Subgroups — C₁₀₉×#12606 #42130600

Order 27904.

466 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2112	C₂	1	2	yes	maximal
3	C₂×#347×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#1375	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×#117×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
9	#405×C₁₀₉	—	2	4	no	—
10	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
14	C₂²⋊C₁₆×C₁₀₉	C₄	1	8	yes	complemented
15	#347×C₁₀₉	—	2	4	no	—
16	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×M₁₆×C₁₀₉	—	2	2	no	—
18	C₂×M₁₆×C₁₀₉	—	2	2	no	—
19	C₂×M₁₆×C₁₀₉	—	2	2	no	—
20	C₂²×C₈×C₁₀₉	—	2	2	no	—
21	C₂²×C₈×C₁₀₉	—	2	2	no	—
22	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
27	#117×C₁₀₉	—	2	2	no	—
28	#117×C₁₀₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₁₀₉	—	2	4	no	—
34	M₃₂×C₁₀₉	—	2	4	no	—
35	C₂×C₁₆×C₁₀₉	C₂×C₄	1	8	yes	—
36	#97×C₁₀₉	—	2	2	no	—
37	#97×C₁₀₉	D₄	1	4	yes	—
38	C₄×C₈×C₁₀₉	—	2	2	no	—
39	C₄×C₈×C₁₀₉	—	2	2	no	—
40	C₂⁴×C₁₀₉	M₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	—	2	1	no	—
42	C₂²×C₄×C₁₀₉	—	2	1	no	—
43	C₂²×C₄×C₁₀₉	—	2	1	no	—
44	C₂²×C₄×C₁₀₉	—	2	1	no	—
45	C₂²×C₄×C₁₀₉	—	2	1	no	—
46	C₂²×C₄×C₁₀₉	—	2	1	no	—
47	C₂²×C₄×C₁₀₉	C₄²	1	2	yes	—
48	C₂²×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
50	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
51	M₁₆×C₁₀₉	—	2	8	no	—
52	C₂×C₈×C₁₀₉	—	2	1	no	—
53	C₂×C₈×C₁₀₉	—	2	1	no	—
54	C₂×C₈×C₁₀₉	—	2	1	no	—
55	C₂×C₈×C₁₀₉	—	2	1	no	—
56	C₂×C₈×C₁₀₉	—	2	1	no	—
57	C₂×C₈×C₁₀₉	—	2	1	no	—
58	C₂×C₈×C₁₀₉	—	2	8	no	—
59	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
60	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₀₉	M₁₆	1	2	yes	—
62	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₁₀₉	#41	1	2	yes	—
65	C₄⋊C₄×C₁₀₉	—	2	4	no	—
66	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
67	C₄²×C₁₀₉	—	2	1	no	—
68	C₄²×C₁₀₉	—	2	1	no	—
69	C₄²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₁₀₉	M₁₆	1	2	yes	—
71	C₁₆×C₁₀₉	—	2	8	no	—
72	C₂³×C₁₀₉	—	2	1	no	—
73	C₂³×C₁₀₉	—	2	1	no	—
74	C₂³×C₁₀₉	—	4	2	no	—
75	C₂³×C₁₀₉	#97	1	2	yes	—
76	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₀₉	—	2	1	no	—
78	C₂×C₄×C₁₀₉	—	2	1	no	—
79	C₂×C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄×C₁₀₉	—	2	2	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	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	4	2	no	—
86	C₂×C₄×C₁₀₉	—	4	2	no	—
87	C₂×C₄×C₁₀₉	#97	1	2	yes	—
88	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
94	C₈×C₁₀₉	—	2	2	no	—
95	C₈×C₁₀₉	—	2	2	no	—
96	C₈×C₁₀₉	—	2	2	no	—
97	C₈×C₁₀₉	C₄×D₄	1	4	yes	—
98	C₂²×C₁₀₉	—	2	2	no	—
99	C₂²×C₁₀₉	—	2	4	no	—
100	C₂²×C₁₀₉	—	2	2	no	—
101	C₂²×C₁₀₉	—	4	2	no	—
102	C₂²×C₁₀₉	—	4	2	no	—
103	C₂²×C₁₀₉	C₂×#97	1	1	yes	characteristic
104	C₂²×C₁₀₉	#407	1	1	yes	characteristic
105	C₂²×C₁₀₉	#432	1	1	yes	characteristic
106	C₄×C₁₀₉	—	2	2	no	—
107	C₄×C₁₀₉	—	4	2	no	—
108	C₄×C₁₀₉	—	4	2	no	—
109	C₄×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
110	C₄×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
111	C₄×C₁₀₉	#407	1	1	yes	characteristic
112	C₄×C₁₀₉	#432	1	1	yes	characteristic
113	#12606	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂×C₁₀₉	—	2	1	no	—
115	C₂×C₁₀₉	—	2	1	no	—
116	C₂×C₁₀₉	—	4	2	no	—
117	C₂×C₁₀₉	#1749	1	1	yes	characteristic
118	C₂×C₁₀₉	#2110	1	2	yes	—
119	C₂×#405	C₂×C₁₀₉	1	1	yes	characteristic
120	#2112	C₂×C₁₀₉	1	2	yes	—
121	C₂×#347	C₂×C₁₀₉	1	2	yes	complemented
122	#1375	C₂×C₁₀₉	1	2	yes	complemented
123	C₁₀₉	#12606	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×M₁₆	C₄×C₁₀₉	1	1	yes	characteristic
