Subgroups — C₁₀₉×#7469 #42125463

Order 27904.

174 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2108	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₀₉	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
6	#405×C₁₀₉	C₄	1	1	yes	characteristic
7	M₆₄×C₁₀₉	—	2	1	no	—
8	M₆₄×C₁₀₉	—	2	1	no	—
9	M₆₄×C₁₀₉	—	2	1	no	—
10	M₆₄×C₁₀₉	C₄	1	2	yes	complemented
11	C₂×C₃₂×C₁₀₉	—	2	1	no	—
12	C₂×C₃₂×C₁₀₉	C₄	1	2	yes	complemented
13	#346×C₁₀₉	—	2	1	no	—
14	C₄×C₁₆×C₁₀₉	—	2	1	no	—
15	C₂×M₁₆×C₁₀₉	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	#117×C₁₀₉	C₈	1	1	yes	characteristic
18	M₃₂×C₁₀₉	—	2	1	no	—
19	M₃₂×C₁₀₉	C₈	1	2	yes	—
20	C₂×C₁₆×C₁₀₉	—	2	1	no	—
21	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₁₀₉	Q₈	1	1	yes	characteristic
27	#97×C₁₀₉	—	2	1	no	—
28	C₄×C₈×C₁₀₉	—	2	1	no	—
29	C₃₂×C₁₀₉	—	2	2	no	—
30	C₃₂×C₁₀₉	—	2	2	no	—
31	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
32	M₁₆×C₁₀₉	—	2	1	no	—
33	M₁₆×C₁₀₉	C₁₆	1	2	yes	—
34	C₂×C₈×C₁₀₉	—	2	1	no	—
35	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
41	C₄⋊C₄×C₁₀₉	—	2	1	no	—
42	C₂²⋊C₄×C₁₀₉	C₁₆	1	2	yes	—
43	C₄²×C₁₀₉	—	2	1	no	—
44	C₁₆×C₁₀₉	—	2	2	no	—
45	C₁₆×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₆×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₁₆×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
48	C₁₆×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	—	2	2	no	—
51	C₂×C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	M₃₂	1	1	yes	characteristic
57	C₈×C₁₀₉	—	2	2	no	—
58	C₈×C₁₀₉	#95	1	1	yes	characteristic
59	C₈×C₁₀₉	#97	1	1	yes	characteristic
60	C₈×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
61	C₈×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂²×C₁₀₉	—	4	1	no	—
63	C₂²×C₁₀₉	C₄×C₁₆	1	1	yes	characteristic
64	C₂²×C₁₀₉	C₂²⋊C₁₆	1	1	yes	characteristic
65	C₂²×C₁₀₉	C₄⋊C₁₆	1	1	yes	characteristic
66	C₄×C₁₀₉	—	4	2	no	—
67	C₄×C₁₀₉	#336	1	1	yes	characteristic
68	C₄×C₁₀₉	#346	1	1	yes	characteristic
69	C₄×C₁₀₉	C₂²⋊C₁₆	1	1	yes	characteristic
70	C₄×C₁₀₉	C₄⋊C₁₆	1	1	yes	characteristic
71	#7469	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₁₀₉	—	2	1	no	—
73	C₂×C₁₀₉	—	2	1	no	—
74	C₂×C₁₀₉	—	2	1	no	—
75	C₂×C₁₀₉	#1374	1	1	yes	characteristic
76	C₂×M₆₄	C₂×C₁₀₉	1	1	yes	characteristic
77	C₂×M₆₄	C₂×C₁₀₉	1	1	yes	characteristic
78	#2108	C₂×C₁₀₉	1	1	yes	characteristic
79	C₁₀₉	#7469	1	1	yes	characteristic, direct product factor, Sylow
80	C₂×M₃₂	C₄×C₁₀₉	1	1	yes	characteristic
81	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
82	#405	C₄×C₁₀₉	1	1	yes	characteristic
83	M₆₄	—	2	1	no	—
84	M₆₄	—	2	1	no	—
85	M₆₄	—	2	1	no	—
86	M₆₄	C₄×C₁₀₉	1	2	yes	complemented
87	C₂×C₃₂	—	2	1	no	—
88	C₂×C₃₂	C₄×C₁₀₉	1	2	yes	complemented
89	#346	—	2	1	no	—
90	C₄×C₁₆	—	2	1	no	—
91	C₂×M₁₆	C₈×C₁₀₉	1	1	yes	characteristic
