Subgroups — C₂×C₁₀₉×#1744 #42138355

Order 27904.

1438 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₁₀₉	C₂	1	2	yes	maximal
2	C₂×C₄³×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1744	C₂	1	8	yes	maximal, direct product factor
4	C₂×#336×C₁₀₉	C₂	1	4	yes	maximal
5	C₂³×C₈×C₁₀₉	C₂²	1	4	yes	—
6	C₂²×C₄²×C₁₀₉	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₀₉	C₄	1	2	yes	—
8	C₂²×C₄²×C₁₀₉	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₀₉	C₄	1	16	yes	complemented
11	C₂×C₄×C₈×C₁₀₉	C₂²	1	8	yes	—
12	C₄³×C₁₀₉	C₄	1	4	yes	—
13	C₄³×C₁₀₉	C₂²	1	4	yes	—
14	#336×C₁₀₉	C₂²	1	16	yes	—
15	C₂³×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	—	2	16	no	—
20	C₂²×C₈×C₁₀₉	C₂×C₄	1	16	yes	—
21	C₂²×C₈×C₁₀₉	C₂³	1	8	yes	—
22	C₂×C₄²×C₁₀₉	—	2	8	no	—
23	C₂×C₄²×C₁₀₉	—	2	8	no	—
24	C₂×C₄²×C₁₀₉	C₈	1	16	yes	complemented
25	C₂×C₄²×C₁₀₉	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₀₉	C₂×C₄	1	8	yes	—
29	C₂×C₄²×C₁₀₉	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
31	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
32	C₄×C₈×C₁₀₉	C₂×C₄	1	32	yes	complemented
33	C₂⁴×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₉	—	2	8	no	—
35	C₂²×C₄×C₁₀₉	—	2	8	no	—
36	C₂²×C₄×C₁₀₉	C₄²	1	4	yes	—
37	C₂²×C₄×C₁₀₉	C₄²	1	4	yes	—
38	C₂²×C₄×C₁₀₉	C₂×C₈	1	16	yes	—
39	C₂²×C₄×C₁₀₉	C₂×C₈	1	8	yes	—
40	C₂²×C₄×C₁₀₉	M₁₆	1	8	yes	—
41	C₂²×C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
42	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₁₀₉	#41	1	8	yes	—
47	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
48	C₂×C₈×C₁₀₉	—	2	32	no	—
49	C₂×C₈×C₁₀₉	—	2	16	no	—
50	C₂×C₈×C₁₀₉	C₂²×C₄	1	16	yes	—
51	C₄²×C₁₀₉	—	2	8	no	—
52	C₄²×C₁₀₉	—	2	16	no	—
53	C₄²×C₁₀₉	—	2	8	no	—
54	C₄²×C₁₀₉	C₄²	1	8	yes	—
55	C₄²×C₁₀₉	C₂×C₈	1	32	yes	complemented
56	C₄²×C₁₀₉	C₂²×C₄	1	8	yes	—
57	C₂³×C₁₀₉	C₂×C₄²	1	2	yes	—
58	C₂³×C₁₀₉	#117	1	4	yes	—
59	C₂³×C₁₀₉	C₂²×C₈	1	4	yes	—
60	C₂³×C₁₀₉	C₂×M₁₆	1	4	yes	—
61	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	—	2	8	no	—
63	C₂×C₄×C₁₀₉	—	2	8	no	—
64	C₂×C₄×C₁₀₉	—	2	32	no	—
65	C₂×C₄×C₁₀₉	C₄×C₈	1	16	yes	—
66	C₂×C₄×C₁₀₉	#97	1	16	yes	—
67	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
68	C₂×C₄×C₁₀₉	C₂×C₄²	1	4	yes	—
69	C₂×C₄×C₁₀₉	C₂×C₄²	1	4	yes	—
70	C₂×C₄×C₁₀₉	#117	1	16	yes	—
71	C₂×C₄×C₁₀₉	#117	1	4	yes	—
72	C₂×C₄×C₁₀₉	C₂²×C₈	1	16	yes	—
73	C₂×C₄×C₁₀₉	C₂²×C₈	1	8	yes	—
74	C₂×C₄×C₁₀₉	C₂×M₁₆	1	8	yes	—
75	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
