Subgroups — C₇₁×#2841 #22585403

Order 9088.

396 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400×C₇₁	C₂	1	2	yes	maximal, complemented
2	#399×C₇₁	C₂	1	1	yes	maximal, characteristic
3	#398×C₇₁	C₂	1	2	yes	maximal, complemented
4	#398×C₇₁	C₂	1	2	yes	maximal, complemented
5	#396×C₇₁	C₂	1	2	yes	maximal
6	#395×C₇₁	C₂	1	2	yes	maximal, complemented
7	#393×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#381×C₇₁	C₂	1	1	yes	maximal, characteristic
9	#380×C₇₁	C₂	1	2	yes	maximal
10	C₂²×Q₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₇₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₇₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×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	2	yes	—
26	#95×C₇₁	C₂²	1	2	yes	—
27	#95×C₇₁	C₂²	1	1	yes	characteristic
28	#95×C₇₁	C₂²	1	2	yes	—
29	#95×C₇₁	C₂²	1	2	yes	—
30	#95×C₇₁	C₂²	1	1	yes	characteristic
31	C₂⁴×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₇₁	—	4	1	no	—
33	C₂×Q₈×C₇₁	—	4	2	no	—
34	C₂²×C₄×C₇₁	—	4	1	no	—
35	C₂²×C₄×C₇₁	—	4	1	no	—
36	C₂²×C₄×C₇₁	Q₈	1	4	yes	complemented
37	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
45	C₄⋊C₄×C₇₁	—	4	2	no	—
46	C₄⋊C₄×C₇₁	—	4	2	no	—
47	C₄⋊C₄×C₇₁	—	4	2	no	—
48	C₄⋊C₄×C₇₁	—	4	2	no	—
49	C₄⋊C₄×C₇₁	—	4	1	no	—
50	C₄⋊C₄×C₇₁	—	4	2	no	—
51	C₄⋊C₄×C₇₁	—	4	2	no	—
52	C₂²⋊C₄×C₇₁	—	4	2	no	—
53	C₂²⋊C₄×C₇₁	—	4	1	no	—
54	C₂²⋊C₄×C₇₁	—	4	2	no	—
55	C₂²⋊C₄×C₇₁	—	4	1	no	—
56	C₂³×C₇₁	—	2	2	no	—
57	C₂³×C₇₁	—	2	1	no	—
58	C₂³×C₇₁	—	2	1	no	—
59	C₂³×C₇₁	—	2	2	no	—
60	C₂³×C₇₁	C₂×Q₈	1	2	yes	—
61	C₂³×C₇₁	C₂⁴	1	1	yes	characteristic
62	Q₈×C₇₁	—	4	4	no	—
63	C₂×C₄×C₇₁	—	2	2	no	—
64	C₂×C₄×C₇₁	—	2	2	no	—
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	—	2	2	no	—
67	C₂×C₄×C₇₁	—	2	2	no	—
68	C₂×C₄×C₇₁	—	2	2	no	—
69	C₂×C₄×C₇₁	—	2	1	no	—
70	C₂×C₄×C₇₁	—	2	1	no	—
71	C₂×C₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	—	2	2	no	—
73	C₂×C₄×C₇₁	—	2	2	no	—
74	C₂×C₄×C₇₁	—	2	2	no	—
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	—	2	2	no	—
78	C₂×C₄×C₇₁	—	2	2	no	—
79	C₂×C₄×C₇₁	—	2	2	no	—
80	C₂×C₄×C₇₁	—	2	2	no	—
81	C₂×C₄×C₇₁	—	2	2	no	—
82	C₂×C₄×C₇₁	—	2	2	no	—
83	C₂×C₄×C₇₁	—	2	2	no	—
84	C₂×C₄×C₇₁	—	4	4	no	—
85	C₂×C₄×C₇₁	C₂×Q₈	1	4	yes	—
86	C₂²×C₇₁	—	2	2	no	—
87	C₂²×C₇₁	—	2	4	no	—
88	C₂²×C₇₁	—	4	1	no	—
89	C₂²×C₇₁	—	4	2	no	—
90	C₂²×C₇₁	—	4	1	no	—
91	C₂²×C₇₁	C₂²×Q₈	1	1	yes	characteristic
92	C₂²×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
93	C₂²×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂²×C₇₁	H(5,𝔽₂)	1	2	yes	—
95	C₂²×C₇₁	2₋¹⁺⁴	1	2	yes	—
96	C₄×C₇₁	—	2	4	no	—
97	C₄×C₇₁	—	4	2	no	—
98	C₄×C₇₁	—	4	1	no	—
99	C₄×C₇₁	—	4	2	no	—
100	C₄×C₇₁	—	4	2	no	—
101	C₄×C₇₁	—	4	2	no	—
102	C₄×C₇₁	—	4	2	no	—
103	C₄×C₇₁	—	4	1	no	—
104	C₂×C₇₁	—	4	2	no	—
105	C₂×C₇₁	#543	1	1	yes	characteristic
106	C₂×C₇₁	#544	1	2	yes	—
107	C₂×C₇₁	#560	1	1	yes	characteristic
108	C₂×C₇₁	#562	1	2	yes	—
109	C₂×C₇₁	#563	1	1	yes	characteristic
110	#2841	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₇₁	#2841	1	1	yes	characteristic, direct product factor, Sylow
112	#400	C₂×C₇₁	1	2	yes	complemented
113	#399	C₂×C₇₁	1	1	yes	characteristic
114	#398	C₂×C₇₁	1	2	yes	complemented
