Subgroups — C₂³×#14556109 #32904098

Order 18176.

15150 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14556109	C₂	1	56	yes	maximal, direct product factor
3	C₂³×Dic₇₁⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂³×C₂²⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₂²⋊Dic₇₁	C₂	1	2	yes	maximal, complemented
6	C₂⁴×D₄×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₇₁	—	2	16	no	—
8	C₂⁴×Dic₇₁	C₂²	1	2	yes	—
9	C₂⁴×Dic₇₁	C₂²	1	2	yes	—
10	C₂⁴×Dic₇₁	C₂²	1	28	yes	complemented
11	C₂×#14556109	C₂²	1	448	yes	direct product factor
12	C₂²×Dic₇₁⋊C₄	C₂²	1	56	yes	complemented
13	C₂²×C₂²⋊Dic₇₁	C₂²	1	56	yes	complemented
14	C₂²×C₂²⋊Dic₇₁	C₂²	1	28	yes	complemented
15	C₂⁶×C₇₁	C₂²	1	1	yes	characteristic
16	C₂⁶×C₇₁	C₂²	1	1	yes	characteristic
17	C₂³×D₄×C₇₁	—	2	16	no	—
18	C₂³×D₄×C₇₁	C₂²	1	28	yes	—
19	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
20	C₂³×Dic₇₁	—	2	16	no	—
21	C₂³×Dic₇₁	—	2	224	no	—
22	C₂³×Dic₇₁	—	2	16	no	—
23	C₂³×Dic₇₁	C₂³	1	28	yes	—
24	C₂³×Dic₇₁	C₂³	1	28	yes	—
25	C₂³×Dic₇₁	C₂³	1	112	yes	complemented
26	#14556109	C₂³	1	512	yes	direct product factor
27	C₂×Dic₇₁⋊C₄	C₂³	1	224	yes	complemented
28	C₂×C₂²⋊Dic₇₁	C₂³	1	224	yes	complemented
29	C₂×C₂²⋊Dic₇₁	C₂³	1	112	yes	complemented
30	C₂⁵×C₇₁	—	2	16	no	—
31	C₂⁵×C₇₁	—	2	8	no	—
32	C₂⁵×C₇₁	—	2	16	no	—
33	C₂⁵×C₇₁	D₄	1	16	yes	—
34	C₂⁵×C₇₁	C₂³	1	14	yes	—
35	C₂⁵×C₇₁	C₂³	1	14	yes	—
36	C₂⁵×C₇₁	C₂³	1	1	yes	characteristic
37	C₂²×D₄×C₇₁	—	2	224	no	—
38	C₂²×D₄×C₇₁	C₂³	1	112	yes	—
39	C₂³×C₄×C₇₁	—	2	8	no	—
40	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
41	C₂²×Dic₇₁	—	2	112	no	—
42	C₂²×Dic₇₁	—	2	448	no	—
43	C₂²×Dic₇₁	—	2	112	no	—
44	C₂²×Dic₇₁	C₂⁴	1	56	yes	—
45	C₂²×Dic₇₁	C₂⁴	1	56	yes	—
46	C₂²×Dic₇₁	C₂⁴	1	64	yes	complemented
47	Dic₇₁⋊C₄	C₂⁴	1	128	yes	complemented
48	C₂²⋊Dic₇₁	C₂⁴	1	64	yes	complemented
49	C₂²⋊Dic₇₁	C₂⁴	1	128	yes	complemented
50	C₂⁴×C₇₁	—	2	128	no	—
51	C₂⁴×C₇₁	—	2	112	no	—
52	C₂⁴×C₇₁	—	2	56	no	—
53	C₂⁴×C₇₁	—	2	112	no	—
54	C₂⁴×C₇₁	—	4	64	no	—
55	C₂⁴×C₇₁	C₂×D₄	1	112	yes	—
56	C₂⁴×C₇₁	C₂×D₄	1	8	yes	—
57	C₂⁴×C₇₁	#41	1	16	yes	—
58	C₂⁴×C₇₁	C₂⁴	1	28	yes	—
59	C₂⁴×C₇₁	C₂⁴	1	28	yes	—
60	C₂⁴×C₇₁	C₂⁴	1	7	yes	—
61	C₂×D₄×C₇₁	—	2	448	no	—
62	C₂×D₄×C₇₁	C₂⁴	1	64	yes	—
63	C₂²×C₄×C₇₁	—	2	56	no	—
64	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
65	C₂³×C₇₁	—	2	448	no	—
66	C₂³×C₇₁	—	2	112	no	—
67	C₂³×C₇₁	—	2	56	no	—
68	C₂³×C₇₁	—	2	112	no	—
69	C₂³×C₇₁	—	4	224	no	—
70	C₂³×C₇₁	#123	1	64	yes	complemented
71	C₂³×C₇₁	C₂²×D₄	1	112	yes	—
72	C₂³×C₇₁	C₂²×D₄	1	28	yes	—
73	C₂³×C₇₁	C₂×#41	1	56	yes	—
74	C₂³×C₇₁	C₂⁵	1	8	yes	—
