Subgroups — #30647433

Order 16256.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861215	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550387	C₂	1	1	yes	maximal, characteristic
3	#22549952	C₂	1	4	yes	maximal, complemented
4	#22549945	C₂	1	4	yes	maximal, complemented
5	#22549902	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549842	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549840	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549831	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14523248	C₂²	1	1	yes	characteristic
11	#16861215	—	2	4	no	—
12	C₂×#14523232	C₂²	1	1	yes	characteristic
13	C₂×#14523232	C₂²	1	1	yes	characteristic
14	C₂×#14523231	C₂²	1	2	yes	complemented
15	C₂×C₈×D₁₂₇	C₂²	1	2	yes	complemented
16	#16861192	C₄	1	2	yes	—
17	#16861188	C₄	1	2	yes	—
18	#16861183	C₄	1	2	yes	—
19	#16861178	C₄	1	2	yes	—
20	C₄×C₂²⋊D₁₂₇	C₂²	1	4	yes	—
21	#16861168	C₂²	1	1	yes	characteristic
22	Dic₁₂₇⋊C₈	C₂²	1	4	yes	complemented
23	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
24	(C₂×D₁₂₇)⋊C₈	C₂²	1	4	yes	complemented
25	#16861119	C₂²	1	2	yes	complemented
26	C₈×Dic₁₂₇	C₂²	1	2	yes	complemented
27	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₁₂₇	—	2	4	no	—
29	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
30	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
31	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
33	#14523248	—	2	2	no	—
34	#14523248	—	2	2	no	—
35	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
36	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	4	yes	—
39	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
40	#14523232	—	2	2	no	—
41	#14523232	—	2	2	no	—
42	#14523231	—	2	4	no	—
43	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
44	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
45	C₈×D₁₂₇	—	2	4	no	—
46	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
47	Dic₁₂₇⋊C₄	C₂×C₄	1	4	yes	—
48	C₂²⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
49	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
50	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	4	yes	—
51	C₄×Dic₁₂₇	C₂³	1	2	yes	—
52	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	—	2	2	no	—
54	C₂²×C₄×C₁₂₇	—	2	2	no	—
55	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
56	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
59	M₁₆×C₁₂₇	—	2	4	no	—
60	M₁₆×C₁₂₇	—	2	4	no	—
61	C₂×C₈×C₁₂₇	—	2	2	no	—
62	C₂×C₈×C₁₂₇	D₄	1	4	yes	complemented
63	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
64	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
66	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
67	Dic₂₅₄	—	2	2	no	—
68	D₅₀₈	—	2	2	no	—
69	C₂²⋊D₁₂₇	—	2	4	no	—
70	C₂²⋊D₁₂₇	—	4	4	no	—
71	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
72	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
73	C₄×D₁₂₇	—	2	2	no	—
74	C₄×D₁₂₇	—	2	2	no	—
75	C₁₂₇⋊C₈	—	2	2	no	—
76	C₁₂₇⋊C₈	—	2	2	no	—
77	C₂³×C₁₂₇	—	2	2	no	—
78	C₂³×C₁₂₇	—	4	2	no	—
79	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
80	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	—	2	2	no	—
82	C₂×C₄×C₁₂₇	—	2	4	no	—
83	C₂×C₄×C₁₂₇	—	2	2	no	—
84	C₂×C₄×C₁₂₇	—	2	2	no	—
85	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₂₇	#41	1	2	yes	—
89	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
90	C₈×C₁₂₇	—	2	2	no	—
91	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
92	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
93	C₂²×C₁₂₇	—	2	2	no	—
94	C₂²×C₁₂₇	—	2	2	no	—
95	C₂²×C₁₂₇	—	2	2	no	—
96	C₂²×C₁₂₇	—	2	2	no	—
97	C₂²×C₁₂₇	—	4	4	no	—
98	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
99	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×D₁₂₇	—	2	2	no	—
101	C₂×D₁₂₇	—	2	4	no	—
102	C₄×C₁₂₇	—	2	2	no	—
103	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
104	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
106	Dic₁₂₇	—	2	2	no	—
107	Dic₁₂₇	—	2	4	no	—
108	C₂×C₁₂₇	—	2	2	no	—
109	C₂×C₁₂₇	—	4	2	no	—
110	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₁₂₇	#568	1	2	yes	—
