Subgroups — C₁₀₉×#12792 #42130786

Order 27904.

440 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#364×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#2117	C₂	1	2	yes	maximal
5	C₁₀₉×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×#131×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₄×#41×C₁₀₉	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
10	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
11	#405×C₁₀₉	—	2	1	no	—
12	#405×C₁₀₉	—	2	1	no	—
13	#405×C₁₀₉	C₄	1	1	yes	characteristic
14	#405×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₁₀₉	—	2	1	no	—
16	C₄×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₀₉	—	2	1	no	—
18	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	#364×C₁₀₉	—	2	1	no	—
20	#364×C₁₀₉	—	2	1	no	—
21	#364×C₁₀₉	—	2	1	no	—
22	#364×C₁₀₉	—	2	1	no	—
23	#350×C₁₀₉	—	2	4	no	—
24	C₂×#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	#131×C₁₀₉	—	2	2	no	—
26	#131×C₁₀₉	—	2	1	no	—
27	#131×C₁₀₉	—	2	1	no	—
28	#131×C₁₀₉	C₈	1	4	yes	—
29	#131×C₁₀₉	D₄	1	2	yes	—
30	#131×C₁₀₉	Q₈	1	2	yes	—
31	C₂×M₁₆×C₁₀₉	—	2	1	no	—
32	C₂×M₁₆×C₁₀₉	—	2	1	no	—
33	C₂×M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₀₉	—	2	1	no	—
36	C₂²×C₈×C₁₀₉	—	2	1	no	—
37	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₁₀₉	—	2	1	no	—
40	C₄×D₄×C₁₀₉	—	2	1	no	—
41	C₄×D₄×C₁₀₉	—	2	2	no	—
42	#117×C₁₀₉	—	2	1	no	—
43	#117×C₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₀₉	—	2	1	no	—
45	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
46	M₃₂×C₁₀₉	—	2	2	no	—
47	M₃₂×C₁₀₉	—	2	2	no	—
48	M₃₂×C₁₀₉	—	2	2	no	—
49	M₃₂×C₁₀₉	—	2	2	no	—
50	C₂×C₁₆×C₁₀₉	—	2	2	no	—
51	C₂×C₁₆×C₁₀₉	—	2	2	no	—
52	#97×C₁₀₉	—	2	1	no	—
53	#97×C₁₀₉	—	2	1	no	—
54	#97×C₁₀₉	—	2	1	no	—
55	C₄×C₈×C₁₀₉	—	2	1	no	—
56	C₄×C₈×C₁₀₉	—	2	1	no	—
57	C₄×C₈×C₁₀₉	—	2	1	no	—
58	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
59	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
62	#41×C₁₀₉	—	2	1	no	—
63	#41×C₁₀₉	—	2	1	no	—
64	#41×C₁₀₉	C₄⋊C₄	1	2	yes	—
65	#41×C₁₀₉	C₂×C₈	1	2	yes	—
66	C₂×Q₈×C₁₀₉	M₁₆	1	1	yes	characteristic
67	C₂×D₄×C₁₀₉	—	2	1	no	—
68	C₂×D₄×C₁₀₉	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	—	2	1	no	—
70	C₂²×C₄×C₁₀₉	—	2	2	no	—
71	C₂²×C₄×C₁₀₉	—	2	1	no	—
72	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
73	M₁₆×C₁₀₉	—	2	1	no	—
74	M₁₆×C₁₀₉	—	2	1	no	—
75	M₁₆×C₁₀₉	—	2	4	no	—
76	M₁₆×C₁₀₉	—	2	2	no	—
77	M₁₆×C₁₀₉	—	2	2	no	—
78	M₁₆×C₁₀₉	C₄⋊C₄	1	2	yes	—
79	M₁₆×C₁₀₉	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₁₀₉	—	2	1	no	—
81	C₂×C₈×C₁₀₉	—	2	1	no	—
82	C₂×C₈×C₁₀₉	—	2	4	no	—
83	C₂×C₈×C₁₀₉	—	2	2	no	—
84	C₂×C₈×C₁₀₉	—	2	2	no	—
85	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
88	C₂×C₈×C₁₀₉	M₁₆	1	2	yes	—
89	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
94	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
95	C₄⋊C₄×C₁₀₉	—	2	2	no	—
96	C₄⋊C₄×C₁₀₉	—	2	1	no	—
97	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
98	C₂²⋊C₄×C₁₀₉	M₁₆	1	2	yes	—
99	C₄²×C₁₀₉	—	2	2	no	—
100	C₄²×C₁₀₉	—	2	1	no	—
101	C₄²×C₁₀₉	—	2	1	no	—
102	C₄²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
