Subgroups — C₂³×#14556049 #32860960

Order 18176.

7646 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14556049	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×D₄×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₇₁	—	2	1	no	—
6	C₂³×C₇₁⋊C₈	—	2	8	no	—
7	C₂³×C₇₁⋊C₈	C₂²	1	14	yes	complemented
8	C₂²×C₄⋊Dic₇₁	—	2	7	no	—
9	C₂²×C₄⋊Dic₇₁	C₂²	1	14	yes	complemented
10	C₂×#14556049	C₂²	1	112	yes	direct product factor
11	C₂⁶×C₇₁	—	2	1	no	—
12	C₂³×D₄×C₇₁	—	2	7	no	—
13	C₂³×D₄×C₇₁	—	2	8	no	—
14	C₂³×D₄×C₇₁	C₄	1	16	yes	complemented
15	C₂³×D₄×C₇₁	C₂²	1	14	yes	—
16	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
17	C₂³×Dic₇₁	—	2	14	no	—
18	C₂³×Dic₇₁	—	4	8	no	—
19	C₂²×C₇₁⋊C₈	—	2	56	no	—
20	C₂²×C₇₁⋊C₈	C₂³	1	28	yes	complemented
21	C₂×C₄⋊Dic₇₁	—	2	42	no	—
22	C₂×C₄⋊Dic₇₁	C₂³	1	28	yes	complemented
23	#14556049	C₂³	1	64	yes	direct product factor
24	C₂⁵×C₇₁	—	2	16	no	—
25	C₂⁵×C₇₁	—	2	14	no	—
26	C₂⁵×C₇₁	—	4	8	no	—
27	C₂⁵×C₇₁	—	4	8	no	—
28	C₂⁵×C₇₁	D₄	1	1	yes	characteristic
29	C₂²×D₄×C₇₁	—	2	112	no	—
30	C₂²×D₄×C₇₁	—	2	42	no	—
31	C₂²×D₄×C₇₁	—	2	56	no	—
32	C₂²×D₄×C₇₁	C₂×C₄	1	112	yes	complemented
33	C₂²×D₄×C₇₁	C₂³	1	28	yes	—
34	C₂³×C₄×C₇₁	C₂×C₄	1	8	yes	—
35	C₂³×C₄×C₇₁	D₄	1	7	yes	—
36	C₂³×C₄×C₇₁	D₄	1	8	yes	—
37	C₂³×C₄×C₇₁	C₂³	1	7	yes	—
38	C₂²×Dic₇₁	—	2	28	no	—
39	C₂²×Dic₇₁	—	4	56	no	—
40	C₂×C₇₁⋊C₈	—	2	56	no	—
41	C₂×C₇₁⋊C₈	C₂⁴	1	8	yes	complemented
42	C₄⋊Dic₇₁	—	2	28	no	—
43	C₄⋊Dic₇₁	C₂⁴	1	8	yes	complemented
44	C₂⁴×C₇₁	—	2	112	no	—
45	C₂⁴×C₇₁	—	2	28	no	—
46	C₂⁴×C₇₁	—	4	128	no	—
47	C₂⁴×C₇₁	—	4	56	no	—
48	C₂⁴×C₇₁	—	4	56	no	—
49	C₂⁴×C₇₁	C₂²⋊C₄	1	8	yes	—
50	C₂⁴×C₇₁	D₈	1	8	yes	—
51	C₂⁴×C₇₁	SD₁₆	1	8	yes	—
52	C₂⁴×C₇₁	C₂×D₄	1	7	yes	—
53	C₂×D₄×C₇₁	—	2	336	no	—
54	C₂×D₄×C₇₁	—	2	28	no	—
55	C₂×D₄×C₇₁	—	2	56	no	—
56	C₂×D₄×C₇₁	C₂²×C₄	1	112	yes	complemented
57	C₂×D₄×C₇₁	C₂⁴	1	8	yes	—
58	C₂²×C₄×C₇₁	C₂²⋊C₄	1	56	yes	—
59	C₂²×C₄×C₇₁	C₂²×C₄	1	28	yes	—
60	C₂²×C₄×C₇₁	C₂×D₄	1	21	yes	—
