Subgroups — C₂×C₃×#6393 #13055539

Order 1344.

212 conjugacy classes of subgroups.

The subgroup lattice has 132 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	C₂×C₃×#1136	C₂	1	2	yes	maximal
3	C₃×#6393	C₂	1	4	yes	maximal, direct product factor
4	C₂×#6393	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₇×C₈	C₄	1	2	yes	—
7	C₂×C₃×C₇×C₈	C₄	1	2	yes	—
8	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
9	C₃×#1136	—	2	2	no	—
10	C₃×#1136	C₂²	1	4	yes	—
11	C₂×C₃×C₇⋊C₈	—	2	2	no	—
12	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
13	C₂×#1136	C₂×C₃	1	2	yes	—
14	#6393	C₂×C₃	1	4	yes	direct product factor
15	C₂×C₃×#108	—	7	1	no	maximal, Hall
16	C₂³×C₃×C₇	Q₈	1	1	yes	characteristic
17	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂×C₃×C₄×C₇	Q₈	1	1	yes	characteristic
22	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₃×C₇×C₈	C₂×C₄	1	4	yes	—
24	C₃×C₇⋊C₈	—	2	4	no	—
25	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
26	C₂×C₇×C₈	C₃×C₄	1	2	yes	—
27	C₂×C₇×C₈	C₃×C₄	1	2	yes	—
28	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
29	#1136	—	2	2	no	—
30	#1136	C₂²×C₃	1	4	yes	—
31	C₂×C₇⋊C₈	—	2	2	no	—
32	C₂×C₃×M₁₆	—	7	2	no	—
33	C₂²×C₃×C₈	D₇	1	1	yes	characteristic
34	C₃×#108	—	7	4	no	—
35	C₂²×C₃×C₇	—	2	2	no	—
36	C₂²×C₃×C₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₃×C₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₃×C₇	C₂×Q₈	1	1	yes	characteristic
39	C₃×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
40	C₃×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
41	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
42	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
43	C₂×#108	—	7	1	no	Sylow
44	C₂³×C₇	C₃×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
46	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
47	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇	C₃×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
51	C₇×C₈	C₂×C₃×C₄	1	4	yes	—
52	C₇⋊C₈	—	2	4	no	—
53	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
54	C₃×M₁₆	—	7	4	no	—
55	C₃×M₁₆	—	14	2	no	—
56	C₂×C₃×C₈	—	14	2	no	—
57	C₂×C₃×C₈	Dic₇	1	2	yes	—
58	C₂×C₃×C₈	Dic₇	1	2	yes	—
59	C₂×C₃×C₈	C₂×D₇	1	2	yes	—
60	C₂×C₃×C₇	—	2	1	no	—
61	C₂×C₃×C₇	—	2	1	no	—
62	C₂×C₃×C₇	#108	1	2	yes	complemented
63	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×M₁₆	—	7	2	no	—
65	C₂²×C₈	C₃×D₇	1	1	yes	characteristic
66	#108	—	7	4	no	—
67	C₂²×C₇	—	2	2	no	—
68	C₂²×C₇	C₃×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₇	C₃×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
71	C₄×C₇	C₃×C₄⋊C₄	1	1	yes	characteristic
72	C₄×C₇	C₃×C₄⋊C₄	1	1	yes	characteristic
73	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
74	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
75	C₂³×C₃	Dic₁₄	1	1	yes	characteristic
76	C₂×C₃×C₄	Dic₁₄	1	1	yes	characteristic
