Subgroups — C₂×#22586476 #31917637

Order 18176.

222 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985833	C₂	1	2	yes	maximal
2	#22586476	C₂	1	4	yes	maximal, direct product factor
3	C₂×#346×C₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985833	C₄	1	4	yes	complemented
5	#17985833	C₂²	1	4	yes	—
6	C₂×C₇₁⋊C₃₂	C₄	1	4	yes	complemented
7	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
10	#346×C₇₁	C₄	1	2	yes	—
11	#346×C₇₁	C₂²	1	2	yes	—
12	C₇₁⋊C₃₂	C₂×C₄	1	8	yes	complemented
13	C₂²×C₈×C₇₁	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₇₁	C₈	1	1	yes	characteristic
16	C₂×C₁₆×C₇₁	C₈	1	2	yes	—
17	C₂×C₁₆×C₇₁	C₈	1	2	yes	—
18	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
22	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₇₁	C₈	1	2	yes	—
24	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₇₁	—	2	1	no	—
26	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₇₁	—	2	1	no	—
28	C₂×C₈×C₇₁	—	2	1	no	—
29	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
30	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
31	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₇₁	—	2	1	no	—
37	C₄²×C₇₁	C₂×C₈	1	2	yes	—
38	C₁₆×C₇₁	C₄²	1	2	yes	—
39	C₁₆×C₇₁	C₂×C₈	1	4	yes	—
40	C₁₆×C₇₁	C₂²×C₄	1	2	yes	—
41	C₂³×C₇₁	M₃₂	1	1	yes	characteristic
42	C₂×C₄×C₇₁	—	2	2	no	—
43	C₂×C₄×C₇₁	—	4	2	no	—
44	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
46	C₂×C₄×C₇₁	M₃₂	1	1	yes	characteristic
47	C₂×C₄×C₇₁	M₃₂	1	1	yes	characteristic
48	C₂×C₄×C₇₁	M₃₂	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₇₁	—	2	2	no	—
51	C₈×C₇₁	C₄×C₈	1	1	yes	characteristic
52	C₈×C₇₁	C₄×C₈	1	1	yes	characteristic
53	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
54	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
55	C₂²×C₇₁	—	2	2	no	—
56	C₂²×C₇₁	#346	1	1	yes	characteristic
57	C₂²×C₇₁	#346	1	1	yes	characteristic
58	C₂²×C₇₁	C₂×M₃₂	1	1	yes	characteristic
59	C₄×C₇₁	—	4	2	no	—
60	C₄×C₇₁	#346	1	1	yes	characteristic
61	C₄×C₇₁	#346	1	1	yes	characteristic
62	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
64	C₂×#1398	—	71	1	no	maximal, Sylow
65	C₂×C₇₁	—	2	1	no	—
66	C₂×C₇₁	—	2	1	no	—
67	C₂×C₇₁	#1398	1	2	yes	complemented
68	C₂×C₇₁	C₂×#346	1	1	yes	characteristic
69	C₂×M₆₄	—	71	2	no	—
70	C₂×#346	D₇₁	1	1	yes	characteristic
71	#1398	—	71	4	no	—
72	C₇₁	C₂×#1398	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₁₆	Dic₇₁	1	1	yes	characteristic
74	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
75	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
76	M₆₄	—	71	4	no	—
77	M₆₄	—	71	4	no	—
78	C₂×C₃₂	—	71	4	no	—
79	#346	Dic₇₁	1	2	yes	—
80	#346	C₂×D₇₁	1	2	yes	—
81	C₂²×C₈	C₇₁⋊C₈	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
83	C₂×C₄²	C₇₁⋊C₈	1	1	yes	characteristic
84	C₂×C₁₆	C₇₁⋊C₈	1	2	yes	—
85	C₂×C₁₆	C₇₁⋊C₈	1	2	yes	—
86	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
87	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
88	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
89	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
90	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
91	C₄×C₈	C₇₁⋊C₈	1	2	yes	—
92	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
93	C₃₂	—	71	8	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
99	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
100	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
101	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
102	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
103	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
104	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
105	C₄²	—	2	1	no	—
106	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
107	C₁₆	C₄×Dic₇₁	1	2	yes	—
108	C₁₆	C₂×C₇₁⋊C₈	1	4	yes	—
109	C₁₆	C₂×C₄×D₇₁	1	2	yes	—
110	C₂³	#14556068	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄×C₇₁⋊C₈	1	1	yes	characteristic
114	C₂×C₄	C₄×C₇₁⋊C₈	1	1	yes	characteristic
115	C₂×C₄	#14556068	1	1	yes	characteristic
116	C₂×C₄	#14556068	1	1	yes	characteristic
117	C₂×C₄	#14556068	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	C₄×C₇₁⋊C₈	1	1	yes	characteristic
121	C₈	C₄×C₇₁⋊C₈	1	1	yes	characteristic
122	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
123	C₈	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
124	C₂²	—	2	2	no	—
125	C₂²	#17985775	1	1	yes	characteristic
126	C₂²	#17985775	1	1	yes	characteristic
127	C₂²	C₂×#14556068	1	1	yes	characteristic
128	C₄	—	4	2	no	—
129	C₄	#17985775	1	1	yes	characteristic
130	C₄	#17985775	1	1	yes	characteristic
131	C₄	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
132	C₄	C₂×#14556068	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	2	1	no	—
135	C₂	#22586476	1	2	yes	direct product factor
136	C₂	C₂×#17985775	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₂×#346×C₇₁	C₂
socle	C₂²×C₇₁	#346
prime radical	C₂³×C₇₁	M₃₂
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₂×#346	D₇₁
hypocenter	C₇₁	C₂×#1398
LCS	G ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. #22586476	C₂	direct (N × H)
4. #17985833	C₄	semidirect
6. C₂×C₇₁⋊C₃₂	C₄	semidirect
12. C₇₁⋊C₃₂	C₂×C₄	semidirect
67. C₂×C₇₁	#1398	semidirect
72. C₇₁	C₂×#1398	semidirect
135. C₂	#22586476	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×#17985833	2
#22586476	2
C₂×#346×C₇₁	2
C₂×#1398	71

Subgroups — C2×#22586476 #31917637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586476 #31917637