Subgroups — C₂³×#14556084 #32904009

Order 18176.

14334 conjugacy classes of subgroups.

The subgroup lattice has 184 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
2	C₂²×#14556084	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
5	C₂³×C₄×Dic₇₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₂²⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₇₁	C₄	1	32	yes	complemented
8	C₂⁴×Dic₇₁	C₂²	1	2	yes	—
9	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₇₁	C₂²	1	28	yes	—
12	C₂×#14556084	C₂²	1	448	yes	direct product factor
13	C₂²×Dic₇₁⋊C₄	C₂²	1	56	yes	complemented
14	C₂²×C₂²⋊Dic₇₁	C₂²	1	28	yes	—
15	C₂²×C₄×Dic₇₁	C₂²	1	56	yes	complemented
16	C₂⁶×C₇₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₇₁	C₂²	1	2	yes	—
18	C₂²×C₂²⋊C₄×C₇₁	C₂²	1	28	yes	—
19	C₂³×Dic₇₁	—	2	16	no	—
20	C₂³×Dic₇₁	C₂×C₄	1	16	yes	—
21	C₂³×Dic₇₁	C₂×C₄	1	16	yes	—
22	C₂³×Dic₇₁	C₂×C₄	1	448	yes	complemented
23	C₂³×Dic₇₁	C₂³	1	28	yes	—
24	C₂³×Dic₇₁	C₂³	1	14	yes	—
25	C₂³×Dic₇₁	C₂³	1	14	yes	—
26	C₂³×Dic₇₁	C₂³	1	112	yes	—
27	#14556084	C₂³	1	512	yes	direct product factor
28	C₂×Dic₇₁⋊C₄	C₂³	1	224	yes	complemented
29	C₂×C₂²⋊Dic₇₁	C₂³	1	112	yes	—
30	C₂×C₄×Dic₇₁	C₂³	1	224	yes	complemented
31	C₂⁵×C₇₁	—	2	16	no	—
32	C₂⁵×C₇₁	C₂×C₄	1	16	yes	—
33	C₂⁵×C₇₁	C₂³	1	14	yes	—
34	C₂⁵×C₇₁	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₇₁	—	2	16	no	—
36	C₂³×C₄×C₇₁	C₂³	1	28	yes	—
37	C₂×C₂²⋊C₄×C₇₁	C₂³	1	112	yes	—
38	C₂²×Dic₇₁	—	2	112	no	—
39	C₂²×Dic₇₁	C₂²×C₄	1	112	yes	—
40	C₂²×Dic₇₁	C₂²×C₄	1	112	yes	—
41	C₂²×Dic₇₁	C₂²×C₄	1	896	yes	complemented
42	C₂²×Dic₇₁	C₂⁴	1	56	yes	—
43	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
44	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
45	C₂²×Dic₇₁	C₂⁴	1	64	yes	—
46	Dic₇₁⋊C₄	C₂⁴	1	128	yes	complemented
47	C₂²⋊Dic₇₁	C₂⁴	1	64	yes	—
48	C₄×Dic₇₁	C₂⁴	1	128	yes	complemented
49	C₂⁴×C₇₁	—	2	128	no	—
50	C₂⁴×C₇₁	—	2	112	no	—
51	C₂⁴×C₇₁	C₂²×C₄	1	112	yes	—
52	C₂⁴×C₇₁	C₂²×C₄	1	8	yes	—
53	C₂⁴×C₇₁	#41	1	16	yes	—
54	C₂⁴×C₇₁	C₂⁴	1	28	yes	—
55	C₂⁴×C₇₁	C₂⁴	1	7	yes	—
56	C₂²×C₄×C₇₁	—	2	112	no	—
57	C₂²×C₄×C₇₁	C₂⁴	1	56	yes	—
58	C₂²⋊C₄×C₇₁	C₂⁴	1	64	yes	—
59	C₂³×C₇₁	—	2	448	no	—
60	C₂³×C₇₁	—	2	112	no	—
61	C₂³×C₇₁	#117	1	64	yes	complemented
62	C₂³×C₇₁	C₂³×C₄	1	112	yes	—
63	C₂³×C₇₁	C₂³×C₄	1	28	yes	—
64	C₂³×C₇₁	C₂×#41	1	56	yes	—
65	C₂³×C₇₁	C₂⁵	1	8	yes	—
