Subgroups — C₁₀₉×#13554 #42131548

Order 27904.

298 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×C₂²⋊C₃₂	C₂	1	4	yes	maximal, complemented
5	C₂×#131×C₁₀₉	C₄	1	1	yes	characteristic
6	#504×C₁₀₉	—	2	2	no	—
7	#504×C₁₀₉	—	2	2	no	—
8	C₂×M₃₂×C₁₀₉	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₁₀₉	C₄	1	2	yes	—
10	C₂²×C₁₆×C₁₀₉	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
12	M₆₄×C₁₀₉	—	2	2	no	—
13	C₂×C₃₂×C₁₀₉	—	2	2	no	—
14	C₂×C₃₂×C₁₀₉	C₂²	1	2	yes	—
15	C₂×C₃₂×C₁₀₉	C₂²	1	2	yes	—
16	C₂×#41×C₁₀₉	C₈	1	1	yes	characteristic
17	#131×C₁₀₉	—	2	2	no	—
18	#131×C₁₀₉	—	2	2	no	—
19	C₂×M₁₆×C₁₀₉	C₈	1	1	yes	characteristic
20	C₂×M₁₆×C₁₀₉	C₈	1	2	yes	—
21	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	M₃₂×C₁₀₉	—	2	2	no	—
24	M₃₂×C₁₀₉	—	2	4	no	—
25	C₂×C₁₆×C₁₀₉	—	2	2	no	—
26	C₂×C₁₆×C₁₀₉	—	2	2	no	—
27	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
30	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
31	C₂×C₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₃₂×C₁₀₉	—	2	2	no	—
33	C₃₂×C₁₀₉	—	2	2	no	—
34	#41×C₁₀₉	—	2	2	no	—
35	#41×C₁₀₉	—	2	2	no	—
36	C₂×Q₈×C₁₀₉	C₁₆	1	1	yes	characteristic
37	C₂×D₄×C₁₀₉	C₁₆	1	2	yes	—
38	C₂×D₄×C₁₀₉	C₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
41	M₁₆×C₁₀₉	—	2	2	no	—
42	M₁₆×C₁₀₉	—	2	4	no	—
43	C₂×C₈×C₁₀₉	—	2	2	no	—
44	C₂×C₈×C₁₀₉	—	2	2	no	—
45	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₀₉	M₁₆	1	2	yes	—
49	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₁₆×C₁₀₉	—	2	2	no	—
51	C₁₆×C₁₀₉	C₂²⋊C₄	1	2	yes	—
52	C₁₆×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₁₆×C₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂³×C₁₀₉	C₂×C₁₆	1	2	yes	—
55	C₂³×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
56	Q₈×C₁₀₉	—	2	2	no	—
57	D₄×C₁₀₉	—	2	4	no	—
58	D₄×C₁₀₉	—	2	2	no	—
59	C₂×C₄×C₁₀₉	—	2	2	no	—
60	C₂×C₄×C₁₀₉	—	2	2	no	—
61	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₁₀₉	C₂×C₁₆	1	2	yes	—
63	C₂×C₄×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
64	C₂×C₄×C₁₀₉	M₃₂	1	2	yes	—
65	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
66	C₈×C₁₀₉	—	2	2	no	—
67	C₈×C₁₀₉	C₂²⋊C₈	1	2	yes	—
68	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₈×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
70	C₂²×C₁₀₉	—	2	2	no	—
71	C₂²×C₁₀₉	—	2	2	no	—
72	C₂²×C₁₀₉	—	2	4	no	—
73	C₂²×C₁₀₉	C₂²⋊C₁₆	1	2	yes	—
74	C₂²×C₁₀₉	C₂²×C₁₆	1	1	yes	characteristic
75	C₄×C₁₀₉	—	2	2	no	—
76	C₄×C₁₀₉	C₂²⋊C₁₆	1	2	yes	—
77	C₄×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
78	C₄×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
79	#13554	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₁₀₉	—	2	2	no	—
81	C₂×C₁₀₉	—	4	2	no	—
82	C₂×C₁₀₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
83	C₂×C₁₀₉	#2258	1	2	yes	—
84	C₂×#504	C₂×C₁₀₉	1	1	yes	characteristic
85	C₂×M₆₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
86	C₂²×C₃₂	C₂×C₁₀₉	1	1	yes	characteristic, complemented
87	C₂²⋊C₃₂	C₂×C₁₀₉	1	4	yes	complemented
88	C₁₀₉	#13554	1	1	yes	characteristic, direct product factor, Sylow
89	C₂×#131	C₄×C₁₀₉	1	1	yes	characteristic
90	#504	—	2	2	no	—
