Subgroups — C₂×#22549637 #30638437

Order 16256.

488 conjugacy classes of subgroups.

The subgroup lattice has 214 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	C₂²×C₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549637	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₂₇	—	2	1	no	—
6	#16861168	—	2	1	no	—
7	#16861168	—	2	1	no	—
8	#16861168	C₄	1	2	yes	complemented
9	#16861168	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₂₇	C₄	1	2	yes	complemented
11	C₂×C₂²⋊Dic₁₂₇	—	2	1	no	—
12	C₂×C₄×Dic₁₂₇	—	2	2	no	—
13	C₂×C₄×Dic₁₂₇	—	2	1	no	—
14	C₂×C₄×Dic₁₂₇	—	2	1	no	—
15	C₂×C₄×Dic₁₂₇	C₄	1	4	yes	complemented
16	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
17	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₂₇	—	2	2	no	—
19	C₂×M₁₆×C₁₂₇	—	2	1	no	—
20	C₂×M₁₆×C₁₂₇	C₄	1	4	yes	complemented
21	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
22	C₂²×C₈×C₁₂₇	—	2	1	no	—
23	C₂²×Dic₁₂₇	—	2	8	no	—
24	C₂²×Dic₁₂₇	—	2	4	no	—
25	C₂²×Dic₁₂₇	—	2	2	no	—
26	C₂²×Dic₁₂₇	—	2	2	no	—
27	C₄⋊Dic₁₂₇	—	2	2	no	—
28	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	complemented
29	C₂²⋊Dic₁₂₇	—	2	2	no	—
30	C₂²⋊Dic₁₂₇	—	2	2	no	—
31	C₄×Dic₁₂₇	—	2	4	no	—
32	C₄×Dic₁₂₇	—	2	2	no	—
33	C₄×Dic₁₂₇	—	2	2	no	—
34	C₄×Dic₁₂₇	—	2	2	no	—
35	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	complemented
36	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
42	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	Q₈	1	2	yes	—
46	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
47	M₁₆×C₁₂₇	—	2	4	no	—
48	M₁₆×C₁₂₇	—	2	2	no	—
49	M₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
50	C₂×C₈×C₁₂₇	—	2	4	no	—
51	C₂×C₈×C₁₂₇	—	2	2	no	—
52	C₂×Dic₁₂₇	—	2	16	no	—
53	C₂×Dic₁₂₇	—	2	8	no	—
54	C₂×Dic₁₂₇	—	2	4	no	—
55	C₂×Dic₁₂₇	—	2	4	no	—
56	C₂×Dic₁₂₇	—	4	4	no	—
57	C₂³×C₁₂₇	—	2	4	no	—
58	C₂³×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
60	C₂³×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
62	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
64	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
65	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
69	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
70	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
81	C₈×C₁₂₇	—	2	4	no	—
82	C₂²×C₁₂₇	—	2	4	no	—
83	C₂²×C₁₂₇	—	2	4	no	—
84	C₂²×C₁₂₇	—	2	2	no	—
85	C₂²×C₁₂₇	—	2	2	no	—
86	C₂²×C₁₂₇	#95	1	2	yes	—
87	C₂²×C₁₂₇	#95	1	2	yes	—
88	C₂²×C₁₂₇	#104	1	4	yes	—
89	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₁₂₇	#95	1	2	yes	—
93	C₄×C₁₂₇	#95	1	2	yes	—
94	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
95	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	Dic₁₂₇	—	2	8	no	—
99	Dic₁₂₇	—	4	4	no	—
100	C₂×C₁₂₇	—	2	1	no	—
101	C₂×C₁₂₇	—	2	2	no	—
102	C₂×C₁₂₇	—	2	1	no	—
103	C₂×C₁₂₇	#339	1	4	yes	complemented
104	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
105	C₂×C₁₂₇	C₂×#104	1	2	yes	—
106	C₂×#339	—	127	1	no	maximal, Sylow
107	C₁₂₇	C₂×#339	1	1	yes	characteristic, complemented, Sylow
108	C₂²×M₁₆	D₁₂₇	1	1	yes	characteristic
109	C₂×#117	—	127	1	no	—
110	C₂²×C₄²	—	127	1	no	—
111	#339	—	127	4	no	—
112	C₂³×C₄	—	254	1	no	—
113	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	2	1	no	—
116	C₂×M₁₆	Dic₁₂₇	1	4	yes	complemented
117	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
118	C₂²×C₈	—	2	1	no	—
119	#117	—	127	2	no	—
120	#117	—	127	2	no	—
121	#117	—	254	1	no	—
122	#117	—	254	1	no	—
123	C₂×C₄⋊C₄	—	127	2	no	—
124	C₂×C₂²⋊C₄	—	254	1	no	—
125	C₂×C₄²	—	127	4	no	—
126	C₂×C₄²	—	127	2	no	—
127	C₂×C₄²	—	254	2	no	—
128	C₂×C₄²	—	254	1	no	—
129	C₂×C₄²	—	254	1	no	—
130	C₂⁴	D₅₀₈	1	1	yes	characteristic
131	C₂²×C₄	—	254	2	no	—
132	C₂²×C₄	—	254	8	no	—
133	C₂²×C₄	—	254	4	no	—
134	C₂²×C₄	—	254	2	no	—
135	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
