Subgroups — C₁₀₉×#9668 #42127662

Order 27904.

510 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#1478	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#1460	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
8	C₄×D₈×C₁₀₉	—	2	1	no	—
9	C₄×D₈×C₁₀₉	—	2	1	no	—
10	C₄×D₈×C₁₀₉	—	2	1	no	—
11	C₄×D₈×C₁₀₉	—	2	1	no	—
12	#431×C₁₀₉	C₂²	1	2	yes	—
13	#431×C₁₀₉	C₂²	1	1	yes	characteristic
14	#431×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×#107×C₁₀₉	C₄	1	1	yes	characteristic
16	C₂×#102×C₁₀₉	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	#335×C₁₀₉	—	2	1	no	—
19	#334×C₁₀₉	—	2	1	no	—
20	D₄⋊C₈×C₁₀₉	—	2	4	no	—
21	#323×C₁₀₉	—	2	4	no	—
22	#322×C₁₀₉	—	2	1	no	—
23	C₈²×C₁₀₉	—	2	1	no	—
24	C₂²×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₀₉	—	2	2	no	—
26	C₂×D₈×C₁₀₉	—	2	1	no	—
27	C₂×D₈×C₁₀₉	—	2	1	no	—
28	C₂×D₈×C₁₀₉	—	2	2	no	—
29	C₂×D₈×C₁₀₉	C₈	1	4	yes	complemented
30	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₄×D₄×C₁₀₉	—	2	2	no	—
33	C₄×D₄×C₁₀₉	—	2	2	no	—
34	C₄×D₄×C₁₀₉	—	2	2	no	—
35	C₄×D₄×C₁₀₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
38	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
39	#107×C₁₀₉	—	2	1	no	—
40	#107×C₁₀₉	—	2	1	no	—
41	C₄⋊C₈×C₁₀₉	—	2	2	no	—
42	C₄⋊C₈×C₁₀₉	—	2	1	no	—
43	C₄⋊C₈×C₁₀₉	D₄	1	2	yes	—
44	#102×C₁₀₉	—	2	2	no	—
45	#102×C₁₀₉	—	2	2	no	—
46	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
47	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
48	C₂²⋊C₈×C₁₀₉	D₄	1	2	yes	—
49	C₄×C₈×C₁₀₉	—	2	1	no	—
50	C₄×C₈×C₁₀₉	—	2	1	no	—
51	C₄×C₈×C₁₀₉	—	2	2	no	—
52	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
56	C₂⁴×C₁₀₉	—	2	2	no	—
57	C₂×D₄×C₁₀₉	—	2	2	no	—
58	C₂×D₄×C₁₀₉	—	2	2	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	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₁₀₉	—	2	2	no	—
64	C₂²×C₄×C₁₀₉	—	4	2	no	—
65	C₂²×C₄×C₁₀₉	—	4	2	no	—
66	C₂²×C₄×C₁₀₉	—	8	4	no	—
67	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
70	D₈×C₁₀₉	—	2	4	no	—
71	D₈×C₁₀₉	—	2	4	no	—
72	C₂×C₈×C₁₀₉	—	2	1	no	—
73	C₂×C₈×C₁₀₉	—	2	1	no	—
74	C₂×C₈×C₁₀₉	—	2	2	no	—
75	C₂×C₈×C₁₀₉	—	2	2	no	—
76	C₂×C₈×C₁₀₉	—	4	2	no	—
77	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
82	C₂×C₈×C₁₀₉	M₁₆	1	2	yes	—
83	C₄⋊C₄×C₁₀₉	—	2	2	no	—
84	C₄⋊C₄×C₁₀₉	—	2	2	no	—
85	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
86	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
87	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₀₉	#41	1	1	yes	characteristic
91	C₂³×C₁₀₉	—	2	4	no	—
92	C₂³×C₁₀₉	—	4	2	no	—
93	C₂³×C₁₀₉	—	4	2	no	—
94	C₂³×C₁₀₉	—	8	2	no	—
95	C₂³×C₁₀₉	—	8	2	no	—
96	C₂³×C₁₀₉	C₄×D₄	1	1	yes	characteristic
97	D₄×C₁₀₉	—	2	4	no	—
98	D₄×C₁₀₉	—	2	4	no	—
99	D₄×C₁₀₉	—	4	2	no	—
100	D₄×C₁₀₉	—	4	2	no	—
101	C₂×C₄×C₁₀₉	—	2	2	no	—
102	C₂×C₄×C₁₀₉	—	4	1	no	—
103	C₂×C₄×C₁₀₉	—	4	2	no	—
104	C₂×C₄×C₁₀₉	—	8	4	no	—
105	C₂×C₄×C₁₀₉	—	8	4	no	—
106	C₂×C₄×C₁₀₉	—	8	2	no	—
