Subgroups — #30647573

Order 16256.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861240	C₂	1	1	yes	maximal, characteristic
2	#22550356	C₂	1	1	yes	maximal, characteristic
3	#22549981	C₂	1	2	yes	maximal, complemented
4	#22549974	C₂	1	2	yes	maximal, complemented
5	C₄×#14523238	C₂	1	2	yes	maximal, complemented
6	C₄×#14523235	C₂	1	2	yes	maximal, complemented
7	#22549919	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549870	C₂	1	1	yes	maximal, characteristic
9	C₂×#16861133	C₂	1	1	yes	maximal, characteristic, complemented
10	#22549832	C₂	1	1	yes	maximal, characteristic, complemented
11	#519×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	#16861240	C₄	1	8	yes	complemented
14	C₂×#14523238	C₂²	1	2	yes	—
15	C₂×#14523235	C₂²	1	2	yes	—
16	C₂×#14523232	C₂²	1	1	yes	characteristic
17	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₅₄	C₂²	1	2	yes	—
19	C₄×D₅₀₈	C₂²	1	2	yes	—
20	(C₂²⋊D₁₂₇)⋊C₄	—	2	2	no	—
21	#16861167	—	2	1	no	—
22	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₁₂₇	—	2	1	no	—
24	#16861142	C₂²	1	2	yes	—
25	#16861140	C₂²	1	2	yes	—
26	#16861133	C₂²	1	2	yes	—
27	#16861133	C₂²	1	2	yes	—
28	#16861132	C₂²	1	1	yes	characteristic
29	#16861132	C₂²	1	1	yes	characteristic
30	#16861129	C₂²	1	1	yes	characteristic
31	#16861129	C₂²	1	1	yes	characteristic
32	#16861120	C₂²	1	2	yes	—
33	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
34	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₂₇	—	2	1	no	—
36	C₄×Q₈×C₁₂₇	—	2	1	no	—
37	C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
38	C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
39	#117×C₁₂₇	—	2	2	no	—
40	#117×C₁₂₇	C₂²	1	1	yes	characteristic
41	#117×C₁₂₇	C₂²	1	1	yes	characteristic
42	#14523248	—	2	2	no	—
43	#14523248	C₂×C₄	1	4	yes	—
44	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
45	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
47	C₂²×Dic₁₂₇	—	2	1	no	—
48	C₂×C₄×D₁₂₇	—	2	1	no	—
49	C₂×C₄×D₁₂₇	—	2	2	no	—
50	#14523238	C₂×C₄	1	8	yes	complemented
51	#14523235	C₂×C₄	1	8	yes	complemented
52	#14523232	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₂₇⋊C₈	—	2	2	no	—
54	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
56	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
57	C₄⋊Dic₁₂₇	—	2	1	no	—
58	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
59	Dic₁₂₇⋊C₄	—	2	2	no	—
60	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
61	C₄×Dic₁₂₇	—	2	2	no	—
62	C₂×Q₈×C₁₂₇	—	2	1	no	—
63	C₂×Q₈×C₁₂₇	—	2	2	no	—
64	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
65	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₂₇	—	2	1	no	—
67	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₂₇	—	2	2	no	—
69	C₄⋊C₄×C₁₂₇	—	2	2	no	—
70	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
75	C₂²⋊C₄×C₁₂₇	D₄	1	2	yes	—
76	C₄²×C₁₂₇	—	2	2	no	—
77	C₄²×C₁₂₇	—	2	2	no	—
78	C₄²×C₁₂₇	—	2	1	no	—
79	C₄²×C₁₂₇	C₂³	1	2	yes	—
