Subgroups — C₂×#22550433 #30652021

Order 16256.

1168 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550433	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16861209	C₂	1	2	yes	maximal, complemented
3	C₂×#16861201	C₂	1	2	yes	maximal, complemented
4	C₂×#16861188	C₂	1	4	yes	maximal
5	C₂×C₄×Dic₂₅₄	C₂	1	4	yes	maximal, complemented
6	C₂×#16861168	C₂	1	2	yes	maximal
7	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₅₄	C₂²	1	4	yes	—
9	#16861209	C₂²	1	16	yes	complemented
10	#16861201	C₂²	1	16	yes	complemented
11	#16861188	C₂²	1	32	yes	—
12	C₄×Dic₂₅₄	C₂²	1	32	yes	complemented
13	#16861168	C₂²	1	16	yes	—
14	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
16	C₂×Dic₁₂₇⋊C₄	C₂²	1	8	yes	—
17	C₂×C₂²⋊Dic₁₂₇	C₂²	1	4	yes	—
18	C₂×C₄×Dic₁₂₇	C₂²	1	4	yes	—
19	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₂₇	C₂²	1	8	yes	—
25	C₂×Dic₂₅₄	—	2	16	no	—
26	C₂×Dic₂₅₄	C₂³	1	16	yes	—
27	C₂²×Dic₁₂₇	C₂³	1	8	yes	—
28	C₄⋊Dic₁₂₇	C₂³	1	16	yes	—
29	C₄⋊Dic₁₂₇	C₂³	1	16	yes	—
30	Dic₁₂₇⋊C₄	C₂³	1	32	yes	—
31	C₂²⋊Dic₁₂₇	C₂³	1	16	yes	—
32	C₄×Dic₁₂₇	C₂³	1	16	yes	—
33	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₂₇	—	2	8	no	—
35	C₂²×C₄×C₁₂₇	Q₈	1	8	yes	complemented
36	C₂²×C₄×C₁₂₇	C₂³	1	8	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
40	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₄²×C₁₂₇	C₂³	1	4	yes	—
43	C₄²×C₁₂₇	C₂³	1	4	yes	—
44	C₄²×C₁₂₇	C₂³	1	8	yes	—
45	Dic₂₅₄	—	2	32	no	—
46	C₂×Dic₁₂₇	—	2	16	no	—
47	C₂×Dic₁₂₇	C₂⁴	1	16	yes	—
48	C₂³×C₁₂₇	—	2	4	no	—
49	C₂³×C₁₂₇	C₂×Q₈	1	4	yes	—
50	C₂³×C₁₂₇	C₂⁴	1	2	yes	—
51	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	—	2	4	no	—
53	C₂×C₄×C₁₂₇	—	2	16	no	—
54	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
55	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
56	C₂×C₄×C₁₂₇	C₂×Q₈	1	16	yes	complemented
57	C₂×C₄×C₁₂₇	#41	1	8	yes	—
58	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
59	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
60	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
61	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
62	C₂²×C₁₂₇	—	2	8	no	—
63	C₂²×C₁₂₇	—	2	4	no	—
64	C₂²×C₁₂₇	C₂²×Q₈	1	4	yes	—
65	C₂²×C₁₂₇	C₂²×Q₈	1	2	yes	—
66	C₂²×C₁₂₇	C₂×#41	1	4	yes	—
67	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
68	C₄×C₁₂₇	—	2	4	no	—
69	C₄×C₁₂₇	C₂²×Q₈	1	4	yes	—
70	C₄×C₁₂₇	C₂²×Q₈	1	4	yes	—
71	C₄×C₁₂₇	C₂×#41	1	8	yes	—
72	Dic₁₂₇	—	2	16	no	—
73	C₂×C₁₂₇	—	2	4	no	—
74	C₂×C₁₂₇	#533	1	4	yes	complemented
75	C₂×C₁₂₇	C₂³×Q₈	1	1	yes	characteristic
76	C₂×C₁₂₇	C₂²×#41	1	2	yes	—
77	C₂×#533	—	127	1	no	maximal, Sylow
78	C₁₂₇	C₂×#533	1	1	yes	characteristic, complemented, Sylow
79	#533	—	127	16	no	—
80	C₂×C₄⋊Q₈	—	127	2	no	—
81	C₂×#125	—	127	2	no	—
82	C₂×C₂²⋊Q₈	—	127	4	no	—
83	C₂×C₄×Q₈	—	127	4	no	—
84	C₂×#117	—	127	2	no	—
85	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic
86	C₂²×Q₈	—	127	4	no	—
87	C₂³×C₄	C₂×D₁₂₇	1	2	yes	—
88	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
89	C₄⋊Q₈	—	127	16	no	—
90	#125	—	127	16	no	—
91	C₂²⋊Q₈	—	127	32	no	—
92	C₄×Q₈	—	127	32	no	—
93	#117	—	127	16	no	—
94	C₂×C₄⋊C₄	—	127	4	no	—
95	C₂×C₄⋊C₄	—	127	8	no	—
