Subgroups — C₂³×#14556102 #32904089

Order 18176.

19998 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	C₂⁴×D₂₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₂²⋊D₇₁	C₂	1	2	yes	maximal, complemented
3	C₂²×#14556102	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×(C₂×D₇₁)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂⁵×C₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₇₁	C₂²	1	2	yes	—
8	C₂³×D₂₈₄	—	2	16	no	—
9	C₂³×D₂₈₄	C₂²	1	28	yes	complemented
10	C₂³×C₂²⋊D₇₁	—	2	32	no	—
11	C₂³×C₂²⋊D₇₁	C₂²	1	56	yes	complemented
12	C₂⁴×Dic₇₁	C₂²	1	2	yes	—
13	C₂×#14556102	C₂²	1	448	yes	direct product factor
14	C₂²×C₄⋊Dic₇₁	C₂²	1	28	yes	complemented
15	C₂²×(C₂×D₇₁)⋊C₄	C₂²	1	56	yes	complemented
16	C₂⁶×C₇₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₇₁	—	2	16	no	—
18	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄×C₇₁	C₂²	1	28	yes	complemented
21	C₂⁴×D₇₁	—	2	16	no	—
22	C₂⁴×D₇₁	—	2	16	no	—
23	C₂⁴×D₇₁	—	2	16	no	—
24	C₂⁴×D₇₁	C₂³	1	28	yes	—
25	C₂²×D₂₈₄	—	2	224	no	—
26	C₂²×D₂₈₄	C₂³	1	112	yes	complemented
27	C₂²×C₂²⋊D₇₁	—	2	448	no	—
28	C₂²×C₂²⋊D₇₁	C₂³	1	224	yes	complemented
29	C₂³×Dic₇₁	—	2	16	no	—
30	C₂³×Dic₇₁	C₂³	1	28	yes	—
31	#14556102	C₂³	1	512	yes	direct product factor
32	C₂×C₄⋊Dic₇₁	C₂³	1	112	yes	complemented
33	C₂×(C₂×D₇₁)⋊C₄	C₂³	1	224	yes	complemented
34	C₂⁵×C₇₁	—	2	8	no	—
35	C₂⁵×C₇₁	—	2	8	no	—
36	C₂⁵×C₇₁	D₄	1	16	yes	complemented
37	C₂⁵×C₇₁	C₂³	1	14	yes	—
38	C₂⁵×C₇₁	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₇₁	—	2	8	no	—
40	C₂³×C₄×C₇₁	—	2	224	no	—
41	C₂³×C₄×C₇₁	D₄	1	16	yes	complemented
42	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
43	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
44	C₂³×C₄×C₇₁	C₂³	1	112	yes	complemented
45	C₂³×D₇₁	—	2	112	no	—
46	C₂³×D₇₁	—	2	112	no	—
47	C₂³×D₇₁	—	2	112	no	—
48	C₂³×D₇₁	—	4	128	no	—
49	C₂³×D₇₁	C₂⁴	1	56	yes	—
50	C₂×D₂₈₄	—	2	448	no	—
51	C₂×D₂₈₄	C₂⁴	1	64	yes	complemented
52	C₂×C₂²⋊D₇₁	—	2	896	no	—
53	C₂×C₂²⋊D₇₁	C₂⁴	1	128	yes	complemented
54	C₂²×Dic₇₁	—	2	112	no	—
55	C₂²×Dic₇₁	C₂⁴	1	56	yes	—
56	C₄⋊Dic₇₁	C₂⁴	1	64	yes	complemented
57	(C₂×D₇₁)⋊C₄	C₂⁴	1	128	yes	complemented
58	C₂⁴×C₇₁	—	2	128	no	—
59	C₂⁴×C₇₁	—	2	56	no	—
60	C₂⁴×C₇₁	—	2	56	no	—
61	C₂⁴×C₇₁	C₂×D₄	1	8	yes	—
62	C₂⁴×C₇₁	C₂×D₄	1	8	yes	—
63	C₂⁴×C₇₁	C₂×D₄	1	112	yes	complemented
64	C₂⁴×C₇₁	#41	1	8	yes	—
65	C₂⁴×C₇₁	C₂⁴	1	28	yes	—
66	C₂⁴×C₇₁	C₂⁴	1	7	yes	—
67	C₂²×C₄×C₇₁	—	2	56	no	—
68	C₂²×C₄×C₇₁	—	2	448	no	—
69	C₂²×C₄×C₇₁	C₂×D₄	1	112	yes	complemented
