Subgroups — C₂³×#14556123 #32904124

Order 18176.

9486 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556123	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×Dic₇₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×(C₂×D₇₁)⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂³×(C₂×D₇₁)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₂³×(C₂×D₇₁)⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₇₁	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
13	C₂×#14556123	C₂²	1	448	yes	direct product factor
14	C₂²×C₄⋊Dic₇₁	C₂²	1	28	yes	complemented
15	C₂²×Dic₇₁⋊C₄	C₂²	1	28	yes	complemented
16	C₂²×(C₂×D₇₁)⋊C₄	C₂²	1	28	yes	—
17	C₂²×(C₂×D₇₁)⋊C₄	C₂²	1	28	yes	—
18	C₂²×(C₂×D₇₁)⋊C₄	C₂²	1	28	yes	—
19	C₂²×C₄×Dic₇₁	C₂²	1	28	yes	complemented
20	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
21	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
22	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₇₁	C₂²	1	28	yes	complemented
24	C₂⁴×D₇₁	—	2	8	no	—
25	C₂⁴×D₇₁	—	2	8	no	—
26	C₂⁴×D₇₁	—	2	8	no	—
27	C₂⁴×D₇₁	C₂³	1	14	yes	—
28	C₂³×Dic₇₁	—	2	8	no	—
29	C₂³×Dic₇₁	—	2	8	no	—
30	C₂³×Dic₇₁	—	2	8	no	—
31	C₂³×Dic₇₁	C₂³	1	14	yes	—
32	C₂³×Dic₇₁	C₂³	1	14	yes	—
33	C₂³×Dic₇₁	C₂³	1	14	yes	—
34	#14556123	C₂³	1	512	yes	direct product factor
35	C₂×C₄⋊Dic₇₁	C₂³	1	112	yes	complemented
36	C₂×Dic₇₁⋊C₄	C₂³	1	112	yes	complemented
37	C₂×(C₂×D₇₁)⋊C₄	C₂³	1	112	yes	—
38	C₂×(C₂×D₇₁)⋊C₄	C₂³	1	112	yes	—
39	C₂×(C₂×D₇₁)⋊C₄	C₂³	1	112	yes	—
40	C₂×C₄×Dic₇₁	C₂³	1	112	yes	complemented
41	C₂⁵×C₇₁	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₇₁	—	2	8	no	—
43	C₂³×C₄×C₇₁	—	2	8	no	—
44	C₂³×C₄×C₇₁	—	2	8	no	—
45	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
46	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
47	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
48	C₂×C₄⋊C₄×C₇₁	C₂³	1	112	yes	complemented
49	C₂³×D₇₁	—	2	56	no	—
50	C₂³×D₇₁	—	2	56	no	—
51	C₂³×D₇₁	—	2	56	no	—
52	C₂³×D₇₁	—	4	64	no	—
53	C₂³×D₇₁	C₂⁴	1	28	yes	—
54	C₂²×Dic₇₁	—	2	56	no	—
55	C₂²×Dic₇₁	—	2	56	no	—
56	C₂²×Dic₇₁	—	2	56	no	—
57	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
58	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
59	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
60	C₄⋊Dic₇₁	C₂⁴	1	64	yes	complemented
61	Dic₇₁⋊C₄	C₂⁴	1	64	yes	complemented
62	(C₂×D₇₁)⋊C₄	C₂⁴	1	64	yes	—
