Subgroups — C₁₂₇×#20831 #44584504

Order 32512.

758 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2920	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1848	C₂	1	2	yes	maximal, complemented
4	C₁₂₇×#1837	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1811	C₂	1	4	yes	maximal
6	C₁₂₇×#1778	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₁₂₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
9	#432×C₁₂₇	—	2	4	no	—
10	#431×C₁₂₇	—	2	4	no	—
11	C₂×#108×C₁₂₇	C₂²	1	8	yes	—
12	#423×C₁₂₇	C₂²	1	4	yes	—
13	#407×C₁₂₇	C₂²	1	4	yes	—
14	C₄×M₁₆×C₁₂₇	C₄	1	4	yes	—
15	C₄×M₁₆×C₁₂₇	C₄	1	4	yes	—
16	C₄×M₁₆×C₁₂₇	C₂²	1	4	yes	—
17	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	#341×C₁₂₇	C₂²	1	8	yes	—
21	C₂³×C₄×C₁₂₇	Q₈	1	2	yes	—
22	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₁₂₇	—	2	8	no	—
24	C₂×M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
25	C₂×M₁₆×C₁₂₇	C₂³	1	8	yes	—
26	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
27	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
28	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
31	C₂×C₄²×C₁₂₇	Q₈	1	2	yes	—
32	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
34	#108×C₁₂₇	—	2	16	no	—
35	C₄⋊C₈×C₁₂₇	—	2	8	no	—
36	C₂²⋊C₈×C₁₂₇	—	2	8	no	—
37	#97×C₁₂₇	—	2	4	no	—
38	#97×C₁₂₇	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₁₂₇	—	2	4	no	—
40	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
44	C₂⁴×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	—	2	4	no	—
46	C₂²×C₄×C₁₂₇	—	4	4	no	—
47	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
53	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
55	M₁₆×C₁₂₇	—	2	16	no	—
56	M₁₆×C₁₂₇	—	2	16	no	—
57	C₂×C₈×C₁₂₇	—	2	4	no	—
58	C₂×C₈×C₁₂₇	—	2	8	no	—
59	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
60	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₁₂₇	#41	1	8	yes	—
62	C₄²×C₁₂₇	C₄⋊C₄	1	4	yes	—
63	C₄²×C₁₂₇	C₄⋊C₄	1	4	yes	—
64	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
68	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
69	C₂³×C₁₂₇	—	2	2	no	—
70	C₂³×C₁₂₇	—	4	2	no	—
71	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	—	2	4	no	—
75	C₂×C₄×C₁₂₇	—	2	4	no	—
76	C₂×C₄×C₁₂₇	—	2	8	no	—
77	C₂×C₄×C₁₂₇	—	2	4	no	—
78	C₂×C₄×C₁₂₇	—	4	4	no	—
79	C₂×C₄×C₁₂₇	—	4	2	no	—
80	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	C₂×#41	1	4	yes	—
87	C₈×C₁₂₇	—	2	4	no	—
88	C₈×C₁₂₇	—	4	8	no	—
89	C₈×C₁₂₇	#117	1	8	yes	—
90	C₂²×C₁₂₇	—	2	2	no	—
91	C₂²×C₁₂₇	—	2	4	no	—
92	C₂²×C₁₂₇	—	2	2	no	—
93	C₂²×C₁₂₇	—	4	4	no	—
94	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₁₂₇	#533	1	2	yes	—
96	C₄×C₁₂₇	—	2	4	no	—
97	C₄×C₁₂₇	—	2	2	no	—
98	C₄×C₁₂₇	—	2	2	no	—
99	C₄×C₁₂₇	#430	1	4	yes	—
100	C₄×C₁₂₇	C₂×#117	1	2	yes	—
101	C₄×C₁₂₇	#532	1	2	yes	—
102	#20831	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₁₂₇	—	2	2	no	—
104	C₂×C₁₂₇	—	4	2	no	—
105	C₂×C₁₂₇	#2285	1	1	yes	characteristic
106	C₂×C₁₂₇	C₂×#430	1	2	yes	—
107	#2920	C₂×C₁₂₇	1	2	yes	—
108	C₂×C₄×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
109	#1848	C₂×C₁₂₇	1	2	yes	complemented
110	#1837	C₂×C₁₂₇	1	2	yes	complemented
111	#1811	C₂×C₁₂₇	1	4	yes	—
112	#1778	C₂×C₁₂₇	1	4	yes	complemented
113	C₁₂₇	#20831	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×M₁₆	C₂²×C₁₂₇	1	2	yes	—
115	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
116	#432	—	2	4	no	—
117	#431	—	2	4	no	—
118	C₂×#108	C₂²×C₁₂₇	1	8	yes	—
119	#423	C₂²×C₁₂₇	1	4	yes	—
120	#407	C₂²×C₁₂₇	1	4	yes	—
121	C₄×M₁₆	C₄×C₁₂₇	1	4	yes	—
122	C₄×M₁₆	C₄×C₁₂₇	1	4	yes	—
123	C₄×M₁₆	C₂²×C₁₂₇	1	4	yes	—
124	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
