Subgroups — C₁₀₉×#12662 #42130656

Order 27904.

374 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	C₂×C₄⋊C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#346×C₁₀₉	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1374	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₀₉	C₂²	1	2	yes	—
8	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
13	C₄⋊C₁₆×C₁₀₉	C₄	1	4	yes	complemented
14	#346×C₁₀₉	—	2	4	no	—
15	C₄×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
16	#336×C₁₀₉	C₄	1	2	yes	—
17	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₀₉	C₈	1	2	yes	—
21	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₀₉	—	2	2	no	—
24	C₂×C₁₆×C₁₀₉	—	2	1	no	—
25	C₂×C₁₆×C₁₀₉	—	2	1	no	—
26	C₂×C₁₆×C₁₀₉	—	2	2	no	—
27	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
29	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
31	C₂×C₁₆×C₁₀₉	Q₈	1	2	yes	—
32	C₄⋊C₈×C₁₀₉	C₈	1	4	yes	—
33	C₄×C₈×C₁₀₉	—	2	2	no	—
34	C₄×C₈×C₁₀₉	—	2	2	no	—
35	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
37	#95×C₁₀₉	C₈	1	2	yes	—
38	C₂²×C₄×C₁₀₉	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₀₉	—	2	2	no	—
43	C₂×C₈×C₁₀₉	—	2	1	no	—
44	C₂×C₈×C₁₀₉	—	2	1	no	—
45	C₂×C₈×C₁₀₉	—	2	2	no	—
46	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
47	C₂×C₈×C₁₀₉	C₂×C₈	1	4	yes	—
48	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₁₀₉	—	2	4	no	—
55	C₄²×C₁₀₉	—	2	2	no	—
56	C₄²×C₁₀₉	—	2	2	no	—
57	C₄²×C₁₀₉	C₄²	1	2	yes	—
58	C₄²×C₁₀₉	C₂×C₈	1	2	yes	—
59	C₁₆×C₁₀₉	—	2	2	no	—
60	C₁₆×C₁₀₉	—	2	4	no	—
61	C₁₆×C₁₀₉	C₄⋊C₄	1	4	yes	complemented
62	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	—	2	2	no	—
64	C₂×C₄×C₁₀₉	—	2	4	no	—
65	C₂×C₄×C₁₀₉	—	2	4	no	—
66	C₂×C₄×C₁₀₉	—	2	4	no	—
67	C₂×C₄×C₁₀₉	C₄×C₈	1	2	yes	—
68	C₂×C₄×C₁₀₉	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₁₀₉	M₃₂	1	4	yes	—
70	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₀₉	#131	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	#131	1	1	yes	characteristic
76	C₈×C₁₀₉	—	2	2	no	—
77	C₈×C₁₀₉	—	2	4	no	—
78	C₈×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₁₀₉	#117	1	1	yes	characteristic
80	C₈×C₁₀₉	C₄×D₄	1	1	yes	characteristic
81	C₈×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
82	C₂²×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₁₀₉	C₈×D₄	1	1	yes	characteristic
84	C₂²×C₁₀₉	C₈×Q₈	1	1	yes	characteristic
85	C₂²×C₁₀₉	C₂×M₃₂	1	2	yes	—
86	C₂²×C₁₀₉	#504	1	2	yes	—
87	C₄×C₁₀₉	—	2	2	no	—
88	C₄×C₁₀₉	—	4	4	no	—
89	C₄×C₁₀₉	#346	1	4	yes	complemented
90	C₄×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₁₀₉	#405	1	1	yes	characteristic
92	C₄×C₁₀₉	C₈×D₄	1	1	yes	characteristic
93	C₄×C₁₀₉	C₈×Q₈	1	1	yes	characteristic
94	#12662	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₁₀₉	C₈×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₁₀₉	C₂×#346	1	1	yes	characteristic
97	C₂×C₁₀₉	#2108	1	1	yes	characteristic
98	C₂×C₁₀₉	#2169	1	2	yes	—
99	C₂×C₁₀₉	#2183	1	2	yes	—
100	C₂×C₄⋊C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
101	C₂×#346	C₂×C₁₀₉	1	2	yes	—
102	C₂×C₄×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
103	C₈×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic
104	#1374	C₂×C₁₀₉	1	2	yes	—
105	C₁₀₉	#12662	1	1	yes	characteristic, direct product factor, Sylow
106	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
107	C₂²×C₁₆	C₂²×C₁₀₉	1	2	yes	—
108	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
109	C₂×C₄⋊C₈	C₄×C₁₀₉	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
