Subgroups — C₇₁×#23201 #31844073

Order 18176.

1014 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	C₂×#493×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2948	C₂	1	4	yes	maximal, complemented
3	C₇₁×#1978	C₂	1	4	yes	maximal, complemented
4	C₇₁×#1887	C₂	1	4	yes	maximal, complemented
5	C₇₁×#1772	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₇₁	C₂²	1	4	yes	—
7	C₂×#127×C₇₁	C₂²	1	2	yes	—
8	#518×C₇₁	C₂²	1	4	yes	—
9	#493×C₇₁	C₄	1	8	yes	complemented
10	#442×C₇₁	—	2	8	no	—
11	C₄×D₈×C₇₁	—	2	8	no	—
12	C₂×#108×C₇₁	C₂²	1	2	yes	—
13	#427×C₇₁	C₂²	1	2	yes	—
14	#418×C₇₁	C₂²	1	8	yes	—
15	C₂×#100×C₇₁	C₂²	1	4	yes	—
16	#405×C₇₁	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	#339×C₇₁	C₂²	1	4	yes	—
19	C₂²×D₄×C₇₁	—	4	2	no	—
20	C₂²×D₄×C₇₁	C₂³	1	4	yes	—
21	C₂×D₈×C₇₁	—	2	4	no	—
22	C₂×D₈×C₇₁	—	2	4	no	—
23	C₂×D₈×C₇₁	—	2	8	no	—
24	C₂×D₈×C₇₁	C₂×C₄	1	16	yes	complemented
25	C₂×M₁₆×C₇₁	C₂³	1	4	yes	—
26	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
27	#127×C₇₁	—	4	2	no	—
28	#127×C₇₁	C₂×C₄	1	8	yes	complemented
29	C₄×D₄×C₇₁	—	2	16	no	—
30	#117×C₇₁	C₂³	1	4	yes	—
31	C₂×C₂²⋊C₄×C₇₁	—	4	4	no	—
32	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	#108×C₇₁	—	2	4	no	—
34	#107×C₇₁	D₄	1	4	yes	complemented
35	#106×C₇₁	D₄	1	4	yes	complemented
36	#102×C₇₁	—	2	16	no	—
37	#100×C₇₁	—	4	4	no	—
38	#97×C₇₁	—	2	4	no	—
39	C₄×C₈×C₇₁	—	2	4	no	—
40	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	complemented
41	C₂⁴×C₇₁	—	2	4	no	—
42	C₂×D₄×C₇₁	—	2	8	no	—
43	C₂×D₄×C₇₁	—	2	8	no	—
44	C₂×D₄×C₇₁	—	2	4	no	—
45	C₂×D₄×C₇₁	—	4	8	no	—
46	C₂×D₄×C₇₁	—	8	8	no	—
47	C₂×D₄×C₇₁	C₂²×C₄	1	8	yes	—
48	C₂²×C₄×C₇₁	—	2	1	no	—
49	C₂²×C₄×C₇₁	—	4	8	no	—
50	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
51	D₈×C₇₁	—	2	16	no	—
52	D₈×C₇₁	—	2	16	no	—
53	M₁₆×C₇₁	—	2	4	no	—
54	M₁₆×C₇₁	—	2	4	no	—
55	C₂×C₈×C₇₁	—	2	4	no	—
56	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₇₁	#41	1	4	yes	—
60	C₄⋊C₄×C₇₁	C₂×D₄	1	8	yes	—
61	C₂²⋊C₄×C₇₁	—	2	4	no	—
62	C₂²⋊C₄×C₇₁	—	4	8	no	—
63	C₄²×C₇₁	—	2	4	no	—
64	C₄²×C₇₁	—	2	1	no	—
65	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
66	C₂³×C₇₁	—	2	8	no	—
67	C₂³×C₇₁	—	4	4	no	—
68	C₂³×C₇₁	—	4	4	no	—
69	C₂³×C₇₁	—	8	8	no	—
70	C₂³×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
71	D₄×C₇₁	—	2	16	no	—
72	D₄×C₇₁	—	4	8	no	—
73	D₄×C₇₁	—	8	16	no	—
74	C₂×C₄×C₇₁	—	2	4	no	—
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	—	2	8	no	—
