Subgroups — C₄×#16861232 #30647502

Order 16256.

500 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861232	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523249	C₂	1	2	yes	maximal, complemented
3	Q₁₆×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#14523235	C₂	1	2	yes	maximal, complemented
5	#22549961	C₂	1	2	yes	maximal, complemented
6	C₄×D₁₀₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549958	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549913	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549871	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523249	C₂²	1	2	yes	—
13	#16861232	C₄	1	8	yes	direct product factor
14	C₂×#14523235	C₂²	1	2	yes	—
15	C₂×D₁₀₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
18	Q₈×Dic₁₂₇	C₂²	1	2	yes	—
19	C₄×D₅₀₈	—	2	2	no	—
20	C₄×D₅₀₈	C₂²	1	2	yes	—
21	#16861171	—	2	2	no	—
22	#16861171	C₂²	1	2	yes	—
23	#16861167	—	2	2	no	—
24	#16861167	C₂²	1	2	yes	—
25	C₄²×D₁₂₇	—	2	2	no	—
26	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
27	#16861143	C₂²	1	1	yes	characteristic
28	#16861140	C₂²	1	2	yes	—
29	#16861133	C₂²	1	2	yes	—
30	#16861131	C₂²	1	2	yes	—
31	#16861129	C₂²	1	2	yes	—
32	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
33	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
37	#107×C₁₂₇	C₂²	1	1	yes	characteristic
38	#103×C₁₂₇	C₂²	1	2	yes	—
39	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
40	#14523249	—	2	4	no	—
41	#14523249	C₂×C₄	1	8	yes	complemented
42	C₂×D₅₀₈	—	2	2	no	—
43	C₂×D₅₀₈	C₂³	1	2	yes	—
44	C₂×C₄×D₁₂₇	—	2	2	no	—
45	C₂×C₄×D₁₂₇	—	2	2	no	—
46	C₂×C₄×D₁₂₇	—	2	2	no	—
47	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
48	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
49	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
50	#14523235	C₂×C₄	1	8	yes	complemented
51	D₁₀₁₆	C₂×C₄	1	4	yes	complemented
52	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
54	C₈×D₁₂₇	—	2	2	no	—
55	C₈×D₁₂₇	C₂×C₄	1	4	yes	complemented
56	C₄⋊Dic₁₂₇	—	2	2	no	—
57	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
58	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
59	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
60	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
61	C₄×Dic₁₂₇	—	2	2	no	—
62	C₄×Dic₁₂₇	—	2	2	no	—
63	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
64	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
65	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
67	Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
68	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₂₇	—	2	2	no	—
71	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
72	C₄²×C₁₂₇	—	2	2	no	—
73	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
74	C₂²×D₁₂₇	—	2	2	no	—
75	C₂²×D₁₂₇	C₂×D₄	1	1	yes	characteristic
76	D₅₀₈	—	2	8	no	—
77	D₅₀₈	—	2	2	no	—
78	D₅₀₈	C₂²×C₄	1	4	yes	—
