Subgroups — C₇₁×#2552 #22585114

Order 9088.

452 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×C₇₁	C₂	1	2	yes	maximal
2	#398×C₇₁	C₂	1	4	yes	maximal
3	#389×C₇₁	C₂	1	2	yes	maximal
4	#389×C₇₁	C₂	1	2	yes	maximal
5	#384×C₇₁	C₂	1	2	yes	maximal
6	#384×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
9	#125×C₇₁	—	2	8	no	—
10	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₇₁	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₇₁	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₇₁	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₇₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₇₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
23	#95×C₇₁	C₂²	1	2	yes	—
24	#95×C₇₁	C₂²	1	2	yes	—
25	#95×C₇₁	C₂²	1	4	yes	—
26	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
27	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
28	C₂²×C₄×C₇₁	C₂³	1	4	yes	—
29	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
30	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
32	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₇₁	—	2	8	no	—
35	C₄⋊C₄×C₇₁	—	2	8	no	—
36	C₄⋊C₄×C₇₁	—	2	4	no	—
37	C₄⋊C₄×C₇₁	—	2	2	no	—
38	C₄⋊C₄×C₇₁	—	2	2	no	—
39	C₄⋊C₄×C₇₁	—	2	4	no	—
40	C₄⋊C₄×C₇₁	—	4	4	no	—
41	C₄⋊C₄×C₇₁	—	4	2	no	—
42	C₄⋊C₄×C₇₁	D₄	1	4	yes	—
43	C₄⋊C₄×C₇₁	Q₈	1	4	yes	—
44	C₄²×C₇₁	—	2	4	no	—
45	C₄²×C₇₁	—	2	4	no	—
46	C₄²×C₇₁	Q₈	1	4	yes	—
47	C₂³×C₇₁	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₇₁	—	2	2	no	—
49	C₂×C₄×C₇₁	—	2	2	no	—
50	C₂×C₄×C₇₁	—	2	1	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	—	2	1	no	—
53	C₂×C₄×C₇₁	—	2	2	no	—
54	C₂×C₄×C₇₁	—	2	2	no	—
55	C₂×C₄×C₇₁	—	2	2	no	—
56	C₂×C₄×C₇₁	—	2	4	no	—
57	C₂×C₄×C₇₁	—	2	2	no	—
58	C₂×C₄×C₇₁	—	2	2	no	—
59	C₂×C₄×C₇₁	—	2	4	no	—
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	2	4	no	—
62	C₂×C₄×C₇₁	—	2	2	no	—
63	C₂×C₄×C₇₁	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
66	C₂×C₄×C₇₁	C₂×Q₈	1	4	yes	—
67	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
69	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₇₁	#41	1	2	yes	—
71	C₂×C₄×C₇₁	#41	1	2	yes	—
72	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₇₁	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₇₁	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₇₁	C₂×#41	1	1	yes	characteristic
76	C₂²×C₇₁	C₂×#41	1	1	yes	characteristic
77	C₂²×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂²×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
79	C₄×C₇₁	—	2	2	no	—
80	C₄×C₇₁	—	2	4	no	—
81	C₄×C₇₁	—	2	4	no	—
82	C₄×C₇₁	—	4	2	no	—
83	C₄×C₇₁	—	4	2	no	—
84	C₄×C₇₁	—	4	4	no	—
85	C₄×C₇₁	#125	1	4	yes	complemented
86	C₄×C₇₁	C₄⋊Q₈	1	4	yes	—
87	C₂×C₇₁	C₂×#125	1	1	yes	characteristic
88	C₂×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₇₁	#544	1	1	yes	characteristic
90	C₂×C₇₁	#547	1	1	yes	characteristic
91	C₂×C₇₁	#550	1	1	yes	characteristic
92	C₂×C₇₁	#554	1	1	yes	characteristic
93	C₂×C₇₁	#557	1	1	yes	characteristic
