Subgroups — #30648450

Order 16256.

452 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	#22550674	C₂	1	2	yes	maximal, complemented
2	C₂×#16861231	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550151	C₂	1	2	yes	maximal, complemented
4	#22550146	C₂	1	2	yes	maximal, complemented
5	#22550139	C₂	1	4	yes	maximal, complemented
6	C₂×#16861145	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549858	C₂	1	2	yes	maximal, complemented
8	C₂×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523248	C₂²	1	2	yes	—
10	#16861240	—	2	2	no	—
11	#16861239	C₂²	1	4	yes	—
12	#16861235	—	2	2	no	—
13	#16861234	C₂²	1	4	yes	—
14	#16861231	—	2	1	no	—
15	#16861231	—	2	1	no	—
16	#16861231	—	2	1	no	—
17	#16861231	—	2	1	no	—
18	#16861229	—	2	4	no	—
19	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic, complemented
20	C₂×#14523237	C₂²	1	2	yes	complemented
21	C₂×D₁₀₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₂×#14523232	C₂²	1	2	yes	—
23	#16861145	C₂²	1	2	yes	complemented
24	#16861145	C₂²	1	2	yes	complemented
25	#16861127	C₂²	1	4	yes	complemented
26	#16861125	C₂²	1	4	yes	complemented
27	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
28	#131×C₁₂₇	—	2	1	no	—
29	#131×C₁₂₇	—	2	1	no	—
30	#131×C₁₂₇	C₂²	1	2	yes	—
31	#131×C₁₂₇	C₂²	1	2	yes	—
32	C₂×M₁₆×C₁₂₇	—	2	1	no	—
33	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₂²×C₈×C₁₂₇	—	2	1	no	—
35	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
36	#14523248	—	2	2	no	—
37	#14523248	—	2	2	no	—
38	#14523248	—	2	2	no	—
39	#14523248	—	2	2	no	—
40	C₂×Dic₂₅₄	C₂³	1	2	yes	—
41	C₂×D₅₀₈	C₂³	1	2	yes	—
42	C₂×C₂²⋊D₁₂₇	—	2	2	no	—
43	C₂×C₄×D₁₂₇	—	2	2	no	—
44	Dic₅₀₈	—	2	1	no	—
45	Dic₅₀₈	—	2	1	no	—
46	#14523238	—	2	4	no	—
47	#14523237	—	2	2	no	—
48	#14523237	—	2	2	no	—
49	#14523236	—	2	4	no	—
50	#14523235	—	2	4	no	—
51	D₁₀₁₆	—	2	1	no	—
52	D₁₀₁₆	—	2	1	no	—
53	#14523233	—	2	4	no	—
54	#14523232	—	2	2	no	—
55	#14523232	C₂³	1	4	yes	—
56	C₂×C₁₂₇⋊C₈	—	2	2	no	—
57	#41×C₁₂₇	—	2	1	no	—
58	#41×C₁₂₇	—	2	1	no	—
59	#41×C₁₂₇	D₄	1	2	yes	—
60	#41×C₁₂₇	C₂³	1	2	yes	—
61	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
62	C₂×D₄×C₁₂₇	—	2	1	no	—
63	C₂×D₄×C₁₂₇	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₂₇	—	2	1	no	—
65	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
66	M₁₆×C₁₂₇	—	2	1	no	—
67	M₁₆×C₁₂₇	—	2	2	no	—
68	M₁₆×C₁₂₇	—	2	2	no	—
69	M₁₆×C₁₂₇	C₂³	1	2	yes	—
70	C₂×C₈×C₁₂₇	—	2	1	no	—
71	C₂×C₈×C₁₂₇	—	2	2	no	—
72	C₂×C₈×C₁₂₇	—	2	2	no	—
73	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
79	C₂²×D₁₂₇	—	2	2	no	—
80	Dic₂₅₄	—	2	2	no	—
81	Dic₂₅₄	—	2	2	no	—
82	D₅₀₈	—	2	2	no	—
83	D₅₀₈	—	2	2	no	—
84	C₂²⋊D₁₂₇	—	4	2	no	—
85	C₂²⋊D₁₂₇	—	4	2	no	—
86	C₂×Dic₁₂₇	—	2	2	no	—
87	C₄×D₁₂₇	—	4	2	no	—
88	C₄×D₁₂₇	—	4	2	no	—
89	C₁₂₇⋊C₈	—	2	4	no	—
90	C₂³×C₁₂₇	—	2	1	no	—
91	C₂³×C₁₂₇	#41	1	1	yes	characteristic
