Subgroups — C₁₀₉×#12666 #42130660

Order 27904.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂×#347×C₁₀₉	C₂	1	2	yes	maximal
3	C₄×M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1770	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1375	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
8	C₂×M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
10	C₂×#97×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	2	yes	complemented
14	C₄⋊C₁₆×C₁₀₉	C₄	1	2	yes	complemented
15	#347×C₁₀₉	—	2	4	no	—
16	#346×C₁₀₉	C₄	1	2	yes	complemented
17	C₄×C₁₆×C₁₀₉	C₄	1	2	yes	complemented
18	#336×C₁₀₉	C₄	1	2	yes	—
19	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
23	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	M₃₂×C₁₀₉	—	2	2	no	—
26	M₃₂×C₁₀₉	—	2	1	no	—
27	M₃₂×C₁₀₉	—	2	1	no	—
28	M₃₂×C₁₀₉	—	2	2	no	—
29	M₃₂×C₁₀₉	D₄	1	2	yes	—
30	M₃₂×C₁₀₉	Q₈	1	2	yes	—
31	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₄⋊C₈×C₁₀₉	—	2	1	no	—
35	C₄⋊C₈×C₁₀₉	—	2	1	no	—
36	#97×C₁₀₉	—	2	2	no	—
37	#97×C₁₀₉	—	2	2	no	—
38	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
42	#95×C₁₀₉	—	2	1	no	—
43	C₂²×C₄×C₁₀₉	—	2	2	no	—
44	C₂²×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₀₉	—	2	1	no	—
48	C₂×C₈×C₁₀₉	—	2	1	no	—
49	C₂×C₈×C₁₀₉	—	2	2	no	—
50	C₂×C₈×C₁₀₉	—	2	2	no	—
51	C₂×C₈×C₁₀₉	—	2	2	no	—
52	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
59	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₁₀₉	—	4	2	no	—
61	C₄²×C₁₀₉	—	2	2	no	—
62	C₄²×C₁₀₉	—	2	2	no	—
63	C₄²×C₁₀₉	C₄²	1	1	yes	characteristic
64	C₄²×C₁₀₉	C₄²	1	1	yes	characteristic
65	C₄²×C₁₀₉	M₁₆	1	1	yes	characteristic
66	C₄²×C₁₀₉	M₁₆	1	1	yes	characteristic
67	C₁₆×C₁₀₉	—	2	2	no	—
68	C₁₆×C₁₀₉	—	2	4	no	—
69	C₁₆×C₁₀₉	C₄⋊C₄	1	4	yes	complemented
70	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	—	2	2	no	—
72	C₂×C₄×C₁₀₉	—	4	1	no	—
73	C₂×C₄×C₁₀₉	—	4	2	no	—
74	C₂×C₄×C₁₀₉	—	4	2	no	—
75	C₂×C₄×C₁₀₉	—	4	2	no	—
76	C₂×C₄×C₁₀₉	#97	1	1	yes	characteristic
77	C₂×C₄×C₁₀₉	#97	1	1	yes	characteristic
78	C₂×C₄×C₁₀₉	#97	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	#97	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	#131	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	#131	1	1	yes	characteristic
86	C₈×C₁₀₉	—	2	1	no	—
87	C₈×C₁₀₉	—	2	1	no	—
88	C₈×C₁₀₉	—	4	2	no	—
89	C₈×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₈×C₁₀₉	#117	1	1	yes	characteristic
91	C₈×C₁₀₉	C₄×D₄	1	1	yes	characteristic
92	C₈×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
93	C₂²×C₁₀₉	—	2	1	no	—
94	C₂²×C₁₀₉	—	2	1	no	—
95	C₂²×C₁₀₉	C₂×#97	1	1	yes	characteristic
96	C₂²×C₁₀₉	#436	1	1	yes	characteristic
97	C₂²×C₁₀₉	#446	1	1	yes	characteristic
98	C₄×C₁₀₉	—	2	1	no	—
99	C₄×C₁₀₉	—	2	1	no	—
100	C₄×C₁₀₉	—	2	1	no	—
101	C₄×C₁₀₉	—	8	2	no	—
102	C₄×C₁₀₉	#347	1	2	yes	complemented
103	C₄×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₄×C₁₀₉	#405	1	1	yes	characteristic
105	C₄×C₁₀₉	#436	1	1	yes	characteristic
106	C₄×C₁₀₉	#446	1	1	yes	characteristic
107	#12666	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₁₀₉	—	2	1	no	—
109	C₂×C₁₀₉	—	2	1	no	—
110	C₂×C₁₀₉	#1770	1	1	yes	characteristic
111	C₂×C₁₀₉	C₂×#347	1	1	yes	characteristic
112	C₂×C₁₀₉	#2110	1	1	yes	characteristic
113	#2150	C₂×C₁₀₉	1	1	yes	characteristic
114	C₂×#347	C₂×C₁₀₉	1	2	yes	—
115	C₄×M₃₂	C₂×C₁₀₉	1	1	yes	characteristic
116	#1770	C₂×C₁₀₉	1	1	yes	characteristic
117	#1375	C₂×C₁₀₉	1	2	yes	—
118	C₁₀₉	#12666	1	1	yes	characteristic, direct product factor, Sylow
119	C₂×M₃₂	C₂²×C₁₀₉	1	1	yes	characteristic
120	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
121	C₂×M₃₂	C₂²×C₁₀₉	1	1	yes	characteristic
122	C₂×C₄⋊C₈	C₄×C₁₀₉	1	1	yes	characteristic
123	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₄×C₁₀₉	1	1	yes	characteristic
126	C₄⋊C₁₆	C₄×C₁₀₉	1	2	yes	complemented
127	C₄⋊C₁₆	C₄×C₁₀₉	1	2	yes	complemented
128	#347	—	2	4	no	—
