Subgroups — #30647440

Order 16256.

500 conjugacy classes of subgroups.

The subgroup lattice has 232 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	#22550422	C₂	1	1	yes	maximal, characteristic
3	#22549952	C₂	1	4	yes	maximal, complemented
4	C₈×C₂²⋊D₁₂₇	C₂	1	4	yes	maximal, complemented
5	#22549899	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549844	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549843	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×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×#14523231	C₂²	1	2	yes	—
14	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
16	#16861207	C₄	1	1	yes	characteristic
17	#16861205	C₄	1	1	yes	characteristic
18	#16861198	C₄	1	1	yes	characteristic
19	#16861196	C₄	1	1	yes	characteristic
20	#16861181	C₄	1	4	yes	—
21	C₄×C₂²⋊D₁₂₇	C₂²	1	4	yes	—
22	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
23	Dic₁₂₇⋊C₈	C₂²	1	4	yes	—
24	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
25	(C₂×D₁₂₇)⋊C₈	C₂²	1	4	yes	—
26	C₈×Dic₁₂₇	C₂²	1	2	yes	—
27	C₈×Dic₁₂₇	C₂²	1	2	yes	—
28	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
29	#131×C₁₂₇	—	2	4	no	—
30	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
32	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
33	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
34	#14523248	—	2	2	no	—
35	#14523248	—	2	2	no	—
36	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
37	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
40	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
42	#14523232	—	2	2	no	—
43	#14523231	—	2	4	no	—
44	C₂×C₁₂₇⋊C₈	—	2	2	no	—
45	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
46	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
47	C₈×D₁₂₇	—	2	4	no	—
48	Dic₁₂₇⋊C₄	C₂×C₄	1	4	yes	—
49	C₂²⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
50	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	4	yes	—
51	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
52	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₁₂₇	C₂³	1	2	yes	—
54	#41×C₁₂₇	—	2	2	no	—
55	#41×C₁₂₇	—	2	2	no	—
56	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
57	C₂×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
58	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
59	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
60	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
61	M₁₆×C₁₂₇	—	2	2	no	—
62	M₁₆×C₁₂₇	—	2	4	no	—
63	C₂×C₈×C₁₂₇	—	2	4	no	—
64	C₂×C₈×C₁₂₇	D₄	1	4	yes	complemented
65	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
66	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
69	Dic₂₅₄	—	2	2	no	—
70	D₅₀₈	—	2	2	no	—
71	C₂²⋊D₁₂₇	—	2	4	no	—
72	C₂²⋊D₁₂₇	—	2	8	no	—
73	C₂×Dic₁₂₇	—	2	4	no	—
74	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
77	C₄×D₁₂₇	—	2	2	no	—
78	C₄×D₁₂₇	—	2	2	no	—
79	C₁₂₇⋊C₈	—	2	2	no	—
80	C₁₂₇⋊C₈	—	2	2	no	—
81	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
82	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
83	Q₈×C₁₂₇	—	2	2	no	—
84	D₄×C₁₂₇	—	2	4	no	—
85	D₄×C₁₂₇	—	2	2	no	—
86	C₂×C₄×C₁₂₇	—	2	2	no	—
87	C₂×C₄×C₁₂₇	—	2	2	no	—
88	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₂₇	#41	1	2	yes	—
92	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
93	C₈×C₁₂₇	—	2	2	no	—
94	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
95	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
96	C₂²×C₁₂₇	—	2	2	no	—
97	C₂²×C₁₂₇	—	2	2	no	—
98	C₂²×C₁₂₇	—	2	4	no	—
99	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
100	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₁₂₇	—	2	2	no	—
102	C₂×D₁₂₇	—	2	4	no	—
103	C₄×C₁₂₇	—	2	2	no	—
104	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
105	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
107	Dic₁₂₇	—	2	2	no	—
108	Dic₁₂₇	—	2	2	no	—
109	Dic₁₂₇	#131	1	4	yes	complemented
110	C₂×C₁₂₇	—	2	2	no	—
111	C₂×C₁₂₇	—	4	2	no	—
112	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂×C₁₂₇	C₂×#131	1	2	yes	—
114	D₁₂₇	—	4	2	no	—
