	characteristic
155	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
156	C₂×C₈	SD₁₆×C₁₂₇	1	1	yes	characteristic
157	C₂×C₈	Q₁₆×C₁₂₇	1	1	yes	characteristic
158	C₄⋊C₄	—	4	2	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	C₄²×C₁₂₇	1	1	yes	characteristic
162	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
163	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
164	C₄²	M₁₆×C₁₂₇	1	1	yes	characteristic
165	C₁₆	—	2	4	no	—
166	C₁₆	—	4	2	no	—
167	C₂³	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	#95×C₁₂₇	1	1	yes	characteristic
174	C₂×C₄	C₄×C₈×C₁₂₇	1	1	yes	characteristic
175	C₂×C₄	#97×C₁₂₇	1	1	yes	characteristic
176	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
177	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
178	C₂×C₄	#103×C₁₂₇	1	1	yes	characteristic
179	C₂×C₄	#104×C₁₂₇	1	1	yes	characteristic
180	C₂×C₄	#104×C₁₂₇	1	1	yes	characteristic
181	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
182	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	4	2	no	—
186	C₈	#102×C₁₂₇	1	1	yes	characteristic
187	C₈	#103×C₁₂₇	1	1	yes	characteristic
188	C₈	#106×C₁₂₇	1	1	yes	characteristic
189	C₈	#107×C₁₂₇	1	1	yes	characteristic
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	D₄⋊C₈×C₁₂₇	1	1	yes	characteristic
193	C₂²	Q₈⋊C₈×C₁₂₇	1	1	yes	characteristic
194	C₂²	#336×C₁₂₇	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	8	2	no	—
199	C₄	D₄⋊C₈×C₁₂₇	1	1	yes	characteristic
200	C₄	Q₈⋊C₈×C₁₂₇	1	1	yes	characteristic
201	C₄	#339×C₁₂₇	1	1	yes	characteristic
202	C₄	#340×C₁₂₇	1	1	yes	characteristic
203	C₄	#347×C₁₂₇	1	2	yes	—
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	C₁₂₇×#1279	1	1	yes	characteristic
207	C₂	C₁₂₇×#1375	1	1	yes	characteristic
208	C₂	C₁₂₇×#1378	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#95
commutator	C₂×C₄	C₄×C₈×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₂₇
socle	C₂²×C₁₂₇	#336
prime radical	C₂³×C₁₂₇	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₄⋊C₁₆×C₁₂₇	C₄	semidirect
13. C₄×C₁₆×C₁₂₇	C₄	semidirect
99. #7168	C₁₂₇	direct (N × H)
108. C₁₂₇	#7168	direct (N × H)
116. C₄⋊C₁₆	C₄×C₁₂₇	semidirect
118. C₄×C₁₆	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2150	2
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1770	2
#7168	127

Subgroups — C127×#7168 #44570841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7168 #44570841