Subgroups — C₁₀₉×#7667 #42125661

Order 27904.

568 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2857	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1797	C₂	1	1	yes	maximal, characteristic
3	C₂×#336×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1272	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₁₀₉	—	2	1	no	—
6	C₂×C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×C₂²⋊Q₈×C₁₀₉	C₄	1	2	yes	—
8	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₁₀₉	—	2	1	no	—
10	C₂×C₂²⋊C₈×C₁₀₉	—	2	1	no	—
11	#391×C₁₀₉	C₂²	1	2	yes	—
12	#389×C₁₀₉	—	2	1	no	—
13	#389×C₁₀₉	—	2	1	no	—
14	#384×C₁₀₉	C₄	1	1	yes	characteristic
15	#380×C₁₀₉	—	2	1	no	—
16	#380×C₁₀₉	—	2	1	no	—
17	#336×C₁₀₉	—	2	1	no	—
18	#336×C₁₀₉	C₂²	1	2	yes	—
19	#336×C₁₀₉	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₉	—	2	4	no	—
25	C₂²×C₈×C₁₀₉	—	2	2	no	—
26	C₂²×C₈×C₁₀₉	—	2	2	no	—
27	C₂²×C₈×C₁₀₉	—	4	2	no	—
28	C₂²×C₈×C₁₀₉	—	4	2	no	—
29	C₄⋊Q₈×C₁₀₉	—	4	2	no	—
30	C₂²⋊Q₈×C₁₀₉	—	4	4	no	—
31	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
38	C₂×C₄²×C₁₀₉	—	4	2	no	—
39	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₁₀₉	—	4	2	no	—
44	C₂²⋊C₈×C₁₀₉	—	4	2	no	—
45	#95×C₁₀₉	—	2	2	no	—
46	#95×C₁₀₉	—	2	2	no	—
47	C₂⁴×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Q₈×C₁₀₉	—	2	4	no	—
49	C₂×Q₈×C₁₀₉	—	4	2	no	—
50	C₂×Q₈×C₁₀₉	—	4	2	no	—
51	C₂²×C₄×C₁₀₉	—	2	2	no	—
52	C₂²×C₄×C₁₀₉	—	2	2	no	—
53	C₂²×C₄×C₁₀₉	—	2	2	no	—
54	C₂²×C₄×C₁₀₉	—	2	2	no	—
55	C₂²×C₄×C₁₀₉	—	2	2	no	—
56	C₂²×C₄×C₁₀₉	—	2	2	no	—
57	C₂²×C₄×C₁₀₉	—	4	1	no	—
58	C₂²×C₄×C₁₀₉	—	4	1	no	—
59	C₂²×C₄×C₁₀₉	—	4	1	no	—
60	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₁₀₉	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₁₀₉	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₁₀₉	Q₁₆	1	2	yes	—
67	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₀₉	—	2	4	no	—
71	C₂×C₈×C₁₀₉	—	4	2	no	—
72	C₂×C₈×C₁₀₉	—	4	8	no	—
73	C₂×C₈×C₁₀₉	—	4	2	no	—
74	C₂×C₈×C₁₀₉	—	4	2	no	—
75	C₂×C₈×C₁₀₉	—	4	2	no	—
76	C₂×C₈×C₁₀₉	—	4	2	no	—
77	C₄⋊C₄×C₁₀₉	—	4	2	no	—
78	C₄⋊C₄×C₁₀₉	—	4	2	no	—
79	C₄⋊C₄×C₁₀₉	—	4	4	no	—
80	C₄⋊C₄×C₁₀₉	—	4	2	no	—
81	C₄⋊C₄×C₁₀₉	—	4	2	no	—
82	C₄⋊C₄×C₁₀₉	—	4	2	no	—
83	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
84	C₄²×C₁₀₉	—	2	2	no	—
85	C₄²×C₁₀₉	—	2	2	no	—
86	C₄²×C₁₀₉	—	4	1	no	—
87	C₄²×C₁₀₉	—	4	1	no	—
88	C₂³×C₁₀₉	—	2	1	no	—
89	C₂³×C₁₀₉	—	2	1	no	—
90	C₂³×C₁₀₉	—	2	1	no	—
91	C₂³×C₁₀₉	—	2	1	no	—
92	C₂³×C₁₀₉	#100	1	2	yes	—
93	C₂³×C₁₀₉	#103	1	2	yes	—
94	C₂³×C₁₀₉	#103	1	2	yes	—
95	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	Q₈×C₁₀₉	—	8	4	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	—	2	1	no	—
102	C₂×C₄×C₁₀₉	—	2	1	no	—
