Subgroups — #30647600

Order 16256.

484 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861228	C₂	1	1	yes	maximal, characteristic
2	#22550355	C₂	1	2	yes	maximal, complemented
3	#22549976	C₂	1	2	yes	maximal, complemented
4	#22549973	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549970	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#107×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549874	C₂	1	2	yes	maximal, complemented
10	#22549830	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523247	C₂²	1	2	yes	—
13	#16861228	C₄	1	8	yes	complemented
14	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
15	C₂×#14523236	C₂²	1	2	yes	—
16	C₂×#14523231	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₅₄	C₂²	1	2	yes	—
19	#16861174	C₂²	1	2	yes	—
20	D₄×Dic₁₂₇	C₂²	1	2	yes	—
21	C₄×C₂²⋊D₁₂₇	—	2	2	no	—
22	(C₂²⋊D₁₂₇)⋊C₄	—	2	2	no	—
23	#16861166	C₂²	1	1	yes	characteristic
24	#16861165	—	2	2	no	—
25	C₂×Dic₁₂₇⋊C₄	—	2	2	no	—
26	C₄⋊C₄×D₁₂₇	C₂²	1	2	yes	—
27	#16861143	C₂²	1	1	yes	characteristic
28	#16861142	C₂²	1	2	yes	—
29	#16861134	C₂²	1	2	yes	—
30	#16861132	C₂²	1	2	yes	—
31	#16861130	C₂²	1	2	yes	—
32	#16861120	C₂²	1	1	yes	characteristic
33	#16861119	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
35	C₂×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
37	#107×C₁₂₇	C₂²	1	1	yes	characteristic
38	#102×C₁₂₇	C₂²	1	2	yes	—
39	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
40	#14523247	—	2	4	no	—
41	#14523247	C₂×C₄	1	8	yes	complemented
42	C₂×Dic₂₅₄	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₁₂₇	—	2	2	no	—
44	C₂²×Dic₁₂₇	—	2	2	no	—
45	C₂²×Dic₁₂₇	—	2	2	no	—
46	C₂×C₄×D₁₂₇	—	2	2	no	—
47	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
48	Dic₅₀₈	C₂×C₄	1	4	yes	complemented
49	#14523236	C₂×C₄	1	8	yes	complemented
50	#14523231	—	2	2	no	—
51	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
53	C₈×D₁₂₇	C₂×C₄	1	4	yes	complemented
54	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
55	Dic₁₂₇⋊C₄	—	2	2	no	—
56	Dic₁₂₇⋊C₄	—	2	2	no	—
57	Dic₁₂₇⋊C₄	—	2	2	no	—
58	Dic₁₂₇⋊C₄	D₄	1	1	yes	characteristic
59	Dic₁₂₇⋊C₄	D₄	1	1	yes	characteristic
60	C₂²⋊Dic₁₂₇	—	2	2	no	—
61	(C₂×D₁₂₇)⋊C₄	D₄	1	1	yes	characteristic
62	(C₂×D₁₂₇)⋊C₄	D₄	1	1	yes	characteristic
63	C₄×Dic₁₂₇	—	2	2	no	—
64	C₄×Dic₁₂₇	—	2	2	no	—
65	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₂₇	—	2	2	no	—
68	D₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
69	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
73	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
74	C₂²×D₁₂₇	C₂×D₄	1	1	yes	characteristic
75	Dic₂₅₄	—	2	2	no	—
76	Dic₂₅₄	C₂²×C₄	1	4	yes	—
77	C₂²⋊D₁₂₇	—	2	8	no	—
78	C₂×Dic₁₂₇	—	2	2	no	—
79	C₂×Dic₁₂₇	—	2	8	no	—
