Subgroups — C₁₂₇×#20460 #44584133

Order 32512.

994 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	C₂²×#108×C₁₂₇	C₂	1	2	yes	maximal
2	C₁₂₇×#2873	C₂	1	2	yes	maximal
3	C₂×#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1778	C₂	1	4	yes	maximal, complemented
5	C₄×#108×C₁₂₇	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1738	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₁₂₇	C₂²	1	4	yes	—
8	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×#108×C₁₂₇	C₄	1	16	yes	complemented
12	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
13	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
14	#405×C₁₂₇	—	2	8	no	—
15	#405×C₁₂₇	C₄	1	8	yes	—
16	C₄×M₁₆×C₁₂₇	C₂²	1	8	yes	—
17	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
19	#341×C₁₂₇	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₂₇	—	2	4	no	—
22	C₂×M₁₆×C₁₂₇	—	2	8	no	—
23	C₂×M₁₆×C₁₂₇	—	2	4	no	—
24	C₂×M₁₆×C₁₂₇	C₂×C₄	1	16	yes	—
25	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₁₂₇	C₂³	1	8	yes	—
27	C₂²×C₈×C₁₂₇	—	2	4	no	—
28	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
33	#117×C₁₂₇	C₂×C₄	1	4	yes	—
34	#117×C₁₂₇	D₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₂₇	Q₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₁₂₇	Q₈	1	2	yes	—
37	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
38	#108×C₁₂₇	C₂×C₄	1	16	yes	—
39	#108×C₁₂₇	C₂×C₄	1	16	yes	—
40	#97×C₁₂₇	—	2	8	no	—
41	#97×C₁₂₇	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₁₂₇	—	2	8	no	—
43	C₄×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
44	C₂⁴×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
51	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
52	M₁₆×C₁₂₇	—	2	8	no	—
53	M₁₆×C₁₂₇	—	2	16	no	—
54	M₁₆×C₁₂₇	—	2	8	no	—
55	M₁₆×C₁₂₇	C₂²×C₄	1	16	yes	—
56	C₂×C₈×C₁₂₇	—	2	16	no	—
57	C₂×C₈×C₁₂₇	—	2	8	no	—
58	C₂×C₈×C₁₂₇	—	2	4	no	—
59	C₂×C₈×C₁₂₇	C₄²	1	16	yes	—
60	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
65	C₄⋊C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
66	C₂²⋊C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
67	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
68	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
69	C₂³×C₁₂₇	—	4	2	no	—
70	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₁₂₇	C₄×Q₈	1	4	yes	—
72	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	—	4	4	no	—
74	C₂×C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
75	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	8	yes	—
76	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₁₂₇	#117	1	4	yes	—
78	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
79	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
80	C₂×C₄×C₁₂₇	C₄×Q₈	1	4	yes	—
81	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
83	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
85	C₈×C₁₂₇	—	2	4	no	—
86	C₈×C₁₂₇	—	2	16	no	—
87	C₈×C₁₂₇	C₂×C₄²	1	4	yes	—
88	C₈×C₁₂₇	C₂×C₄²	1	4	yes	—
89	C₂²×C₁₂₇	—	2	2	no	—
90	C₂²×C₁₂₇	—	2	4	no	—
91	C₂²×C₁₂₇	—	4	4	no	—
92	C₂²×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
93	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₁₂₇	C₂×C₄×Q₈	1	2	yes	—
95	C₄×C₁₂₇	—	2	8	no	—
96	C₄×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
97	C₄×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
98	C₄×C₁₂₇	C₂×#117	1	1	yes	characteristic
99	C₄×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
100	#20460	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂×C₁₂₇	—	2	2	no	—
102	C₂×C₁₂₇	—	2	4	no	—
103	C₂×C₁₂₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₁₂₇	#2916	1	2	yes	—
105	C₂²×#108	C₂×C₁₂₇	1	2	yes	—
106	#2873	C₂×C₁₂₇	1	2	yes	—
107	C₂×#405	C₂×C₁₂₇	1	1	yes	characteristic
108	#1778	C₂×C₁₂₇	1	4	yes	complemented
109	C₄×#108	C₂×C₁₂₇	1	4	yes	complemented
110	#1738	C₂×C₁₂₇	1	2	yes	—
111	C₁₂₇	#20460	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×M₁₆	C₂²×C₁₂₇	1	4	yes	—
113	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
114	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
115	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
116	C₂×#108	C₄×C₁₂₇	1	16	yes	complemented
117	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
118	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
119	#405	—	2	8	no	—
120	#405	C₄×C₁₂₇	1	8	yes	—
121	C₄×M₁₆	C₂²×C₁₂₇	1	8	yes	—
