Subgroups — #31906433

Order 18176.

330 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556068	C₂	1	1	yes	maximal, characteristic
2	#22586623	C₂	1	1	yes	maximal, characteristic
3	#22586256	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₇₁⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₄	1	2	yes	—
7	#17986052	C₄	1	2	yes	—
8	#17985775	C₄	1	2	yes	—
9	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
10	#17985720	C₄	1	2	yes	—
11	C₈×C₇₁⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₇₁	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₁	C₄	1	2	yes	—
15	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	#14556068	C₈	1	8	yes	complemented
18	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
19	C₄⋊(C₇₁⋊C₈)	C₈	1	4	yes	—
20	C₂²⋊(C₇₁⋊C₈)	C₈	1	4	yes	—
21	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₁	C₈	1	8	yes	complemented
26	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₇₁⋊C₈	C₂×C₈	1	8	yes	—
33	C₇₁⋊C₁₆	C₂×C₈	1	8	yes	—
34	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
40	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₇₁	C₄²	1	1	yes	characteristic
43	C₄²×C₇₁	C₂×C₈	1	2	yes	—
44	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₇₁	C₂×C₈	1	4	yes	—
46	C₁₆×C₇₁	C₂×C₈	1	4	yes	—
47	C₂³×C₇₁	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₁	—	2	4	no	—
49	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
53	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
55	C₈×C₇₁	C₄×C₈	1	2	yes	—
56	C₈×C₇₁	C₄×C₈	1	2	yes	—
57	C₈×C₇₁	C₂²×C₈	1	2	yes	—
58	C₈×C₇₁	C₂²×C₈	1	2	yes	—
59	C₇₁⋊C₈	—	2	8	no	—
60	C₂²×C₇₁	—	2	2	no	—
61	C₂²×C₇₁	C₈²	1	2	yes	—
62	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₇₁	—	2	2	no	—
64	C₄×C₇₁	C₈²	1	2	yes	—
65	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₇₁	#504	1	4	yes	—
68	#11399	—	71	1	no	maximal, Sylow
69	C₂×C₇₁	—	2	2	no	—
70	C₂×C₇₁	C₂×C₈²	1	1	yes	characteristic
71	C₂×C₇₁	#2108	1	2	yes	—
72	C₄×M₃₂	—	71	1	no	—
73	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
74	#1450	—	71	1	no	—
75	#1311	—	71	2	no	—
76	C₈×C₁₆	—	71	2	no	—
77	C₇₁	#11399	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₃₂	—	71	2	no	—
79	C₂²×C₁₆	Dic₇₁	1	2	yes	—
80	#431	—	71	2	no	—
81	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
82	#346	—	71	2	no	—
83	C₄×C₁₆	—	71	2	no	—
84	C₄×C₁₆	Dic₇₁	1	2	yes	—
85	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
86	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
87	#322	—	71	2	no	—
88	C₈²	—	71	2	no	—
89	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
91	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
92	M₃₂	—	71	8	no	—
93	C₂×C₁₆	—	71	4	no	—
94	C₂×C₁₆	C₇₁⋊C₈	1	8	yes	complemented
95	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
96	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
97	C₄⋊C₈	—	71	4	no	—
98	C₂²⋊C₈	—	71	4	no	—
99	C₄×C₈	—	71	4	no	—
100	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
101	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
104	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
105	C₂²×C₄	C₈×D₇₁	1	2	yes	—
106	C₂×C₈	—	71	8	no	—
107	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
108	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
109	C₂×C₈	C₈×D₇₁	1	4	yes	—
110	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
111	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
112	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
113	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
114	C₄²	C₈×D₇₁	1	2	yes	—
115	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
116	C₁₆	—	71	8	no	—
117	C₁₆	C₂×C₇₁⋊C₈	1	4	yes	—
118	C₁₆	C₂×C₇₁⋊C₈	1	4	yes	—
119	C₂³	C₈×Dic₇₁	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
122	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
123	C₂×C₄	C₈×Dic₇₁	1	1	yes	characteristic
124	C₂×C₄	C₄×C₇₁⋊C₈	1	2	yes	—
125	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
126	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
127	C₈	—	142	8	no	—
128	C₈	C₈×Dic₇₁	1	2	yes	—
129	C₈	C₄×C₇₁⋊C₈	1	2	yes	—
130	C₈	C₂×C₈×D₇₁	1	2	yes	—
131	C₈	C₂²×C₇₁⋊C₈	1	2	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	C₈×C₇₁⋊C₈	1	2	yes	—
134	C₂²	C₂×C₈×Dic₇₁	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	C₈×C₇₁⋊C₈	1	2	yes	—
137	C₄	C₂×C₈×Dic₇₁	1	1	yes	characteristic
138	C₄	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
139	C₄	#17986415	1	4	yes	—
140	C₂	—	2	2	no	—
141	C₂	C₂×C₈×C₇₁⋊C₈	1	1	yes	characteristic
142	C₂	#22589936	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₂×D₇₁
commutator	C₂×C₇₁	C₂×C₈²
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄×C₈
prime radical	C₂³×C₇₁	C₄×C₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#11399
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #22586256	C₂	semidirect
4. C₁₆×C₇₁⋊C₈	C₂	semidirect
17. #14556068	C₈	semidirect
25. C₂×C₁₆×C₇₁	C₈	semidirect
77. C₇₁	#11399	semidirect
94. C₂×C₁₆	C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556068	2
#22586623	2
#22586256	2
C₁₆×C₇₁⋊C₈	2
C₂×C₄×C₁₆×C₇₁	2
#11399	71