Subgroups — C₁₀₉×#11229 #42129223

Order 27904.

654 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1740	C₂	1	2	yes	maximal
4	C₁₀₉×#1735	C₂	1	1	yes	maximal, characteristic
5	C₂×H(3,ℤ/4)×C₁₀₉	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
9	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
10	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
11	#427×C₁₀₉	—	2	4	no	—
12	C₂×#107×C₁₀₉	—	2	1	no	—
13	C₂×#106×C₁₀₉	—	2	1	no	—
14	#424×C₁₀₉	—	2	2	no	—
15	#424×C₁₀₉	—	2	1	no	—
16	#424×C₁₀₉	—	2	1	no	—
17	C₂×C₄⋊C₈×C₁₀₉	C₄	1	4	yes	complemented
18	#342×C₁₀₉	—	2	4	no	—
19	#340×C₁₀₉	—	2	2	no	—
20	H(3,ℤ/4)×C₁₀₉	—	2	4	no	—
21	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₀₉	—	2	2	no	—
23	C₂×M₁₆×C₁₀₉	—	2	1	no	—
24	C₂×M₁₆×C₁₀₉	—	2	1	no	—
25	C₂×M₁₆×C₁₀₉	—	2	2	no	—
26	C₂²×C₈×C₁₀₉	—	2	2	no	—
27	C₂²×C₈×C₁₀₉	—	2	2	no	—
28	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
31	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
33	#117×C₁₀₉	—	2	1	no	—
34	#117×C₁₀₉	—	2	1	no	—
35	#117×C₁₀₉	—	2	1	no	—
36	#117×C₁₀₉	—	2	1	no	—
37	#117×C₁₀₉	—	2	2	no	—
38	#117×C₁₀₉	—	2	2	no	—
39	#117×C₁₀₉	—	2	2	no	—
40	#117×C₁₀₉	—	2	2	no	—
41	#117×C₁₀₉	—	2	2	no	—
42	#117×C₁₀₉	—	2	2	no	—
43	#117×C₁₀₉	D₄	1	4	yes	—
44	#117×C₁₀₉	Q₈	1	4	yes	—
45	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₀₉	Q₈	1	2	yes	—
48	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
51	C₂×C₄²×C₁₀₉	—	2	2	no	—
52	C₂×C₄²×C₁₀₉	—	2	2	no	—
53	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
54	#107×C₁₀₉	—	2	4	no	—
55	#106×C₁₀₉	—	2	4	no	—
56	C₄⋊C₈×C₁₀₉	—	2	4	no	—
57	C₄⋊C₈×C₁₀₉	—	2	4	no	—
58	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	—	4	2	no	—
60	C₂²×C₄×C₁₀₉	—	4	2	no	—
61	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
71	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
72	M₁₆×C₁₀₉	—	4	1	no	—
73	M₁₆×C₁₀₉	—	4	2	no	—
74	M₁₆×C₁₀₉	—	4	1	no	—
75	C₂×C₈×C₁₀₉	—	2	4	no	—
76	C₂×C₈×C₁₀₉	—	2	1	no	—
77	C₂×C₈×C₁₀₉	—	2	1	no	—
78	C₂×C₈×C₁₀₉	—	2	4	no	—
79	C₂×C₈×C₁₀₉	—	2	1	no	—
80	C₂×C₈×C₁₀₉	—	2	1	no	—
81	C₂×C₈×C₁₀₉	—	2	2	no	—
82	C₂×C₈×C₁₀₉	—	2	4	no	—
83	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
85	C₄⋊C₄×C₁₀₉	—	2	2	no	—
86	C₄⋊C₄×C₁₀₉	—	2	2	no	—
87	C₄⋊C₄×C₁₀₉	—	2	4	no	—
88	C₄⋊C₄×C₁₀₉	—	2	4	no	—
89	C₄⋊C₄×C₁₀₉	—	4	2	no	—
90	C₄⋊C₄×C₁₀₉	—	4	2	no	—
91	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
92	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
93	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
94	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
95	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
96	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
97	C₄²×C₁₀₉	—	2	2	no	—
98	C₄²×C₁₀₉	—	2	2	no	—
99	C₄²×C₁₀₉	—	2	4	no	—
100	C₄²×C₁₀₉	—	2	4	no	—
101	C₄²×C₁₀₉	—	4	2	no	—