129	#346	C₄×C₁₀₉	1	2	yes	complemented
130	C₄×C₁₆	C₄×C₁₀₉	1	2	yes	complemented
131	#336	C₄×C₁₀₉	1	2	yes	—
132	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
133	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
137	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
138	M₃₂	—	2	2	no	—
139	M₃₂	—	2	1	no	—
140	M₃₂	—	2	1	no	—
141	M₃₂	—	2	2	no	—
142	M₃₂	D₄×C₁₀₉	1	2	yes	—
143	M₃₂	Q₈×C₁₀₉	1	2	yes	—
144	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
145	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
146	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
147	C₄⋊C₈	—	2	1	no	—
148	C₄⋊C₈	—	2	1	no	—
149	#97	—	2	2	no	—
150	#97	—	2	2	no	—
151	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
152	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
153	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
154	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
155	#95	—	2	1	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	M₁₆×C₁₀₉	1	1	yes	characteristic
158	C₂²×C₄	M₁₆×C₁₀₉	1	1	yes	characteristic
159	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
166	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
167	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
168	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
169	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
170	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
171	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
172	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
173	C₄⋊C₄	—	4	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₄²×C₁₀₉	1	1	yes	characteristic
177	C₄²	C₄²×C₁₀₉	1	1	yes	characteristic
178	C₄²	M₁₆×C₁₀₉	1	1	yes	characteristic
179	C₄²	M₁₆×C₁₀₉	1	1	yes	characteristic
180	C₁₆	—	2	2	no	—
181	C₁₆	—	2	4	no	—
182	C₁₆	C₄⋊C₄×C₁₀₉	1	4	yes	complemented
183	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	#97×C₁₀₉	1	1	yes	characteristic
190	C₂×C₄	#97×C₁₀₉	1	1	yes	characteristic
191	C₂×C₄	#97×C₁₀₉	1	1	yes	characteristic
192	C₂×C₄	#97×C₁₀₉	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
194	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
195	C₂×C₄	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
197	C₂×C₄	#131×C₁₀₉	1	1	yes	characteristic
198	C₂×C₄	#131×C₁₀₉	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	4	2	no	—
202	C₈	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
203	C₈	#117×C₁₀₉	1	1	yes	characteristic
204	C₈	C₄×D₄×C₁₀₉	1	1	yes	characteristic
205	C₈	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×#97×C₁₀₉	1	1	yes	characteristic
209	C₂²	#436×C₁₀₉	1	1	yes	characteristic
210	C₂²	#446×C₁₀₉	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	#347×C₁₀₉	1	2	yes	complemented
216	C₄	C₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
217	C₄	#405×C₁₀₉	1	1	yes	characteristic
218	C₄	#436×C₁₀₉	1	1	yes	characteristic
219	C₄	#446×C₁₀₉	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	C₁₀₉×#1770	1	1	yes	characteristic
223	C₂	C₂×#347×C₁₀₉	1	1	yes	characteristic
224	C₂	C₁₀₉×#2110	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×#97
prime radical	C₂³×C₁₀₉	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄⋊C₁₆×C₁₀₉	C₄	semidirect
14. C₄⋊C₁₆×C₁₀₉	C₄	semidirect
16. #346×C₁₀₉	C₄	semidirect
17. C₄×C₁₆×C₁₀₉	C₄	semidirect
69. C₁₆×C₁₀₉	C₄⋊C₄	semidirect
102. C₄×C₁₀₉	#347	semidirect
107. #12666	C₁₀₉	direct (N × H)
118. C₁₀₉	#12666	direct (N × H)
126. C₄⋊C₁₆	C₄×C₁₀₉	semidirect
127. C₄⋊C₁₆	C₄×C₁₀₉	semidirect
129. #346	C₄×C₁₀₉	semidirect
130. C₄×C₁₆	C₄×C₁₀₉	semidirect
182. C₁₆	C₄⋊C₄×C₁₀₉	semidirect
215. C₄	#347×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2150	2
C₂×#347×C₁₀₉	2
C₄×M₃₂×C₁₀₉	2
C₁₀₉×#1770	2
C₁₀₉×#1375	2
#12666	109

Subgroups — C109×#12666 #42130660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12666 #42130660