Subgroups — #30638468

Order 16256.

388 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523232	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861168	C₂	1	1	yes	maximal, characteristic
3	#22549644	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523232	—	2	2	no	—
6	C₂×#14523232	—	2	1	no	—
7	C₂×#14523232	C₄	1	4	yes	complemented
8	C₂×#14523232	C₂²	1	2	yes	—
9	C₂²×C₁₂₇⋊C₈	—	2	1	no	—
10	#16861168	—	2	1	no	—
11	#16861168	—	2	1	no	—
12	#16861168	—	2	1	no	—
13	#16861168	—	2	1	no	—
14	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₂₇	C₄	1	2	yes	—
16	C₂×C₄×Dic₁₂₇	C₄	1	2	yes	—
17	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₂₇	—	2	2	no	—
19	C₂×M₁₆×C₁₂₇	—	2	1	no	—
20	C₂×M₁₆×C₁₂₇	C₄	1	4	yes	complemented
21	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
22	C₂²×C₈×C₁₂₇	—	2	1	no	—
23	C₂²×Dic₁₂₇	C₂×C₄	1	4	yes	—
24	#14523232	—	2	4	no	—
25	#14523232	—	2	2	no	—
26	#14523232	C₂×C₄	1	4	yes	—
27	C₂×C₁₂₇⋊C₈	—	2	4	no	—
28	C₂×C₁₂₇⋊C₈	—	2	2	no	—
29	C₄⋊Dic₁₂₇	—	2	2	no	—
30	C₄⋊Dic₁₂₇	—	2	2	no	—
31	C₂²⋊Dic₁₂₇	—	2	2	no	—
32	C₂²⋊Dic₁₂₇	—	2	2	no	—
33	C₄×Dic₁₂₇	—	2	2	no	—
34	C₄×Dic₁₂₇	—	2	2	no	—
35	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
40	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₂₇	Q₈	1	2	yes	—
45	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
46	M₁₆×C₁₂₇	—	2	4	no	—
47	M₁₆×C₁₂₇	—	2	2	no	—
48	M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
49	C₂×C₈×C₁₂₇	—	2	4	no	—
50	C₂×C₈×C₁₂₇	—	2	2	no	—
51	C₂×Dic₁₂₇	—	2	4	no	—
52	C₂×Dic₁₂₇	—	4	4	no	—
53	C₁₂₇⋊C₈	—	2	4	no	—
54	C₂³×C₁₂₇	—	4	2	no	—
55	C₂³×C₁₂₇	C₄²	1	2	yes	—
56	C₂³×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂³×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
59	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
61	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
67	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
78	C₈×C₁₂₇	—	2	4	no	—
79	C₂²×C₁₂₇	—	2	2	no	—
80	C₂²×C₁₂₇	—	2	2	no	—
81	C₂²×C₁₂₇	—	2	2	no	—
82	C₂²×C₁₂₇	—	4	2	no	—
83	C₂²×C₁₂₇	—	4	2	no	—
84	C₂²×C₁₂₇	#95	1	2	yes	—
85	C₂²×C₁₂₇	#95	1	2	yes	—
86	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
87	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₁₂₇	#95	1	2	yes	—
90	C₄×C₁₂₇	#95	1	2	yes	—
91	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	Dic₁₂₇	—	4	4	no	—
96	C₂×C₁₂₇	—	2	2	no	—
97	C₂×C₁₂₇	—	2	2	no	—
98	C₂×C₁₂₇	—	2	1	no	—
99	C₂×C₁₂₇	—	2	1	no	—
100	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
101	C₂×C₁₂₇	#413	1	2	yes	—
102	#1742	—	127	1	no	maximal, Sylow
103	C₁₂₇	#1742	1	1	yes	characteristic, complemented, Sylow
104	C₂²×M₁₆	—	127	1	no	—
105	C₂²×M₁₆	D₁₂₇	1	1	yes	characteristic
106	C₂×#117	—	127	1	no	—
107	#343	—	127	4	no	—
108	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂×M₁₆	—	127	4	no	—
112	C₂×M₁₆	—	127	2	no	—
113	C₂×M₁₆	—	254	2	no	—
114	C₂×M₁₆	—	254	1	no	—
115	C₂×M₁₆	Dic₁₂₇	1	4	yes	complemented
116	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	254	1	no	—
119	#117	—	254	1	no	—
120	#117	—	254	1	no	—
121	#117	—	254	1	no	—
122	#117	—	254	1	no	—
123	C₂×C₄⋊C₄	—	127	2	no	—
124	C₂×C₂²⋊C₄	—	127	2	no	—
125	C₂×C₄²	—	127	2	no	—
126	C₂⁴	C₄×D₁₂₇	1	1	yes	characteristic
127	C₂²×C₄	—	127	4	no	—
128	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
129	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
130	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
131	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
132	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
133	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
134	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
135	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
136	C₂²×C₄	Dic₂₅₄	1	2	yes	—
137	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
138	M₁₆	—	2	4	no	—
139	M₁₆	—	2	2	no	—
140	M₁₆	—	127	4	no	—
141	M₁₆	—	254	4	no	—
142	M₁₆	—	254	2	no	—
143	M₁₆	C₂×Dic₁₂₇	1	4	yes	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	254	4	no	—
147	C₂×C₈	—	254	2	no	—
148	C₄⋊C₄	—	254	2	no	—
149	C₄⋊C₄	—	254	2	no	—
150	C₂²⋊C₄	—	254	2	no	—
151	C₂²⋊C₄	—	254	2	no	—
152	C₄²	—	254	2	no	—
153	C₄²	—	254	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	C₄×Dic₁₂₇	1	2	yes	—
156	C₂³	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
157	C₂³	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
158	C₂³	Dic₁₂₇⋊C₄	1	2	yes	—
159	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
160	C₂×C₄	—	508	4	no	—
161	C₂×C₄	—	254	4	no	—
162	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
163	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
164	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
165	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
166	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
167	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
168	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
169	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
170	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
171	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
172	C₂×C₄	C₄⋊Dic₁₂₇	1	2	yes	—
173	C₂×C₄	C₄⋊Dic₁₂₇	1	2	yes	—
174	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
175	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
176	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
177	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
178	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	254	4	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	4	2	no	—
187	C₂²	C₁₂₇⋊#95	1	2	yes	—
188	C₂²	C₁₂₇⋊#95	1	2	yes	—
189	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
190	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
191	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
192	C₄	—	508	4	no	—
193	C₄	C₁₂₇⋊#95	1	2	yes	—
194	C₄	C₁₂₇⋊#95	1	2	yes	—
195	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
196	C₄	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
197	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
198	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	—	2	2	no	—
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	C₂×C₁₂₇⋊#95	1	1	yes	characteristic
204	C₂	#22549858	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	(C₂×D₁₂₇)⋊C₄
commutator	C₂²×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂²×M₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂²×M₁₆	D₁₂₇
hypocenter	C₁₂₇	#1742
LCS	G ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22549644	C₂	semidirect
7. C₂×#14523232	C₄	semidirect
20. C₂×M₁₆×C₁₂₇	C₄	semidirect
103. C₁₂₇	#1742	semidirect
115. C₂×M₁₆	Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523232	2
C₂×#16861168	2
#22549644	2
C₂²×M₁₆×C₁₂₇	2
#1742	127