63	(C₂×D₇₁)⋊C₄	C₂⁴	1	64	yes	—
64	(C₂×D₇₁)⋊C₄	C₂⁴	1	64	yes	—
65	C₄×Dic₇₁	C₂⁴	1	64	yes	complemented
66	C₂⁴×C₇₁	#41	1	8	yes	—
67	C₂⁴×C₇₁	#41	1	8	yes	—
68	C₂⁴×C₇₁	#41	1	8	yes	—
69	C₂⁴×C₇₁	C₂⁴	1	7	yes	—
70	C₂²×C₄×C₇₁	—	2	56	no	—
71	C₂²×C₄×C₇₁	—	2	56	no	—
72	C₂²×C₄×C₇₁	—	2	56	no	—
73	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
74	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
75	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
76	C₄⋊C₄×C₇₁	C₂⁴	1	64	yes	complemented
77	C₂³×C₇₁	#126	1	64	yes	complemented
78	C₂³×C₇₁	C₂×#41	1	28	yes	—
79	C₂³×C₇₁	C₂×#41	1	28	yes	—
80	C₂³×C₇₁	C₂×#41	1	28	yes	—
81	C₂³×C₇₁	C₂⁵	1	7	yes	—
82	C₂²×D₇₁	—	2	56	no	—
83	C₂²×D₇₁	—	2	56	no	—
84	C₂²×D₇₁	—	2	56	no	—
85	C₂²×D₇₁	—	4	224	no	—
86	C₂²×D₇₁	C₂⁵	1	8	yes	—
87	C₂×C₄×C₇₁	—	2	56	no	—
88	C₂×C₄×C₇₁	—	2	56	no	—
89	C₂×C₄×C₇₁	—	2	56	no	—
90	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
91	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
92	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
93	C₂×Dic₇₁	—	2	56	no	—
94	C₂×Dic₇₁	—	2	56	no	—
95	C₂×Dic₇₁	—	2	56	no	—
96	C₂×Dic₇₁	C₂⁵	1	8	yes	—
97	C₂×Dic₇₁	C₂⁵	1	8	yes	—
98	C₂×Dic₇₁	C₂⁵	1	8	yes	—
99	C₂²×C₇₁	C₂×#126	1	112	yes	complemented
100	C₂²×C₇₁	C₂²×#41	1	14	yes	—
101	C₂²×C₇₁	C₂²×#41	1	14	yes	—
102	C₂²×C₇₁	C₂²×#41	1	14	yes	—
103	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
104	C₂×D₇₁	—	2	8	no	—
105	C₂×D₇₁	—	2	8	no	—
106	C₂×D₇₁	—	2	8	no	—
107	C₂×D₇₁	—	4	112	no	—
108	C₄×C₇₁	—	2	8	no	—
109	C₄×C₇₁	—	2	8	no	—
110	C₄×C₇₁	—	2	8	no	—
111	Dic₇₁	—	2	8	no	—
112	Dic₇₁	—	2	8	no	—
113	Dic₇₁	—	2	8	no	—
114	C₂³×#126	—	71	1	no	maximal, Sylow
115	C₂×C₇₁	C₂²×#126	1	28	yes	complemented
116	C₂×C₇₁	C₂³×#41	1	1	yes	characteristic
117	C₂×C₇₁	C₂³×#41	1	1	yes	characteristic
118	C₂×C₇₁	C₂³×#41	1	1	yes	characteristic
119	D₇₁	—	4	8	no	—
120	C₂²×#126	—	71	56	no	—
121	C₂³×C₄⋊C₄	—	71	1	no	—
122	C₂³×C₄⋊C₄	—	71	1	no	—
123	C₂³×C₄⋊C₄	D₇₁	1	1	yes	characteristic, complemented
124	C₂³×C₂²⋊C₄	—	71	1	no	—
125	C₂³×C₂²⋊C₄	—	71	1	no	—
126	C₂³×C₂²⋊C₄	—	71	1	no	—
127	C₂³×C₄²	—	71	1	no	—
128	C₇₁	C₂³×#126	1	1	yes	characteristic, complemented, Sylow
129	C₂⁶	—	71	1	no	—
130	C₂⁴×C₄	—	71	1	no	—
131	C₂⁴×C₄	—	71	1	no	—
132	C₂⁴×C₄	—	71	1	no	—
133	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
134	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
