Subgroups — C₂×C₁₀₉×#1380 #42130549

Order 27904.

694 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#107×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#1380	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
5	C₂²×C₄⋊C₄×C₁₀₉	—	2	2	no	—
6	C₂²×C₁₆×C₁₀₉	—	2	4	no	—
7	C₂²×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
8	C₂²×C₁₆×C₁₀₉	C₂²	1	2	yes	—
9	C₂×#107×C₁₀₉	—	2	2	no	—
10	C₂×#107×C₁₀₉	—	2	4	no	—
11	C₂×#107×C₁₀₉	C₄	1	8	yes	complemented
12	C₂×#107×C₁₀₉	C₂²	1	4	yes	—
13	C₂³×C₄×C₁₀₉	—	4	2	no	—
14	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
17	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₉	D₄	1	4	yes	—
19	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
20	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₀₉	—	2	8	no	—
22	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
23	C₂×C₄⋊C₄×C₁₀₉	—	4	2	no	—
24	C₂×C₄⋊C₄×C₁₀₉	—	4	4	no	—
25	C₂×C₁₆×C₁₀₉	—	2	8	no	—
26	C₂×C₁₆×C₁₀₉	—	2	4	no	—
27	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
28	#107×C₁₀₉	—	2	8	no	—
29	#107×C₁₀₉	—	2	4	no	—
30	#107×C₁₀₉	C₂×C₄	1	8	yes	complemented
31	C₂⁴×C₁₀₉	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	—	4	8	no	—
33	C₂²×C₄×C₁₀₉	—	4	4	no	—
34	C₂²×C₄×C₁₀₉	—	8	4	no	—
35	C₂²×C₄×C₁₀₉	—	8	4	no	—
36	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
37	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	SD₁₆	1	4	yes	—
40	C₂²×C₄×C₁₀₉	Q₁₆	1	2	yes	—
41	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₀₉	C₄²	1	4	yes	—
43	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
44	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₀₉	C₄⋊C₄	1	8	yes	—
47	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
51	C₄⋊C₄×C₁₀₉	—	2	8	no	—
52	C₄⋊C₄×C₁₀₉	—	4	8	no	—
53	C₄⋊C₄×C₁₀₉	—	4	4	no	—
54	C₁₆×C₁₀₉	—	2	4	no	—
55	C₂³×C₁₀₉	#102	1	4	yes	—
56	C₂³×C₁₀₉	#107	1	2	yes	—
57	C₂³×C₁₀₉	D₁₆	1	2	yes	—
58	C₂³×C₁₀₉	SD₃₂	1	4	yes	—
59	C₂³×C₁₀₉	Q₃₂	1	2	yes	—
60	C₂³×C₁₀₉	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	—	4	8	no	—
62	C₂×C₄×C₁₀₉	—	8	16	no	—
63	C₂×C₄×C₁₀₉	—	8	4	no	—
64	C₂×C₄×C₁₀₉	—	8	4	no	—
65	C₂×C₄×C₁₀₉	#95	1	4	yes	—
66	C₂×C₄×C₁₀₉	#102	1	4	yes	—
67	C₂×C₄×C₁₀₉	#103	1	8	yes	—
68	C₂×C₄×C₁₀₉	#106	1	4	yes	—
69	C₂×C₄×C₁₀₉	#107	1	2	yes	—
70	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
71	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
74	C₈×C₁₀₉	#95	1	4	yes	—
75	C₈×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
76	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₈×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
78	C₂²×C₁₀₉	#340	1	4	yes	—
79	C₂²×C₁₀₉	#357	1	8	yes	—
80	C₂²×C₁₀₉	#358	1	8	yes	—
81	C₂²×C₁₀₉	#366	1	4	yes	—
82	C₂²×C₁₀₉	#367	1	4	yes	—
83	C₂²×C₁₀₉	C₂×#102	1	2	yes	—
84	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
85	C₂²×C₁₀₉	C₂×D₁₆	1	1	yes	characteristic
86	C₂²×C₁₀₉	C₂×SD₃₂	1	2	yes	—
87	C₂²×C₁₀₉	C₂×Q₃₂	1	1	yes	characteristic
88	C₄×C₁₀₉	—	8	8	no	—
89	C₄×C₁₀₉	#340	1	4	yes	—
90	C₄×C₁₀₉	C₂×#95	1	1	yes	characteristic
91	C₄×C₁₀₉	C₂×#103	1	2	yes	—
92	C₄×C₁₀₉	C₂×#106	1	1	yes	characteristic
93	C₂×#1380	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×C₁₀₉	#1380	1	8	yes	direct product factor
95	C₂×C₁₀₉	C₂×#340	1	1	yes	characteristic
96	C₂×C₁₀₉	C₂×#357	1	2	yes	—
97	C₂×C₁₀₉	C₂×#358	1	2	yes	—
98	C₂×C₁₀₉	C₂×#366	1	1	yes	characteristic
99	C₂×C₁₀₉	C₂×#367	1	1	yes	characteristic
100	C₂³×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
101	C₂²×#107	C₂×C₁₀₉	1	2	yes	—
102	#1380	C₂×C₁₀₉	1	4	yes	direct product factor
103	C₁₀₉	C₂×#1380	1	1	yes	characteristic, direct product factor, Sylow
104	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
105	C₂²×C₄⋊C₄	—	2	2	no	—
106	C₂²×C₁₆	—	2	4	no	—
107	C₂²×C₁₆	C₄×C₁₀₉	1	4	yes	complemented
108	C₂²×C₁₆	C₂²×C₁₀₉	1	2	yes	—
109	C₂×#107	—	2	2	no	—
110	C₂×#107	—	2	4	no	—
111	C₂×#107	C₄×C₁₀₉	1	8	yes	complemented
112	C₂×#107	C₂²×C₁₀₉	1	4	yes	—
113	C₂³×C₄	—	4	2	no	—
114	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
