Subgroups — C₁₀₉×#12664 #42130658

Order 27904.

334 conjugacy classes of subgroups.

The subgroup lattice has 192 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	1	yes	maximal, characteristic
2	C₁₀₉×#2108	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1772	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1379	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#108×C₁₀₉	C₄	1	1	yes	characteristic
10	#427×C₁₀₉	C₄	1	1	yes	characteristic
11	#405×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	#364×C₁₀₉	C₄	1	4	yes	complemented
14	#346×C₁₀₉	—	2	2	no	—
15	C₄×C₁₆×C₁₀₉	—	2	2	no	—
16	C₄×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
17	#339×C₁₀₉	C₄	1	2	yes	—
18	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
21	#117×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	M₃₂×C₁₀₉	—	2	2	no	—
24	M₃₂×C₁₀₉	C₂×C₄	1	4	yes	—
25	C₂×C₁₆×C₁₀₉	—	2	2	no	—
26	C₂×C₁₆×C₁₀₉	—	2	1	no	—
27	C₂×C₁₆×C₁₀₉	—	2	1	no	—
28	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₀₉	Q₈	1	2	yes	—
33	#108×C₁₀₉	C₈	1	4	yes	—
34	#107×C₁₀₉	C₈	1	2	yes	—
35	#106×C₁₀₉	C₈	1	2	yes	—
36	#97×C₁₀₉	—	2	2	no	—
37	C₄×C₈×C₁₀₉	—	2	2	no	—
38	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₁₀₉	—	2	1	no	—
41	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	M₁₆×C₁₀₉	—	2	2	no	—
43	M₁₆×C₁₀₉	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₁₀₉	—	2	2	no	—
45	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
46	C₂×C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
54	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₁₀₉	C₂×C₈	1	4	yes	—
56	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
57	C₄²×C₁₀₉	—	2	2	no	—
58	C₄²×C₁₀₉	—	2	1	no	—
59	C₄²×C₁₀₉	C₄²	1	2	yes	—
60	C₁₆×C₁₀₉	—	2	2	no	—
61	C₁₆×C₁₀₉	—	2	4	no	—
62	C₁₆×C₁₀₉	C₄⋊C₄	1	2	yes	—
63	C₁₆×C₁₀₉	C₄⋊C₄	1	2	yes	—
64	C₂³×C₁₀₉	#131	1	1	yes	characteristic
65	C₂×C₄×C₁₀₉	—	2	4	no	—
66	C₂×C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄×C₁₀₉	—	2	4	no	—
68	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	#131	1	1	yes	characteristic
74	C₈×C₁₀₉	—	2	4	no	—
75	C₈×C₁₀₉	C₄×C₈	1	2	yes	—
76	C₈×C₁₀₉	C₄×C₈	1	2	yes	—
77	C₈×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₈×C₁₀₉	#117	1	1	yes	characteristic
79	C₈×C₁₀₉	C₄×D₄	1	1	yes	characteristic
80	C₈×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
81	C₂²×C₁₀₉	—	2	2	no	—
82	C₂²×C₁₀₉	#405	1	1	yes	characteristic
83	C₂²×C₁₀₉	C₈×D₄	1	1	yes	characteristic
84	C₂²×C₁₀₉	C₈×Q₈	1	1	yes	characteristic
85	C₄×C₁₀₉	—	2	4	no	—
86	C₄×C₁₀₉	—	4	4	no	—
87	C₄×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
89	C₄×C₁₀₉	C₈×D₄	1	1	yes	characteristic
90	C₄×C₁₀₉	C₈×Q₈	1	1	yes	characteristic
91	#12664	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₁₀₉	—	2	1	no	—
93	C₂×C₁₀₉	—	2	1	no	—
94	C₂×C₁₀₉	C₈×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₁₀₉	#2170	1	2	yes	—
96	C₂×#364	C₂×C₁₀₉	1	1	yes	characteristic
97	#2108	C₂×C₁₀₉	1	2	yes	—
98	C₂×C₄×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
99	#1772	C₂×C₁₀₉	1	1	yes	characteristic
100	#1379	C₂×C₁₀₉	1	2	yes	—
101	C₁₀₉	#12664	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
103	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
104	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
105	C₂×#108	C₄×C₁₀₉	1	1	yes	characteristic
106	#427	C₄×C₁₀₉	1	1	yes	characteristic
107	#405	C₂²×C₁₀₉	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
109	#364	C₄×C₁₀₉	1	4	yes	complemented
110	#346	—	2	2	no	—
