Subgroups — C₂×C₄×#14523248 #30651910

Order 16256.

1728 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523248	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523248	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₂₅₄	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₅₀₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₂²⋊D₁₂₇	C₂	1	4	yes	maximal, complemented
6	C₂×#16861168	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16861166	C₂	1	2	yes	maximal, complemented
8	C₂×C₄²×D₁₂₇	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14523248	C₄	1	16	yes	direct product factor
11	C₂×#14523248	C₂²	1	8	yes	—
12	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
13	C₂²×D₅₀₈	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₂₇	C₂²	1	4	yes	—
16	C₂²×C₄×D₁₂₇	C₂²	1	2	yes	—
17	C₄×Dic₂₅₄	C₂²	1	16	yes	complemented
18	C₄×D₅₀₈	C₂²	1	16	yes	complemented
19	C₄×C₂²⋊D₁₂₇	C₂²	1	32	yes	complemented
20	#16861168	C₂²	1	8	yes	complemented
21	#16861166	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
23	C₂×Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
24	C₂×C₂²⋊Dic₁₂₇	C₂²	1	2	yes	—
25	C₂×(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
26	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₂₇	C₂²	1	4	yes	—
28	C₄²×D₁₂₇	C₂²	1	16	yes	complemented
29	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
30	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₂₇	C₂²	1	8	yes	complemented
35	C₂³×D₁₂₇	C₂³	1	2	yes	—
36	#14523248	C₂×C₄	1	64	yes	direct product factor
37	C₂×Dic₂₅₄	C₂×C₄	1	8	yes	complemented
38	C₂×Dic₂₅₄	C₂³	1	4	yes	—
39	C₂×D₅₀₈	C₂×C₄	1	8	yes	complemented
40	C₂×D₅₀₈	C₂³	1	4	yes	—
41	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	16	yes	complemented
42	C₂×C₂²⋊D₁₂₇	C₂³	1	8	yes	—
43	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
44	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
45	C₂×C₄×D₁₂₇	—	2	16	no	—
46	C₂×C₄×D₁₂₇	C₂×C₄	1	16	yes	complemented
47	C₂×C₄×D₁₂₇	C₂³	1	16	yes	—
48	C₂×C₄×D₁₂₇	C₂³	1	8	yes	—
49	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
50	Dic₁₂₇⋊C₄	C₂³	1	16	yes	—
51	C₂²⋊Dic₁₂₇	C₂³	1	8	yes	—
52	(C₂×D₁₂₇)⋊C₄	C₂³	1	16	yes	—
53	C₄×Dic₁₂₇	C₂³	1	8	yes	—
54	C₄×Dic₁₂₇	C₂³	1	16	yes	—
55	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	—	2	8	no	—
57	C₂²×C₄×C₁₂₇	C₂×C₄	1	8	yes	complemented
58	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₂₇	C₂³	1	8	yes	—
60	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
63	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
64	C₄²×C₁₂₇	C₂³	1	4	yes	—
65	C₄²×C₁₂₇	C₂³	1	4	yes	—
66	C₄²×C₁₂₇	C₂³	1	8	yes	—
67	C₂²×D₁₂₇	—	2	8	no	—
68	C₂²×D₁₂₇	C₂²×C₄	1	8	yes	—
69	C₂²×D₁₂₇	C₂⁴	1	4	yes	—
70	Dic₂₅₄	C₂²×C₄	1	16	yes	complemented
71	D₅₀₈	C₂²×C₄	1	16	yes	complemented
72	C₂²⋊D₁₂₇	C₂²×C₄	1	32	yes	complemented
73	C₂×Dic₁₂₇	—	2	8	no	—
74	C₂×Dic₁₂₇	C₂²×C₄	1	8	yes	—
75	C₂×Dic₁₂₇	C₂⁴	1	8	yes	—
76	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
77	C₄×D₁₂₇	—	2	32	no	—
78	C₄×D₁₂₇	C₂²×C₄	1	32	yes	complemented
79	C₂³×C₁₂₇	—	2	4	no	—
80	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
81	C₂³×C₁₂₇	C₂⁴	1	2	yes	—
82	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	—	2	4	no	—
