Subgroups — C₂×#22549934 #30647353

Order 16256.

412 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861168	C₂	1	1	yes	maximal, characteristic
2	#22549934	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂×#16861119	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	#16861168	C₄	1	4	yes	—
12	#16861168	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊Dic₁₂₇	C₄	1	1	yes	characteristic
16	C₂×C₂²⋊Dic₁₂₇	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
19	Dic₁₂₇⋊C₈	C₂²	1	4	yes	complemented
20	Dic₁₂₇⋊C₈	C₂²	1	4	yes	complemented
21	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
22	#16861119	C₂²	1	4	yes	complemented
23	C₈×Dic₁₂₇	C₂²	1	4	yes	complemented
24	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	—
28	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₂₇⋊C₈	—	2	2	no	—
33	C₂×C₁₂₇⋊C₈	—	2	2	no	—
34	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
35	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
36	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
37	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
38	C₂²⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
39	C₂²⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
40	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₂₇	C₂³	1	2	yes	—
43	C₄×Dic₁₂₇	C₂³	1	2	yes	—
44	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	—	2	2	no	—
46	C₂²×C₄×C₁₂₇	—	2	2	no	—
47	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
49	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	—	2	2	no	—
52	C₂×C₈×C₁₂₇	—	2	2	no	—
53	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
54	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
55	C₂×Dic₁₂₇	—	2	2	no	—
56	C₂×Dic₁₂₇	—	2	2	no	—
57	C₂×Dic₁₂₇	—	2	2	no	—
58	C₂×Dic₁₂₇	—	2	2	no	—
59	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
60	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
61	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
62	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
63	C₁₂₇⋊C₈	—	2	2	no	—
64	C₁₂₇⋊C₈	—	2	2	no	—
65	C₂³×C₁₂₇	—	2	2	no	—
66	C₂³×C₁₂₇	—	2	2	no	—
67	C₂³×C₁₂₇	—	2	2	no	—
68	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
69	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	—	2	2	no	—
71	C₂×C₄×C₁₂₇	—	2	4	no	—
72	C₂×C₄×C₁₂₇	—	2	2	no	—
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	#41	1	2	yes	—
77	C₂×C₄×C₁₂₇	#41	1	2	yes	—
78	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
79	C₈×C₁₂₇	—	2	2	no	—
80	C₈×C₁₂₇	—	2	2	no	—
81	C₂²×C₁₂₇	—	2	2	no	—
82	C₂²×C₁₂₇	—	2	2	no	—
83	C₂²×C₁₂₇	—	2	2	no	—
84	C₂²×C₁₂₇	—	4	4	no	—
85	C₂²×C₁₂₇	#117	1	2	yes	—
86	C₂²×C₁₂₇	#131	1	2	yes	—
87	C₂²×C₁₂₇	#131	1	2	yes	—
88	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
89	C₄×C₁₂₇	—	2	2	no	—
90	C₄×C₁₂₇	#117	1	2	yes	—
91	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
92	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
93	Dic₁₂₇	—	2	2	no	—
94	Dic₁₂₇	—	2	2	no	—
95	Dic₁₂₇	—	2	2	no	—
96	Dic₁₂₇	—	2	2	no	—
97	C₂×C₁₂₇	—	4	2	no	—
98	C₂×C₁₂₇	#433	1	4	yes	complemented
99	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
100	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
101	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
102	C₂×#433	—	127	1	no	maximal, Sylow
103	C₁₂₇	C₂×#433	1	1	yes	characteristic, complemented, Sylow
104	C₂×#117	—	127	1	no	—
105	#433	—	127	8	no	—
106	C₂×C₄⋊C₈	—	127	1	no	—
107	C₂×C₄⋊C₈	—	127	1	no	—