103	C₂×C₄×C₁₀₉	—	2	1	no	—
104	C₂×C₄×C₁₀₉	—	2	1	no	—
105	C₂×C₄×C₁₀₉	—	2	4	no	—
106	C₂×C₄×C₁₀₉	—	2	4	no	—
107	C₂×C₄×C₁₀₉	—	2	1	no	—
108	C₂×C₄×C₁₀₉	—	2	1	no	—
109	C₂×C₄×C₁₀₉	—	2	1	no	—
110	C₂×C₄×C₁₀₉	—	4	2	no	—
111	C₂×C₄×C₁₀₉	—	4	2	no	—
112	C₂×C₄×C₁₀₉	—	4	4	no	—
113	C₂×C₄×C₁₀₉	—	4	2	no	—
114	C₂×C₄×C₁₀₉	—	4	2	no	—
115	C₂×C₄×C₁₀₉	—	4	2	no	—
116	C₂×C₄×C₁₀₉	—	4	2	no	—
117	C₂×C₄×C₁₀₉	—	4	2	no	—
118	C₂×C₄×C₁₀₉	—	4	2	no	—
119	C₂×C₄×C₁₀₉	#99	1	2	yes	—
120	C₂×C₄×C₁₀₉	#103	1	2	yes	—
121	C₂×C₄×C₁₀₉	#103	1	2	yes	—
122	C₂×C₄×C₁₀₉	#104	1	2	yes	—
123	C₂×C₄×C₁₀₉	#131	1	2	yes	—
124	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
128	C₈×C₁₀₉	—	4	4	no	—
129	C₈×C₁₀₉	—	8	2	no	—
130	C₂²×C₁₀₉	—	2	2	no	—
131	C₂²×C₁₀₉	—	2	2	no	—
132	C₂²×C₁₀₉	—	2	2	no	—
133	C₂²×C₁₀₉	—	2	2	no	—
134	C₂²×C₁₀₉	—	2	2	no	—
135	C₂²×C₁₀₉	—	2	2	no	—
136	C₂²×C₁₀₉	#333	1	4	yes	—
137	C₂²×C₁₀₉	#408	1	1	yes	characteristic
138	C₂²×C₁₀₉	C₂×#99	1	1	yes	characteristic
139	C₂²×C₁₀₉	C₂×#100	1	1	yes	characteristic
140	C₂²×C₁₀₉	C₂×#103	1	1	yes	characteristic
141	C₂²×C₁₀₉	C₂×#103	1	1	yes	characteristic
142	C₂²×C₁₀₉	C₂×#104	1	1	yes	characteristic
143	C₂²×C₁₀₉	#421	1	1	yes	characteristic
144	C₄×C₁₀₉	—	2	2	no	—
145	C₄×C₁₀₉	—	2	1	no	—
146	C₄×C₁₀₉	—	2	1	no	—
147	C₄×C₁₀₉	—	2	1	no	—
148	C₄×C₁₀₉	—	2	2	no	—
149	C₄×C₁₀₉	—	2	1	no	—
150	C₄×C₁₀₉	—	4	1	no	—
151	C₄×C₁₀₉	—	8	2	no	—
152	C₄×C₁₀₉	—	8	2	no	—
153	C₄×C₁₀₉	#328	1	4	yes	—
154	#7667	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂×C₁₀₉	—	2	4	no	—
156	C₂×C₁₀₉	#1471	1	1	yes	characteristic
157	C₂×C₁₀₉	#1481	1	1	yes	characteristic
158	C₂×C₁₀₉	#1493	1	1	yes	characteristic
159	C₂×C₁₀₉	C₂×#328	1	1	yes	characteristic
160	C₂×C₁₀₉	#1520	1	1	yes	characteristic
161	C₂×C₁₀₉	#1530	1	1	yes	characteristic
162	C₂×C₁₀₉	C₂×#333	1	1	yes	characteristic
163	#2857	C₂×C₁₀₉	1	1	yes	characteristic
164	#1797	C₂×C₁₀₉	1	1	yes	characteristic
165	C₂×#336	C₂×C₁₀₉	1	1	yes	characteristic
166	#1272	C₂×C₁₀₉	1	4	yes	complemented
167	C₁₀₉	#7667	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³×C₈	—	2	1	no	—
169	C₂×C₄⋊Q₈	C₄×C₁₀₉	1	1	yes	characteristic
170	C₂×C₂²⋊Q₈	C₄×C₁₀₉	1	2	yes	—
171	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
172	C₂×C₄⋊C₈	—	2	1	no	—
173	C₂×C₂²⋊C₈	—	2	1	no	—
174	#391	C₂²×C₁₀₉	1	2	yes	—
175	#389	—	2	1	no	—
176	#389	—	2	1	no	—
177	#384	C₄×C₁₀₉	1	1	yes	characteristic
178	#380	—	2	1	no	—
179	#380	—	2	1	no	—
180	#336	—	2	1	no	—
181	#336	C₂²×C₁₀₉	1	2	yes	—
182	#336	C₂²×C₁₀₉	1	2	yes	—
183	C₂²×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
184	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
185	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