115	#398	C₂×C₇₁	1	2	yes	complemented
116	#396	C₂×C₇₁	1	2	yes	—
117	#395	C₂×C₇₁	1	2	yes	complemented
118	#393	C₂×C₇₁	1	1	yes	characteristic
119	#381	C₂×C₇₁	1	1	yes	characteristic
120	#380	C₂×C₇₁	1	2	yes	—
121	C₂²×Q₈	C₂²×C₇₁	1	1	yes	characteristic
122	C₂³×C₄	C₂²×C₇₁	1	1	yes	characteristic
123	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
128	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
129	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
130	C₂×C₂²⋊C₄	C₂²×C₇₁	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₇₁	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
134	#95	C₂²×C₇₁	1	2	yes	—
135	#95	C₂²×C₇₁	1	2	yes	—
136	#95	C₂²×C₇₁	1	2	yes	—
137	#95	C₂²×C₇₁	1	2	yes	—
138	#95	C₂²×C₇₁	1	1	yes	characteristic
139	#95	C₂²×C₇₁	1	2	yes	—
140	#95	C₂²×C₇₁	1	2	yes	—
141	#95	C₂²×C₇₁	1	1	yes	characteristic
142	C₂⁴	C₂³×C₇₁	1	1	yes	characteristic
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	2	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	Q₈×C₇₁	1	4	yes	complemented
148	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
150	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
151	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
152	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
153	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
155	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	1	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	4	2	no	—
164	C₂²⋊C₄	—	4	1	no	—
165	C₂²⋊C₄	—	4	2	no	—
166	C₂²⋊C₄	—	4	1	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	1	no	—
169	C₂³	—	2	1	no	—
170	C₂³	—	2	2	no	—
171	C₂³	C₂×Q₈×C₇₁	1	2	yes	—
172	C₂³	C₂⁴×C₇₁	1	1	yes	characteristic
173	Q₈	—	4	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂×Q₈×C₇₁	1	4	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	4	1	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	1	no	—
202	C₂²	C₂²×Q₈×C₇₁	1	1	yes	characteristic
203	C₂²	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
204	C₂²	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
205	C₂²	H(5,𝔽₂)×C₇₁	1	2	yes	—
206	C₂²	2₋¹⁺⁴×C₇₁	1	2	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	4	1	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	1	no	—
215	C₂	—	4	2	no	—
216	C₂	#543×C₇₁	1	1	yes	characteristic
217	C₂	#544×C₇₁	1	2	yes	—
218	C₂	#560×C₇₁	1	1	yes	characteristic
219	C₂	#562×C₇₁	1	2	yes	—
220	C₂	#563×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇₁
prime radical	C₂⁴×C₇₁	C₂³
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
1. #400×C₇₁	C₂	semidirect
3. #398×C₇₁	C₂	semidirect
4. #398×C₇₁	C₂	semidirect
6. #395×C₇₁	C₂	semidirect
36. C₂²×C₄×C₇₁	Q₈	semidirect
110. #2841	C₇₁	direct (N × H)
111. C₇₁	#2841	direct (N × H)
112. #400	C₂×C₇₁	semidirect
114. #398	C₂×C₇₁	semidirect
115. #398	C₂×C₇₁	semidirect
117. #395	C₂×C₇₁	semidirect
147. C₂²×C₄	Q₈×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
#400×C₇₁	2
#399×C₇₁	2
#398×C₇₁	2
#396×C₇₁	2
#395×C₇₁	2
#393×C₇₁	2
#381×C₇₁	2
#380×C₇₁	2
#2841	71

Subgroups — C71×#2841 #22585403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#2841 #22585403