Subgroups — C₂³×C₄×#127 #72289379

Order 1024.

253365 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	C₂⁴×#127	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₄	C₂	1	6	yes	maximal, complemented
3	C₂²×C₄×#127	C₂	1	112	yes	maximal, direct product factor
4	C₂³×#390	C₂	1	6	yes	maximal, complemented
5	C₂³×#384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄³	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₄	C₂²	1	6	yes	—
8	C₂⁶×C₄	C₂²	1	4	yes	—
9	C₂³×#127	C₄	1	64	yes	direct product factor
10	C₂³×#127	C₂²	1	56	yes	—
11	C₂³×C₄×D₄	—	2	192	no	—
12	C₂³×C₄×D₄	C₂²	1	336	yes	complemented
13	C₂⁴×C₄⋊C₄	C₂²	1	6	yes	—
14	C₂⁴×C₂²⋊C₄	C₂²	1	12	yes	—
15	C₂⁴×C₄²	C₂²	1	3	yes	—
16	C₂⁴×C₄²	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄²	C₂²	1	3	yes	—
18	C₂×C₄×#127	C₂²	1	1792	yes	direct product factor
19	C₂²×#390	C₂²	1	336	yes	complemented
20	C₂²×#384	C₂²	1	56	yes	complemented
21	C₂²×C₄³	C₂²	1	56	yes	complemented
22	C₂⁷	C₂³	1	4	yes	—
23	C₂⁴×D₄	—	2	96	no	—
24	C₂⁴×D₄	—	2	96	no	—
25	C₂⁴×D₄	C₂×C₄	1	192	yes	complemented
26	C₂⁴×D₄	C₂³	1	168	yes	—
27	C₂⁵×C₄	—	2	192	no	—
28	C₂⁵×C₄	C₂³	1	4	yes	—
29	C₂⁵×C₄	C₂³	1	112	yes	—
30	C₂⁵×C₄	C₂³	1	3	yes	—
31	C₂⁵×C₄	C₂³	1	3	yes	—
32	C₂⁵×C₄	C₂³	1	1	yes	characteristic
33	C₂²×#127	C₂×C₄	1	1792	yes	direct product factor
34	C₂²×#127	C₂³	1	448	yes	—
35	C₂²×C₄×D₄	—	2	5376	no	—
36	C₂²×C₄×D₄	C₂³	1	2688	yes	complemented
37	C₂³×C₄⋊C₄	—	2	96	no	—
38	C₂³×C₄⋊C₄	C₂³	1	168	yes	—
39	C₂³×C₂²⋊C₄	—	2	192	no	—
40	C₂³×C₂²⋊C₄	C₂³	1	336	yes	—
41	C₂³×C₄²	—	2	48	no	—
42	C₂³×C₄²	C₂×C₄	1	32	yes	complemented
43	C₂³×C₄²	D₄	1	96	yes	complemented
44	C₂³×C₄²	C₂³	1	84	yes	—
45	C₂³×C₄²	C₂³	1	28	yes	—
46	C₂³×C₄²	C₂³	1	84	yes	—
47	C₄×#127	C₂³	1	4096	yes	direct product factor
48	C₂×#390	C₂³	1	2688	yes	complemented
49	C₂×#384	C₂³	1	448	yes	complemented
50	C₂×C₄³	C₂³	1	448	yes	complemented
51	C₂⁶	—	2	96	no	—
52	C₂⁶	—	2	96	no	—
53	C₂⁶	C₂²×C₄	1	64	yes	—
54	C₂⁶	C₂⁴	1	56	yes	—
55	C₂⁶	C₂⁴	1	1	yes	characteristic
56	C₂³×D₄	—	2	3072	no	—
57	C₂³×D₄	—	2	1344	no	—
58	C₂³×D₄	—	2	1344	no	—
59	C₂³×D₄	C₂²×C₄	1	2688	yes	complemented
60	C₂³×D₄	C₂⁴	1	672	yes	—
61	C₂⁴×C₄	—	2	48	no	—
62	C₂⁴×C₄	—	2	96	no	—