61	C₂²×C₄×C₇₁	C₂×D₄	1	28	yes	—
62	C₂²×C₄×C₇₁	C₂⁴	1	7	yes	—
63	C₂³×C₇₁	—	2	112	no	—
64	C₂³×C₇₁	—	2	8	no	—
65	C₂³×C₇₁	—	4	56	no	—
66	C₂³×C₇₁	—	4	56	no	—
67	C₂³×C₇₁	—	4	448	no	—
68	C₂³×C₇₁	#102	1	64	yes	complemented
69	C₂³×C₇₁	C₂×C₂²⋊C₄	1	28	yes	—
70	C₂³×C₇₁	C₂×D₈	1	28	yes	—
71	C₂³×C₇₁	C₂×SD₁₆	1	28	yes	—
72	C₂³×C₇₁	C₂²×D₄	1	7	yes	—
73	D₄×C₇₁	—	2	112	no	—
74	D₄×C₇₁	—	2	8	no	—
75	D₄×C₇₁	C₂³×C₄	1	16	yes	complemented
76	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	84	yes	—
77	C₂×C₄×C₇₁	C₂³×C₄	1	14	yes	—
78	C₂×C₄×C₇₁	C₂²×D₄	1	7	yes	—
79	C₂×C₄×C₇₁	C₂²×D₄	1	14	yes	—
80	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
81	C₂×Dic₇₁	—	2	8	no	—
82	C₂×Dic₇₁	—	4	56	no	—
83	C₇₁⋊C₈	—	2	8	no	—
84	C₂²×C₇₁	—	2	16	no	—
85	C₂²×C₇₁	—	4	224	no	—
86	C₂²×C₇₁	—	4	8	no	—
87	C₂²×C₇₁	—	4	8	no	—
88	C₂²×C₇₁	C₂×#102	1	112	yes	complemented
89	C₂²×C₇₁	C₂²×C₂²⋊C₄	1	14	yes	—
90	C₂²×C₇₁	C₂²×D₈	1	14	yes	—
91	C₂²×C₇₁	C₂²×SD₁₆	1	14	yes	—
92	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
93	C₄×C₇₁	C₂²×C₂²⋊C₄	1	14	yes	—
94	C₄×C₇₁	C₂⁴×C₄	1	1	yes	characteristic
95	C₄×C₇₁	C₂³×D₄	1	1	yes	characteristic
96	Dic₇₁	—	4	8	no	—
97	C₂³×#102	—	71	1	no	maximal, Sylow
98	C₂×C₇₁	—	4	16	no	—
99	C₂×C₇₁	C₂²×#102	1	28	yes	complemented
100	C₂×C₇₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₇₁	C₂³×D₈	1	1	yes	characteristic
102	C₂×C₇₁	C₂³×SD₁₆	1	1	yes	characteristic
103	C₂⁴×D₄	D₇₁	1	1	yes	characteristic
104	C₂⁴×C₈	—	71	1	no	—
105	C₂³×C₄⋊C₄	—	71	1	no	—
106	C₂²×#102	—	71	28	no	—
107	C₇₁	C₂³×#102	1	1	yes	characteristic, complemented, Sylow
108	C₂⁶	—	2	1	no	—
109	C₂³×D₄	—	2	7	no	—
110	C₂³×D₄	—	2	8	no	—
111	C₂³×D₄	Dic₇₁	1	16	yes	complemented
112	C₂³×D₄	C₂×D₇₁	1	14	yes	—
113	C₂⁴×C₄	—	142	1	no	—
114	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
115	C₂³×C₈	—	71	14	no	—
116	C₂³×C₈	—	142	8	no	—
117	C₂²×C₄⋊C₄	—	71	14	no	—
118	C₂²×C₄⋊C₄	—	142	7	no	—
119	C₂×#102	—	71	112	no	—
