Subgroups — C₄×#16861240 #30647549

Order 16256.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 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	C₄×#14523248	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	#22549916	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549885	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549875	C₂	1	1	yes	maximal, characteristic
10	C₄×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	#16861240	C₄	1	8	yes	direct product factor
14	C₂×#14523238	C₂²	1	2	yes	—
15	C₂×#14523235	C₂²	1	2	yes	—
16	C₂×#14523232	C₂²	1	1	yes	characteristic
17	C₂×#14523232	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
19	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
20	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
21	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₂₇	—	2	1	no	—
23	C₄×C₂²⋊D₁₂₇	—	2	1	no	—
24	#16861168	C₂²	1	1	yes	characteristic
25	#16861166	—	2	1	no	—
26	C₄²×D₁₂₇	—	2	1	no	—
27	#16861142	C₂²	1	2	yes	—
28	#16861140	C₂²	1	2	yes	—
29	#16861133	C₂²	1	2	yes	—
30	#16861133	C₂²	1	2	yes	—
31	#16861132	C₂²	1	2	yes	—
32	#16861129	C₂²	1	2	yes	—
33	#16861120	C₂²	1	2	yes	—
34	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₂₇	—	2	1	no	—
37	C₄×Q₈×C₁₂₇	—	2	1	no	—
38	C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
39	C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₁₂₇	—	2	1	no	—
43	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
44	#14523248	—	2	2	no	—
45	#14523248	C₂×C₄	1	4	yes	—
46	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
47	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
49	C₂×C₄×D₁₂₇	—	2	1	no	—
50	C₂×C₄×D₁₂₇	—	2	1	no	—
51	C₂×C₄×D₁₂₇	—	2	1	no	—
52	#14523238	C₂×C₄	1	8	yes	complemented
53	#14523235	C₂×C₄	1	8	yes	complemented
54	#14523232	—	2	2	no	—
55	#14523232	C₂×C₄	1	4	yes	complemented
56	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
57	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
58	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
60	Dic₁₂₇⋊C₄	—	2	1	no	—
61	Dic₁₂₇⋊C₄	—	2	1	no	—
62	C₂²⋊Dic₁₂₇	—	2	1	no	—
63	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
64	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
65	C₄×Dic₁₂₇	—	2	1	no	—
66	C₄×Dic₁₂₇	—	2	1	no	—
67	C₄×Dic₁₂₇	—	2	1	no	—
68	C₂×Q₈×C₁₂₇	—	2	1	no	—
69	C₂×Q₈×C₁₂₇	—	2	2	no	—
70	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
71	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
72	C₂²×C₄×C₁₂₇	—	2	1	no	—
73	C₂²×C₄×C₁₂₇	—	2	1	no	—
74	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₂₇	—	2	2	no	—
78	C₄⋊C₄×C₁₂₇	—	2	2	no	—
79	C₄⋊C₄×C₁₂₇	—	2	1	no	—
80	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
81	C₄²×C₁₂₇	—	2	2	no	—
82	C₄²×C₁₂₇	—	2	2	no	—
83	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
84	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
85	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
86	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
87	C₂²×D₁₂₇	—	2	1	no	—
88	Dic₂₅₄	—	2	1	no	—
89	Dic₂₅₄	C₂²×C₄	1	2	yes	—
90	D₅₀₈	—	2	1	no	—
