Subgroups — C₁₂₇×#17314 #44580987

Order 32512.

914 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1450	C₂	1	4	yes	maximal, complemented
4	C₈×M₁₆×C₁₂₇	C₂	1	8	yes	maximal, complemented
5	C₂²×M₁₆×C₁₂₇	C₄	1	2	yes	—
6	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
7	#431×C₁₂₇	C₄	1	4	yes	—
8	#431×C₁₂₇	C₄	1	4	yes	—
9	#431×C₁₂₇	C₂²	1	4	yes	—
10	#431×C₁₂₇	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈×C₁₂₇	C₄	1	4	yes	—
12	C₂×C₂²⋊C₈×C₁₂₇	C₄	1	4	yes	—
13	C₄×M₁₆×C₁₂₇	C₄	1	4	yes	—
14	C₄×M₁₆×C₁₂₇	C₂²	1	4	yes	—
15	C₂×#97×C₁₂₇	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₂₇	C₂²	1	4	yes	—
18	#322×C₁₂₇	C₂²	1	8	yes	complemented
19	C₈²×C₁₂₇	C₂²	1	8	yes	complemented
20	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₁₂₇	C₈	1	16	yes	complemented
23	C₂×M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₂₇	C₂×C₄	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	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
32	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
34	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
35	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
36	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
37	#97×C₁₂₇	C₂×C₄	1	8	yes	—
38	C₄×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
40	C₄×C₈×C₁₂₇	C₂³	1	4	yes	—
41	C₄×C₈×C₁₂₇	C₂³	1	8	yes	—
42	C₂⁴×C₁₂₇	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	—	4	4	no	—
44	C₂²×C₄×C₁₂₇	C₄²	1	2	yes	—
45	C₂²×C₄×C₁₂₇	C₄²	1	2	yes	—
46	C₂²×C₄×C₁₂₇	C₄²	1	2	yes	—
47	C₂²×C₄×C₁₂₇	C₄²	1	2	yes	—
48	C₂²×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	C₂×C₈	1	8	yes	—
50	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
55	M₁₆×C₁₂₇	C₂×C₈	1	32	yes	complemented
56	C₂×C₈×C₁₂₇	—	4	8	no	—
57	C₂×C₈×C₁₂₇	C₂×C₈	1	16	yes	complemented
58	C₂×C₈×C₁₂₇	C₂²×C₄	1	16	yes	—
59	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
60	C₄²×C₁₂₇	C₄²	1	1	yes	characteristic
61	C₄²×C₁₂₇	C₄²	1	1	yes	characteristic
62	C₄²×C₁₂₇	C₄²	1	2	yes	—
63	C₄²×C₁₂₇	C₄²	1	2	yes	—
64	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
66	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
70	C₂³×C₁₂₇	—	4	2	no	—
71	C₂³×C₁₂₇	C₄×C₈	1	4	yes	—
72	C₂³×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
73	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
74	C₂×C₄×C₁₂₇	—	2	4	no	—
75	C₂×C₄×C₁₂₇	—	2	4	no	—
76	C₂×C₄×C₁₂₇	—	2	4	no	—
77	C₂×C₄×C₁₂₇	—	4	2	no	—
78	C₂×C₄×C₁₂₇	—	4	2	no	—
79	C₂×C₄×C₁₂₇	—	4	2	no	—
80	C₂×C₄×C₁₂₇	C₄×C₈	1	8	yes	—
81	C₂×C₄×C₁₂₇	C₄×C₈	1	4	yes	—
82	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
83	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂²×C₈	1	4	yes	—
88	C₂×C₄×C₁₂₇	C₂²×C₈	1	8	yes	—
89	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
91	C₈×C₁₂₇	—	2	16	no	—
92	C₈×C₁₂₇	C₂²×C₈	1	16	yes	—
93	C₂²×C₁₂₇	—	2	2	no	—
94	C₂²×C₁₂₇	—	2	2	no	—
95	C₂²×C₁₂₇	—	2	2	no	—
96	C₂²×C₁₂₇	—	4	4	no	—
97	C₂²×C₁₂₇	C₂×C₄×C₈	1	2	yes	—
98	C₂²×C₁₂₇	C₂×C₄×C₈	1	4	yes	—
99	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
100	C₄×C₁₂₇	—	2	2	no	—
101	C₄×C₁₂₇	—	2	2	no	—
102	C₄×C₁₂₇	—	2	2	no	—
103	C₄×C₁₂₇	C₂×C₄×C₈	1	2	yes	—
104	C₄×C₁₂₇	C₂×C₄×C₈	1	4	yes	—
105	C₄×C₁₂₇	C₂³×C₈	1	2	yes	—
106	C₄×C₁₂₇	#568	1	4	yes	—
107	#17314	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₁₂₇	—	2	4	no	—
109	C₂×C₁₂₇	—	2	2	no	—
110	C₂×C₁₂₇	C₂²×C₄×C₈	1	1	yes	characteristic
111	C₂×C₁₂₇	#2873	1	2	yes	—
112	C₂×#431	C₂×C₁₂₇	1	2	yes	—
113	C₂×C₄×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
114	#1450	C₂×C₁₂₇	1	4	yes	complemented
115	C₈×M₁₆	C₂×C₁₂₇	1	8	yes	complemented
116	C₁₂₇	#17314	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×M₁₆	C₄×C₁₂₇	1	2	yes	—
118	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
119	#431	C₄×C₁₂₇	1	4	yes	—
120	#431	C₄×C₁₂₇	1	4	yes	—
121	#431	C₂²×C₁₂₇	1	4	yes	—
122	#431	C₂²×C₁₂₇	1	4	yes	—
123	C₂×C₄⋊C₈	C₄×C₁₂₇	1	4	yes	—
124	C₂×C₂²⋊C₈	C₄×C₁₂₇	1	4	yes	—
125	C₄×M₁₆	C₄×C₁₂₇	1	4	yes	—
126	C₄×M₁₆	C₂²×C₁₂₇	1	4	yes	—
127	C₂×#97	C₄×C₁₂₇	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
129	C₂×C₄×C₈	C₂²×C₁₂₇	1	4	yes	—
130	#322	C₂²×C₁₂₇	1	8	yes	complemented
131	C₈²	C₂²×C₁₂₇	1	8	yes	complemented
132	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
133	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
134	C₂×M₁₆	C₈×C₁₂₇	1	16	yes	complemented
135	C₂×M₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
