Subgroups — #339×D₁₂₇ #30638434

Order 16256.

488 conjugacy classes of subgroups.

The subgroup lattice has 256 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	#117×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×D₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#22549637	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549636	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549635	C₂	1	1	yes	maximal, characteristic, complemented
7	#339×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₁₂₇	—	2	1	no	—
9	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×#14523232	C₂²	1	1	yes	characteristic
11	C₂×#14523231	—	2	1	no	—
12	M₁₆×D₁₂₇	—	2	2	no	—
13	M₁₆×D₁₂₇	C₄	1	4	yes	complemented
14	C₂×C₈×D₁₂₇	—	2	1	no	—
15	#16861168	C₂²	1	1	yes	characteristic
16	#16861167	C₄	1	2	yes	complemented
17	#16861166	—	2	1	no	—
18	#16861165	—	2	1	no	—
19	C₄⋊C₄×D₁₂₇	C₄	1	2	yes	complemented
20	C₂²⋊C₄×D₁₂₇	—	2	1	no	—
21	C₂×C₄×Dic₁₂₇	—	2	1	no	—
22	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
23	C₄²×D₁₂₇	—	2	2	no	—
24	C₄²×D₁₂₇	—	2	1	no	—
25	C₄²×D₁₂₇	C₄	1	4	yes	complemented
26	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
27	#117×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₂³×D₁₂₇	D₄	1	1	yes	characteristic
30	C₂²×Dic₁₂₇	—	2	1	no	—
31	C₂²×Dic₁₂₇	D₄	1	1	yes	characteristic
32	C₂×C₄×D₁₂₇	—	2	8	no	—
33	C₂×C₄×D₁₂₇	—	2	4	no	—
34	C₂×C₄×D₁₂₇	—	2	2	no	—
35	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
40	C₂×C₄×D₁₂₇	D₄	1	2	yes	—
41	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
42	C₂×C₄×D₁₂₇	Q₈	1	2	yes	—
43	#14523232	—	2	1	no	—
44	#14523232	C₂×C₄	1	2	yes	complemented
45	#14523231	—	2	4	no	—
46	C₂×C₁₂₇⋊C₈	—	2	1	no	—
47	C₈×D₁₂₇	—	2	4	no	—
48	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	complemented
49	Dic₁₂₇⋊C₄	—	2	2	no	—
50	C₂²⋊Dic₁₂₇	—	2	1	no	—
51	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
52	C₄×Dic₁₂₇	—	2	4	no	—
53	C₄×Dic₁₂₇	—	2	2	no	—
54	C₄×Dic₁₂₇	—	2	1	no	—
55	C₄×Dic₁₂₇	—	2	1	no	—
56	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	complemented
57	C₂²×C₄×C₁₂₇	—	2	1	no	—
58	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
59	M₁₆×C₁₂₇	—	2	1	no	—
60	M₁₆×C₁₂₇	C₂×C₄	1	2	yes	complemented
61	C₂×C₈×C₁₂₇	—	2	1	no	—
62	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	complemented
63	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
64	C₄²×C₁₂₇	—	2	1	no	—
65	C₄²×C₁₂₇	—	2	1	no	—
66	C₄²×C₁₂₇	C₂×C₄	1	2	yes	complemented
67	C₂²×D₁₂₇	—	2	4	no	—
68	C₂²×D₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×D₁₂₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×D₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×D₁₂₇	C₄⋊C₄	1	2	yes	—
72	C₂×Dic₁₂₇	—	2	4	no	—
73	C₂×Dic₁₂₇	—	2	2	no	—
74	C₂×Dic₁₂₇	—	2	2	no	—
75	C₂×Dic₁₂₇	C₂²⋊C₄	1	2	yes	—
76	C₂×Dic₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×Dic₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×Dic₁₂₇	C₄⋊C₄	1	2	yes	—
79	C₄×D₁₂₇	—	2	16	no	—
80	C₄×D₁₂₇	—	4	4	no	—
81	C₄×D₁₂₇	C₄²	1	2	yes	—
82	C₄×D₁₂₇	C₄²	1	2	yes	—
83	C₄×D₁₂₇	C₂²⋊C₄	1	2	yes	—
84	C₄×D₁₂₇	C₂²⋊C₄	1	2	yes	—
85	C₄×D₁₂₇	C₄⋊C₄	1	2	yes	—
86	C₄×D₁₂₇	C₄⋊C₄	1	2	yes	—
87	C₄×D₁₂₇	C₄⋊C₄	1	2	yes	—
88	C₄×D₁₂₇	C₄⋊C₄	1	2	yes	—
89	C₁₂₇⋊C₈	—	2	2	no	—
90	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₂₇	—	2	4	no	—
92	C₂×C₄×C₁₂₇	—	2	2	no	—
93	C₂×C₄×C₁₂₇	—	2	2	no	—
94	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
100	C₈×C₁₂₇	—	2	2	no	—
101	C₂²×C₁₂₇	—	2	2	no	—
102	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×D₁₂₇	—	2	2	no	—
106	C₂×D₁₂₇	—	2	4	no	—
107	C₂×D₁₂₇	—	2	4	no	—
108	C₂×D₁₂₇	#95	1	2	yes	—
109	C₂×D₁₂₇	#95	1	2	yes	—
110	C₂×D₁₂₇	#104	1	4	yes	—
111	C₄×C₁₂₇	—	2	4	no	—
112	C₄×C₁₂₇	—	4	2	no	—
113	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
114	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	Dic₁₂₇	—	2	4	no	—
118	Dic₁₂₇	—	4	2	no	—
119	Dic₁₂₇	#95	1	2	yes	—
120	Dic₁₂₇	#95	1	2	yes	—
121	C₂×C₁₂₇	—	2	1	no	—
