Subgroups — C₂×C₁₀₇×#1398 #41039514

Order 27392.

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×M₆₄×C₁₀₇	C₂	1	2	yes	maximal
2	C₂×#346×C₁₀₇	C₂	1	1	yes	maximal, characteristic
3	C₁₀₇×#1398	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₀₇	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₀₇	C₄	1	1	yes	characteristic
7	M₆₄×C₁₀₇	C₄	1	4	yes	complemented
8	M₆₄×C₁₀₇	C₂²	1	4	yes	—
9	C₂×C₃₂×C₁₀₇	C₄	1	4	yes	complemented
10	#346×C₁₀₇	C₄	1	2	yes	—
11	#346×C₁₀₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₀₇	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₀₇	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₁₀₇	C₈	1	2	yes	—
16	C₂×C₁₆×C₁₀₇	C₈	1	2	yes	—
17	C₂×C₁₆×C₁₀₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₁₀₇	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₁₀₇	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₁₀₇	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₁₀₇	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₁₀₇	C₈	1	2	yes	—
23	C₄×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
24	C₃₂×C₁₀₇	C₂×C₄	1	8	yes	complemented
25	C₂²×C₄×C₁₀₇	—	2	1	no	—
26	C₂²×C₄×C₁₀₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₁₀₇	—	2	1	no	—
28	C₂×C₈×C₁₀₇	—	2	1	no	—
29	C₂×C₈×C₁₀₇	C₄²	1	1	yes	characteristic
30	C₂×C₈×C₁₀₇	C₄²	1	1	yes	characteristic
31	C₂×C₈×C₁₀₇	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₁₀₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₀₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₀₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₁₀₇	—	2	1	no	—
37	C₄²×C₁₀₇	C₂×C₈	1	2	yes	—
38	C₁₆×C₁₀₇	C₄²	1	2	yes	—
39	C₁₆×C₁₀₇	C₂×C₈	1	4	yes	—
40	C₁₆×C₁₀₇	C₂²×C₄	1	2	yes	—
41	C₂³×C₁₀₇	M₃₂	1	1	yes	characteristic
42	C₂×C₄×C₁₀₇	—	2	2	no	—
43	C₂×C₄×C₁₀₇	—	4	2	no	—
44	C₂×C₄×C₁₀₇	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₀₇	C₄×C₈	1	1	yes	characteristic
46	C₂×C₄×C₁₀₇	M₃₂	1	1	yes	characteristic
47	C₂×C₄×C₁₀₇	M₃₂	1	1	yes	characteristic
48	C₂×C₄×C₁₀₇	M₃₂	1	1	yes	characteristic
49	C₂×C₄×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₁₀₇	—	2	2	no	—
51	C₈×C₁₀₇	C₄×C₈	1	1	yes	characteristic
52	C₈×C₁₀₇	C₄×C₈	1	1	yes	characteristic
53	C₈×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
54	C₈×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
55	C₂²×C₁₀₇	—	2	2	no	—
56	C₂²×C₁₀₇	#346	1	1	yes	characteristic
57	C₂²×C₁₀₇	#346	1	1	yes	characteristic
58	C₂²×C₁₀₇	C₂×M₃₂	1	1	yes	characteristic
59	C₄×C₁₀₇	—	4	2	no	—
60	C₄×C₁₀₇	#346	1	1	yes	characteristic
61	C₄×C₁₀₇	#346	1	1	yes	characteristic
62	C₄×C₁₀₇	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₁₀₇	C₂×M₃₂	1	1	yes	characteristic
64	C₂×#1398	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₁₀₇	—	2	1	no	—
66	C₂×C₁₀₇	—	2	1	no	—
67	C₂×C₁₀₇	#1398	1	2	yes	direct product factor
68	C₂×C₁₀₇	C₂×#346	1	1	yes	characteristic
69	C₂×M₆₄	C₂×C₁₀₇	1	2	yes	—
70	C₂×#346	C₂×C₁₀₇	1	1	yes	characteristic
71	#1398	C₂×C₁₀₇	1	4	yes	direct product factor
72	C₁₀₇	C₂×#1398	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×C₁₆	C₄×C₁₀₇	1	1	yes	characteristic
