Subgroups — #424×D₁₂₇ #30647286

Order 16256.

512 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#117×D₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#22549904	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549903	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549902	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549901	C₂	1	4	yes	maximal, complemented
8	C₄⋊C₈×D₁₂₇	C₂	1	4	yes	maximal, complemented
9	#424×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×#14523232	C₂²	1	1	yes	characteristic
12	C₂×#14523231	C₂²	1	2	yes	—
13	M₁₆×D₁₂₇	—	2	4	no	—
14	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₁₂₇	—	2	4	no	—
16	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
17	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
18	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
19	#16861168	C₂²	1	1	yes	characteristic
20	#16861167	C₄	1	2	yes	—
21	#16861166	C₂²	1	2	yes	—
22	#16861165	C₄	1	2	yes	—
23	C₄⋊C₄×D₁₂₇	C₄	1	2	yes	—
24	C₂²⋊C₄×D₁₂₇	C₄	1	2	yes	—
25	C₄²×D₁₂₇	C₂²	1	2	yes	—
26	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	complemented
27	Dic₁₂₇⋊C₈	C₂²	1	4	yes	complemented
28	Dic₁₂₇⋊C₈	C₂²	1	4	yes	complemented
29	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
31	#117×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	complemented
33	C₂³×D₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₂₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₂₇	D₄	1	4	yes	—
39	C₂×C₄×D₁₂₇	Q₈	1	4	yes	—
40	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
42	#14523232	—	2	2	no	—
43	#14523231	—	2	4	no	—
44	C₂×C₁₂₇⋊C₈	—	2	2	no	—
45	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
46	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
47	C₈×D₁₂₇	—	2	4	no	—
48	C₈×D₁₂₇	—	2	4	no	—
49	C₈×D₁₂₇	—	2	4	no	—
50	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
51	Dic₁₂₇⋊C₄	C₂×C₄	1	4	yes	—
52	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
53	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	4	yes	—
54	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	—
55	C₄×Dic₁₂₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
57	M₁₆×C₁₂₇	—	2	2	no	—
58	C₂×C₈×C₁₂₇	—	2	2	no	—
59	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
60	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
61	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
62	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
63	C₄²×C₁₂₇	C₂³	1	2	yes	—
64	C₂²×D₁₂₇	—	2	4	no	—
65	C₂²×D₁₂₇	C₄⋊C₄	1	4	yes	—
66	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×Dic₁₂₇	C₄⋊C₄	1	4	yes	—
69	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
70	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
72	C₄×D₁₂₇	—	2	8	no	—
73	C₄×D₁₂₇	C₄⋊C₄	1	4	yes	—
74	C₄×D₁₂₇	C₄⋊C₄	1	4	yes	—
75	C₄×D₁₂₇	C₂×D₄	1	2	yes	—
76	C₄×D₁₂₇	C₂×D₄	1	2	yes	—
77	C₄×D₁₂₇	C₂×Q₈	1	2	yes	—
78	C₄×D₁₂₇	C₂×Q₈	1	2	yes	—
79	C₁₂₇⋊C₈	—	2	2	no	—
80	C₁₂₇⋊C₈	—	2	2	no	—
81	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
83	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
87	C₈×C₁₂₇	—	2	2	no	—
88	C₈×C₁₂₇	—	2	2	no	—
89	C₂²×C₁₂₇	—	2	2	no	—
90	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×D₁₂₇	—	2	2	no	—
93	C₂×D₁₂₇	—	2	8	no	—
94	C₂×D₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×D₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×D₁₂₇	#131	1	4	yes	—
97	C₄×C₁₂₇	—	2	4	no	—
98	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
100	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
101	Dic₁₂₇	—	2	4	no	—
102	Dic₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
103	Dic₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₁₂₇	—	2	2	no	—
105	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₁₂₇	C₂×#131	1	2	yes	—
107	D₁₂₇	—	2	2	no	—
108	D₁₂₇	#424	1	4	yes	direct product factor
109	C₂×#424	—	127	1	no	maximal, Sylow
110	C₁₂₇	C₂×#424	1	1	yes	characteristic, complemented, Sylow
111	C₂²×M₁₆	—	127	1	no	—
112	C₂³×C₈	—	127	1	no	—