80	C₂×Dic₁₂₇	—	2	2	no	—
81	C₂×Dic₁₂₇	—	8	2	no	—
82	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
84	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
85	C₄×D₁₂₇	—	8	2	no	—
86	C₄×D₁₂₇	C₂²×C₄	1	2	yes	—
87	C₄×D₁₂₇	#41	1	2	yes	—
88	C₁₂₇⋊C₈	—	2	1	no	—
89	C₁₂₇⋊C₈	C₂²×C₄	1	2	yes	—
90	C₂³×C₁₂₇	—	2	2	no	—
91	D₄×C₁₂₇	—	2	2	no	—
92	D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
93	C₂×C₄×C₁₂₇	—	2	2	no	—
94	C₂×C₄×C₁₂₇	—	8	2	no	—
95	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
98	C₈×C₁₂₇	—	2	1	no	—
99	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₁₂₇	—	2	4	no	—
101	C₂²×C₁₂₇	—	4	2	no	—
102	C₂²×C₁₂₇	—	4	2	no	—
103	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×D₁₂₇	—	2	1	no	—
105	C₂×D₁₂₇	—	2	1	no	—
106	C₂×D₁₂₇	C₄×D₄	1	2	yes	—
107	C₄×C₁₂₇	—	2	1	no	—
108	C₄×C₁₂₇	—	2	1	no	—
109	C₄×C₁₂₇	—	4	2	no	—
110	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
111	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
112	Dic₁₂₇	—	2	4	no	—
113	Dic₁₂₇	—	2	1	no	—
114	Dic₁₂₇	—	2	1	no	—
115	Dic₁₂₇	—	4	2	no	—
116	Dic₁₂₇	C₄×D₄	1	2	yes	—
117	C₂×C₁₂₇	—	4	4	no	—
118	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₁₂₇	#576	1	1	yes	characteristic
120	C₂×C₁₂₇	#578	1	1	yes	characteristic
121	D₁₂₇	—	2	2	no	—
122	#2951	—	127	1	no	maximal, Sylow
123	C₁₂₇	#2951	1	1	yes	characteristic, complemented, Sylow
124	C₂×#135	—	127	1	no	—
125	#520	—	127	2	no	—
126	#440	—	127	2	no	—
127	#440	—	127	2	no	—
128	C₄×Q₁₆	—	127	1	no	—
129	C₄×Q₁₆	—	127	1	no	—
130	C₄×D₈	—	127	1	no	—
131	C₄×D₈	D₁₂₇	1	1	yes	characteristic, complemented
132	C₂×#107	—	127	1	no	—
133	#417	—	127	2	no	—
134	#405	—	127	1	no	—
135	C₂×#41	—	127	2	no	—
136	#135	—	127	8	no	—
137	C₂×Q₁₆	—	127	1	no	—
138	C₂×SD₁₆	—	127	2	no	—
139	C₂×D₈	C₂×D₁₂₇	1	1	yes	characteristic
140	C₂×M₁₆	—	127	1	no	—
141	C₂²×C₈	—	127	1	no	—
142	C₄×Q₈	—	127	2	no	—
143	C₄×Q₈	—	127	2	no	—
144	C₄×D₄	—	127	2	no	—
145	C₄×D₄	—	254	2	no	—
146	C₄×D₄	—	254	2	no	—
147	C₄×D₄	C₂×D₁₂₇	1	2	yes	—
148	#117	—	127	1	no	—
149	#117	—	254	2	no	—
150	C₂×C₄⋊C₄	—	127	2	no	—
151	C₂×C₄⋊C₄	—	254	2	no	—
152	#107	—	127	1	no	—
153	#107	—	127	2	no	—
154	#107	C₂×D₁₂₇	1	1	yes	characteristic
155	#103	—	127	2	no	—
156	#103	—	127	2	no	—
157	#102	—	127	2	no	—
158	#102	C₂×D₁₂₇	1	2	yes	—
159	#97	—	127	1	no	—
160	#97	—	127	1	no	—
161	C₄×C₈	—	127	1	no	—
162	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
163	#41	—	127	8	no	—
164	#41	—	254	4	no	—
165	C₂×Q₈	—	127	2	no	—
166	C₂×D₄	—	254	2	no	—
167	C₂×D₄	C₂²×D₁₂₇	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	127	1	no	—
170	C₂²×C₄	—	254	2	no	—
171	C₂²×C₄	—	254	2	no	—
172	C₂²×C₄	—	254	2	no	—
173	Q₁₆	—	127	4	no	—
174	SD₁₆	—	127	8	no	—
175	D₈	C₄×D₁₂₇	1	4	yes	complemented
