Subgroups — #31912286

Order 18176.

350 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	#22589944	C₂	1	2	yes	maximal
2	#22589939	C₂	1	2	yes	maximal
3	#22589937	C₂	1	1	yes	maximal, characteristic
4	#22588095	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#2110	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	1	yes	characteristic
7	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	#17985954	C₂²	1	2	yes	—
9	#17985776	C₄	1	8	yes	complemented
10	#17985775	C₄	1	2	yes	complemented
11	#17985774	C₄	1	4	yes	complemented
12	C₄×C₇₁⋊C₁₆	C₄	1	2	yes	complemented
13	C₁₆×Dic₇₁	C₄	1	4	yes	complemented
14	C₇₁⋊#338	C₄	1	1	yes	characteristic
15	#17985756	C₄	1	1	yes	characteristic
16	#17985755	C₄	1	2	yes	—
17	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
18	#405×C₇₁	C₂²	1	1	yes	characteristic
19	#347×C₇₁	C₄	1	4	yes	complemented
20	C₂×#11765849	C₂×C₄	1	2	yes	—
21	#14556087	C₂×C₄	1	2	yes	—
22	#14556068	C₂×C₄	1	4	yes	—
23	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
28	#14556038	C₂×C₄	1	4	yes	—
29	C₈×Dic₇₁	C₂×C₄	1	4	yes	—
30	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₇₁	C₂×C₄	1	4	yes	—
34	M₃₂×C₇₁	C₂×C₄	1	4	yes	—
35	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
36	#97×C₇₁	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
38	#11765849	—	2	2	no	—
39	#11765849	—	2	2	no	—
40	C₂×C₇₁⋊C₈	C₄²	1	4	yes	—
41	C₄⋊Dic₇₁	—	2	2	no	—
42	C₂²⋊Dic₇₁	—	2	2	no	—
43	C₄×Dic₇₁	C₄²	1	4	yes	—
44	C₇₁⋊C₁₆	C₄²	1	1	yes	characteristic
45	C₇₁⋊C₁₆	C₄²	1	2	yes	—
46	C₇₁⋊C₁₆	C₄²	1	1	yes	characteristic
47	C₇₁⋊C₁₆	C₄²	1	4	yes	—
48	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₇₁	—	2	2	no	—
50	C₂×C₈×C₇₁	C₄²	1	2	yes	—
51	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₇₁	—	2	1	no	—
56	C₂²⋊C₄×C₇₁	—	2	1	no	—
57	C₄²×C₇₁	C₄²	1	2	yes	—
58	C₁₆×C₇₁	C₄²	1	8	yes	—
59	C₂³×C₇₁	#131	1	1	yes	characteristic
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	C₂×C₄²	1	2	yes	—
62	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
64	C₂×C₄×C₇₁	#131	1	2	yes	—
65	C₂×Dic₇₁	—	2	4	no	—
66	C₈×C₇₁	—	2	2	no	—
67	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
68	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₇₁	C₂×C₄²	1	2	yes	—
70	C₇₁⋊C₈	—	2	4	no	—
71	C₂²×C₇₁	—	4	1	no	—
72	C₂²×C₇₁	#405	1	2	yes	—
73	C₂²×C₇₁	#405	1	1	yes	characteristic
74	C₄×C₇₁	—	4	2	no	—
75	C₄×C₇₁	C₄³	1	1	yes	characteristic
76	C₄×C₇₁	#405	1	2	yes	—
77	C₄×C₇₁	#405	1	1	yes	characteristic
78	Dic₇₁	—	4	4	no	—
79	#12542	—	71	1	no	maximal, Sylow
80	C₂×C₇₁	—	2	2	no	—
81	C₂×C₇₁	—	2	1	no	—
82	C₂×C₇₁	#1726	1	1	yes	characteristic
83	#2110	—	71	2	no	—
84	#2110	D₇₁	1	1	yes	characteristic
85	#2108	—	71	2	no	—
86	#2108	—	71	1	no	—
87	#1747	—	71	1	no	—
88	C₇₁	#12542	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	—	71	1	no	—