94	#2552	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₇₁	#2552	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×#125	C₂×C₇₁	1	2	yes	—
97	#398	C₂×C₇₁	1	4	yes	—
98	#389	C₂×C₇₁	1	2	yes	—
99	#389	C₂×C₇₁	1	2	yes	—
100	#384	C₂×C₇₁	1	2	yes	—
101	#384	C₂×C₇₁	1	1	yes	characteristic
102	C₄×C₄⋊C₄	C₂×C₇₁	1	1	yes	characteristic
103	C₄×C₄⋊C₄	C₂×C₇₁	1	1	yes	characteristic
104	#125	—	2	8	no	—
105	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
106	C₂×C₄⋊C₄	C₂²×C₇₁	1	4	yes	—
107	C₂×C₄⋊C₄	C₂²×C₇₁	1	4	yes	—
108	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
109	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₇₁	1	4	yes	—
112	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
115	C₂×C₄²	C₂²×C₇₁	1	2	yes	—
116	C₂×C₄²	C₂²×C₇₁	1	2	yes	—
117	C₂×C₄²	C₂²×C₇₁	1	1	yes	characteristic
118	#95	C₂²×C₇₁	1	2	yes	—
119	#95	C₂²×C₇₁	1	2	yes	—
120	#95	C₂²×C₇₁	1	4	yes	—
121	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
122	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
123	C₂²×C₄	C₂³×C₇₁	1	4	yes	—
124	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
125	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
127	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
128	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
129	C₄⋊C₄	—	2	8	no	—
130	C₄⋊C₄	—	2	8	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	4	4	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	D₄×C₇₁	1	4	yes	—
138	C₄⋊C₄	Q₈×C₇₁	1	4	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	4	no	—
141	C₄²	Q₈×C₇₁	1	4	yes	—
142	C₂³	C₂⁴×C₇₁	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₂×D₄×C₇₁	1	4	yes	—
159	C₂×C₄	C₂×D₄×C₇₁	1	2	yes	—
160	C₂×C₄	C₂×Q₈×C₇₁	1	2	yes	—
161	C₂×C₄	C₂×Q₈×C₇₁	1	4	yes	—
162	C₂×C₄	C₂×Q₈×C₇₁	1	2	yes	—
163	C₂×C₄	C₂×Q₈×C₇₁	1	2	yes	—
164	C₂×C₄	C₂×Q₈×C₇₁	1	2	yes	—
165	C₂×C₄	#41×C₇₁	1	2	yes	—
166	C₂×C₄	#41×C₇₁	1	2	yes	—
167	C₂²	C₂²×D₄×C₇₁	1	1	yes	characteristic
168	C₂²	C₂²×Q₈×C₇₁	1	1	yes	characteristic
169	C₂²	C₂²×Q₈×C₇₁	1	1	yes	characteristic
170	C₂²	C₂×#41×C₇₁	1	1	yes	characteristic
171	C₂²	C₂×#41×C₇₁	1	1	yes	characteristic
172	C₂²	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
173	C₂²	2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	2	4	no	—
176	C₄	—	2	4	no	—
177	C₄	—	4	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	4	4	no	—
180	C₄	#125×C₇₁	1	4	yes	complemented
181	C₄	C₄⋊Q₈×C₇₁	1	4	yes	—
182	C₂	C₂×#125×C₇₁	1	1	yes	characteristic
183	C₂	C₂×C₄⋊Q₈×C₇₁	1	1	yes	characteristic
184	C₂	#544×C₇₁	1	1	yes	characteristic
185	C₂	#547×C₇₁	1	1	yes	characteristic
186	C₂	#550×C₇₁	1	1	yes	characteristic
187	C₂	#554×C₇₁	1	1	yes	characteristic
188	C₂	#557×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
85. C₄×C₇₁	#125	semidirect
94. #2552	C₇₁	direct (N × H)
95. C₇₁	#2552	direct (N × H)
180. C₄	#125×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#125×C₇₁	2
#398×C₇₁	2
#389×C₇₁	2
#384×C₇₁	2
C₄×C₄⋊C₄×C₇₁	2
#2552	71

Subgroups — C71×#2552 #22585114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#2552 #22585114