Subgroups — C₁₂₇×#12053 #44575726

Order 32512.

206 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2183	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1370	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₁₂₇×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₁₂₇×#1311	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₄×Q₁₆×C₁₂₇	C₄	1	1	yes	characteristic
11	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
12	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
13	C₄⋊C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
15	#346×C₁₂₇	—	2	1	no	—
16	C₄×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	Q₈⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
21	Q₈⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
22	C₈²×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆×C₁₂₇	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₁₂₇	—	2	1	no	—
27	C₂×C₁₆×C₁₂₇	—	2	1	no	—
28	C₂×C₁₆×C₁₂₇	—	2	1	no	—
29	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
31	#107×C₁₂₇	C₈	1	1	yes	characteristic
32	C₄⋊C₈×C₁₂₇	—	2	1	no	—
33	C₄⋊C₈×C₁₂₇	—	2	1	no	—
34	C₄⋊C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	#103×C₁₂₇	C₈	1	1	yes	characteristic
37	#103×C₁₂₇	C₈	1	1	yes	characteristic
38	C₄×C₈×C₁₂₇	—	2	1	no	—
39	C₄×C₈×C₁₂₇	—	2	1	no	—
40	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
44	C₂×Q₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
45	Q₁₆×C₁₂₇	—	2	2	no	—
46	C₂×C₈×C₁₂₇	—	2	1	no	—
47	C₂×C₈×C₁₂₇	—	2	1	no	—
48	C₂×C₈×C₁₂₇	—	2	1	no	—
49	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₁₂₇	—	2	1	no	—
54	C₄⋊C₄×C₁₂₇	—	2	1	no	—
55	C₄⋊C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
56	C₄⋊C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₁₂₇	—	2	1	no	—
58	C₄²×C₁₂₇	—	2	1	no	—
59	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
60	C₁₆×C₁₂₇	—	2	2	no	—
61	C₁₆×C₁₂₇	—	2	1	no	—
62	C₁₆×C₁₂₇	—	4	1	no	—
63	C₁₆×C₁₂₇	—	4	1	no	—
64	C₁₆×C₁₂₇	Q₁₆	1	2	yes	complemented
65	Q₈×C₁₂₇	—	2	1	no	—
66	Q₈×C₁₂₇	—	2	1	no	—
67	Q₈×C₁₂₇	—	2	1	no	—
68	Q₈×C₁₂₇	M₃₂	1	2	yes	—
69	C₂×C₄×C₁₂₇	—	2	1	no	—
70	C₂×C₄×C₁₂₇	—	2	1	no	—
71	C₂×C₄×C₁₂₇	—	2	1	no	—
72	C₂×C₄×C₁₂₇	—	2	1	no	—
73	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	#131	1	1	yes	characteristic
76	C₈×C₁₂₇	—	2	1	no	—
77	C₈×C₁₂₇	—	2	1	no	—
78	C₈×C₁₂₇	—	2	1	no	—
79	C₈×C₁₂₇	—	4	1	no	—
80	C₈×C₁₂₇	—	4	1	no	—
81	C₈×C₁₂₇	—	4	1	no	—
82	C₈×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₈×C₁₂₇	#135	1	1	yes	characteristic
84	C₂²×C₁₂₇	C₈×D₄	1	1	yes	characteristic
85	C₄×C₁₂₇	—	2	2	no	—
86	C₄×C₁₂₇	—	2	1	no	—
87	C₄×C₁₂₇	—	4	1	no	—
88	C₄×C₁₂₇	—	4	1	no	—
89	C₄×C₁₂₇	—	4	1	no	—
90	C₄×C₁₂₇	C₄×Q₁₆	1	1	yes	characteristic
91	C₄×C₁₂₇	#443	1	1	yes	characteristic
92	C₄×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
93	C₄×C₁₂₇	#504	1	1	yes	characteristic
94	#12053	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₁₂₇	C₈×Q₁₆	1	1	yes	characteristic
96	C₂×C₁₂₇	#2168	1	1	yes	characteristic
97	C₂×C₁₂₇	#2171	1	1	yes	characteristic
98	#2183	C₂×C₁₂₇	1	1	yes	characteristic
99	Q₈×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
100	C₈×Q₁₆	C₂×C₁₂₇	1	1	yes	characteristic
101	#1370	C₂×C₁₂₇	1	1	yes	characteristic
102	Q₈⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
103	Q₈⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
