Subgroups — C₄×#17989519 #34014206

Order 18688.

250 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×#17989519	C₂	1	2	yes	maximal
2	#22625038	C₂	1	2	yes	maximal, complemented
3	C₄×#14556641	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#11924003	C₂	1	1	yes	maximal, characteristic
5	#17989519	C₄	1	8	yes	direct product factor
6	C₂×#14556641	C₂²	1	4	yes	—
7	C₂²×#11924003	C₄	1	1	yes	characteristic
8	C₂²×#11924003	C₂²	1	1	yes	characteristic
9	C₄×#11924003	C₄	1	1	yes	characteristic
10	C₄×#11924003	C₄	1	1	yes	characteristic
11	C₄×#11924003	C₂²	1	1	yes	characteristic
12	C₄×#11924003	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
14	#14556641	C₂×C₄	1	8	yes	complemented
15	C₂×#11924003	C₈	1	4	yes	—
16	C₂×#11924003	C₂×C₄	1	4	yes	—
17	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
18	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
19	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
20	C₂×#11924003	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
22	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
24	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
25	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
26	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	#11924003	C₄²	1	2	yes	—
29	#11924003	C₂×C₈	1	2	yes	—
30	#11924003	C₂×C₈	1	2	yes	—
31	#11924003	C₂²×C₄	1	2	yes	—
32	C₂²×C₄×C₇₃	C₁₆	1	2	yes	—
33	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₇₃	C₁₆	1	4	yes	—
35	C₂×C₈×C₇₃	C₄²	1	2	yes	—
36	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₇₃	C₁₆	1	2	yes	—
42	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₃	—	2	4	no	—
46	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
51	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₇₃	C₄×C₈	1	2	yes	—
53	C₈×C₇₃	C₂×C₁₆	1	2	yes	—
54	C₈×C₇₃	C₂×C₁₆	1	2	yes	—
55	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
56	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
57	C₂²×C₇₃	—	2	2	no	—
58	C₂²×C₇₃	C₄×C₁₆	1	2	yes	—
59	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
60	C₄×C₇₃	—	2	2	no	—
61	C₄×C₇₃	C₄×C₁₆	1	2	yes	—
62	C₄×C₇₃	M₆₄	1	4	yes	complemented
63	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
65	C₄×M₆₄	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	—	2	2	no	—
67	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
68	C₂×C₇₃	C₂×M₆₄	1	2	yes	—
69	C₂×M₆₄	—	73	2	no	—
70	C₂×C₄×C₁₆	—	73	1	no	—
71	#1397	—	73	2	no	—
72	C₄×C₃₂	—	73	2	no	—
73	C₇₃	C₄×M₆₄	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₁₆	—	73	1	no	—
75	C₂²×C₁₆	—	73	1	no	—
76	C₂×C₄×C₈	#64408	1	1	yes	characteristic
77	M₆₄	—	73	8	no	—
78	C₂×C₃₂	—	73	4	no	—
79	C₄×C₁₆	—	73	1	no	—
80	C₄×C₁₆	—	73	1	no	—
81	C₄×C₁₆	—	73	1	no	—
82	C₄×C₁₆	—	73	1	no	—
83	C₂²×C₈	#10596833	1	1	yes	characteristic
84	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
85	C₂×C₄²	#10596833	1	1	yes	characteristic
86	C₂×C₁₆	—	73	4	no	—
87	C₂×C₁₆	—	73	4	no	—
88	C₂×C₁₆	—	73	1	no	—
89	C₂×C₁₆	—	73	1	no	—
90	C₂×C₁₆	—	73	1	no	—
91	C₂×C₁₆	—	73	1	no	—
92	C₄×C₈	#10596833	1	1	yes	characteristic
93	C₄×C₈	#10596833	1	1	yes	characteristic
94	C₄×C₈	C₂×#64408	1	1	yes	characteristic
95	C₄×C₈	C₂×#64408	1	1	yes	characteristic
96	C₃₂	—	73	8	no	—
97	C₂²×C₄	#11924031	1	2	yes	—
98	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
99	C₂×C₈	#11924031	1	4	yes	—
100	C₂×C₈	C₄×#64408	1	2	yes	—
101	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
102	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
103	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
104	C₂×C₈	#11924036	1	2	yes	—
105	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
106	C₄²	#11924031	1	2	yes	—
107	C₄²	C₂×#10596833	1	1	yes	characteristic
108	C₄²	C₂×#10596833	1	1	yes	characteristic
109	C₁₆	—	73	2	no	—
110	C₁₆	—	73	2	no	—
111	C₁₆	—	73	2	no	—
112	C₁₆	—	73	2	no	—
113	C₂³	C₂×#11924031	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	C₄×#10596833	1	2	yes	—
116	C₂×C₄	C₂×#11924031	1	2	yes	—
117	C₂×C₄	C₂×#11924031	1	2	yes	—
118	C₂×C₄	C₂×#11924031	1	1	yes	characteristic
119	C₂×C₄	C₂×#11924031	1	2	yes	—
120	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
121	C₈	C₄×#10596833	1	2	yes	—
122	C₈	C₂×#11924031	1	2	yes	—
123	C₈	C₂×#11924031	1	2	yes	—
124	C₈	C₂×C₄×#64408	1	1	yes	characteristic
125	C₈	C₂×#11924036	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	C₄×#11924031	1	2	yes	—
128	C₂²	C₂²×#11924031	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	C₄×#11924031	1	2	yes	—
131	C₄	#17989519	1	4	yes	direct product factor
132	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
133	C₄	C₂²×#11924031	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂×C₄×#11924031	1	1	yes	characteristic
136	C₂	C₂×#17989519	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂×#64408
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₁₆
prime radical	C₂³×C₇₃	C₂×C₁₆
cosocle	C₂×#11924003	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₄×M₆₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22625038	C₂	semidirect
3. C₄×#14556641	C₂	semidirect
5. #17989519	C₄	direct (N × H)
14. #14556641	C₂×C₄	semidirect
62. C₄×C₇₃	M₆₄	semidirect
73. C₇₃	C₄×M₆₄	semidirect
131. C₄	#17989519	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×#17989519	2
#22625038	2
C₄×#14556641	2
C₂×C₄×#11924003	2
C₄×M₆₄	73

Subgroups — C4×#17989519 #34014206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17989519 #34014206