Subgroups — #30650473

Order 16256.

482 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861231	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550535	C₂	1	2	yes	maximal, complemented
4	#22550260	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550117	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550112	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550099	C₂	1	2	yes	maximal, complemented
8	#22550040	C₂	1	2	yes	maximal, complemented
9	#22550017	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	2	yes	—
13	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
14	#16861231	—	2	4	no	—
15	C₂×Dic₅₀₈	—	2	4	no	—
16	C₂×Dic₅₀₈	C₂²	1	2	yes	—
17	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
18	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
19	C₂×#14523237	C₂²	1	2	yes	—
20	C₂×#14523237	C₂²	1	2	yes	—
21	C₂×D₁₀₁₆	C₂²	1	1	yes	characteristic
22	#16861197	C₂²	1	2	yes	—
23	#16861195	—	2	2	no	—
24	#16861188	C₂²	1	2	yes	—
25	#16861187	—	2	2	no	—
26	#16861183	C₂²	1	2	yes	—
27	#16861181	—	2	2	no	—
28	C₄×Dic₂₅₄	C₂²	1	2	yes	—
29	(C₂²⋊D₁₂₇)⋊C₄	—	2	2	no	—
30	#16861143	C₂²	1	1	yes	characteristic
31	#16861134	C₂²	1	2	yes	—
32	#16861134	C₂²	1	2	yes	—
33	#16861131	C₂²	1	2	yes	—
34	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
41	#14523248	—	2	4	no	—
42	#14523248	—	4	2	no	—
43	C₂×Dic₂₅₄	—	2	4	no	—
44	C₂×Dic₂₅₄	—	2	2	no	—
45	C₂×Dic₂₅₄	—	4	2	no	—
46	C₂×Dic₂₅₄	C₂³	1	2	yes	—
47	C₂×Dic₂₅₄	C₂³	1	2	yes	—
48	C₂×D₅₀₈	C₂³	1	2	yes	—
49	C₂×C₂²⋊D₁₂₇	—	2	2	no	—
50	C₂×C₂²⋊D₁₂₇	—	2	2	no	—
51	C₂²×Dic₁₂₇	—	2	2	no	—
52	C₂×C₄×D₁₂₇	—	2	2	no	—
53	Dic₅₀₈	—	2	4	no	—
54	Dic₅₀₈	—	2	2	no	—
55	Dic₅₀₈	D₄	1	4	yes	complemented
56	#14523237	—	2	4	no	—
57	#14523237	—	4	2	no	—
58	D₁₀₁₆	—	2	2	no	—
59	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
60	Dic₁₂₇⋊C₄	—	2	2	no	—
61	Dic₁₂₇⋊C₄	—	2	2	no	—
62	C₂²⋊Dic₁₂₇	—	2	2	no	—
63	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
64	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
65	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
66	C₄×Dic₁₂₇	—	2	2	no	—
67	C₂×D₄×C₁₂₇	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₂₇	—	2	2	no	—
71	C₂×C₈×C₁₂₇	—	2	2	no	—
72	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
79	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
80	C₂²×D₁₂₇	—	2	2	no	—
81	Dic₂₅₄	—	2	2	no	—
82	Dic₂₅₄	—	2	2	no	—
83	Dic₂₅₄	—	2	2	no	—
84	Dic₂₅₄	—	4	4	no	—
85	Dic₂₅₄	C₂×D₄	1	4	yes	—
86	D₅₀₈	—	2	2	no	—
87	D₅₀₈	—	2	2	no	—
88	C₂²⋊D₁₂₇	—	4	4	no	—
89	C₂²⋊D₁₂₇	—	4	4	no	—
90	C₂×Dic₁₂₇	—	2	2	no	—
91	C₂×Dic₁₂₇	—	2	2	no	—
92	C₂×Dic₁₂₇	—	2	2	no	—
93	C₂×Dic₁₂₇	—	4	4	no	—
94	C₄×D₁₂₇	—	4	4	no	—
95	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
96	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
