Subgroups — #30649300

Order 16256.

380 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550410	C₂	1	2	yes	maximal, complemented
2	#22550278	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550260	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550231	C₂	1	2	yes	maximal, complemented
5	#22550117	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550112	C₂	1	1	yes	maximal, characteristic
7	#22550105	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14523237	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₀₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
13	C₂×#14523237	C₂²	1	2	yes	—
14	C₂×D₁₀₁₆	C₂²	1	1	yes	characteristic
15	#16861203	—	2	2	no	—
16	#16861201	C₂²	1	2	yes	—
17	#16861197	C₂²	1	2	yes	—
18	#16861192	—	2	2	no	—
19	#16861188	C₂²	1	2	yes	—
20	#16861183	C₂²	1	2	yes	—
21	C₄×Dic₂₅₄	C₂²	1	2	yes	—
22	C₄×D₅₀₈	C₂²	1	2	yes	—
23	C₄×C₂²⋊D₁₂₇	—	2	2	no	—
24	#16861166	—	2	2	no	—
25	#16861143	C₂²	1	1	yes	characteristic
26	#16861143	C₂²	1	1	yes	characteristic
27	#16861141	C₂²	1	2	yes	—
28	#16861134	C₂²	1	2	yes	—
29	#16861134	C₂²	1	2	yes	—
30	#16861131	C₂²	1	2	yes	—
31	#16861131	C₂²	1	2	yes	—
32	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
36	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₅₄	C₂³	1	2	yes	—
40	C₂×D₅₀₈	C₂³	1	2	yes	—
41	C₂×C₂²⋊D₁₂₇	—	2	2	no	—
42	C₂×C₄×D₁₂₇	—	2	2	no	—
43	Dic₅₀₈	—	2	2	no	—
44	#14523237	—	2	4	no	—
45	D₁₀₁₆	—	2	2	no	—
46	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
47	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
48	Dic₁₂₇⋊C₄	—	2	2	no	—
49	Dic₁₂₇⋊C₄	—	2	2	no	—
50	Dic₁₂₇⋊C₄	—	2	2	no	—
51	Dic₁₂₇⋊C₄	—	2	2	no	—
52	C₂²⋊Dic₁₂₇	—	2	2	no	—
53	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
54	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
55	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
56	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
57	C₄×Dic₁₂₇	—	2	2	no	—
58	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₁₂₇	—	2	2	no	—
62	C₂×C₈×C₁₂₇	—	2	2	no	—
63	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
67	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
68	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
69	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
71	C₂²×D₁₂₇	—	2	2	no	—
72	Dic₂₅₄	—	2	2	no	—
73	Dic₂₅₄	—	2	2	no	—
74	D₅₀₈	—	2	2	no	—
75	D₅₀₈	—	2	2	no	—
76	C₂²⋊D₁₂₇	—	4	4	no	—
77	C₂×Dic₁₂₇	—	2	2	no	—
78	C₂×Dic₁₂₇	—	2	2	no	—
79	C₂×Dic₁₂₇	—	2	2	no	—
80	C₄×D₁₂₇	—	4	4	no	—
81	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	—	2	2	no	—
83	C₂×C₄×C₁₂₇	—	2	2	no	—
84	C₂×C₄×C₁₂₇	—	2	1	no	—
85	C₂×C₄×C₁₂₇	—	2	1	no	—
86	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₂₇	—	2	1	no	—
93	C₈×C₁₂₇	—	2	1	no	—
94	C₈×C₁₂₇	#41	1	2	yes	—
95	C₈×C₁₂₇	#41	1	2	yes	—
96	C₂²×C₁₂₇	—	2	1	no	—
97	C₂²×C₁₂₇	—	2	1	no	—
98	C₂²×C₁₂₇	#135	1	2	yes	—
99	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×D₁₂₇	—	4	2	no	—
101	C₂×D₁₂₇	—	4	2	no	—
102	C₂×D₁₂₇	—	4	2	no	—
103	C₄×C₁₂₇	—	2	1	no	—
104	C₄×C₁₂₇	—	2	1	no	—
105	C₄×C₁₂₇	—	2	1	no	—
106	C₄×C₁₂₇	—	2	1	no	—
107	C₄×C₁₂₇	#135	1	2	yes	—
108	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
109	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
110	Dic₁₂₇	—	4	2	no	—
111	Dic₁₂₇	—	4	2	no	—
112	Dic₁₂₇	—	4	2	no	—
113	C₂×C₁₂₇	—	2	2	no	—
114	C₂×C₁₂₇	#532	1	1	yes	characteristic
115	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
116	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
117	D₁₂₇	—	8	2	no	—
118	#3168	—	127	1	no	maximal, Sylow
119	C₁₂₇	#3168	1	1	yes	characteristic, complemented, Sylow
120	#529	—	127	2	no	—
121	#499	—	127	1	no	—
122	#495	—	127	1	no	—
123	#488	—	127	2	no	—
124	#467	—	127	1	no	—
125	#466	—	127	1	no	—
126	#465	—	127	2	no	—
