Subgroups — C₁₀₉×#20539 #42138533

Order 27904.

866 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₁₀₉	C₂	1	2	yes	maximal
2	C₁₀₉×#2285	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1771	C₂	1	8	yes	maximal, complemented
4	C₁₀₉×#1750	C₂	1	4	yes	maximal
5	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
7	#432×C₁₀₉	C₄	1	16	yes	complemented
8	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
9	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	4	yes	—
10	C₂×#97×C₁₀₉	C₂²	1	4	yes	—
11	C₂×#97×C₁₀₉	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₁₀₉	C₂²	1	4	yes	—
13	#384×C₁₀₉	C₄	1	2	yes	—
14	#382×C₁₀₉	C₄	1	2	yes	—
15	#380×C₁₀₉	C₄	1	4	yes	—
16	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
17	#336×C₁₀₉	C₂²	1	8	yes	—
18	C₂³×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₁₀₉	C₂³	1	8	yes	—
21	#117×C₁₀₉	—	4	2	no	—
22	#117×C₁₀₉	—	4	2	no	—
23	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
25	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₀₉	C₂×C₄	1	8	yes	—
29	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
30	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
31	C₄⋊C₈×C₁₀₉	C₂×C₄	1	16	yes	—
32	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	16	yes	—
33	#97×C₁₀₉	—	4	4	no	—
34	#97×C₁₀₉	C₂×C₄	1	16	yes	—
35	C₄×C₈×C₁₀₉	—	4	4	no	—
36	#95×C₁₀₉	C₂×C₄	1	8	yes	—
37	C₂⁴×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	—	2	2	no	—
39	C₂²×C₄×C₁₀₉	—	2	2	no	—
40	C₂²×C₄×C₁₀₉	—	4	4	no	—
41	C₂²×C₄×C₁₀₉	C₄²	1	8	yes	—
42	C₂²×C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
43	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
48	C₂×C₈×C₁₀₉	—	2	8	no	—
49	C₂×C₈×C₁₀₉	—	2	8	no	—
50	C₂×C₈×C₁₀₉	C₂²×C₄	1	16	yes	—
51	C₄⋊C₄×C₁₀₉	—	4	8	no	—
52	C₄⋊C₄×C₁₀₉	—	4	4	no	—
53	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
54	C₄²×C₁₀₉	—	2	4	no	—
55	C₄²×C₁₀₉	—	2	4	no	—
56	C₄²×C₁₀₉	C₄²	1	4	yes	—
57	C₄²×C₁₀₉	C₄²	1	4	yes	—
58	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
59	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
60	C₂³×C₁₀₉	—	2	1	no	—
61	C₂³×C₁₀₉	—	2	1	no	—
62	C₂³×C₁₀₉	—	2	4	no	—
63	C₂³×C₁₀₉	C₂×C₄²	1	2	yes	—
64	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₀₉	—	2	1	no	—
66	C₂×C₄×C₁₀₉	—	2	1	no	—
67	C₂×C₄×C₁₀₉	—	2	4	no	—
68	C₂×C₄×C₁₀₉	—	2	4	no	—
69	C₂×C₄×C₁₀₉	—	2	8	no	—
70	C₂×C₄×C₁₀₉	—	2	8	no	—
71	C₂×C₄×C₁₀₉	—	2	16	no	—
72	C₂×C₄×C₁₀₉	—	4	8	no	—
73	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
74	C₂×C₄×C₁₀₉	C₂×C₄²	1	4	yes	—
75	C₂×C₄×C₁₀₉	C₂×C₄²	1	4	yes	—
76	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
77	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
78	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
79	C₈×C₁₀₉	—	2	16	no	—
80	C₂²×C₁₀₉	—	2	4	no	—
81	C₂²×C₁₀₉	—	2	2	no	—
82	C₂²×C₁₀₉	—	4	2	no	—
83	C₂²×C₁₀₉	—	4	2	no	—
84	C₂²×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
85	C₂²×C₁₀₉	#518	1	1	yes	characteristic
86	C₂²×C₁₀₉	#519	1	1	yes	characteristic
87	C₂²×C₁₀₉	#568	1	4	yes	—
88	C₄×C₁₀₉	—	2	4	no	—
89	C₄×C₁₀₉	—	2	4	no	—
90	C₄×C₁₀₉	—	2	2	no	—
91	C₄×C₁₀₉	—	4	8	no	—
92	C₄×C₁₀₉	#520	1	4	yes	—
93	#20539	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×C₁₀₉	—	4	2	no	—
95	C₂×C₁₀₉	#2274	1	1	yes	characteristic
96	C₂×C₁₀₉	#2873	1	2	yes	—
97	C₂×C₁₀₉	#2980	1	2	yes	—
98	C₂×C₁₀₉	#2987	1	2	yes	—
99	C₂×#432	C₂×C₁₀₉	1	2	yes	—
100	#2285	C₂×C₁₀₉	1	1	yes	characteristic
101	#1771	C₂×C₁₀₉	1	8	yes	complemented
102	#1750	C₂×C₁₀₉	1	4	yes	—
103	C₁₀₉	#20539	1	1	yes	characteristic, direct product factor, Sylow
104	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
