Subgroups — #30647691

Order 16256.

412 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523248	C₂	1	1	yes	maximal, characteristic
2	#22550003	C₂	1	2	yes	maximal, complemented
3	#22549953	C₂	1	2	yes	maximal, complemented
4	#22549944	C₂	1	4	yes	maximal, complemented
5	#22549927	C₂	1	2	yes	maximal, complemented
6	C₄×#14523231	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#431×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523248	C₄	1	1	yes	characteristic
10	C₂×#14523231	C₂²	1	4	yes	—
11	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
12	C₄×Dic₂₅₄	C₄	1	1	yes	characteristic
13	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
14	C₄×D₅₀₈	C₄	1	1	yes	characteristic
15	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊D₁₂₇	C₄	1	2	yes	—
17	C₄×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
18	#16861168	C₄	1	1	yes	characteristic
19	#16861166	C₄	1	2	yes	—
20	C₄²×D₁₂₇	C₂²	1	2	yes	—
21	Dic₁₂₇⋊C₈	C₂²	1	4	yes	—
22	(C₂×D₁₂₇)⋊C₈	C₂²	1	4	yes	—
23	#16861119	C₂²	1	4	yes	—
24	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
25	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
29	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
30	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
31	#14523248	—	2	4	no	—
32	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
33	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
38	#14523231	—	2	8	no	—
39	C₂×C₁₂₇⋊C₈	—	2	4	no	—
40	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
41	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
42	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
44	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
45	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
46	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
47	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
48	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₂₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
56	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
59	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
60	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
61	Dic₂₅₄	—	2	2	no	—
62	D₅₀₈	—	2	2	no	—
63	C₂²⋊D₁₂₇	—	2	4	no	—
64	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
65	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
66	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
67	C₄×D₁₂₇	—	2	4	no	—
68	C₄×D₁₂₇	—	2	4	no	—
69	C₄×D₁₂₇	—	2	4	no	—
70	C₁₂₇⋊C₈	—	2	2	no	—
71	C₁₂₇⋊C₈	#41	1	4	yes	complemented
72	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	—	2	2	no	—
75	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	complemented
76	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
82	C₈×C₁₂₇	—	2	4	no	—
83	C₂²×C₁₂₇	—	2	1	no	—
84	C₂²×C₁₂₇	—	2	1	no	—
85	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
86	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₁₂₇	—	2	2	no	—
88	C₂×D₁₂₇	—	2	2	no	—
89	C₂×D₁₂₇	—	2	2	no	—
90	C₄×C₁₂₇	—	2	1	no	—
91	C₄×C₁₂₇	—	2	1	no	—
92	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
93	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
94	C₄×C₁₂₇	#131	1	2	yes	—
95	C₄×C₁₂₇	C₂×#41	1	2	yes	—
96	Dic₁₂₇	—	2	2	no	—
97	Dic₁₂₇	—	2	2	no	—
98	Dic₁₂₇	—	2	2	no	—
99	C₂×C₁₂₇	—	2	2	no	—
100	C₂×C₁₂₇	C₄×#41	1	1	yes	characteristic
101	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
102	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
103	D₁₂₇	—	4	2	no	—
104	#2965	—	127	1	no	maximal, Sylow
