Subgroups — #30647575

Order 16256.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861242	C₂	1	1	yes	maximal, characteristic
2	#22550356	C₂	1	1	yes	maximal, characteristic
3	#22550353	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549984	C₂	1	2	yes	maximal, complemented
5	#22549978	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₅₀₈	C₂	1	2	yes	maximal, complemented
7	C₄×#14523237	C₂	1	2	yes	maximal, complemented
8	#22549918	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549873	C₂	1	1	yes	maximal, characteristic
10	C₂×#16861134	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
14	#16861242	C₄	1	8	yes	complemented
15	C₂×Dic₅₀₈	C₂²	1	2	yes	—
16	C₂×#14523237	C₂²	1	2	yes	—
17	C₄×Dic₂₅₄	C₂²	1	2	yes	—
18	C₄×Dic₂₅₄	C₂²	1	2	yes	—
19	C₄×Dic₂₅₄	C₂²	1	2	yes	—
20	#16861174	—	2	1	no	—
21	#16861174	—	2	1	no	—
22	C₄×D₅₀₈	C₂²	1	2	yes	—
23	(C₂²⋊D₁₂₇)⋊C₄	—	2	2	no	—
24	#16861168	C₂²	1	1	yes	characteristic
25	#16861167	—	2	1	no	—
26	#16861165	—	2	2	no	—
27	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×D₁₂₇	—	2	1	no	—
29	#16861143	C₂²	1	2	yes	—
30	#16861141	C₂²	1	2	yes	—
31	#16861134	C₂²	1	2	yes	—
32	#16861134	C₂²	1	2	yes	—
33	#16861134	C₂²	1	1	yes	characteristic
34	#16861134	C₂²	1	1	yes	characteristic
35	#16861131	C₂²	1	1	yes	characteristic
36	#16861131	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
39	#117×C₁₂₇	C₂²	1	1	yes	characteristic
40	#97×C₁₂₇	C₂²	1	2	yes	—
41	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
42	#14523248	—	2	2	no	—
43	#14523248	C₂×C₄	1	4	yes	—
44	C₂×Dic₂₅₄	—	2	1	no	—
45	C₂×Dic₂₅₄	—	2	2	no	—
46	C₂×Dic₂₅₄	C₂×C₄	1	4	yes	complemented
47	C₂×Dic₂₅₄	C₂³	1	2	yes	—
48	C₂×Dic₂₅₄	C₂³	1	1	yes	characteristic
49	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
51	C₂²×Dic₁₂₇	—	2	1	no	—
52	C₂²×Dic₁₂₇	—	2	1	no	—
53	C₂×C₄×D₁₂₇	—	2	1	no	—
54	C₂×C₄×D₁₂₇	—	2	2	no	—
55	Dic₅₀₈	C₂×C₄	1	8	yes	complemented
56	#14523237	C₂×C₄	1	8	yes	complemented
57	C₄⋊Dic₁₂₇	—	2	1	no	—
58	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
60	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
61	Dic₁₂₇⋊C₄	—	2	2	no	—
62	Dic₁₂₇⋊C₄	—	2	2	no	—
63	Dic₁₂₇⋊C₄	—	2	2	no	—
64	C₂²⋊Dic₁₂₇	—	2	1	no	—
65	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
66	C₄×Dic₁₂₇	—	2	2	no	—
67	C₄×Dic₁₂₇	—	2	2	no	—
68	C₄×Dic₁₂₇	—	2	2	no	—
69	C₄×Dic₁₂₇	—	2	1	no	—
70	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
71	M₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
72	C₂×C₈×C₁₂₇	—	2	2	no	—
73	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
76	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₂₇	D₄	1	2	yes	—
79	C₄²×C₁₂₇	C₂³	1	2	yes	—
80	C₂²×D₁₂₇	—	2	1	no	—
81	Dic₂₅₄	—	2	4	no	—
82	Dic₂₅₄	—	2	2	no	—
83	Dic₂₅₄	—	2	1	no	—
84	Dic₂₅₄	C₂²×C₄	1	4	yes	—
85	Dic₂₅₄	C₂²×C₄	1	2	yes	—
86	D₅₀₈	—	2	1	no	—
87	D₅₀₈	C₂²×C₄	1	2	yes	—
88	C₂²⋊D₁₂₇	—	2	4	no	—
