Subgroups — C₁₀₉×#12642 #42130636

Order 27904.

430 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#443×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#359×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2138	C₂	1	1	yes	maximal, characteristic
4	C₂×#346×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₉×#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₉×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#131×C₁₀₉	—	2	1	no	—
10	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
12	#443×C₁₀₉	—	2	2	no	—
13	#443×C₁₀₉	C₄	1	4	yes	complemented
14	C₂×#108×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×#108×C₁₀₉	C₂²	1	1	yes	characteristic
16	#427×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#104×C₁₀₉	—	2	1	no	—
18	#405×C₁₀₉	—	2	1	no	—
19	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	#359×C₁₀₉	C₄	1	4	yes	complemented
21	#358×C₁₀₉	C₄	1	2	yes	complemented
22	#357×C₁₀₉	C₄	1	2	yes	complemented
23	#346×C₁₀₉	—	2	2	no	—
24	#339×C₁₀₉	—	2	1	no	—
25	C₂×#41×C₁₀₉	—	2	1	no	—
26	#135×C₁₀₉	—	2	1	no	—
27	#135×C₁₀₉	—	2	1	no	—
28	#135×C₁₀₉	C₂×C₄	1	2	yes	—
29	#135×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×SD₁₆×C₁₀₉	—	2	1	no	—
32	C₂×D₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	#131×C₁₀₉	—	2	4	no	—
34	#131×C₁₀₉	—	4	2	no	—
35	C₂×M₁₆×C₁₀₉	—	2	1	no	—
36	C₂×M₁₆×C₁₀₉	—	2	1	no	—
37	C₂×M₁₆×C₁₀₉	—	4	1	no	—
38	C₂²×C₈×C₁₀₉	—	4	1	no	—
39	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
41	#117×C₁₀₉	—	2	1	no	—
42	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₁₀₉	—	2	1	no	—
44	C₂×C₁₆×C₁₀₉	—	2	1	no	—
45	C₂×C₁₆×C₁₀₉	—	2	1	no	—
46	C₂×C₁₆×C₁₀₉	—	2	1	no	—
47	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
48	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
49	#108×C₁₀₉	—	2	1	no	—
50	#108×C₁₀₉	—	2	1	no	—
51	#108×C₁₀₉	C₂×C₄	1	2	yes	—
52	#108×C₁₀₉	C₂×C₄	1	2	yes	—
53	#107×C₁₀₉	C₂×C₄	1	2	yes	—
54	#106×C₁₀₉	—	2	1	no	—
55	#104×C₁₀₉	—	2	4	no	—
56	#97×C₁₀₉	—	4	1	no	—
57	C₄×C₈×C₁₀₉	—	4	1	no	—
58	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
60	#41×C₁₀₉	—	2	2	no	—
61	#41×C₁₀₉	—	2	2	no	—
62	#41×C₁₀₉	—	4	1	no	—
63	#41×C₁₀₉	—	4	1	no	—
64	C₂×Q₈×C₁₀₉	—	2	1	no	—
65	C₂×D₄×C₁₀₉	—	2	1	no	—
66	C₂×D₄×C₁₀₉	—	4	1	no	—
67	C₂²×C₄×C₁₀₉	—	2	1	no	—
68	C₂²×C₄×C₁₀₉	—	4	1	no	—
69	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
70	Q₁₆×C₁₀₉	—	2	1	no	—
71	Q₁₆×C₁₀₉	C₄²	1	2	yes	—
72	SD₁₆×C₁₀₉	—	2	2	no	—
73	SD₁₆×C₁₀₉	—	2	2	no	—
74	D₈×C₁₀₉	—	2	1	no	—
75	D₈×C₁₀₉	C₄²	1	2	yes	—
76	M₁₆×C₁₀₉	—	2	2	no	—
77	M₁₆×C₁₀₉	—	2	2	no	—
78	M₁₆×C₁₀₉	—	4	1	no	—
79	M₁₆×C₁₀₉	—	4	1	no	—
80	M₁₆×C₁₀₉	—	4	4	no	—
81	C₂×C₈×C₁₀₉	—	4	1	no	—
82	C₂×C₈×C₁₀₉	—	4	1	no	—
83	C₂×C₈×C₁₀₉	—	4	4	no	—
84	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₁₀₉	D₈	1	1	yes	characteristic
86	C₂×C₈×C₁₀₉	D₈	1	1	yes	characteristic
87	C₂×C₈×C₁₀₉	SD₁₆	1	2	yes	—
88	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
93	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
94	C₄⋊C₄×C₁₀₉	—	2	2	no	—
95	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
96	C₄²×C₁₀₉	—	2	1	no	—
97	C₄²×C₁₀₉	—	4	1	no	—
98	C₄²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
99	C₁₆×C₁₀₉	—	2	2	no	—
100	C₁₆×C₁₀₉	—	2	2	no	—
101	C₂³×C₁₀₉	—	4	1	no	—