97	D₄×C₁₂₇	—	2	2	no	—
98	C₂×C₄×C₁₂₇	—	2	1	no	—
99	C₂×C₄×C₁₂₇	—	2	1	no	—
100	C₂×C₄×C₁₂₇	—	2	1	no	—
101	C₂×C₄×C₁₂₇	—	2	1	no	—
102	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
107	C₈×C₁₂₇	—	2	1	no	—
108	C₈×C₁₂₇	—	2	1	no	—
109	C₈×C₁₂₇	—	2	1	no	—
110	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
111	C₂²×C₁₂₇	—	2	1	no	—
112	C₂²×C₁₂₇	—	2	1	no	—
113	C₂²×C₁₂₇	—	2	1	no	—
114	C₂²×C₁₂₇	—	2	1	no	—
115	C₂²×C₁₂₇	—	2	1	no	—
116	C₂²×C₁₂₇	#135	1	2	yes	complemented
117	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₂₇	—	4	2	no	—
119	C₂×D₁₂₇	—	4	2	no	—
120	C₂×D₁₂₇	—	4	2	no	—
121	C₄×C₁₂₇	—	2	1	no	—
122	C₄×C₁₂₇	—	2	1	no	—
123	C₄×C₁₂₇	—	2	1	no	—
124	C₄×C₁₂₇	—	2	1	no	—
125	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₁₂₇	—	2	4	no	—
128	Dic₁₂₇	—	4	2	no	—
129	Dic₁₂₇	—	4	2	no	—
130	C₂×C₁₂₇	—	2	2	no	—
131	C₂×C₁₂₇	—	4	1	no	—
132	C₂×C₁₂₇	D₄²	1	1	yes	characteristic
133	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
134	C₂×C₁₂₇	#578	1	1	yes	characteristic
135	D₁₂₇	—	8	2	no	—
136	#3285	—	127	1	no	maximal, Sylow
137	C₁₂₇	#3285	1	1	yes	characteristic, complemented, Sylow
138	C₂×#135	—	127	1	no	—
139	C₂²×Q₁₆	—	127	1	no	—
140	#548	—	127	2	no	—
141	#495	—	127	1	no	—
142	#467	—	127	1	no	—
143	#466	—	127	1	no	—
144	#464	—	127	2	no	—
145	#452	—	127	2	no	—
146	#448	—	127	2	no	—
147	C₄×Q₁₆	—	127	1	no	—
148	C₈×D₄	D₁₂₇	1	1	yes	characteristic, complemented
149	C₂×#41	—	127	2	no	—
150	C₂²×Q₈	—	127	2	no	—
151	#135	—	254	4	no	—
152	C₂×Q₁₆	—	127	1	no	—
153	C₂×Q₁₆	—	127	2	no	—
154	C₂×Q₁₆	—	127	1	no	—
155	C₂×Q₁₆	—	254	4	no	—
156	C₂×SD₁₆	—	127	2	no	—
157	C₂×SD₁₆	—	127	2	no	—
158	C₂×D₈	—	127	1	no	—
159	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
160	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
161	#124	—	127	2	no	—
162	#124	—	254	2	no	—
163	C₂²⋊Q₈	—	127	2	no	—
164	C₂²⋊Q₈	—	254	2	no	—
165	#121	—	127	2	no	—
166	#121	—	254	2	no	—
167	C₄×Q₈	—	127	2	no	—
168	C₄×D₄	—	254	2	no	—
169	C₄×D₄	C₂×D₁₂₇	1	1	yes	characteristic
170	#107	—	127	1	no	—
171	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
172	#103	—	127	2	no	—
173	#103	—	127	2	no	—
174	#102	—	127	2	no	—
175	C₂²⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
176	C₂²⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
177	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
178	#41	—	508	2	no	—
179	#41	—	254	4	no	—
180	C₂×Q₈	—	127	2	no	—
181	C₂×Q₈	—	127	2	no	—
182	C₂×Q₈	—	508	2	no	—
183	C₂×Q₈	—	254	4	no	—
184	C₂×Q₈	—	254	2	no	—
185	C₂×D₄	—	127	2	no	—
186	C₂×D₄	—	254	2	no	—
187	C₂×D₄	—	254	2	no	—
188	C₂×D₄	D₅₀₈	1	1	yes	characteristic
189	C₂²×C₄	—	254	2	no	—
190	C₂²×C₄	—	254	2	no	—
191	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
193	Q₁₆	—	127	4	no	—