107	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	#131	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	#131	1	1	yes	characteristic
117	C₈×C₁₀₉	—	2	1	no	—
118	C₈×C₁₀₉	—	2	1	no	—
119	C₈×C₁₀₉	—	4	2	no	—
120	C₈×C₁₀₉	—	4	2	no	—
121	C₈×C₁₀₉	—	8	2	no	—
122	C₈×C₁₀₉	C₂²⋊C₈	1	2	yes	—
123	C₈×C₁₀₉	C₂²⋊C₈	1	2	yes	—
124	C₂²×C₁₀₉	—	2	1	no	—
125	C₂²×C₁₀₉	—	2	1	no	—
126	C₂²×C₁₀₉	—	4	4	no	—
127	C₂²×C₁₀₉	—	8	4	no	—
128	C₂²×C₁₀₉	—	8	4	no	—
129	C₂²×C₁₀₉	—	8	2	no	—
130	C₂²×C₁₀₉	—	8	2	no	—
131	C₂²×C₁₀₉	#390	1	1	yes	characteristic
132	C₂²×C₁₀₉	C₈×D₄	1	1	yes	characteristic
133	C₂²×C₁₀₉	#436	1	1	yes	characteristic
134	C₄×C₁₀₉	—	2	1	no	—
135	C₄×C₁₀₉	—	2	1	no	—
136	C₄×C₁₀₉	—	2	1	no	—
137	C₄×C₁₀₉	—	8	2	no	—
138	C₄×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₁₀₉	#408	1	1	yes	characteristic
140	C₄×C₁₀₉	C₈×D₄	1	1	yes	characteristic
141	C₄×C₁₀₉	#436	1	1	yes	characteristic
142	C₄×C₁₀₉	#472	1	2	yes	—
143	#9668	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₁₀₉	—	2	1	no	—
145	C₂×C₁₀₉	—	2	1	no	—
146	C₂×C₁₀₉	—	8	4	no	—
147	C₂×C₁₀₉	#1954	1	1	yes	characteristic
148	C₂×C₁₀₉	#1976	1	1	yes	characteristic
149	C₂×C₁₀₉	#1978	1	1	yes	characteristic
150	C₂×C₄×D₈	C₂×C₁₀₉	1	1	yes	characteristic
151	#1564	C₂×C₁₀₉	1	1	yes	characteristic, complemented
152	#1478	C₂×C₁₀₉	1	2	yes	complemented
153	#1460	C₂×C₁₀₉	1	2	yes	complemented
154	#1450	C₂×C₁₀₉	1	1	yes	characteristic, complemented
155	C₁₀₉	#9668	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×D₈	C₄×C₁₀₉	1	1	yes	characteristic
157	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
158	C₄×D₈	—	2	1	no	—
159	C₄×D₈	—	2	1	no	—
160	C₄×D₈	—	2	1	no	—
161	C₄×D₈	—	2	1	no	—
162	#431	C₂²×C₁₀₉	1	2	yes	—
163	#431	C₂²×C₁₀₉	1	1	yes	characteristic
164	#431	C₂²×C₁₀₉	1	1	yes	characteristic
165	C₂×#107	C₄×C₁₀₉	1	1	yes	characteristic
166	C₂×#102	C₄×C₁₀₉	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
168	#335	—	2	1	no	—
169	#334	—	2	1	no	—
170	D₄⋊C₈	—	2	4	no	—
171	#323	—	2	4	no	—
172	#322	—	2	1	no	—
173	C₈²	—	2	1	no	—
174	C₂²×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
175	C₂³×C₄	—	2	2	no	—
176	C₂×D₈	—	2	1	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	2	2	no	—
179	C₂×D₈	C₈×C₁₀₉	1	4	yes	complemented
180	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
181	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	2	2	no	—
184	C₄×D₄	—	2	2	no	—
185	C₄×D₄	—	2	2	no	—
186	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
189	#107	—	2	1	no	—
190	#107	—	2	1	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	D₄×C₁₀₉	1	2	yes	—
194	#102	—	2	2	no	—
195	#102	—	2	2	no	—
196	C₂²⋊C₈	—	2	1	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	C₂²⋊C₈	D₄×C₁₀₉	1	2	yes	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	2	2	no	—
202	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
203	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
204	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