92	Q₈×C₁₂₇	—	2	1	no	—
93	Q₈×C₁₂₇	C₂×D₄	1	2	yes	—
94	D₄×C₁₂₇	—	2	2	no	—
95	D₄×C₁₂₇	—	2	2	no	—
96	D₄×C₁₂₇	—	2	1	no	—
97	D₄×C₁₂₇	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₂₇	—	2	1	no	—
99	C₂×C₄×C₁₂₇	—	2	2	no	—
100	C₂×C₄×C₁₂₇	—	2	2	no	—
101	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
107	C₈×C₁₂₇	—	2	2	no	—
108	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
109	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
110	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
111	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₁₂₇	—	2	2	no	—
113	C₂²×C₁₂₇	—	4	1	no	—
114	C₂²×C₁₂₇	—	4	2	no	—
115	C₂²×C₁₂₇	#121	1	1	yes	characteristic
116	C₂²×C₁₂₇	#121	1	1	yes	characteristic
117	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
118	C₂×D₁₂₇	—	4	2	no	—
119	C₂×D₁₂₇	—	8	2	no	—
120	C₄×C₁₂₇	—	2	2	no	—
121	C₄×C₁₂₇	#121	1	1	yes	characteristic
122	C₄×C₁₂₇	#121	1	1	yes	characteristic
123	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
124	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
125	Dic₁₂₇	—	4	2	no	—
126	C₂×C₁₂₇	—	2	1	no	—
127	C₂×C₁₂₇	—	2	1	no	—
128	C₂×C₁₂₇	—	2	1	no	—
129	C₂×C₁₂₇	—	4	2	no	—
130	C₂×C₁₂₇	C₂×#121	1	1	yes	characteristic
131	D₁₂₇	—	8	2	no	—
132	#3067	—	127	1	no	maximal, Sylow
133	C₁₂₇	#3067	1	1	yes	characteristic, complemented, Sylow
134	#575	—	127	2	no	—
135	C₂×#135	—	127	1	no	—
136	C₂×#131	D₁₂₇	1	1	yes	characteristic, complemented
137	#473	—	127	2	no	—
138	#472	—	127	2	no	—
139	#471	—	127	4	no	—
140	C₂×#108	—	127	1	no	—
141	#413	—	127	2	no	—
142	C₂×#41	—	127	2	no	—
143	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
144	#137	—	127	4	no	—
145	#137	—	254	2	no	—
146	AGL(1,ℤ/8)	—	127	4	no	—
147	AGL(1,ℤ/8)	—	254	2	no	—
148	#135	—	254	4	no	—
149	#135	—	254	1	no	—
150	#135	—	254	1	no	—
151	#135	—	254	1	no	—
152	#135	—	254	1	no	—
153	C₂×Q₁₆	—	127	1	no	—
154	C₂×SD₁₆	—	127	2	no	—
155	C₂×D₈	—	127	1	no	—
156	#131	—	2	1	no	—
157	#131	—	2	1	no	—
158	#131	C₂×D₁₂₇	1	2	yes	—
159	#131	C₂×D₁₂₇	1	2	yes	—
160	C₂×M₁₆	—	2	1	no	—
161	C₂×M₁₆	—	127	2	no	—
162	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
165	#108	—	127	2	no	—
166	#108	—	127	2	no	—
167	#101	—	127	4	no	—
168	#100	—	127	4	no	—
169	#41	—	2	1	no	—
170	#41	—	2	1	no	—
171	#41	—	254	2	no	—
172	#41	—	254	2	no	—
173	#41	—	254	2	no	—
174	#41	—	254	2	no	—
175	#41	D₅₀₈	1	2	yes	—
176	#41	C₂²×D₁₂₇	1	2	yes	—
177	C₂×Q₈	—	127	2	no	—
178	C₂×Q₈	D₅₀₈	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	127	2	no	—
181	C₂×D₄	—	254	2	no	—
182	C₂×D₄	D₅₀₈	1	1	yes	characteristic
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	254	2	no	—
185	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
186	Q₁₆	—	254	4	no	—
187	Q₁₆	—	254	1	no	—
188	Q₁₆	—	254	1	no	—
189	SD₁₆	—	254	4	no	—
190	SD₁₆	—	254	4	no	—