91	D₅₀₈	C₂²×C₄	1	2	yes	—
92	C₂²⋊D₁₂₇	—	2	4	no	—
93	C₂×Dic₁₂₇	—	2	1	no	—
94	C₂×Dic₁₂₇	—	2	1	no	—
95	C₂×Dic₁₂₇	—	2	1	no	—
96	C₄×D₁₂₇	—	2	4	no	—
97	C₄×D₁₂₇	—	4	2	no	—
98	C₄×D₁₂₇	—	4	2	no	—
99	C₁₂₇⋊C₈	—	2	2	no	—
100	C₁₂₇⋊C₈	C₂²×C₄	1	4	yes	—
101	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
102	Q₈×C₁₂₇	—	2	4	no	—
103	Q₈×C₁₂₇	—	2	2	no	—
104	Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
105	C₂×C₄×C₁₂₇	—	2	4	no	—
106	C₂×C₄×C₁₂₇	—	2	2	no	—
107	C₂×C₄×C₁₂₇	—	2	2	no	—
108	C₂×C₄×C₁₂₇	—	2	2	no	—
109	C₂×C₄×C₁₂₇	—	4	1	no	—
110	C₂×C₄×C₁₂₇	—	4	2	no	—
111	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
112	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₂₇	#41	1	2	yes	—
118	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
119	C₂²×C₁₂₇	—	2	1	no	—
120	C₂²×C₁₂₇	—	2	1	no	—
121	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
122	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₁₂₇	—	2	2	no	—
124	C₂×D₁₂₇	—	4	1	no	—
125	C₂×D₁₂₇	—	4	1	no	—
126	C₄×C₁₂₇	—	2	4	no	—
127	C₄×C₁₂₇	—	2	1	no	—
128	C₄×C₁₂₇	—	2	1	no	—
129	C₄×C₁₂₇	—	2	1	no	—
130	C₄×C₁₂₇	—	4	2	no	—
131	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
132	C₄×C₁₂₇	#137	1	2	yes	complemented
133	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
135	Dic₁₂₇	—	2	2	no	—
136	Dic₁₂₇	—	4	1	no	—
137	Dic₁₂₇	—	4	1	no	—
138	C₂×C₁₂₇	—	2	2	no	—
139	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₁₂₇	C₂×#137	1	1	yes	characteristic
141	C₂×C₁₂₇	#575	1	1	yes	characteristic
142	D₁₂₇	—	4	2	no	—
143	C₄×#137	—	127	1	no	maximal, Sylow
144	C₁₂₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
145	C₂×#137	—	127	1	no	—
146	C₄×#41	—	127	1	no	—
147	C₂×C₄×Q₈	D₁₂₇	1	1	yes	characteristic, complemented
148	#441	—	127	2	no	—
149	#440	—	127	2	no	—
150	C₄×Q₁₆	—	127	2	no	—
151	C₄×SD₁₆	—	127	2	no	—
152	#428	—	127	1	no	—
153	#419	—	127	1	no	—
154	#417	—	127	1	no	—
155	C₄×M₁₆	—	127	1	no	—
156	C₂×#41	—	127	1	no	—
157	C₂²×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
158	#137	—	127	8	no	—
159	C₂×Q₁₆	—	127	2	no	—
160	C₂×SD₁₆	—	127	2	no	—
161	C₂×M₁₆	—	127	1	no	—
162	C₂×M₁₆	—	127	1	no	—
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	—	2	1	no	—
165	C₄×Q₈	—	127	1	no	—
166	C₄×Q₈	—	127	1	no	—
167	C₄×Q₈	C₂×D₁₂₇	1	2	yes	—
168	C₄×Q₈	C₂×D₁₂₇	1	2	yes	—
169	C₄×D₄	—	127	1	no	—
170	C₄×D₄	—	127	1	no	—
171	C₄×D₄	—	254	1	no	—
172	C₄×D₄	—	254	1	no	—
173	#117	—	127	1	no	—
174	#117	—	254	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	—	254	1	no	—
179	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
180	#107	—	127	2	no	—
181	#106	—	127	2	no	—
182	#103	—	127	2	no	—
183	#103	—	127	2	no	—
184	#103	—	127	2	no	—
185	#102	—	127	2	no	—
186	#97	—	127	2	no	—
187	C₄×C₈	—	127	2	no	—
188	#41	—	127	4	no	—
189	#41	—	254	2	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	127	1	no	—
193	C₂×Q₈	C₄×D₁₂₇	1	4	yes	complemented
194	C₂×Q₈	C₂²×D₁₂₇	1	2	yes	—
195	C₂×D₄	—	127	1	no	—
196	C₂×D₄	—	254	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	254	1	no	—
200	C₂²×C₄	—	254	1	no	—
201	C₂²×C₄	—	254	1	no	—