70	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
71	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
72	C₂²×C₄×C₇₁	C₂⁴	1	64	yes	complemented
73	C₂³×C₇₁	—	2	448	no	—
74	C₂³×C₇₁	—	2	56	no	—
75	C₂³×C₇₁	—	2	56	no	—
76	C₂³×C₇₁	#121	1	64	yes	complemented
77	C₂³×C₇₁	C₂²×D₄	1	28	yes	—
78	C₂³×C₇₁	C₂²×D₄	1	28	yes	—
79	C₂³×C₇₁	C₂²×D₄	1	112	yes	complemented
80	C₂³×C₇₁	C₂×#41	1	28	yes	—
81	C₂³×C₇₁	C₂⁵	1	8	yes	—
82	C₂³×C₇₁	C₂⁵	1	7	yes	—
83	C₂²×D₇₁	—	2	112	no	—
84	C₂²×D₇₁	—	2	112	no	—
85	C₂²×D₇₁	—	2	112	no	—
86	C₂²×D₇₁	—	4	448	no	—
87	C₂²×D₇₁	C₂⁵	1	16	yes	—
88	C₂×C₄×C₇₁	—	2	56	no	—
89	C₂×C₄×C₇₁	—	2	128	no	—
90	C₂×C₄×C₇₁	C₂²×D₄	1	112	yes	complemented
91	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
92	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
93	C₂²⋊D₇₁	—	2	256	no	—
94	C₂×Dic₇₁	—	2	112	no	—
95	C₂×Dic₇₁	C₂⁵	1	16	yes	—
96	D₂₈₄	—	2	128	no	—
97	C₂²×C₇₁	—	2	8	no	—
98	C₂²×C₇₁	—	2	8	no	—
99	C₂²×C₇₁	—	2	224	no	—
100	C₂²×C₇₁	C₂×#121	1	112	yes	complemented
101	C₂²×C₇₁	C₂³×D₄	1	14	yes	—
102	C₂²×C₇₁	C₂³×D₄	1	14	yes	—
103	C₂²×C₇₁	C₂³×D₄	1	16	yes	complemented
104	C₂²×C₇₁	C₂²×#41	1	14	yes	—
105	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
106	C₂×D₇₁	—	2	16	no	—
107	C₂×D₇₁	—	2	16	no	—
108	C₂×D₇₁	—	2	16	no	—
109	C₂×D₇₁	—	4	224	no	—
110	C₄×C₇₁	—	2	8	no	—
111	C₄×C₇₁	C₂³×D₄	1	16	yes	complemented
112	Dic₇₁	—	2	16	no	—
113	C₂³×#121	—	71	1	no	maximal, Sylow
114	C₂×C₇₁	—	2	16	no	—
115	C₂×C₇₁	C₂²×#121	1	28	yes	complemented
116	C₂×C₇₁	C₂⁴×D₄	1	1	yes	characteristic
117	C₂×C₇₁	C₂⁴×D₄	1	1	yes	characteristic
118	C₂×C₇₁	C₂³×#41	1	1	yes	characteristic
119	D₇₁	—	4	16	no	—
120	C₂⁴×D₄	—	71	2	no	—
121	C₂⁴×D₄	—	71	1	no	—
122	C₂⁵×C₄	D₇₁	1	1	yes	characteristic, complemented
123	C₂²×#121	—	71	56	no	—
124	C₂³×C₄⋊C₄	—	71	1	no	—
125	C₂³×C₂²⋊C₄	—	71	2	no	—
126	C₇₁	C₂³×#121	1	1	yes	characteristic, complemented, Sylow
127	C₂⁶	—	71	2	no	—
128	C₂⁶	C₂×D₇₁	1	1	yes	characteristic
129	C₂³×D₄	—	71	56	no	—
130	C₂³×D₄	—	71	28	no	—
131	C₂³×D₄	—	142	32	no	—
132	C₂³×D₄	—	142	16	no	—
133	C₂⁴×C₄	—	2	16	no	—
134	C₂⁴×C₄	—	71	2	no	—
135	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
136	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
137	C₂⁴×C₄	C₂×D₇₁	1	28	yes	complemented
138	C₂×#121	—	71	448	no	—
139	C₂²×C₄⋊C₄	—	71	28	no	—
140	C₂²×C₂²⋊C₄	—	71	56	no	—
141	C₂⁵	—	2	8	no	—
142	C₂⁵	—	2	8	no	—
143	C₂⁵	—	71	28	no	—
144	C₂⁵	—	142	16	no	—
145	C₂⁵	—	142	16	no	—