89	C₂×Dic₁₂₇	—	2	4	no	—
90	C₂×Dic₁₂₇	—	2	1	no	—
91	C₂×Dic₁₂₇	—	2	2	no	—
92	C₂×Dic₁₂₇	—	2	1	no	—
93	C₂×Dic₁₂₇	—	2	1	no	—
94	C₂×Dic₁₂₇	—	2	2	no	—
95	C₂×Dic₁₂₇	—	8	1	no	—
96	C₄×D₁₂₇	—	2	4	no	—
97	C₄×D₁₂₇	—	8	2	no	—
98	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	#41	1	2	yes	—
104	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
105	C₈×C₁₂₇	—	2	1	no	—
106	C₈×C₁₂₇	—	2	1	no	—
107	C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
108	C₂²×C₁₂₇	—	2	1	no	—
109	C₂²×C₁₂₇	—	2	1	no	—
110	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
111	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×D₁₂₇	—	2	2	no	—
113	C₂×D₁₂₇	—	4	1	no	—
114	C₂×D₁₂₇	—	4	1	no	—
115	C₄×C₁₂₇	—	2	2	no	—
116	C₄×C₁₂₇	—	2	2	no	—
117	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
118	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
120	Dic₁₂₇	—	2	2	no	—
121	Dic₁₂₇	—	2	4	no	—
122	Dic₁₂₇	—	4	2	no	—
123	Dic₁₂₇	—	4	1	no	—
124	Dic₁₂₇	—	4	1	no	—
125	C₂×C₁₂₇	—	2	2	no	—
126	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×C₁₂₇	#577	1	1	yes	characteristic
128	C₂×C₁₂₇	#578	1	1	yes	characteristic
129	D₁₂₇	—	4	2	no	—
130	#2947	—	127	1	no	maximal, Sylow
131	C₁₂₇	#2947	1	1	yes	characteristic, complemented, Sylow
132	C₂×#137	—	127	1	no	—
133	#520	—	127	1	no	—
134	#519	—	127	1	no	—
135	#441	—	127	2	no	—
136	#440	—	127	2	no	—
137	C₄×Q₁₆	—	127	2	no	—
138	C₄×SD₁₆	—	127	2	no	—
139	#428	—	127	1	no	—
140	#416	—	127	1	no	—
141	C₂×#103	—	127	1	no	—
142	#405	D₁₂₇	1	1	yes	characteristic, complemented
143	C₂×#41	—	127	1	no	—
144	C₂²×Q₈	—	127	1	no	—
145	#137	—	127	8	no	—
146	C₂×Q₁₆	—	127	2	no	—
147	C₂×SD₁₆	—	127	2	no	—
148	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
149	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
150	C₄×Q₈	—	127	2	no	—
151	C₄×Q₈	—	127	2	no	—
152	C₄×Q₈	—	127	2	no	—
153	C₄×Q₈	—	254	1	no	—
154	C₄×Q₈	—	254	1	no	—
155	C₄×D₄	—	127	2	no	—
156	C₄×D₄	—	254	2	no	—
157	#117	—	127	1	no	—
158	#117	—	254	1	no	—
159	#117	—	254	2	no	—
160	#117	C₂×D₁₂₇	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	127	1	no	—
162	C₂×C₄⋊C₄	—	254	1	no	—
163	#107	—	127	2	no	—
164	#106	—	127	2	no	—
165	#103	—	127	2	no	—
166	#103	—	127	2	no	—
167	#103	—	127	1	no	—
168	#103	—	127	1	no	—
169	#102	—	127	1	no	—
170	#102	—	127	1	no	—
171	#97	C₂×D₁₂₇	1	2	yes	—
172	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
173	#41	—	127	4	no	—
174	#41	—	254	2	no	—
175	C₂×Q₈	—	127	4	no	—
176	C₂×Q₈	—	127	2	no	—
177	C₂×Q₈	—	127	1	no	—
178	C₂×Q₈	—	254	1	no	—
179	C₂×Q₈	—	254	2	no	—
180	C₂×D₄	—	127	1	no	—
181	C₂×D₄	—	254	1	no	—
182	C₂²×C₄	—	254	1	no	—
183	C₂²×C₄	—	254	2	no	—
184	C₂²×C₄	—	254	1	no	—
185	C₂²×C₄	—	254	1	no	—
186	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
187	Q₁₆	—	127	8	no	—
188	SD₁₆	—	127	8	no	—
189	M₁₆	C₄×D₁₂₇	1	4	yes	complemented