120	C₂⁵	—	2	16	no	—
121	C₂⁵	—	2	14	no	—
122	C₂⁵	—	4	8	no	—
123	C₂⁵	—	4	8	no	—
124	C₂⁵	C₂²⋊D₇₁	1	1	yes	characteristic
125	C₂²×D₄	—	2	112	no	—
126	C₂²×D₄	—	2	56	no	—
127	C₂²×D₄	—	2	42	no	—
128	C₂²×D₄	C₂×Dic₇₁	1	112	yes	complemented
129	C₂²×D₄	C₂²×D₇₁	1	28	yes	—
130	C₂³×C₄	—	284	8	no	—
131	C₂³×C₄	—	142	14	no	—
132	C₂³×C₄	C₂×Dic₇₁	1	8	yes	—
133	C₂³×C₄	C₂²⋊D₇₁	1	7	yes	—
134	C₂³×C₄	C₂²⋊D₇₁	1	8	yes	—
135	C₂³×C₄	C₂²×D₇₁	1	7	yes	—
136	C₂²×C₈	—	71	28	no	—
137	C₂²×C₈	—	142	56	no	—
138	C₂×C₄⋊C₄	—	71	28	no	—
139	C₂×C₄⋊C₄	—	142	42	no	—
140	#102	—	71	64	no	—
141	C₂⁴	—	2	28	no	—
142	C₂⁴	—	2	112	no	—
143	C₂⁴	—	4	128	no	—
144	C₂⁴	—	4	56	no	—
145	C₂⁴	—	4	56	no	—
146	C₂⁴	C₂²⋊Dic₇₁	1	8	yes	—
147	C₂⁴	#11765850	1	8	yes	—
148	C₂⁴	#11765853	1	8	yes	—
149	C₂⁴	C₂×C₂²⋊D₇₁	1	7	yes	—
150	C₂×D₄	—	2	56	no	—
151	C₂×D₄	—	2	28	no	—
152	C₂×D₄	—	2	336	no	—
153	C₂×D₄	C₂²×Dic₇₁	1	112	yes	complemented
154	C₂×D₄	C₂³×D₇₁	1	8	yes	—
155	C₂²×C₄	—	284	56	no	—
156	C₂²×C₄	—	142	28	no	—
157	C₂²×C₄	C₂²⋊Dic₇₁	1	56	yes	—
158	C₂²×C₄	C₂²×Dic₇₁	1	28	yes	—
159	C₂²×C₄	C₂×C₂²⋊D₇₁	1	28	yes	—
160	C₂²×C₄	C₂×C₂²⋊D₇₁	1	21	yes	—
161	C₂²×C₄	C₂³×D₇₁	1	7	yes	—
162	C₂×C₈	—	71	8	no	—
163	C₂×C₈	—	142	56	no	—
164	C₄⋊C₄	—	71	8	no	—
165	C₄⋊C₄	—	142	28	no	—
166	C₂³	—	2	8	no	—
167	C₂³	—	2	112	no	—
168	C₂³	—	4	448	no	—
169	C₂³	—	4	56	no	—
170	C₂³	—	4	56	no	—
171	C₂³	#14556049	1	64	yes	direct product factor
172	C₂³	C₂×C₂²⋊Dic₇₁	1	28	yes	—
173	C₂³	C₂×#11765850	1	28	yes	—
174	C₂³	C₂×#11765853	1	28	yes	—
175	C₂³	C₂²×C₂²⋊D₇₁	1	7	yes	—
176	D₄	—	2	8	no	—
177	D₄	—	2	112	no	—
178	D₄	C₂³×Dic₇₁	1	16	yes	complemented
179	C₂×C₄	—	284	56	no	—
180	C₂×C₄	—	142	8	no	—
181	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	84	yes	—
182	C₂×C₄	C₂³×Dic₇₁	1	14	yes	—
183	C₂×C₄	C₂²×C₂²⋊D₇₁	1	14	yes	—
184	C₂×C₄	C₂²×C₂²⋊D₇₁	1	7	yes	—
185	C₂×C₄	C₂⁴×D₇₁	1	1	yes	characteristic
186	C₈	—	142	8	no	—
187	C₂²	—	2	16	no	—
188	C₂²	—	4	8	no	—