75	C₂³×C₇₁	C₂⁵	1	8	yes	—
76	C₂³×C₇₁	C₂⁵	1	7	yes	—
77	D₄×C₇₁	—	2	128	no	—
78	C₂×C₄×C₇₁	—	2	56	no	—
79	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
80	C₂×Dic₇₁	—	2	112	no	—
81	C₂×Dic₇₁	—	2	128	no	—
82	C₂×Dic₇₁	—	2	112	no	—
83	C₂×Dic₇₁	C₂⁵	1	16	yes	—
84	C₂×Dic₇₁	C₂⁵	1	16	yes	—
85	C₂²×C₇₁	—	2	16	no	—
86	C₂²×C₇₁	—	2	224	no	—
87	C₂²×C₇₁	—	2	8	no	—
88	C₂²×C₇₁	—	2	16	no	—
89	C₂²×C₇₁	—	4	112	no	—
90	C₂²×C₇₁	C₂×#123	1	112	yes	complemented
91	C₂²×C₇₁	C₂³×D₄	1	16	yes	—
92	C₂²×C₇₁	C₂³×D₄	1	14	yes	—
93	C₂²×C₇₁	C₂²×#41	1	28	yes	—
94	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
95	C₄×C₇₁	—	2	8	no	—
96	Dic₇₁	—	2	16	no	—
97	Dic₇₁	—	2	16	no	—
98	C₂³×#123	—	71	1	no	maximal, Sylow
99	C₂×C₇₁	—	2	16	no	—
100	C₂×C₇₁	—	4	8	no	—
101	C₂×C₇₁	C₂²×#123	1	28	yes	complemented
102	C₂×C₇₁	C₂⁴×D₄	1	1	yes	characteristic
103	C₂×C₇₁	C₂³×#41	1	2	yes	—
104	C₂⁴×D₄	D₇₁	1	1	yes	characteristic
105	C₂⁵×C₄	—	71	1	no	—
106	C₂²×#123	—	71	56	no	—
107	C₂³×C₄⋊C₄	—	71	2	no	—
108	C₂³×C₂²⋊C₄	—	71	2	no	—
109	C₂³×C₂²⋊C₄	—	71	1	no	—
110	C₇₁	C₂³×#123	1	1	yes	characteristic, complemented, Sylow
111	C₂⁶	C₂×D₇₁	1	1	yes	characteristic
112	C₂⁶	C₂×D₇₁	1	1	yes	characteristic
113	C₂³×D₄	—	2	16	no	—
114	C₂³×D₄	C₂×D₇₁	1	28	yes	—
115	C₂⁴×C₄	—	71	28	no	—
116	C₂⁴×C₄	—	71	2	no	—
117	C₂⁴×C₄	—	71	2	no	—
118	C₂⁴×C₄	—	142	16	no	—
119	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
120	C₂×#123	—	71	448	no	—
121	C₂²×C₄⋊C₄	—	71	56	no	—
122	C₂²×C₂²⋊C₄	—	71	28	no	—
123	C₂²×C₂²⋊C₄	—	71	56	no	—
124	C₂⁵	—	2	16	no	—
125	C₂⁵	—	2	8	no	—
126	C₂⁵	—	2	16	no	—
127	C₂⁵	C₂²⋊D₇₁	1	16	yes	—
128	C₂⁵	C₂²×D₇₁	1	14	yes	—
129	C₂⁵	C₂²×D₇₁	1	14	yes	—
130	C₂⁵	C₂²×D₇₁	1	1	yes	characteristic
131	C₂²×D₄	—	2	224	no	—
132	C₂²×D₄	C₂²×D₇₁	1	112	yes	—
133	C₂³×C₄	—	2	8	no	—
134	C₂³×C₄	—	71	28	no	—
135	C₂³×C₄	—	71	28	no	—
136	C₂³×C₄	—	71	112	no	—
137	C₂³×C₄	—	142	224	no	—
138	C₂³×C₄	—	142	16	no	—
139	C₂³×C₄	—	142	16	no	—
140	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
141	#123	—	71	512	no	—
142	C₂×C₄⋊C₄	—	71	224	no	—
143	C₂×C₂²⋊C₄	—	71	112	no	—
144	C₂×C₂²⋊C₄	—	71	224	no	—
145	C₂⁴	—	2	112	no	—
146	C₂⁴	—	2	128	no	—
147	C₂⁴	—	2	56	no	—
148	C₂⁴	—	2	112	no	—
149	C₂⁴	—	4	64	no	—
150	C₂⁴	C₂×C₂²⋊D₇₁	1	112	yes	—
151	C₂⁴	C₂×C₂²⋊D₇₁	1	8	yes	—
152	C₂⁴	#11765864	1	16	yes	—
153	C₂⁴	C₂³×D₇₁	1	28	yes	—
154	C₂⁴	C₂³×D₇₁	1	28	yes	—
155	C₂⁴	C₂³×D₇₁	1	7	yes	—
156	C₂×D₄	—	2	448	no	—
157	C₂×D₄	C₂³×D₇₁	1	64	yes	—
158	C₂²×C₄	—	2	56	no	—
159	C₂²×C₄	—	71	56	no	—
160	C₂²×C₄	—	71	56	no	—
161	C₂²×C₄	—	71	64	no	—
162	C₂²×C₄	—	142	448	no	—
163	C₂²×C₄	—	142	112	no	—