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
192	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
194	C₄⋊C₄	—	127	1	no	—
195	C₄⋊C₄	—	127	1	no	—
196	C₄⋊C₄	—	127	2	no	—
197	C₄⋊C₄	—	254	2	no	—
198	C₄⋊C₄	—	254	2	no	—
199	C₄⋊C₄	—	254	2	no	—
200	C₄⋊C₄	—	254	1	no	—
201	C₄⋊C₄	D₅₀₈	1	1	yes	characteristic
202	C₄⋊C₄	D₅₀₈	1	1	yes	characteristic
203	C₂²⋊C₄	—	254	1	no	—
204	C₂²⋊C₄	—	254	2	no	—
205	C₂²⋊C₄	D₅₀₈	1	2	yes	—
206	C₄²	—	254	1	no	—
207	C₄²	—	254	2	no	—
208	C₄²	—	254	2	no	—
209	C₄²	—	254	2	no	—
210	C₄²	C₂²×D₁₂₇	1	2	yes	—
211	C₂³	—	254	1	no	—
212	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
213	Q₈	—	127	4	no	—
214	Q₈	—	127	2	no	—
215	Q₈	—	254	2	no	—
216	Q₈	—	254	1	no	—
217	Q₈	—	254	4	no	—
218	D₄	—	127	2	no	—
219	D₄	—	254	4	no	—
220	D₄	—	254	1	no	—
221	C₂×C₄	—	1016	2	no	—
222	C₂×C₄	—	1016	1	no	—
223	C₂×C₄	—	254	1	no	—
224	C₂×C₄	—	254	1	no	—
225	C₂×C₄	—	254	2	no	—
226	C₂×C₄	—	254	1	no	—
227	C₂×C₄	—	254	2	no	—
228	C₂×C₄	—	254	4	no	—
229	C₂×C₄	—	254	4	no	—
230	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
231	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
232	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
233	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
234	C₂×C₄	#14523248	1	2	yes	—
235	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	C₂×C₄×D₁₂₇	1	4	yes	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	508	1	no	—
242	C₂²	—	508	1	no	—
243	C₂²	—	254	2	no	—
244	C₂²	C₄×D₅₀₈	1	2	yes	—
245	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	508	2	no	—
249	C₄	—	508	1	no	—
250	C₄	—	508	1	no	—
251	C₄	—	254	2	no	—
252	C₄	—	254	4	no	—
253	C₄	C₄×D₅₀₈	1	2	yes	—
254	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
255	C₄	C₂×#14523248	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	508	2	no	—
258	C₂	C₂×C₄×D₅₀₈	1	1	yes	characteristic
259	C₂	#22550688	1	1	yes	characteristic
260	C₂	#22550693	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₀₈
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#405×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#14523248	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#405	D₁₂₇
hypocenter	C₁₂₇	#2947
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
3. #22550353	C₂	semidirect
4. #22549984	C₂	semidirect
5. #22549978	C₂	semidirect
6. C₄×Dic₅₀₈	C₂	semidirect
7. C₄×#14523237	C₂	semidirect
8. #22549918	C₂	semidirect
10. C₂×#16861134	C₂	semidirect
11. #405×C₁₂₇	C₂	semidirect
14. #16861242	C₄	semidirect
46. C₂×Dic₂₅₄	C₂×C₄	semidirect
55. Dic₅₀₈	C₂×C₄	semidirect
56. #14523237	C₂×C₄	semidirect
71. M₁₆×C₁₂₇	C₂×C₄	semidirect
131. C₁₂₇	#2947	semidirect
142. #405	D₁₂₇	semidirect
189. M₁₆	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×#16861242	2
#22550356	2
#22550353	2
#22549984	2
#22549978	2
C₄×Dic₅₀₈	2
C₄×#14523237	2
#22549918	2
#22549873	2
C₂×#16861134	2
#405×C₁₂₇	2
#2947	127