102	C₂³×C₁₀₉	#135	1	1	yes	characteristic
103	Q₈×C₁₀₉	—	2	2	no	—
104	Q₈×C₁₀₉	—	4	1	no	—
105	D₄×C₁₀₉	—	2	2	no	—
106	D₄×C₁₀₉	—	4	2	no	—
107	D₄×C₁₀₉	—	4	2	no	—
108	D₄×C₁₀₉	—	4	1	no	—
109	C₂×C₄×C₁₀₉	—	2	2	no	—
110	C₂×C₄×C₁₀₉	—	4	2	no	—
111	C₂×C₄×C₁₀₉	—	4	1	no	—
112	C₂×C₄×C₁₀₉	—	4	2	no	—
113	C₂×C₄×C₁₀₉	—	4	2	no	—
114	C₂×C₄×C₁₀₉	—	4	2	no	—
115	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₀₉	#135	1	1	yes	characteristic
121	C₈×C₁₀₉	—	4	2	no	—
122	C₈×C₁₀₉	—	4	2	no	—
123	C₈×C₁₀₉	#102	1	2	yes	—
124	C₈×C₁₀₉	#102	1	2	yes	—
125	C₈×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
126	C₈×C₁₀₉	#117	1	1	yes	characteristic
127	C₈×C₁₀₉	C₄×D₄	1	1	yes	characteristic
128	C₈×C₁₀₉	C₄×D₄	1	1	yes	characteristic
129	C₂²×C₁₀₉	—	2	2	no	—
130	C₂²×C₁₀₉	—	4	2	no	—
131	C₂²×C₁₀₉	—	8	2	no	—
132	C₂²×C₁₀₉	#416	1	1	yes	characteristic
133	C₂²×C₁₀₉	C₄×D₈	1	1	yes	characteristic
134	C₂²×C₁₀₉	C₄×SD₁₆	1	1	yes	characteristic
135	C₄×C₁₀₉	—	4	2	no	—
136	C₄×C₁₀₉	—	4	2	no	—
137	C₄×C₁₀₉	—	8	2	no	—
138	C₄×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₁₀₉	C₂×#102	1	1	yes	characteristic
140	C₄×C₁₀₉	C₄×D₈	1	1	yes	characteristic
141	C₄×C₁₀₉	C₄×SD₁₆	1	1	yes	characteristic
142	#12642	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₁₀₉	—	2	1	no	—
144	C₂×C₁₀₉	—	2	1	no	—
145	C₂×C₁₀₉	—	8	2	no	—
146	C₂×C₁₀₉	C₄×#102	1	1	yes	characteristic
147	C₂×C₁₀₉	#2179	1	2	yes	—
148	C₂×#443	C₂×C₁₀₉	1	1	yes	characteristic
149	C₂×#359	C₂×C₁₀₉	1	1	yes	characteristic
150	#2138	C₂×C₁₀₉	1	1	yes	characteristic
151	C₂×#346	C₂×C₁₀₉	1	1	yes	characteristic, complemented
152	#1772	C₂×C₁₀₉	1	1	yes	characteristic, complemented
153	#1382	C₂×C₁₀₉	1	1	yes	characteristic, complemented
154	#1381	C₂×C₁₀₉	1	1	yes	characteristic, complemented
155	C₁₀₉	#12642	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×#135	C₂²×C₁₀₉	1	1	yes	characteristic
157	C₂×#131	—	2	1	no	—
158	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
159	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
160	#443	—	2	2	no	—
161	#443	C₄×C₁₀₉	1	4	yes	complemented
162	C₂×#108	C₂²×C₁₀₉	1	1	yes	characteristic
163	C₂×#108	C₂²×C₁₀₉	1	1	yes	characteristic
164	#427	C₂²×C₁₀₉	1	1	yes	characteristic
165	C₂×#104	—	2	1	no	—
166	#405	—	2	1	no	—
167	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
168	#359	C₄×C₁₀₉	1	4	yes	complemented
169	#358	C₄×C₁₀₉	1	2	yes	complemented
170	#357	C₄×C₁₀₉	1	2	yes	complemented
171	#346	—	2	2	no	—
172	#339	—	2	1	no	—
173	C₂×#41	—	2	1	no	—
174	#135	—	2	1	no	—
175	#135	—	2	1	no	—
176	#135	C₂×C₄×C₁₀₉	1	2	yes	—
177	#135	C₂×C₄×C₁₀₉	1	2	yes	—
178	C₂×Q₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
179	C₂×SD₁₆	—	2	1	no	—
180	C₂×D₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
181	#131	—	2	4	no	—
182	#131	—	4	2	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	4	1	no	—
186	C₂²×C₈	—	4	1	no	—
187	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
189	#117	—	2	1	no	—
190	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
196	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
197	#108	—	2	1	no	—
198	#108	—	2	1	no	—
199	#108	C₂×C₄×C₁₀₉	1	2	yes	—
200	#108	C₂×C₄×C₁₀₉	1	2	yes	—
201	#107	C₂×C₄×C₁₀₉	1	2	yes	—
202	#106	—	2	1	no	—
203	#104	—	2	4	no	—
204	#97	—	4	1	no	—
205	C₄×C₈	—	4	1	no	—
