Subgroups — C₃×#89873 #13055534

Order 1344.

260 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#6489	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#1136	C₂	1	1	yes	maximal, characteristic
4	C₃×#6389	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×Dic₇	C₂	1	2	yes	maximal, complemented
6	#89873	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₇×C₈	C₄	1	4	yes	complemented
9	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₂²⋊Dic₇	C₄	1	2	yes	—
12	C₃×C₄⋊Dic₇	C₄	1	2	yes	—
13	C₃×C₄×Dic₇	C₂²	1	2	yes	—
14	C₃×#1136	C₄	1	4	yes	complemented
15	C₂×C₃×C₇⋊C₈	C₂²	1	2	yes	—
16	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
17	#6489	C₂×C₃	1	1	yes	characteristic
18	C₂×#1136	C₂×C₃	1	1	yes	characteristic
19	#6389	C₂×C₃	1	2	yes	complemented
20	C₈×Dic₇	C₂×C₃	1	2	yes	complemented
21	C₃×#405	—	7	1	no	maximal, Hall
22	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×C₃×Dic₇	C₂×C₄	1	4	yes	—
28	C₃×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₃×C₇×C₈	C₂×C₄	1	2	yes	—
30	C₃×C₇⋊C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
32	C₂×C₇×C₈	C₃×C₄	1	4	yes	complemented
33	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
34	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
35	C₂²⋊Dic₇	C₃×C₄	1	2	yes	—
36	C₄⋊Dic₇	C₃×C₄	1	2	yes	—
37	C₄×Dic₇	C₂²×C₃	1	2	yes	—
38	#1136	C₃×C₄	1	4	yes	complemented
39	C₂×C₇⋊C₈	C₂²×C₃	1	2	yes	—
40	C₂×C₃×M₁₆	—	7	1	no	—
41	C₂²×C₃×C₈	D₇	1	1	yes	characteristic
42	C₃×#117	—	7	1	no	—
43	C₃×#97	—	7	2	no	—
44	C₃×C₄×C₈	—	7	2	no	—
45	C₂²×C₃×C₇	—	2	2	no	—
46	C₂²×C₃×C₇	C₄²	1	2	yes	—
47	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₄×C₇	C₄²	1	2	yes	—
49	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
50	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₃×Dic₇	—	2	4	no	—
52	#405	—	7	1	no	Sylow
53	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
54	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
55	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
56	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
57	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
58	C₂×Dic₇	C₂×C₃×C₄	1	4	yes	—
59	C₇×C₈	C₂×C₃×C₄	1	2	yes	—
60	C₇×C₈	C₂×C₃×C₄	1	2	yes	—
61	C₇⋊C₈	C₂×C₃×C₄	1	4	yes	—
62	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
63	C₃×M₁₆	—	7	4	no	—
64	C₂×C₃×C₈	—	7	2	no	—
65	C₂×C₃×C₈	Dic₇	1	4	yes	complemented
66	C₂×C₃×C₈	C₂×D₇	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂×D₇	1	1	yes	characteristic
68	C₃×C₄⋊C₄	—	7	2	no	—
69	C₃×C₂²⋊C₄	—	7	2	no	—
70	C₃×C₄²	—	7	2	no	—
71	C₂×C₃×C₇	—	2	2	no	—
72	C₂×C₃×C₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₃×C₇	#131	1	2	yes	—
74	C₂×M₁₆	—	7	1	no	—
75	C₂²×C₈	C₃×D₇	1	1	yes	characteristic
76	#117	—	7	1	no	—
77	#97	—	7	2	no	—
78	C₄×C₈	—	7	2	no	—
79	C₂²×C₇	—	2	2	no	—
80	C₂²×C₇	C₃×C₄²	1	2	yes	—
81	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
82	C₄×C₇	C₃×C₄²	1	2	yes	—
