Subgroups — C₁₀₉×#12912 #42130906

Order 27904.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 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	C₂×C₄⋊C₁₆×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₈×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1374	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₀₉	—	2	2	no	—
8	C₂²×C₁₆×C₁₀₉	—	2	4	no	—
9	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₄×C₂²⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
15	C₄⋊C₁₆×C₁₀₉	—	2	4	no	—
16	C₂²⋊C₁₆×C₁₀₉	—	2	2	no	—
17	#336×C₁₀₉	C₄	1	2	yes	—
18	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	—	2	2	no	—
20	C₂²×C₈×C₁₀₉	—	2	2	no	—
21	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
24	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₀₉	C₈	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₁₀₉	C₈	1	1	yes	characteristic
28	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₁₀₉	—	2	8	no	—
30	C₂×C₁₆×C₁₀₉	—	2	2	no	—
31	C₂×C₁₆×C₁₀₉	—	2	2	no	—
32	C₂×C₁₆×C₁₀₉	—	2	1	no	—
33	C₂×C₁₆×C₁₀₉	—	2	1	no	—
34	C₂×C₁₆×C₁₀₉	—	2	2	no	—
35	C₂×C₁₆×C₁₀₉	—	2	2	no	—
36	C₂×C₁₆×C₁₀₉	D₄	1	4	yes	—
37	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
38	C₄×C₈×C₁₀₉	—	2	2	no	—
39	C₄×C₈×C₁₀₉	—	2	2	no	—
40	#95×C₁₀₉	C₈	1	2	yes	—
41	C₂⁴×C₁₀₉	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	—	2	2	no	—
43	C₂²×C₄×C₁₀₉	—	2	2	no	—
44	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
49	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₀₉	—	2	2	no	—
51	C₂×C₈×C₁₀₉	—	2	2	no	—
52	C₂×C₈×C₁₀₉	—	2	4	no	—
53	C₂×C₈×C₁₀₉	—	2	2	no	—
54	C₂×C₈×C₁₀₉	—	2	2	no	—
55	C₂×C₈×C₁₀₉	—	2	2	no	—
56	C₂×C₈×C₁₀₉	—	2	2	no	—
57	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₁₀₉	#41	1	2	yes	—
62	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
63	C₂²⋊C₄×C₁₀₉	C₁₆	1	4	yes	complemented
64	C₄²×C₁₀₉	—	2	2	no	—
65	C₄²×C₁₀₉	—	2	2	no	—
66	C₁₆×C₁₀₉	—	2	4	no	—
67	C₁₆×C₁₀₉	—	4	1	no	—
68	C₁₆×C₁₀₉	—	4	1	no	—
69	C₂³×C₁₀₉	—	2	2	no	—
70	C₂³×C₁₀₉	—	2	2	no	—
71	C₂³×C₁₀₉	C₄⋊C₈	1	2	yes	—
72	C₂³×C₁₀₉	C₂×C₁₆	1	2	yes	—
73	C₂³×C₁₀₉	M₃₂	1	2	yes	—
74	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	—	2	2	no	—
76	C₂×C₄×C₁₀₉	—	2	2	no	—
77	C₂×C₄×C₁₀₉	—	2	2	no	—
78	C₂×C₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄×C₁₀₉	—	2	4	no	—
81	C₂×C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₁₀₉	C₂×C₁₆	1	4	yes	complemented
86	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	#131	1	2	yes	—
90	C₈×C₁₀₉	—	2	2	no	—
91	C₈×C₁₀₉	—	4	2	no	—
92	C₈×C₁₀₉	#120	1	1	yes	characteristic
93	C₈×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
94	C₈×C₁₀₉	#123	1	1	yes	characteristic
95	C₂²×C₁₀₉	—	2	2	no	—
96	C₂²×C₁₀₉	—	2	2	no	—
97	C₂²×C₁₀₉	—	2	4	no	—
98	C₂²×C₁₀₉	—	2	4	no	—
99	C₂²×C₁₀₉	C₄⋊C₁₆	1	4	yes	complemented
100	C₂²×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₂²×C₁₀₉	C₈×D₄	1	2	yes	—
102	C₂²×C₁₀₉	C₂²×C₁₆	1	1	yes	characteristic
103	C₂²×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
104	C₂²×C₁₀₉	#504	1	2	yes	—
105	C₄×C₁₀₉	—	2	2	no	—
106	C₄×C₁₀₉	—	2	4	no	—
107	C₄×C₁₀₉	—	4	2	no	—
108	C₄×C₁₀₉	#385	1	1	yes	characteristic
109	C₄×C₁₀₉	#424	1	1	yes	characteristic
110	C₄×C₁₀₉	#435	1	2	yes	—
111	#12912	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₁₀₉	—	2	4	no	—
113	C₂×C₁₀₉	#1850	1	1	yes	characteristic
114	C₂×C₁₀₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
115	C₂×C₁₀₉	#2151	1	1	yes	characteristic
116	C₂×C₁₀₉	D₄×C₁₆	1	2	yes	—
117	C₂×C₁₀₉	#2168	1	2	yes	—
118	C₂³×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
119	C₂×C₄⋊C₁₆	C₂×C₁₀₉	1	2	yes	complemented
120	C₂×C₂²⋊C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
121	C₈×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic
122	#1374	C₂×C₁₀₉	1	2	yes	complemented
123	C₁₀₉	#12912	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
125	C₂²×C₁₆	—	2	2	no	—
126	C₂²×C₁₆	—	2	4	no	—
127	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