79	C₂×Dic₁₂₇	—	2	2	no	—
80	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
81	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
83	C₄×D₁₂₇	—	2	2	no	—
84	C₄×D₁₂₇	—	2	8	no	—
85	C₄×D₁₂₇	—	2	2	no	—
86	C₄×D₁₂₇	—	4	4	no	—
87	C₄×D₁₂₇	—	4	4	no	—
88	C₄×D₁₂₇	C₂²×C₄	1	2	yes	—
89	C₄×D₁₂₇	#41	1	2	yes	—
90	C₁₂₇⋊C₈	—	2	1	no	—
91	C₁₂₇⋊C₈	C₂²×C₄	1	2	yes	—
92	Q₈×C₁₂₇	—	2	2	no	—
93	Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₁₂₇	—	2	2	no	—
95	C₂×C₄×C₁₂₇	—	2	2	no	—
96	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
99	C₈×C₁₂₇	—	2	1	no	—
100	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×D₁₂₇	—	2	1	no	—
103	C₂×D₁₂₇	—	2	1	no	—
104	C₂×D₁₂₇	—	2	4	no	—
105	C₂×D₁₂₇	—	4	2	no	—
106	C₂×D₁₂₇	—	4	2	no	—
107	C₂×D₁₂₇	C₄×D₄	1	2	yes	—
108	C₄×C₁₂₇	—	2	4	no	—
109	C₄×C₁₂₇	—	2	1	no	—
110	C₄×C₁₂₇	—	4	2	no	—
111	C₄×C₁₂₇	#135	1	2	yes	complemented
112	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
113	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
114	Dic₁₂₇	—	2	1	no	—
115	Dic₁₂₇	—	4	2	no	—
116	Dic₁₂₇	C₄×D₄	1	2	yes	—
117	Dic₁₂₇	#135	1	2	yes	complemented
118	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
120	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
121	D₁₂₇	—	2	2	no	—
122	D₁₂₇	—	4	4	no	—
123	C₄×#135	—	127	1	no	maximal, Sylow
124	C₁₂₇	C₄×#135	1	1	yes	characteristic, complemented, Sylow
125	C₂×#135	—	127	1	no	—
126	C₄×#41	—	127	2	no	—
127	C₄×Q₁₆	—	127	1	no	—
128	C₄×Q₁₆	D₁₂₇	1	1	yes	characteristic, complemented
129	C₄×SD₁₆	—	127	2	no	—
130	C₄×SD₁₆	—	127	2	no	—
131	C₄×D₈	—	127	1	no	—
132	C₄×D₈	—	127	1	no	—
133	#427	—	127	1	no	—
134	#416	—	127	2	no	—
135	C₂×C₄×C₈	—	127	1	no	—
136	C₂×#41	—	127	2	no	—
137	#135	—	127	8	no	—
138	C₂×Q₁₆	C₂×D₁₂₇	1	1	yes	characteristic
139	C₂×SD₁₆	—	127	2	no	—
140	C₂×D₈	—	127	1	no	—
141	C₂²×C₈	—	127	1	no	—
142	C₂²×C₈	—	127	1	no	—
143	C₄×Q₈	—	127	2	no	—
144	C₄×Q₈	C₂×D₁₂₇	1	2	yes	—
145	C₄×D₄	—	127	2	no	—
146	C₄×D₄	—	127	2	no	—
147	C₄×D₄	—	254	2	no	—
148	C₄×D₄	—	254	2	no	—
149	#117	—	127	2	no	—
150	#117	—	254	2	no	—
151	C₂×C₄²	—	127	1	no	—
152	C₂×C₄²	—	254	2	no	—
153	#107	—	127	1	no	—
154	#107	C₂×D₁₂₇	1	1	yes	characteristic
155	#106	—	127	2	no	—
156	#103	—	127	2	no	—
157	#103	C₂×D₁₂₇	1	2	yes	—
158	#102	—	127	2	no	—
159	#102	—	127	2	no	—
160	C₄×C₈	—	127	1	no	—
161	C₄×C₈	—	127	1	no	—
162	C₄×C₈	—	127	1	no	—
163	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
164	#41	—	127	8	no	—
165	#41	—	254	4	no	—
166	C₂×Q₈	C₂²×D₁₂₇	1	2	yes	—
167	C₂×D₄	—	127	2	no	—
168	C₂×D₄	—	254	2	no	—
169	C₂²×C₄	—	127	1	no	—
170	C₂²×C₄	—	127	1	no	—
171	C₂²×C₄	—	127	1	no	—
172	C₂²×C₄	—	254	2	no	—
173	C₂²×C₄	—	254	2	no	—
174	C₂²×C₄	—	254	2	no	—
175	Q₁₆	C₄×D₁₂₇	1	4	yes	complemented
176	SD₁₆	—	127	8	no	—
177	D₈	—	127	4	no	—
178	C₂×C₈	—	127	1	no	—
179	C₂×C₈	—	127	4	no	—
180	C₂×C₈	—	127	1	no	—
181	C₂×C₈	—	254	2	no	—
182	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	127	2	no	—
