Subgroups — #30647387

Order 16256.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#22549936	C₂	1	2	yes	maximal, complemented
3	#22549936	C₂	1	2	yes	maximal, complemented
4	#22549932	C₂	1	2	yes	maximal, complemented
5	#22549928	C₂	1	2	yes	maximal, complemented
6	#22549903	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549839	C₂	1	1	yes	maximal, characteristic
8	C₂×(C₂×D₁₂₇)⋊C₈	C₂	1	1	yes	maximal, characteristic
9	#22549834	C₂	1	1	yes	maximal, characteristic, complemented
10	#22549831	C₂	1	1	yes	maximal, characteristic, complemented
11	#424×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×#14523232	C₂²	1	1	yes	characteristic
14	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	#16861168	C₂²	1	1	yes	characteristic
16	#16861167	C₄	1	1	yes	characteristic
17	#16861167	C₄	1	1	yes	characteristic
18	#16861166	C₂²	1	2	yes	—
19	#16861165	C₄	1	2	yes	—
20	C₄⋊C₄×D₁₂₇	C₄	1	1	yes	characteristic
21	C₄⋊C₄×D₁₂₇	C₄	1	1	yes	characteristic
22	C₂²⋊C₄×D₁₂₇	C₄	1	2	yes	—
23	C₄²×D₁₂₇	C₂²	1	2	yes	—
24	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	complemented
25	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	complemented
26	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
27	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
28	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
32	#16861119	C₂²	1	2	yes	complemented
33	#16861119	C₂²	1	2	yes	complemented
34	C₈×Dic₁₂₇	C₂²	1	2	yes	complemented
35	C₈×Dic₁₂₇	C₂²	1	2	yes	complemented
36	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
38	#117×C₁₂₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	complemented
40	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	complemented
41	C₂³×D₁₂₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₂₇	—	2	2	no	—
44	C₂×C₄×D₁₂₇	—	2	2	no	—
45	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
51	#14523232	—	2	2	no	—
52	C₂×C₁₂₇⋊C₈	—	2	2	no	—
53	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
56	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
58	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
59	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
60	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
61	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
62	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₂₇	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
67	M₁₆×C₁₂₇	—	2	2	no	—
68	C₂×C₈×C₁₂₇	—	2	2	no	—
69	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
72	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
75	C₄²×C₁₂₇	C₂³	1	2	yes	—
76	C₂²×D₁₂₇	—	2	2	no	—
77	C₂²×D₁₂₇	—	2	2	no	—
78	C₂²×D₁₂₇	—	4	1	no	—
79	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₂₇	—	2	2	no	—
82	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
86	C₄×D₁₂₇	—	2	4	no	—
87	C₄×D₁₂₇	—	2	2	no	—
88	C₄×D₁₂₇	—	2	2	no	—
89	C₄×D₁₂₇	—	2	1	no	—
90	C₄×D₁₂₇	—	2	1	no	—
91	C₄×D₁₂₇	—	2	1	no	—
92	C₄×D₁₂₇	—	2	1	no	—
93	C₄×D₁₂₇	—	4	2	no	—