122	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
123	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
124	#341	C₂²×C₁₂₇	1	8	yes	—
125	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
126	C₂×M₁₆	—	2	4	no	—
127	C₂×M₁₆	—	2	8	no	—
128	C₂×M₁₆	—	2	4	no	—
129	C₂×M₁₆	C₂×C₄×C₁₂₇	1	16	yes	—
130	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
131	C₂×M₁₆	C₂³×C₁₂₇	1	8	yes	—
132	C₂²×C₈	—	2	4	no	—
133	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
134	C₂²×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
135	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
136	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
137	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
138	#117	C₂×C₄×C₁₂₇	1	4	yes	—
139	#117	D₄×C₁₂₇	1	4	yes	—
140	C₂×C₄⋊C₄	Q₈×C₁₂₇	1	2	yes	—
141	C₂×C₂²⋊C₄	Q₈×C₁₂₇	1	2	yes	—
142	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
143	#108	C₂×C₄×C₁₂₇	1	16	yes	—
144	#108	C₂×C₄×C₁₂₇	1	16	yes	—
145	#97	—	2	8	no	—
146	#97	C₂×C₄×C₁₂₇	1	8	yes	—
147	C₄×C₈	—	2	8	no	—
148	C₄×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
149	C₂⁴	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
151	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
152	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
153	C₂²×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
154	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
155	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
156	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
157	M₁₆	—	2	8	no	—
158	M₁₆	—	2	16	no	—
159	M₁₆	—	2	8	no	—
160	M₁₆	C₂²×C₄×C₁₂₇	1	16	yes	—
161	C₂×C₈	—	2	16	no	—
162	C₂×C₈	—	2	8	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	C₄²×C₁₂₇	1	16	yes	—
165	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
166	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
167	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
168	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
169	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
170	C₄⋊C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
171	C₂²⋊C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
172	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
173	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
174	C₂³	—	4	2	no	—
175	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
176	C₂³	C₄×Q₈×C₁₂₇	1	4	yes	—
177	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
180	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	8	yes	—
181	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
182	C₂×C₄	#117×C₁₂₇	1	4	yes	—
183	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
184	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
185	C₂×C₄	C₄×Q₈×C₁₂₇	1	4	yes	—
186	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
187	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
188	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
189	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
190	C₈	—	2	4	no	—
191	C₈	—	2	16	no	—
192	C₈	C₂×C₄²×C₁₂₇	1	4	yes	—
193	C₈	C₂×C₄²×C₁₂₇	1	4	yes	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	4	4	no	—
197	C₂²	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
198	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
199	C₂²	C₂×C₄×Q₈×C₁₂₇	1	2	yes	—
200	C₄	—	2	8	no	—
201	C₄	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
202	C₄	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
203	C₄	C₂×#117×C₁₂₇	1	1	yes	characteristic
204	C₄	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
205	C₂	—	2	2	no	—
206	C₂	—	2	4	no	—
207	C₂	C₂×C₄×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
208	C₂	C₁₂₇×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₂₇	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×#117×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1778	C₂	semidirect
5. C₄×#108×C₁₂₇	C₂	semidirect
11. C₂×#108×C₁₂₇	C₄	semidirect
100. #20460	C₁₂₇	direct (N × H)
108. #1778	C₂×C₁₂₇	semidirect
109. C₄×#108	C₂×C₁₂₇	semidirect
111. C₁₂₇	#20460	direct (N × H)
116. C₂×#108	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#108×C₁₂₇	2
C₁₂₇×#2873	2
C₂×#405×C₁₂₇	2
C₁₂₇×#1778	2
C₄×#108×C₁₂₇	2
C₁₂₇×#1738	2
#20460	127

Subgroups — C127×#20460 #44584133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20460 #44584133