191	SD₁₆	—	254	2	no	—
192	SD₁₆	—	254	2	no	—
193	D₈	—	254	4	no	—
194	D₈	—	254	1	no	—
195	D₈	—	254	1	no	—
196	M₁₆	—	2	1	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	—	2	2	no	—
199	M₁₆	—	127	4	no	—
200	M₁₆	—	254	2	no	—
201	M₁₆	C₂²×D₁₂₇	1	2	yes	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	254	2	no	—
206	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
209	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	C₂³	—	254	2	no	—
214	C₂³	#14523248	1	1	yes	characteristic
215	Q₈	—	2	1	no	—
216	Q₈	—	254	2	no	—
217	Q₈	—	254	2	no	—
218	Q₈	C₂×D₅₀₈	1	2	yes	—
219	D₄	—	2	2	no	—
220	D₄	—	2	2	no	—
221	D₄	—	2	1	no	—
222	D₄	—	508	2	no	—
223	D₄	—	508	2	no	—
224	D₄	—	254	2	no	—
225	D₄	—	254	2	no	—
226	D₄	C₂×D₅₀₈	1	2	yes	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	508	2	no	—
231	C₂×C₄	—	508	2	no	—
232	C₂×C₄	—	254	2	no	—
233	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
235	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
236	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
237	C₂×C₄	#14523248	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	254	4	no	—
241	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
242	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
243	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
244	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
245	C₂²	—	2	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	1016	2	no	—
249	C₂²	—	508	2	no	—
250	C₂²	#16861183	1	1	yes	characteristic
251	C₂²	#16861183	1	1	yes	characteristic
252	C₂²	C₂×#14523248	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	508	2	no	—
255	C₄	#16861183	1	1	yes	characteristic
256	C₄	#16861183	1	1	yes	characteristic
257	C₄	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
258	C₄	C₂²×D₅₀₈	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	1016	2	no	—
264	C₂	C₂×#16861183	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16861183
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#131×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×#121
hypercenter	C₂×#131	D₁₂₇
hypocenter	C₁₂₇	#3067
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #22550674	C₂	semidirect
2. C₂×#16861231	C₂	semidirect
3. #22550151	C₂	semidirect
4. #22550146	C₂	semidirect
5. #22550139	C₂	semidirect
6. C₂×#16861145	C₂	semidirect
7. #22549858	C₂	semidirect
8. C₂×#131×C₁₂₇	C₂	semidirect
19. C₂×Dic₅₀₈	C₂²	semidirect
20. C₂×#14523237	C₂²	semidirect
21. C₂×D₁₀₁₆	C₂²	semidirect
23. #16861145	C₂²	semidirect
24. #16861145	C₂²	semidirect
25. #16861127	C₂²	semidirect
26. #16861125	C₂²	semidirect
133. C₁₂₇	#3067	semidirect
136. C₂×#131	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550674	2
C₂×#16861231	2
#22550151	2
#22550146	2
#22550139	2
C₂×#16861145	2
#22549858	2
C₂×#131×C₁₂₇	2
#3067	127