94	C₁₂₇⋊C₈	—	2	1	no	—
95	C₁₂₇⋊C₈	—	2	1	no	—
96	C₁₂₇⋊C₈	—	2	2	no	—
97	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	#41	1	2	yes	—
102	C₂×C₄×C₁₂₇	#41	1	2	yes	—
103	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
104	C₈×C₁₂₇	—	2	1	no	—
105	C₈×C₁₂₇	—	2	1	no	—
106	C₈×C₁₂₇	—	2	2	no	—
107	C₂²×C₁₂₇	—	2	1	no	—
108	C₂²×C₁₂₇	—	2	1	no	—
109	C₂²×C₁₂₇	#117	1	2	yes	—
110	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×D₁₂₇	—	2	1	no	—
112	C₂×D₁₂₇	—	2	1	no	—
113	C₂×D₁₂₇	—	2	1	no	—
114	C₂×D₁₂₇	—	2	1	no	—
115	C₂×D₁₂₇	—	2	1	no	—
116	C₂×D₁₂₇	—	4	4	no	—
117	C₂×D₁₂₇	#131	1	2	yes	—
118	C₄×C₁₂₇	—	2	2	no	—
119	C₄×C₁₂₇	—	2	2	no	—
120	C₄×C₁₂₇	#117	1	2	yes	—
121	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
122	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
123	Dic₁₂₇	—	2	1	no	—
124	Dic₁₂₇	—	2	1	no	—
125	Dic₁₂₇	—	2	2	no	—
126	Dic₁₂₇	—	2	2	no	—
127	C₂×C₁₂₇	—	2	2	no	—
128	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
129	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
130	C₂×C₁₂₇	#568	1	1	yes	characteristic
131	D₁₂₇	—	2	2	no	—
132	D₁₂₇	—	4	1	no	—
133	#2924	—	127	1	no	maximal, Sylow
134	C₁₂₇	#2924	1	1	yes	characteristic, complemented, Sylow
135	C₂×#117	—	127	1	no	—
136	#433	—	127	2	no	—
137	#433	—	127	2	no	—
138	#432	—	127	2	no	—
139	#431	—	127	2	no	—
140	#424	—	127	1	no	—
141	#424	D₁₂₇	1	1	yes	characteristic, complemented
142	#407	—	127	1	no	—
143	C₂×C₂²⋊C₈	—	127	1	no	—
144	#405	—	127	1	no	—
145	#405	—	127	1	no	—
146	C₂³×C₄	—	127	1	no	—
147	C₂×M₁₆	—	127	1	no	—
148	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
149	C₂²×C₈	—	127	1	no	—
150	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
151	#117	—	127	1	no	—
152	#117	—	127	1	no	—
153	#117	—	127	1	no	—
154	#117	—	127	2	no	—
155	#117	—	127	2	no	—
156	#117	C₂×D₁₂₇	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	127	1	no	—
158	C₂×C₄⋊C₄	—	127	1	no	—
159	C₂×C₂²⋊C₄	—	127	2	no	—
160	C₂×C₄²	—	127	2	no	—
161	C₄⋊C₈	—	127	2	no	—
162	C₄⋊C₈	—	127	2	no	—
163	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	complemented
164	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	complemented
165	C₂²⋊C₈	—	127	1	no	—
166	C₂²⋊C₈	—	127	1	no	—
167	C₂²⋊C₈	—	127	1	no	—
168	C₂²⋊C₈	—	127	1	no	—
169	C₂²⋊C₈	—	127	2	no	—
170	C₂²⋊C₈	—	127	2	no	—
171	#97	—	127	2	no	—
172	#97	—	127	2	no	—
173	C₄×C₈	—	127	2	no	—
174	C₄×C₈	—	127	2	no	—
175	C₂⁴	—	127	1	no	—
176	C₂²×C₄	—	127	1	no	—
177	C₂²×C₄	—	127	2	no	—
178	C₂²×C₄	—	127	2	no	—
179	C₂²×C₄	—	127	1	no	—
180	C₂²×C₄	—	127	1	no	—
181	C₂²×C₄	—	127	1	no	—
182	C₂²×C₄	—	127	1	no	—
183	C₂²×C₄	—	254	2	no	—
184	C₂²×C₄	—	254	2	no	—
185	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	254	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	127	1	no	—
190	C₂×C₈	—	127	2	no	—
191	C₂×C₈	—	127	1	no	—
192	C₂×C₈	—	254	2	no	—
193	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
194	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