136	C₂²×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
137	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
138	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
139	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
140	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
141	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
142	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
143	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
144	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
145	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
146	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
147	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
148	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
149	#97	C₂×C₄×C₁₂₇	1	8	yes	—
150	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
151	C₄×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
152	C₄×C₈	C₂³×C₁₂₇	1	4	yes	—
153	C₄×C₈	C₂³×C₁₂₇	1	8	yes	—
154	C₂⁴	C₄²×C₁₂₇	1	1	yes	characteristic
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	C₄²×C₁₂₇	1	2	yes	—
157	C₂²×C₄	C₄²×C₁₂₇	1	2	yes	—
158	C₂²×C₄	C₄²×C₁₂₇	1	2	yes	—
159	C₂²×C₄	C₄²×C₁₂₇	1	2	yes	—
160	C₂²×C₄	C₄²×C₁₂₇	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₈×C₁₂₇	1	8	yes	—
162	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
163	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
164	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
165	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
166	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
167	M₁₆	C₂×C₈×C₁₂₇	1	32	yes	complemented
168	C₂×C₈	—	4	8	no	—
169	C₂×C₈	C₂×C₈×C₁₂₇	1	16	yes	complemented
170	C₂×C₈	C₂²×C₄×C₁₂₇	1	16	yes	—
171	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
172	C₄²	C₄²×C₁₂₇	1	1	yes	characteristic
173	C₄²	C₄²×C₁₂₇	1	1	yes	characteristic
174	C₄²	C₄²×C₁₂₇	1	2	yes	—
175	C₄²	C₄²×C₁₂₇	1	2	yes	—
176	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
177	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
178	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
179	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
180	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
181	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
182	C₂³	—	4	2	no	—
183	C₂³	C₄×C₈×C₁₂₇	1	4	yes	—
184	C₂³	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
185	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	C₄×C₈×C₁₂₇	1	8	yes	—
193	C₂×C₄	C₄×C₈×C₁₂₇	1	4	yes	—
194	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
195	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
196	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
197	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
198	C₂×C₄	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₈×C₁₂₇	1	4	yes	—
200	C₂×C₄	C₂²×C₈×C₁₂₇	1	8	yes	—
201	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
202	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
203	C₈	—	2	16	no	—
204	C₈	C₂²×C₈×C₁₂₇	1	16	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	4	no	—
209	C₂²	C₂×C₄×C₈×C₁₂₇	1	2	yes	—
210	C₂²	C₂×C₄×C₈×C₁₂₇	1	4	yes	—
211	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	C₂×C₄×C₈×C₁₂₇	1	2	yes	—
216	C₄	C₂×C₄×C₈×C₁₂₇	1	4	yes	—
217	C₄	C₂³×C₈×C₁₂₇	1	2	yes	—
218	C₄	#568×C₁₂₇	1	4	yes	—
219	C₂	—	2	4	no	—
220	C₂	—	2	2	no	—
221	C₂	C₂²×C₄×C₈×C₁₂₇	1	1	yes	characteristic
222	C₂	C₁₂₇×#2873	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₂₇	C₂⁴
commutator	C₂	C₂²×C₄×C₈×C₁₂₇
Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂²×C₄²
prime radical	C₂⁴×C₁₂₇	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1450	C₂	semidirect
4. C₈×M₁₆×C₁₂₇	C₂	semidirect
18. #322×C₁₂₇	C₂²	semidirect
19. C₈²×C₁₂₇	C₂²	semidirect
22. C₂×M₁₆×C₁₂₇	C₈	semidirect
55. M₁₆×C₁₂₇	C₂×C₈	semidirect
57. C₂×C₈×C₁₂₇	C₂×C₈	semidirect
107. #17314	C₁₂₇	direct (N × H)
114. #1450	C₂×C₁₂₇	semidirect
115. C₈×M₁₆	C₂×C₁₂₇	semidirect
116. C₁₂₇	#17314	direct (N × H)
130. #322	C₂²×C₁₂₇	semidirect
131. C₈²	C₂²×C₁₂₇	semidirect
134. C₂×M₁₆	C₈×C₁₂₇	semidirect
167. M₁₆	C₂×C₈×C₁₂₇	semidirect
169. C₂×C₈	C₂×C₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₁₂₇	2
C₂×C₄×M₁₆×C₁₂₇	2
C₁₂₇×#1450	2
C₈×M₁₆×C₁₂₇	2
#17314	127

Subgroups — C127×#17314 #44580987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#17314 #44580987