105	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
106	#432	C₄×C₁₀₉	1	16	yes	complemented
107	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	4	yes	—
108	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	4	yes	—
109	C₂×#97	C₂²×C₁₀₉	1	4	yes	—
110	C₂×#97	C₂²×C₁₀₉	1	4	yes	—
111	C₂×C₄×C₈	C₂²×C₁₀₉	1	4	yes	—
112	#384	C₄×C₁₀₉	1	2	yes	—
113	#382	C₄×C₁₀₉	1	2	yes	—
114	#380	C₄×C₁₀₉	1	4	yes	—
115	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
116	#336	C₂²×C₁₀₉	1	8	yes	—
117	C₂³×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
118	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
119	C₂²×C₈	C₂³×C₁₀₉	1	8	yes	—
120	#117	—	4	2	no	—
121	#117	—	4	2	no	—
122	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
123	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	—
124	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
125	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
126	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
127	C₂×C₄²	C₂×C₄×C₁₀₉	1	8	yes	—
128	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
129	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
130	C₄⋊C₈	C₂×C₄×C₁₀₉	1	16	yes	—
131	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	16	yes	—
132	#97	—	4	4	no	—
133	#97	C₂×C₄×C₁₀₉	1	16	yes	—
134	C₄×C₈	—	4	4	no	—
135	#95	C₂×C₄×C₁₀₉	1	8	yes	—
136	C₂⁴	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	4	4	no	—
140	C₂²×C₄	C₄²×C₁₀₉	1	8	yes	—
141	C₂²×C₄	C₂²×C₄×C₁₀₉	1	8	yes	—
142	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
143	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
145	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
146	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	C₂²×C₄×C₁₀₉	1	16	yes	—
150	C₄⋊C₄	—	4	8	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₂²⋊C₄	—	4	4	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	2	4	no	—
155	C₄²	C₄²×C₁₀₉	1	4	yes	—
156	C₄²	C₄²×C₁₀₉	1	4	yes	—
157	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
158	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
159	C₂³	—	2	1	no	—
160	C₂³	—	2	1	no	—
161	C₂³	—	2	4	no	—
162	C₂³	C₂×C₄²×C₁₀₉	1	2	yes	—
163	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	16	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
173	C₂×C₄	C₂×C₄²×C₁₀₉	1	4	yes	—
174	C₂×C₄	C₂×C₄²×C₁₀₉	1	4	yes	—
175	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
176	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
177	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
178	C₈	—	2	16	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	2	4	no	—
181	C₂²	—	4	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	C₂²×C₄²×C₁₀₉	1	1	yes	characteristic
184	C₂²	#518×C₁₀₉	1	1	yes	characteristic
185	C₂²	#519×C₁₀₉	1	1	yes	characteristic
186	C₂²	#568×C₁₀₉	1	4	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	4	no	—
190	C₄	—	4	8	no	—
191	C₄	#520×C₁₀₉	1	4	yes	—
192	C₂	—	4	2	no	—
193	C₂	C₁₀₉×#2274	1	1	yes	characteristic
194	C₂	C₁₀₉×#2873	1	2	yes	—
195	C₂	C₁₀₉×#2980	1	2	yes	—
196	C₂	C₁₀₉×#2987	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle	C₂³×C₁₀₉	C₂³×C₄
prime radical	C₂⁴×C₁₀₉	C₂²×C₄
LCS	G ▷ C₂² ▷ 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×#1771	C₂	semidirect
7. #432×C₁₀₉	C₄	semidirect
93. #20539	C₁₀₉	direct (N × H)
101. #1771	C₂×C₁₀₉	semidirect
103. C₁₀₉	#20539	direct (N × H)
106. #432	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×#432×C₁₀₉	2
C₁₀₉×#2285	2
C₁₀₉×#1771	2
C₁₀₉×#1750	2
#20539	109

Subgroups — C109×#20539 #42138533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#20539 #42138533