Subgroups — #30638451

Order 16256.

448 conjugacy classes of subgroups.

The subgroup lattice has 188 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	C₂²×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549638	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₂₇	—	2	1	no	—
6	C₂×#14523232	—	2	2	no	—
7	C₂×#14523232	—	2	2	no	—
8	C₂×#14523232	C₄	1	4	yes	complemented
9	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
11	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
12	C₂×C₄⋊Dic₁₂₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	#117×C₁₂₇	—	2	2	no	—
16	#117×C₁₂₇	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
19	C₂×C₄²×C₁₂₇	—	2	1	no	—
20	C₂²×Dic₁₂₇	—	2	4	no	—
21	C₂²×Dic₁₂₇	—	2	2	no	—
22	C₂²×Dic₁₂₇	—	4	4	no	—
23	#14523232	—	2	8	no	—
24	C₂×C₁₂₇⋊C₈	—	2	2	no	—
25	C₂×C₁₂₇⋊C₈	—	2	2	no	—
26	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	complemented
27	C₄⋊Dic₁₂₇	—	2	4	no	—
28	C₄⋊Dic₁₂₇	—	2	2	no	—
29	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
30	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	—	2	2	no	—
32	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
37	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
38	C₂²×C₄×C₁₂₇	Q₈	1	2	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₂₇	—	2	2	no	—
41	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	complemented
42	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
43	C₄²×C₁₂₇	—	2	4	no	—
44	C₂×Dic₁₂₇	—	2	4	no	—
45	C₂×Dic₁₂₇	—	4	8	no	—
46	C₂×Dic₁₂₇	—	4	4	no	—
47	C₁₂₇⋊C₈	—	2	4	no	—
48	C₂³×C₁₂₇	—	2	2	no	—
49	C₂³×C₁₂₇	—	2	2	no	—
50	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
51	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
52	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
53	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	—	2	4	no	—
55	C₂×C₄×C₁₂₇	—	4	2	no	—
56	C₂×C₄×C₁₂₇	—	4	2	no	—
57	C₂×C₄×C₁₂₇	C₄²	1	4	yes	—
58	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
60	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
63	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
64	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₁₂₇	—	2	4	no	—
71	C₂²×C₁₂₇	—	2	2	no	—
72	C₂²×C₁₂₇	—	2	2	no	—
73	C₂²×C₁₂₇	—	2	4	no	—
74	C₂²×C₁₂₇	#95	1	4	yes	—
75	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₂²×C₁₂₇	#137	1	2	yes	—
80	C₄×C₁₂₇	—	4	4	no	—
81	C₄×C₁₂₇	#95	1	4	yes	—
82	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
83	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	Dic₁₂₇	—	4	4	no	—
87	C₂×C₁₂₇	—	2	4	no	—
88	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
89	C₂×C₁₂₇	#417	1	2	yes	—
90	C₂×C₁₂₇	#418	1	1	yes	characteristic
91	C₂×C₁₂₇	#419	1	1	yes	characteristic
92	C₂×C₁₂₇	#428	1	2	yes	—
93	#1736	—	127	1	no	maximal, Sylow
94	C₁₂₇	#1736	1	1	yes	characteristic, complemented, Sylow
95	C₂²×M₁₆	—	127	1	no	—
96	C₂×#117	D₁₂₇	1	1	yes	characteristic
97	C₂²×C₄⋊C₄	—	127	1	no	—
98	#340	—	127	4	no	—
99	C₂³×C₄	—	254	1	no	—
100	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
101	C₂×M₁₆	—	127	4	no	—
102	C₂×M₁₆	—	254	2	no	—
103	C₂×M₁₆	—	254	2	no	—
104	C₂²×C₈	—	127	2	no	—
105	#117	—	2	2	no	—
106	#117	Dic₁₂₇	1	4	yes	complemented
107	C₂×C₄⋊C₄	—	127	4	no	—
108	C₂×C₄⋊C₄	—	127	2	no	—
109	C₂×C₄⋊C₄	—	254	1	no	—
110	C₂×C₄⋊C₄	—	254	2	no	—
111	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
112	C₂×C₂²⋊C₄	—	2	1	no	—
113	C₂×C₄²	—	2	1	no	—
114	C₂⁴	C₂²⋊D₁₂₇	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	508	4	no	—
117	C₂²×C₄	—	254	4	no	—
118	C₂²×C₄	—	254	2	no	—
119	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
120	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
121	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
122	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
123	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
124	C₂²×C₄	D₅₀₈	1	2	yes	—
125	C₂²×C₄	Dic₂₅₄	1	2	yes	—
126	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
127	M₁₆	—	254	8	no	—
128	C₂×C₈	—	127	4	no	—
129	C₂×C₈	—	254	2	no	—
130	C₂×C₈	—	254	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	127	4	no	—
133	C₄⋊C₄	—	254	4	no	—
134	C₄⋊C₄	—	254	2	no	—
135	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	complemented
136	C₂²⋊C₄	—	2	4	no	—
137	C₄²	—	2	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
141	C₂³	C₂²⋊Dic₁₂₇	1	2	yes	—
142	C₂³	Dic₁₂₇⋊C₄	1	2	yes	—
143	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	508	8	no	—
148	C₂×C₄	—	508	4	no	—
149	C₂×C₄	—	254	4	no	—
150	C₂×C₄	C₄×Dic₁₂₇	1	4	yes	—
151	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
152	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	4	yes	—
153	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
154	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
155	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
156	C₂×C₄	C₄⋊Dic₁₂₇	1	4	yes	—
157	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
159	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
160	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
161	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
162	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
163	C₈	—	254	4	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	C₁₂₇⋊#95	1	4	yes	—
169	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
170	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
171	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
172	C₂²	#16861234	1	2	yes	—
173	C₂²	#16861239	1	2	yes	—
174	C₄	—	4	4	no	—
175	C₄	—	508	4	no	—
176	C₄	C₁₂₇⋊#95	1	4	yes	—
177	C₄	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
178	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
179	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
180	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
181	C₂	—	2	4	no	—
182	C₂	C₂×C₁₂₇⋊#95	1	1	yes	characteristic
183	C₂	#22549877	1	2	yes	—
184	C₂	#22549880	1	1	yes	characteristic
185	C₂	#22549884	1	1	yes	characteristic
186	C₂	#22549919	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#117×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂×D₄
prime radical	C₂⁴×C₁₂₇	D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#117	D₁₂₇
hypocenter	C₁₂₇	#1736
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #22549638	C₂	semidirect
8. C₂×#14523232	C₄	semidirect
12. C₂×C₄⋊Dic₁₂₇	C₄	semidirect
16. #117×C₁₂₇	C₄	semidirect
26. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
41. C₄⋊C₄×C₁₂₇	C₂×C₄	semidirect
94. C₁₂₇	#1736	semidirect
106. #117	Dic₁₂₇	semidirect
135. C₄⋊C₄	C₂×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
C₂²×C₄⋊Dic₁₂₇	2
#22549638	2
C₂×#117×C₁₂₇	2
#1736	127