164	C₂²×C₄	—	142	112	no	—
165	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
166	C₄⋊C₄	—	71	128	no	—
167	C₂²⋊C₄	—	71	64	no	—
168	C₂²⋊C₄	—	71	128	no	—
169	C₂³	—	2	112	no	—
170	C₂³	—	2	448	no	—
171	C₂³	—	2	56	no	—
172	C₂³	—	2	112	no	—
173	C₂³	—	4	224	no	—
174	C₂³	#14556109	1	64	yes	direct product factor
175	C₂³	C₂²×C₂²⋊D₇₁	1	112	yes	—
176	C₂³	C₂²×C₂²⋊D₇₁	1	28	yes	—
177	C₂³	C₂×#11765864	1	56	yes	—
178	C₂³	C₂⁴×D₇₁	1	8	yes	—
179	C₂³	C₂⁴×D₇₁	1	8	yes	—
180	C₂³	C₂⁴×D₇₁	1	7	yes	—
181	D₄	—	2	128	no	—
182	C₂×C₄	—	2	56	no	—
183	C₂×C₄	—	71	16	no	—
184	C₂×C₄	—	71	16	no	—
185	C₂×C₄	—	142	128	no	—
186	C₂×C₄	—	142	112	no	—
187	C₂×C₄	—	142	112	no	—
188	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
189	C₂²	—	2	224	no	—
190	C₂²	—	2	16	no	—
191	C₂²	—	2	8	no	—
192	C₂²	—	2	16	no	—
193	C₂²	—	4	112	no	—
194	C₂²	C₂×#14556109	1	112	yes	direct product factor
195	C₂²	C₂³×C₂²⋊D₇₁	1	16	yes	—
196	C₂²	C₂³×C₂²⋊D₇₁	1	14	yes	—
197	C₂²	C₂²×#11765864	1	28	yes	—
198	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
199	C₄	—	2	8	no	—
200	C₄	—	142	16	no	—
201	C₄	—	142	16	no	—
202	C₂	—	2	16	no	—
203	C₂	—	4	8	no	—
204	C₂	C₂²×#14556109	1	28	yes	direct product factor
205	C₂	C₂⁴×C₂²⋊D₇₁	1	1	yes	characteristic
206	C₂	C₂³×#11765864	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₇₁
commutator, cosocle	C₂²×C₇₁	C₂⁶
Frattini	C₂²	C₂⁵×D₇₁
Fitting, prime radical	C₂⁴×D₄×C₇₁	C₂
socle	C₂⁵×C₇₁	C₂³
hypercenter	C₂⁴×D₄	D₇₁
hypocenter	C₇₁	C₂³×#123
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×Dic₇₁	C₂	semidirect
2. C₂²×#14556109	C₂	direct (N × H)
3. C₂³×Dic₇₁⋊C₄	C₂	semidirect
4. C₂³×C₂²⋊Dic₇₁	C₂	semidirect
5. C₂³×C₂²⋊Dic₇₁	C₂	semidirect
10. C₂⁴×Dic₇₁	C₂²	semidirect
11. C₂×#14556109	C₂²	direct (N × H)
12. C₂²×Dic₇₁⋊C₄	C₂²	semidirect
13. C₂²×C₂²⋊Dic₇₁	C₂²	semidirect
14. C₂²×C₂²⋊Dic₇₁	C₂²	semidirect
25. C₂³×Dic₇₁	C₂³	semidirect
26. #14556109	C₂³	direct (N × H)
27. C₂×Dic₇₁⋊C₄	C₂³	semidirect
28. C₂×C₂²⋊Dic₇₁	C₂³	semidirect
29. C₂×C₂²⋊Dic₇₁	C₂³	semidirect
46. C₂²×Dic₇₁	C₂⁴	semidirect
47. Dic₇₁⋊C₄	C₂⁴	semidirect
48. C₂²⋊Dic₇₁	C₂⁴	semidirect
49. C₂²⋊Dic₇₁	C₂⁴	semidirect
70. C₂³×C₇₁	#123	semidirect
90. C₂²×C₇₁	C₂×#123	semidirect
101. C₂×C₇₁	C₂²×#123	semidirect
110. C₇₁	C₂³×#123	semidirect
174. C₂³	#14556109	direct (N × H)
194. C₂²	C₂×#14556109	direct (N × H)
204. C₂	C₂²×#14556109	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×Dic₇₁	2
C₂²×#14556109	2
C₂³×Dic₇₁⋊C₄	2
C₂³×C₂²⋊Dic₇₁	2
C₂⁴×D₄×C₇₁	2
C₂³×#123	71

Subgroups — C23×#14556109 #32904098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556109 #32904098