63	C₂⁴×C₄	—	2	2688	no	—
64	C₂⁴×C₄	—	4	1024	no	—
65	C₂⁴×C₄	C₂²×C₄	1	48	yes	—
66	C₂⁴×C₄	C₂×D₄	1	48	yes	—
67	C₂⁴×C₄	C₂×D₄	1	48	yes	—
68	C₂⁴×C₄	C₂×D₄	1	48	yes	—
69	C₂⁴×C₄	#41	1	48	yes	—
70	C₂⁴×C₄	C₂⁴	1	56	yes	—
71	C₂⁴×C₄	C₂⁴	1	448	yes	—
72	C₂⁴×C₄	C₂⁴	1	42	yes	—
73	C₂⁴×C₄	C₂⁴	1	42	yes	—
74	C₂⁴×C₄	C₂⁴	1	14	yes	—
75	C₂×#127	C₂²×C₄	1	7168	yes	direct product factor
76	C₂×#127	C₂⁴	1	512	yes	—
77	C₂×C₄×D₄	—	2	21504	no	—
78	C₂×C₄×D₄	C₂⁴	1	3072	yes	complemented
79	C₂²×C₄⋊C₄	—	2	1344	no	—
80	C₂²×C₄⋊C₄	C₂⁴	1	672	yes	—
81	C₂²×C₂²⋊C₄	—	2	2688	no	—
82	C₂²×C₂²⋊C₄	C₂⁴	1	1344	yes	—
83	C₂²×C₄²	—	2	672	no	—
84	C₂²×C₄²	C₂²×C₄	1	448	yes	complemented
85	C₂²×C₄²	C₂×D₄	1	1344	yes	complemented
86	C₂²×C₄²	C₂⁴	1	336	yes	—
87	C₂²×C₄²	C₂⁴	1	112	yes	—
88	C₂²×C₄²	C₂⁴	1	336	yes	—
89	#390	C₂⁴	1	3072	yes	complemented
90	#384	C₂⁴	1	512	yes	complemented
91	C₄³	C₂⁴	1	512	yes	complemented
92	C₂⁵	—	2	1536	no	—
93	C₂⁵	—	2	672	no	—
94	C₂⁵	—	2	672	no	—
95	C₂⁵	—	4	256	no	—
96	C₂⁵	—	4	768	no	—
97	C₂⁵	C₂³×C₄	1	448	yes	—
98	C₂⁵	C₂³×C₄	1	8	yes	—
99	C₂⁵	C₂²×D₄	1	24	yes	—
100	C₂⁵	C₂×#41	1	24	yes	—
101	C₂⁵	C₂⁵	1	112	yes	—
102	C₂⁵	C₂⁵	1	7	yes	—
103	C₂²×D₄	—	2	21504	no	—
104	C₂²×D₄	—	2	2688	no	—
105	C₂²×D₄	—	2	2688	no	—
106	C₂²×D₄	C₂³×C₄	1	5376	yes	complemented
107	C₂²×D₄	C₂⁵	1	384	yes	—
108	C₂³×C₄	—	2	384	no	—
109	C₂³×C₄	—	2	336	no	—
110	C₂³×C₄	—	2	672	no	—
111	C₂³×C₄	—	2	5376	no	—
112	C₂³×C₄	—	4	256	no	—
113	C₂³×C₄	—	4	7168	no	—
114	C₂³×C₄	C₄×D₄	1	768	yes	complemented
115	C₂³×C₄	#127	1	256	yes	direct product factor
116	C₂³×C₄	C₂³×C₄	1	336	yes	—
117	C₂³×C₄	C₂²×D₄	1	336	yes	—
118	C₂³×C₄	C₂²×D₄	1	336	yes	—
119	C₂³×C₄	C₂²×D₄	1	336	yes	—
120	C₂³×C₄	C₂×#41	1	336	yes	—
121	C₂³×C₄	C₂⁵	1	112	yes	—
122	C₂³×C₄	C₂⁵	1	256	yes	—
123	C₂³×C₄	C₂⁵	1	84	yes	—
124	C₂³×C₄	C₂⁵	1	84	yes	—
125	C₂³×C₄	C₂⁵	1	28	yes	—
126	#127	C₂³×C₄	1	4096	yes	direct product factor
127	C₄×D₄	—	2	12288	no	—
128	C₂×C₄⋊C₄	—	2	2688	no	—
129	C₂×C₄⋊C₄	C₂⁵	1	384	yes	—
130	C₂×C₂²⋊C₄	—	2	5376	no	—
131	C₂×C₂²⋊C₄	C₂⁵	1	768	yes	—
132	C₂×C₄²	—	2	1344	no	—
133	C₂×C₄²	C₂³×C₄	1	896	yes	complemented
134	C₂×C₄²	C₂²×D₄	1	2688	yes	complemented
135	C₂×C₄²	C₂⁵	1	192	yes	—
136	C₂×C₄²	C₂⁵	1	64	yes	—
