Subgroups — #30647854

Order 16256.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523249	C₂	1	1	yes	maximal, characteristic
2	#22550007	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549949	C₂	1	2	yes	maximal, complemented
4	#22549946	C₂	1	2	yes	maximal, complemented
5	C₈×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549941	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549931	C₂	1	2	yes	maximal, complemented
8	#22549927	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549898	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₁₂₇	C₂	1	2	yes	maximal, complemented
11	#446×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523249	C₄	1	1	yes	characteristic
13	C₂×#14523231	C₂²	1	2	yes	—
14	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
15	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
16	Q₈×Dic₁₂₇	C₄	1	1	yes	characteristic
17	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
18	C₄×D₅₀₈	C₂²	1	2	yes	—
19	#16861171	C₄	1	2	yes	—
20	#16861171	C₄	1	1	yes	characteristic
21	#16861167	C₄	1	2	yes	—
22	#16861167	C₄	1	1	yes	characteristic
23	C₄²×D₁₂₇	C₂²	1	2	yes	—
24	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
25	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
26	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
27	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
28	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
29	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
34	#16861120	C₂²	1	2	yes	—
35	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
38	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
39	#97×C₁₂₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
41	#14523249	—	2	4	no	—
42	C₂×D₅₀₈	C₂×C₄	1	2	yes	—
43	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
49	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
50	#14523231	—	4	2	no	—
51	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
53	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
54	C₈×D₁₂₇	—	4	2	no	—
55	C₈×D₁₂₇	—	4	2	no	—
56	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
58	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
69	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
72	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₄²×C₁₂₇	C₂³	1	2	yes	—
74	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
76	D₅₀₈	—	2	4	no	—
77	D₅₀₈	M₁₆	1	4	yes	complemented
78	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₁₂₇	—	2	4	no	—
83	C₄×D₁₂₇	—	2	2	no	—
84	C₄×D₁₂₇	—	2	2	no	—
85	C₄×D₁₂₇	—	2	1	no	—
86	C₄×D₁₂₇	—	2	2	no	—
87	C₄×D₁₂₇	—	2	1	no	—
88	C₄×D₁₂₇	—	2	4	no	—
89	C₄×D₁₂₇	—	2	2	no	—
90	C₁₂₇⋊C₈	—	2	1	no	—
91	C₁₂₇⋊C₈	—	2	2	no	—
92	C₁₂₇⋊C₈	—	2	1	no	—
93	Q₈×C₁₂₇	—	2	2	no	—
94	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
99	C₈×C₁₂₇	—	2	1	no	—
100	C₈×C₁₂₇	—	2	2	no	—
101	C₈×C₁₂₇	—	2	1	no	—
102	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₂₇	—	2	2	no	—
104	C₂×D₁₂₇	—	2	2	no	—
105	C₂×D₁₂₇	—	2	1	no	—
106	C₂×D₁₂₇	—	2	1	no	—
107	C₂×D₁₂₇	—	2	1	no	—
108	C₂×D₁₂₇	C₂×M₁₆	1	4	yes	—
109	C₄×C₁₂₇	—	2	2	no	—
110	C₄×C₁₂₇	—	2	1	no	—
111	C₄×C₁₂₇	—	2	2	no	—
112	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
113	C₄×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
114	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
115	Dic₁₂₇	—	2	2	no	—
116	Dic₁₂₇	—	2	1	no	—
117	Dic₁₂₇	—	2	1	no	—
118	Dic₁₂₇	#131	1	2	yes	—
119	C₂×C₁₂₇	#520	1	1	yes	characteristic
120	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
121	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
122	D₁₂₇	—	2	4	no	—
123	D₁₂₇	—	4	1	no	—
124	#2990	—	127	1	no	maximal, Sylow
125	C₁₂₇	#2990	1	1	yes	characteristic, complemented, Sylow
126	C₄×#41	—	127	1	no	—
127	#446	—	127	1	no	—