186	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	4	2	no	—
191	C₂²×C₈	—	4	2	no	—
192	C₄⋊Q₈	—	4	2	no	—
193	C₂²⋊Q₈	—	4	4	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₄²	—	4	2	no	—
202	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
203	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
204	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
206	C₄⋊C₈	—	4	2	no	—
207	C₂²⋊C₈	—	4	2	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	C₂⁴	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
226	C₂²×C₄	SD₁₆×C₁₀₉	1	2	yes	—
227	C₂²×C₄	SD₁₆×C₁₀₉	1	2	yes	—
228	C₂²×C₄	Q₁₆×C₁₀₉	1	2	yes	—
229	C₂²×C₄	Q₁₆×C₁₀₉	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	8	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	4	1	no	—
250	C₄²	—	4	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	#100×C₁₀₉	1	2	yes	—
256	C₂³	#103×C₁₀₉	1	2	yes	—
257	C₂³	#103×C₁₀₉	1	2	yes	—
258	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
259	Q₈	—	8	4	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	#99×C₁₀₉	1	2	yes	—
283	C₂×C₄	#103×C₁₀₉	1	2	yes	—
284	C₂×C₄	#103×C₁₀₉	1	2	yes	—
285	C₂×C₄	#104×C₁₀₉	1	2	yes	—
286	C₂×C₄	#131×C₁₀₉	1	2	yes	—
287	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
288	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
289	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
290	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
291	C₈	—	4	4	no	—
292	C₈	—	8	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	#333×C₁₀₉	1	4	yes	—
300	C₂²	#408×C₁₀₉	1	1	yes	characteristic
301	C₂²	C₂×#99×C₁₀₉	1	1	yes	characteristic
302	C₂²	C₂×#100×C₁₀₉	1	1	yes	characteristic
303	C₂²	C₂×#103×C₁₀₉	1	1	yes	characteristic
304	C₂²	C₂×#103×C₁₀₉	1	1	yes	characteristic
305	C₂²	C₂×#104×C₁₀₉	1	1	yes	characteristic
306	C₂²	#421×C₁₀₉	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	#328×C₁₀₉	1	4	yes	—
317	C₂	—	2	4	no	—
318	C₂	C₁₀₉×#1471	1	1	yes	characteristic
319	C₂	C₁₀₉×#1481	1	1	yes	characteristic
320	C₂	C₁₀₉×#1493	1	1	yes	characteristic
321	C₂	C₂×#328×C₁₀₉	1	1	yes	characteristic
322	C₂	C₁₀₉×#1520	1	1	yes	characteristic
323	C₂	C₁₀₉×#1530	1	1	yes	characteristic
324	C₂	C₂×#333×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂⁴×C₁₀₉	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂²×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#1272	C₂	semidirect
154. #7667	C₁₀₉	direct (N × H)
166. #1272	C₂×C₁₀₉	semidirect
167. C₁₀₉	#7667	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×#2857	2
C₁₀₉×#1797	2
C₂×#336×C₁₀₉	2
C₁₀₉×#1272	2
#7667	109

Subgroups — C109×#7667 #42125661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#7667 #42125661