77	C₂×C₄×C₇₁	—	16	8	no	—
78	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
80	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
81	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
82	C₈×C₇₁	—	4	4	no	—
83	C₈×C₇₁	C₄×D₄	1	4	yes	—
84	C₈×C₇₁	C₄×D₄	1	4	yes	—
85	C₂²×C₇₁	—	2	2	no	—
86	C₂²×C₇₁	—	4	8	no	—
87	C₂²×C₇₁	—	8	8	no	—
88	C₂²×C₇₁	—	8	8	no	—
89	C₂²×C₇₁	—	16	4	no	—
90	C₂²×C₇₁	#518	1	1	yes	characteristic
91	C₂²×C₇₁	D₄²	1	1	yes	characteristic
92	C₂²×C₇₁	#559	1	1	yes	characteristic
93	C₄×C₇₁	—	2	4	no	—
94	C₄×C₇₁	—	4	8	no	—
95	C₄×C₇₁	C₂×C₄×D₄	1	2	yes	—
96	C₄×C₇₁	#550	1	2	yes	—
97	#23201	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₇₁	—	2	1	no	—
99	C₂×C₇₁	—	2	1	no	—
100	C₂×C₇₁	—	8	8	no	—
101	C₂×C₇₁	#2366	1	1	yes	characteristic
102	C₂×C₇₁	#3292	1	2	yes	—
103	C₂×#493	C₂×C₇₁	1	1	yes	characteristic
104	#2948	C₂×C₇₁	1	4	yes	complemented
105	#1978	C₂×C₇₁	1	4	yes	complemented
106	#1887	C₂×C₇₁	1	4	yes	complemented
107	#1772	C₂×C₇₁	1	2	yes	complemented
108	C₇₁	#23201	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²×D₈	C₂²×C₇₁	1	4	yes	—
110	C₂×#127	C₂²×C₇₁	1	2	yes	—
111	#518	C₂²×C₇₁	1	4	yes	—
112	#493	C₄×C₇₁	1	8	yes	complemented
113	#442	—	2	8	no	—
114	C₄×D₈	—	2	8	no	—
115	C₂×#108	C₂²×C₇₁	1	2	yes	—
116	#427	C₂²×C₇₁	1	2	yes	—
117	#418	C₂²×C₇₁	1	8	yes	—
118	C₂×#100	C₂²×C₇₁	1	4	yes	—
119	#405	C₂²×C₇₁	1	4	yes	—
120	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
121	#339	C₂²×C₇₁	1	4	yes	—
122	C₂²×D₄	—	4	2	no	—
123	C₂²×D₄	C₂³×C₇₁	1	4	yes	—
124	C₂×D₈	—	2	4	no	—
125	C₂×D₈	—	2	4	no	—
126	C₂×D₈	—	2	8	no	—
127	C₂×D₈	C₂×C₄×C₇₁	1	16	yes	complemented
128	C₂×M₁₆	C₂³×C₇₁	1	4	yes	—
129	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
130	#127	—	4	2	no	—
131	#127	C₂×C₄×C₇₁	1	8	yes	complemented
132	C₄×D₄	—	2	16	no	—
133	#117	C₂³×C₇₁	1	4	yes	—
134	C₂×C₂²⋊C₄	—	4	4	no	—
135	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
136	#108	—	2	4	no	—
137	#107	D₄×C₇₁	1	4	yes	complemented
138	#106	D₄×C₇₁	1	4	yes	complemented
139	#102	—	2	16	no	—
140	#100	—	4	4	no	—
141	#97	—	2	4	no	—
142	C₄×C₈	—	2	4	no	—
143	C₄×C₈	C₂×C₄×C₇₁	1	4	yes	complemented
144	C₂⁴	—	2	4	no	—
145	C₂×D₄	—	2	8	no	—
146	C₂×D₄	—	2	8	no	—
147	C₂×D₄	—	2	4	no	—
148	C₂×D₄	—	4	8	no	—
149	C₂×D₄	—	8	8	no	—
150	C₂×D₄	C₂²×C₄×C₇₁	1	8	yes	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	4	8	no	—
153	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
154	D₈	—	2	16	no	—