128	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
129	C₂²×C₁₆	C₂²×C₁₀₉	1	2	yes	—
130	C₂×C₂²⋊C₈	C₄×C₁₀₉	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
132	C₄×C₂²⋊C₄	C₄×C₁₀₉	1	1	yes	characteristic
133	C₄⋊C₁₆	—	2	4	no	—
134	C₂²⋊C₁₆	—	2	2	no	—
135	#336	C₄×C₁₀₉	1	2	yes	—
136	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
140	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
141	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
142	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
143	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	C₈×C₁₀₉	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	C₈×C₁₀₉	1	1	yes	characteristic
146	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
147	C₂×C₁₆	—	2	8	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	1	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	D₄×C₁₀₉	1	4	yes	—
155	C₂²⋊C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	#95	C₈×C₁₀₉	1	2	yes	—
159	C₂⁴	C₂×C₈×C₁₀₉	1	1	yes	characteristic
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
163	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
164	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
166	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
176	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
177	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
178	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
179	C₂×C₈	#41×C₁₀₉	1	2	yes	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	C₁₆×C₁₀₉	1	4	yes	complemented
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₁₆	—	2	4	no	—
185	C₁₆	—	4	1	no	—
186	C₁₆	—	4	1	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	C₄⋊C₈×C₁₀₉	1	2	yes	—
190	C₂³	C₂×C₁₆×C₁₀₉	1	2	yes	—
191	C₂³	M₃₂×C₁₀₉	1	2	yes	—
192	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	C₄⋊C₈×C₁₀₉	1	2	yes	—
203	C₂×C₄	C₂×C₁₆×C₁₀₉	1	4	yes	complemented
204	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
205	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
206	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
207	C₂×C₄	#131×C₁₀₉	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	4	2	no	—
210	C₈	#120×C₁₀₉	1	1	yes	characteristic
211	C₈	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
212	C₈	#123×C₁₀₉	1	1	yes	characteristic
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	C₄⋊C₁₆×C₁₀₉	1	4	yes	complemented
218	C₂²	C₂×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
219	C₂²	C₈×D₄×C₁₀₉	1	2	yes	—
220	C₂²	C₂²×C₁₆×C₁₀₉	1	1	yes	characteristic
221	C₂²	C₂×M₃₂×C₁₀₉	1	1	yes	characteristic
222	C₂²	#504×C₁₀₉	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	4	no	—
225	C₄	—	4	2	no	—
226	C₄	#385×C₁₀₉	1	1	yes	characteristic
227	C₄	#424×C₁₀₉	1	1	yes	characteristic
228	C₄	#435×C₁₀₉	1	2	yes	—
229	C₂	—	2	4	no	—
230	C₂	C₁₀₉×#1850	1	1	yes	characteristic
231	C₂	C₂×C₄⋊C₁₆×C₁₀₉	1	1	yes	characteristic
232	C₂	C₁₀₉×#2151	1	1	yes	characteristic
233	C₂	D₄×C₁₆×C₁₀₉	1	2	yes	—
234	C₂	C₁₀₉×#2168	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₀₉	C₂³
commutator	C₂²	C₂²×C₁₆×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂³×C₁₀₉	C₂²×C₈
prime radical	C₂⁴×C₁₀₉	C₂×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
2. C₂×C₄⋊C₁₆×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1374	C₂	semidirect
63. C₂²⋊C₄×C₁₀₉	C₁₆	semidirect
85. C₂×C₄×C₁₀₉	C₂×C₁₆	semidirect
99. C₂²×C₁₀₉	C₄⋊C₁₆	semidirect
111. #12912	C₁₀₉	direct (N × H)
119. C₂×C₄⋊C₁₆	C₂×C₁₀₉	semidirect
122. #1374	C₂×C₁₀₉	semidirect
123. C₁₀₉	#12912	direct (N × H)
181. C₂²⋊C₄	C₁₆×C₁₀₉	semidirect
203. C₂×C₄	C₂×C₁₆×C₁₀₉	semidirect
217. C₂²	C₄⋊C₁₆×C₁₀₉	semidirect

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
C₂×C₄⋊C₁₆×C₁₀₉	2
C₂×C₂²⋊C₁₆×C₁₀₉	2
C₈×C₂²⋊C₄×C₁₀₉	2
C₁₀₉×#1374	2
#12912	109

Subgroups — C109×#12912 #42130906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12912 #42130906