Subgroups — C₁₀₉×#12901 #42130895

Order 27904.

342 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×#364×C₁₀₉	C₂	1	4	yes	maximal
2	C₁₀₉×#2150	C₂	1	2	yes	maximal
3	C₁₀₉×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
5	C₂×#97×C₁₀₉	C₄	1	2	yes	—
6	C₂×#97×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	#376×C₁₀₉	C₄	1	1	yes	characteristic
10	#364×C₁₀₉	—	2	8	no	—
11	C₄⋊C₁₆×C₁₀₉	—	2	4	no	—
12	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
15	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	M₃₂×C₁₀₉	—	2	8	no	—
18	C₂×C₁₆×C₁₀₉	—	2	4	no	—
19	#97×C₁₀₉	—	2	4	no	—
20	#97×C₁₀₉	C₈	1	4	yes	—
21	C₄×C₈×C₁₀₉	D₄	1	4	yes	—
22	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
23	C₄×C₈×C₁₀₉	Q₈	1	4	yes	—
24	C₄×C₈×C₁₀₉	Q₈	1	2	yes	—
25	#95×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	—	2	2	no	—
27	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
28	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
32	C₂×C₈×C₁₀₉	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₁₀₉	M₁₆	1	4	yes	—
34	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
35	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
38	C₄²×C₁₀₉	—	2	2	no	—
39	C₄²×C₁₀₉	C₄⋊C₄	1	4	yes	—
40	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
41	C₄²×C₁₀₉	C₂×C₈	1	2	yes	—
42	C₁₆×C₁₀₉	—	4	4	no	—
43	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄×C₁₀₉	—	4	2	no	—
46	C₂×C₄×C₁₀₉	—	4	4	no	—
47	C₂×C₄×C₁₀₉	C₄⋊C₈	1	4	yes	—
48	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
52	C₈×C₁₀₉	—	2	2	no	—
53	C₈×C₁₀₉	C₄⋊C₈	1	8	yes	—
54	C₈×C₁₀₉	#127	1	1	yes	characteristic
55	C₈×C₁₀₉	C₄⋊Q₈	1	2	yes	—
56	C₈×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₁₀₉	—	2	2	no	—
58	C₂²×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂²×C₁₀₉	#423	1	2	yes	—
60	C₄×C₁₀₉	—	2	4	no	—
61	C₄×C₁₀₉	—	4	4	no	—
62	C₄×C₁₀₉	#384	1	1	yes	characteristic
63	C₄×C₁₀₉	C₂×C₄⋊C₈	1	2	yes	—
64	C₄×C₁₀₉	#423	1	1	yes	characteristic
65	#12901	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₁₀₉	—	2	2	no	—
67	C₂×C₁₀₉	#1842	1	1	yes	characteristic
68	C₂×C₁₀₉	#2153	1	2	yes	—
69	C₂×#364	C₂×C₁₀₉	1	4	yes	—
70	#2150	C₂×C₁₀₉	1	2	yes	—
71	#1726	C₂×C₁₀₉	1	1	yes	characteristic
72	C₁₀₉	#12901	1	1	yes	characteristic, direct product factor, Sylow
73	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
74	C₂×#97	C₄×C₁₀₉	1	2	yes	—
75	C₂×#97	C₄×C₁₀₉	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
78	#376	C₄×C₁₀₉	1	1	yes	characteristic
79	#364	—	2	8	no	—
80	C₄⋊C₁₆	—	2	4	no	—
81	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
82	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
83	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
84	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
85	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
86	M₃₂	—	2	8	no	—
87	C₂×C₁₆	—	2	4	no	—
88	#97	—	2	4	no	—
89	#97	C₈×C₁₀₉	1	4	yes	—
90	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
91	C₄×C₈	D₄×C₁₀₉	1	4	yes	—
92	C₄×C₈	Q₈×C₁₀₉	1	4	yes	—
93	C₄×C₈	Q₈×C₁₀₉	1	2	yes	—
94	#95	—	2	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
97	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
100	C₂×C₈	C₄⋊C₄×C₁₀₉	1	4	yes	—
101	C₂×C₈	C₂×C₈×C₁₀₉	1	4	yes	—
102	C₂×C₈	M₁₆×C₁₀₉	1	4	yes	—
103	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
104	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
105	C₂×C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
106	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
107	C₄²	—	2	2	no	—
108	C₄²	C₄⋊C₄×C₁₀₉	1	4	yes	—
109	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
110	C₄²	C₂×C₈×C₁₀₉	1	2	yes	—
111	C₁₆	—	4	4	no	—
112	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	C₄⋊C₈×C₁₀₉	1	4	yes	—
117	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
119	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
120	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
121	C₈	—	2	2	no	—
122	C₈	C₄⋊C₈×C₁₀₉	1	8	yes	—
123	C₈	#127×C₁₀₉	1	1	yes	characteristic
124	C₈	C₄⋊Q₈×C₁₀₉	1	2	yes	—
125	C₈	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	C₂×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
128	C₂²	#423×C₁₀₉	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	#384×C₁₀₉	1	1	yes	characteristic
132	C₄	C₂×C₄⋊C₈×C₁₀₉	1	2	yes	—
133	C₄	#423×C₁₀₉	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₁₀₉×#1842	1	1	yes	characteristic
136	C₂	C₁₀₉×#2153	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₀₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₄⋊C₈
prime radical	C₂³×C₁₀₉	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
65. #12901	C₁₀₉	direct (N × H)
72. C₁₀₉	#12901	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×#364×C₁₀₉	2
C₁₀₉×#2150	2
C₁₀₉×#1726	2
#12901	109

Subgroups — C109×#12901 #42130895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12901 #42130895