83	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
84	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
85	Dic₇	—	2	4	no	—
86	C₂³×C₃	C₄×D₇	1	1	yes	characteristic
87	C₂×C₃×C₄	—	7	4	no	—
88	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
89	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
90	C₂×C₃×C₄	C₂×Dic₇	1	2	yes	—
91	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
92	C₃×C₈	—	7	4	no	—
93	C₃×C₈	C₂×Dic₇	1	2	yes	—
94	C₃×C₈	C₂×Dic₇	1	2	yes	—
95	C₃×C₇	#405	1	1	yes	characteristic, complemented, Hall
96	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
97	M₁₆	—	7	4	no	—
98	C₂×C₈	—	7	2	no	—
99	C₂×C₈	C₃×Dic₇	1	4	yes	complemented
100	C₂×C₈	C₂×C₃×D₇	1	1	yes	characteristic
101	C₂×C₈	C₂×C₃×D₇	1	1	yes	characteristic
102	C₄⋊C₄	—	7	2	no	—
103	C₂²⋊C₄	—	7	2	no	—
104	C₄²	—	7	2	no	—
105	C₂×C₇	—	2	2	no	—
106	C₂×C₇	C₂×C₃×C₄²	1	1	yes	characteristic
107	C₂×C₇	C₃×#131	1	2	yes	—
108	C₂²×C₃	—	2	2	no	—
109	C₂²×C₃	C₄×Dic₇	1	2	yes	—
110	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
111	C₃×C₄	—	14	4	no	—
112	C₃×C₄	C₄×Dic₇	1	2	yes	—
113	C₃×C₄	C₂×C₄×D₇	1	1	yes	characteristic
114	C₃×C₄	C₂²×Dic₇	1	1	yes	characteristic
115	C₂³	C₃×C₄×D₇	1	1	yes	characteristic
116	C₂×C₄	—	7	4	no	—
117	C₂×C₄	C₃×C₄×D₇	1	1	yes	characteristic
118	C₂×C₄	C₃×C₄×D₇	1	2	yes	—
119	C₂×C₄	C₂×C₃×Dic₇	1	2	yes	—
120	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
121	C₈	—	7	4	no	—
122	C₈	C₂×C₃×Dic₇	1	2	yes	—
123	C₈	C₂×C₃×Dic₇	1	2	yes	—
124	C₇	C₃×#405	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₃	—	2	2	no	—
126	C₂×C₃	#6464	1	2	yes	—
127	C₂×C₃	C₂×C₄×Dic₇	1	1	yes	characteristic
128	C₂²	—	2	2	no	—
129	C₂²	C₃×C₄×Dic₇	1	2	yes	—
130	C₂²	C₂×C₃×C₄×D₇	1	1	yes	characteristic
131	C₄	—	14	4	no	—
132	C₄	C₃×C₄×Dic₇	1	2	yes	—
133	C₄	C₂×C₃×C₄×D₇	1	1	yes	characteristic
134	C₄	C₂²×C₃×Dic₇	1	1	yes	characteristic
135	C₃	#89873	1	1	yes	characteristic, direct product factor, Sylow
136	C₂	—	2	2	no	—
137	C₂	C₃×#6464	1	2	yes	—
138	C₂	C₂×C₃×C₄×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂×D₇
commutator	C₂×C₇	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₂²×C₃×C₇×C₈	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂³×C₃×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₂²×C₃×C₈	D₇
hypocenter	C₇	C₃×#405
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#6389	C₂	semidirect
5. C₃×C₈×Dic₇	C₂	semidirect
6. #89873	C₃	direct (N × H)
8. C₂×C₃×C₇×C₈	C₄	semidirect
14. C₃×#1136	C₄	semidirect
19. #6389	C₂×C₃	semidirect
20. C₈×Dic₇	C₂×C₃	semidirect
32. C₂×C₇×C₈	C₃×C₄	semidirect
38. #1136	C₃×C₄	semidirect
65. C₂×C₃×C₈	Dic₇	semidirect
95. C₃×C₇	#405	semidirect
99. C₂×C₈	C₃×Dic₇	semidirect
124. C₇	C₃×#405	semidirect
135. C₃	#89873	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₇×C₈	2
C₃×#6489	2
C₂×C₃×#1136	2
C₃×#6389	2
C₃×C₈×Dic₇	2
#89873	3
C₃×#405	7

Subgroups — C3×#89873 #13055534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#89873 #13055534