Subgroups — C₇₁×#13267 #31834139

Order 18176.

334 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#2960	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×#346×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1379	C₂	1	4	yes	maximal
5	#533×C₇₁	C₄	1	1	yes	characteristic
6	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
8	#446×C₇₁	—	2	2	no	—
9	C₈×Q₈×C₇₁	—	2	2	no	—
10	#424×C₇₁	C₄	1	2	yes	—
11	#423×C₇₁	C₄	1	1	yes	characteristic
12	#405×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₇₁	—	2	2	no	—
15	#346×C₇₁	—	2	2	no	—
16	C₂×M₁₆×C₇₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
19	C₄⋊Q₈×C₇₁	C₈	1	2	yes	—
20	#125×C₇₁	C₈	1	2	yes	—
21	C₂²⋊Q₈×C₇₁	—	2	2	no	—
22	C₄×Q₈×C₇₁	—	2	2	no	—
23	#117×C₇₁	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
25	M₃₂×C₇₁	—	2	4	no	—
26	C₂×C₁₆×C₇₁	—	2	2	no	—
27	C₂×C₁₆×C₇₁	—	2	2	no	—
28	C₂×C₁₆×C₇₁	—	2	2	no	—
29	C₂×C₁₆×C₇₁	Q₈	1	4	yes	complemented
30	C₄⋊C₈×C₇₁	—	2	4	no	—
31	C₄⋊C₈×C₇₁	—	2	2	no	—
32	#97×C₇₁	—	2	2	no	—
33	C₄×C₈×C₇₁	—	2	2	no	—
34	C₄×C₈×C₇₁	D₄	1	2	yes	—
35	C₄×C₈×C₇₁	D₄	1	2	yes	—
36	C₂×Q₈×C₇₁	—	2	2	no	—
37	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
38	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
39	M₁₆×C₇₁	—	2	4	no	—
40	C₂×C₈×C₇₁	—	2	2	no	—
41	C₂×C₈×C₇₁	—	2	2	no	—
42	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₇₁	#41	1	2	yes	—
47	C₄⋊C₄×C₇₁	—	2	4	no	—
48	C₄⋊C₄×C₇₁	—	2	2	no	—
49	C₄⋊C₄×C₇₁	C₂×C₈	1	4	yes	—
50	C₂²⋊C₄×C₇₁	—	2	2	no	—
51	C₄²×C₇₁	—	2	2	no	—
52	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
55	C₁₆×C₇₁	—	2	4	no	—
56	C₁₆×C₇₁	—	4	2	no	—
57	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
58	Q₈×C₇₁	—	4	4	no	—
59	C₂×C₄×C₇₁	—	2	4	no	—
60	C₂×C₄×C₇₁	—	4	2	no	—
61	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#131	1	2	yes	—
67	C₈×C₇₁	—	2	2	no	—
68	C₈×C₇₁	—	4	2	no	—
69	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
70	C₈×C₇₁	#124	1	1	yes	characteristic
71	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
72	C₂²×C₇₁	—	2	2	no	—
73	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₂²×C₇₁	#446	1	2	yes	—
75	C₄×C₇₁	—	2	2	no	—
76	C₄×C₇₁	—	2	2	no	—
77	C₄×C₇₁	—	4	4	no	—
78	C₄×C₇₁	#391	1	1	yes	characteristic
79	C₄×C₇₁	#408	1	1	yes	characteristic
80	C₄×C₇₁	C₈×Q₈	1	2	yes	—
81	#13267	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₇₁	—	2	2	no	—
83	C₂×C₇₁	#1984	1	1	yes	characteristic
84	C₂×C₇₁	#2117	1	2	yes	—
85	#2960	C₂×C₇₁	1	1	yes	characteristic
86	#2150	C₂×C₇₁	1	1	yes	characteristic
87	C₂×#346	C₂×C₇₁	1	1	yes	characteristic
88	#1379	C₂×C₇₁	1	4	yes	—
89	C₇₁	#13267	1	1	yes	characteristic, direct product factor, Sylow
90	#533	C₄×C₇₁	1	1	yes	characteristic
91	C₂×M₃₂	C₂²×C₇₁	1	2	yes	—
92	C₂²×C₁₆	C₂²×C₇₁	1	2	yes	—
93	#446	—	2	2	no	—
94	C₈×Q₈	—	2	2	no	—
95	#424	C₄×C₇₁	1	2	yes	—
96	#423	C₄×C₇₁	1	1	yes	characteristic
97	#405	C₂²×C₇₁	1	2	yes	—
98	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
99	C₄⋊C₁₆	—	2	2	no	—
100	#346	—	2	2	no	—
101	C₂×M₁₆	C₂×C₄×C₇₁	1	2	yes	—
102	C₂²×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
103	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
104	C₄⋊Q₈	C₈×C₇₁	1	2	yes	—
105	#125	C₈×C₇₁	1	2	yes	—
106	C₂²⋊Q₈	—	2	2	no	—
107	C₄×Q₈	—	2	2	no	—
108	#117	C₂×C₄×C₇₁	1	2	yes	—
109	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
110	M₃₂	—	2	4	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	Q₈×C₇₁	1	4	yes	complemented
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	#97	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	D₄×C₇₁	1	2	yes	—
120	C₄×C₈	D₄×C₇₁	1	2	yes	—
121	C₂×Q₈	—	2	2	no	—
122	C₂²×C₄	M₁₆×C₇₁	1	2	yes	—
123	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	C₂²⋊C₄×C₇₁	1	2	yes	—
128	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
129	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₈×C₇₁	1	2	yes	—
131	C₂×C₈	#41×C₇₁	1	2	yes	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	C₂×C₈×C₇₁	1	4	yes	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
138	C₄²	C₂×C₈×C₇₁	1	1	yes	characteristic
139	C₄²	C₂×C₈×C₇₁	1	1	yes	characteristic
140	C₁₆	—	2	4	no	—
141	C₁₆	—	4	2	no	—
142	C₂³	C₂×M₁₆×C₇₁	1	1	yes	characteristic
143	Q₈	—	4	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	C₂²⋊C₈×C₇₁	1	2	yes	—
147	C₂×C₄	C₂²⋊C₈×C₇₁	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
149	C₂×C₄	C₄×Q₈×C₇₁	1	2	yes	—
150	C₂×C₄	C₂²×C₈×C₇₁	1	1	yes	characteristic
151	C₂×C₄	#131×C₇₁	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	4	2	no	—
154	C₈	C₂²⋊Q₈×C₇₁	1	2	yes	—
155	C₈	#124×C₇₁	1	1	yes	characteristic
156	C₈	C₄⋊Q₈×C₇₁	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×C₂²⋊C₈×C₇₁	1	1	yes	characteristic
159	C₂²	#446×C₇₁	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	4	4	no	—
163	C₄	#391×C₇₁	1	1	yes	characteristic
164	C₄	#408×C₇₁	1	1	yes	characteristic
165	C₄	C₈×Q₈×C₇₁	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	C₇₁×#1984	1	1	yes	characteristic
168	C₂	C₇₁×#2117	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₇₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₇₁
Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
socle	C₂²×C₇₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₇₁	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
29. C₂×C₁₆×C₇₁	Q₈	semidirect
81. #13267	C₇₁	direct (N × H)
89. C₇₁	#13267	direct (N × H)
114. C₂×C₁₆	Q₈×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#2960	2
C₇₁×#2150	2
C₂×#346×C₇₁	2
C₇₁×#1379	2
#13267	71

Subgroups — C71×#13267 #31834139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#13267 #31834139