91	#504	—	2	2	no	—
92	C₂×M₃₂	C₄×C₁₀₉	1	1	yes	characteristic
93	C₂×M₃₂	C₄×C₁₀₉	1	2	yes	—
94	C₂²×C₁₆	C₂²×C₁₀₉	1	2	yes	—
95	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
96	M₆₄	—	2	2	no	—
97	C₂×C₃₂	—	2	2	no	—
98	C₂×C₃₂	C₂²×C₁₀₉	1	2	yes	—
99	C₂×C₃₂	C₂²×C₁₀₉	1	2	yes	—
100	C₂×#41	C₈×C₁₀₉	1	1	yes	characteristic
101	#131	—	2	2	no	—
102	#131	—	2	2	no	—
103	C₂×M₁₆	C₈×C₁₀₉	1	1	yes	characteristic
104	C₂×M₁₆	C₈×C₁₀₉	1	2	yes	—
105	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
106	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	2	4	no	—
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
112	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
113	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
114	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
115	C₂×C₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
116	C₃₂	—	2	2	no	—
117	C₃₂	—	2	2	no	—
118	#41	—	2	2	no	—
119	#41	—	2	2	no	—
120	C₂×Q₈	C₁₆×C₁₀₉	1	1	yes	characteristic
121	C₂×D₄	C₁₆×C₁₀₉	1	2	yes	—
122	C₂×D₄	C₁₆×C₁₀₉	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
124	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
125	M₁₆	—	2	2	no	—
126	M₁₆	—	2	4	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
130	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
131	C₂×C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
132	C₂×C₈	M₁₆×C₁₀₉	1	2	yes	—
133	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
134	C₁₆	—	2	2	no	—
135	C₁₆	C₂²⋊C₄×C₁₀₉	1	2	yes	—
136	C₁₆	C₂×D₄×C₁₀₉	1	1	yes	characteristic
137	C₁₆	C₂×D₄×C₁₀₉	1	1	yes	characteristic
138	C₂³	C₂×C₁₆×C₁₀₉	1	2	yes	—
139	C₂³	C₂×C₁₆×C₁₀₉	1	1	yes	characteristic
140	Q₈	—	2	2	no	—
141	D₄	—	2	4	no	—
142	D₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	2	yes	—
146	C₂×C₄	C₂×C₁₆×C₁₀₉	1	2	yes	—
147	C₂×C₄	C₂×C₁₆×C₁₀₉	1	1	yes	characteristic
148	C₂×C₄	M₃₂×C₁₀₉	1	2	yes	—
149	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	C₂²⋊C₈×C₁₀₉	1	2	yes	—
152	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
153	C₈	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂²⋊C₁₆×C₁₀₉	1	2	yes	—
158	C₂²	C₂²×C₁₆×C₁₀₉	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	C₂²⋊C₁₆×C₁₀₉	1	2	yes	—
161	C₄	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
162	C₄	C₂×M₃₂×C₁₀₉	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	4	2	no	—
165	C₂	C₂×C₂²⋊C₁₆×C₁₀₉	1	1	yes	characteristic
166	C₂	C₁₀₉×#2258	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₁₀₉	C₂³
commutator	C₂²	C₂²×C₁₆×C₁₀₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂²×C₁₆
prime radical	C₂×#41×C₁₀₉	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×M₆₄×C₁₀₉	C₂	semidirect
3. C₂²×C₃₂×C₁₀₉	C₂	semidirect
4. C₁₀₉×C₂²⋊C₃₂	C₂	semidirect
79. #13554	C₁₀₉	direct (N × H)
85. C₂×M₆₄	C₂×C₁₀₉	semidirect
86. C₂²×C₃₂	C₂×C₁₀₉	semidirect
87. C₂²⋊C₃₂	C₂×C₁₀₉	semidirect
88. C₁₀₉	#13554	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×#504×C₁₀₉	2
C₂×M₆₄×C₁₀₉	2
C₂²×C₃₂×C₁₀₉	2
C₁₀₉×C₂²⋊C₃₂	2
#13554	109

Subgroups — C109×#13554 #42131548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#13554 #42131548