66	C₂³×C₇₁	C₂⁵	1	7	yes	—
67	C₂×C₄×C₇₁	—	2	112	no	—
68	C₂×C₄×C₇₁	C₂⁵	1	16	yes	—
69	C₂×Dic₇₁	—	2	112	no	—
70	C₂×Dic₇₁	C₂³×C₄	1	112	yes	—
71	C₂×Dic₇₁	C₂³×C₄	1	112	yes	—
72	C₂×Dic₇₁	C₂³×C₄	1	256	yes	complemented
73	C₂×Dic₇₁	C₂⁵	1	16	yes	—
74	C₂×Dic₇₁	C₂⁵	1	8	yes	—
75	C₂×Dic₇₁	C₂⁵	1	8	yes	—
76	C₂²×C₇₁	—	2	16	no	—
77	C₂²×C₇₁	—	2	224	no	—
78	C₂²×C₇₁	C₂×#117	1	112	yes	complemented
79	C₂²×C₇₁	C₂⁴×C₄	1	16	yes	—
80	C₂²×C₇₁	C₂⁴×C₄	1	14	yes	—
81	C₂²×C₇₁	C₂²×#41	1	28	yes	—
82	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
83	C₄×C₇₁	—	2	16	no	—
84	Dic₇₁	—	2	16	no	—
85	Dic₇₁	C₂⁴×C₄	1	16	yes	—
86	Dic₇₁	C₂⁴×C₄	1	16	yes	—
87	C₂³×#117	—	71	1	no	maximal, Sylow
88	C₂×C₇₁	—	2	16	no	—
89	C₂×C₇₁	C₂²×#117	1	28	yes	complemented
90	C₂×C₇₁	C₂⁵×C₄	1	1	yes	characteristic
91	C₂×C₇₁	C₂³×#41	1	2	yes	—
92	C₂⁵×C₄	—	71	1	no	—
93	C₂²×#117	—	71	56	no	—
94	C₂³×C₄⋊C₄	—	71	2	no	—
95	C₂³×C₂²⋊C₄	—	71	1	no	—
96	C₂³×C₂²⋊C₄	D₇₁	1	1	yes	characteristic
97	C₂³×C₄²	—	71	2	no	—
98	C₇₁	C₂³×#117	1	1	yes	characteristic, complemented, Sylow
99	C₂⁶	C₂×D₇₁	1	1	yes	characteristic
100	C₂⁴×C₄	—	71	32	no	—
101	C₂⁴×C₄	—	71	28	no	—
102	C₂⁴×C₄	—	71	1	no	—
103	C₂⁴×C₄	—	71	1	no	—
104	C₂⁴×C₄	—	71	2	no	—
105	C₂⁴×C₄	C₂×D₇₁	1	2	yes	—
106	C₂×#117	—	71	448	no	—
107	C₂²×C₄⋊C₄	—	71	56	no	—
108	C₂²×C₂²⋊C₄	—	71	28	no	—
109	C₂²×C₂²⋊C₄	C₂×D₇₁	1	28	yes	—
110	C₂²×C₄²	—	71	56	no	—
111	C₂⁵	—	2	16	no	—
112	C₂⁵	C₄×D₇₁	1	16	yes	—
113	C₂⁵	C₂²×D₇₁	1	14	yes	—
114	C₂⁵	C₂²×D₇₁	1	1	yes	characteristic
115	C₂³×C₄	—	2	16	no	—
116	C₂³×C₄	—	71	16	no	—
117	C₂³×C₄	—	71	16	no	—
118	C₂³×C₄	—	71	448	no	—
119	C₂³×C₄	—	71	14	no	—
120	C₂³×C₄	—	71	112	no	—
121	C₂³×C₄	—	71	14	no	—
122	C₂³×C₄	—	71	28	no	—
123	C₂³×C₄	—	142	16	no	—
124	C₂³×C₄	C₂²×D₇₁	1	28	yes	—
125	#117	—	71	512	no	—
126	C₂×C₄⋊C₄	—	71	224	no	—
127	C₂×C₂²⋊C₄	—	71	112	no	—
128	C₂×C₂²⋊C₄	C₂²×D₇₁	1	112	yes	—
129	C₂×C₄²	—	71	224	no	—
130	C₂⁴	—	2	112	no	—
131	C₂⁴	—	2	128	no	—
132	C₂⁴	C₂×C₄×D₇₁	1	112	yes	—
133	C₂⁴	C₂×C₄×D₇₁	1	8	yes	—
134	C₂⁴	#11765864	1	16	yes	—
135	C₂⁴	C₂³×D₇₁	1	28	yes	—
136	C₂⁴	C₂³×D₇₁	1	7	yes	—
137	C₂²×C₄	—	2	112	no	—
138	C₂²×C₄	—	71	112	no	—
139	C₂²×C₄	—	71	112	no	—
140	C₂²×C₄	—	71	896	no	—
141	C₂²×C₄	—	71	28	no	—
142	C₂²×C₄	—	71	28	no	—
143	C₂²×C₄	—	71	64	no	—
144	C₂²×C₄	—	71	56	no	—
145	C₂²×C₄	—	142	112	no	—
146	C₂²×C₄	C₂³×D₇₁	1	56	yes	—