105	C₁₂₇	#2965	1	1	yes	characteristic, complemented, Sylow
106	C₄×#41	—	127	1	no	—
107	#446	—	127	2	no	—
108	#436	—	127	2	no	—
109	#435	—	127	4	no	—
110	#431	—	127	2	no	—
111	#431	D₁₂₇	1	1	yes	characteristic, complemented
112	C₄×M₁₆	—	127	2	no	—
113	C₂×C₄×C₈	—	127	1	no	—
114	C₂×#41	—	127	1	no	—
115	C₂×M₁₆	—	127	4	no	—
116	C₂²×C₈	—	127	2	no	—
117	C₄×Q₈	—	127	1	no	—
118	C₄×Q₈	—	127	1	no	—
119	C₄×D₄	—	127	2	no	—
120	C₄×D₄	—	127	2	no	—
121	C₄×D₄	—	127	1	no	—
122	C₄×D₄	—	127	1	no	—
123	#117	—	127	1	no	—
124	#117	—	127	2	no	—
125	C₂×C₄²	—	127	2	no	—
126	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
127	C₄⋊C₈	—	127	4	no	—
128	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
129	C₂²⋊C₈	—	127	4	no	—
130	C₂²⋊C₈	C₂×D₁₂₇	1	2	yes	—
131	#97	—	127	4	no	—
132	C₄×C₈	—	127	2	no	—
133	C₄×C₈	—	127	1	no	—
134	C₄×C₈	—	127	1	no	—
135	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
136	#41	—	254	4	no	—
137	C₂×Q₈	—	127	1	no	—
138	C₂×D₄	—	127	2	no	—
139	C₂×D₄	—	127	1	no	—
140	C₂²×C₄	—	127	2	no	—
141	C₂²×C₄	—	127	2	no	—
142	C₂²×C₄	—	127	2	no	—
143	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
146	M₁₆	—	254	8	no	—
147	C₂×C₈	—	127	2	no	—
148	C₂×C₈	—	127	2	no	—
149	C₂×C₈	—	254	4	no	—
150	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
151	C₄⋊C₄	—	127	2	no	—
152	C₄⋊C₄	—	127	2	no	—
153	C₄⋊C₄	—	127	1	no	—
154	C₄⋊C₄	—	127	1	no	—
155	C₂²⋊C₄	—	127	2	no	—
156	C₂²⋊C₄	—	127	2	no	—
157	C₂²⋊C₄	—	127	2	no	—
158	C₄²	—	127	2	no	—
159	C₄²	—	127	2	no	—
160	C₄²	—	127	2	no	—
161	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
162	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
163	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
164	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
165	C₂³	—	127	2	no	—
166	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
167	Q₈	—	254	2	no	—
168	D₄	—	254	4	no	—
169	D₄	—	254	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	127	2	no	—
173	C₂×C₄	—	127	2	no	—
174	C₂×C₄	—	127	2	no	—
175	C₂×C₄	—	254	4	no	—
176	C₂×C₄	—	254	4	no	—
177	C₂×C₄	—	254	4	no	—
178	C₂×C₄	#14523231	1	4	yes	complemented
179	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	127	4	no	—
187	C₈	—	254	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	254	2	no	—
191	C₂²	—	254	2	no	—
192	C₂²	—	254	2	no	—
193	C₂²	C₂×#14523231	1	2	yes	—
194	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	254	2	no	—
198	C₄	—	254	2	no	—
199	C₄	—	254	2	no	—
200	C₄	C₂×#14523231	1	2	yes	—
201	C₄	C₂×#14523231	1	2	yes	—
202	C₄	#16861217	1	2	yes	—
203	C₄	#41×D₁₂₇	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	508	2	no	—
206	C₂	#22550344	1	1	yes	characteristic
207	C₂	C₂²×#14523231	1	1	yes	characteristic
208	C₂	C₂×#16861217	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#431×C₁₂₇	C₂
prime radical	C₂×#14523248	C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#431	D₁₂₇
hypocenter	C₁₂₇	#2965
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
2. #22550003	C₂	semidirect
3. #22549953	C₂	semidirect
4. #22549944	C₂	semidirect
5. #22549927	C₂	semidirect
6. C₄×#14523231	C₂	semidirect
7. C₂×C₄×C₁₂₇⋊C₈	C₂	semidirect
8. #431×C₁₂₇	C₂	semidirect
71. C₁₂₇⋊C₈	#41	semidirect
75. C₂×C₄×C₁₂₇	M₁₆	semidirect
105. C₁₂₇	#2965	semidirect
111. #431	D₁₂₇	semidirect
178. C₂×C₄	#14523231	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523248	2
#22550003	2
#22549953	2
#22549944	2
#22549927	2
C₄×#14523231	2
C₂×C₄×C₁₂₇⋊C₈	2
#431×C₁₂₇	2
#2965	127