104	C₁₆×C₁₀₉	—	4	2	no	—
105	C₁₆×C₁₀₉	—	4	2	no	—
106	C₂³×C₁₀₉	—	2	1	no	—
107	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
108	Q₈×C₁₀₉	—	2	1	no	—
109	Q₈×C₁₀₉	C₄⋊C₈	1	2	yes	—
110	D₄×C₁₀₉	—	2	2	no	—
111	D₄×C₁₀₉	—	2	2	no	—
112	D₄×C₁₀₉	—	2	1	no	—
113	D₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	2	no	—
117	C₂×C₄×C₁₀₉	—	2	2	no	—
118	C₂×C₄×C₁₀₉	—	2	2	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	4	2	no	—
121	C₂×C₄×C₁₀₉	—	4	4	no	—
122	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
128	C₈×C₁₀₉	—	2	2	no	—
129	C₈×C₁₀₉	—	2	2	no	—
130	C₈×C₁₀₉	C₂²⋊C₈	1	2	yes	—
131	C₈×C₁₀₉	C₂²⋊C₈	1	2	yes	—
132	C₈×C₁₀₉	#121	1	1	yes	characteristic
133	C₈×C₁₀₉	#121	1	1	yes	characteristic
134	C₈×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
135	C₈×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂²×C₁₀₉	—	2	2	no	—
137	C₂²×C₁₀₉	—	4	1	no	—
138	C₂²×C₁₀₉	—	4	2	no	—
139	C₂²×C₁₀₉	#407	1	1	yes	characteristic
140	C₂²×C₁₀₉	#423	1	1	yes	characteristic
141	C₂²×C₁₀₉	#431	1	1	yes	characteristic
142	C₄×C₁₀₉	—	2	2	no	—
143	C₄×C₁₀₉	—	2	2	no	—
144	C₄×C₁₀₉	—	4	2	no	—
145	C₄×C₁₀₉	—	4	1	no	—
146	C₄×C₁₀₉	#380	1	1	yes	characteristic
147	C₄×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
148	C₄×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
149	C₄×C₁₀₉	#432	1	1	yes	characteristic
150	#12792	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₁₀₉	—	2	1	no	—
152	C₂×C₁₀₉	—	2	1	no	—
153	C₂×C₁₀₉	—	2	1	no	—
154	C₂×C₁₀₉	—	4	2	no	—
155	C₂×C₁₀₉	#1797	1	1	yes	characteristic
156	C₄×#131	C₂×C₁₀₉	1	1	yes	characteristic
157	C₂×#364	C₂×C₁₀₉	1	1	yes	characteristic, complemented
158	C₂×#364	C₂×C₁₀₉	1	1	yes	characteristic, complemented
159	#2117	C₂×C₁₀₉	1	2	yes	—
160	#1378	C₂×C₁₀₉	1	2	yes	complemented
161	C₁₀₉	#12792	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×#131	C₄×C₁₀₉	1	1	yes	characteristic
163	C₂×#131	C₂²×C₁₀₉	1	1	yes	characteristic
164	C₄×#41	C₄×C₁₀₉	1	1	yes	characteristic
165	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
166	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
167	#405	—	2	1	no	—
168	#405	—	2	1	no	—
169	#405	C₄×C₁₀₉	1	1	yes	characteristic
170	#405	C₂²×C₁₀₉	1	1	yes	characteristic
171	C₄×M₁₆	—	2	1	no	—
172	C₄×M₁₆	C₄×C₁₀₉	1	1	yes	characteristic
173	C₂×C₄×C₈	—	2	1	no	—
174	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
175	#364	—	2	1	no	—
176	#364	—	2	1	no	—
177	#364	—	2	1	no	—
178	#364	—	2	1	no	—
179	#350	—	2	4	no	—
180	C₂×#41	C₂×C₄×C₁₀₉	1	1	yes	characteristic
181	#131	—	2	2	no	—
182	#131	—	2	1	no	—
183	#131	—	2	1	no	—
184	#131	C₈×C₁₀₉	1	4	yes	—
185	#131	D₄×C₁₀₉	1	2	yes	—
186	#131	Q₈×C₁₀₉	1	2	yes	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
190	C₂×M₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
195	C₄×Q₈	—	2	1	no	—
196	C₄×D₄	—	2	1	no	—
197	C₄×D₄	—	2	2	no	—
198	#117	—	2	1	no	—
199	#117	C₂×C₄×C₁₀₉	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
202	M₃₂	—	2	2	no	—
203	M₃₂	—	2	2	no	—
204	M₃₂	—	2	2	no	—
205	M₃₂	—	2	2	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	2	2	no	—
208	#97	—	2	1	no	—
209	#97	—	2	1	no	—