146	C₂⁵	—	142	16	no	—
147	C₂⁵	D₂₈₄	1	16	yes	complemented
148	C₂⁵	C₂²×D₇₁	1	14	yes	—
149	C₂⁵	C₂²×D₇₁	1	1	yes	characteristic
150	C₂²×D₄	—	71	224	no	—
151	C₂²×D₄	—	71	112	no	—
152	C₂²×D₄	—	142	448	no	—
153	C₂²×D₄	—	142	224	no	—
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	—	2	224	no	—
156	C₂³×C₄	—	71	28	no	—
157	C₂³×C₄	—	142	16	no	—
158	C₂³×C₄	C₂²⋊D₇₁	1	16	yes	complemented
159	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
160	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
161	C₂³×C₄	C₂²×D₇₁	1	112	yes	complemented
162	#121	—	71	512	no	—
163	C₂×C₄⋊C₄	—	71	112	no	—
164	C₂×C₂²⋊C₄	—	71	224	no	—
165	C₂⁴	—	2	56	no	—
166	C₂⁴	—	2	56	no	—
167	C₂⁴	—	2	128	no	—
168	C₂⁴	—	284	128	no	—
169	C₂⁴	—	71	56	no	—
170	C₂⁴	—	142	112	no	—
171	C₂⁴	—	142	112	no	—
172	C₂⁴	—	142	112	no	—
173	C₂⁴	C₂×C₂²⋊D₇₁	1	8	yes	—
174	C₂⁴	C₂×D₂₈₄	1	8	yes	—
175	C₂⁴	C₂×D₂₈₄	1	112	yes	complemented
176	C₂⁴	#11765865	1	8	yes	—
177	C₂⁴	C₂³×D₇₁	1	28	yes	—
178	C₂⁴	C₂³×D₇₁	1	7	yes	—
179	C₂×D₄	—	71	128	no	—
180	C₂×D₄	—	71	64	no	—
181	C₂×D₄	—	142	896	no	—
182	C₂×D₄	—	142	448	no	—
183	C₂²×C₄	—	2	56	no	—
184	C₂²×C₄	—	2	448	no	—
185	C₂²×C₄	—	71	56	no	—
186	C₂²×C₄	—	142	112	no	—
187	C₂²×C₄	C₂×C₂²⋊D₇₁	1	112	yes	complemented
188	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
189	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
190	C₂²×C₄	C₂³×D₇₁	1	64	yes	complemented
191	C₄⋊C₄	—	71	64	no	—
192	C₂²⋊C₄	—	71	128	no	—
193	C₂³	—	2	56	no	—
194	C₂³	—	2	56	no	—
195	C₂³	—	2	448	no	—
196	C₂³	—	284	448	no	—
197	C₂³	—	71	16	no	—
198	C₂³	—	142	112	no	—
199	C₂³	—	142	112	no	—
200	C₂³	—	142	112	no	—
201	C₂³	#14556102	1	64	yes	direct product factor
202	C₂³	C₂²×C₂²⋊D₇₁	1	28	yes	—
203	C₂³	C₂²×D₂₈₄	1	28	yes	—
204	C₂³	C₂²×D₂₈₄	1	112	yes	complemented
205	C₂³	C₂×#11765865	1	28	yes	—
206	C₂³	C₂⁴×D₇₁	1	8	yes	—
207	C₂³	C₂⁴×D₇₁	1	7	yes	—
208	D₄	—	142	256	no	—
209	D₄	—	142	128	no	—
210	C₂×C₄	—	2	56	no	—
211	C₂×C₄	—	2	128	no	—
212	C₂×C₄	—	71	16	no	—
213	C₂×C₄	—	142	112	no	—
214	C₂×C₄	C₂²×C₂²⋊D₇₁	1	112	yes	complemented
215	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
216	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
217	C₂²	—	2	224	no	—
218	C₂²	—	2	8	no	—
219	C₂²	—	2	8	no	—
220	C₂²	—	284	224	no	—
221	C₂²	—	142	16	no	—
222	C₂²	—	142	16	no	—
223	C₂²	—	142	16	no	—
224	C₂²	C₂×#14556102	1	112	yes	direct product factor
225	C₂²	C₂³×C₂²⋊D₇₁	1	14	yes	—
226	C₂²	C₂³×D₂₈₄	1	14	yes	—
227	C₂²	C₂³×D₂₈₄	1	16	yes	complemented