155	D₈	—	2	16	no	—
156	M₁₆	—	2	4	no	—
157	M₁₆	—	2	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	C₂²×C₄×C₇₁	1	4	yes	—
160	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
161	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
162	C₂×C₈	#41×C₇₁	1	4	yes	—
163	C₄⋊C₄	C₂×D₄×C₇₁	1	8	yes	—
164	C₂²⋊C₄	—	2	4	no	—
165	C₂²⋊C₄	—	4	8	no	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
169	C₂³	—	2	8	no	—
170	C₂³	—	4	4	no	—
171	C₂³	—	4	4	no	—
172	C₂³	—	8	8	no	—
173	C₂³	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
174	D₄	—	2	16	no	—
175	D₄	—	4	8	no	—
176	D₄	—	8	16	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	16	8	no	—
181	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
182	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
183	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
184	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
185	C₈	—	4	4	no	—
186	C₈	C₄×D₄×C₇₁	1	4	yes	—
187	C₈	C₄×D₄×C₇₁	1	4	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	—	4	8	no	—
190	C₂²	—	8	8	no	—
191	C₂²	—	8	8	no	—
192	C₂²	—	16	4	no	—
193	C₂²	#518×C₇₁	1	1	yes	characteristic
194	C₂²	D₄²×C₇₁	1	1	yes	characteristic
195	C₂²	#559×C₇₁	1	1	yes	characteristic
196	C₄	—	2	4	no	—
197	C₄	—	4	8	no	—
198	C₄	C₂×C₄×D₄×C₇₁	1	2	yes	—
199	C₄	#550×C₇₁	1	2	yes	—
200	C₂	—	2	1	no	—
201	C₂	—	2	1	no	—
202	C₂	—	8	8	no	—
203	C₂	C₇₁×#2366	1	1	yes	characteristic
204	C₂	C₇₁×#3292	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
prime radical	C₂×#493×C₇₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#2948	C₂	semidirect
3. C₇₁×#1978	C₂	semidirect
4. C₇₁×#1887	C₂	semidirect
5. C₇₁×#1772	C₂	semidirect
9. #493×C₇₁	C₄	semidirect
24. C₂×D₈×C₇₁	C₂×C₄	semidirect
28. #127×C₇₁	C₂×C₄	semidirect
34. #107×C₇₁	D₄	semidirect
35. #106×C₇₁	D₄	semidirect
40. C₄×C₈×C₇₁	C₂×C₄	semidirect
97. #23201	C₇₁	direct (N × H)
104. #2948	C₂×C₇₁	semidirect
105. #1978	C₂×C₇₁	semidirect
106. #1887	C₂×C₇₁	semidirect
107. #1772	C₂×C₇₁	semidirect
108. C₇₁	#23201	direct (N × H)
112. #493	C₄×C₇₁	semidirect
127. C₂×D₈	C₂×C₄×C₇₁	semidirect
131. #127	C₂×C₄×C₇₁	semidirect
137. #107	D₄×C₇₁	semidirect
138. #106	D₄×C₇₁	semidirect
143. C₄×C₈	C₂×C₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₇₁	2
C₇₁×#2948	2
C₇₁×#1978	2
C₇₁×#1887	2
C₇₁×#1772	2
#23201	71

Subgroups — C71×#23201 #31844073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23201 #31844073