Subgroups — C₁₀₉×#12647 #42130641

Order 27904.

486 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#442×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#361×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2140	C₂	1	1	yes	maximal, characteristic
4	C₂×#347×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1773	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₉×#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₉×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₁₀₉	—	2	1	no	—
10	C₂×M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
12	#442×C₁₀₉	—	2	2	no	—
13	#442×C₁₀₉	C₄	1	4	yes	complemented
14	C₂×#108×C₁₀₉	C₂²	1	1	yes	characteristic
15	#427×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×#106×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#102×C₁₀₉	—	2	1	no	—
18	#405×C₁₀₉	—	2	1	no	—
19	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
20	#361×C₁₀₉	C₄	1	4	yes	complemented
21	#357×C₁₀₉	C₄	1	4	yes	complemented
22	#347×C₁₀₉	—	2	2	no	—
23	#339×C₁₀₉	—	2	1	no	—
24	C₂²×D₄×C₁₀₉	—	2	1	no	—
25	C₂³×C₄×C₁₀₉	—	4	1	no	—
26	C₂×D₈×C₁₀₉	—	2	1	no	—
27	C₂×D₈×C₁₀₉	—	2	1	no	—
28	C₂×D₈×C₁₀₉	—	2	1	no	—
29	C₂×D₈×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×D₈×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₂×D₈×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₁₀₉	—	2	1	no	—
33	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₁₀₉	—	2	4	no	—
36	C₄×D₄×C₁₀₉	—	4	2	no	—
37	#117×C₁₀₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₁₀₉	—	4	1	no	—
40	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
41	M₃₂×C₁₀₉	—	2	1	no	—
42	M₃₂×C₁₀₉	—	2	1	no	—
43	M₃₂×C₁₀₉	—	2	1	no	—
44	M₃₂×C₁₀₉	C₂×C₄	1	2	yes	complemented
45	C₂×C₁₆×C₁₀₉	—	2	1	no	—
46	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	complemented
47	#108×C₁₀₉	—	2	1	no	—
48	#108×C₁₀₉	C₂×C₄	1	2	yes	complemented
49	#107×C₁₀₉	C₂×C₄	1	2	yes	complemented
50	#106×C₁₀₉	—	2	1	no	—
51	#106×C₁₀₉	—	2	1	no	—
52	#106×C₁₀₉	C₂×C₄	1	2	yes	—
53	#102×C₁₀₉	—	2	4	no	—
54	#97×C₁₀₉	—	4	1	no	—
55	#97×C₁₀₉	C₂×C₄	1	2	yes	—
56	#97×C₁₀₉	D₄	1	2	yes	—
57	C₄×C₈×C₁₀₉	—	4	1	no	—
58	C₂⁴×C₁₀₉	—	4	1	no	—
59	C₂×D₄×C₁₀₉	—	2	2	no	—
60	C₂×D₄×C₁₀₉	—	2	2	no	—
61	C₂×D₄×C₁₀₉	—	2	2	no	—
62	C₂×D₄×C₁₀₉	—	4	1	no	—
63	C₂×D₄×C₁₀₉	—	4	1	no	—
64	C₂×D₄×C₁₀₉	—	4	1	no	—
65	C₂²×C₄×C₁₀₉	—	2	1	no	—
66	C₂²×C₄×C₁₀₉	—	4	4	no	—
67	C₂²×C₄×C₁₀₉	—	4	1	no	—
68	C₂²×C₄×C₁₀₉	—	8	4	no	—
69	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
70	D₈×C₁₀₉	—	2	2	no	—
71	D₈×C₁₀₉	—	2	2	no	—
72	D₈×C₁₀₉	—	2	2	no	—
73	D₈×C₁₀₉	C₄²	1	4	yes	—
74	M₁₆×C₁₀₉	—	2	2	no	—
75	M₁₆×C₁₀₉	—	4	1	no	—
76	C₂×C₈×C₁₀₉	—	2	1	no	—
77	C₂×C₈×C₁₀₉	—	2	1	no	—
78	C₂×C₈×C₁₀₉	—	4	1	no	—
79	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
86	C₄⋊C₄×C₁₀₉	—	2	2	no	—
87	C₄⋊C₄×C₁₀₉	—	2	2	no	—
88	C₄⋊C₄×C₁₀₉	—	4	1	no	—
89	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
90	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
91	C₄²×C₁₀₉	—	2	1	no	—
92	C₄²×C₁₀₉	—	4	1	no	—
93	C₄²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
94	C₁₆×C₁₀₉	—	2	2	no	—
95	C₁₆×C₁₀₉	—	2	2	no	—
96	C₂³×C₁₀₉	—	4	2	no	—
97	C₂³×C₁₀₉	—	4	2	no	—