147	C₄⋊C₄	—	71	128	no	—
148	C₂²⋊C₄	—	71	64	no	—
149	C₂²⋊C₄	C₂³×D₇₁	1	64	yes	—
150	C₄²	—	71	128	no	—
151	C₂³	—	2	112	no	—
152	C₂³	—	2	448	no	—
153	C₂³	#14556084	1	64	yes	direct product factor
154	C₂³	C₂²×C₄×D₇₁	1	112	yes	—
155	C₂³	C₂²×C₄×D₇₁	1	28	yes	—
156	C₂³	C₂×#11765864	1	56	yes	—
157	C₂³	C₂⁴×D₇₁	1	8	yes	—
158	C₂³	C₂⁴×D₇₁	1	7	yes	—
159	C₂×C₄	—	2	112	no	—
160	C₂×C₄	—	71	112	no	—
161	C₂×C₄	—	71	112	no	—
162	C₂×C₄	—	71	256	no	—
163	C₂×C₄	—	71	8	no	—
164	C₂×C₄	—	71	8	no	—
165	C₂×C₄	—	71	16	no	—
166	C₂×C₄	—	142	112	no	—
167	C₂×C₄	C₂⁴×D₇₁	1	16	yes	—
168	C₂²	—	2	224	no	—
169	C₂²	—	2	16	no	—
170	C₂²	C₂×#14556084	1	112	yes	direct product factor
171	C₂²	C₂³×C₄×D₇₁	1	16	yes	—
172	C₂²	C₂³×C₄×D₇₁	1	14	yes	—
173	C₂²	C₂²×#11765864	1	28	yes	—
174	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
175	C₄	—	2	16	no	—
176	C₄	—	71	16	no	—
177	C₄	—	71	16	no	—
178	C₄	—	142	16	no	—
179	C₂	—	2	16	no	—
180	C₂	C₂²×#14556084	1	28	yes	direct product factor
181	C₂	C₂⁴×C₄×D₇₁	1	1	yes	characteristic
182	C₂	C₂³×#11765864	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₇₁
commutator	C₂×C₇₁	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₇₁
Fitting	C₂³×C₂²⋊C₄×C₇₁	C₂
socle	C₂⁵×C₇₁	C₂³
prime radical	C₂⁶×C₇₁	C₂²
cosocle	C₂²×C₇₁	C₂⁶
hypercenter	C₂³×C₂²⋊C₄	D₇₁
hypocenter	C₇₁	C₂³×#117
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#14556084	C₂	direct (N × H)
3. C₂³×Dic₇₁⋊C₄	C₂	semidirect
5. C₂³×C₄×Dic₇₁	C₂	semidirect
7. C₂⁴×Dic₇₁	C₄	semidirect
12. C₂×#14556084	C₂²	direct (N × H)
13. C₂²×Dic₇₁⋊C₄	C₂²	semidirect
15. C₂²×C₄×Dic₇₁	C₂²	semidirect
22. C₂³×Dic₇₁	C₂×C₄	semidirect
27. #14556084	C₂³	direct (N × H)
28. C₂×Dic₇₁⋊C₄	C₂³	semidirect
30. C₂×C₄×Dic₇₁	C₂³	semidirect
41. C₂²×Dic₇₁	C₂²×C₄	semidirect
46. Dic₇₁⋊C₄	C₂⁴	semidirect
48. C₄×Dic₇₁	C₂⁴	semidirect
61. C₂³×C₇₁	#117	semidirect
72. C₂×Dic₇₁	C₂³×C₄	semidirect
78. C₂²×C₇₁	C₂×#117	semidirect
89. C₂×C₇₁	C₂²×#117	semidirect
98. C₇₁	C₂³×#117	semidirect
153. C₂³	#14556084	direct (N × H)
170. C₂²	C₂×#14556084	direct (N × H)
180. C₂	C₂²×#14556084	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×#14556084	2
C₂³×Dic₇₁⋊C₄	2
C₂³×C₂²⋊Dic₇₁	2
C₂³×C₄×Dic₇₁	2
C₂³×C₂²⋊C₄×C₇₁	2
C₂³×#117	71

Subgroups — C23×#14556084 #32904009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556084 #32904009