Subgroups — C₇₁×#12024 #31832896

Order 18176.

226 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1568	C₂	1	2	yes	maximal
3	C₇₁×#1372	C₂	1	2	yes	maximal, complemented
4	C₇₁×#1371	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
7	#432×C₇₁	—	2	2	no	—
8	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₇₁	C₄	1	1	yes	characteristic
10	C₄×C₁₆×C₇₁	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	#335×C₇₁	C₂²	1	4	yes	—
14	#334×C₇₁	—	2	2	no	—
15	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₁	—	2	2	no	—
19	C₂×C₁₆×C₇₁	—	2	2	no	—
20	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₄⋊C₈×C₇₁	—	2	4	no	—
25	C₂²⋊C₈×C₇₁	—	4	2	no	—
26	#97×C₇₁	—	4	2	no	—
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
30	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
34	C₂×C₈×C₇₁	—	2	2	no	—
35	C₂×C₈×C₇₁	—	2	1	no	—
36	C₂×C₈×C₇₁	—	2	1	no	—
37	C₂×C₈×C₇₁	—	4	4	no	—
38	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₇₁	D₈	1	1	yes	characteristic
44	C₄²×C₇₁	Q₁₆	1	1	yes	characteristic
45	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₇₁	—	2	1	no	—
47	C₁₆×C₇₁	—	2	1	no	—
48	C₁₆×C₇₁	M₁₆	1	2	yes	—
49	C₁₆×C₇₁	M₁₆	1	2	yes	—
50	C₂³×C₇₁	#107	1	1	yes	characteristic
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	—	2	1	no	—
53	C₂×C₄×C₇₁	—	2	1	no	—
54	C₂×C₄×C₇₁	—	2	2	no	—
55	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₈×C₇₁	—	2	1	no	—
62	C₈×C₇₁	—	2	1	no	—
63	C₈×C₇₁	—	8	4	no	—
64	C₈×C₇₁	#108	1	2	yes	—
65	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
67	C₂²×C₇₁	—	2	1	no	—
68	C₂²×C₇₁	—	2	1	no	—
69	C₂²×C₇₁	#368	1	2	yes	—
70	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
71	C₄×C₇₁	—	2	1	no	—
72	C₄×C₇₁	—	2	1	no	—
73	C₄×C₇₁	—	2	1	no	—
74	C₄×C₇₁	—	2	1	no	—
75	C₄×C₇₁	#423	1	1	yes	characteristic
76	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
77	C₄×C₇₁	#508	1	2	yes	—
78	#12024	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×C₇₁	—	2	2	no	—
80	C₂×C₇₁	#1567	1	1	yes	characteristic
81	C₂×C₇₁	#2158	1	1	yes	characteristic
82	C₂×C₇₁	C₂×#368	1	1	yes	characteristic
83	C₂×C₄×C₁₆	C₂×C₇₁	1	1	yes	characteristic
84	#1568	C₂×C₇₁	1	2	yes	—
85	#1372	C₂×C₇₁	1	2	yes	complemented
86	#1371	C₂×C₇₁	1	2	yes	complemented
87	C₇₁	#12024	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²×C₁₆	C₄×C₇₁	1	1	yes	characteristic
89	C₂²×C₁₆	C₄×C₇₁	1	1	yes	characteristic
90	#432	—	2	2	no	—
91	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
92	C₄×C₁₆	C₄×C₇₁	1	1	yes	characteristic
93	C₄×C₁₆	C₄×C₇₁	1	1	yes	characteristic
94	C₄×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
95	C₄×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
96	#335	C₂²×C₇₁	1	4	yes	—
97	#334	—	2	2	no	—
98	C₂²×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
99	C₂²×C₈	Q₈×C₇₁	1	1	yes	characteristic
100	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	C₂×C₄×C₇₁	1	1	yes	characteristic
104	C₂×C₁₆	C₂×C₄×C₇₁	1	1	yes	characteristic
105	C₂×C₁₆	C₂×C₄×C₇₁	1	1	yes	characteristic
106	C₂×C₁₆	C₂×C₄×C₇₁	1	1	yes	characteristic
107	C₄⋊C₈	—	2	4	no	—
108	C₂²⋊C₈	—	4	2	no	—
109	#97	—	4	2	no	—
110	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
111	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
112	C₄×C₈	Q₈×C₇₁	1	1	yes	characteristic
113	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊C₄×C₇₁	1	1	yes	characteristic
115	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
116	C₂²×C₄	Q₁₆×C₇₁	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	4	4	no	—
121	C₂×C₈	C₄⋊C₄×C₇₁	1	1	yes	characteristic
122	C₂×C₈	C₄⋊C₄×C₇₁	1	1	yes	characteristic
123	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
124	C₂×C₈	C₂×Q₈×C₇₁	1	1	yes	characteristic
125	C₄²	C₄⋊C₄×C₇₁	1	1	yes	characteristic
126	C₄²	D₈×C₇₁	1	1	yes	characteristic
127	C₄²	Q₁₆×C₇₁	1	1	yes	characteristic
128	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	1	no	—
131	C₁₆	M₁₆×C₇₁	1	2	yes	—
132	C₁₆	M₁₆×C₇₁	1	2	yes	—
133	C₂³	#107×C₇₁	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	#107×C₇₁	1	1	yes	characteristic
139	C₂×C₄	#107×C₇₁	1	1	yes	characteristic
140	C₂×C₄	#107×C₇₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
142	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
143	C₂×C₄	C₂×Q₁₆×C₇₁	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	4	no	—
147	C₈	#108×C₇₁	1	2	yes	—
148	C₈	C₂×M₁₆×C₇₁	1	1	yes	characteristic
149	C₈	C₂×M₁₆×C₇₁	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#368×C₇₁	1	2	yes	—
153	C₂²	C₂×#107×C₇₁	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#423×C₇₁	1	1	yes	characteristic
159	C₄	C₂×#108×C₇₁	1	1	yes	characteristic
160	C₄	#508×C₇₁	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	C₇₁×#1567	1	1	yes	characteristic
163	C₂	C₇₁×#2158	1	1	yes	characteristic
164	C₂	C₂×#368×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇₁
Frattini, cosocle	C₄×C₈	C₂³×C₇₁
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇₁×#1372	C₂	semidirect
4. C₇₁×#1371	C₂	semidirect
78. #12024	C₇₁	direct (N × H)
85. #1372	C₂×C₇₁	semidirect
86. #1371	C₂×C₇₁	semidirect
87. C₇₁	#12024	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆×C₇₁	2
C₇₁×#1568	2
C₇₁×#1372	2
C₇₁×#1371	2
#12024	71

Subgroups — C71×#12024 #31832896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#12024 #31832896