98	C₂³×C₁₀₉	—	4	2	no	—
99	C₂³×C₁₀₉	—	8	2	no	—
100	C₂³×C₁₀₉	—	8	2	no	—
101	C₂³×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
102	D₄×C₁₀₉	—	2	4	no	—
103	D₄×C₁₀₉	—	4	2	no	—
104	D₄×C₁₀₉	—	4	2	no	—
105	D₄×C₁₀₉	—	4	2	no	—
106	C₂×C₄×C₁₀₉	—	2	2	no	—
107	C₂×C₄×C₁₀₉	—	4	2	no	—
108	C₂×C₄×C₁₀₉	—	4	2	no	—
109	C₂×C₄×C₁₀₉	—	4	2	no	—
110	C₂×C₄×C₁₀₉	—	8	8	no	—
111	C₂×C₄×C₁₀₉	—	8	2	no	—
112	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	#137	1	1	yes	characteristic
118	C₈×C₁₀₉	—	2	2	no	—
119	C₈×C₁₀₉	—	4	2	no	—
120	C₈×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
121	C₈×C₁₀₉	#117	1	1	yes	characteristic
122	C₈×C₁₀₉	C₄×D₄	1	1	yes	characteristic
123	C₈×C₁₀₉	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₁₀₉	—	2	2	no	—
125	C₂²×C₁₀₉	—	4	4	no	—
126	C₂²×C₁₀₉	—	8	2	no	—
127	C₂²×C₁₀₉	—	8	2	no	—
128	C₂²×C₁₀₉	—	8	4	no	—
129	C₂²×C₁₀₉	—	8	4	no	—
130	C₂²×C₁₀₉	#418	1	1	yes	characteristic
131	C₂²×C₁₀₉	#440	1	1	yes	characteristic
132	C₂²×C₁₀₉	#442	1	1	yes	characteristic
133	C₄×C₁₀₉	—	4	2	no	—
134	C₄×C₁₀₉	—	8	2	no	—
135	C₄×C₁₀₉	—	8	2	no	—
136	C₄×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₁₀₉	#417	1	1	yes	characteristic
138	C₄×C₁₀₉	#440	1	1	yes	characteristic
139	C₄×C₁₀₉	#442	1	1	yes	characteristic
140	#12647	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₁₀₉	—	2	1	no	—
142	C₂×C₁₀₉	—	2	1	no	—
143	C₂×C₁₀₉	—	8	4	no	—
144	C₂×C₁₀₉	#1762	1	1	yes	characteristic
145	C₂×C₁₀₉	AGL(1,ℤ/16)	1	2	yes	—
146	C₂×#442	C₂×C₁₀₉	1	1	yes	characteristic
147	C₂×#361	C₂×C₁₀₉	1	1	yes	characteristic
148	#2140	C₂×C₁₀₉	1	1	yes	characteristic
149	C₂×#347	C₂×C₁₀₉	1	1	yes	characteristic, complemented
150	#1773	C₂×C₁₀₉	1	1	yes	characteristic, complemented
151	#1386	C₂×C₁₀₉	1	1	yes	characteristic, complemented
152	#1383	C₂×C₁₀₉	1	1	yes	characteristic, complemented
153	C₁₀₉	#12647	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
155	C₂×C₄×D₄	—	2	1	no	—
156	C₂×M₃₂	C₂²×C₁₀₉	1	1	yes	characteristic
157	C₂×M₃₂	C₂²×C₁₀₉	1	1	yes	characteristic
158	#442	—	2	2	no	—
159	#442	C₄×C₁₀₉	1	4	yes	complemented
160	C₂×#108	C₂²×C₁₀₉	1	1	yes	characteristic
161	#427	C₂²×C₁₀₉	1	1	yes	characteristic
162	C₂×#106	C₂²×C₁₀₉	1	1	yes	characteristic
163	C₂×#102	—	2	1	no	—
164	#405	—	2	1	no	—
165	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
166	#361	C₄×C₁₀₉	1	4	yes	complemented
167	#357	C₄×C₁₀₉	1	4	yes	complemented
168	#347	—	2	2	no	—
169	#339	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂³×C₄	—	4	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	C₂×C₄×C₁₀₉	1	2	yes	—
176	C₂×D₈	C₂×C₄×C₁₀₉	1	2	yes	—
177	C₂×D₈	C₂×C₄×C₁₀₉	1	2	yes	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
181	C₄×D₄	—	2	4	no	—
182	C₄×D₄	—	4	2	no	—
183	#117	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₂²⋊C₄	—	4	1	no	—
186	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	1	no	—
190	M₃₂	C₂×C₄×C₁₀₉	1	2	yes	complemented
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	complemented
193	#108	—	2	1	no	—
194	#108	C₂×C₄×C₁₀₉	1	2	yes	complemented
195	#107	C₂×C₄×C₁₀₉	1	2	yes	complemented
196	#106	—	2	1	no	—
197	#106	—	2	1	no	—
198	#106	C₂×C₄×C₁₀₉	1	2	yes	—
199	#102	—	2	4	no	—