108	C₂×C₂²⋊C₈	—	127	1	no	—
109	C₂×C₂²⋊C₈	D₁₂₇	1	1	yes	characteristic
110	C₂×#97	—	127	1	no	—
111	C₂×C₄×C₈	—	127	1	no	—
112	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
113	C₂²×C₈	—	127	1	no	—
114	C₂²×C₈	—	127	1	no	—
115	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
116	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
117	#117	—	127	4	no	—
118	#117	—	127	4	no	—
119	C₂×C₄⋊C₄	—	127	1	no	—
120	C₂×C₄⋊C₄	—	127	1	no	—
121	C₂×C₂²⋊C₄	—	127	1	no	—
122	C₂×C₂²⋊C₄	—	127	1	no	—
123	C₂×C₄²	—	127	1	no	—
124	C₂×C₄²	—	127	1	no	—
125	C₄⋊C₈	—	127	4	no	—
126	C₄⋊C₈	—	127	4	no	—
127	C₂²⋊C₈	—	127	4	no	—
128	C₂²⋊C₈	C₂×D₁₂₇	1	4	yes	—
129	#97	—	127	4	no	—
130	C₄×C₈	—	127	4	no	—
131	C₂⁴	C₄×D₁₂₇	1	1	yes	characteristic
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	127	1	no	—
135	C₂²×C₄	—	127	1	no	—
136	C₂²×C₄	—	127	1	no	—
137	C₂²×C₄	—	127	1	no	—
138	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
139	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
140	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
141	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	127	2	no	—
145	C₂×C₈	—	127	2	no	—
146	C₂×C₈	—	254	2	no	—
147	C₂×C₈	—	254	2	no	—
148	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
149	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
150	C₄⋊C₄	—	127	4	no	—
151	C₄⋊C₄	—	127	4	no	—
152	C₂²⋊C₄	—	127	4	no	—
153	C₂²⋊C₄	—	127	4	no	—
154	C₄²	—	127	2	no	—
155	C₄²	—	127	2	no	—
156	C₄²	—	127	2	no	—
157	C₄²	—	127	2	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	C₂×C₄×D₁₂₇	1	2	yes	—
162	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	127	2	no	—
168	C₂×C₄	—	127	2	no	—
169	C₂×C₄	—	127	2	no	—
170	C₂×C₄	—	127	2	no	—
171	C₂×C₄	—	254	2	no	—
172	C₂×C₄	—	254	2	no	—
173	C₂×C₄	—	254	2	no	—
174	C₂×C₄	—	254	2	no	—
175	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
176	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
177	C₂×C₄	#14523247	1	2	yes	—
178	C₂×C₄	#14523247	1	2	yes	—
179	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	254	2	no	—
183	C₈	—	254	2	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	4	4	no	—
188	C₂²	#16861165	1	2	yes	—
189	C₂²	#16861215	1	2	yes	—
190	C₂²	#16861217	1	2	yes	—
191	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	254	2	no	—
194	C₄	—	254	2	no	—
195	C₄	—	254	2	no	—
196	C₄	—	254	2	no	—
197	C₄	#16861165	1	2	yes	—
198	C₄	C₂×#14523247	1	1	yes	characteristic
199	C₄	C₂×#14523247	1	1	yes	characteristic
200	C₂	—	4	2	no	—
201	C₂	#22549934	1	4	yes	direct product factor
202	C₂	C₂×#16861165	1	1	yes	characteristic
203	C₂	C₂×#16861215	1	1	yes	characteristic
204	C₂	C₂×#16861217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₂²⋊C₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#433
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. #22549934	C₂	direct (N × H)
3. C₂×Dic₁₂₇⋊C₈	C₂	semidirect
4. C₂×Dic₁₂₇⋊C₈	C₂	semidirect
6. C₂×#16861119	C₂	semidirect
7. C₂×C₈×Dic₁₂₇	C₂	semidirect
19. Dic₁₂₇⋊C₈	C₂²	semidirect
20. Dic₁₂₇⋊C₈	C₂²	semidirect
22. #16861119	C₂²	semidirect
23. C₈×Dic₁₂₇	C₂²	semidirect
98. C₂×C₁₂₇	#433	semidirect
103. C₁₂₇	C₂×#433	semidirect
201. C₂	#22549934	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16861168	2
#22549934	2
C₂×Dic₁₂₇⋊C₈	2
C₂×C₂²⋊(C₁₂₇⋊C₈)	2
C₂×#16861119	2
C₂×C₈×Dic₁₂₇	2
C₂×C₂²⋊C₈×C₁₂₇	2
C₂×#433	127

Subgroups — C2×#22549934 #30647353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549934 #30647353