Subgroups — C₁₀₉×#20791 #42138785

Order 27904.

1350 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×#423×C₁₀₉	C₂	1	6	yes	maximal
2	C₂×C₄³×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1842	C₂	1	8	yes	maximal, complemented
4	C₂²×M₁₆×C₁₀₉	C₂²	1	4	yes	—
5	C₂²×C₄²×C₁₀₉	C₄	1	4	yes	—
6	C₂²×C₄²×C₁₀₉	C₂²	1	3	yes	—
7	#423×C₁₀₉	—	2	24	no	—
8	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	24	yes	—
9	C₄³×C₁₀₉	C₄	1	3	yes	—
10	C₄³×C₁₀₉	C₄	1	1	yes	characteristic
11	C₄³×C₁₀₉	C₂²	1	4	yes	—
12	C₂³×C₄×C₁₀₉	C₂×C₄	1	6	yes	—
13	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
14	C₂×M₁₆×C₁₀₉	—	2	24	no	—
15	C₂²×C₈×C₁₀₉	C₂³	1	8	yes	—
16	C₂×C₄²×C₁₀₉	—	2	16	no	—
17	C₂×C₄²×C₁₀₉	C₂×C₄	1	12	yes	—
18	C₂×C₄²×C₁₀₉	C₂×C₄	1	3	yes	—
19	C₂×C₄²×C₁₀₉	C₂×C₄	1	3	yes	—
20	C₂×C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₁₀₉	D₄	1	12	yes	—
22	C₂×C₄²×C₁₀₉	Q₈	1	12	yes	—
23	C₂×C₄²×C₁₀₉	C₂³	1	6	yes	—
24	C₄⋊C₈×C₁₀₉	—	2	48	no	—
25	C₂⁴×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₉	—	2	24	no	—
27	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	24	yes	—
28	C₂²×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
29	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
31	C₂²×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
32	C₂²×C₄×C₁₀₉	C₂×Q₈	1	6	yes	—
33	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
34	M₁₆×C₁₀₉	—	4	16	no	—
35	C₂×C₈×C₁₀₉	—	2	24	no	—
36	C₄²×C₁₀₉	—	2	24	no	—
37	C₄²×C₁₀₉	C₄⋊C₄	1	12	yes	—
38	C₄²×C₁₀₉	C₄⋊C₄	1	12	yes	—
39	C₄²×C₁₀₉	M₁₆	1	16	yes	complemented
40	C₄²×C₁₀₉	C₂×D₄	1	12	yes	—
41	C₄²×C₁₀₉	C₂×Q₈	1	12	yes	—
42	C₂³×C₁₀₉	—	2	4	no	—
43	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
44	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	—	2	12	no	—
46	C₂×C₄×C₁₀₉	—	2	48	no	—
47	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	12	yes	—
48	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
49	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	12	yes	—
50	C₂×C₄×C₁₀₉	#127	1	4	yes	—
51	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	12	yes	—
52	C₂×C₄×C₁₀₉	C₂×M₁₆	1	24	yes	—
53	C₂×C₄×C₁₀₉	C₂²×D₄	1	3	yes	—
54	C₂×C₄×C₁₀₉	C₂²×Q₈	1	3	yes	—
55	C₈×C₁₀₉	—	4	8	no	—
56	C₂²×C₁₀₉	—	2	12	no	—
57	C₂²×C₁₀₉	#384	1	4	yes	—
58	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	3	yes	—
59	C₂²×C₁₀₉	C₂²×M₁₆	1	4	yes	—
60	C₄×C₁₀₉	—	2	12	no	—
61	C₄×C₁₀₉	#384	1	4	yes	—
62	C₄×C₁₀₉	#423	1	24	yes	complemented
63	C₄×C₁₀₉	C₂×#127	1	1	yes	characteristic
64	C₄×C₁₀₉	C₂×C₄⋊Q₈	1	3	yes	—
65	#20791	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₁₀₉	—	2	4	no	—
67	C₂×C₁₀₉	C₂×#384	1	1	yes	characteristic
68	C₂×C₁₀₉	C₂×#423	1	6	yes	—
69	C₂×#423	C₂×C₁₀₉	1	6	yes	—
70	C₂×C₄³	C₂×C₁₀₉	1	1	yes	characteristic
71	#1842	C₂×C₁₀₉	1	8	yes	complemented
72	C₁₀₉	#20791	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×M₁₆	C₂²×C₁₀₉	1	4	yes	—