76	C₂×C₄×C₁₀₉	C₂×#41	1	4	yes	—
77	C₈×C₁₀₉	—	2	16	no	—
78	C₂²×C₁₀₉	#376	1	4	yes	—
79	C₂²×C₁₀₉	C₂×C₄×C₈	1	4	yes	—
80	C₂²×C₁₀₉	C₂×#97	1	4	yes	—
81	C₂²×C₁₀₉	#431	1	16	yes	—
82	C₂²×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
83	C₂²×C₁₀₉	C₂×#117	1	2	yes	—
84	C₂²×C₁₀₉	C₂³×C₈	1	2	yes	—
85	C₂²×C₁₀₉	C₂²×M₁₆	1	2	yes	—
86	C₄×C₁₀₉	—	2	16	no	—
87	C₄×C₁₀₉	#376	1	4	yes	—
88	C₄×C₁₀₉	C₂×C₄×C₈	1	8	yes	—
89	C₄×C₁₀₉	C₂×#97	1	8	yes	—
90	C₄×C₁₀₉	C₂×#117	1	4	yes	—
91	C₂×#1744	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₁₀₉	#1744	1	8	yes	direct product factor
93	C₂×C₁₀₉	C₂×#376	1	1	yes	characteristic
94	C₂×C₁₀₉	C₂²×C₄×C₈	1	1	yes	characteristic
95	C₂×C₁₀₉	C₂²×#97	1	1	yes	characteristic
96	C₂×C₁₀₉	C₂×#431	1	4	yes	—
97	C₂²×C₄×C₈	C₂×C₁₀₉	1	2	yes	—
98	C₂×C₄³	C₂×C₁₀₉	1	1	yes	characteristic
99	#1744	C₂×C₁₀₉	1	8	yes	direct product factor
100	C₂×#336	C₂×C₁₀₉	1	4	yes	—
101	C₁₀₉	C₂×#1744	1	1	yes	characteristic, direct product factor, Sylow
102	C₂³×C₈	C₂²×C₁₀₉	1	4	yes	—
103	C₂²×C₄²	C₄×C₁₀₉	1	2	yes	—
104	C₂²×C₄²	C₄×C₁₀₉	1	2	yes	—
105	C₂²×C₄²	C₂²×C₁₀₉	1	2	yes	—
106	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
107	C₂×C₄×C₈	C₄×C₁₀₉	1	16	yes	complemented
108	C₂×C₄×C₈	C₂²×C₁₀₉	1	8	yes	—
109	C₄³	C₄×C₁₀₉	1	4	yes	—
110	C₄³	C₂²×C₁₀₉	1	4	yes	—
111	#336	C₂²×C₁₀₉	1	16	yes	—
112	C₂³×C₄	C₂×C₄×C₁₀₉	1	4	yes	—
113	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
114	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
115	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
116	C₂²×C₈	—	2	16	no	—
117	C₂²×C₈	C₂×C₄×C₁₀₉	1	16	yes	—
118	C₂²×C₈	C₂³×C₁₀₉	1	8	yes	—
119	C₂×C₄²	—	2	8	no	—
120	C₂×C₄²	—	2	8	no	—
121	C₂×C₄²	C₈×C₁₀₉	1	16	yes	complemented
122	C₂×C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
123	C₂×C₄²	C₂×C₄×C₁₀₉	1	8	yes	—
124	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
125	C₂×C₄²	C₂×C₄×C₁₀₉	1	8	yes	—
126	C₂×C₄²	C₂×C₄×C₁₀₉	1	8	yes	—
127	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
128	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
129	C₄×C₈	C₂×C₄×C₁₀₉	1	32	yes	complemented
130	C₂⁴	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
131	C₂²×C₄	—	2	8	no	—
132	C₂²×C₄	—	2	8	no	—
133	C₂²×C₄	C₄²×C₁₀₉	1	4	yes	—
134	C₂²×C₄	C₄²×C₁₀₉	1	4	yes	—
135	C₂²×C₄	C₂×C₈×C₁₀₉	1	16	yes	—
136	C₂²×C₄	C₂×C₈×C₁₀₉	1	8	yes	—
137	C₂²×C₄	M₁₆×C₁₀₉	1	8	yes	—
138	C₂²×C₄	C₂²×C₄×C₁₀₉	1	8	yes	—
139	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