122	C₂×C₁₂₇	—	2	1	no	—
123	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
124	C₂×C₁₂₇	C₂×#104	1	2	yes	—
125	D₁₂₇	—	2	2	no	—
126	D₁₂₇	#339	1	4	yes	direct product factor
127	C₂×#339	—	127	1	no	maximal, Sylow
128	C₁₂₇	C₂×#339	1	1	yes	characteristic, complemented, Sylow
129	C₂²×M₁₆	—	127	1	no	—
130	C₂×#117	—	127	1	no	—
131	C₂²×C₄²	—	127	1	no	—
132	#339	—	127	1	no	—
133	#339	—	127	1	no	—
134	#339	—	127	1	no	—
135	#339	D₁₂₇	1	1	yes	characteristic, direct product factor
136	C₂³×C₄	—	127	1	no	—
137	C₂³×C₄	—	254	1	no	—
138	C₂×M₁₆	—	127	1	no	—
139	C₂×M₁₆	—	127	4	no	—
140	C₂×M₁₆	—	254	1	no	—
141	C₂×M₁₆	—	254	2	no	—
142	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
143	C₂²×C₈	—	254	1	no	—
144	#117	—	127	1	no	—
145	#117	—	127	2	no	—
146	#117	—	254	1	no	—
147	#117	—	254	1	no	—
148	#117	C₂×D₁₂₇	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	127	2	no	—
150	C₂×C₂²⋊C₄	—	254	1	no	—
151	C₂×C₄²	—	127	1	no	—
152	C₂×C₄²	—	127	4	no	—
153	C₂×C₄²	—	254	1	no	—
154	C₂×C₄²	—	254	2	no	—
155	C₂×C₄²	—	254	1	no	—
156	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
157	C₂⁴	—	127	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	127	1	no	—
160	C₂²×C₄	—	127	2	no	—
161	C₂²×C₄	—	127	1	no	—
162	C₂²×C₄	—	127	1	no	—
163	C₂²×C₄	—	127	2	no	—
164	C₂²×C₄	—	127	2	no	—
165	C₂²×C₄	—	127	1	no	—
166	C₂²×C₄	—	127	1	no	—
167	C₂²×C₄	—	127	2	no	—
168	C₂²×C₄	—	254	2	no	—
169	C₂²×C₄	—	254	4	no	—
170	C₂²×C₄	—	254	8	no	—
171	C₂²×C₄	—	254	1	no	—
172	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
173	M₁₆	—	2	1	no	—
174	M₁₆	—	127	2	no	—
175	M₁₆	—	254	4	no	—
176	M₁₆	—	254	1	no	—
177	M₁₆	C₄×D₁₂₇	1	2	yes	complemented
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	254	4	no	—
180	C₂×C₈	—	254	1	no	—
181	C₄⋊C₄	—	127	2	no	—
182	C₄⋊C₄	—	254	2	no	—
183	C₄⋊C₄	C₄×D₁₂₇	1	2	yes	complemented
184	C₂²⋊C₄	—	2	1	no	—
185	C₂²⋊C₄	—	254	2	no	—
186	C₂²⋊C₄	—	254	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	127	2	no	—
190	C₄²	—	254	4	no	—
191	C₄²	—	254	2	no	—
192	C₄²	—	254	1	no	—
193	C₄²	—	254	1	no	—
194	C₄²	C₄×D₁₂₇	1	2	yes	complemented
195	C₂³	—	127	1	no	—
196	C₂³	—	127	1	no	—
197	C₂³	—	127	2	no	—
198	C₂³	—	127	2	no	—
199	C₂³	—	254	4	no	—
200	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	127	2	no	—
205	C₂×C₄	—	127	1	no	—
206	C₂×C₄	—	127	1	no	—
207	C₂×C₄	—	127	2	no	—
208	C₂×C₄	—	127	2	no	—
209	C₂×C₄	—	127	2	no	—
210	C₂×C₄	—	127	2	no	—
211	C₂×C₄	—	127	2	no	—
212	C₂×C₄	—	127	2	no	—
213	C₂×C₄	—	127	2	no	—
214	C₂×C₄	—	127	2	no	—
215	C₂×C₄	—	127	2	no	—
216	C₂×C₄	—	508	4	no	—
217	C₂×C₄	—	254	16	no	—
218	C₂×C₄	—	254	2	no	—
219	C₂×C₄	—	254	4	no	—
220	C₂×C₄	—	254	2	no	—
221	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
224	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
225	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
226	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	254	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	127	4	no	—
231	C₂²	—	127	2	no	—
232	C₂²	—	127	2	no	—
233	C₂²	—	254	4	no	—
234	C₂²	—	254	4	no	—
235	C₂²	—	254	2	no	—
236	C₂²	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
237	C₂²	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
238	C₂²	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	4	2	no	—
241	C₄	—	127	2	no	—
242	C₄	—	127	2	no	—
243	C₄	—	508	2	no	—
244	C₄	—	254	4	no	—
245	C₄	C₄²×D₁₂₇	1	1	yes	characteristic
246	C₄	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
247	C₄	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
248	C₄	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	—	127	4	no	—
252	C₂	—	254	2	no	—
253	C₂	#95×D₁₂₇	1	1	yes	characteristic
254	C₂	#104×D₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#339×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₄
prime radical	C₂³×D₁₂₇	D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#339	D₁₂₇
hypocenter	C₁₂₇	C₂×#339
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

