Subgroups — C₁₀₉×#7669 #42125663

Order 27904.

384 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2863	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1831	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1744	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1272	C₂	1	2	yes	maximal
5	C₁₀₉×#1271	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₁₀₉	—	2	1	no	—
7	C₂×C₄×C₈×C₁₀₉	—	2	1	no	—
8	#391×C₁₀₉	C₄	1	1	yes	characteristic
9	#391×C₁₀₉	C₄	1	1	yes	characteristic
10	#391×C₁₀₉	C₂²	1	1	yes	characteristic
11	#389×C₁₀₉	C₄	1	1	yes	characteristic
12	#389×C₁₀₉	C₄	1	1	yes	characteristic
13	#389×C₁₀₉	C₂²	1	1	yes	characteristic
14	#387×C₁₀₉	—	2	1	no	—
15	#387×C₁₀₉	—	2	1	no	—
16	#387×C₁₀₉	—	2	1	no	—
17	#383×C₁₀₉	—	2	1	no	—
18	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
19	#336×C₁₀₉	—	2	1	no	—
20	#336×C₁₀₉	—	2	1	no	—
21	#336×C₁₀₉	C₂²	1	2	yes	—
22	#336×C₁₀₉	C₂²	1	2	yes	—
23	C₂²×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₉	—	2	2	no	—
25	C₂²×C₈×C₁₀₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₁₀₉	—	4	2	no	—
40	C₄×C₈×C₁₀₉	—	2	4	no	—
41	#95×C₁₀₉	—	2	1	no	—
42	#95×C₁₀₉	—	2	1	no	—
43	#95×C₁₀₉	—	2	1	no	—
44	#95×C₁₀₉	—	2	1	no	—
45	#95×C₁₀₉	—	2	1	no	—
46	#95×C₁₀₉	—	2	1	no	—
47	#95×C₁₀₉	—	2	1	no	—
48	#95×C₁₀₉	—	2	1	no	—
49	#95×C₁₀₉	—	2	1	no	—
50	C₂×Q₈×C₁₀₉	—	2	2	no	—
51	C₂×Q₈×C₁₀₉	—	4	1	no	—
52	C₂×Q₈×C₁₀₉	—	4	1	no	—
53	C₂²×C₄×C₁₀₉	—	2	1	no	—
54	C₂²×C₄×C₁₀₉	—	2	1	no	—
55	C₂²×C₄×C₁₀₉	—	2	1	no	—
56	C₂²×C₄×C₁₀₉	—	2	1	no	—
57	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₀₉	—	2	4	no	—
65	C₂×C₈×C₁₀₉	—	4	2	no	—
66	C₂×C₈×C₁₀₉	—	4	2	no	—
67	C₂×C₈×C₁₀₉	—	4	2	no	—
68	C₂×C₈×C₁₀₉	—	4	2	no	—
69	C₂×C₈×C₁₀₉	—	4	2	no	—
70	C₄⋊C₄×C₁₀₉	—	2	2	no	—
71	C₄⋊C₄×C₁₀₉	—	4	1	no	—
72	C₄⋊C₄×C₁₀₉	—	4	1	no	—
73	C₄⋊C₄×C₁₀₉	—	4	2	no	—
74	C₄⋊C₄×C₁₀₉	—	4	2	no	—
75	C₄⋊C₄×C₁₀₉	—	4	2	no	—
76	C₄²×C₁₀₉	—	2	2	no	—
77	C₄²×C₁₀₉	—	2	2	no	—
78	C₄²×C₁₀₉	—	2	2	no	—
79	C₄²×C₁₀₉	—	4	1	no	—
80	C₄²×C₁₀₉	—	4	1	no	—
81	C₄²×C₁₀₉	—	4	1	no	—
82	C₄²×C₁₀₉	SD₁₆	1	2	yes	—
83	C₄²×C₁₀₉	Q₁₆	1	2	yes	—
84	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₁₀₉	—	8	2	no	—
86	C₂×C₄×C₁₀₉	—	2	1	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	—	2	1	no	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	2	1	no	—
91	C₂×C₄×C₁₀₉	—	2	1	no	—
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	—	2	1	no	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	—	2	1	no	—
96	C₂×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₄×C₁₀₉	—	2	1	no	—
99	C₂×C₄×C₁₀₉	—	2	1	no	—
100	C₂×C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄×C₁₀₉	—	4	1	no	—
102	C₂×C₄×C₁₀₉	—	4	1	no	—
103	C₂×C₄×C₁₀₉	—	4	1	no	—
104	C₂×C₄×C₁₀₉	—	4	1	no	—
105	C₂×C₄×C₁₀₉	—	4	1	no	—
106	C₂×C₄×C₁₀₉	—	4	1	no	—
107	C₂×C₄×C₁₀₉	—	4	1	no	—
108	C₂×C₄×C₁₀₉	—	4	1	no	—
109	C₂×C₄×C₁₀₉	—	4	1	no	—
110	C₂×C₄×C₁₀₉	—	4	1	no	—
111	C₂×C₄×C₁₀₉	—	4	1	no	—
112	C₂×C₄×C₁₀₉	—	4	1	no	—
113	C₂×C₄×C₁₀₉	#103	1	2	yes	—
114	C₂×C₄×C₁₀₉	#103	1	2	yes	—
115	C₂×C₄×C₁₀₉	#104	1	2	yes	—
116	C₂×C₄×C₁₀₉	#131	1	2	yes	—
117	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	#135	1	2	yes	—
120	C₈×C₁₀₉	—	4	4	no	—
121	C₈×C₁₀₉	—	8	2	no	—
122	C₂²×C₁₀₉	#408	1	1	yes	characteristic
123	C₂²×C₁₀₉	#410	1	1	yes	characteristic
124	C₂²×C₁₀₉	#413	1	1	yes	characteristic
125	C₂²×C₁₀₉	C₂×#103	1	1	yes	characteristic
126	C₂²×C₁₀₉	#416	1	1	yes	characteristic
127	C₂²×C₁₀₉	C₂×#104	1	1	yes	characteristic
128	C₂²×C₁₀₉	#421	1	1	yes	characteristic
129	C₄×C₁₀₉	—	2	2	no	—
130	C₄×C₁₀₉	—	2	2	no	—
131	C₄×C₁₀₉	—	2	2	no	—
132	C₄×C₁₀₉	—	2	2	no	—
133	C₄×C₁₀₉	—	2	1	no	—