80	C₂²×D₁₂₇	—	2	1	no	—
81	Dic₂₅₄	—	2	1	no	—
82	Dic₂₅₄	C₂²×C₄	1	2	yes	—
83	D₅₀₈	—	2	1	no	—
84	D₅₀₈	C₂²×C₄	1	2	yes	—
85	C₂²⋊D₁₂₇	—	2	4	no	—
86	C₂×Dic₁₂₇	—	2	1	no	—
87	C₂×Dic₁₂₇	—	2	2	no	—
88	C₂×Dic₁₂₇	—	8	1	no	—
89	C₄×D₁₂₇	—	2	4	no	—
90	C₄×D₁₂₇	—	8	2	no	—
91	C₁₂₇⋊C₈	—	2	1	no	—
92	C₁₂₇⋊C₈	—	2	1	no	—
93	C₁₂₇⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
95	Q₈×C₁₂₇	—	2	4	no	—
96	Q₈×C₁₂₇	—	2	2	no	—
97	Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
98	C₂×C₄×C₁₂₇	—	2	4	no	—
99	C₂×C₄×C₁₂₇	—	2	2	no	—
100	C₂×C₄×C₁₂₇	—	2	1	no	—
101	C₂×C₄×C₁₂₇	—	2	1	no	—
102	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	#41	1	2	yes	—
107	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
108	C₂²×C₁₂₇	—	2	1	no	—
109	C₂²×C₁₂₇	—	2	1	no	—
110	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
111	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×D₁₂₇	—	2	2	no	—
113	C₂×D₁₂₇	—	4	1	no	—
114	C₂×D₁₂₇	—	4	1	no	—
115	C₄×C₁₂₇	—	2	4	no	—
116	C₄×C₁₂₇	—	2	2	no	—
117	C₄×C₁₂₇	—	2	2	no	—
118	C₄×C₁₂₇	—	4	1	no	—
119	C₄×C₁₂₇	—	4	1	no	—
120	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
121	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
123	Dic₁₂₇	—	2	2	no	—
124	Dic₁₂₇	—	4	2	no	—
125	C₂×C₁₂₇	—	2	2	no	—
126	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×C₁₂₇	#577	1	1	yes	characteristic
128	C₂×C₁₂₇	#578	1	1	yes	characteristic
129	D₁₂₇	—	4	2	no	—
130	#2947	—	127	1	no	maximal, Sylow
131	C₁₂₇	#2947	1	1	yes	characteristic, complemented, Sylow
132	C₂×#137	—	127	1	no	—
133	#520	—	127	1	no	—
134	#519	D₁₂₇	1	1	yes	characteristic, complemented
135	#441	—	127	2	no	—
136	#440	—	127	2	no	—
137	C₄×Q₁₆	—	127	2	no	—
138	C₄×SD₁₆	—	127	2	no	—
139	#428	—	127	1	no	—
140	#416	—	127	1	no	—
141	C₂×#103	—	127	1	no	—
142	#405	—	127	1	no	—
143	C₂×#41	—	127	1	no	—
144	C₂²×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
145	#137	—	127	8	no	—
146	C₂×Q₁₆	—	127	2	no	—
147	C₂×SD₁₆	—	127	2	no	—
148	C₂×M₁₆	—	127	1	no	—
149	C₂²×C₈	—	127	1	no	—
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	—	2	1	no	—
152	C₄×Q₈	—	127	2	no	—
153	C₄×Q₈	C₂×D₁₂₇	1	2	yes	—
154	C₄×Q₈	C₂×D₁₂₇	1	2	yes	—
155	C₄×D₄	—	127	2	no	—
156	C₄×D₄	—	254	2	no	—
157	#117	—	2	2	no	—
158	#117	—	254	1	no	—
159	#117	C₂×D₁₂₇	1	1	yes	characteristic
160	#117	C₂×D₁₂₇	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	127	1	no	—
162	C₂×C₄⋊C₄	—	254	1	no	—
163	#107	—	127	2	no	—
164	#106	—	127	2	no	—
165	#103	—	127	2	no	—
166	#103	—	127	2	no	—
167	#103	—	127	1	no	—
168	#103	—	127	1	no	—
169	#102	—	127	1	no	—
170	#102	—	127	1	no	—
171	#97	—	127	2	no	—
172	C₄×C₈	—	127	2	no	—
173	#41	—	127	4	no	—
174	#41	—	254	2	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	127	1	no	—
178	C₂×Q₈	C₄×D₁₂₇	1	4	yes	complemented
179	C₂×Q₈	C₂²×D₁₂₇	1	2	yes	—
180	C₂×D₄	—	127	1	no	—
181	C₂×D₄	—	254	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	254	1	no	—
