Subgroups — #30647528

Order 16256.

492 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861229	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523248	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549986	C₂	1	2	yes	maximal, complemented
4	#22549981	C₂	1	2	yes	maximal, complemented
5	#22549976	C₂	1	2	yes	maximal, complemented
6	#22549974	C₂	1	2	yes	maximal, complemented
7	#22549912	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549872	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549870	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16861120	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	#16861229	C₄	1	8	yes	complemented
14	C₂×#14523238	C₂²	1	1	yes	characteristic
15	C₂×#14523236	C₂²	1	1	yes	characteristic
16	C₂×#14523235	C₂²	1	1	yes	characteristic
17	C₂×#14523233	C₂²	1	1	yes	characteristic
18	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₅₄	C₂²	1	2	yes	—
21	C₄×D₅₀₈	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₁₂₇	—	2	2	no	—
23	#16861168	C₂²	1	1	yes	characteristic
24	#16861166	—	2	1	no	—
25	C₄²×D₁₂₇	—	2	1	no	—
26	#16861142	C₂²	1	2	yes	—
27	#16861140	C₂²	1	2	yes	—
28	#16861133	C₂²	1	2	yes	—
29	#16861132	C₂²	1	2	yes	—
30	#16861130	C₂²	1	2	yes	—
31	#16861129	C₂²	1	2	yes	—
32	#16861120	C₂²	1	2	yes	—
33	#16861120	C₂²	1	2	yes	—
34	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
36	C₄×D₄×C₁₂₇	—	2	2	no	—
37	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
38	#117×C₁₂₇	—	2	1	no	—
39	#117×C₁₂₇	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₁₂₇	—	2	1	no	—
41	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
42	#14523248	—	2	2	no	—
43	#14523248	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
45	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
47	C₂×C₄×D₁₂₇	—	2	1	no	—
48	C₂×C₄×D₁₂₇	—	2	2	no	—
49	#14523238	C₂×C₄	1	4	yes	complemented
50	#14523236	C₂×C₄	1	4	yes	complemented
51	#14523235	C₂×C₄	1	4	yes	complemented
52	#14523233	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₂₇⋊C₈	—	2	2	no	—
54	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
56	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
57	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
59	Dic₁₂₇⋊C₄	—	2	2	no	—
60	C₂²⋊Dic₁₂₇	—	2	1	no	—
61	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
62	C₄×Dic₁₂₇	—	2	1	no	—
63	C₄×Dic₁₂₇	—	2	2	no	—
64	#41×C₁₂₇	—	2	2	no	—
65	#41×C₁₂₇	C₂×C₄	1	4	yes	complemented
66	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₂₇	—	2	1	no	—
68	C₂×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₂₇	—	2	1	no	—
70	C₂²×C₄×C₁₂₇	—	2	2	no	—
71	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
72	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₂₇	—	2	2	no	—
74	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
76	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
77	C₄²×C₁₂₇	—	2	1	no	—
78	C₄²×C₁₂₇	—	2	2	no	—
79	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
80	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
81	C₄²×C₁₂₇	C₂³	1	2	yes	—
82	C₂²×D₁₂₇	—	2	1	no	—
83	Dic₂₅₄	—	2	1	no	—
84	Dic₂₅₄	C₂²×C₄	1	2	yes	—
85	D₅₀₈	—	2	1	no	—
86	D₅₀₈	C₂²×C₄	1	2	yes	—
87	C₂²⋊D₁₂₇	—	2	4	no	—
88	C₂×Dic₁₂₇	—	2	1	no	—
89	C₂×Dic₁₂₇	—	2	2	no	—
90	C₄×D₁₂₇	—	2	4	no	—
91	C₄×D₁₂₇	—	4	4	no	—
92	C₁₂₇⋊C₈	—	2	2	no	—
93	C₁₂₇⋊C₈	C₂²×C₄	1	2	yes	—
94	C₁₂₇⋊C₈	C₂²×C₄	1	2	yes	—
95	C₂³×C₁₂₇	—	2	1	no	—
96	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₁₂₇	—	2	1	no	—
98	Q₈×C₁₂₇	C₂²×C₄	1	2	yes	—
99	D₄×C₁₂₇	—	2	4	no	—
100	D₄×C₁₂₇	—	2	1	no	—
101	D₄×C₁₂₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₂₇	—	2	1	no	—
103	C₂×C₄×C₁₂₇	—	2	4	no	—
104	C₂×C₄×C₁₂₇	—	2	2	no	—
105	C₂×C₄×C₁₂₇	—	4	2	no	—
106	C₂×C₄×C₁₂₇	—	4	4	no	—
107	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₂₇	#41	1	2	yes	—
112	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
113	C₂²×C₁₂₇	—	2	2	no	—
114	C₂²×C₁₂₇	—	2	2	no	—
115	C₂²×C₁₂₇	—	4	2	no	—
116	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
117	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₂₇	—	2	2	no	—
119	C₂×D₁₂₇	—	4	2	no	—
120	C₄×C₁₂₇	—	2	2	no	—
121	C₄×C₁₂₇	—	2	2	no	—
122	C₄×C₁₂₇	—	2	2	no	—
123	C₄×C₁₂₇	—	4	2	no	—
124	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
125	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
127	Dic₁₂₇	—	2	2	no	—
128	Dic₁₂₇	—	4	2	no	—
129	C₂×C₁₂₇	—	2	2	no	—
130	C₂×C₁₂₇	—	4	2	no	—
131	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₂₇	#575	1	2	yes	—
133	D₁₂₇	—	4	2	no	—