134	C₄×C₁₀₉	—	2	1	no	—
135	C₄×C₁₀₉	—	2	2	no	—
136	C₄×C₁₀₉	—	4	1	no	—
137	C₄×C₁₀₉	—	8	1	no	—
138	C₄×C₁₀₉	—	8	1	no	—
139	C₄×C₁₀₉	—	8	1	no	—
140	C₄×C₁₀₉	—	8	1	no	—
141	#7669	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₁₀₉	#1473	1	1	yes	characteristic
143	C₂×C₁₀₉	#1482	1	1	yes	characteristic
144	C₂×C₁₀₉	#1493	1	1	yes	characteristic
145	C₂×C₁₀₉	#1500	1	1	yes	characteristic
146	C₂×C₁₀₉	#1518	1	1	yes	characteristic
147	C₂×C₁₀₉	#1532	1	1	yes	characteristic
148	C₂×C₁₀₉	#1542	1	1	yes	characteristic
149	#2863	C₂×C₁₀₉	1	1	yes	characteristic
150	#1831	C₂×C₁₀₉	1	1	yes	characteristic
151	#1744	C₂×C₁₀₉	1	1	yes	characteristic
152	#1272	C₂×C₁₀₉	1	2	yes	—
153	#1271	C₂×C₁₀₉	1	2	yes	—
154	C₁₀₉	#7669	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×C₄⋊C₈	—	2	1	no	—
156	C₂×C₄×C₈	—	2	1	no	—
157	#391	C₄×C₁₀₉	1	1	yes	characteristic
158	#391	C₄×C₁₀₉	1	1	yes	characteristic
159	#391	C₂²×C₁₀₉	1	1	yes	characteristic
160	#389	C₄×C₁₀₉	1	1	yes	characteristic
161	#389	C₄×C₁₀₉	1	1	yes	characteristic
162	#389	C₂²×C₁₀₉	1	1	yes	characteristic
163	#387	—	2	1	no	—
164	#387	—	2	1	no	—
165	#387	—	2	1	no	—
166	#383	—	2	1	no	—
167	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
168	#336	—	2	1	no	—
169	#336	—	2	1	no	—
170	#336	C₂²×C₁₀₉	1	2	yes	—
171	#336	C₂²×C₁₀₉	1	2	yes	—
172	C₂²×Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
179	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
182	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
184	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
185	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
186	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
188	C₄⋊C₈	—	4	2	no	—
189	C₄×C₈	—	2	4	no	—
190	#95	—	2	1	no	—
191	#95	—	2	1	no	—
192	#95	—	2	1	no	—
193	#95	—	2	1	no	—
194	#95	—	2	1	no	—
195	#95	—	2	1	no	—
196	#95	—	2	1	no	—
197	#95	—	2	1	no	—
198	#95	—	2	1	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	4	1	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₄²	—	4	1	no	—
231	C₄²	SD₁₆×C₁₀₉	1	2	yes	—
232	C₄²	Q₁₆×C₁₀₉	1	2	yes	—
233	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
234	Q₈	—	8	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	#103×C₁₀₉	1	2	yes	—
263	C₂×C₄	#103×C₁₀₉	1	2	yes	—
264	C₂×C₄	#104×C₁₀₉	1	2	yes	—
265	C₂×C₄	#131×C₁₀₉	1	2	yes	—
266	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
267	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
268	C₂×C₄	#135×C₁₀₉	1	2	yes	—
269	C₈	—	4	4	no	—
270	C₈	—	8	2	no	—
271	C₂²	#408×C₁₀₉	1	1	yes	characteristic
272	C₂²	#410×C₁₀₉	1	1	yes	characteristic
273	C₂²	#413×C₁₀₉	1	1	yes	characteristic
274	C₂²	C₂×#103×C₁₀₉	1	1	yes	characteristic
275	C₂²	#416×C₁₀₉	1	1	yes	characteristic
276	C₂²	C₂×#104×C₁₀₉	1	1	yes	characteristic
277	C₂²	#421×C₁₀₉	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	1	no	—
286	C₄	—	8	1	no	—
287	C₄	—	8	1	no	—
288	C₄	—	8	1	no	—
289	C₄	—	8	1	no	—
290	C₂	C₁₀₉×#1473	1	1	yes	characteristic
291	C₂	C₁₀₉×#1482	1	1	yes	characteristic
292	C₂	C₁₀₉×#1493	1	1	yes	characteristic
293	C₂	C₁₀₉×#1500	1	1	yes	characteristic
294	C₂	C₁₀₉×#1518	1	1	yes	characteristic
295	C₂	C₁₀₉×#1532	1	1	yes	characteristic
296	C₂	C₁₀₉×#1542	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₄³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
141. #7669	C₁₀₉	direct (N × H)
154. C₁₀₉	#7669	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#2863	2
C₁₀₉×#1831	2
C₁₀₉×#1744	2
C₁₀₉×#1272	2
C₁₀₉×#1271	2
#7669	109

Subgroups — C109×#7669 #42125663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#7669 #42125663