184	C₂²×C₄	—	254	2	no	—
185	C₂²×C₄	—	254	1	no	—
186	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
187	Q₁₆	—	127	8	no	—
188	SD₁₆	—	127	8	no	—
189	M₁₆	—	127	4	no	—
190	C₂×C₈	—	127	2	no	—
191	C₂×C₈	—	127	1	no	—
192	C₂×C₈	—	127	1	no	—
193	C₂×C₈	—	254	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	127	2	no	—
197	C₄⋊C₄	—	254	2	no	—
198	C₄⋊C₄	—	254	1	no	—
199	C₄⋊C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
200	C₄⋊C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
201	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	254	2	no	—
205	C₂²⋊C₄	C₂²⋊D₁₂₇	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	254	2	no	—
210	C₄²	C₂²×D₁₂₇	1	2	yes	—
211	C₂³	—	254	1	no	—
212	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
213	Q₈	—	2	2	no	—
214	Q₈	—	2	4	no	—
215	Q₈	—	127	2	no	—
216	Q₈	—	254	1	no	—
217	Q₈	C₂×C₄×D₁₂₇	1	4	yes	—
218	D₄	—	127	2	no	—
219	D₄	—	254	4	no	—
220	D₄	—	254	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	1016	2	no	—
226	C₂×C₄	—	1016	1	no	—
227	C₂×C₄	—	254	2	no	—
228	C₂×C₄	—	254	1	no	—
229	C₂×C₄	—	254	4	no	—
230	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
233	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
234	C₂×C₄	#14523248	1	2	yes	—
235	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
236	C₈	—	127	4	no	—
237	C₈	—	254	1	no	—
238	C₈	—	254	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	508	1	no	—
242	C₂²	—	508	1	no	—
243	C₂²	—	254	2	no	—
244	C₂²	C₄×C₂²⋊D₁₂₇	1	2	yes	—
245	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	508	2	no	—
252	C₄	—	254	2	no	—
253	C₄	C₄×C₂²⋊D₁₂₇	1	2	yes	—
254	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
255	C₄	C₂×#14523248	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	508	2	no	—
258	C₂	C₂×C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
259	C₂	#22550684	1	1	yes	characteristic
260	C₂	#22550691	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#519×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#14523248	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#519	D₁₂₇
hypocenter	C₁₂₇	#2947
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #519

Complements

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

N	H ≅ G/N	type
3. #22549981	C₂	semidirect
4. #22549974	C₂	semidirect
5. C₄×#14523238	C₂	semidirect
6. C₄×#14523235	C₂	semidirect
7. #22549919	C₂	semidirect
9. C₂×#16861133	C₂	semidirect
10. #22549832	C₂	semidirect
11. #519×C₁₂₇	C₂	semidirect
13. #16861240	C₄	semidirect
50. #14523238	C₂×C₄	semidirect
51. #14523235	C₂×C₄	semidirect
52. #14523232	C₂×C₄	semidirect
64. C₂×Q₈×C₁₂₇	C₂×C₄	semidirect
131. C₁₂₇	#2947	semidirect
134. #519	D₁₂₇	semidirect
178. C₂×Q₈	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×#16861240	2
#22550356	2
#22549981	2
#22549974	2
C₄×#14523238	2
C₄×#14523235	2
#22549919	2
#22549870	2
C₂×#16861133	2
#22549832	2
#519×C₁₂₇	2
#2947	127