202	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
205	Q₁₆	—	127	8	no	—
206	SD₁₆	—	127	8	no	—
207	M₁₆	—	127	4	no	—
208	M₁₆	—	254	2	no	—
209	C₂×C₈	—	127	2	no	—
210	C₂×C₈	—	127	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	127	1	no	—
215	C₄⋊C₄	—	127	1	no	—
216	C₄⋊C₄	—	254	1	no	—
217	C₄⋊C₄	—	254	1	no	—
218	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
219	C₂²⋊C₄	—	254	1	no	—
220	C₂²⋊C₄	—	254	1	no	—
221	C₂²⋊C₄	—	254	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	254	1	no	—
225	C₄²	—	254	1	no	—
226	C₄²	—	254	1	no	—
227	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
228	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
229	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
230	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
231	C₂³	—	254	1	no	—
232	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
233	Q₈	—	2	2	no	—
234	Q₈	—	2	4	no	—
235	Q₈	—	127	2	no	—
236	Q₈	—	254	1	no	—
237	Q₈	C₂×C₄×D₁₂₇	1	4	yes	—
238	D₄	—	127	2	no	—
239	D₄	—	254	4	no	—
240	D₄	—	254	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	508	2	no	—
248	C₂×C₄	—	508	2	no	—
249	C₂×C₄	—	254	1	no	—
250	C₂×C₄	—	254	1	no	—
251	C₂×C₄	—	254	1	no	—
252	C₂×C₄	—	254	4	no	—
253	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
254	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
259	C₂×C₄	#14523248	1	2	yes	—
260	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
261	C₈	—	127	4	no	—
262	C₈	—	254	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	508	1	no	—
266	C₂²	—	508	1	no	—
267	C₂²	—	254	2	no	—
268	C₂²	C₄×C₂²⋊D₁₂₇	1	2	yes	—
269	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	2	no	—
275	C₄	—	508	1	no	—
276	C₄	—	508	1	no	—
277	C₄	—	254	2	no	—
278	C₄	C₄×C₂²⋊D₁₂₇	1	2	yes	—
279	C₄	#16861240	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
281	C₄	C₂×#14523248	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	508	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
285	C₂	C₂×#16861240	1	1	yes	characteristic
286	C₂	#22550674	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄×Q₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#14523248	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₂₇
hypocenter	C₁₂₇	C₄×#137
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

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. #22549916	C₂	semidirect
8. #22549885	C₂	semidirect
10. C₄×#14523232	C₂	semidirect
11. C₂×C₄×Q₈×C₁₂₇	C₂	semidirect
13. #16861240	C₄	direct (N × H)
52. #14523238	C₂×C₄	semidirect
53. #14523235	C₂×C₄	semidirect
55. #14523232	C₂×C₄	semidirect
70. C₂×Q₈×C₁₂₇	C₂×C₄	semidirect
132. C₄×C₁₂₇	#137	semidirect
144. C₁₂₇	C₄×#137	semidirect
147. C₂×C₄×Q₈	D₁₂₇	semidirect
193. C₂×Q₈	C₄×D₁₂₇	semidirect
279. C₄	#16861240	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16861240	2
C₄×#14523248	2
#22549981	2
#22549974	2
C₄×#14523238	2
C₄×#14523235	2
#22549916	2
#22549885	2
#22549875	2
C₄×#14523232	2
C₂×C₄×Q₈×C₁₂₇	2
C₄×#137	127

Subgroups — C4×#16861240 #30647549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16861240 #30647549