195	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
196	C₄⋊C₄	—	127	2	no	—
197	C₄⋊C₄	—	127	2	no	—
198	C₄⋊C₄	—	127	1	no	—
199	C₄⋊C₄	—	127	1	no	—
200	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
201	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
202	C₂²⋊C₄	—	127	2	no	—
203	C₂²⋊C₄	—	127	2	no	—
204	C₂²⋊C₄	—	127	2	no	—
205	C₂²⋊C₄	C₄×D₁₂₇	1	2	yes	—
206	C₄²	—	127	2	no	—
207	C₄²	—	127	2	no	—
208	C₄²	—	127	2	no	—
209	C₄²	C₂²×D₁₂₇	1	2	yes	—
210	C₂³	—	127	1	no	—
211	C₂³	—	127	1	no	—
212	C₂³	—	508	1	no	—
213	C₂³	—	254	2	no	—
214	C₂³	—	254	2	no	—
215	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
216	C₂×C₄	—	127	1	no	—
217	C₂×C₄	—	127	2	no	—
218	C₂×C₄	—	127	2	no	—
219	C₂×C₄	—	127	1	no	—
220	C₂×C₄	—	508	2	no	—
221	C₂×C₄	—	254	4	no	—
222	C₂×C₄	—	254	2	no	—
223	C₂×C₄	—	254	1	no	—
224	C₂×C₄	—	254	1	no	—
225	C₂×C₄	—	254	1	no	—
226	C₂×C₄	—	254	1	no	—
227	C₂×C₄	—	254	2	no	—
228	C₂×C₄	—	254	2	no	—
229	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
230	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
231	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
232	C₂×C₄	#14523247	1	2	yes	—
233	C₂×C₄	#14523249	1	2	yes	—
234	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	254	1	no	—
239	C₈	—	254	1	no	—
240	C₈	—	254	2	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	127	2	no	—
244	C₂²	—	508	4	no	—
245	C₂²	—	254	1	no	—
246	C₂²	—	254	1	no	—
247	C₂²	—	254	1	no	—
248	C₂²	—	254	1	no	—
249	C₂²	—	254	1	no	—
250	C₂²	#16861167	1	2	yes	—
251	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	254	1	no	—
255	C₄	—	254	1	no	—
256	C₄	—	254	2	no	—
257	C₄	—	254	2	no	—
258	C₄	#16861167	1	2	yes	—
259	C₄	C₂×#14523247	1	1	yes	characteristic
260	C₄	C₂×#14523249	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	508	1	no	—
263	C₂	—	254	2	no	—
264	C₂	C₂×#16861167	1	1	yes	characteristic
265	C₂	#131×D₁₂₇	1	1	yes	characteristic
266	C₂	#22550641	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#424×C₁₂₇	C₂
prime radical	C₂³×D₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#424	D₁₂₇
hypocenter	C₁₂₇	#2924
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #22549936	C₂	semidirect
3. #22549936	C₂	semidirect
4. #22549932	C₂	semidirect
5. #22549928	C₂	semidirect
6. #22549903	C₂	semidirect
9. #22549834	C₂	semidirect
10. #22549831	C₂	semidirect
11. #424×C₁₂₇	C₂	semidirect
24. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
25. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
32. #16861119	C₂²	semidirect
33. #16861119	C₂²	semidirect
34. C₈×Dic₁₂₇	C₂²	semidirect
35. C₈×Dic₁₂₇	C₂²	semidirect
39. C₄⋊C₈×C₁₂₇	C₂²	semidirect
40. C₄⋊C₈×C₁₂₇	C₂²	semidirect
134. C₁₂₇	#2924	semidirect
141. #424	D₁₂₇	semidirect
163. C₄⋊C₈	C₂×D₁₂₇	semidirect
164. C₄⋊C₈	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#117×D₁₂₇	2
#22549936	2
#22549932	2
#22549928	2
#22549903	2
#22549839	2
C₂×(C₂×D₁₂₇)⋊C₈	2
#22549834	2
#22549831	2
#424×C₁₂₇	2
#2924	127