134	#2943	—	127	1	no	maximal, Sylow
135	C₁₂₇	#2943	1	1	yes	characteristic, complemented, Sylow
136	C₂×#135	—	127	1	no	—
137	C₄×#41	—	127	1	no	—
138	C₄×#41	D₁₂₇	1	1	yes	characteristic, complemented
139	#442	—	127	2	no	—
140	#441	—	127	2	no	—
141	#440	—	127	2	no	—
142	#440	—	127	2	no	—
143	#427	—	127	1	no	—
144	#416	—	127	1	no	—
145	#416	—	127	1	no	—
146	C₂×#97	—	127	1	no	—
147	C₂×#41	—	127	1	no	—
148	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
149	#135	—	127	8	no	—
150	C₂×Q₁₆	—	127	1	no	—
151	C₂×SD₁₆	—	127	1	no	—
152	C₂×SD₁₆	—	127	1	no	—
153	C₂×D₈	—	127	1	no	—
154	C₂²×C₈	—	127	1	no	—
155	C₂²×C₈	—	127	1	no	—
156	C₄×Q₈	—	127	2	no	—
157	C₄×Q₈	C₂×D₁₂₇	1	2	yes	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	127	2	no	—
160	C₄×D₄	—	254	2	no	—
161	C₄×D₄	C₂×D₁₂₇	1	2	yes	—
162	#117	—	2	1	no	—
163	#117	—	127	1	no	—
164	#117	—	254	1	no	—
165	#117	C₂×D₁₂₇	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	254	1	no	—
168	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
169	#107	—	127	2	no	—
170	#106	—	127	2	no	—
171	#103	—	127	2	no	—
172	#103	—	127	2	no	—
173	#102	—	127	2	no	—
174	#102	—	127	2	no	—
175	#97	—	127	2	no	—
176	#97	—	127	2	no	—
177	#41	—	2	2	no	—
178	#41	—	127	4	no	—
179	#41	—	254	2	no	—
180	#41	C₄×D₁₂₇	1	4	yes	complemented
181	C₂×Q₈	—	127	1	no	—
182	C₂×Q₈	C₂²×D₁₂₇	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	127	1	no	—
185	C₂×D₄	—	254	1	no	—
186	C₂×D₄	C₂²×D₁₂₇	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	254	2	no	—
190	C₂²×C₄	—	254	1	no	—
191	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
192	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
193	Q₁₆	—	127	4	no	—
194	SD₁₆	—	127	4	no	—
195	SD₁₆	—	127	4	no	—
196	D₈	—	127	4	no	—
197	C₂×C₈	—	127	4	no	—
198	C₂×C₈	—	127	1	no	—
199	C₂×C₈	—	127	1	no	—
200	C₂×C₈	—	127	2	no	—
201	C₂×C₈	—	254	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	127	2	no	—
204	C₄⋊C₄	—	254	2	no	—
205	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	254	1	no	—
209	C₂²⋊C₄	—	254	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	254	2	no	—
213	C₄²	—	254	1	no	—
214	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
215	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
216	C₄²	C₂²×D₁₂₇	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	254	1	no	—
219	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	127	2	no	—
222	Q₈	—	254	1	no	—
223	Q₈	C₂×C₄×D₁₂₇	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	127	2	no	—
227	D₄	—	254	4	no	—
228	D₄	—	254	1	no	—
229	D₄	C₂×C₄×D₁₂₇	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	508	4	no	—
236	C₂×C₄	—	254	2	no	—
237	C₂×C₄	—	254	1	no	—
238	C₂×C₄	—	254	4	no	—
239	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
242	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
243	C₂×C₄	#14523248	1	2	yes	—
244	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
245	C₈	—	127	2	no	—
246	C₈	—	127	2	no	—
247	C₈	—	254	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	508	2	no	—
252	C₂²	—	254	2	no	—
253	C₂²	C₄×C₂²⋊D₁₂₇	1	2	yes	—
254	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	508	2	no	—
260	C₄	—	254	2	no	—
261	C₄	C₄×C₂²⋊D₁₂₇	1	2	yes	—
262	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
263	C₄	C₂×#14523248	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	508	2	no	—
267	C₂	C₂×C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
268	C₂	#22550674	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₄×#41×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#16861229	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₄×#41	D₁₂₇
hypocenter	C₁₂₇	#2943
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#14523248	C₂	semidirect
3. #22549986	C₂	semidirect
4. #22549981	C₂	semidirect
5. #22549976	C₂	semidirect
6. #22549974	C₂	semidirect
7. #22549912	C₂	semidirect
8. #22549872	C₂	semidirect
9. #22549870	C₂	semidirect
10. C₂×#16861120	C₂	semidirect
11. C₄×#41×C₁₂₇	C₂	semidirect
13. #16861229	C₄	semidirect
43. #14523248	C₂×C₄	semidirect
49. #14523238	C₂×C₄	semidirect
50. #14523236	C₂×C₄	semidirect
51. #14523235	C₂×C₄	semidirect
52. #14523233	C₂×C₄	semidirect
54. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
65. #41×C₁₂₇	C₂×C₄	semidirect
135. C₁₂₇	#2943	semidirect
138. C₄×#41	D₁₂₇	semidirect
180. #41	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×#16861229	2
C₄×#14523248	2
#22549986	2
#22549981	2
#22549976	2
#22549974	2
#22549912	2
#22549872	2
#22549870	2
C₂×#16861120	2
C₄×#41×C₁₂₇	2
#2943	127