135	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
136	C₂×#126	—	71	448	no	—
137	C₂²×C₄⋊C₄	—	71	28	no	—
138	C₂²×C₄⋊C₄	—	71	28	no	—
139	C₂²×C₄⋊C₄	C₂×D₇₁	1	28	yes	complemented
140	C₂²×C₂²⋊C₄	—	71	28	no	—
141	C₂²×C₂²⋊C₄	—	71	28	no	—
142	C₂²×C₂²⋊C₄	—	71	28	no	—
143	C₂²×C₄²	—	71	28	no	—
144	C₂⁵	—	71	14	no	—
145	C₂⁵	—	142	8	no	—
146	C₂⁵	—	142	8	no	—
147	C₂⁵	—	142	8	no	—
148	C₂⁵	C₂²×D₇₁	1	1	yes	characteristic
149	C₂³×C₄	—	2	8	no	—
150	C₂³×C₄	—	2	8	no	—
151	C₂³×C₄	—	2	8	no	—
152	C₂³×C₄	—	71	14	no	—
153	C₂³×C₄	—	71	14	no	—
154	C₂³×C₄	—	71	14	no	—
155	C₂³×C₄	—	142	8	no	—
156	C₂³×C₄	—	142	8	no	—
157	C₂³×C₄	—	142	8	no	—
158	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
159	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
160	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
161	#126	—	71	512	no	—
162	C₂×C₄⋊C₄	—	71	112	no	—
163	C₂×C₄⋊C₄	—	71	112	no	—
164	C₂×C₄⋊C₄	C₂²×D₇₁	1	112	yes	complemented
165	C₂×C₂²⋊C₄	—	71	112	no	—
166	C₂×C₂²⋊C₄	—	71	112	no	—
167	C₂×C₂²⋊C₄	—	71	112	no	—
168	C₂×C₄²	—	71	112	no	—
169	C₂⁴	—	284	64	no	—
170	C₂⁴	—	71	28	no	—
171	C₂⁴	—	142	56	no	—
172	C₂⁴	—	142	56	no	—
173	C₂⁴	—	142	56	no	—
174	C₂⁴	#11765864	1	8	yes	—
175	C₂⁴	#11765865	1	8	yes	—
176	C₂⁴	#11765866	1	8	yes	—
177	C₂⁴	C₂³×D₇₁	1	7	yes	—
178	C₂²×C₄	—	2	56	no	—
179	C₂²×C₄	—	2	56	no	—
180	C₂²×C₄	—	2	56	no	—
181	C₂²×C₄	—	71	28	no	—
182	C₂²×C₄	—	71	28	no	—
183	C₂²×C₄	—	71	28	no	—
184	C₂²×C₄	—	142	56	no	—
185	C₂²×C₄	—	142	56	no	—
186	C₂²×C₄	—	142	56	no	—
187	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
188	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
189	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
190	C₄⋊C₄	—	71	64	no	—
191	C₄⋊C₄	—	71	64	no	—
192	C₄⋊C₄	C₂³×D₇₁	1	64	yes	complemented
193	C₂²⋊C₄	—	71	64	no	—
194	C₂²⋊C₄	—	71	64	no	—
195	C₂²⋊C₄	—	71	64	no	—
196	C₄²	—	71	64	no	—
197	C₂³	—	284	224	no	—
198	C₂³	—	71	8	no	—
199	C₂³	—	142	56	no	—
200	C₂³	—	142	56	no	—
201	C₂³	—	142	56	no	—
202	C₂³	#14556123	1	64	yes	direct product factor
203	C₂³	C₂×#11765864	1	28	yes	—
204	C₂³	C₂×#11765865	1	28	yes	—
205	C₂³	C₂×#11765866	1	28	yes	—
206	C₂³	C₂⁴×D₇₁	1	7	yes	—
207	C₂×C₄	—	2	56	no	—
208	C₂×C₄	—	2	56	no	—
209	C₂×C₄	—	2	56	no	—
210	C₂×C₄	—	71	8	no	—
211	C₂×C₄	—	71	8	no	—
212	C₂×C₄	—	71	8	no	—
213	C₂×C₄	—	142	56	no	—