111	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
112	C₄×C₄⋊C₄	C₄×C₁₀₉	1	1	yes	characteristic
113	C₄⋊C₁₆	C₄×C₁₀₉	1	4	yes	complemented
114	#346	—	2	4	no	—
115	C₄×C₁₆	C₄×C₁₀₉	1	4	yes	complemented
116	#336	C₄×C₁₀₉	1	2	yes	—
117	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
118	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
119	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₈×C₁₀₉	1	2	yes	—
121	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
122	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
128	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
129	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
130	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
131	C₂×C₁₆	Q₈×C₁₀₉	1	2	yes	—
132	C₄⋊C₈	C₈×C₁₀₉	1	4	yes	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
136	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
137	#95	C₈×C₁₀₉	1	2	yes	—
138	C₂²×C₄	C₂×C₈×C₁₀₉	1	4	yes	—
139	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
141	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
147	C₂×C₈	C₂×C₈×C₁₀₉	1	4	yes	—
148	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
149	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
150	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
152	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
153	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	C₄²×C₁₀₉	1	2	yes	—
158	C₄²	C₂×C₈×C₁₀₉	1	2	yes	—
159	C₁₆	—	2	2	no	—
160	C₁₆	—	2	4	no	—
161	C₁₆	C₄⋊C₄×C₁₀₉	1	4	yes	complemented
162	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	C₄×C₈×C₁₀₉	1	2	yes	—
168	C₂×C₄	C₄×C₈×C₁₀₉	1	2	yes	—
169	C₂×C₄	M₃₂×C₁₀₉	1	4	yes	—
170	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
171	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
172	C₂×C₄	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
173	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
174	C₂×C₄	#131×C₁₀₉	1	1	yes	characteristic
175	C₂×C₄	#131×C₁₀₉	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	4	no	—
178	C₈	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
179	C₈	#117×C₁₀₉	1	1	yes	characteristic
180	C₈	C₄×D₄×C₁₀₉	1	1	yes	characteristic
181	C₈	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
182	C₂²	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
183	C₂²	C₈×D₄×C₁₀₉	1	1	yes	characteristic
184	C₂²	C₈×Q₈×C₁₀₉	1	1	yes	characteristic
185	C₂²	C₂×M₃₂×C₁₀₉	1	2	yes	—
186	C₂²	#504×C₁₀₉	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	4	4	no	—
189	C₄	#346×C₁₀₉	1	4	yes	complemented
190	C₄	C₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
191	C₄	#405×C₁₀₉	1	1	yes	characteristic
192	C₄	C₈×D₄×C₁₀₉	1	1	yes	characteristic
193	C₄	C₈×Q₈×C₁₀₉	1	1	yes	characteristic
194	C₂	C₈×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
195	C₂	C₂×#346×C₁₀₉	1	1	yes	characteristic
196	C₂	C₁₀₉×#2108	1	1	yes	characteristic
197	C₂	C₁₀₉×#2169	1	2	yes	—
198	C₂	C₁₀₉×#2183	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
13. C₄⋊C₁₆×C₁₀₉	C₄	semidirect
15. C₄×C₁₆×C₁₀₉	C₄	semidirect
61. C₁₆×C₁₀₉	C₄⋊C₄	semidirect
89. C₄×C₁₀₉	#346	semidirect
94. #12662	C₁₀₉	direct (N × H)
105. C₁₀₉	#12662	direct (N × H)
113. C₄⋊C₁₆	C₄×C₁₀₉	semidirect
115. C₄×C₁₆	C₄×C₁₀₉	semidirect
161. C₁₆	C₄⋊C₄×C₁₀₉	semidirect
189. C₄	#346×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₁₆×C₁₀₉	2
C₂×#346×C₁₀₉	2
C₂×C₄×C₁₆×C₁₀₉	2
C₈×C₄⋊C₄×C₁₀₉	2
C₁₀₉×#1374	2
#12662	109

Subgroups — C109×#12662 #42130656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12662 #42130656