137	C₂×C₄²	C₂⁵	1	192	yes	—
138	C₂⁴	—	2	672	no	—
139	C₂⁴	—	2	672	no	—
140	C₂⁴	—	2	5376	no	—
141	C₂⁴	—	4	4096	no	—
142	C₂⁴	—	4	896	no	—
143	C₂⁴	—	4	2688	no	—
144	C₂⁴	C₂×C₄×D₄	1	192	yes	—
145	C₂⁴	C₂×#127	1	64	yes	—
146	C₂⁴	#532	1	192	yes	—
147	C₂⁴	C₂⁴×C₄	1	448	yes	—
148	C₂⁴	C₂⁴×C₄	1	28	yes	—
149	C₂⁴	C₂³×D₄	1	84	yes	—
150	C₂⁴	C₂²×#41	1	84	yes	—
151	C₂⁴	C₂⁶	1	32	yes	—
152	C₂⁴	C₂⁶	1	7	yes	—
153	C₂×D₄	—	2	768	no	—
154	C₂×D₄	—	2	768	no	—
155	C₂×D₄	—	2	21504	no	—
156	C₂×D₄	C₂⁴×C₄	1	1536	yes	complemented
157	C₂²×C₄	—	2	336	no	—
158	C₂²×C₄	—	2	1344	no	—
159	C₂²×C₄	—	2	672	no	—
160	C₂²×C₄	—	2	1536	no	—
161	C₂²×C₄	—	4	896	no	—
162	C₂²×C₄	—	4	7168	no	—
163	C₂²×C₄	C₂×C₄×D₄	1	2688	yes	complemented
164	C₂²×C₄	C₂×#127	1	896	yes	direct product factor
165	C₂²×C₄	C₂⁴×C₄	1	336	yes	—
166	C₂²×C₄	C₂³×D₄	1	336	yes	—
167	C₂²×C₄	C₂³×D₄	1	336	yes	—
168	C₂²×C₄	C₂³×D₄	1	336	yes	—
169	C₂²×C₄	C₂²×#41	1	336	yes	—
170	C₂²×C₄	C₂⁶	1	32	yes	—
171	C₂²×C₄	C₂⁶	1	24	yes	—
172	C₂²×C₄	C₂⁶	1	24	yes	—
173	C₂²×C₄	C₂⁶	1	8	yes	—
174	C₄⋊C₄	—	2	768	no	—
175	C₂²⋊C₄	—	2	1536	no	—
176	C₄²	—	2	384	no	—
177	C₄²	C₂⁴×C₄	1	256	yes	complemented
178	C₄²	C₂³×D₄	1	768	yes	complemented
179	C₂³	—	2	96	no	—
180	C₂³	—	2	96	no	—
181	C₂³	—	2	2688	no	—
182	C₂³	—	4	448	no	—
183	C₂³	—	4	1344	no	—
184	C₂³	—	4	7168	no	—
185	C₂³	C₄×#127	1	512	yes	direct product factor
186	C₂³	C₂²×C₄×D₄	1	336	yes	—
187	C₂³	C₂²×#127	1	112	yes	—
188	C₂³	C₂×#532	1	336	yes	—
189	C₂³	C₂⁵×C₄	1	64	yes	—
190	C₂³	C₂⁵×C₄	1	14	yes	—
191	C₂³	C₂⁴×D₄	1	42	yes	—
192	C₂³	C₂³×#41	1	42	yes	—
193	C₂³	C₂⁷	1	1	yes	characteristic
194	D₄	—	2	3072	no	—
195	C₂×C₄	—	2	48	no	—
196	C₂×C₄	—	2	672	no	—
197	C₂×C₄	—	2	96	no	—
198	C₂×C₄	—	4	448	no	—
199	C₂×C₄	—	4	1024	no	—
200	C₂×C₄	C₂²×C₄×D₄	1	1344	yes	complemented
201	C₂×C₄	C₂²×#127	1	448	yes	direct product factor
202	C₂×C₄	C₂⁵×C₄	1	48	yes	—
203	C₂×C₄	C₂⁴×D₄	1	48	yes	—
204	C₂×C₄	C₂⁴×D₄	1	48	yes	—
205	C₂×C₄	C₂⁴×D₄	1	48	yes	—
206	C₂×C₄	C₂³×#41	1	48	yes	—
207	C₂²	—	2	192	no	—
208	C₂²	—	4	32	no	—
209	C₂²	—	4	1792	no	—
210	C₂²	—	4	96	no	—
211	C₂²	C₂×C₄×#127	1	448	yes	direct product factor
212	C₂²	C₂³×C₄×D₄	1	84	yes	—
213	C₂²	C₂³×#127	1	28	yes	—