214	C₂×C₄	—	142	56	no	—
215	C₂×C₄	—	142	56	no	—
216	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
217	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
218	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
219	C₂²	—	284	112	no	—
220	C₂²	—	142	8	no	—
221	C₂²	—	142	8	no	—
222	C₂²	—	142	8	no	—
223	C₂²	C₂×#14556123	1	112	yes	direct product factor
224	C₂²	C₂²×#11765864	1	14	yes	—
225	C₂²	C₂²×#11765865	1	14	yes	—
226	C₂²	C₂²×#11765866	1	14	yes	—
227	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
228	C₄	—	2	8	no	—
229	C₄	—	2	8	no	—
230	C₄	—	2	8	no	—
231	C₄	—	142	8	no	—
232	C₄	—	142	8	no	—
233	C₄	—	142	8	no	—
234	C₂	—	284	8	no	—
235	C₂	C₂²×#14556123	1	28	yes	direct product factor
236	C₂	C₂³×#11765864	1	1	yes	characteristic
237	C₂	C₂³×#11765865	1	1	yes	characteristic
238	C₂	C₂³×#11765866	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₇₁
commutator, cosocle	C₂²×C₇₁	C₂⁶
Frattini	C₂²	C₂⁵×D₇₁
Fitting	C₂³×C₄⋊C₄×C₇₁	C₂
socle	C₂⁵×C₇₁	C₂³
prime radical	C₂⁵×D₇₁	C₂²
hypercenter	C₂³×C₄⋊C₄	D₇₁
hypocenter	C₇₁	C₂³×#126
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14556123	C₂	direct (N × H)
2. C₂³×C₄⋊Dic₇₁	C₂	semidirect
3. C₂³×Dic₇₁⋊C₄	C₂	semidirect
7. C₂³×C₄×Dic₇₁	C₂	semidirect
8. C₂³×C₄⋊C₄×C₇₁	C₂	semidirect
13. C₂×#14556123	C₂²	direct (N × H)
14. C₂²×C₄⋊Dic₇₁	C₂²	semidirect
15. C₂²×Dic₇₁⋊C₄	C₂²	semidirect
19. C₂²×C₄×Dic₇₁	C₂²	semidirect
23. C₂²×C₄⋊C₄×C₇₁	C₂²	semidirect
34. #14556123	C₂³	direct (N × H)
35. C₂×C₄⋊Dic₇₁	C₂³	semidirect
36. C₂×Dic₇₁⋊C₄	C₂³	semidirect
40. C₂×C₄×Dic₇₁	C₂³	semidirect
48. C₂×C₄⋊C₄×C₇₁	C₂³	semidirect
60. C₄⋊Dic₇₁	C₂⁴	semidirect
61. Dic₇₁⋊C₄	C₂⁴	semidirect
65. C₄×Dic₇₁	C₂⁴	semidirect
76. C₄⋊C₄×C₇₁	C₂⁴	semidirect
77. C₂³×C₇₁	#126	semidirect
99. C₂²×C₇₁	C₂×#126	semidirect
115. C₂×C₇₁	C₂²×#126	semidirect
123. C₂³×C₄⋊C₄	D₇₁	semidirect
128. C₇₁	C₂³×#126	semidirect
139. C₂²×C₄⋊C₄	C₂×D₇₁	semidirect
164. C₂×C₄⋊C₄	C₂²×D₇₁	semidirect
192. C₄⋊C₄	C₂³×D₇₁	semidirect
202. C₂³	#14556123	direct (N × H)
223. C₂²	C₂×#14556123	direct (N × H)
235. C₂	C₂²×#14556123	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×#14556123	2
C₂³×C₄⋊Dic₇₁	2
C₂³×Dic₇₁⋊C₄	2
C₂³×(C₂×D₇₁)⋊C₄	2
C₂³×C₄×Dic₇₁	2
C₂³×C₄⋊C₄×C₇₁	2
C₂³×#126	71

Subgroups — C23×#14556123 #32904124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556123 #32904124