Subgroups — #30647147

Order 16256.

564 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₂₇	C₂	1	2	yes	maximal
2	C₂×#16861166	C₂	1	1	yes	maximal, characteristic
3	#22549833	C₂	1	4	yes	maximal, complemented
4	#22549830	C₂	1	4	yes	maximal, complemented
5	C₄×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
6	M₁₆×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
7	C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×#14523232	C₂²	1	2	yes	—
10	C₂×#14523231	C₂²	1	4	yes	—
11	M₁₆×D₁₂₇	C₄	1	16	yes	complemented
12	C₂×C₈×D₁₂₇	C₂²	1	4	yes	—
13	#16861166	C₄	1	4	yes	—
14	#16861166	C₂²	1	2	yes	—
15	#16861166	C₂²	1	2	yes	—
16	C₂×Dic₁₂₇⋊C₄	C₄	1	2	yes	—
17	C₂×(C₂×D₁₂₇)⋊C₄	C₄	1	2	yes	—
18	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
19	#16861120	C₂²	1	2	yes	complemented
20	#16861119	C₂²	1	4	yes	complemented
21	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	complemented
22	C₈×Dic₁₂₇	C₂²	1	4	yes	complemented
23	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
25	#97×C₁₂₇	C₂²	1	2	yes	complemented
26	C₄×C₈×C₁₂₇	C₂²	1	2	yes	complemented
27	C₂³×D₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
29	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₁₂₇	C₂×C₄	1	8	yes	—
32	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
33	#14523232	C₂×C₄	1	8	yes	—
34	#14523231	C₂×C₄	1	16	yes	complemented
35	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
36	C₈×D₁₂₇	C₂×C₄	1	16	yes	complemented
37	Dic₁₂₇⋊C₄	C₂×C₄	1	4	yes	—
38	Dic₁₂₇⋊C₄	C₂×C₄	1	4	yes	—
39	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	4	yes	—
40	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₂₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
45	M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
46	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
47	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
48	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
49	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
50	C₂²×D₁₂₇	—	4	2	no	—
51	C₂²×D₁₂₇	C₄²	1	4	yes	—
52	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
53	C₂×Dic₁₂₇	—	4	2	no	—
54	C₂×Dic₁₂₇	C₄²	1	4	yes	—
55	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
56	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
57	C₄×D₁₂₇	C₄²	1	8	yes	—
58	C₄×D₁₂₇	C₂²×C₄	1	4	yes	—
59	C₄×D₁₂₇	C₂²×C₄	1	4	yes	—
60	C₁₂₇⋊C₈	C₂²×C₄	1	8	yes	—
61	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	—	4	2	no	—
63	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
67	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
68	C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
69	C₂²×C₁₂₇	—	2	2	no	—
70	C₂²×C₁₂₇	C₂×C₄²	1	2	yes	—
71	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×D₁₂₇	—	2	4	no	—
73	C₂×D₁₂₇	—	4	4	no	—
74	C₂×D₁₂₇	C₂×C₄²	1	4	yes	—
75	C₄×C₁₂₇	—	2	2	no	—
76	C₄×C₁₂₇	—	2	2	no	—
77	C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
78	C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
79	Dic₁₂₇	—	2	4	no	—
80	Dic₁₂₇	C₂×C₄²	1	4	yes	—
81	C₂×C₁₂₇	—	2	2	no	—
82	C₂×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
83	C₂×C₁₂₇	#568	1	2	yes	—
84	D₁₂₇	—	2	4	no	—
85	#2873	—	127	1	no	maximal, Sylow
86	C₁₂₇	#2873	1	1	yes	characteristic, complemented, Sylow
87	C₂²×M₁₆	—	127	2	no	—
88	C₂×#117	—	127	1	no	—
89	#405	—	127	4	no	—
90	#405	—	127	4	no	—