74	C₂²×C₄²	C₄×C₁₀₉	1	4	yes	—
75	C₂²×C₄²	C₂²×C₁₀₉	1	3	yes	—
76	#423	—	2	24	no	—
77	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	24	yes	—
78	C₄³	C₄×C₁₀₉	1	3	yes	—
79	C₄³	C₄×C₁₀₉	1	1	yes	characteristic
80	C₄³	C₂²×C₁₀₉	1	4	yes	—
81	C₂³×C₄	C₂×C₄×C₁₀₉	1	6	yes	—
82	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
83	C₂×M₁₆	—	2	24	no	—
84	C₂²×C₈	C₂³×C₁₀₉	1	8	yes	—
85	C₂×C₄²	—	2	16	no	—
86	C₂×C₄²	C₂×C₄×C₁₀₉	1	12	yes	—
87	C₂×C₄²	C₂×C₄×C₁₀₉	1	3	yes	—
88	C₂×C₄²	C₂×C₄×C₁₀₉	1	3	yes	—
89	C₂×C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
90	C₂×C₄²	D₄×C₁₀₉	1	12	yes	—
91	C₂×C₄²	Q₈×C₁₀₉	1	12	yes	—
92	C₂×C₄²	C₂³×C₁₀₉	1	6	yes	—
93	C₄⋊C₈	—	2	48	no	—
94	C₂⁴	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
95	C₂²×C₄	—	2	24	no	—
96	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	24	yes	—
97	C₂²×C₄	C₂²×C₄×C₁₀₉	1	6	yes	—
98	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
99	C₂²×C₄	C₂²×C₄×C₁₀₉	1	6	yes	—
100	C₂²×C₄	C₂×D₄×C₁₀₉	1	6	yes	—
101	C₂²×C₄	C₂×Q₈×C₁₀₉	1	6	yes	—
102	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
103	M₁₆	—	4	16	no	—
104	C₂×C₈	—	2	24	no	—
105	C₄²	—	2	24	no	—
106	C₄²	C₄⋊C₄×C₁₀₉	1	12	yes	—
107	C₄²	C₄⋊C₄×C₁₀₉	1	12	yes	—
108	C₄²	M₁₆×C₁₀₉	1	16	yes	complemented
109	C₄²	C₂×D₄×C₁₀₉	1	12	yes	—
110	C₄²	C₂×Q₈×C₁₀₉	1	12	yes	—
111	C₂³	—	2	4	no	—
112	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	6	yes	—
113	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
114	C₂×C₄	—	2	12	no	—
115	C₂×C₄	—	2	48	no	—
116	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	12	yes	—
117	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	6	yes	—
118	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	12	yes	—
119	C₂×C₄	#127×C₁₀₉	1	4	yes	—
120	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	12	yes	—
121	C₂×C₄	C₂×M₁₆×C₁₀₉	1	24	yes	—
122	C₂×C₄	C₂²×D₄×C₁₀₉	1	3	yes	—
123	C₂×C₄	C₂²×Q₈×C₁₀₉	1	3	yes	—
124	C₈	—	4	8	no	—
125	C₂²	—	2	12	no	—
126	C₂²	#384×C₁₀₉	1	4	yes	—
127	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	3	yes	—
128	C₂²	C₂²×M₁₆×C₁₀₉	1	4	yes	—
129	C₄	—	2	12	no	—
130	C₄	#384×C₁₀₉	1	4	yes	—
131	C₄	#423×C₁₀₉	1	24	yes	complemented
132	C₄	C₂×#127×C₁₀₉	1	1	yes	characteristic
133	C₄	C₂×C₄⋊Q₈×C₁₀₉	1	3	yes	—
134	C₂	—	2	4	no	—
135	C₂	C₂×#384×C₁₀₉	1	1	yes	characteristic
136	C₂	C₂×#423×C₁₀₉	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle	C₂³×C₁₀₉	C₂³×C₄
prime radical	C₂⁴×C₁₀₉	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1842	C₂	semidirect
39. C₄²×C₁₀₉	M₁₆	semidirect
62. C₄×C₁₀₉	#423	semidirect
65. #20791	C₁₀₉	direct (N × H)
71. #1842	C₂×C₁₀₉	semidirect
72. C₁₀₉	#20791	direct (N × H)
108. C₄²	M₁₆×C₁₀₉	semidirect
131. C₄	#423×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#423×C₁₀₉	2
C₂×C₄³×C₁₀₉	2
C₁₀₉×#1842	2
#20791	109

Subgroups — C109×#20791 #42138785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#20791 #42138785