Subgroups — C₂×#22549736 #30641103

Order 16256.

1232 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×C₄⋊Dic₁₂₇	C₂	1	2	yes	maximal, complemented
2	C₂²×C₂²⋊Dic₁₂₇	C₂	1	2	yes	maximal
3	#22549736	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₂₇⋊#95	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₂₇	C₂²	1	4	yes	—
7	C₂×C₄⋊Dic₁₂₇	—	2	8	no	—
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	8	yes	complemented
9	C₂×C₂²⋊Dic₁₂₇	—	2	8	no	—
10	C₂×C₂²⋊Dic₁₂₇	C₂²	1	8	yes	—
11	C₁₂₇⋊#95	C₂²	1	8	yes	complemented
12	C₂⁵×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₂₇	—	2	8	no	—
14	C₂³×C₄×C₁₂₇	C₄	1	8	yes	complemented
15	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₂₇	C₂²	1	4	yes	—
17	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₂₇	—	2	8	no	—
19	C₂²×Dic₁₂₇	—	2	8	no	—
20	C₂²×Dic₁₂₇	—	2	8	no	—
21	C₂²×Dic₁₂₇	C₂³	1	8	yes	—
22	C₄⋊Dic₁₂₇	—	2	16	no	—
23	C₂²⋊Dic₁₂₇	—	2	16	no	—
24	C₂⁴×C₁₂₇	—	2	4	no	—
25	C₂⁴×C₁₂₇	—	2	4	no	—
26	C₂⁴×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₂⁴×C₁₂₇	D₄	1	4	yes	—
28	C₂⁴×C₁₂₇	Q₈	1	4	yes	—
29	C₂⁴×C₁₂₇	C₂³	1	2	yes	—
30	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	—	2	4	no	—
32	C₂²×C₄×C₁₂₇	—	2	32	no	—
33	C₂²×C₄×C₁₂₇	—	2	16	no	—
34	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₁₂₇	C₂×C₄	1	16	yes	complemented
37	C₂²×C₄×C₁₂₇	D₄	1	8	yes	—
38	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
40	C₂×Dic₁₂₇	—	2	8	no	—
41	C₂×Dic₁₂₇	—	2	8	no	—
42	C₂×Dic₁₂₇	—	2	8	no	—
43	C₂×Dic₁₂₇	—	4	16	no	—
44	C₂³×C₁₂₇	—	2	8	no	—
45	C₂³×C₁₂₇	—	2	8	no	—
46	C₂³×C₁₂₇	—	2	16	no	—
47	C₂³×C₁₂₇	—	2	16	no	—
48	C₂³×C₁₂₇	—	2	4	no	—
49	C₂³×C₁₂₇	—	2	4	no	—
50	C₂³×C₁₂₇	C₄⋊C₄	1	16	yes	complemented
51	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
52	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
53	C₂³×C₁₂₇	C₂×D₄	1	4	yes	—
54	C₂³×C₁₂₇	C₂×D₄	1	4	yes	—
55	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
56	C₂³×C₁₂₇	C₂×Q₈	1	4	yes	—
57	C₂³×C₁₂₇	C₂×Q₈	1	2	yes	—
58	C₂³×C₁₂₇	#41	1	4	yes	—
59	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	—	2	8	no	—
61	C₂×C₄×C₁₂₇	—	2	4	no	—
62	C₂×C₄×C₁₂₇	—	2	32	no	—
63	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	16	yes	complemented
64	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
65	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
66	C₂×C₄×C₁₂₇	C₂×D₄	1	8	yes	—
67	C₂²×C₁₂₇	—	2	4	no	—
68	C₂²×C₁₂₇	—	2	4	no	—
69	C₂²×C₁₂₇	—	2	8	no	—
70	C₂²×C₁₂₇	—	2	8	no	—