214	C₂²	C₂²×#532	1	84	yes	—
215	C₂²	C₂⁶×C₄	1	1	yes	characteristic
216	C₂²	C₂⁵×D₄	1	3	yes	—
217	C₂²	C₂⁴×#41	1	3	yes	—
218	C₄	—	2	48	no	—
219	C₄	—	4	32	no	—
220	C₄	C₂³×C₄×D₄	1	96	yes	complemented
221	C₄	C₂³×#127	1	32	yes	direct product factor
222	C₂	—	4	64	no	—
223	C₂	C₂²×C₄×#127	1	56	yes	direct product factor
224	C₂	C₂⁴×C₄×D₄	1	3	yes	—
225	C₂	C₂⁴×#127	1	1	yes	characteristic
226	C₂	C₂³×#532	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₄	C₂³
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁴×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₄	C₂	semidirect
3. C₂²×C₄×#127	C₂	direct (N × H)
4. C₂³×#390	C₂	semidirect
5. C₂³×#384	C₂	semidirect
6. C₂³×C₄³	C₂	semidirect
9. C₂³×#127	C₄	direct (N × H)
12. C₂³×C₄×D₄	C₂²	semidirect
18. C₂×C₄×#127	C₂²	direct (N × H)
19. C₂²×#390	C₂²	semidirect
20. C₂²×#384	C₂²	semidirect
21. C₂²×C₄³	C₂²	semidirect
25. C₂⁴×D₄	C₂×C₄	semidirect
33. C₂²×#127	C₂×C₄	direct (N × H)
36. C₂²×C₄×D₄	C₂³	semidirect
42. C₂³×C₄²	C₂×C₄	semidirect
43. C₂³×C₄²	D₄	semidirect
47. C₄×#127	C₂³	direct (N × H)
48. C₂×#390	C₂³	semidirect
49. C₂×#384	C₂³	semidirect
50. C₂×C₄³	C₂³	semidirect
59. C₂³×D₄	C₂²×C₄	semidirect
75. C₂×#127	C₂²×C₄	direct (N × H)
78. C₂×C₄×D₄	C₂⁴	semidirect
84. C₂²×C₄²	C₂²×C₄	semidirect
85. C₂²×C₄²	C₂×D₄	semidirect
89. #390	C₂⁴	semidirect
90. #384	C₂⁴	semidirect
91. C₄³	C₂⁴	semidirect
106. C₂²×D₄	C₂³×C₄	semidirect
114. C₂³×C₄	C₄×D₄	semidirect
115. C₂³×C₄	#127	direct (N × H)
126. #127	C₂³×C₄	direct (N × H)
133. C₂×C₄²	C₂³×C₄	semidirect
134. C₂×C₄²	C₂²×D₄	semidirect
156. C₂×D₄	C₂⁴×C₄	semidirect
163. C₂²×C₄	C₂×C₄×D₄	semidirect
164. C₂²×C₄	C₂×#127	direct (N × H)
177. C₄²	C₂⁴×C₄	semidirect
178. C₄²	C₂³×D₄	semidirect
185. C₂³	C₄×#127	direct (N × H)
200. C₂×C₄	C₂²×C₄×D₄	semidirect
201. C₂×C₄	C₂²×#127	direct (N × H)
211. C₂²	C₂×C₄×#127	direct (N × H)
220. C₄	C₂³×C₄×D₄	semidirect
221. C₄	C₂³×#127	direct (N × H)
223. C₂	C₂²×C₄×#127	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#127	2
C₂⁴×C₄×D₄	2
C₂²×C₄×#127	2
C₂³×#390	2
C₂³×#384	2
C₂³×C₄³	2

Subgroups — C23×C4×#127 #72289379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×#127 #72289379