84	C₂×C₄×C₁₂₇	—	2	16	no	—
85	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
86	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
87	C₂×C₄×C₁₂₇	C₂²×C₄	1	16	yes	complemented
88	C₂×C₄×C₁₂₇	#41	1	8	yes	complemented
89	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
90	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
91	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
92	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
93	C₂²×C₁₂₇	—	2	8	no	—
94	C₂²×C₁₂₇	—	2	4	no	—
95	C₂²×C₁₂₇	C₂³×C₄	1	4	yes	—
96	C₂²×C₁₂₇	C₂³×C₄	1	2	yes	—
97	C₂²×C₁₂₇	C₂×#41	1	4	yes	—
98	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
99	C₂×D₁₂₇	—	2	16	no	—
100	C₂×D₁₂₇	—	2	8	no	—
101	C₂×D₁₂₇	C₂³×C₄	1	8	yes	—
102	C₄×C₁₂₇	—	2	4	no	—
103	C₄×C₁₂₇	C₂³×C₄	1	4	yes	—
104	C₄×C₁₂₇	C₂³×C₄	1	4	yes	—
105	C₄×C₁₂₇	C₂×#41	1	8	yes	complemented
106	Dic₁₂₇	—	2	8	no	—
107	Dic₁₂₇	C₂³×C₄	1	8	yes	—
108	C₂×C₁₂₇	—	2	4	no	—
109	C₂×C₁₂₇	C₄×#41	1	4	yes	complemented
110	C₂×C₁₂₇	C₂⁴×C₄	1	1	yes	characteristic
111	C₂×C₁₂₇	C₂²×#41	1	2	yes	—
112	D₁₂₇	—	2	8	no	—
113	C₂×C₄×#41	—	127	1	no	maximal, Sylow
114	C₁₂₇	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
115	C₂²×#41	—	127	1	no	—
116	C₄×#41	—	127	16	no	—
117	C₂×C₄×Q₈	—	127	2	no	—
118	C₂×C₄×D₄	—	127	4	no	—
119	C₂×C₄×D₄	—	127	2	no	—
120	C₂×#117	—	127	1	no	—
121	C₂×#117	—	127	2	no	—
122	C₂²×C₄²	—	127	2	no	—
123	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic, complemented
124	C₂×#41	—	127	16	no	—
125	C₂×#41	—	127	8	no	—
126	C₂²×Q₈	—	127	1	no	—
127	C₂²×D₄	—	127	2	no	—
128	C₂²×D₄	—	127	1	no	—
129	C₂³×C₄	—	127	2	no	—
130	C₂³×C₄	—	127	4	no	—
131	C₂³×C₄	C₂×D₁₂₇	1	2	yes	—
132	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
133	C₄×Q₈	—	127	16	no	—
134	C₄×D₄	—	127	32	no	—
135	C₄×D₄	—	127	16	no	—
136	#117	—	127	8	no	—
137	#117	—	127	16	no	—
138	C₂×C₄⋊C₄	—	127	4	no	—
139	C₂×C₄⋊C₄	—	127	2	no	—
140	C₂×C₂²⋊C₄	—	127	2	no	—
141	C₂×C₂²⋊C₄	—	127	4	no	—
142	C₂×C₄²	—	127	2	no	—
143	C₂×C₄²	—	127	4	no	—
144	C₂×C₄²	—	127	16	no	—
145	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
146	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
147	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
148	C₂×C₄²	C₂×D₁₂₇	1	8	yes	complemented
149	C₂⁴	—	127	2	no	—
150	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
151	#41	—	127	64	no	—
152	C₂×Q₈	—	127	8	no	—
153	C₂×Q₈	—	127	4	no	—
154	C₂×D₄	—	127	16	no	—
155	C₂×D₄	—	127	8	no	—
156	C₂×D₄	—	127	8	no	—
157	C₂×D₄	—	127	4	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	—	127	2	no	—
160	C₂²×C₄	—	127	16	no	—
161	C₂²×C₄	—	127	8	no	—
162	C₂²×C₄	—	127	4	no	—
163	C₂²×C₄	—	127	16	no	—
164	C₂²×C₄	—	254	16	no	—
165	C₂²×C₄	C₄×D₁₂₇	1	8	yes	complemented
166	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
167	C₂²×C₄	C₂²×D₁₂₇	1	8	yes	—
168	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
171	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
172	C₄⋊C₄	—	127	16	no	—
173	C₄⋊C₄	—	127	8	no	—
174	C₂²⋊C₄	—	127	8	no	—
175	C₂²⋊C₄	—	127	16	no	—
176	C₄²	—	127	8	no	—
177	C₄²	—	127	16	no	—
178	C₄²	C₂²×D₁₂₇	1	4	yes	—
179	C₄²	C₂²×D₁₂₇	1	4	yes	—
180	C₄²	C₂²×D₁₂₇	1	8	yes	—