127	C₄×Q₁₆	—	127	1	no	—
128	C₄×SD₁₆	—	127	2	no	—
129	C₄×D₈	—	127	1	no	—
130	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic, complemented
131	C₂×Q₁₆	—	127	1	no	—
132	C₂×SD₁₆	—	127	2	no	—
133	C₂×D₈	—	127	1	no	—
134	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
136	#126	—	254	2	no	—
137	#125	—	127	2	no	—
138	#124	—	127	2	no	—
139	#123	—	254	2	no	—
140	C₂²⋊Q₈	—	127	2	no	—
141	#121	—	127	2	no	—
142	C₄×Q₈	—	127	2	no	—
143	C₄×D₄	—	127	2	no	—
144	C₄×D₄	—	254	2	no	—
145	#117	—	254	2	no	—
146	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
147	#107	—	127	1	no	—
148	#107	—	127	1	no	—
149	#106	—	127	2	no	—
150	#103	—	127	2	no	—
151	#103	—	127	2	no	—
152	#102	—	127	2	no	—
153	#102	—	127	2	no	—
154	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
155	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
156	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
157	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
158	C₂×Q₈	—	127	2	no	—
159	C₂×D₄	—	127	2	no	—
160	C₂×D₄	—	254	2	no	—
161	C₂²×C₄	—	254	2	no	—
162	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
163	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
165	Q₁₆	—	254	2	no	—
166	SD₁₆	—	254	4	no	—
167	D₈	—	254	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
174	C₄⋊C₄	—	127	2	no	—
175	C₄⋊C₄	—	127	2	no	—
176	C₄⋊C₄	—	254	2	no	—
177	C₄⋊C₄	—	254	2	no	—
178	C₄⋊C₄	—	254	2	no	—
179	C₄⋊C₄	—	254	2	no	—
180	C₂²⋊C₄	—	254	2	no	—
181	C₂²⋊C₄	—	254	2	no	—
182	C₂²⋊C₄	—	254	2	no	—
183	C₂²⋊C₄	—	254	2	no	—
184	C₂²⋊C₄	—	254	2	no	—
185	C₄²	—	254	2	no	—
186	C₄²	D₅₀₈	1	1	yes	characteristic
187	C₄²	D₅₀₈	1	1	yes	characteristic
188	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
189	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
190	C₂³	—	254	2	no	—
191	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
192	Q₈	—	254	2	no	—
193	Q₈	—	254	2	no	—
194	D₄	—	508	4	no	—
195	D₄	—	254	2	no	—
196	D₄	—	254	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	508	4	no	—
202	C₂×C₄	—	254	2	no	—
203	C₂×C₄	—	254	2	no	—
204	C₂×C₄	—	254	2	no	—
205	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
206	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
207	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
208	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
209	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	#14523248	1	2	yes	—
214	C₈	#14523248	1	2	yes	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	508	2	no	—
218	C₂²	—	508	2	no	—
219	C₂²	—	508	2	no	—
220	C₂²	#16861231	1	2	yes	—
221	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	508	2	no	—
227	C₄	—	508	2	no	—
228	C₄	—	508	2	no	—
229	C₄	#16861231	1	2	yes	—
230	C₄	C₂×#14523248	1	1	yes	characteristic
231	C₄	C₂×#14523248	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	—	1016	2	no	—
234	C₂	#22550426	1	1	yes	characteristic
235	C₂	C₂×#16861231	1	1	yes	characteristic
236	C₂	C₂×#16861231	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₀₈
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	#22550112	C₂
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#3168
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #22550410	C₂	semidirect
2. #22550278	C₂	semidirect
3. #22550260	C₂	semidirect
4. #22550231	C₂	semidirect
5. #22550117	C₂	semidirect
7. #22550105	C₂	semidirect
8. C₄×Dic₅₀₈	C₂	semidirect
9. C₄×#14523237	C₂	semidirect
10. C₄×D₁₀₁₆	C₂	semidirect
11. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
119. C₁₂₇	#3168	semidirect
130. C₂×C₄×C₈	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550410	2
#22550278	2
#22550260	2
#22550231	2
#22550117	2
#22550112	2
#22550105	2
C₄×Dic₅₀₈	2
C₄×#14523237	2
C₄×D₁₀₁₆	2
C₂×C₄×C₈×C₁₂₇	2
#3168	127