Subgroups — #30647155

Order 16256.

556 conjugacy classes of subgroups.

The subgroup lattice has 174 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	2	yes	maximal
2	C₄×#14523248	C₂	1	1	yes	maximal, characteristic
3	#22549833	C₂	1	4	yes	maximal, complemented
4	#22549831	C₂	1	2	yes	maximal, complemented
5	C₄×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549828	C₂	1	2	yes	maximal, complemented
7	#97×D₁₂₇	C₂	1	2	yes	maximal, complemented
8	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523248	C₂²	1	1	yes	characteristic
10	#16861215	C₄	1	16	yes	complemented
11	C₂×#14523232	C₂²	1	2	yes	—
12	C₂×#14523231	C₂²	1	4	yes	—
13	C₂×C₈×D₁₂₇	C₂²	1	4	yes	—
14	C₄×Dic₂₅₄	C₄	1	2	yes	—
15	C₄×D₅₀₈	C₄	1	2	yes	—
16	C₄×C₂²⋊D₁₂₇	C₄	1	4	yes	—
17	#16861168	C₂²	1	1	yes	characteristic
18	#16861166	C₂²	1	2	yes	—
19	C₄²×D₁₂₇	C₂²	1	2	yes	—
20	#16861120	C₂²	1	2	yes	—
21	#16861119	C₂²	1	4	yes	—
22	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
23	C₈×Dic₁₂₇	C₂²	1	4	yes	—
24	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
26	#97×C₁₂₇	C₂²	1	1	yes	characteristic
27	#97×C₁₂₇	C₂²	1	1	yes	characteristic
28	#97×C₁₂₇	C₂²	1	2	yes	—
29	#14523248	C₂×C₄	1	8	yes	—
30	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
31	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₁₂₇	C₂×C₄	1	4	yes	—
34	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
35	#14523232	C₂×C₄	1	8	yes	complemented
36	#14523231	C₂×C₄	1	16	yes	complemented
37	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
38	C₈×D₁₂₇	C₂×C₄	1	16	yes	complemented
39	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
40	Dic₁₂₇⋊C₄	C₂×C₄	1	4	yes	—
41	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
42	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	4	yes	—
43	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	—
44	C₄×Dic₁₂₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
48	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
49	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
50	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
51	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
52	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
53	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
54	Dic₂₅₄	C₄²	1	4	yes	—
55	D₅₀₈	C₄²	1	4	yes	—
56	C₂²⋊D₁₂₇	C₄²	1	8	yes	—
57	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
58	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
59	C₄×D₁₂₇	—	4	4	no	—
60	C₄×D₁₂₇	C₂²×C₄	1	8	yes	—
61	C₁₂₇⋊C₈	C₂²×C₄	1	8	yes	—
62	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	—	4	2	no	—
64	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
69	C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
70	C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₁₂₇	—	2	2	no	—
72	C₂²×C₁₂₇	C₂×C₄²	1	2	yes	—
73	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₁₂₇	—	2	4	no	—
75	C₂×D₁₂₇	C₂×C₄²	1	4	yes	—
76	C₄×C₁₂₇	—	2	2	no	—
77	C₄×C₁₂₇	—	2	2	no	—
78	C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
79	C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
80	Dic₁₂₇	—	2	4	no	—
81	Dic₁₂₇	C₂×C₄²	1	4	yes	—
82	C₂×C₁₂₇	—	2	2	no	—
83	C₂×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
84	C₂×C₁₂₇	#568	1	2	yes	—
85	D₁₂₇	—	2	4	no	—
86	#2875	—	127	1	no	maximal, Sylow
87	C₁₂₇	#2875	1	1	yes	characteristic, complemented, Sylow
88	C₂×#131	—	127	2	no	—