200	#97	—	4	1	no	—
201	#97	C₂×C₄×C₁₀₉	1	2	yes	—
202	#97	D₄×C₁₀₉	1	2	yes	—
203	C₄×C₈	—	4	1	no	—
204	C₂⁴	—	4	1	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	8	4	no	—
215	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
216	D₈	—	2	2	no	—
217	D₈	—	2	2	no	—
218	D₈	—	2	2	no	—
219	D₈	C₄²×C₁₀₉	1	4	yes	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	4	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
228	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
230	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
231	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₂²⋊C₄	—	4	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂²⋊C₄×C₁₀₉	1	2	yes	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	2	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	—	8	2	no	—
247	C₂³	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
248	D₄	—	2	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
259	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
260	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
261	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
262	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
263	C₂×C₄	#137×C₁₀₉	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
267	C₈	#117×C₁₀₉	1	1	yes	characteristic
268	C₈	C₄×D₄×C₁₀₉	1	1	yes	characteristic
269	C₈	C₄×D₄×C₁₀₉	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	#418×C₁₀₉	1	1	yes	characteristic
277	C₂²	#440×C₁₀₉	1	1	yes	characteristic
278	C₂²	#442×C₁₀₉	1	1	yes	characteristic
279	C₄	—	4	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	C₄×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
283	C₄	#417×C₁₀₉	1	1	yes	characteristic
284	C₄	#440×C₁₀₉	1	1	yes	characteristic
285	C₄	#442×C₁₀₉	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	8	4	no	—
289	C₂	C₁₀₉×#1762	1	1	yes	characteristic
290	C₂	C₁₀₉×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#442
commutator	C₈	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
prime radical	C₂²×D₈×C₁₀₉	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂×#97×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×#347×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1773	C₂	semidirect
6. C₁₀₉×#1386	C₂	semidirect
7. C₁₀₉×#1383	C₂	semidirect
13. #442×C₁₀₉	C₄	semidirect
20. #361×C₁₀₉	C₄	semidirect
21. #357×C₁₀₉	C₄	semidirect
44. M₃₂×C₁₀₉	C₂×C₄	semidirect
46. C₂×C₁₆×C₁₀₉	C₂×C₄	semidirect
48. #108×C₁₀₉	C₂×C₄	semidirect
49. #107×C₁₀₉	C₂×C₄	semidirect
140. #12647	C₁₀₉	direct (N × H)
149. C₂×#347	C₂×C₁₀₉	semidirect
150. #1773	C₂×C₁₀₉	semidirect
151. #1386	C₂×C₁₀₉	semidirect
152. #1383	C₂×C₁₀₉	semidirect
153. C₁₀₉	#12647	direct (N × H)
159. #442	C₄×C₁₀₉	semidirect
166. #361	C₄×C₁₀₉	semidirect
167. #357	C₄×C₁₀₉	semidirect
190. M₃₂	C₂×C₄×C₁₀₉	semidirect
192. C₂×C₁₆	C₂×C₄×C₁₀₉	semidirect
194. #108	C₂×C₄×C₁₀₉	semidirect
195. #107	C₂×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×#442×C₁₀₉	2
C₂×#361×C₁₀₉	2
C₁₀₉×#2140	2
C₂×#347×C₁₀₉	2
C₁₀₉×#1773	2
C₁₀₉×#1386	2
C₁₀₉×#1383	2
#12647	109

Subgroups — C109×#12647 #42130641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12647 #42130641