Subgroups — C₁₂₇×#13190 #44576863

Order 32512.

522 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊C₁₆×C₁₂₇	C₂	1	4	yes	maximal, complemented
4	C₂×#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₈×C₁₂₇	C₂²	1	2	yes	—
6	C₂×C₄×D₄×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
8	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
9	C₈×D₄×C₁₂₇	—	2	2	no	—
10	C₈×D₄×C₁₂₇	—	2	2	no	—
11	C₂×C₄⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊C₈×C₁₂₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
15	C₂²⋊C₁₆×C₁₂₇	—	2	8	no	—
16	#346×C₁₂₇	—	2	2	no	—
17	C₂²×D₄×C₁₂₇	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₂₇	—	2	4	no	—
20	C₂²×C₈×C₁₂₇	—	2	2	no	—
21	C₂²×C₈×C₁₂₇	—	2	2	no	—
22	C₂²×C₈×C₁₂₇	—	2	4	no	—
23	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₄×D₄×C₁₂₇	—	2	2	no	—
27	C₄×D₄×C₁₂₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₂₇	C₈	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₂₇	C₈	1	2	yes	—
30	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₁₂₇	—	2	2	no	—
32	C₂×C₁₆×C₁₂₇	—	2	2	no	—
33	C₂×C₁₆×C₁₂₇	—	2	4	no	—
34	C₂×C₁₆×C₁₂₇	—	2	2	no	—
35	C₂×C₁₆×C₁₂₇	D₄	1	4	yes	complemented
36	C₄⋊C₈×C₁₂₇	—	2	2	no	—
37	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
38	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
39	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
40	C₂⁴×C₁₂₇	C₂×C₈	1	2	yes	—
41	C₂×D₄×C₁₂₇	—	2	4	no	—
42	C₂×D₄×C₁₂₇	—	2	2	no	—
43	C₂²×C₄×C₁₂₇	—	2	2	no	—
44	C₂²×C₄×C₁₂₇	—	2	4	no	—
45	C₂²×C₄×C₁₂₇	—	2	2	no	—
46	C₂²×C₄×C₁₂₇	—	2	4	no	—
47	C₂²×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	—	2	2	no	—
52	C₂×C₈×C₁₂₇	—	2	2	no	—
53	C₂×C₈×C₁₂₇	—	2	4	no	—
54	C₂×C₈×C₁₂₇	—	4	4	no	—
55	C₂×C₈×C₁₂₇	—	4	2	no	—
56	C₂×C₈×C₁₂₇	—	4	2	no	—
57	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₂₇	#41	1	2	yes	—
62	C₄⋊C₄×C₁₂₇	—	2	2	no	—
63	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
64	C₄²×C₁₂₇	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₁₂₇	M₁₆	1	2	yes	—
66	C₁₆×C₁₂₇	—	2	4	no	—
67	C₁₆×C₁₂₇	—	4	2	no	—
68	C₂³×C₁₂₇	—	2	2	no	—
69	C₂³×C₁₂₇	—	2	4	no	—
70	C₂³×C₁₂₇	—	2	2	no	—
71	C₂³×C₁₂₇	—	2	4	no	—
72	C₂³×C₁₂₇	M₃₂	1	4	yes	—
73	C₂³×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
74	D₄×C₁₂₇	—	2	8	no	—
75	C₂×C₄×C₁₂₇	—	2	2	no	—
76	C₂×C₄×C₁₂₇	—	2	2	no	—
77	C₂×C₄×C₁₂₇	—	2	2	no	—
78	C₂×C₄×C₁₂₇	—	2	4	no	—
79	C₂×C₄×C₁₂₇	—	4	2	no	—
80	C₂×C₄×C₁₂₇	—	4	2	no	—
81	C₂×C₄×C₁₂₇	—	4	4	no	—
82	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
84	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	#131	1	2	yes	—
88	C₈×C₁₂₇	—	2	2	no	—
89	C₈×C₁₂₇	—	4	2	no	—
90	C₈×C₁₂₇	#121	1	2	yes	—
91	C₈×C₁₂₇	#124	1	1	yes	characteristic
92	C₈×C₁₂₇	#127	1	1	yes	characteristic
93	C₂²×C₁₂₇	—	2	4	no	—
94	C₂²×C₁₂₇	—	2	8	no	—
95	C₂²×C₁₂₇	—	4	2	no	—
96	C₂²×C₁₂₇	—	4	4	no	—
97	C₂²×C₁₂₇	—	4	2	no	—
98	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
99	C₂²×C₁₂₇	C₈×D₄	1	2	yes	—
100	C₂²×C₁₂₇	C₂×M₃₂	1	2	yes	—
101	C₂²×C₁₂₇	#504	1	2	yes	—
102	C₄×C₁₂₇	—	2	2	no	—
103	C₄×C₁₂₇	—	2	2	no	—
104	C₄×C₁₂₇	—	4	2	no	—
105	C₄×C₁₂₇	#390	1	1	yes	characteristic
106	C₄×C₁₂₇	#408	1	1	yes	characteristic
107	C₄×C₁₂₇	#436	1	2	yes	—
108	#13190	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₁₂₇	—	4	4	no	—
110	C₂×C₁₂₇	#1954	1	1	yes	characteristic
111	C₂×C₁₂₇	#2112	1	1	yes	characteristic
112	C₂×C₁₂₇	#2113	1	1	yes	characteristic
113	C₂×C₁₂₇	#2168	1	4	yes	—
114	C₂×C₈×D₄	C₂×C₁₂₇	1	1	yes	characteristic
115	C₂×C₄⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
116	C₂×C₂²⋊C₁₆	C₂×C₁₂₇	1	4	yes	complemented
117	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic, complemented
118	C₁₂₇	#13190	1	1	yes	characteristic, direct product factor, Sylow
119	C₂³×C₈	C₂²×C₁₂₇	1	2	yes	—