77	C₂×C₃×C₄	D₂₈	1	1	yes	characteristic
78	C₂×C₃×C₄	D₂₈	1	1	yes	characteristic
79	C₂×C₃×C₄	C₂×Dic₇	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×Dic₇	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
82	C₃×C₈	—	14	4	no	—
83	C₃×C₈	C₂×Dic₇	1	4	yes	—
84	C₃×C₇	C₂×#108	1	1	yes	characteristic, complemented, Hall
85	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
86	M₁₆	—	7	4	no	—
87	M₁₆	—	14	2	no	—
88	C₂×C₈	—	14	2	no	—
89	C₂×C₈	C₃×Dic₇	1	2	yes	—
90	C₂×C₈	C₃×Dic₇	1	2	yes	—
91	C₂×C₈	C₂×C₃×D₇	1	2	yes	—
92	C₂×C₇	—	2	1	no	—
93	C₂×C₇	—	2	1	no	—
94	C₂×C₇	C₃×#108	1	2	yes	complemented
95	C₂×C₇	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₃	—	2	2	no	—
97	C₂²×C₃	C₄⋊Dic₇	1	1	yes	characteristic
98	C₂²×C₃	C₄⋊Dic₇	1	1	yes	characteristic
99	C₂²×C₃	C₂×Dic₁₄	1	1	yes	characteristic
100	C₃×C₄	C₄⋊Dic₇	1	1	yes	characteristic
101	C₃×C₄	C₄⋊Dic₇	1	1	yes	characteristic
102	C₃×C₄	C₂×D₂₈	1	1	yes	characteristic
103	C₃×C₄	C₂²×Dic₇	1	1	yes	characteristic
104	C₂³	C₃×Dic₁₄	1	1	yes	characteristic
105	C₂×C₄	C₃×Dic₁₄	1	1	yes	characteristic
106	C₂×C₄	C₃×D₂₈	1	1	yes	characteristic
107	C₂×C₄	C₃×D₂₈	1	1	yes	characteristic
108	C₂×C₄	C₂×C₃×Dic₇	1	1	yes	characteristic
109	C₂×C₄	C₂×C₃×Dic₇	1	1	yes	characteristic
110	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
111	C₈	—	14	4	no	—
112	C₈	C₂×C₃×Dic₇	1	4	yes	—
113	C₇	C₂×C₃×#108	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₃	—	2	1	no	—
115	C₂×C₃	—	2	1	no	—
116	C₂×C₃	#6393	1	2	yes	direct product factor
117	C₂×C₃	C₂×C₄⋊Dic₇	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	C₃×C₄⋊Dic₇	1	1	yes	characteristic
120	C₂²	C₃×C₄⋊Dic₇	1	1	yes	characteristic
121	C₂²	C₂×C₃×Dic₁₄	1	1	yes	characteristic
122	C₄	C₃×C₄⋊Dic₇	1	1	yes	characteristic
123	C₄	C₃×C₄⋊Dic₇	1	1	yes	characteristic
124	C₄	C₂×C₃×D₂₈	1	1	yes	characteristic
125	C₄	C₂²×C₃×Dic₇	1	1	yes	characteristic
126	C₃	C₂×#6393	1	1	yes	characteristic, direct product factor, Sylow
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	C₃×#6393	1	2	yes	direct product factor
130	C₂	C₂×C₃×C₄⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₂₈
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₂²×C₃×C₇×C₈	C₂
socle	C₂²×C₃×C₇	C₄⋊C₄
prime radical	C₂³×C₃×C₇	Q₈
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₂²×C₃×C₈	D₇
hypocenter	C₇	C₂×C₃×#108
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×#6393	C₂	direct (N × H)
4. C₂×#6393	C₃	direct (N × H)
14. #6393	C₂×C₃	direct (N × H)
62. C₂×C₃×C₇	#108	semidirect
84. C₃×C₇	C₂×#108	semidirect
94. C₂×C₇	C₃×#108	semidirect
113. C₇	C₂×C₃×#108	semidirect
116. C₂×C₃	#6393	direct (N × H)
126. C₃	C₂×#6393	direct (N × H)
129. C₂	C₃×#6393	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
C₂×C₃×#1136	2
C₃×#6393	2
C₂×#6393	3
C₂×C₃×#108	7

Subgroups — C2×C3×#6393 #13055539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#6393 #13055539