Subgroups — #30647324

Order 16256.

476 conjugacy classes of subgroups.

The subgroup lattice has 252 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	#22550357	C₂	1	1	yes	maximal, characteristic
3	#22550356	C₂	1	1	yes	maximal, characteristic
4	#22549918	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549917	C₂	1	2	yes	maximal, complemented
6	#22549915	C₂	1	2	yes	maximal, complemented
7	#22549913	C₂	1	2	yes	maximal, complemented
8	#22549912	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16861140	C₂	1	1	yes	maximal, characteristic, complemented
10	#106×D₁₂₇	C₂	1	2	yes	maximal, complemented
11	#428×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	#16861217	C₄	1	8	yes	complemented
14	C₂×#14523231	C₂²	1	2	yes	—
15	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
17	C₄×Dic₂₅₄	—	2	1	no	—
18	#16861174	—	2	1	no	—
19	C₄×D₅₀₈	—	2	1	no	—
20	#16861171	—	2	1	no	—
21	C₄×C₂²⋊D₁₂₇	—	2	2	no	—
22	(C₂²⋊D₁₂₇)⋊C₄	—	2	2	no	—
23	#16861168	C₂²	1	1	yes	characteristic
24	#16861167	C₂²	1	2	yes	—
25	#16861167	C₂²	1	2	yes	—
26	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₁₂₇	C₂²	1	2	yes	—
28	C₄⋊C₄×D₁₂₇	C₂²	1	2	yes	—
29	#16861143	C₂²	1	2	yes	—
30	#16861142	C₂²	1	1	yes	characteristic
31	#16861142	C₂²	1	1	yes	characteristic
32	#16861141	C₂²	1	2	yes	—
33	#16861140	C₂²	1	1	yes	characteristic
34	#16861140	C₂²	1	2	yes	—
35	#16861140	C₂²	1	2	yes	—
36	#16861140	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
38	#117×C₁₂₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
40	#107×C₁₂₇	C₂²	1	2	yes	—
41	#106×C₁₂₇	C₂²	1	2	yes	—
42	#14523248	C₂×C₄	1	4	yes	—
43	#14523248	Q₈	1	4	yes	—
44	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
45	C₂×D₅₀₈	D₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₂₇	D₄	1	2	yes	—
47	C₂²×Dic₁₂₇	—	2	1	no	—
48	C₂×C₄×D₁₂₇	—	2	2	no	—
49	C₂×C₄×D₁₂₇	—	2	2	no	—
50	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
51	#14523231	C₂×C₄	1	8	yes	complemented
52	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
55	C₈×D₁₂₇	C₂×C₄	1	8	yes	complemented
56	C₄⋊Dic₁₂₇	—	2	1	no	—
57	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
58	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
60	Dic₁₂₇⋊C₄	—	2	2	no	—
61	Dic₁₂₇⋊C₄	—	2	2	no	—
62	C₂²⋊Dic₁₂₇	—	2	1	no	—
63	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
64	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
65	C₄×Dic₁₂₇	—	2	2	no	—
66	C₄×Dic₁₂₇	—	2	2	no	—
67	C₄×Dic₁₂₇	—	2	1	no	—
68	C₂²×C₄×C₁₂₇	—	2	1	no	—
69	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
70	M₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
71	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
72	C₄⋊C₄×C₁₂₇	—	2	1	no	—
73	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
75	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
77	C₄²×C₁₂₇	—	2	1	no	—
78	C₂²×D₁₂₇	C₂×D₄	1	2	yes	—
79	Dic₂₅₄	C₄⋊C₄	1	4	yes	—
80	D₅₀₈	C₄⋊C₄	1	4	yes	—
81	C₂²⋊D₁₂₇	C₄⋊C₄	1	8	yes	—
82	C₂×Dic₁₂₇	—	2	1	no	—
83	C₂×Dic₁₂₇	—	2	2	no	—
84	C₂×Dic₁₂₇	—	2	1	no	—
85	C₂×Dic₁₂₇	—	8	1	no	—
86	C₂×Dic₁₂₇	C₂×D₄	1	2	yes	—
87	C₄×D₁₂₇	—	8	2	no	—
88	C₄×D₁₂₇	—	8	2	no	—
89	C₄×D₁₂₇	C₂²×C₄	1	4	yes	—
90	C₄×D₁₂₇	C₂×Q₈	1	4	yes	—
91	C₁₂₇⋊C₈	C₂²×C₄	1	2	yes	—
92	C₁₂₇⋊C₈	C₂²×C₄	1	2	yes	—
93	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₂₇	—	2	1	no	—
95	C₂×C₄×C₁₂₇	—	2	1	no	—
96	C₂×C₄×C₁₂₇	—	2	2	no	—
97	C₂×C₄×C₁₂₇	—	8	1	no	—
98	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
101	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
102	C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
103	C₂²×C₁₂₇	—	2	1	no	—
104	C₂²×C₁₂₇	—	2	1	no	—
105	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×D₁₂₇	—	2	2	no	—
108	C₂×D₁₂₇	—	2	2	no	—
109	C₂×D₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
110	C₄×C₁₂₇	—	4	1	no	—
111	C₄×C₁₂₇	—	4	2	no	—
112	C₄×C₁₂₇	—	4	1	no	—
113	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
115	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
116	Dic₁₂₇	—	4	1	no	—
117	Dic₁₂₇	—	4	2	no	—
118	Dic₁₂₇	—	4	1	no	—
119	Dic₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₁₂₇	—	2	2	no	—
121	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₁₂₇	#577	1	1	yes	characteristic
123	C₂×C₁₂₇	#577	1	1	yes	characteristic
124	D₁₂₇	—	2	4	no	—
125	#2912	—	127	1	no	maximal, Sylow
126	C₁₂₇	#2912	1	1	yes	characteristic, complemented, Sylow