96	C₂×C₄⋊C₄	—	127	4	no	—
97	C₂×C₂²⋊C₄	—	127	4	no	—
98	C₂×C₄²	—	127	4	no	—
99	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₁₂₇	1	8	yes	—
102	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
103	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
104	C₂×Q₈	—	127	16	no	—
105	C₂×Q₈	—	254	16	no	—
106	C₂²×C₄	—	2	8	no	—
107	C₂²×C₄	—	127	8	no	—
108	C₂²×C₄	Dic₂₅₄	1	8	yes	complemented
109	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
110	C₂²×C₄	C₂²×D₁₂₇	1	8	yes	—
111	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
112	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
113	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
114	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
115	C₄⋊C₄	—	127	32	no	—
116	C₄⋊C₄	—	127	16	no	—
117	C₄⋊C₄	—	127	16	no	—
118	C₂²⋊C₄	—	127	16	no	—
119	C₄²	—	127	16	no	—
120	C₄²	C₂²×D₁₂₇	1	8	yes	—
121	C₄²	C₂²×D₁₂₇	1	4	yes	—
122	C₄²	C₂²×D₁₂₇	1	4	yes	—
123	C₂³	—	2	4	no	—
124	C₂³	C₂×Dic₂₅₄	1	4	yes	—
125	C₂³	C₂³×D₁₂₇	1	2	yes	—
126	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
127	Q₈	—	254	32	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	16	no	—
130	C₂×C₄	—	127	16	no	—
131	C₂×C₄	—	254	16	no	—
132	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
133	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
134	C₂×C₄	C₂×Dic₂₅₄	1	16	yes	complemented
135	C₂×C₄	#14523248	1	8	yes	—
136	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
137	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
138	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
139	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
140	C₂²	—	2	8	no	—
141	C₂²	—	2	4	no	—
142	C₂²	C₂²×Dic₂₅₄	1	4	yes	—
143	C₂²	C₂²×Dic₂₅₄	1	2	yes	—
144	C₂²	C₂×#14523248	1	4	yes	—
145	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	—	254	16	no	—
148	C₄	C₂²×Dic₂₅₄	1	4	yes	—
149	C₄	C₂²×Dic₂₅₄	1	4	yes	—
150	C₄	C₂×#14523248	1	8	yes	—
151	C₂	—	2	4	no	—
152	C₂	#22550433	1	4	yes	direct product factor
153	C₂	C₂³×Dic₂₅₄	1	1	yes	characteristic
154	C₂	C₂²×#14523248	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₂₇
commutator, cosocle	C₂²×C₁₂₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂²×C₄²×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂⁴
prime radical	C₂⁴×C₁₂₇	C₂³
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	C₂×#533
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #22550433	C₂	direct (N × H)
2. C₂×#16861209	C₂	semidirect
3. C₂×#16861201	C₂	semidirect
5. C₂×C₄×Dic₂₅₄	C₂	semidirect
9. #16861209	C₂²	semidirect
10. #16861201	C₂²	semidirect
12. C₄×Dic₂₅₄	C₂²	semidirect
35. C₂²×C₄×C₁₂₇	Q₈	semidirect
56. C₂×C₄×C₁₂₇	C₂×Q₈	semidirect
74. C₂×C₁₂₇	#533	semidirect
78. C₁₂₇	C₂×#533	semidirect
108. C₂²×C₄	Dic₂₅₄	semidirect
134. C₂×C₄	C₂×Dic₂₅₄	semidirect
152. C₂	#22550433	direct (N × H)

Maximal subgroups

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

subgroup	index
#22550433	2
C₂×#16861209	2
C₂×#16861201	2
C₂×#16861188	2
C₂×C₄×Dic₂₅₄	2
C₂×#16861168	2
C₂²×C₄²×C₁₂₇	2
C₂×#533	127

Subgroups — C2×#22550433 #30652021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550433 #30652021