71	C₂²×C₁₂₇	—	2	32	no	—
72	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
73	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
74	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	8	yes	complemented
75	C₂²×C₁₂₇	#117	1	4	yes	—
76	C₂²×C₁₂₇	#121	1	8	yes	—
77	C₂²×C₁₂₇	C₂²⋊Q₈	1	8	yes	—
78	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
79	C₂²×C₁₂₇	C₂²×D₄	1	2	yes	—
80	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
83	C₄×C₁₂₇	—	2	4	no	—
84	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	8	yes	complemented
85	Dic₁₂₇	—	4	8	no	—
86	C₂×C₁₂₇	—	2	8	no	—
87	C₂×C₁₂₇	#380	1	8	yes	complemented
88	C₂×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
91	C₂×C₁₂₇	C₂×#121	1	2	yes	—
92	C₂×C₁₂₇	C₂×C₂²⋊Q₈	1	2	yes	—
93	C₂×#380	—	127	1	no	maximal, Sylow
94	C₁₂₇	C₂×#380	1	1	yes	characteristic, complemented, Sylow
95	C₂⁴×C₄	D₁₂₇	1	1	yes	characteristic
96	C₂²×C₄⋊C₄	—	127	2	no	—
97	C₂²×C₂²⋊C₄	—	127	2	no	—
98	#380	—	127	8	no	—
99	C₂×#95	—	127	2	no	—
100	C₂⁵	C₂×D₁₂₇	1	1	yes	characteristic
101	C₂³×C₄	—	2	8	no	—
102	C₂³×C₄	—	127	4	no	—
103	C₂³×C₄	Dic₁₂₇	1	8	yes	complemented
104	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
105	C₂³×C₄	C₂×D₁₂₇	1	4	yes	—
106	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	127	8	no	—
108	C₂×C₄⋊C₄	—	254	8	no	—
109	C₂×C₂²⋊C₄	—	127	8	no	—
110	C₂×C₂²⋊C₄	—	254	8	no	—
111	#95	—	127	8	no	—
112	C₂⁴	—	2	4	no	—
113	C₂⁴	—	2	4	no	—
114	C₂⁴	C₂×Dic₁₂₇	1	4	yes	—
115	C₂⁴	D₅₀₈	1	4	yes	—
116	C₂⁴	Dic₂₅₄	1	4	yes	—
117	C₂⁴	C₂²×D₁₂₇	1	2	yes	—
118	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
119	C₂²×C₄	—	2	4	no	—
120	C₂²×C₄	—	2	32	no	—
121	C₂²×C₄	—	2	16	no	—
122	C₂²×C₄	—	127	8	no	—
123	C₂²×C₄	—	254	8	no	—
124	C₂²×C₄	—	254	8	no	—
125	C₂²×C₄	—	254	8	no	—
126	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
127	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
128	C₂²×C₄	C₂×Dic₁₂₇	1	16	yes	complemented
129	C₂²×C₄	C₂²⋊D₁₂₇	1	8	yes	—
130	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
131	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
132	C₄⋊C₄	—	254	16	no	—
133	C₂²⋊C₄	—	254	16	no	—
134	C₂³	—	2	16	no	—
135	C₂³	—	2	4	no	—
136	C₂³	—	2	4	no	—
137	C₂³	—	2	8	no	—
138	C₂³	—	2	8	no	—
139	C₂³	—	2	16	no	—
140	C₂³	C₄⋊Dic₁₂₇	1	16	yes	complemented
141	C₂³	C₂²×Dic₁₂₇	1	4	yes	—
142	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
143	C₂³	C₂×C₂²⋊D₁₂₇	1	4	yes	—
144	C₂³	C₂×D₅₀₈	1	4	yes	—
145	C₂³	C₂×D₅₀₈	1	2	yes	—
146	C₂³	C₂×Dic₂₅₄	1	4	yes	—