210	#97	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
215	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
216	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
217	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
218	#41	—	2	1	no	—
219	#41	—	2	1	no	—
220	#41	C₄⋊C₄×C₁₀₉	1	2	yes	—
221	#41	C₂×C₈×C₁₀₉	1	2	yes	—
222	C₂×Q₈	M₁₆×C₁₀₉	1	1	yes	characteristic
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	M₁₆×C₁₀₉	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
229	M₁₆	—	2	1	no	—
230	M₁₆	—	2	1	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	C₄⋊C₄×C₁₀₉	1	2	yes	—
235	M₁₆	C₂×C₈×C₁₀₉	1	2	yes	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
242	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
243	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
244	C₂×C₈	M₁₆×C₁₀₉	1	2	yes	—
245	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
247	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
248	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
249	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
250	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	M₁₆×C₁₀₉	1	2	yes	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
259	C₄²	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
260	C₁₆	—	4	2	no	—
261	C₁₆	—	4	2	no	—
262	C₂³	—	2	1	no	—
263	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	C₄⋊C₈×C₁₀₉	1	2	yes	—
266	D₄	—	2	2	no	—
267	D₄	—	2	2	no	—
268	D₄	—	2	1	no	—
269	D₄	C₄⋊C₈×C₁₀₉	1	2	yes	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
280	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
282	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
283	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	C₂²⋊C₈×C₁₀₉	1	2	yes	—
287	C₈	C₂²⋊C₈×C₁₀₉	1	2	yes	—
288	C₈	#121×C₁₀₉	1	1	yes	characteristic
289	C₈	#121×C₁₀₉	1	1	yes	characteristic
290	C₈	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
291	C₈	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	#407×C₁₀₉	1	1	yes	characteristic
296	C₂²	#423×C₁₀₉	1	1	yes	characteristic
297	C₂²	#431×C₁₀₉	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	1	no	—
302	C₄	#380×C₁₀₉	1	1	yes	characteristic
303	C₄	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
304	C₄	C₂×C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
305	C₄	#432×C₁₀₉	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	C₁₀₉×#1797	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₀₉	#121
commutator	C₂×C₄	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#1797
prime radical	C₂×#41×C₁₀₉	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#364×C₁₀₉	C₂	semidirect
3. C₂×#364×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1378	C₂	semidirect
150. #12792	C₁₀₉	direct (N × H)
157. C₂×#364	C₂×C₁₀₉	semidirect
158. C₂×#364	C₂×C₁₀₉	semidirect
160. #1378	C₂×C₁₀₉	semidirect
161. C₁₀₉	#12792	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#131×C₁₀₉	2
C₂×#364×C₁₀₉	2
C₁₀₉×#2117	2
C₁₀₉×#1378	2
#12792	109

Subgroups — C109×#12792 #42130786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12792 #42130786