228	C₂²	C₂²×#11765865	1	14	yes	—
229	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
230	C₄	—	2	8	no	—
231	C₄	—	142	16	no	—
232	C₄	C₂³×C₂²⋊D₇₁	1	16	yes	complemented
233	C₂	—	2	16	no	—
234	C₂	—	284	16	no	—
235	C₂	C₂²×#14556102	1	28	yes	direct product factor
236	C₂	C₂⁴×C₂²⋊D₇₁	1	1	yes	characteristic
237	C₂	C₂⁴×D₂₈₄	1	1	yes	characteristic
238	C₂	C₂³×#11765865	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₇₁
commutator, cosocle	C₂²×C₇₁	C₂⁶
Frattini	C₂²	C₂⁵×D₇₁
Fitting	C₂⁵×C₄×C₇₁	C₂
socle	C₂⁵×C₇₁	C₂³
hypercenter	C₂⁵×C₄	D₇₁
hypocenter	C₇₁	C₂³×#121
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₂₈₄	C₂	semidirect
2. C₂⁴×C₂²⋊D₇₁	C₂	semidirect
3. C₂²×#14556102	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₇₁	C₂	semidirect
5. C₂³×(C₂×D₇₁)⋊C₄	C₂	semidirect
6. C₂⁵×C₄×C₇₁	C₂	semidirect
9. C₂³×D₂₈₄	C₂²	semidirect
11. C₂³×C₂²⋊D₇₁	C₂²	semidirect
13. C₂×#14556102	C₂²	direct (N × H)
14. C₂²×C₄⋊Dic₇₁	C₂²	semidirect
15. C₂²×(C₂×D₇₁)⋊C₄	C₂²	semidirect
20. C₂⁴×C₄×C₇₁	C₂²	semidirect
26. C₂²×D₂₈₄	C₂³	semidirect
28. C₂²×C₂²⋊D₇₁	C₂³	semidirect
31. #14556102	C₂³	direct (N × H)
32. C₂×C₄⋊Dic₇₁	C₂³	semidirect
33. C₂×(C₂×D₇₁)⋊C₄	C₂³	semidirect
36. C₂⁵×C₇₁	D₄	semidirect
41. C₂³×C₄×C₇₁	D₄	semidirect
44. C₂³×C₄×C₇₁	C₂³	semidirect
51. C₂×D₂₈₄	C₂⁴	semidirect
53. C₂×C₂²⋊D₇₁	C₂⁴	semidirect
56. C₄⋊Dic₇₁	C₂⁴	semidirect
57. (C₂×D₇₁)⋊C₄	C₂⁴	semidirect
63. C₂⁴×C₇₁	C₂×D₄	semidirect
69. C₂²×C₄×C₇₁	C₂×D₄	semidirect
72. C₂²×C₄×C₇₁	C₂⁴	semidirect
76. C₂³×C₇₁	#121	semidirect
79. C₂³×C₇₁	C₂²×D₄	semidirect
90. C₂×C₄×C₇₁	C₂²×D₄	semidirect
100. C₂²×C₇₁	C₂×#121	semidirect
103. C₂²×C₇₁	C₂³×D₄	semidirect
111. C₄×C₇₁	C₂³×D₄	semidirect
115. C₂×C₇₁	C₂²×#121	semidirect
122. C₂⁵×C₄	D₇₁	semidirect
126. C₇₁	C₂³×#121	semidirect
137. C₂⁴×C₄	C₂×D₇₁	semidirect
147. C₂⁵	D₂₈₄	semidirect
158. C₂³×C₄	C₂²⋊D₇₁	semidirect
161. C₂³×C₄	C₂²×D₇₁	semidirect
175. C₂⁴	C₂×D₂₈₄	semidirect
187. C₂²×C₄	C₂×C₂²⋊D₇₁	semidirect
190. C₂²×C₄	C₂³×D₇₁	semidirect
201. C₂³	#14556102	direct (N × H)
204. C₂³	C₂²×D₂₈₄	semidirect
214. C₂×C₄	C₂²×C₂²⋊D₇₁	semidirect
224. C₂²	C₂×#14556102	direct (N × H)
227. C₂²	C₂³×D₂₈₄	semidirect
232. C₄	C₂³×C₂²⋊D₇₁	semidirect
235. C₂	C₂²×#14556102	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×D₂₈₄	2
C₂⁴×C₂²⋊D₇₁	2
C₂²×#14556102	2
C₂³×C₄⋊Dic₇₁	2
C₂³×(C₂×D₇₁)⋊C₄	2
C₂⁵×C₄×C₇₁	2
C₂³×#121	71

Subgroups — C23×#14556102 #32904089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556102 #32904089