91	C₄×M₁₆	—	127	2	no	—
92	C₄×M₁₆	—	127	1	no	—
93	C₄×M₁₆	D₁₂₇	1	1	yes	characteristic, complemented
94	C₂³×C₄	—	127	1	no	—
95	C₂×M₁₆	—	127	2	no	—
96	C₂×M₁₆	—	127	16	no	—
97	C₂×M₁₆	—	127	4	no	—
98	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
99	C₂²×C₈	—	127	4	no	—
100	#117	—	127	4	no	—
101	#117	—	127	2	no	—
102	#117	—	127	2	no	—
103	C₂×C₄⋊C₄	—	127	2	no	—
104	C₂×C₂²⋊C₄	—	127	2	no	—
105	C₂×C₄²	—	127	1	no	—
106	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
107	#97	—	127	4	no	—
108	#97	—	127	2	no	—
109	#97	C₂×D₁₂₇	1	2	yes	complemented
110	C₄×C₈	—	127	4	no	—
111	C₄×C₈	—	127	2	no	—
112	C₄×C₈	C₂×D₁₂₇	1	2	yes	complemented
113	C₂⁴	—	127	1	no	—
114	C₂²×C₄	—	127	1	no	—
115	C₂²×C₄	—	127	2	no	—
116	C₂²×C₄	—	127	8	no	—
117	C₂²×C₄	—	127	2	no	—
118	C₂²×C₄	—	127	2	no	—
119	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
120	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
121	M₁₆	—	127	8	no	—
122	M₁₆	—	127	16	no	—
123	M₁₆	C₄×D₁₂₇	1	8	yes	—
124	C₂×C₈	—	127	4	no	—
125	C₂×C₈	—	127	16	no	—
126	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
127	C₄⋊C₄	—	127	4	no	—
128	C₄⋊C₄	—	127	4	no	—
129	C₂²⋊C₄	—	127	4	no	—
130	C₂²⋊C₄	—	127	4	no	—
131	C₄²	—	127	2	no	—
132	C₄²	—	127	2	no	—
133	C₄²	C₄×D₁₂₇	1	2	yes	—
134	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
135	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
136	C₂³	—	127	4	no	—
137	C₂³	—	127	2	no	—
138	C₂³	—	508	2	no	—
139	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	127	2	no	—
142	C₂×C₄	—	127	4	no	—
143	C₂×C₄	—	127	4	no	—
144	C₂×C₄	—	127	4	no	—
145	C₂×C₄	—	127	8	no	—
146	C₂×C₄	—	127	4	no	—
147	C₂×C₄	—	508	2	no	—
148	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
149	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
150	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
151	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
152	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
153	C₈	—	127	8	no	—
154	C₈	C₂×C₄×D₁₂₇	1	8	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	—	127	4	no	—
157	C₂²	—	508	4	no	—
158	C₂²	—	254	4	no	—
159	C₂²	C₄²×D₁₂₇	1	2	yes	—
160	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	2	no	—
163	C₄	—	127	4	no	—
164	C₄	—	254	4	no	—
165	C₄	C₄²×D₁₂₇	1	2	yes	—
166	C₄	C₂²×C₄×D₁₂₇	1	2	yes	—
167	C₂	—	2	2	no	—
168	C₂	—	254	4	no	—
169	C₂	C₂×C₄²×D₁₂₇	1	1	yes	characteristic
170	C₂	#22550642	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₄²
Fitting	C₄×M₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂³×C₄
prime radical	C₂³×D₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₄×M₁₆	D₁₂₇
hypocenter	C₁₂₇	#2873
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22549833	C₂	semidirect
4. #22549830	C₂	semidirect
5. C₄×#14523232	C₂	semidirect
6. M₁₆×Dic₁₂₇	C₂	semidirect
7. C₄×M₁₆×C₁₂₇	C₂	semidirect
11. M₁₆×D₁₂₇	C₄	semidirect
19. #16861120	C₂²	semidirect
20. #16861119	C₂²	semidirect
21. C₄×C₁₂₇⋊C₈	C₂²	semidirect
22. C₈×Dic₁₂₇	C₂²	semidirect
25. #97×C₁₂₇	C₂²	semidirect
26. C₄×C₈×C₁₂₇	C₂²	semidirect
34. #14523231	C₂×C₄	semidirect
36. C₈×D₁₂₇	C₂×C₄	semidirect
86. C₁₂₇	#2873	semidirect
93. C₄×M₁₆	D₁₂₇	semidirect
109. #97	C₂×D₁₂₇	semidirect
112. C₄×C₈	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₂₇	2
C₂×#16861166	2
#22549833	2
#22549830	2
C₄×#14523232	2
M₁₆×Dic₁₂₇	2
C₄×M₁₆×C₁₂₇	2
#2873	127