140	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
141	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
142	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
143	C₂²×C₄	#41×C₁₀₉	1	8	yes	—
144	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
145	C₂×C₈	—	2	32	no	—
146	C₂×C₈	—	2	16	no	—
147	C₂×C₈	C₂²×C₄×C₁₀₉	1	16	yes	—
148	C₄²	—	2	8	no	—
149	C₄²	—	2	16	no	—
150	C₄²	—	2	8	no	—
151	C₄²	C₄²×C₁₀₉	1	8	yes	—
152	C₄²	C₂×C₈×C₁₀₉	1	32	yes	complemented
153	C₄²	C₂²×C₄×C₁₀₉	1	8	yes	—
154	C₂³	C₂×C₄²×C₁₀₉	1	2	yes	—
155	C₂³	#117×C₁₀₉	1	4	yes	—
156	C₂³	C₂²×C₈×C₁₀₉	1	4	yes	—
157	C₂³	C₂×M₁₆×C₁₀₉	1	4	yes	—
158	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	32	no	—
162	C₂×C₄	C₄×C₈×C₁₀₉	1	16	yes	—
163	C₂×C₄	#97×C₁₀₉	1	16	yes	—
164	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
165	C₂×C₄	C₂×C₄²×C₁₀₉	1	4	yes	—
166	C₂×C₄	C₂×C₄²×C₁₀₉	1	4	yes	—
167	C₂×C₄	#117×C₁₀₉	1	16	yes	—
168	C₂×C₄	#117×C₁₀₉	1	4	yes	—
169	C₂×C₄	C₂²×C₈×C₁₀₉	1	16	yes	—
170	C₂×C₄	C₂²×C₈×C₁₀₉	1	8	yes	—
171	C₂×C₄	C₂×M₁₆×C₁₀₉	1	8	yes	—
172	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
173	C₂×C₄	C₂×#41×C₁₀₉	1	4	yes	—
174	C₈	—	2	16	no	—
175	C₂²	#376×C₁₀₉	1	4	yes	—
176	C₂²	C₂×C₄×C₈×C₁₀₉	1	4	yes	—
177	C₂²	C₂×#97×C₁₀₉	1	4	yes	—
178	C₂²	#431×C₁₀₉	1	16	yes	—
179	C₂²	C₂²×C₄²×C₁₀₉	1	1	yes	characteristic
180	C₂²	C₂×#117×C₁₀₉	1	2	yes	—
181	C₂²	C₂³×C₈×C₁₀₉	1	2	yes	—
182	C₂²	C₂²×M₁₆×C₁₀₉	1	2	yes	—
183	C₄	—	2	16	no	—
184	C₄	#376×C₁₀₉	1	4	yes	—
185	C₄	C₂×C₄×C₈×C₁₀₉	1	8	yes	—
186	C₄	C₂×#97×C₁₀₉	1	8	yes	—
187	C₄	C₂×#117×C₁₀₉	1	4	yes	—
188	C₂	C₁₀₉×#1744	1	8	yes	direct product factor
189	C₂	C₂×#376×C₁₀₉	1	1	yes	characteristic
190	C₂	C₂²×C₄×C₈×C₁₀₉	1	1	yes	characteristic
191	C₂	C₂²×#97×C₁₀₉	1	1	yes	characteristic
192	C₂	C₂×#431×C₁₀₉	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1744	C₂	direct (N × H)
10. C₂×C₄×C₈×C₁₀₉	C₄	semidirect
24. C₂×C₄²×C₁₀₉	C₈	semidirect
32. C₄×C₈×C₁₀₉	C₂×C₄	semidirect
55. C₄²×C₁₀₉	C₂×C₈	semidirect
91. C₂×#1744	C₁₀₉	direct (N × H)
92. C₂×C₁₀₉	#1744	direct (N × H)
99. #1744	C₂×C₁₀₉	direct (N × H)
101. C₁₀₉	C₂×#1744	direct (N × H)
107. C₂×C₄×C₈	C₄×C₁₀₉	semidirect
121. C₂×C₄²	C₈×C₁₀₉	semidirect
129. C₄×C₈	C₂×C₄×C₁₀₉	semidirect
152. C₄²	C₂×C₈×C₁₀₉	semidirect
188. C₂	C₁₀₉×#1744	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₁₀₉	2
C₂×C₄³×C₁₀₉	2
C₁₀₉×#1744	2
C₂×#336×C₁₀₉	2
C₂×#1744	109

Subgroups — C2×C109×#1744 #42138355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#1744 #42138355