112	D₁₂₇	—	4	2	no	—
113	#2930	—	127	1	no	maximal, Sylow
114	C₁₂₇	#2930	1	1	yes	characteristic, complemented, Sylow
115	C₂×#131	—	127	1	no	—
116	C₂²×M₁₆	D₁₂₇	1	1	yes	characteristic, complemented
117	#525	—	127	1	no	—
118	#436	—	127	4	no	—
119	#435	—	127	4	no	—
120	#424	—	127	1	no	—
121	#408	—	127	1	no	—
122	#407	—	127	1	no	—
123	#405	—	127	1	no	—
124	C₂×#41	—	127	1	no	—
125	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
126	#131	—	254	4	no	—
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	—	127	1	no	—
129	C₂×M₁₆	—	127	1	no	—
130	C₂×M₁₆	—	127	2	no	—
131	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
132	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
133	C₂²×C₈	—	127	2	no	—
134	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
136	#123	—	127	2	no	—
137	C₂²⋊Q₈	—	127	2	no	—
138	#121	—	127	2	no	—
139	#120	—	127	2	no	—
140	C₄×D₄	—	127	4	no	—
141	#117	—	127	1	no	—
142	C₄⋊C₈	—	127	4	no	—
143	C₂²⋊C₈	—	127	4	no	—
144	C₂²⋊C₈	—	127	4	no	—
145	#97	—	127	2	no	—
146	C₄×C₈	—	127	2	no	—
147	C₂⁴	C₄×D₁₂₇	1	1	yes	characteristic
148	#41	—	254	2	no	—
149	#41	—	254	2	no	—
150	C₂×Q₈	—	127	1	no	—
151	C₂×D₄	—	127	4	no	—
152	C₂×D₄	—	127	2	no	—
153	C₂×D₄	—	127	1	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	127	2	no	—
157	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
158	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
160	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
161	M₁₆	—	2	4	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	254	2	no	—
164	M₁₆	—	254	2	no	—
165	M₁₆	—	254	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	127	2	no	—
168	C₂×C₈	—	127	2	no	—
169	C₂×C₈	—	254	4	no	—
170	C₂×C₈	C₂²⋊D₁₂₇	1	4	yes	complemented
171	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
172	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
174	C₄⋊C₄	—	127	4	no	—
175	C₄⋊C₄	—	127	2	no	—
176	C₂²⋊C₄	—	127	2	no	—
177	C₂²⋊C₄	—	127	4	no	—
178	C₂²⋊C₄	—	127	4	no	—
179	C₄²	—	127	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	4	2	no	—
182	C₂³	—	127	2	no	—
183	C₂³	C₂×C₄×D₁₂₇	1	2	yes	—
184	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
185	Q₈	—	254	2	no	—
186	D₄	—	508	4	no	—
187	D₄	—	254	4	no	—
188	D₄	—	254	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	127	4	no	—
194	C₂×C₄	—	127	2	no	—
195	C₂×C₄	—	254	2	no	—
196	C₂×C₄	—	254	2	no	—
197	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
199	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
200	C₂×C₄	#14523248	1	2	yes	—
201	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	254	2	no	—
204	C₈	—	254	2	no	—
205	C₈	C₂×C₂²⋊D₁₂₇	1	2	yes	—
206	C₈	C₂×C₂²⋊D₁₂₇	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	—	254	2	no	—
213	C₂²	—	254	4	no	—
214	C₂²	C₄×C₂²⋊D₁₂₇	1	2	yes	—
215	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	254	2	no	—
218	C₄	—	254	4	no	—
219	C₄	C₄×C₂²⋊D₁₂₇	1	2	yes	—
220	C₄	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
221	C₄	C₂×#14523248	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	508	2	no	—
225	C₂	C₂×C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
226	C₂	#22550642	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	C₂²×M₁₆×C₁₂₇	C₂
prime radical	#22550387	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂²×M₁₆	D₁₂₇
hypocenter	C₁₂₇	#2930
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16861215	C₂	semidirect
3. #22549952	C₂	semidirect
4. #22549945	C₂	semidirect
5. #22549902	C₂	semidirect
6. #22549842	C₂	semidirect
7. #22549840	C₂	semidirect
8. #22549831	C₂	semidirect
9. C₂²×M₁₆×C₁₂₇	C₂	semidirect
14. C₂×#14523231	C₂²	semidirect
15. C₂×C₈×D₁₂₇	C₂²	semidirect
22. Dic₁₂₇⋊C₈	C₂²	semidirect
24. (C₂×D₁₂₇)⋊C₈	C₂²	semidirect
25. #16861119	C₂²	semidirect
26. C₈×Dic₁₂₇	C₂²	semidirect
62. C₂×C₈×C₁₂₇	D₄	semidirect
114. C₁₂₇	#2930	semidirect
116. C₂²×M₁₆	D₁₂₇	semidirect
170. C₂×C₈	C₂²⋊D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861215	2
#22550387	2
#22549952	2
#22549945	2
#22549902	2
#22549842	2
#22549840	2
#22549831	2
C₂²×M₁₆×C₁₂₇	2
#2930	127