89	C₄×#41	—	127	1	no	—
90	#405	—	127	2	no	—
91	#405	—	127	4	no	—
92	C₄×M₁₆	—	127	1	no	—
93	C₄×M₁₆	—	127	2	no	—
94	C₂×#97	—	127	2	no	—
95	C₂×#97	D₁₂₇	1	1	yes	characteristic, complemented
96	C₂×#41	—	127	1	no	—
97	#131	—	127	16	no	—
98	C₂×M₁₆	—	127	2	no	—
99	C₂×M₁₆	—	127	4	no	—
100	C₂²×C₈	—	127	4	no	—
101	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
102	C₄×Q₈	—	127	2	no	—
103	C₄×D₄	—	127	4	no	—
104	C₄×D₄	—	127	2	no	—
105	#117	—	127	1	no	—
106	#117	—	127	2	no	—
107	C₂×C₄²	—	127	2	no	—
108	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
109	#97	—	127	4	no	—
110	#97	—	127	2	no	—
111	#97	C₂×D₁₂₇	1	1	yes	characteristic
112	#97	C₂×D₁₂₇	1	1	yes	characteristic
113	#97	C₂×D₁₂₇	1	2	yes	—
114	C₄×C₈	—	127	4	no	—
115	C₄×C₈	—	127	2	no	—
116	#41	—	127	8	no	—
117	C₂×Q₈	—	127	1	no	—
118	C₂×D₄	—	127	2	no	—
119	C₂×D₄	—	127	1	no	—
120	C₂²×C₄	—	127	2	no	—
121	C₂²×C₄	—	127	4	no	—
122	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
123	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
124	M₁₆	—	127	8	no	—
125	M₁₆	—	127	16	no	—
126	C₂×C₈	—	127	4	no	—
127	C₂×C₈	—	127	16	no	—
128	C₂×C₈	C₄×D₁₂₇	1	8	yes	complemented
129	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
130	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
131	C₄⋊C₄	—	127	4	no	—
132	C₄⋊C₄	—	127	2	no	—
133	C₂²⋊C₄	—	127	2	no	—
134	C₂²⋊C₄	—	127	4	no	—
135	C₄²	—	127	4	no	—
136	C₄²	—	127	2	no	—
137	C₄²	C₄×D₁₂₇	1	2	yes	—
138	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
139	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
140	C₂³	—	127	2	no	—
141	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
142	Q₈	—	127	4	no	—
143	D₄	—	127	8	no	—
144	D₄	—	127	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	127	2	no	—
147	C₂×C₄	—	127	8	no	—
148	C₂×C₄	—	127	4	no	—
149	C₂×C₄	—	508	4	no	—
150	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
152	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
153	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
154	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
155	C₈	—	127	8	no	—
156	C₈	C₂×C₄×D₁₂₇	1	4	yes	—
157	C₈	C₂×C₄×D₁₂₇	1	4	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	127	4	no	—
160	C₂²	—	254	4	no	—
161	C₂²	C₄²×D₁₂₇	1	2	yes	—
162	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	127	4	no	—
166	C₄	—	254	4	no	—
167	C₄	C₄²×D₁₂₇	1	2	yes	—
168	C₄	C₂²×C₄×D₁₂₇	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	—	254	4	no	—
171	C₂	C₂×C₄²×D₁₂₇	1	1	yes	characteristic
172	C₂	#22550642	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₄²
Fitting	C₂×#97×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂³×C₄
prime radical	C₂×#14523248	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#97	D₁₂₇
hypocenter	C₁₂₇	#2875
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
3. #22549833	C₂	semidirect
4. #22549831	C₂	semidirect
5. C₄×#14523232	C₂	semidirect
6. #22549828	C₂	semidirect
7. #97×D₁₂₇	C₂	semidirect
8. C₂×#97×C₁₂₇	C₂	semidirect
10. #16861215	C₄	semidirect
35. #14523232	C₂×C₄	semidirect
36. #14523231	C₂×C₄	semidirect
38. C₈×D₁₂₇	C₂×C₄	semidirect
47. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
87. C₁₂₇	#2875	semidirect
95. C₂×#97	D₁₂₇	semidirect
128. C₂×C₈	C₄×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
C₄×#14523248	2
#22549833	2
#22549831	2
C₄×#14523232	2
#22549828	2
#97×D₁₂₇	2
C₂×#97×C₁₂₇	2
#2875	127