194	Q₁₆	—	254	4	no	—
195	Q₁₆	—	254	2	no	—
196	SD₁₆	—	508	2	no	—
197	SD₁₆	—	254	4	no	—
198	D₈	—	254	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
204	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
205	C₄⋊C₄	—	127	2	no	—
206	C₄⋊C₄	—	254	2	no	—
207	C₄⋊C₄	—	254	2	no	—
208	C₄⋊C₄	D₅₀₈	1	1	yes	characteristic
209	C₂²⋊C₄	—	254	2	no	—
210	C₂²⋊C₄	—	254	2	no	—
211	C₂²⋊C₄	—	254	2	no	—
212	C₂²⋊C₄	—	254	2	no	—
213	C₂²⋊C₄	D₅₀₈	1	1	yes	characteristic
214	C₂²⋊C₄	D₅₀₈	1	1	yes	characteristic
215	C₄²	—	254	2	no	—
216	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
217	C₂³	—	254	2	no	—
218	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
219	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
220	Q₈	—	127	4	no	—
221	Q₈	—	508	4	no	—
222	Q₈	—	254	2	no	—
223	Q₈	—	254	2	no	—
224	Q₈	—	254	2	no	—
225	D₄	—	2	2	no	—
226	D₄	—	508	4	no	—
227	D₄	—	508	4	no	—
228	D₄	—	254	2	no	—
229	D₄	—	254	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	508	4	no	—
235	C₂×C₄	—	508	4	no	—
236	C₂×C₄	—	254	2	no	—
237	C₂×C₄	—	254	2	no	—
238	C₂×C₄	—	254	2	no	—
239	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
240	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
241	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
242	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	D₄×D₁₂₇	1	2	yes	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	508	2	no	—
254	C₂²	—	508	2	no	—
255	C₂²	—	508	2	no	—
256	C₂²	#16861231	1	2	yes	complemented
257	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	508	2	no	—
263	C₄	—	508	2	no	—
264	C₄	—	254	4	no	—
265	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
266	C₄	#16861255	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	4	1	no	—
269	C₂	—	1016	2	no	—
270	C₂	D₄×D₅₀₈	1	1	yes	characteristic
271	C₂	C₂×#16861231	1	1	yes	characteristic
272	C₂	#22550693	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₀₈
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₈×D₄×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
hypercenter	C₈×D₄	D₁₂₇
hypocenter	C₁₂₇	#3285
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16861231	C₂	semidirect
2. C₂²×Dic₅₀₈	C₂	semidirect
3. #22550535	C₂	semidirect
4. #22550260	C₂	semidirect
5. #22550117	C₂	semidirect
6. #22550112	C₂	semidirect
7. #22550099	C₂	semidirect
8. #22550040	C₂	semidirect
9. #22550017	C₂	semidirect
10. C₄×Dic₅₀₈	C₂	semidirect
11. C₈×D₄×C₁₂₇	C₂	semidirect
55. Dic₅₀₈	D₄	semidirect
116. C₂²×C₁₂₇	#135	semidirect
137. C₁₂₇	#3285	semidirect
148. C₈×D₄	D₁₂₇	semidirect
256. C₂²	#16861231	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861231	2
C₂²×Dic₅₀₈	2
#22550535	2
#22550260	2
#22550117	2
#22550112	2
#22550099	2
#22550040	2
#22550017	2
C₄×Dic₅₀₈	2
C₈×D₄×C₁₂₇	2
#3285	127