Subgroups — #30647297

Order 16256.

484 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861217	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861215	C₂	1	1	yes	maximal, characteristic
3	#22550344	C₂	1	1	yes	maximal, characteristic
4	#22549903	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549901	C₂	1	2	yes	maximal, complemented
6	#22549901	C₂	1	2	yes	maximal, complemented
7	#22549899	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549898	C₂	1	2	yes	maximal, complemented
9	C₂×Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₁₂₇	C₂	1	2	yes	maximal, complemented
11	#424×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	#16861217	—	2	4	no	—
14	#16861215	—	2	4	no	—
15	C₂×#14523232	C₂²	1	1	yes	characteristic
16	C₂×#14523231	C₂²	1	2	yes	—
17	C₂×#14523231	C₂²	1	2	yes	—
18	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
20	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
21	#16861174	C₄	1	1	yes	characteristic
22	#16861174	C₄	1	1	yes	characteristic
23	#16861171	C₄	1	1	yes	characteristic
24	#16861171	C₄	1	1	yes	characteristic
25	(C₂²⋊D₁₂₇)⋊C₄	C₄	1	2	yes	—
26	(C₂²⋊D₁₂₇)⋊C₄	C₄	1	2	yes	—
27	#16861166	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
29	C₄²×D₁₂₇	C₂²	1	2	yes	—
30	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
31	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
32	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
33	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
34	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
35	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
36	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
38	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
39	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
40	#117×C₁₂₇	C₂²	1	1	yes	characteristic
41	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
42	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
43	#14523248	D₄	1	4	yes	—
44	#14523248	Q₈	1	4	yes	—
45	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
46	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
48	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
53	#14523232	—	2	2	no	—
54	#14523231	—	2	4	no	—
55	#14523231	—	2	4	no	—
56	C₂×C₁₂₇⋊C₈	—	2	2	no	—
57	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
58	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
59	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
60	C₈×D₁₂₇	—	2	4	no	—
61	C₈×D₁₂₇	—	2	4	no	—
62	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
64	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
65	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
66	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
71	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
73	M₁₆×C₁₂₇	—	2	2	no	—
74	C₂×C₈×C₁₂₇	—	2	2	no	—
75	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
77	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
81	C₄²×C₁₂₇	C₂³	1	2	yes	—
82	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
83	Dic₂₅₄	C₄⋊C₄	1	4	yes	—
84	D₅₀₈	C₄⋊C₄	1	4	yes	—
85	C₂²⋊D₁₂₇	C₄⋊C₄	1	8	yes	—
86	C₂×Dic₁₂₇	—	2	2	no	—
87	C₂×Dic₁₂₇	—	4	1	no	—
88	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
92	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₁₂₇	—	2	4	no	—
94	C₄×D₁₂₇	—	4	2	no	—
95	C₄×D₁₂₇	C₂×D₄	1	4	yes	—
96	C₄×D₁₂₇	C₂×Q₈	1	4	yes	—
97	C₁₂₇⋊C₈	—	2	1	no	—
98	C₁₂₇⋊C₈	—	2	2	no	—
99	C₁₂₇⋊C₈	—	2	1	no	—
100	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
106	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
107	C₈×C₁₂₇	—	2	1	no	—
108	C₈×C₁₂₇	—	2	1	no	—
109	C₈×C₁₂₇	—	2	2	no	—
110	C₂²×C₁₂₇	—	2	1	no	—
111	C₂²×C₁₂₇	—	2	1	no	—
112	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×D₁₂₇	—	2	2	no	—
115	C₂×D₁₂₇	—	2	2	no	—
116	C₂×D₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
117	C₄×C₁₂₇	—	2	2	no	—
118	C₄×C₁₂₇	—	2	2	no	—
119	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
122	Dic₁₂₇	—	2	1	no	—
123	Dic₁₂₇	—	2	1	no	—
124	Dic₁₂₇	—	2	1	no	—
125	Dic₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
126	Dic₁₂₇	#131	1	2	yes	complemented
127	C₂×C₁₂₇	—	2	2	no	—
128	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
130	C₂×C₁₂₇	#568	1	1	yes	characteristic
131	D₁₂₇	—	2	4	no	—
132	#2906	—	127	1	no	maximal, Sylow
133	C₁₂₇	#2906	1	1	yes	characteristic, complemented, Sylow
134	C₂×#131	—	127	1	no	—
135	C₂×#131	—	127	1	no	—