205	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
206	C₂⁴	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	C₂×C₈×C₁₀₉	1	4	yes	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	8	4	no	—
217	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
219	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
220	D₈	—	2	4	no	—
221	D₈	—	2	4	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
231	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
232	C₂×C₈	M₁₆×C₁₀₉	1	2	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
238	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
239	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
240	C₄²	#41×C₁₀₉	1	1	yes	characteristic
241	C₂³	—	2	4	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	8	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	C₄×D₄×C₁₀₉	1	1	yes	characteristic
247	D₄	—	2	4	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
258	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
259	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
260	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
261	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
262	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
264	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
265	C₂×C₄	#131×C₁₀₉	1	1	yes	characteristic
266	C₂×C₄	#131×C₁₀₉	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	8	2	no	—
272	C₈	C₂²⋊C₈×C₁₀₉	1	2	yes	—
273	C₈	C₂²⋊C₈×C₁₀₉	1	2	yes	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	8	4	no	—
278	C₂²	—	8	4	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	#390×C₁₀₉	1	1	yes	characteristic
282	C₂²	C₈×D₄×C₁₀₉	1	1	yes	characteristic
283	C₂²	#436×C₁₀₉	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	8	2	no	—
288	C₄	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
289	C₄	#408×C₁₀₉	1	1	yes	characteristic
290	C₄	C₈×D₄×C₁₀₉	1	1	yes	characteristic
291	C₄	#436×C₁₀₉	1	1	yes	characteristic
292	C₄	#472×C₁₀₉	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	4	no	—
296	C₂	C₁₀₉×#1954	1	1	yes	characteristic
297	C₂	C₁₀₉×#1976	1	1	yes	characteristic
298	C₂	C₁₀₉×#1978	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#121
commutator	C₂×C₄	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#390
prime radical	C₂²×D₈×C₁₀₉	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#1564	C₂	semidirect
3. C₁₀₉×#1478	C₂	semidirect
4. C₁₀₉×#1460	C₂	semidirect
5. C₁₀₉×#1450	C₂	semidirect
29. C₂×D₈×C₁₀₉	C₈	semidirect
143. #9668	C₁₀₉	direct (N × H)
151. #1564	C₂×C₁₀₉	semidirect
152. #1478	C₂×C₁₀₉	semidirect
153. #1460	C₂×C₁₀₉	semidirect
154. #1450	C₂×C₁₀₉	semidirect
155. C₁₀₉	#9668	direct (N × H)
179. C₂×D₈	C₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₁₀₉	2
C₁₀₉×#1564	2
C₁₀₉×#1478	2
C₁₀₉×#1460	2
C₁₀₉×#1450	2
#9668	109

Subgroups — C109×#9668 #42127662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9668 #42127662