Subgroups — #30647152

Order 16256.

572 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×#16861217	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861215	C₂	1	1	yes	maximal, characteristic
3	#22550344	C₂	1	1	yes	maximal, characteristic
4	#22549833	C₂	1	2	yes	maximal, complemented
5	#22549832	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549830	C₂	1	2	yes	maximal, complemented
7	#22549828	C₂	1	2	yes	maximal, complemented
8	M₁₆×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×D₁₂₇	C₂	1	2	yes	maximal, complemented
11	#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	#16861217	C₄	1	8	yes	complemented
14	#16861215	C₄	1	8	yes	complemented
15	C₂×#14523232	C₂²	1	1	yes	characteristic
16	C₂×#14523231	C₂²	1	2	yes	—
17	C₂×#14523231	C₂²	1	2	yes	—
18	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
20	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
21	#16861174	C₄	1	2	yes	—
22	#16861171	C₄	1	2	yes	—
23	(C₂²⋊D₁₂₇)⋊C₄	C₄	1	4	yes	—
24	#16861166	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
26	C₄²×D₁₂₇	C₂²	1	2	yes	—
27	#16861120	C₂²	1	2	yes	—
28	#16861119	C₂²	1	2	yes	—
29	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
30	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
31	C₈×Dic₁₂₇	C₂²	1	2	yes	—
32	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
33	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
35	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
37	#117×C₁₂₇	C₂²	1	1	yes	characteristic
38	#97×C₁₂₇	C₂²	1	2	yes	—
39	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
40	#14523248	C₂×C₄	1	4	yes	—
41	#14523248	C₂×C₄	1	4	yes	—
42	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
43	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₂₇	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
48	#14523232	C₂×C₄	1	4	yes	complemented
49	#14523231	C₂×C₄	1	8	yes	complemented
50	#14523231	C₂×C₄	1	8	yes	complemented
51	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	complemented
52	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
54	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
55	C₈×D₁₂₇	C₂×C₄	1	8	yes	complemented
56	C₈×D₁₂₇	C₂×C₄	1	8	yes	complemented
57	Dic₁₂₇⋊C₄	C₂×C₄	1	4	yes	—
58	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	4	yes	—
59	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	—
60	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
62	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
64	M₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
65	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
66	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
67	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
70	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
71	C₄²×C₁₂₇	C₂³	1	2	yes	—
72	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
73	Dic₂₅₄	C₄²	1	4	yes	—
74	D₅₀₈	C₄²	1	4	yes	—
75	C₂²⋊D₁₂₇	C₄²	1	8	yes	—
76	C₂×Dic₁₂₇	—	2	4	no	—
77	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
80	C₄×D₁₂₇	—	2	8	no	—
81	C₄×D₁₂₇	C₂²×C₄	1	4	yes	—
82	C₄×D₁₂₇	C₂²×C₄	1	4	yes	—
83	C₁₂₇⋊C₈	C₂²×C₄	1	2	yes	—
84	C₁₂₇⋊C₈	C₂²×C₄	1	4	yes	—
85	C₁₂₇⋊C₈	C₂²×C₄	1	2	yes	—
86	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
88	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
93	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
94	C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
95	C₂²×C₁₂₇	—	2	2	no	—
96	C₂²×C₁₂₇	C₂×C₄²	1	2	yes	—
97	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₂₇	—	2	4	no	—
99	C₂×D₁₂₇	C₂×C₄²	1	4	yes	—
100	C₄×C₁₂₇	—	2	4	no	—
101	C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
102	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
103	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
104	Dic₁₂₇	—	2	2	no	—
105	Dic₁₂₇	C₂×C₄²	1	4	yes	—
106	Dic₁₂₇	#131	1	4	yes	complemented
107	C₂×C₁₂₇	—	2	2	no	—
108	C₂×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
109	C₂×C₁₂₇	C₂×#131	1	2	yes	—
110	D₁₂₇	—	2	4	no	—
111	C₄×#131	—	127	1	no	maximal, Sylow
112	C₁₂₇	C₄×#131	1	1	yes	characteristic, complemented, Sylow
113	C₂×#131	—	127	1	no	—