90	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
91	C₂²×C₁₆	—	71	1	no	—
92	#405	—	71	2	no	—
93	#405	C₂×D₇₁	1	1	yes	characteristic
94	#347	—	71	8	no	—
95	#347	Dic₇₁	1	4	yes	complemented
96	#346	—	71	4	no	—
97	#346	—	71	2	no	—
98	C₄×C₁₆	—	71	4	no	—
99	C₄×C₁₆	—	71	2	no	—
100	#338	—	71	1	no	—
101	H(3,ℤ/4)	—	71	1	no	—
102	H(3,ℤ/4)	—	71	2	no	—
103	C₂×M₁₆	—	71	2	no	—
104	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
106	#117	—	71	2	no	—
107	#117	C₄×D₇₁	1	1	yes	characteristic
108	M₃₂	—	71	4	no	—
109	M₃₂	C₄×D₇₁	1	4	yes	—
110	M₃₂	C₂×Dic₇₁	1	4	yes	—
111	C₂×C₁₆	—	71	1	no	—
112	C₂×C₁₆	—	71	2	no	—
113	C₂×C₁₆	—	71	2	no	—
114	C₂×C₁₆	—	71	1	no	—
115	C₂×C₁₆	—	71	2	no	—
116	C₂×C₁₆	C₄×D₇₁	1	4	yes	—
117	#97	—	71	4	no	—
118	#97	C₂×Dic₇₁	1	2	yes	—
119	C₄×C₈	—	71	4	no	—
120	C₄×C₈	C₄×D₇₁	1	2	yes	—
121	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	—	142	2	no	—
124	M₁₆	—	142	2	no	—
125	C₂×C₈	—	71	4	no	—
126	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
127	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
130	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	142	2	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	142	2	no	—
135	C₄²	—	71	4	no	—
136	C₄²	C₄×Dic₇₁	1	2	yes	—
137	C₁₆	—	71	4	no	—
138	C₁₆	—	71	1	no	—
139	C₁₆	—	71	2	no	—
140	C₁₆	—	71	1	no	—
141	C₁₆	C₄×Dic₇₁	1	8	yes	—
142	C₂³	#14556136	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	142	4	no	—
145	C₂×C₄	C₄²×D₇₁	1	2	yes	—
146	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
147	C₂×C₄	#14556136	1	1	yes	characteristic
148	C₂×C₄	#14556136	1	2	yes	—
149	C₈	—	2	2	no	—
150	C₈	—	142	4	no	—
151	C₈	C₄²×D₇₁	1	1	yes	characteristic
152	C₈	C₄²×D₇₁	1	1	yes	characteristic
153	C₈	C₂×C₄×Dic₇₁	1	2	yes	—
154	C₂²	—	4	1	no	—
155	C₂²	#17985956	1	2	yes	—
156	C₂²	#17985957	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	—	284	4	no	—
159	C₄	C₄²×Dic₇₁	1	1	yes	characteristic
160	C₄	#17985956	1	2	yes	—
161	C₄	#17985957	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	2	no	—
164	C₂	#22588028	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄²×D₇₁
commutator	C₄×C₇₁	C₄³
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2110	C₂
socle	C₂×C₇₁	#1726
prime radical	C₂³×C₇₁	#131
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2110	D₇₁
hypocenter	C₇₁	#12542
LCS	G ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ #2110

Complements

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

N	H ≅ G/N	type
9. #17985776	C₄	semidirect
10. #17985775	C₄	semidirect
11. #17985774	C₄	semidirect
12. C₄×C₇₁⋊C₁₆	C₄	semidirect
13. C₁₆×Dic₇₁	C₄	semidirect
19. #347×C₇₁	C₄	semidirect
88. C₇₁	#12542	semidirect
95. #347	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22589944	2
#22589939	2
#22589937	2
#22588095	2
C₇₁×#2110	2
#12542	71