136	C₄×#41	—	127	1	no	—
137	#424	—	127	1	no	—
138	#424	—	127	2	no	—
139	#424	—	127	2	no	—
140	#424	D₁₂₇	1	1	yes	characteristic, complemented
141	#423	—	127	1	no	—
142	#423	—	127	2	no	—
143	C₂×C₄⋊C₈	—	127	1	no	—
144	C₂×C₄⋊C₈	—	127	2	no	—
145	C₂×#41	—	127	1	no	—
146	#131	—	254	4	no	—
147	#131	—	254	4	no	—
148	C₂×M₁₆	—	127	1	no	—
149	C₂×M₁₆	—	127	2	no	—
150	C₂×M₁₆	—	127	2	no	—
151	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
152	C₂²×C₈	—	127	1	no	—
153	C₂²×C₈	—	127	2	no	—
154	C₂²×C₈	—	127	2	no	—
155	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
156	C₄×Q₈	—	127	1	no	—
157	C₄×Q₈	—	127	1	no	—
158	C₄×D₄	—	127	1	no	—
159	C₄×D₄	—	127	1	no	—
160	C₄×D₄	—	127	2	no	—
161	C₄×D₄	—	127	2	no	—
162	#117	—	127	2	no	—
163	#117	C₂×D₁₂₇	1	1	yes	characteristic
164	C₂×C₄²	—	127	1	no	—
165	C₂×C₄²	—	127	2	no	—
166	C₄⋊C₈	—	127	1	no	—
167	C₄⋊C₈	—	127	2	no	—
168	C₄⋊C₈	—	127	1	no	—
169	C₄⋊C₈	—	127	1	no	—
170	C₄⋊C₈	—	127	1	no	—
171	C₄⋊C₈	—	127	2	no	—
172	C₄⋊C₈	—	127	2	no	—
173	C₄⋊C₈	—	127	2	no	—
174	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
175	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
176	#41	—	127	4	no	—
177	#41	—	127	4	no	—
178	C₂×Q₈	—	127	1	no	—
179	C₂×D₄	—	127	2	no	—
180	C₂×D₄	—	127	1	no	—
181	C₂²×C₄	—	127	2	no	—
182	C₂²×C₄	—	127	1	no	—
183	C₂²×C₄	—	127	1	no	—
184	C₂²×C₄	—	127	2	no	—
185	C₂²×C₄	—	127	2	no	—
186	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
187	M₁₆	—	2	2	no	—
188	M₁₆	—	254	4	no	—
189	M₁₆	—	254	4	no	—
190	M₁₆	—	254	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	127	1	no	—
193	C₂×C₈	—	127	2	no	—
194	C₂×C₈	—	127	1	no	—
195	C₂×C₈	—	254	4	no	—
196	C₂×C₈	—	254	4	no	—
197	C₂×C₈	—	254	2	no	—
198	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
199	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
200	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
201	C₄⋊C₄	—	127	2	no	—
202	C₄⋊C₄	—	127	2	no	—
203	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
204	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
205	C₂²⋊C₄	—	127	2	no	—
206	C₂²⋊C₄	—	127	2	no	—
207	C₂²⋊C₄	C₄×D₁₂₇	1	2	yes	—
208	C₄²	—	127	2	no	—
209	C₄²	—	127	1	no	—
210	C₄²	—	127	2	no	—
211	C₄²	—	127	1	no	—
212	C₄²	—	127	1	no	—
213	C₄²	—	127	1	no	—
214	C₄²	C₂²×D₁₂₇	1	2	yes	—
215	C₂³	—	127	2	no	—
216	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
217	Q₈	—	127	4	no	—
218	D₄	—	127	8	no	—
219	D₄	—	127	4	no	—
220	C₂×C₄	—	127	1	no	—
221	C₂×C₄	—	127	2	no	—
222	C₂×C₄	—	127	4	no	—
223	C₂×C₄	—	127	4	no	—
224	C₂×C₄	—	127	1	no	—
225	C₂×C₄	—	127	1	no	—
226	C₂×C₄	—	127	1	no	—
227	C₂×C₄	—	508	2	no	—
228	C₂×C₄	—	508	1	no	—
229	C₂×C₄	—	254	4	no	—
230	C₂×C₄	—	254	2	no	—
231	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
232	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
233	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
234	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
235	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
236	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	2	no	—
240	C₈	—	254	1	no	—
241	C₈	—	254	2	no	—
242	C₈	—	254	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	127	4	no	—
246	C₂²	—	254	2	no	—
247	C₂²	—	254	2	no	—
248	C₂²	C₄⋊C₄×D₁₂₇	1	2	yes	—
249	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	127	4	no	—
253	C₄	—	127	2	no	—
254	C₄	—	254	1	no	—
255	C₄	—	254	1	no	—
256	C₄	—	254	1	no	—
257	C₄	C₄⋊C₄×D₁₂₇	1	2	yes	—
258	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
259	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
260	C₂	—	2	2	no	—
261	C₂	—	254	4	no	—
262	C₂	C₂×C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
263	C₂	#131×D₁₂₇	1	1	yes	characteristic
264	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	C₂×#14523248	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#424	D₁₂₇
hypocenter	C₁₂₇	#2906
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
4. #22549903	C₂	semidirect
5. #22549901	C₂	semidirect
6. #22549901	C₂	semidirect
7. #22549899	C₂	semidirect
8. #22549898	C₂	semidirect
9. C₂×Dic₁₂₇⋊C₈	C₂	semidirect
10. C₄⋊C₈×D₁₂₇	C₂	semidirect
11. #424×C₁₂₇	C₂	semidirect
126. Dic₁₂₇	#131	semidirect
133. C₁₂₇	#2906	semidirect
140. #424	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861217	2
C₂×#16861215	2
#22550344	2
#22549903	2
#22549901	2
#22549899	2
#22549898	2
C₂×Dic₁₂₇⋊C₈	2
C₄⋊C₈×D₁₂₇	2
#424×C₁₂₇	2
#2906	127