92	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
93	#117	C₈×C₁₀₉	1	1	yes	characteristic
94	M₃₂	—	2	1	no	—
95	M₃₂	C₈×C₁₀₉	1	2	yes	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
98	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
99	C₂×C₁₆	D₄×C₁₀₉	1	1	yes	characteristic
100	C₂×C₁₆	D₄×C₁₀₉	1	1	yes	characteristic
101	C₂×C₁₆	D₄×C₁₀₉	1	1	yes	characteristic
102	C₂×C₁₆	Q₈×C₁₀₉	1	1	yes	characteristic
103	#97	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₃₂	—	2	2	no	—
106	C₃₂	—	2	2	no	—
107	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	C₁₆×C₁₀₉	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
113	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
114	C₂×C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
115	C₂×C₈	M₁₆×C₁₀₉	1	1	yes	characteristic
116	C₂×C₈	M₁₆×C₁₀₉	1	1	yes	characteristic
117	C₄⋊C₄	—	2	1	no	—
118	C₂²⋊C₄	C₁₆×C₁₀₉	1	2	yes	—
119	C₄²	—	2	1	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
122	C₁₆	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
123	C₁₆	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
124	C₁₆	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
125	C₂³	C₂×C₁₆×C₁₀₉	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄×C₈×C₁₀₉	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
129	C₂×C₄	C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
130	C₂×C₄	C₂×C₁₆×C₁₀₉	1	1	yes	characteristic
131	C₂×C₄	M₃₂×C₁₀₉	1	1	yes	characteristic
132	C₂×C₄	M₃₂×C₁₀₉	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	#95×C₁₀₉	1	1	yes	characteristic
135	C₈	#97×C₁₀₉	1	1	yes	characteristic
136	C₈	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
137	C₈	C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
138	C₂²	—	4	1	no	—
139	C₂²	C₄×C₁₆×C₁₀₉	1	1	yes	characteristic
140	C₂²	C₂²⋊C₁₆×C₁₀₉	1	1	yes	characteristic
141	C₂²	C₄⋊C₁₆×C₁₀₉	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	#336×C₁₀₉	1	1	yes	characteristic
144	C₄	#346×C₁₀₉	1	1	yes	characteristic
145	C₄	C₂²⋊C₁₆×C₁₀₉	1	1	yes	characteristic
146	C₄	C₄⋊C₁₆×C₁₀₉	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	—	2	1	no	—
150	C₂	C₁₀₉×#1374	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₁₀₉	C₂²⋊C₄
commutator	C₂²	C₄×C₁₆×C₁₀₉
Frattini, cosocle	C₂²×C₁₆	C₂²×C₁₀₉
socle	C₂×C₁₀₉	#1374
prime radical	C₂³×C₁₀₉	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₁₀₉ ◁ C₂²×C₁₆×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
10. M₆₄×C₁₀₉	C₄	semidirect
12. C₂×C₃₂×C₁₀₉	C₄	semidirect
71. #7469	C₁₀₉	direct (N × H)
79. C₁₀₉	#7469	direct (N × H)
86. M₆₄	C₄×C₁₀₉	semidirect
88. C₂×C₃₂	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×M₆₄×C₁₀₉	2
C₁₀₉×#2108	2
#7469	109

Subgroups — C109×#7469 #42125463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#7469 #42125463