113	C₂×#117	—	127	1	no	—
114	#424	—	127	1	no	—
115	#424	—	127	1	no	—
116	#424	—	127	1	no	—
117	#424	—	127	4	no	—
118	#424	D₁₂₇	1	1	yes	characteristic, direct product factor
119	C₂×C₄⋊C₈	—	127	4	no	—
120	C₂³×C₄	—	127	1	no	—
121	C₂×M₁₆	—	127	1	no	—
122	C₂×M₁₆	—	127	2	no	—
123	C₂×M₁₆	—	254	4	no	—
124	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
125	C₂²×C₈	—	127	1	no	—
126	C₂²×C₈	—	127	2	no	—
127	C₂²×C₈	—	127	2	no	—
128	C₂²×C₈	—	127	2	no	—
129	C₂²×C₈	—	254	4	no	—
130	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
131	#117	—	127	1	no	—
132	#117	—	127	2	no	—
133	#117	—	127	2	no	—
134	#117	—	127	2	no	—
135	#117	C₂×D₁₂₇	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	127	2	no	—
137	C₂×C₂²⋊C₄	—	127	2	no	—
138	C₂×C₄²	—	127	2	no	—
139	C₄⋊C₈	—	127	4	no	—
140	C₄⋊C₈	—	127	4	no	—
141	C₄⋊C₈	—	127	4	no	—
142	C₄⋊C₈	C₂×D₁₂₇	1	4	yes	complemented
143	C₂⁴	—	127	1	no	—
144	C₂²×C₄	—	127	1	no	—
145	C₂²×C₄	—	127	1	no	—
146	C₂²×C₄	—	127	1	no	—
147	C₂²×C₄	—	127	4	no	—
148	C₂²×C₄	—	127	4	no	—
149	C₂²×C₄	—	127	1	no	—
150	C₂²×C₄	—	127	1	no	—
151	C₂²×C₄	—	127	4	no	—
152	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
153	M₁₆	—	2	2	no	—
154	M₁₆	—	254	4	no	—
155	M₁₆	—	254	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	127	2	no	—
158	C₂×C₈	—	127	2	no	—
159	C₂×C₈	—	254	4	no	—
160	C₂×C₈	—	254	4	no	—
161	C₂×C₈	—	254	4	no	—
162	C₂×C₈	—	254	2	no	—
163	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
164	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
165	C₄⋊C₄	—	127	4	no	—
166	C₄⋊C₄	—	127	2	no	—
167	C₄⋊C₄	C₄×D₁₂₇	1	2	yes	—
168	C₂²⋊C₄	—	127	2	no	—
169	C₂²⋊C₄	—	127	4	no	—
170	C₂²⋊C₄	C₄×D₁₂₇	1	2	yes	—
171	C₄²	—	127	4	no	—
172	C₄²	—	127	2	no	—
173	C₄²	C₂²×D₁₂₇	1	2	yes	—
174	C₂³	—	127	1	no	—
175	C₂³	—	127	1	no	—
176	C₂³	—	127	4	no	—
177	C₂³	—	254	4	no	—
178	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
179	C₂×C₄	—	127	1	no	—
180	C₂×C₄	—	127	1	no	—
181	C₂×C₄	—	127	4	no	—
182	C₂×C₄	—	127	2	no	—
183	C₂×C₄	—	127	2	no	—
184	C₂×C₄	—	127	2	no	—
185	C₂×C₄	—	127	2	no	—
186	C₂×C₄	—	127	4	no	—
187	C₂×C₄	—	127	4	no	—
188	C₂×C₄	—	127	4	no	—
189	C₂×C₄	—	254	8	no	—
190	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
191	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
192	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
193	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
194	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	2	2	no	—
197	C₈	—	254	2	no	—
198	C₈	—	254	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	127	2	no	—
201	C₂²	—	127	4	no	—
202	C₂²	—	127	2	no	—
203	C₂²	—	254	8	no	—
204	C₂²	—	254	2	no	—
205	C₂²	C₄⋊C₄×D₁₂₇	1	2	yes	—
206	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	127	2	no	—
209	C₄	—	127	2	no	—
210	C₄	—	254	4	no	—
211	C₄	C₄⋊C₄×D₁₂₇	1	2	yes	—
212	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
213	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	—	127	4	no	—
216	C₂	—	254	2	no	—
217	C₂	C₂×C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
218	C₂	#131×D₁₂₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#424×C₁₂₇	C₂
prime radical	C₂³×D₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#424	D₁₂₇
hypocenter	C₁₂₇	C₂×#424
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
4. #22549904	C₂	semidirect
5. #22549903	C₂	semidirect
6. #22549902	C₂	semidirect
7. #22549901	C₂	semidirect
8. C₄⋊C₈×D₁₂₇	C₂	semidirect
9. #424×C₁₂₇	C₂	semidirect
26. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
27. Dic₁₂₇⋊C₈	C₂²	semidirect
28. Dic₁₂₇⋊C₈	C₂²	semidirect
32. C₄⋊C₈×C₁₂₇	C₂²	semidirect
108. D₁₂₇	#424	direct (N × H)
110. C₁₂₇	C₂×#424	semidirect
118. #424	D₁₂₇	direct (N × H)
142. C₄⋊C₈	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₂₇	2
C₂²×C₈×D₁₂₇	2
#117×D₁₂₇	2
#22549904	2
#22549903	2
#22549902	2
#22549901	2
C₄⋊C₈×D₁₂₇	2
#424×C₁₂₇	2
C₂×#424	127

Subgroups — #424×D127 #30647286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #424×D₁₂₇ #30647286