115	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
116	C₂²×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
117	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
118	C₂²×C₈	D₄×C₁₀₉	1	4	yes	—
119	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
120	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	—	2	8	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	—	4	2	no	—
124	C₂×C₄⋊C₄	—	4	4	no	—
125	C₂×C₁₆	—	2	8	no	—
126	C₂×C₁₆	—	2	4	no	—
127	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	complemented
128	#107	—	2	8	no	—
129	#107	—	2	4	no	—
130	#107	C₂×C₄×C₁₀₉	1	8	yes	complemented
131	C₂⁴	D₈×C₁₀₉	1	1	yes	characteristic
132	C₂²×C₄	—	4	8	no	—
133	C₂²×C₄	—	4	4	no	—
134	C₂²×C₄	—	8	4	no	—
135	C₂²×C₄	—	8	4	no	—
136	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	4	yes	—
137	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
138	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
139	C₂²×C₄	SD₁₆×C₁₀₉	1	4	yes	—
140	C₂²×C₄	Q₁₆×C₁₀₉	1	2	yes	—
141	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
142	C₂×C₈	C₄²×C₁₀₉	1	4	yes	—
143	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
144	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
145	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
146	C₂×C₈	C₄⋊C₄×C₁₀₉	1	8	yes	—
147	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
148	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
149	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
150	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
151	C₄⋊C₄	—	2	8	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₁₆	—	2	4	no	—
155	C₂³	#102×C₁₀₉	1	4	yes	—
156	C₂³	#107×C₁₀₉	1	2	yes	—
157	C₂³	D₁₆×C₁₀₉	1	2	yes	—
158	C₂³	SD₃₂×C₁₀₉	1	4	yes	—
159	C₂³	Q₃₂×C₁₀₉	1	2	yes	—
160	C₂³	C₂×D₈×C₁₀₉	1	1	yes	characteristic
161	C₂×C₄	—	4	8	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	8	16	no	—
165	C₂×C₄	#95×C₁₀₉	1	4	yes	—
166	C₂×C₄	#102×C₁₀₉	1	4	yes	—
167	C₂×C₄	#103×C₁₀₉	1	8	yes	—
168	C₂×C₄	#106×C₁₀₉	1	4	yes	—
169	C₂×C₄	#107×C₁₀₉	1	2	yes	—
170	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	2	yes	—
171	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
172	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
173	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
174	C₈	#95×C₁₀₉	1	4	yes	—
175	C₈	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
176	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
177	C₈	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
178	C₂²	#340×C₁₀₉	1	4	yes	—
179	C₂²	#357×C₁₀₉	1	8	yes	—
180	C₂²	#358×C₁₀₉	1	8	yes	—
181	C₂²	#366×C₁₀₉	1	4	yes	—
182	C₂²	#367×C₁₀₉	1	4	yes	—
183	C₂²	C₂×#102×C₁₀₉	1	2	yes	—
184	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
185	C₂²	C₂×D₁₆×C₁₀₉	1	1	yes	characteristic
186	C₂²	C₂×SD₃₂×C₁₀₉	1	2	yes	—
187	C₂²	C₂×Q₃₂×C₁₀₉	1	1	yes	characteristic
188	C₄	—	8	8	no	—
189	C₄	#340×C₁₀₉	1	4	yes	—
190	C₄	C₂×#95×C₁₀₉	1	1	yes	characteristic
191	C₄	C₂×#103×C₁₀₉	1	2	yes	—
192	C₄	C₂×#106×C₁₀₉	1	1	yes	characteristic
193	C₂	C₁₀₉×#1380	1	8	yes	direct product factor
194	C₂	C₂×#340×C₁₀₉	1	1	yes	characteristic
195	C₂	C₂×#357×C₁₀₉	1	2	yes	—
196	C₂	C₂×#358×C₁₀₉	1	2	yes	—
197	C₂	C₂×#366×C₁₀₉	1	1	yes	characteristic
198	C₂	C₂×#367×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₀₉	D₈
commutator	C₈	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1380	C₂	direct (N × H)
7. C₂²×C₁₆×C₁₀₉	C₄	semidirect
11. C₂×#107×C₁₀₉	C₄	semidirect
27. C₂×C₁₆×C₁₀₉	C₂×C₄	semidirect
30. #107×C₁₀₉	C₂×C₄	semidirect
93. C₂×#1380	C₁₀₉	direct (N × H)
94. C₂×C₁₀₉	#1380	direct (N × H)
102. #1380	C₂×C₁₀₉	direct (N × H)
103. C₁₀₉	C₂×#1380	direct (N × H)
107. C₂²×C₁₆	C₄×C₁₀₉	semidirect
111. C₂×#107	C₄×C₁₀₉	semidirect
127. C₂×C₁₆	C₂×C₄×C₁₀₉	semidirect
130. #107	C₂×C₄×C₁₀₉	semidirect
193. C₂	C₁₀₉×#1380	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₁₀₉	2
C₂²×#107×C₁₀₉	2
C₁₀₉×#1380	2
C₂×#1380	109

Subgroups — C2×C109×#1380 #42130549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#1380 #42130549