127	#341	C₂²×C₁₂₇	1	8	yes	—
128	C₂³×C₄	Q₈×C₁₂₇	1	2	yes	—
129	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
130	C₂×M₁₆	—	2	8	no	—
131	C₂×M₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
132	C₂×M₁₆	C₂³×C₁₂₇	1	8	yes	—
133	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
134	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
135	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
136	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
137	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
138	C₂×C₄²	Q₈×C₁₂₇	1	2	yes	—
139	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
140	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
141	#108	—	2	16	no	—
142	C₄⋊C₈	—	2	8	no	—
143	C₂²⋊C₈	—	2	8	no	—
144	#97	—	2	4	no	—
145	#97	C₂×C₄×C₁₂₇	1	8	yes	—
146	C₄×C₈	—	2	4	no	—
147	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
148	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
149	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
150	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
151	C₂⁴	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
155	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
156	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
157	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
158	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
159	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
160	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
161	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
162	M₁₆	—	2	16	no	—
163	M₁₆	—	2	16	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	8	no	—
166	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
167	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
168	C₂×C₈	#41×C₁₂₇	1	8	yes	—
169	C₄²	C₄⋊C₄×C₁₂₇	1	4	yes	—
170	C₄²	C₄⋊C₄×C₁₂₇	1	4	yes	—
171	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
172	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
173	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
174	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
175	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	—	4	2	no	—
178	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
179	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
188	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
189	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
190	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
191	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
192	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
193	C₂×C₄	C₂×#41×C₁₂₇	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	4	8	no	—
196	C₈	#117×C₁₂₇	1	8	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	4	no	—
201	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
202	C₂²	#533×C₁₂₇	1	2	yes	—
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	#430×C₁₂₇	1	4	yes	—
207	C₄	C₂×#117×C₁₂₇	1	2	yes	—
208	C₄	#532×C₁₂₇	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	4	2	no	—
211	C₂	C₁₂₇×#2285	1	1	yes	characteristic
212	C₂	C₂×#430×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₂₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1848	C₂	semidirect
4. C₁₂₇×#1837	C₂	semidirect
6. C₁₂₇×#1778	C₂	semidirect
102. #20831	C₁₂₇	direct (N × H)
109. #1848	C₂×C₁₂₇	semidirect
110. #1837	C₂×C₁₂₇	semidirect
112. #1778	C₂×C₁₂₇	semidirect
113. C₁₂₇	#20831	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#2920	2
C₂×C₄×M₁₆×C₁₂₇	2
C₁₂₇×#1848	2
C₁₂₇×#1837	2
C₁₂₇×#1811	2
C₁₂₇×#1778	2
#20831	127

Subgroups — C127×#20831 #44584504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20831 #44584504