Complements

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

N	H ≅ G/N	type
4. #22549637	C₂	semidirect
5. #22549636	C₂	semidirect
6. #22549635	C₂	semidirect
7. #339×C₁₂₇	C₂	semidirect
13. M₁₆×D₁₂₇	C₄	semidirect
16. #16861167	C₄	semidirect
19. C₄⋊C₄×D₁₂₇	C₄	semidirect
25. C₄²×D₁₂₇	C₄	semidirect
44. #14523232	C₂×C₄	semidirect
48. C₄⋊Dic₁₂₇	C₂×C₄	semidirect
56. C₄×Dic₁₂₇	C₂×C₄	semidirect
60. M₁₆×C₁₂₇	C₂×C₄	semidirect
62. C₄⋊C₄×C₁₂₇	C₂×C₄	semidirect
66. C₄²×C₁₂₇	C₂×C₄	semidirect
126. D₁₂₇	#339	direct (N × H)
128. C₁₂₇	C₂×#339	semidirect
135. #339	D₁₂₇	direct (N × H)
177. M₁₆	C₄×D₁₂₇	semidirect
183. C₄⋊C₄	C₄×D₁₂₇	semidirect
194. C₄²	C₄×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
#117×D₁₂₇	2
C₂×C₄²×D₁₂₇	2
#22549637	2
#22549636	2
#22549635	2
#339×C₁₂₇	2
C₂×#339	127

Subgroups — #339×D127 #30638434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #339×D₁₂₇ #30638434