176	M₁₆	—	254	2	no	—
177	C₂×C₈	—	127	1	no	—
178	C₂×C₈	—	127	4	no	—
179	C₂×C₈	—	127	1	no	—
180	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
182	C₄⋊C₄	—	127	1	no	—
183	C₄⋊C₄	—	127	1	no	—
184	C₄⋊C₄	—	127	2	no	—
185	C₄⋊C₄	—	254	2	no	—
186	C₄⋊C₄	—	254	2	no	—
187	C₄⋊C₄	—	254	2	no	—
188	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	127	1	no	—
191	C₂²⋊C₄	—	127	1	no	—
192	C₂²⋊C₄	—	254	2	no	—
193	C₄²	—	127	1	no	—
194	C₄²	—	254	2	no	—
195	C₄²	—	254	2	no	—
196	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
197	C₂³	—	2	2	no	—
198	C₂³	—	127	1	no	—
199	Q₈	—	127	4	no	—
200	Q₈	—	254	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	254	8	no	—
203	D₄	C₂×C₄×D₁₂₇	1	4	yes	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	127	1	no	—
207	C₂×C₄	—	127	2	no	—
208	C₂×C₄	—	127	2	no	—
209	C₂×C₄	—	127	1	no	—
210	C₂×C₄	—	127	1	no	—
211	C₂×C₄	—	1016	2	no	—
212	C₂×C₄	—	1016	2	no	—
213	C₂×C₄	—	254	8	no	—
214	C₂×C₄	—	254	2	no	—
215	C₂×C₄	—	254	2	no	—
216	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
217	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
218	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	127	2	no	—
221	C₈	—	254	1	no	—
222	C₈	C₂×C₄×D₁₂₇	1	2	yes	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	127	2	no	—
227	C₂²	—	254	1	no	—
228	C₂²	—	254	1	no	—
229	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	127	2	no	—
234	C₄	—	508	2	no	—
235	C₄	—	254	4	no	—
236	C₄	—	254	1	no	—
237	C₄	—	254	1	no	—
238	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
239	C₄	#41×D₁₂₇	1	1	yes	characteristic
240	C₂	—	4	4	no	—
241	C₂	—	254	2	no	—
242	C₂	C₄×D₄×D₁₂₇	1	1	yes	characteristic
243	C₂	#22550677	1	1	yes	characteristic
244	C₂	#22550690	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₄×D₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#16861228	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₄×D₈	D₁₂₇
hypocenter	C₁₂₇	#2951
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #22550355	C₂	semidirect
3. #22549976	C₂	semidirect
4. #22549973	C₂	semidirect
5. C₄×Dic₅₀₈	C₂	semidirect
6. #22549970	C₂	semidirect
7. D₈×Dic₁₂₇	C₂	semidirect
8. #107×D₁₂₇	C₂	semidirect
9. #22549874	C₂	semidirect
10. #22549830	C₂	semidirect
11. C₄×D₈×C₁₂₇	C₂	semidirect
13. #16861228	C₄	semidirect
41. #14523247	C₂×C₄	semidirect
48. Dic₅₀₈	C₂×C₄	semidirect
49. #14523236	C₂×C₄	semidirect
53. C₈×D₁₂₇	C₂×C₄	semidirect
68. D₈×C₁₂₇	C₂×C₄	semidirect
123. C₁₂₇	#2951	semidirect
131. C₄×D₈	D₁₂₇	semidirect
175. D₈	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×#16861228	2
#22550355	2
#22549976	2
#22549973	2
C₄×Dic₅₀₈	2
#22549970	2
D₈×Dic₁₂₇	2
#107×D₁₂₇	2
#22549874	2
#22549830	2
C₄×D₈×C₁₂₇	2
#2951	127