120	C₂×C₄×D₄	C₄×C₁₂₇	1	1	yes	characteristic
121	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
122	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
123	C₈×D₄	—	2	2	no	—
124	C₈×D₄	—	2	2	no	—
125	C₂×C₄⋊C₈	C₄×C₁₂₇	1	1	yes	characteristic
126	C₂×C₂²⋊C₈	C₄×C₁₂₇	1	2	yes	—
127	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
128	C₄⋊C₁₆	—	2	2	no	—
129	C₂²⋊C₁₆	—	2	8	no	—
130	#346	—	2	2	no	—
131	C₂²×D₄	C₈×C₁₂₇	1	1	yes	characteristic
132	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
133	C₂²×C₈	—	2	4	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	4	no	—
137	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
138	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
139	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
140	C₄×D₄	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	C₂×C₄⋊C₄	C₈×C₁₂₇	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	C₈×C₁₂₇	1	2	yes	—
144	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	2	4	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	D₄×C₁₂₇	1	4	yes	complemented
150	C₄⋊C₈	—	2	2	no	—
151	C₂²⋊C₈	—	2	4	no	—
152	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
153	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
154	C₂⁴	C₂×C₈×C₁₂₇	1	2	yes	—
155	C₂×D₄	—	2	4	no	—
156	C₂×D₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	C₂×C₈×C₁₂₇	1	2	yes	—
162	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
164	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	4	4	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
172	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
174	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
175	C₂×C₈	#41×C₁₂₇	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	4	no	—
178	C₄²	C₂²⋊C₄×C₁₂₇	1	2	yes	—
179	C₄²	M₁₆×C₁₂₇	1	2	yes	—
180	C₁₆	—	2	4	no	—
181	C₁₆	—	4	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	4	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	4	no	—
186	C₂³	M₃₂×C₁₂₇	1	4	yes	—
187	C₂³	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
188	D₄	—	2	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
197	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
199	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
200	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
201	C₂×C₄	#131×C₁₂₇	1	2	yes	—
202	C₈	—	2	2	no	—
203	C₈	—	4	2	no	—
204	C₈	#121×C₁₂₇	1	2	yes	—
205	C₈	#124×C₁₂₇	1	1	yes	characteristic
206	C₈	#127×C₁₂₇	1	1	yes	characteristic
207	C₂²	—	2	4	no	—
208	C₂²	—	2	8	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
213	C₂²	C₈×D₄×C₁₂₇	1	2	yes	—
214	C₂²	C₂×M₃₂×C₁₂₇	1	2	yes	—
215	C₂²	#504×C₁₂₇	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	4	2	no	—
219	C₄	#390×C₁₂₇	1	1	yes	characteristic
220	C₄	#408×C₁₂₇	1	1	yes	characteristic
221	C₄	#436×C₁₂₇	1	2	yes	—
222	C₂	—	4	4	no	—
223	C₂	C₁₂₇×#1954	1	1	yes	characteristic
224	C₂	C₁₂₇×#2112	1	1	yes	characteristic
225	C₂	C₁₂₇×#2113	1	1	yes	characteristic
226	C₂	C₁₂₇×#2168	1	4	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₂₇	C₂³
commutator	C₂³	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂³×C₁₂₇	C₂²×C₈
prime radical	C₂²×D₄×C₁₂₇	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊C₁₆×C₁₂₇	C₂	semidirect
3. C₂×C₂²⋊C₁₆×C₁₂₇	C₂	semidirect
4. C₂×#346×C₁₂₇	C₂	semidirect
35. C₂×C₁₆×C₁₂₇	D₄	semidirect
108. #13190	C₁₂₇	direct (N × H)
115. C₂×C₄⋊C₁₆	C₂×C₁₂₇	semidirect
116. C₂×C₂²⋊C₁₆	C₂×C₁₂₇	semidirect
117. C₂×#346	C₂×C₁₂₇	semidirect
118. C₁₂₇	#13190	direct (N × H)
149. C₂×C₁₆	D₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₂₇	2
C₂×C₄⋊C₁₆×C₁₂₇	2
C₂×C₂²⋊C₁₆×C₁₂₇	2
C₂×#346×C₁₂₇	2
#13190	127

Subgroups — C127×#13190 #44576863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13190 #44576863