114	C₂×#131	—	127	1	no	—
115	C₄×#41	—	127	1	no	—
116	#405	—	127	1	no	—
117	#405	—	127	2	no	—
118	#405	—	127	2	no	—
119	#405	D₁₂₇	1	1	yes	characteristic, complemented
120	C₄×M₁₆	—	127	1	no	—
121	C₄×M₁₆	—	127	2	no	—
122	C₂×C₄×C₈	—	127	1	no	—
123	C₂×C₄×C₈	—	127	2	no	—
124	C₂×#41	—	127	1	no	—
125	#131	—	127	8	no	—
126	#131	—	127	8	no	—
127	C₂×M₁₆	—	127	1	no	—
128	C₂×M₁₆	—	127	2	no	—
129	C₂×M₁₆	—	127	2	no	—
130	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
131	C₂²×C₈	—	127	1	no	—
132	C₂²×C₈	—	127	2	no	—
133	C₂²×C₈	—	127	2	no	—
134	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
135	C₄×Q₈	—	127	2	no	—
136	C₄×D₄	—	127	2	no	—
137	C₄×D₄	—	127	4	no	—
138	#117	—	127	2	no	—
139	#117	C₂×D₁₂₇	1	1	yes	characteristic
140	C₂×C₄²	—	127	1	no	—
141	C₂×C₄²	—	127	2	no	—
142	#97	—	127	2	no	—
143	#97	—	127	2	no	—
144	#97	C₂×D₁₂₇	1	2	yes	—
145	C₄×C₈	—	127	1	no	—
146	C₄×C₈	—	127	2	no	—
147	C₄×C₈	—	127	1	no	—
148	C₄×C₈	—	127	1	no	—
149	C₄×C₈	—	127	1	no	—
150	C₄×C₈	—	127	2	no	—
151	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
152	#41	—	127	4	no	—
153	#41	—	127	4	no	—
154	C₂×Q₈	—	127	1	no	—
155	C₂×D₄	—	127	2	no	—
156	C₂×D₄	—	127	1	no	—
157	C₂²×C₄	—	127	2	no	—
158	C₂²×C₄	—	127	2	no	—
159	C₂²×C₄	—	127	4	no	—
160	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
161	M₁₆	—	127	4	no	—
162	M₁₆	—	127	8	no	—
163	M₁₆	—	127	8	no	—
164	M₁₆	C₄×D₁₂₇	1	4	yes	complemented
165	C₂×C₈	—	127	4	no	—
166	C₂×C₈	—	127	1	no	—
167	C₂×C₈	—	127	2	no	—
168	C₂×C₈	—	127	1	no	—
169	C₂×C₈	—	127	8	no	—
170	C₂×C₈	—	127	8	no	—
171	C₂×C₈	C₄×D₁₂₇	1	4	yes	complemented
172	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
174	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
175	C₄⋊C₄	—	127	4	no	—
176	C₄⋊C₄	C₄×D₁₂₇	1	2	yes	—
177	C₂²⋊C₄	—	127	4	no	—
178	C₂²⋊C₄	C₄×D₁₂₇	1	2	yes	—
179	C₄²	—	127	4	no	—
180	C₄²	—	127	2	no	—
181	C₄²	—	127	1	no	—
182	C₄²	—	127	1	no	—
183	C₄²	C₂²×D₁₂₇	1	2	yes	—
184	C₂³	—	127	2	no	—
185	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
186	Q₈	—	127	4	no	—
187	D₄	—	127	8	no	—
188	D₄	—	127	4	no	—
189	C₂×C₄	—	127	2	no	—
190	C₂×C₄	—	127	2	no	—
191	C₂×C₄	—	127	4	no	—
192	C₂×C₄	—	127	4	no	—
193	C₂×C₄	—	127	2	no	—
194	C₂×C₄	—	254	8	no	—
195	C₂×C₄	—	254	4	no	—
196	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
197	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
201	C₈	—	127	2	no	—
202	C₈	—	127	2	no	—
203	C₈	—	127	4	no	—
204	C₈	C₂×C₄×D₁₂₇	1	2	yes	—
205	C₈	C₂×C₄×D₁₂₇	1	4	yes	—
206	C₈	C₂×C₄×D₁₂₇	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	127	4	no	—
209	C₂²	—	254	4	no	—
210	C₂²	C₄²×D₁₂₇	1	2	yes	—
211	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	127	4	no	—
214	C₄	—	127	4	no	—
215	C₄	—	254	2	no	—
216	C₄	C₄²×D₁₂₇	1	2	yes	—
217	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
218	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	—	254	4	no	—
221	C₂	C₂×C₄²×D₁₂₇	1	1	yes	characteristic
222	C₂	#131×D₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#405×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂³×C₄
prime radical	C₂×#14523248	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#405	D₁₂₇
hypocenter	C₁₂₇	C₄×#131
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #22549833	C₂	semidirect
5. #22549832	C₂	semidirect
6. #22549830	C₂	semidirect
7. #22549828	C₂	semidirect
8. M₁₆×Dic₁₂₇	C₂	semidirect
9. C₂×C₈×Dic₁₂₇	C₂	semidirect
10. C₄×C₈×D₁₂₇	C₂	semidirect
11. #405×C₁₂₇	C₂	semidirect
13. #16861217	C₄	semidirect
14. #16861215	C₄	semidirect
48. #14523232	C₂×C₄	semidirect
49. #14523231	C₂×C₄	semidirect
50. #14523231	C₂×C₄	semidirect
51. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
55. C₈×D₁₂₇	C₂×C₄	semidirect
56. C₈×D₁₂₇	C₂×C₄	semidirect
64. M₁₆×C₁₂₇	C₂×C₄	semidirect
65. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
106. Dic₁₂₇	#131	semidirect
112. C₁₂₇	C₄×#131	semidirect
119. #405	D₁₂₇	semidirect
164. M₁₆	C₄×D₁₂₇	semidirect
171. C₂×C₈	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×#16861217	2
C₂×#16861215	2
#22550344	2
#22549833	2
#22549832	2
#22549830	2
#22549828	2
M₁₆×Dic₁₂₇	2
C₂×C₈×Dic₁₂₇	2
C₄×C₈×D₁₂₇	2
#405×C₁₂₇	2
C₄×#131	127