206	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
207	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
208	#41	—	2	2	no	—
209	#41	—	2	2	no	—
210	#41	—	4	1	no	—
211	#41	—	4	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
218	Q₁₆	—	2	1	no	—
219	Q₁₆	C₄²×C₁₀₉	1	2	yes	—
220	SD₁₆	—	2	2	no	—
221	SD₁₆	—	2	2	no	—
222	D₈	—	2	1	no	—
223	D₈	C₄²×C₁₀₉	1	2	yes	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	4	1	no	—
227	M₁₆	—	4	1	no	—
228	M₁₆	—	4	4	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	4	no	—
232	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
233	C₂×C₈	D₈×C₁₀₉	1	1	yes	characteristic
234	C₂×C₈	D₈×C₁₀₉	1	1	yes	characteristic
235	C₂×C₈	SD₁₆×C₁₀₉	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
237	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
240	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
241	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	4	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	C₂²⋊C₄×C₁₀₉	1	2	yes	—
247	C₁₆	—	2	2	no	—
248	C₁₆	—	2	2	no	—
249	C₂³	—	4	1	no	—
250	C₂³	#135×C₁₀₉	1	1	yes	characteristic
251	Q₈	—	2	2	no	—
252	Q₈	—	4	1	no	—
253	D₄	—	2	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
264	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
266	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
267	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
268	C₂×C₄	#135×C₁₀₉	1	1	yes	characteristic
269	C₈	—	4	2	no	—
270	C₈	—	4	2	no	—
271	C₈	#102×C₁₀₉	1	2	yes	—
272	C₈	#102×C₁₀₉	1	2	yes	—
273	C₈	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
274	C₈	#117×C₁₀₉	1	1	yes	characteristic
275	C₈	C₄×D₄×C₁₀₉	1	1	yes	characteristic
276	C₈	C₄×D₄×C₁₀₉	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	#416×C₁₀₉	1	1	yes	characteristic
281	C₂²	C₄×D₈×C₁₀₉	1	1	yes	characteristic
282	C₂²	C₄×SD₁₆×C₁₀₉	1	1	yes	characteristic
283	C₄	—	4	2	no	—
284	C₄	—	4	2	no	—
285	C₄	—	8	2	no	—
286	C₄	C₄×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
287	C₄	C₂×#102×C₁₀₉	1	1	yes	characteristic
288	C₄	C₄×D₈×C₁₀₉	1	1	yes	characteristic
289	C₄	C₄×SD₁₆×C₁₀₉	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	8	2	no	—
293	C₂	C₄×#102×C₁₀₉	1	1	yes	characteristic
294	C₂	C₁₀₉×#2179	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×D₈
commutator	C₈	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₄×D₈
prime radical	C₂×#135×C₁₀₉	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×#346×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1772	C₂	semidirect
6. C₁₀₉×#1382	C₂	semidirect
7. C₁₀₉×#1381	C₂	semidirect
13. #443×C₁₀₉	C₄	semidirect
20. #359×C₁₀₉	C₄	semidirect
21. #358×C₁₀₉	C₄	semidirect
22. #357×C₁₀₉	C₄	semidirect
142. #12642	C₁₀₉	direct (N × H)
151. C₂×#346	C₂×C₁₀₉	semidirect
152. #1772	C₂×C₁₀₉	semidirect
153. #1382	C₂×C₁₀₉	semidirect
154. #1381	C₂×C₁₀₉	semidirect
155. C₁₀₉	#12642	direct (N × H)
161. #443	C₄×C₁₀₉	semidirect
168. #359	C₄×C₁₀₉	semidirect
169. #358	C₄×C₁₀₉	semidirect
170. #357	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×#443×C₁₀₉	2
C₂×#359×C₁₀₉	2
C₁₀₉×#2138	2
C₂×#346×C₁₀₉	2
C₁₀₉×#1772	2
C₁₀₉×#1382	2
C₁₀₉×#1381	2
#12642	109

Subgroups — C109×#12642 #42130636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12642 #42130636