Subgroups — #30772268

Order 17152.

374 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863224	C₂	1	4	yes	maximal
2	#22560369	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523977	—	2	2	no	—
5	C₂×#14523977	C₂²	1	4	yes	—
6	#16863266	—	2	2	no	—
7	#16863224	—	2	8	no	—
8	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₆₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₆₇	—	2	4	no	—
12	C₂×C₄⋊Dic₆₇	—	4	4	no	—
13	#14523977	—	2	8	no	—
14	#14523977	—	4	4	no	—
15	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
19	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
20	C₂×C₁₆×C₆₇	Q₈	1	4	yes	—
21	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₆₇	Q₈	1	2	yes	—
23	C₂²×Dic₆₇	—	4	4	no	—
24	C₄⋊Dic₆₇	—	4	4	no	—
25	C₄⋊Dic₆₇	—	4	4	no	—
26	C₄⋊Dic₆₇	—	8	8	no	—
27	C₂²×C₄×C₆₇	D₈	1	2	yes	—
28	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₆₇	D₈	1	2	yes	—
31	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
32	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
34	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
35	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
37	C₄²×C₆₇	Q₁₆	1	2	yes	—
38	C₁₆×C₆₇	C₄⋊C₄	1	8	yes	complemented
39	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₆₇	#107	1	4	yes	—
41	C₂×C₄×C₆₇	SD₃₂	1	8	yes	—
42	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×Dic₆₇	—	8	4	no	—
49	C₂×Dic₆₇	—	8	8	no	—
50	C₈×C₆₇	#107	1	4	yes	—
51	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
52	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
53	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
54	C₂²×C₆₇	#493	1	1	yes	characteristic
55	C₂²×C₆₇	#500	1	1	yes	characteristic
56	C₂²×C₆₇	C₂×SD₃₂	1	4	yes	—
57	C₄×C₆₇	#367	1	8	yes	complemented
58	C₄×C₆₇	#384	1	1	yes	characteristic
59	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
60	C₄×C₆₇	#494	1	1	yes	characteristic
61	C₄×C₆₇	#500	1	1	yes	characteristic
62	Dic₆₇	—	16	4	no	—
63	#12906	—	67	1	no	maximal, Sylow
64	C₂×C₆₇	#1845	1	1	yes	characteristic
65	C₂×C₆₇	C₂×#367	1	2	yes	—
66	C₂×C₆₇	#2248	1	2	yes	—
67	C₂×C₆₇	#2254	1	2	yes	—
68	C₂×#367	—	67	4	no	—
69	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
70	#1845	—	67	2	no	—
71	C₆₇	#12906	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
73	C₂×#107	—	67	4	no	—
74	C₂×#107	—	134	2	no	—
75	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
76	#384	—	134	2	no	—
77	#367	—	134	8	no	—
78	C₄×C₁₆	Dic₆₇	1	4	yes	complemented
79	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
81	C₂×C₄⋊C₄	—	268	4	no	—
82	C₂×C₄⋊C₄	—	134	4	no	—
83	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
84	C₂×C₁₆	Dic₁₃₄	1	4	yes	—
85	C₂×C₁₆	D₂₆₈	1	4	yes	—
86	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
87	#107	—	268	4	no	—
88	#107	—	134	8	no	—
89	C₄×C₈	Dic₁₃₄	1	2	yes	—
90	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
91	C₂²×C₄	—	268	4	no	—
92	C₂²×C₄	D₅₃₆	1	2	yes	—
93	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
94	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
95	C₂×C₈	D₅₃₆	1	2	yes	—
96	C₂×C₈	Dic₂₆₈	1	2	yes	—
97	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
98	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
99	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
100	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
101	C₄⋊C₄	—	268	4	no	—
102	C₄⋊C₄	—	268	4	no	—
103	C₄⋊C₄	—	536	8	no	—
104	C₄²	C₄⋊Dic₆₇	1	2	yes	—
105	C₄²	Dic₂₆₈	1	2	yes	—
106	C₁₆	C₄⋊Dic₆₇	1	8	yes	complemented
107	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
108	C₂×C₄	—	536	4	no	—
109	C₂×C₄	—	536	8	no	—
110	C₂×C₄	#14523977	1	4	yes	—
111	C₂×C₄	#14523988	1	8	yes	—
112	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
113	C₂×C₄	#14524040	1	1	yes	characteristic
114	C₂×C₄	#14524043	1	1	yes	characteristic
115	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
116	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
117	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
118	C₈	#14523977	1	4	yes	—
119	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
120	C₈	#14524043	1	2	yes	—
121	C₂²	C₂×#14523977	1	1	yes	characteristic
122	C₂²	#16863773	1	1	yes	characteristic
123	C₂²	#16863801	1	1	yes	characteristic
124	C₂²	C₂×#14523988	1	4	yes	—
125	C₄	—	1072	4	no	—
126	C₄	#16863224	1	8	yes	complemented
127	C₄	#16863266	1	1	yes	characteristic
128	C₄	C₂×#14523977	1	1	yes	characteristic
129	C₄	#16863776	1	1	yes	characteristic
130	C₄	#16863801	1	1	yes	characteristic
131	C₂	#22560369	1	1	yes	characteristic
132	C₂	C₂×#16863224	1	2	yes	—
133	C₂	#22562465	1	2	yes	—
134	C₂	#22562491	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	#12906
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₆×C₆₇	C₄	semidirect
38. C₁₆×C₆₇	C₄⋊C₄	semidirect
57. C₄×C₆₇	#367	semidirect
71. C₆₇	#12906	semidirect
78. C₄×C₁₆	Dic₆₇	semidirect
106. C₁₆	C₄⋊Dic₆₇	semidirect
126. C₄	#16863224	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863224	2
#22560369	2
C₂×C₄×C₁₆×C₆₇	2
#12906	67