Subgroups — C₂×#22549903 #30647289

Order 16256.

512 conjugacy classes of subgroups.

The subgroup lattice has 140 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	C₂³×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22549903	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	4	yes	maximal, complemented
5	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523232	—	2	4	no	—
7	C₂×#14523232	C₂²	1	4	yes	—
8	C₂²×C₁₂₇⋊C₈	—	2	4	no	—
9	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
10	C₂²×C₁₂₇⋊C₈	C₂²	1	4	yes	—
11	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
12	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	16	yes	complemented
13	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	#117×C₁₂₇	C₄	1	4	yes	—
15	#117×C₁₂₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
17	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
18	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
19	#14523232	—	2	8	no	—
20	C₂×C₁₂₇⋊C₈	—	2	4	no	—
21	C₂×C₁₂₇⋊C₈	—	2	8	no	—
22	C₂×C₁₂₇⋊C₈	—	2	4	no	—
23	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
24	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
25	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
30	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
31	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
34	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
35	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
36	C₄²×C₁₂₇	C₂³	1	4	yes	—
37	C₁₂₇⋊C₈	—	2	4	no	—
38	C₁₂₇⋊C₈	—	2	4	no	—
39	C₂³×C₁₂₇	—	2	4	no	—
40	C₂³×C₁₂₇	C₄⋊C₄	1	4	yes	—
41	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
42	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	—	2	8	no	—
44	C₂×C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
45	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
46	C₂×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
47	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
51	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
52	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
54	C₂²×C₁₂₇	—	2	4	no	—
55	C₂²×C₁₂₇	—	2	8	no	—
56	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
57	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
58	C₂²×C₁₂₇	#131	1	4	yes	—
59	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
60	C₄×C₁₂₇	—	2	8	no	—
61	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
62	C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
63	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₁₂₇	—	2	4	no	—
66	C₂×C₁₂₇	#424	1	4	yes	complemented
67	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₁₂₇	C₂×#131	1	2	yes	—
69	C₂×#424	—	127	1	no	maximal, Sylow
70	C₁₂₇	C₂×#424	1	1	yes	characteristic, complemented, Sylow
71	C₂²×M₁₆	—	127	1	no	—
72	C₂³×C₈	—	127	1	no	—
73	C₂×#117	D₁₂₇	1	1	yes	characteristic
74	#424	—	127	8	no	—
75	C₂×C₄⋊C₈	—	127	4	no	—
76	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
77	C₂×M₁₆	—	127	4	no	—
78	C₂×M₁₆	—	254	4	no	—
79	C₂²×C₈	—	127	4	no	—
80	C₂²×C₈	—	127	2	no	—
81	C₂²×C₈	—	127	2	no	—
82	C₂²×C₈	—	254	4	no	—
83	#117	Dic₁₂₇	1	4	yes	—
84	#117	C₂×D₁₂₇	1	4	yes	—
85	C₂×C₄⋊C₄	Dic₁₂₇	1	2	yes	—
86	C₂×C₂²⋊C₄	Dic₁₂₇	1	2	yes	—
87	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
88	C₄⋊C₈	—	127	16	no	—
89	C₂⁴	C₂×Dic₁₂₇	1	1	yes	characteristic
90	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
91	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
93	C₂²×C₄	D₅₀₈	1	4	yes	—
94	C₂²×C₄	Dic₂₅₄	1	4	yes	—
95	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
97	M₁₆	—	254	8	no	—
98	C₂×C₈	—	127	4	no	—
99	C₂×C₈	—	127	4	no	—
100	C₂×C₈	—	254	4	no	—
101	C₂×C₈	—	254	8	no	—
102	C₂×C₈	—	254	4	no	—
103	C₄⋊C₄	C₂×Dic₁₂₇	1	8	yes	—
104	C₂²⋊C₄	C₂×Dic₁₂₇	1	8	yes	—
105	C₄²	C₂×Dic₁₂₇	1	4	yes	—
106	C₄²	C₂²×D₁₂₇	1	4	yes	—
107	C₂³	—	2	4	no	—
108	C₂³	C₄⋊Dic₁₂₇	1	4	yes	—
109	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
110	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	C₄⋊Dic₁₂₇	1	8	yes	—
113	C₂×C₄	C₄⋊Dic₁₂₇	1	4	yes	—
114	C₂×C₄	C₂²×Dic₁₂₇	1	8	yes	—
115	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
116	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
117	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
118	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
119	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
120	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
121	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
122	C₈	—	254	4	no	—
123	C₈	—	254	4	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	8	no	—
126	C₂²	C₂×C₄⋊Dic₁₂₇	1	4	yes	—
127	C₂²	C₂×C₄⋊Dic₁₂₇	1	2	yes	—
128	C₂²	#16861216	1	4	yes	—
129	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
130	C₄	—	2	8	no	—
131	C₄	C₂×C₄⋊Dic₁₂₇	1	2	yes	—
132	C₄	C₂×C₄⋊Dic₁₂₇	1	4	yes	—
133	C₄	C₂²×D₅₀₈	1	1	yes	characteristic
134	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
135	C₂	—	2	4	no	—
136	C₂	#22549903	1	4	yes	direct product factor
137	C₂	C₂²×C₄⋊Dic₁₂₇	1	1	yes	characteristic
138	C₂	C₂×#16861216	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#117×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#117	D₁₂₇
hypocenter	C₁₂₇	C₂×#424
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #22549903	C₂	direct (N × H)
4. C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
12. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
66. C₂×C₁₂₇	#424	semidirect
70. C₁₂₇	C₂×#424	semidirect
136. C₂	#22549903	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523232	2
C₂³×C₁₂₇⋊C₈	2
#22549903	2
C₂×C₄⋊(C₁₂₇⋊C₈)	2
C₂×#117×C₁₂₇	2
C₂×#424	127

Subgroups — C2×#22549903 #30647289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549903 #30647289