136	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
137	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
138	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
139	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
140	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
141	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
142	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
143	C₂²×C₄	Dic₂₅₄	1	2	yes	—
144	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
145	M₁₆	—	2	4	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	C₂×Dic₁₂₇	1	4	yes	complemented
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₄⋊C₄	—	127	4	no	—
151	C₄⋊C₄	—	254	2	no	—
152	C₂²⋊C₄	—	254	2	no	—
153	C₂²⋊C₄	—	254	2	no	—
154	C₄²	—	127	4	no	—
155	C₄²	—	254	4	no	—
156	C₄²	—	254	2	no	—
157	C₄²	—	254	2	no	—
158	C₄²	—	254	2	no	—
159	C₂³	—	2	4	no	—
160	C₂³	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
161	C₂³	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
162	C₂³	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
163	C₂³	C₄⋊Dic₁₂₇	1	2	yes	—
164	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
165	C₂×C₄	—	508	4	no	—
166	C₂×C₄	—	254	4	no	—
167	C₂×C₄	—	254	8	no	—
168	C₂×C₄	—	254	16	no	—
169	C₂×C₄	—	254	4	no	—
170	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
171	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
172	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
173	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
175	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
176	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
177	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
178	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
179	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
180	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
181	C₂×C₄	C₄⋊Dic₁₂₇	1	2	yes	—
182	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
184	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
185	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
187	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
188	C₈	—	2	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	C₁₂₇⋊#95	1	2	yes	—
194	C₂²	C₁₂₇⋊#95	1	2	yes	—
195	C₂²	#16861135	1	4	yes	—
196	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
197	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
198	C₂²	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
199	C₄	—	508	4	no	—
200	C₄	—	254	8	no	—
201	C₄	C₁₂₇⋊#95	1	2	yes	—
202	C₄	C₁₂₇⋊#95	1	2	yes	—
203	C₄	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
204	C₄	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
205	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
206	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	2	no	—
209	C₂	—	2	1	no	—
210	C₂	#22549637	1	4	yes	direct product factor
211	C₂	C₂×C₁₂₇⋊#95	1	1	yes	characteristic
212	C₂	C₂×#16861135	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂²×M₁₆×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²⋊C₄
prime radical	C₂⁴×C₁₂₇	D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂²×M₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂×#339
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22549637	C₂	direct (N × H)
8. #16861168	C₄	semidirect
10. C₂×C₄⋊Dic₁₂₇	C₄	semidirect
15. C₂×C₄×Dic₁₂₇	C₄	semidirect
20. C₂×M₁₆×C₁₂₇	C₄	semidirect
28. C₄⋊Dic₁₂₇	C₂×C₄	semidirect
35. C₄×Dic₁₂₇	C₂×C₄	semidirect
49. M₁₆×C₁₂₇	C₂×C₄	semidirect
103. C₂×C₁₂₇	#339	semidirect
107. C₁₂₇	C₂×#339	semidirect
116. C₂×M₁₆	Dic₁₂₇	semidirect
147. M₁₆	C₂×Dic₁₂₇	semidirect
210. C₂	#22549637	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
C₂²×C₄×Dic₁₂₇	2
#22549637	2
C₂²×M₁₆×C₁₂₇	2
C₂×#339	127

Subgroups — C2×#22549637 #30638437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549637 #30638437