181	C₂³	—	2	4	no	—
182	C₂³	—	127	8	no	—
183	C₂³	—	127	4	no	—
184	C₂³	—	254	8	no	—
185	C₂³	C₂×C₄×D₁₂₇	1	4	yes	—
186	C₂³	C₂³×D₁₂₇	1	2	yes	—
187	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
188	Q₈	—	127	16	no	—
189	D₄	—	127	32	no	—
190	D₄	—	127	16	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	127	8	no	—
194	C₂×C₄	—	127	4	no	—
195	C₂×C₄	—	127	32	no	—
196	C₂×C₄	—	127	8	no	—
197	C₂×C₄	—	254	8	no	—
198	C₂×C₄	—	254	32	no	—
199	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
200	C₂×C₄	C₂×C₄×D₁₂₇	1	4	yes	—
201	C₂×C₄	C₂×C₄×D₁₂₇	1	16	yes	complemented
202	C₂×C₄	#14523248	1	8	yes	direct product factor
203	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
204	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
205	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
206	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	127	8	no	—
210	C₂²	—	254	16	no	—
211	C₂²	—	254	8	no	—
212	C₂²	C₂²×C₄×D₁₂₇	1	4	yes	—
213	C₂²	C₂²×C₄×D₁₂₇	1	2	yes	—
214	C₂²	C₂×#14523248	1	4	yes	—
215	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	127	8	no	—
218	C₄	—	254	8	no	—
219	C₄	C₂²×C₄×D₁₂₇	1	4	yes	—
220	C₄	C₂²×C₄×D₁₂₇	1	4	yes	—
221	C₄	C₂×#14523248	1	8	yes	direct product factor
222	C₂	—	2	4	no	—
223	C₂	—	254	8	no	—
224	C₂	C₄×#14523248	1	4	yes	direct product factor
225	C₂	C₂³×C₄×D₁₂₇	1	1	yes	characteristic
226	C₂	C₂²×#14523248	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂²×C₄²×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂⁴
prime radical	C₂²×#14523248	C₂
cosocle	C₂²×C₁₂₇	C₂⁵
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	C₂×C₄×#41
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#14523248	C₂	direct (N × H)
3. C₂×C₄×Dic₂₅₄	C₂	semidirect
4. C₂×C₄×D₅₀₈	C₂	semidirect
5. C₂×C₄×C₂²⋊D₁₂₇	C₂	semidirect
6. C₂×#16861168	C₂	semidirect
7. C₂×#16861166	C₂	semidirect
8. C₂×C₄²×D₁₂₇	C₂	semidirect
9. C₂²×C₄²×C₁₂₇	C₂	semidirect
10. C₂×#14523248	C₄	direct (N × H)
17. C₄×Dic₂₅₄	C₂²	semidirect
18. C₄×D₅₀₈	C₂²	semidirect
19. C₄×C₂²⋊D₁₂₇	C₂²	semidirect
20. #16861168	C₂²	semidirect
21. #16861166	C₂²	semidirect
28. C₄²×D₁₂₇	C₂²	semidirect
34. C₂×C₄²×C₁₂₇	C₂²	semidirect
36. #14523248	C₂×C₄	direct (N × H)
37. C₂×Dic₂₅₄	C₂×C₄	semidirect
39. C₂×D₅₀₈	C₂×C₄	semidirect
41. C₂×C₂²⋊D₁₂₇	C₂×C₄	semidirect
46. C₂×C₄×D₁₂₇	C₂×C₄	semidirect
57. C₂²×C₄×C₁₂₇	C₂×C₄	semidirect
70. Dic₂₅₄	C₂²×C₄	semidirect
71. D₅₀₈	C₂²×C₄	semidirect
72. C₂²⋊D₁₂₇	C₂²×C₄	semidirect
78. C₄×D₁₂₇	C₂²×C₄	semidirect
87. C₂×C₄×C₁₂₇	C₂²×C₄	semidirect
88. C₂×C₄×C₁₂₇	#41	semidirect
105. C₄×C₁₂₇	C₂×#41	semidirect
109. C₂×C₁₂₇	C₄×#41	semidirect
114. C₁₂₇	C₂×C₄×#41	semidirect
123. C₂²×C₄²	D₁₂₇	semidirect
148. C₂×C₄²	C₂×D₁₂₇	semidirect
165. C₂²×C₄	C₄×D₁₂₇	semidirect
201. C₂×C₄	C₂×C₄×D₁₂₇	semidirect
202. C₂×C₄	#14523248	direct (N × H)
221. C₄	C₂×#14523248	direct (N × H)
224. C₂	C₄×#14523248	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×#14523248	2
C₄×#14523248	2
C₂×C₄×Dic₂₅₄	2
C₂×C₄×D₅₀₈	2
C₂×C₄×C₂²⋊D₁₂₇	2
C₂×#16861168	2
C₂×#16861166	2
C₂×C₄²×D₁₂₇	2
C₂²×C₄²×C₁₂₇	2
C₂×C₄×#41	127

Subgroups — C2×C4×#14523248 #30651910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14523248 #30651910