74	C₂²×C₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
75	C₂×C₄×C₈	C₄×C₁₀₇	1	1	yes	characteristic
76	M₆₄	C₄×C₁₀₇	1	4	yes	complemented
77	M₆₄	C₂²×C₁₀₇	1	4	yes	—
78	C₂×C₃₂	C₄×C₁₀₇	1	4	yes	complemented
79	#346	C₄×C₁₀₇	1	2	yes	—
80	#346	C₂²×C₁₀₇	1	2	yes	—
81	C₂²×C₈	C₈×C₁₀₇	1	1	yes	characteristic
82	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
83	C₂×C₄²	C₈×C₁₀₇	1	1	yes	characteristic
84	C₂×C₁₆	C₈×C₁₀₇	1	2	yes	—
85	C₂×C₁₆	C₈×C₁₀₇	1	2	yes	—
86	C₂×C₁₆	C₂×C₄×C₁₀₇	1	1	yes	characteristic
87	C₂×C₁₆	C₂×C₄×C₁₀₇	1	2	yes	—
88	C₂×C₁₆	C₂×C₄×C₁₀₇	1	2	yes	—
89	C₂×C₁₆	C₂×C₄×C₁₀₇	1	2	yes	—
90	C₂×C₁₆	C₂³×C₁₀₇	1	1	yes	characteristic
91	C₄×C₈	C₈×C₁₀₇	1	2	yes	—
92	C₄×C₈	C₂×C₄×C₁₀₇	1	2	yes	—
93	C₃₂	C₂×C₄×C₁₀₇	1	8	yes	complemented
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₂×C₈×C₁₀₇	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	C₄²×C₁₀₇	1	1	yes	characteristic
99	C₂×C₈	C₄²×C₁₀₇	1	1	yes	characteristic
100	C₂×C₈	C₂×C₈×C₁₀₇	1	2	yes	—
101	C₂×C₈	C₂×C₈×C₁₀₇	1	1	yes	characteristic
102	C₂×C₈	C₂×C₈×C₁₀₇	1	1	yes	characteristic
103	C₂×C₈	C₂×C₈×C₁₀₇	1	1	yes	characteristic
104	C₂×C₈	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
105	C₄²	—	2	1	no	—
106	C₄²	C₂×C₈×C₁₀₇	1	2	yes	—
107	C₁₆	C₄²×C₁₀₇	1	2	yes	—
108	C₁₆	C₂×C₈×C₁₀₇	1	4	yes	—
109	C₁₆	C₂²×C₄×C₁₀₇	1	2	yes	—
110	C₂³	M₃₂×C₁₀₇	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄×C₈×C₁₀₇	1	1	yes	characteristic
114	C₂×C₄	C₄×C₈×C₁₀₇	1	1	yes	characteristic
115	C₂×C₄	M₃₂×C₁₀₇	1	1	yes	characteristic
116	C₂×C₄	M₃₂×C₁₀₇	1	1	yes	characteristic
117	C₂×C₄	M₃₂×C₁₀₇	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₈×C₁₀₇	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	C₄×C₈×C₁₀₇	1	1	yes	characteristic
121	C₈	C₄×C₈×C₁₀₇	1	1	yes	characteristic
122	C₈	C₂×C₄²×C₁₀₇	1	1	yes	characteristic
123	C₈	C₂²×C₈×C₁₀₇	1	1	yes	characteristic
124	C₂²	—	2	2	no	—
125	C₂²	#346×C₁₀₇	1	1	yes	characteristic
126	C₂²	#346×C₁₀₇	1	1	yes	characteristic
127	C₂²	C₂×M₃₂×C₁₀₇	1	1	yes	characteristic
128	C₄	—	4	2	no	—
129	C₄	#346×C₁₀₇	1	1	yes	characteristic
130	C₄	#346×C₁₀₇	1	1	yes	characteristic
131	C₄	C₂×C₄×C₈×C₁₀₇	1	1	yes	characteristic
132	C₄	C₂×M₃₂×C₁₀₇	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	2	1	no	—
135	C₂	C₁₀₇×#1398	1	2	yes	direct product factor
136	C₂	C₂×#346×C₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₀₇	C₄²
commutator	C₄	C₂×C₄×C₈×C₁₀₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₀₇
socle	C₂²×C₁₀₇	#346
prime radical	C₂³×C₁₀₇	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₇×#1398	C₂	direct (N × H)
7. M₆₄×C₁₀₇	C₄	semidirect
9. C₂×C₃₂×C₁₀₇	C₄	semidirect
24. C₃₂×C₁₀₇	C₂×C₄	semidirect
64. C₂×#1398	C₁₀₇	direct (N × H)
67. C₂×C₁₀₇	#1398	direct (N × H)
71. #1398	C₂×C₁₀₇	direct (N × H)
72. C₁₀₇	C₂×#1398	direct (N × H)
76. M₆₄	C₄×C₁₀₇	semidirect
78. C₂×C₃₂	C₄×C₁₀₇	semidirect
93. C₃₂	C₂×C₄×C₁₀₇	semidirect
135. C₂	C₁₀₇×#1398	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×M₆₄×C₁₀₇	2
C₂×#346×C₁₀₇	2
C₁₀₇×#1398	2
C₂×#1398	107

Subgroups — C2×C107×#1398 #41039514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₇×#1398 #41039514