Subgroups — C₁₀₉×#12907 #42130901

Order 27904.

342 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367×C₁₀₉	C₂	1	2	yes	maximal
2	C₂×#366×C₁₀₉	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1846	C₂	1	2	yes	maximal
5	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×#107×C₁₀₉	C₂²	1	4	yes	—
8	C₂×#106×C₁₀₉	—	2	2	no	—
9	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	#382×C₁₀₉	—	2	2	no	—
11	#367×C₁₀₉	—	2	4	no	—
12	#366×C₁₀₉	—	2	4	no	—
13	C₄×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
14	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
17	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
21	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
22	C₂×C₁₆×C₁₀₉	Q₈	1	2	yes	—
23	C₂×C₁₆×C₁₀₉	Q₈	1	2	yes	—
24	#107×C₁₀₉	—	2	4	no	—
25	#107×C₁₀₉	—	2	4	no	—
26	#106×C₁₀₉	—	4	4	no	—
27	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₀₉	Q₈	1	2	yes	—
29	#95×C₁₀₉	—	4	4	no	—
30	C₂²×C₄×C₁₀₉	—	4	4	no	—
31	C₂²×C₄×C₁₀₉	D₈	1	2	yes	—
32	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
35	C₂×C₈×C₁₀₉	Q₁₆	1	2	yes	—
36	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
42	C₄⋊C₄×C₁₀₉	—	4	4	no	—
43	C₄⋊C₄×C₁₀₉	—	4	4	no	—
44	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
45	C₄²×C₁₀₉	Q₁₆	1	2	yes	—
46	C₁₆×C₁₀₉	C₄⋊C₄	1	4	yes	—
47	C₁₆×C₁₀₉	C₄⋊C₄	1	4	yes	—
48	C₂³×C₁₀₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	—	2	4	no	—
50	C₂×C₄×C₁₀₉	—	8	4	no	—
51	C₂×C₄×C₁₀₉	—	8	8	no	—
52	C₂×C₄×C₁₀₉	#107	1	4	yes	—
53	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₁₀₉	#107	1	4	yes	—
60	C₈×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₈×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
63	C₈×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
65	C₂²×C₁₀₉	#493	1	1	yes	characteristic
66	C₂²×C₁₀₉	#500	1	1	yes	characteristic
67	C₂²×C₁₀₉	#508	1	4	yes	—
68	C₄×C₁₀₉	—	2	4	no	—
69	C₄×C₁₀₉	—	16	4	no	—
70	C₄×C₁₀₉	#384	1	1	yes	characteristic
71	C₄×C₁₀₉	C₂×#107	1	1	yes	characteristic
72	C₄×C₁₀₉	#494	1	1	yes	characteristic
73	C₄×C₁₀₉	#500	1	1	yes	characteristic
74	#12907	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₁₀₉	#1845	1	1	yes	characteristic
76	C₂×C₁₀₉	#2158	1	2	yes	—
77	C₂×C₁₀₉	#2249	1	2	yes	—
78	C₂×C₁₀₉	#2253	1	2	yes	—
79	C₂×#367	C₂×C₁₀₉	1	2	yes	—
80	C₂×#366	C₂×C₁₀₉	1	2	yes	—
81	C₂×C₄×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
82	#1846	C₂×C₁₀₉	1	2	yes	—
83	C₁₀₉	#12907	1	1	yes	characteristic, direct product factor, Sylow
84	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
85	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
86	C₂×#107	C₂²×C₁₀₉	1	4	yes	—
87	C₂×#106	—	2	2	no	—
88	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
89	#382	—	2	2	no	—
90	#367	—	2	4	no	—
91	#366	—	2	4	no	—
92	C₄×C₁₆	C₄×C₁₀₉	1	4	yes	complemented
93	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
94	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	2	4	no	—
96	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
97	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
98	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
99	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
100	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
101	C₂×C₁₆	Q₈×C₁₀₉	1	2	yes	—
102	C₂×C₁₆	Q₈×C₁₀₉	1	2	yes	—
103	#107	—	2	4	no	—
104	#107	—	2	4	no	—
105	#106	—	4	4	no	—
106	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
107	C₄×C₈	Q₈×C₁₀₉	1	2	yes	—
108	#95	—	4	4	no	—
109	C₂²×C₄	—	4	4	no	—
110	C₂²×C₄	D₈×C₁₀₉	1	2	yes	—
111	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
112	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
113	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
114	C₂×C₈	Q₁₆×C₁₀₉	1	2	yes	—
115	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
116	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
117	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
118	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
119	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
120	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
121	C₄⋊C₄	—	4	4	no	—
122	C₄⋊C₄	—	4	4	no	—
123	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
124	C₄²	Q₁₆×C₁₀₉	1	2	yes	—
125	C₁₆	C₄⋊C₄×C₁₀₉	1	4	yes	—
126	C₁₆	C₄⋊C₄×C₁₀₉	1	4	yes	—
127	C₂³	C₂×D₈×C₁₀₉	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	8	8	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	#107×C₁₀₉	1	4	yes	—
132	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
133	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
134	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
135	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
136	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
137	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
138	C₈	#107×C₁₀₉	1	4	yes	—
139	C₈	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
140	C₈	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
141	C₈	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
142	C₈	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
143	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
144	C₂²	#493×C₁₀₉	1	1	yes	characteristic
145	C₂²	#500×C₁₀₉	1	1	yes	characteristic
146	C₂²	#508×C₁₀₉	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	16	4	no	—
149	C₄	#384×C₁₀₉	1	1	yes	characteristic
150	C₄	C₂×#107×C₁₀₉	1	1	yes	characteristic
151	C₄	#494×C₁₀₉	1	1	yes	characteristic
152	C₄	#500×C₁₀₉	1	1	yes	characteristic
153	C₂	C₁₀₉×#1845	1	1	yes	characteristic
154	C₂	C₁₀₉×#2158	1	2	yes	—
155	C₂	C₁₀₉×#2249	1	2	yes	—
156	C₂	C₁₀₉×#2253	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄×C₁₆×C₁₀₉	C₄	semidirect
74. #12907	C₁₀₉	direct (N × H)
83. C₁₀₉	#12907	direct (N × H)
92. C₄×C₁₆	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#367×C₁₀₉	2
C₂×#366×C₁₀₉	2
C₂×C₄×C₁₆×C₁₀₉	2
C₁₀₉×#1846	2
#12907	109

Subgroups — C109×#12907 #42130901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12907 #42130901