111	C₄×C₁₆	—	2	2	no	—
112	C₄×C₁₆	C₄×C₁₀₉	1	4	yes	complemented
113	#339	C₄×C₁₀₉	1	2	yes	—
114	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
115	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
117	#117	C₂×C₄×C₁₀₉	1	2	yes	—
118	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
119	M₃₂	—	2	2	no	—
120	M₃₂	C₂×C₄×C₁₀₉	1	4	yes	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
125	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
126	C₂×C₁₆	D₄×C₁₀₉	1	1	yes	characteristic
127	C₂×C₁₆	D₄×C₁₀₉	1	1	yes	characteristic
128	C₂×C₁₆	Q₈×C₁₀₉	1	2	yes	—
129	#108	C₈×C₁₀₉	1	4	yes	—
130	#107	C₈×C₁₀₉	1	2	yes	—
131	#106	C₈×C₁₀₉	1	2	yes	—
132	#97	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
135	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
138	M₁₆	—	2	2	no	—
139	M₁₆	C₂×C₈×C₁₀₉	1	4	yes	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
142	C₂×C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
143	C₂×C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
144	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
145	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
146	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
147	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
148	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
149	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
150	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
151	C₄⋊C₄	C₂×C₈×C₁₀₉	1	4	yes	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	1	no	—
155	C₄²	C₄²×C₁₀₉	1	2	yes	—
156	C₁₆	—	2	2	no	—
157	C₁₆	—	2	4	no	—
158	C₁₆	C₄⋊C₄×C₁₀₉	1	2	yes	—
159	C₁₆	C₄⋊C₄×C₁₀₉	1	2	yes	—
160	C₂³	#131×C₁₀₉	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
165	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
166	C₂×C₄	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
167	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
168	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
169	C₂×C₄	#131×C₁₀₉	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	C₄×C₈×C₁₀₉	1	2	yes	—
172	C₈	C₄×C₈×C₁₀₉	1	2	yes	—
173	C₈	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
174	C₈	#117×C₁₀₉	1	1	yes	characteristic
175	C₈	C₄×D₄×C₁₀₉	1	1	yes	characteristic
176	C₈	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
177	C₂²	—	2	2	no	—
178	C₂²	#405×C₁₀₉	1	1	yes	characteristic
179	C₂²	C₈×D₄×C₁₀₉	1	1	yes	characteristic
180	C₂²	C₈×Q₈×C₁₀₉	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	—	4	4	no	—
183	C₄	C₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
184	C₄	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
185	C₄	C₈×D₄×C₁₀₉	1	1	yes	characteristic
186	C₄	C₈×Q₈×C₁₀₉	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	C₈×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
190	C₂	C₁₀₉×#2170	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₁₀₉	D₄
commutator	C₄	C₂×C₄×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₈×D₄
prime radical	C₂³×C₁₀₉	#131
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₀₉ ◁ C₂²×C₁₆×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
13. #364×C₁₀₉	C₄	semidirect
16. C₄×C₁₆×C₁₀₉	C₄	semidirect
91. #12664	C₁₀₉	direct (N × H)
101. C₁₀₉	#12664	direct (N × H)
109. #364	C₄×C₁₀₉	semidirect
112. 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×#364×C₁₀₉	2
C₁₀₉×#2108	2
C₂×C₄×C₁₆×C₁₀₉	2
C₁₀₉×#1772	2
C₁₀₉×#1379	2
#12664	109

Subgroups — C109×#12664 #42130658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12664 #42130658