127	C₂×#131	—	127	1	no	—
128	#520	—	127	1	no	—
129	#520	—	127	1	no	—
130	#428	—	127	1	no	—
131	#428	—	127	2	no	—
132	#428	—	127	2	no	—
133	#428	D₁₂₇	1	1	yes	characteristic, complemented
134	#427	—	127	1	no	—
135	#427	—	127	2	no	—
136	C₂×#106	—	127	1	no	—
137	C₂×#106	—	127	2	no	—
138	C₂×#41	—	127	1	no	—
139	#131	—	127	8	no	—
140	C₂×M₁₆	—	127	2	no	—
141	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
142	C₂²×C₈	—	127	1	no	—
143	C₂²×C₈	—	127	2	no	—
144	C₄×Q₈	—	254	1	no	—
145	C₄×Q₈	—	254	1	no	—
146	C₄×D₄	—	254	1	no	—
147	C₄×D₄	—	254	2	no	—
148	C₄×D₄	—	254	1	no	—
149	C₄×D₄	—	254	2	no	—
150	#117	—	127	1	no	—
151	#117	—	127	2	no	—
152	#117	—	127	2	no	—
153	#117	C₂×D₁₂₇	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	127	1	no	—
155	C₂×C₄⋊C₄	—	127	2	no	—
156	C₂×C₄⋊C₄	—	127	2	no	—
157	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
158	#107	—	127	2	no	—
159	#107	—	127	1	no	—
160	#107	—	127	1	no	—
161	#107	C₂×D₁₂₇	1	2	yes	—
162	#106	—	127	2	no	—
163	#106	—	127	2	no	—
164	#106	—	127	1	no	—
165	#106	—	127	2	no	—
166	#106	—	127	1	no	—
167	#106	C₂×D₁₂₇	1	2	yes	—
168	#41	—	127	4	no	—
169	#41	—	127	4	no	—
170	C₂×Q₈	—	127	1	no	—
171	C₂×D₄	—	127	2	no	—
172	C₂×D₄	—	127	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	127	2	no	—
175	C₂²×C₄	—	254	2	no	—
176	C₂²×C₄	—	254	2	no	—
177	C₂²×C₄	—	254	1	no	—
178	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
179	M₁₆	—	127	8	no	—
180	M₁₆	C₄×D₁₂₇	1	4	yes	complemented
181	C₂×C₈	—	127	4	no	—
182	C₂×C₈	—	127	1	no	—
183	C₂×C₈	—	127	1	no	—
184	C₂×C₈	—	127	8	no	—
185	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	127	2	no	—
188	C₄⋊C₄	—	127	1	no	—
189	C₄⋊C₄	—	127	1	no	—
190	C₄⋊C₄	—	254	2	no	—
191	C₄⋊C₄	—	254	2	no	—
192	C₄⋊C₄	—	254	1	no	—
193	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
194	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
195	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	254	2	no	—
198	C₂²⋊C₄	—	254	2	no	—
199	C₂²⋊C₄	—	254	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	254	2	no	—
202	C₄²	—	254	2	no	—
203	C₄²	—	254	1	no	—
204	C₂³	—	127	2	no	—
205	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
206	Q₈	—	127	4	no	—
207	D₄	—	127	8	no	—
208	D₄	—	127	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	8	1	no	—
213	C₂×C₄	—	127	2	no	—
214	C₂×C₄	—	127	4	no	—
215	C₂×C₄	—	127	4	no	—
216	C₂×C₄	—	1016	2	no	—
217	C₂×C₄	—	1016	2	no	—
218	C₂×C₄	—	1016	1	no	—
219	C₂×C₄	—	254	1	no	—
220	C₂×C₄	—	254	2	no	—
221	C₂×C₄	—	254	1	no	—
222	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
223	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
224	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
225	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
226	C₈	—	127	2	no	—
227	C₈	—	127	2	no	—
228	C₈	C₂×C₄×D₁₂₇	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	127	4	no	—
232	C₂²	—	254	2	no	—
233	C₂²	—	254	2	no	—
234	C₂²	C₄⋊C₄×D₁₂₇	1	2	yes	—
235	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
236	C₄	—	4	1	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	1	no	—
239	C₄	—	127	4	no	—
240	C₄	—	508	2	no	—
241	C₄	—	508	1	no	—
242	C₄	—	508	1	no	—
243	C₄	C₄⋊C₄×D₁₂₇	1	2	yes	—
244	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
245	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	254	4	no	—
248	C₂	C₂×C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
249	C₂	#22550685	1	1	yes	characteristic
250	C₂	#22550686	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#428×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#14523248	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#428	D₁₂₇
hypocenter	C₁₂₇	#2912
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
4. #22549918	C₂	semidirect
5. #22549917	C₂	semidirect
6. #22549915	C₂	semidirect
7. #22549913	C₂	semidirect
8. #22549912	C₂	semidirect
9. C₂×#16861140	C₂	semidirect
10. #106×D₁₂₇	C₂	semidirect
11. #428×C₁₂₇	C₂	semidirect
13. #16861217	C₄	semidirect
51. #14523231	C₂×C₄	semidirect
52. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
55. C₈×D₁₂₇	C₂×C₄	semidirect
70. M₁₆×C₁₂₇	C₂×C₄	semidirect
126. C₁₂₇	#2912	semidirect
133. #428	D₁₂₇	semidirect
180. M₁₆	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
#22550357	2
#22550356	2
#22549918	2
#22549917	2
#22549915	2
#22549913	2
#22549912	2
C₂×#16861140	2
#106×D₁₂₇	2
#428×C₁₂₇	2
#2912	127