186	C₄⋊C₄	—	254	2	no	—
187	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
188	C₂²⋊C₄	—	254	2	no	—
189	C₂²⋊C₄	—	254	2	no	—
190	C₂²⋊C₄	—	254	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	127	1	no	—
193	C₄²	—	127	1	no	—
194	C₄²	—	127	1	no	—
195	C₄²	—	254	2	no	—
196	C₄²	—	254	2	no	—
197	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
198	C₂³	—	127	1	no	—
199	C₂³	—	254	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	C₂×C₄×D₁₂₇	1	4	yes	—
202	D₄	—	127	4	no	—
203	D₄	—	254	8	no	—
204	D₄	—	254	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	127	1	no	—
208	C₂×C₄	—	127	1	no	—
209	C₂×C₄	—	127	2	no	—
210	C₂×C₄	—	127	2	no	—
211	C₂×C₄	—	127	1	no	—
212	C₂×C₄	—	508	4	no	—
213	C₂×C₄	—	508	4	no	—
214	C₂×C₄	—	254	2	no	—
215	C₂×C₄	—	254	8	no	—
216	C₂×C₄	—	254	2	no	—
217	C₂×C₄	—	254	2	no	—
218	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
219	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	127	2	no	—
223	C₈	—	254	1	no	—
224	C₈	C₂×C₄×D₁₂₇	1	2	yes	—
225	C₂²	—	127	2	no	—
226	C₂²	—	508	2	no	—
227	C₂²	—	508	2	no	—
228	C₂²	—	254	4	no	—
229	C₂²	—	254	1	no	—
230	C₂²	—	254	1	no	—
231	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
232	C₄	—	2	4	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	127	2	no	—
236	C₄	—	127	2	no	—
237	C₄	—	508	2	no	—
238	C₄	—	254	1	no	—
239	C₄	#16861232	1	2	yes	direct product factor
240	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
241	C₄	#41×D₁₂₇	1	1	yes	characteristic
242	C₂	—	508	4	no	—
243	C₂	—	254	2	no	—
244	C₂	C₄×D₄×D₁₂₇	1	1	yes	characteristic
245	C₂	#135×D₁₂₇	1	1	yes	characteristic
246	C₂	C₂×#16861232	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₄×Q₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#16861232	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₄×Q₁₆	D₁₂₇
hypocenter	C₁₂₇	C₄×#135
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#14523249	C₂	semidirect
3. Q₁₆×Dic₁₂₇	C₂	semidirect
4. C₄×#14523235	C₂	semidirect
5. #22549961	C₂	semidirect
6. C₄×D₁₀₁₆	C₂	semidirect
7. #22549958	C₂	semidirect
8. #22549913	C₂	semidirect
9. #22549871	C₂	semidirect
10. C₄×C₈×D₁₂₇	C₂	semidirect
11. C₄×Q₁₆×C₁₂₇	C₂	semidirect
13. #16861232	C₄	direct (N × H)
41. #14523249	C₂×C₄	semidirect
50. #14523235	C₂×C₄	semidirect
51. D₁₀₁₆	C₂×C₄	semidirect
55. C₈×D₁₂₇	C₂×C₄	semidirect
67. Q₁₆×C₁₂₇	C₂×C₄	semidirect
111. C₄×C₁₂₇	#135	semidirect
117. Dic₁₂₇	#135	semidirect
124. C₁₂₇	C₄×#135	semidirect
128. C₄×Q₁₆	D₁₂₇	semidirect
175. Q₁₆	C₄×D₁₂₇	semidirect
239. C₄	#16861232	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×#16861232	2
C₄×#14523249	2
Q₁₆×Dic₁₂₇	2
C₄×#14523235	2
#22549961	2
C₄×D₁₀₁₆	2
#22549958	2
#22549913	2
#22549871	2
C₄×C₈×D₁₂₇	2
C₄×Q₁₆×C₁₂₇	2
C₄×#135	127

Subgroups — C4×#16861232 #30647502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16861232 #30647502