104	#1311	C₂×C₁₂₇	1	1	yes	characteristic
105	C₁₂₇	#12053	1	1	yes	characteristic, direct product factor, Sylow
106	C₈×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
107	C₈×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
108	C₄×Q₁₆	C₄×C₁₂₇	1	1	yes	characteristic
109	C₄⋊C₁₆	—	2	1	no	—
110	C₄⋊C₁₆	—	2	1	no	—
111	C₄⋊C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
112	C₄⋊C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
113	#346	—	2	1	no	—
114	C₄×C₁₆	—	2	1	no	—
115	C₄×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
116	C₄×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
117	#335	C₄×C₁₂₇	1	1	yes	characteristic
118	Q₈⋊C₈	C₄×C₁₂₇	1	1	yes	characteristic
119	Q₈⋊C₈	C₄×C₁₂₇	1	1	yes	characteristic
120	C₈²	C₂²×C₁₂₇	1	1	yes	characteristic
121	C₂×Q₁₆	C₈×C₁₂₇	1	1	yes	characteristic
122	C₄×Q₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
123	C₄×Q₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
128	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
129	#107	C₈×C₁₂₇	1	1	yes	characteristic
130	C₄⋊C₈	—	2	1	no	—
131	C₄⋊C₈	—	2	1	no	—
132	C₄⋊C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
133	C₄⋊C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
134	#103	C₈×C₁₂₇	1	1	yes	characteristic
135	#103	C₈×C₁₂₇	1	1	yes	characteristic
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
139	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
140	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
141	C₂×Q₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
142	C₂×Q₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
143	Q₁₆	—	2	2	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
148	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
149	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
150	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
154	C₄⋊C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
158	C₁₆	—	2	2	no	—
159	C₁₆	—	2	1	no	—
160	C₁₆	—	4	1	no	—
161	C₁₆	—	4	1	no	—
162	C₁₆	Q₁₆×C₁₂₇	1	2	yes	complemented
163	Q₈	—	2	1	no	—
164	Q₈	—	2	1	no	—
165	Q₈	—	2	1	no	—
166	Q₈	M₃₂×C₁₂₇	1	2	yes	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
172	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
173	C₂×C₄	#131×C₁₂₇	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	4	1	no	—
178	C₈	—	4	1	no	—
179	C₈	—	4	1	no	—
180	C₈	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
181	C₈	#135×C₁₂₇	1	1	yes	characteristic
182	C₂²	C₈×D₄×C₁₂₇	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	1	no	—
187	C₄	—	4	1	no	—
188	C₄	C₄×Q₁₆×C₁₂₇	1	1	yes	characteristic
189	C₄	#443×C₁₂₇	1	1	yes	characteristic
190	C₄	C₂×M₃₂×C₁₂₇	1	1	yes	characteristic
191	C₄	#504×C₁₂₇	1	1	yes	characteristic
192	C₂	C₈×Q₁₆×C₁₂₇	1	1	yes	characteristic
193	C₂	C₁₂₇×#2168	1	1	yes	characteristic
194	C₂	C₁₂₇×#2171	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, prime radical	C₂²×C₁₂₇	C₈×D₄
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
64. C₁₆×C₁₂₇	Q₁₆	semidirect
94. #12053	C₁₂₇	direct (N × H)
105. C₁₂₇	#12053	direct (N × H)
162. C₁₆	Q₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2183	2
Q₈×C₁₆×C₁₂₇	2
C₈×Q₁₆×C₁₂₇	2
C₁₂₇×#1370	2
C₁₂₇×Q₈⋊C₁₆	2
C₁₂₇×#1311	2
#12053	127

Subgroups — C127×#12053 #44575726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12053 #44575726