147	C₂³	C₂×Dic₂₅₄	1	2	yes	—
148	C₂³	#14523248	1	4	yes	—
149	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	8	no	—
152	C₂×C₄	—	2	32	no	—
153	C₂×C₄	—	508	16	no	—
154	C₂×C₄	—	254	8	no	—
155	C₂×C₄	—	254	8	no	—
156	C₂×C₄	—	254	8	no	—
157	C₂×C₄	C₂²⋊Dic₁₂₇	1	16	yes	complemented
158	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
159	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
160	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	8	yes	—
161	C₂²	—	2	4	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	8	no	—
165	C₂²	—	2	32	no	—
166	C₂²	C₂×C₂²⋊Dic₁₂₇	1	4	yes	—
167	C₂²	C₂×C₄⋊Dic₁₂₇	1	4	yes	—
168	C₂²	C₂×C₄⋊Dic₁₂₇	1	8	yes	complemented
169	C₂²	#16861168	1	4	yes	—
170	C₂²	#16861183	1	8	yes	—
171	C₂²	#16861188	1	8	yes	—
172	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
173	C₂²	C₂²×C₂²⋊D₁₂₇	1	2	yes	—
174	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
175	C₂²	C₂²×Dic₂₅₄	1	1	yes	characteristic
176	C₂²	C₂×#14523248	1	2	yes	—
177	C₄	—	2	4	no	—
178	C₄	—	508	8	no	—
179	C₄	C₂×C₂²⋊Dic₁₂₇	1	8	yes	complemented
180	C₂	—	2	8	no	—
181	C₂	#22549736	1	8	yes	direct product factor
182	C₂	C₂²×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
183	C₂	C₂²×C₄⋊Dic₁₂₇	1	1	yes	characteristic
184	C₂	C₂×#16861168	1	1	yes	characteristic
185	C₂	C₂×#16861183	1	2	yes	—
186	C₂	C₂×#16861188	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂³	C₂³×D₁₂₇
Fitting	C₂⁴×C₄×C₁₂₇	C₂
socle	C₂⁴×C₁₂₇	C₂³
prime radical	C₂⁵×C₁₂₇	C₂²
cosocle	C₂³×C₁₂₇	C₂⁴
hypercenter	C₂⁴×C₄	D₁₂₇
hypocenter	C₁₂₇	C₂×#380
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Dic₁₂₇	C₂	semidirect
3. #22549736	C₂	direct (N × H)
4. C₂×C₁₂₇⋊#95	C₂	semidirect
8. C₂×C₄⋊Dic₁₂₇	C₂²	semidirect
11. C₁₂₇⋊#95	C₂²	semidirect
14. C₂³×C₄×C₁₂₇	C₄	semidirect
36. C₂²×C₄×C₁₂₇	C₂×C₄	semidirect
50. C₂³×C₁₂₇	C₄⋊C₄	semidirect
63. C₂×C₄×C₁₂₇	C₂²⋊C₄	semidirect
74. C₂²×C₁₂₇	C₂×C₄⋊C₄	semidirect
84. C₄×C₁₂₇	C₂×C₂²⋊C₄	semidirect
87. C₂×C₁₂₇	#380	semidirect
94. C₁₂₇	C₂×#380	semidirect
103. C₂³×C₄	Dic₁₂₇	semidirect
128. C₂²×C₄	C₂×Dic₁₂₇	semidirect
140. C₂³	C₄⋊Dic₁₂₇	semidirect
157. C₂×C₄	C₂²⋊Dic₁₂₇	semidirect
168. C₂²	C₂×C₄⋊Dic₁₂₇	semidirect
179. C₄	C₂×C₂²⋊Dic₁₂₇	semidirect
181. C₂	#22549736	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×C₄⋊Dic₁₂₇	2
C₂²×C₂²⋊Dic₁₂₇	2
#22549736	2
C₂×C₁₂₇⋊#95	2
C₂⁴×C₄×C₁₂₇	2
C₂×#380	127

Subgroups — C2×#22549736 #30641103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549736 #30641103