128	#446	D₁₂₇	1	1	yes	characteristic, complemented
129	#435	—	127	2	no	—
130	#435	—	127	2	no	—
131	C₈×D₄	—	127	1	no	—
132	C₈×D₄	—	127	1	no	—
133	#432	—	127	2	no	—
134	#431	—	127	1	no	—
135	#423	—	127	1	no	—
136	C₂×C₄⋊C₈	—	127	2	no	—
137	C₂×#41	—	127	1	no	—
138	C₂×M₁₆	—	127	2	no	—
139	C₂²×C₈	—	127	2	no	—
140	C₂²×C₈	—	127	2	no	—
141	C₄×Q₈	—	127	1	no	—
142	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
143	C₄×D₄	—	127	1	no	—
144	C₄×D₄	—	127	2	no	—
145	C₄×D₄	—	127	1	no	—
146	C₄×D₄	—	127	2	no	—
147	#117	—	127	1	no	—
148	#117	—	127	2	no	—
149	C₂×C₄²	—	127	2	no	—
150	C₂×C₄²	—	127	1	no	—
151	C₄⋊C₈	—	127	1	no	—
152	C₄⋊C₈	—	127	2	no	—
153	C₄⋊C₈	—	127	1	no	—
154	C₄⋊C₈	—	127	2	no	—
155	C₄⋊C₈	—	127	1	no	—
156	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
157	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
158	C₂²⋊C₈	—	127	1	no	—
159	C₂²⋊C₈	—	127	1	no	—
160	C₂²⋊C₈	—	127	2	no	—
161	C₂²⋊C₈	—	127	2	no	—
162	#97	—	127	2	no	—
163	#97	C₂×D₁₂₇	1	2	yes	—
164	C₄×C₈	—	127	1	no	—
165	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
166	#41	—	254	4	no	—
167	C₂×Q₈	C₄×D₁₂₇	1	1	yes	characteristic
168	C₂×D₄	—	127	1	no	—
169	C₂×D₄	—	127	2	no	—
170	C₂²×C₄	—	127	2	no	—
171	C₂²×C₄	—	127	1	no	—
172	C₂²×C₄	—	127	2	no	—
173	C₂²×C₄	—	127	1	no	—
174	C₂²×C₄	—	127	2	no	—
175	C₂²×C₄	—	127	1	no	—
176	M₁₆	—	508	2	no	—
177	C₂×C₈	—	127	1	no	—
178	C₂×C₈	—	127	1	no	—
179	C₂×C₈	—	127	2	no	—
180	C₂×C₈	—	508	2	no	—
181	C₂×C₈	—	508	2	no	—
182	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
184	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
185	C₄⋊C₄	—	127	2	no	—
186	C₄⋊C₄	—	127	1	no	—
187	C₄⋊C₄	C₄×D₁₂₇	1	2	yes	—
188	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
189	C₂²⋊C₄	—	127	2	no	—
190	C₂²⋊C₄	—	127	2	no	—
191	C₂²⋊C₄	—	127	2	no	—
192	C₄²	—	127	2	no	—
193	C₄²	—	127	1	no	—
194	C₄²	—	127	2	no	—
195	C₄²	—	127	1	no	—
196	C₄²	—	127	1	no	—
197	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
198	C₄²	C₂²×D₁₂₇	1	2	yes	—
199	C₂³	—	127	2	no	—
200	C₂³	—	127	1	no	—
201	Q₈	—	2	2	no	—
202	D₄	—	127	4	no	—
203	D₄	—	254	4	no	—
204	C₂×C₄	—	127	1	no	—
205	C₂×C₄	—	127	1	no	—
206	C₂×C₄	—	127	2	no	—
207	C₂×C₄	—	127	1	no	—
208	C₂×C₄	—	254	2	no	—
209	C₂×C₄	—	254	4	no	—
210	C₂×C₄	—	254	4	no	—
211	C₂×C₄	—	254	2	no	—
212	C₂×C₄	—	254	2	no	—
213	C₂×C₄	—	254	1	no	—
214	C₂×C₄	—	254	1	no	—
215	C₂×C₄	—	254	2	no	—
216	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	254	1	no	—
225	C₈	—	254	2	no	—
226	C₈	—	254	1	no	—
227	C₂²	—	127	4	no	—
228	C₂²	—	254	1	no	—
229	C₂²	—	254	1	no	—
230	C₂²	—	254	1	no	—
231	C₂²	—	254	2	no	—
232	C₂²	—	254	2	no	—
233	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	2	no	—
237	C₄	—	127	2	no	—
238	C₄	—	254	2	no	—
239	C₄	—	254	1	no	—
240	C₄	—	254	1	no	—
241	C₄	M₁₆×D₁₂₇	1	2	yes	—
242	C₄	#16861255	1	1	yes	characteristic
243	C₄	#16861257	1	1	yes	characteristic
244	C₂	—	508	1	no	—
245	C₂	—	254	4	no	—
246	C₂	#22550358	1	1	yes	characteristic
247	C₂	C₂×M₁₆×D₁₂₇	1	1	yes	characteristic
248	C₂	#131×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#446×C₁₂₇	C₂
prime radical	C₂×#14523249	C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#446	D₁₂₇
hypocenter	C₁₂₇	#2990
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #22550007	C₂	semidirect
3. #22549949	C₂	semidirect
4. #22549946	C₂	semidirect
5. C₈×D₅₀₈	C₂	semidirect
6. #22549941	C₂	semidirect
7. #22549931	C₂	semidirect
8. #22549927	C₂	semidirect
9. #22549898	C₂	semidirect
10. C₄⋊C₈×D₁₂₇	C₂	semidirect
11. #446×C₁₂₇	C₂	semidirect
77. D₅₀₈	M₁₆	semidirect
125. C₁₂₇	#2990	semidirect
128. #446	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523249	2
#22550007	2
#22549949	2
#22549946	2
C₈×D₅₀₈	2
#22549941	2
#22549931	2
#22549927	2
#22549898	2
C₄⋊C₈×D₁₂₇	2
#446×C₁₂₇	2
#2990	127