Subgroups — #30638426

Order 16256.

416 conjugacy classes of subgroups.

The subgroup lattice has 166 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	#22549630	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523232	—	2	2	no	—
6	C₂×#14523232	—	2	2	no	—
7	C₂×#14523232	C₄	1	4	yes	complemented
8	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	#16861168	—	2	2	no	—
10	#16861168	—	2	2	no	—
11	C₂×C₄⋊Dic₁₂₇	C₄	1	2	yes	—
12	C₂×C₂²⋊Dic₁₂₇	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	—	2	2	no	—
16	C₂²×C₈×C₁₂₇	—	2	2	no	—
17	C₂²×C₈×C₁₂₇	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
19	C₂²×Dic₁₂₇	C₂×C₄	1	4	yes	—
20	#14523232	—	2	8	no	—
21	C₂×C₁₂₇⋊C₈	—	2	2	no	—
22	C₂×C₁₂₇⋊C₈	—	2	2	no	—
23	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
24	C₄⋊Dic₁₂₇	—	2	4	no	—
25	C₂²⋊Dic₁₂₇	—	2	4	no	—
26	C₄×Dic₁₂₇	—	2	2	no	—
27	C₄×Dic₁₂₇	—	2	2	no	—
28	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
33	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
34	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
35	C₂²×C₄×C₁₂₇	Q₈	1	2	yes	—
36	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	—	2	8	no	—
38	C₂×C₈×C₁₂₇	—	2	2	no	—
39	C₂×C₈×C₁₂₇	—	2	2	no	—
40	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
41	C₂×Dic₁₂₇	—	2	4	no	—
42	C₂×Dic₁₂₇	—	2	8	no	—
43	C₁₂₇⋊C₈	—	2	4	no	—
44	C₂³×C₁₂₇	—	2	4	no	—
45	C₂³×C₁₂₇	C₄²	1	2	yes	—
46	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
48	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
52	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
62	C₈×C₁₂₇	—	2	4	no	—
63	C₂²×C₁₂₇	—	2	4	no	—
64	C₂²×C₁₂₇	—	2	4	no	—
65	C₂²×C₁₂₇	—	2	4	no	—
66	C₂²×C₁₂₇	#95	1	4	yes	—
67	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₁₂₇	#131	1	4	yes	—
71	C₄×C₁₂₇	#95	1	4	yes	—
72	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	Dic₁₂₇	—	4	4	no	—
77	C₂×C₁₂₇	—	2	4	no	—
78	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
79	C₂×C₁₂₇	#405	1	2	yes	—
80	C₂×C₁₂₇	#408	1	2	yes	—
81	C₂×C₁₂₇	#424	1	2	yes	—
82	#1729	—	127	1	no	maximal, Sylow
83	C₁₂₇	#1729	1	1	yes	characteristic, complemented, Sylow
84	C₂²×M₁₆	—	127	1	no	—
85	C₂³×C₈	D₁₂₇	1	1	yes	characteristic
86	C₂×#117	—	127	1	no	—
87	#336	—	127	4	no	—
88	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
89	C₂×M₁₆	—	127	4	no	—
90	C₂×M₁₆	—	254	2	no	—
91	C₂×M₁₆	—	254	2	no	—
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	—	127	2	no	—
95	C₂²×C₈	Dic₁₂₇	1	4	yes	complemented
96	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
97	#117	—	254	2	no	—
98	#117	—	254	2	no	—
99	C₂×C₄⋊C₄	—	127	2	no	—
100	C₂×C₂²⋊C₄	—	127	2	no	—
101	C₂×C₄²	—	127	2	no	—
102	C₂⁴	C₄×D₁₂₇	1	1	yes	characteristic
103	C₂²×C₄	—	127	4	no	—
104	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
105	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
106	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
107	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
108	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
109	C₂²×C₄	D₅₀₈	1	2	yes	—
110	C₂²×C₄	Dic₂₅₄	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
112	M₁₆	—	254	8	no	—
113	C₂×C₈	—	2	8	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	127	4	no	—
117	C₂×C₈	—	254	2	no	—
118	C₂×C₈	—	254	2	no	—
119	C₂×C₈	C₂×Dic₁₂₇	1	4	yes	—
120	C₄⋊C₄	—	254	4	no	—
121	C₂²⋊C₄	—	254	4	no	—
122	C₄²	—	254	2	no	—
123	C₄²	—	254	2	no	—
124	C₂³	—	2	4	no	—
125	C₂³	C₄×Dic₁₂₇	1	2	yes	—
126	C₂³	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
127	C₂³	Dic₁₂₇⋊C₄	1	2	yes	—
128	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
129	C₂×C₄	—	254	8	no	—
130	C₂×C₄	—	254	4	no	—
131	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
132	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
133	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	4	yes	—
134	C₂×C₄	C₂²⋊Dic₁₂₇	1	4	yes	—
135	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
136	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
137	C₂×C₄	C₄⋊Dic₁₂₇	1	4	yes	—
138	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
139	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
142	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
143	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	254	4	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	2	4	no	—
149	C₂²	C₁₂₇⋊#95	1	4	yes	—
150	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
151	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
152	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
153	C₂²	#16861215	1	4	yes	—
154	C₄	—	508	4	no	—
155	C₄	C₁₂₇⋊#95	1	4	yes	—
156	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
157	C₄	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
158	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
159	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
160	C₂	—	2	4	no	—
161	C₂	C₂×C₁₂₇⋊#95	1	1	yes	characteristic
162	C₂	#22549831	1	2	yes	—
163	C₂	#22549842	1	2	yes	—
164	C₂	#22549902	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂×C₄²
Fitting	C₂³×C₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂³×C₈	D₁₂₇
hypocenter	C₁₂₇	#1729
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. #22549630	C₂	semidirect
7. C₂×#14523232	C₄	semidirect
17. C₂²×C₈×C₁₂₇	C₄	semidirect
83. C₁₂₇	#1729	semidirect
95. C₂²×C₈	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
#22549630	2
C₂³×C₈×C₁₂₇	2
#1729	127