115	#2931	—	127	1	no	maximal, Sylow
116	C₁₂₇	#2931	1	1	yes	characteristic, complemented, Sylow
117	C₂×#131	—	127	1	no	—
118	C₂×#131	D₁₂₇	1	1	yes	characteristic, complemented
119	#532	—	127	1	no	—
120	#436	—	127	4	no	—
121	C₈×D₄	—	127	4	no	—
122	#423	—	127	1	no	—
123	#408	—	127	1	no	—
124	#408	—	127	1	no	—
125	C₂×C₄×C₈	—	127	1	no	—
126	C₂×#41	—	127	1	no	—
127	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
128	#131	—	2	4	no	—
129	#131	—	254	4	no	—
130	C₂×M₁₆	—	127	1	no	—
131	C₂×M₁₆	—	127	2	no	—
132	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
133	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
134	C₂²×C₈	—	127	1	no	—
135	C₂²×C₈	—	127	2	no	—
136	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
137	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	127	1	no	—
139	#127	—	127	1	no	—
140	#124	—	127	1	no	—
141	#124	—	127	1	no	—
142	#121	—	127	4	no	—
143	C₄×D₄	—	127	4	no	—
144	C₂×C₄²	—	127	1	no	—
145	C₄⋊C₈	—	127	4	no	—
146	C₂²⋊C₈	—	127	4	no	—
147	C₂²⋊C₈	—	127	4	no	—
148	C₄×C₈	—	127	2	no	—
149	C₄×C₈	—	127	2	no	—
150	#41	—	2	2	no	—
151	#41	—	2	2	no	—
152	#41	—	254	2	no	—
153	#41	—	254	2	no	—
154	C₂×Q₈	—	127	1	no	—
155	C₂×Q₈	C₄×D₁₂₇	1	1	yes	characteristic
156	C₂×D₄	—	127	4	no	—
157	C₂×D₄	—	127	2	no	—
158	C₂×D₄	—	127	1	no	—
159	C₂×D₄	C₄×D₁₂₇	1	2	yes	—
160	C₂×D₄	C₄×D₁₂₇	1	1	yes	characteristic
161	C₂²×C₄	—	127	2	no	—
162	C₂²×C₄	—	127	2	no	—
163	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
164	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
165	M₁₆	—	2	2	no	—
166	M₁₆	—	2	4	no	—
167	M₁₆	—	254	2	no	—
168	M₁₆	—	254	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	127	2	no	—
171	C₂×C₈	—	127	2	no	—
172	C₂×C₈	—	254	2	no	—
173	C₂×C₈	—	254	4	no	—
174	C₂×C₈	C₂²⋊D₁₂₇	1	4	yes	complemented
175	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
176	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
178	C₄⋊C₄	—	127	4	no	—
179	C₂²⋊C₄	—	127	4	no	—
180	C₂²⋊C₄	—	127	4	no	—
181	C₄²	—	127	1	no	—
182	C₄²	—	127	1	no	—
183	C₄²	—	127	2	no	—
184	C₂³	—	127	2	no	—
185	C₂³	C₂×C₄×D₁₂₇	1	2	yes	—
186	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
187	Q₈	—	2	2	no	—
188	Q₈	—	254	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	2	no	—
191	D₄	—	254	8	no	—
192	D₄	—	254	4	no	—
193	D₄	—	254	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	127	2	no	—
197	C₂×C₄	—	127	2	no	—
198	C₂×C₄	—	127	2	no	—
199	C₂×C₄	—	254	4	no	—
200	C₂×C₄	—	254	2	no	—
201	C₂×C₄	—	254	2	no	—
202	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
204	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
205	C₂×C₄	#14523248	1	2	yes	—
206	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	254	2	no	—
209	C₈	—	254	2	no	—
210	C₈	C₂×C₂²⋊D₁₂₇	1	2	yes	—
211	C₈	C₂×C₂²⋊D₁₂₇	1	2	yes	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	254	2	no	—
216	C₂²	—	254	4	no	—
217	C₂²	C₄×C₂²⋊D₁₂₇	1	2	yes	—
218	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	127	4	no	—
221	C₄	—	254	2	no	—
222	C₄	—	254	2	no	—
223	C₄	C₄×C₂²⋊D₁₂₇	1	2	yes	—
224	C₄	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
225	C₄	C₂×#14523248	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	508	2	no	—
229	C₂	C₂×C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
230	C₂	#131×D₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#131×C₁₂₇	C₂
prime radical	#22550422	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#131	D₁₂₇
hypocenter	C₁₂₇	#2931
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

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. C₈×C₂²⋊D₁₂₇	C₂	semidirect
5. #22549899	C₂	semidirect
6. #22549844	C₂	semidirect
7. #22549843	C₂	semidirect
8. C₂×C₈×Dic₁₂₇	C₂	semidirect
9. C₂×#131×C₁₂₇	C₂	semidirect
64. C₂×C₈×C₁₂₇	D₄	semidirect
109. Dic₁₂₇	#131	semidirect
116. C₁₂₇	#2931	semidirect
118. C₂×#131	D₁₂₇	semidirect
174. 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
#22550422	2
#22549952	2
C₈×C₂²⋊D₁₂₇	2
#22549899	2
#22549844	2
#22549843	2
C₂×C₈×Dic₁₂₇	2
C₂×#131×C₁₂₇	2
#2931	127