125	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
126	C₂×#117	C₄×C₁₀₉	1	1	yes	characteristic
127	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
128	#405	—	2	4	no	—
129	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
130	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
131	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
132	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
133	C₂²⋊C₁₆	C₄×C₁₀₉	1	8	yes	complemented
134	#347	—	2	4	no	—
135	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
136	C₂×M₁₆	—	2	2	no	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
142	C₂²×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
143	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
144	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
146	#117	—	2	2	no	—
147	#117	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₂²⋊C₄	—	2	1	no	—
150	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
151	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
152	M₃₂	—	2	4	no	—
153	M₃₂	—	2	4	no	—
154	C₂×C₁₆	C₂×C₄×C₁₀₉	1	8	yes	—
155	#97	—	2	2	no	—
156	#97	D₄×C₁₀₉	1	4	yes	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₂⁴	M₁₆×C₁₀₉	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	C₄²×C₁₀₉	1	2	yes	—
167	C₂²×C₄	M₁₆×C₁₀₉	1	1	yes	characteristic
168	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
169	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
170	M₁₆	—	2	8	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	8	no	—
178	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
179	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
180	C₂×C₈	M₁₆×C₁₀₉	1	2	yes	—
181	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
182	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
183	C₂×C₈	#41×C₁₀₉	1	2	yes	—
184	C₄⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	C₂²⋊C₄×C₁₀₉	1	2	yes	—
189	C₄²	M₁₆×C₁₀₉	1	2	yes	—
190	C₁₆	—	2	8	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	4	2	no	—
194	C₂³	#97×C₁₀₉	1	2	yes	—
195	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	#97×C₁₀₉	1	2	yes	—
207	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
208	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
209	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
210	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
211	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
212	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	C₄×D₄×C₁₀₉	1	4	yes	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₂×#97×C₁₀₉	1	1	yes	characteristic
223	C₂²	#407×C₁₀₉	1	1	yes	characteristic
224	C₂²	#432×C₁₀₉	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	C₄×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
229	C₄	C₄×M₁₆×C₁₀₉	1	1	yes	characteristic
230	C₄	#407×C₁₀₉	1	1	yes	characteristic
231	C₄	#432×C₁₀₉	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	4	2	no	—
235	C₂	C₁₀₉×#1749	1	1	yes	characteristic
236	C₂	C₁₀₉×#2110	1	2	yes	—

	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	C₂⁴×C₁₀₉	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
3. C₂×#347×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1375	C₂	semidirect
14. C₂²⋊C₁₆×C₁₀₉	C₄	semidirect
113. #12606	C₁₀₉	direct (N × H)
121. C₂×#347	C₂×C₁₀₉	semidirect
122. #1375	C₂×C₁₀₉	semidirect
123. C₁₀₉	#12606	direct (N × H)
133. C₂²⋊C₁₆	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#405×C₁₀₉	2
C₁₀₉×#2112	2
C₂×#347×C₁₀₉	2
C₁₀₉×#1375	2
#12606	109

Subgroups — C109×#12606 #42130600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12606 #42130600