189	C₂²	—	4	8	no	—
190	C₂²	—	4	224	no	—
191	C₂²	C₂×#14556049	1	112	yes	direct product factor
192	C₂²	C₂²×C₂²⋊Dic₇₁	1	14	yes	—
193	C₂²	C₂²×#11765850	1	14	yes	—
194	C₂²	C₂²×#11765853	1	14	yes	—
195	C₂²	C₂³×C₂²⋊D₇₁	1	1	yes	characteristic
196	C₄	—	284	8	no	—
197	C₄	C₂²×C₂²⋊Dic₇₁	1	14	yes	—
198	C₄	C₂⁴×Dic₇₁	1	1	yes	characteristic
199	C₄	C₂³×C₂²⋊D₇₁	1	1	yes	characteristic
200	C₂	—	4	16	no	—
201	C₂	C₂²×#14556049	1	28	yes	direct product factor
202	C₂	C₂³×C₂²⋊Dic₇₁	1	1	yes	characteristic
203	C₂	C₂³×#11765850	1	1	yes	characteristic
204	C₂	C₂³×#11765853	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²⋊D₇₁
commutator	C₄×C₇₁	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇₁
Fitting, prime radical	C₂⁴×D₄×C₇₁	C₂
socle	C₂⁵×C₇₁	D₄
cosocle	C₂×C₄×C₇₁	C₂⁵
hypercenter	C₂⁴×D₄	D₇₁
hypocenter	C₇₁	C₂³×#102
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₇₁⋊C₈	C₂	semidirect
2. C₂³×C₄⋊Dic₇₁	C₂	semidirect
3. C₂²×#14556049	C₂	direct (N × H)
7. C₂³×C₇₁⋊C₈	C₂²	semidirect
9. C₂²×C₄⋊Dic₇₁	C₂²	semidirect
10. C₂×#14556049	C₂²	direct (N × H)
14. C₂³×D₄×C₇₁	C₄	semidirect
20. C₂²×C₇₁⋊C₈	C₂³	semidirect
22. C₂×C₄⋊Dic₇₁	C₂³	semidirect
23. #14556049	C₂³	direct (N × H)
32. C₂²×D₄×C₇₁	C₂×C₄	semidirect
41. C₂×C₇₁⋊C₈	C₂⁴	semidirect
43. C₄⋊Dic₇₁	C₂⁴	semidirect
56. C₂×D₄×C₇₁	C₂²×C₄	semidirect
68. C₂³×C₇₁	#102	semidirect
75. D₄×C₇₁	C₂³×C₄	semidirect
88. C₂²×C₇₁	C₂×#102	semidirect
99. C₂×C₇₁	C₂²×#102	semidirect
107. C₇₁	C₂³×#102	semidirect
111. C₂³×D₄	Dic₇₁	semidirect
128. C₂²×D₄	C₂×Dic₇₁	semidirect
153. C₂×D₄	C₂²×Dic₇₁	semidirect
171. C₂³	#14556049	direct (N × H)
178. D₄	C₂³×Dic₇₁	semidirect
191. C₂²	C₂×#14556049	direct (N × H)
201. C₂	C₂²×#14556049	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×C₇₁⋊C₈	2
C₂³×C₄⋊Dic₇₁	2
C₂²×#14556049	2
C₂⁴×D₄×C₇₁	2
C₂³×#102	71

Subgroups — C23×#14556049 #32860960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556049 #32860960