102	C₄²×C₁₀₉	—	4	2	no	—
103	C₂³×C₁₀₉	—	4	2	no	—
104	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₁₀₉	#121	1	1	yes	characteristic
106	C₂³×C₁₀₉	#121	1	1	yes	characteristic
107	C₂³×C₁₀₉	#121	1	2	yes	—
108	C₂³×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₁₀₉	—	2	4	no	—
110	C₂×C₄×C₁₀₉	—	4	4	no	—
111	C₂×C₄×C₁₀₉	—	4	4	no	—
112	C₂×C₄×C₁₀₉	—	4	4	no	—
113	C₂×C₄×C₁₀₉	—	8	4	no	—
114	C₂×C₄×C₁₀₉	—	8	4	no	—
115	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
120	C₂×C₄×C₁₀₉	#121	1	2	yes	—
121	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	#123	1	2	yes	—
127	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
128	C₂×C₄×C₁₀₉	#125	1	2	yes	—
129	C₂×C₄×C₁₀₉	#125	1	2	yes	—
130	C₂×C₄×C₁₀₉	#126	1	2	yes	—
131	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
133	C₈×C₁₀₉	—	2	2	no	—
134	C₈×C₁₀₉	—	2	2	no	—
135	C₈×C₁₀₉	—	4	2	no	—
136	C₂²×C₁₀₉	—	2	2	no	—
137	C₂²×C₁₀₉	—	2	2	no	—
138	C₂²×C₁₀₉	—	2	2	no	—
139	C₂²×C₁₀₉	—	4	2	no	—
140	C₂²×C₁₀₉	—	4	2	no	—
141	C₂²×C₁₀₉	#380	1	1	yes	characteristic
142	C₂²×C₁₀₉	#389	1	2	yes	—
143	C₂²×C₁₀₉	#394	1	1	yes	characteristic
144	C₂²×C₁₀₉	#396	1	2	yes	—
145	C₂²×C₁₀₉	#398	1	1	yes	characteristic
146	C₄×C₁₀₉	—	4	4	no	—
147	C₄×C₁₀₉	—	8	4	no	—
148	C₄×C₁₀₉	—	8	4	no	—
149	C₄×C₁₀₉	#386	1	1	yes	characteristic
150	C₄×C₁₀₉	#387	1	2	yes	—
151	C₄×C₁₀₉	#391	1	1	yes	characteristic
152	C₄×C₁₀₉	#395	1	1	yes	characteristic
153	C₄×C₁₀₉	#398	1	2	yes	—
154	C₄×C₁₀₉	#400	1	1	yes	characteristic
155	#11229	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₁₀₉	—	2	1	no	—
157	C₂×C₁₀₉	—	2	2	no	—
158	C₂×C₁₀₉	—	2	2	no	—
159	C₂×C₁₀₉	—	2	1	no	—
160	C₂×C₁₀₉	#1441	1	1	yes	characteristic
161	C₂×C₁₀₉	#1812	1	2	yes	—
162	C₂×#427	C₂×C₁₀₉	1	1	yes	characteristic
163	C₂×#424	C₂×C₁₀₉	1	1	yes	characteristic
164	#1740	C₂×C₁₀₉	1	2	yes	—
165	#1735	C₂×C₁₀₉	1	1	yes	characteristic
166	C₂×H(3,ℤ/4)	C₂×C₁₀₉	1	2	yes	—
167	C₁₀₉	#11229	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
169	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
170	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
171	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
172	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
173	#427	—	2	4	no	—
174	C₂×#107	—	2	1	no	—
175	C₂×#106	—	2	1	no	—
176	#424	—	2	2	no	—
177	#424	—	2	1	no	—
178	#424	—	2	1	no	—
179	C₂×C₄⋊C₈	C₄×C₁₀₉	1	4	yes	complemented
180	#342	—	2	4	no	—
181	#340	—	2	2	no	—
182	H(3,ℤ/4)	—	2	4	no	—
183	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
191	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
192	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
193	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
194	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
195	#117	—	2	1	no	—
196	#117	—	2	1	no	—
197	#117	—	2	1	no	—
198	#117	—	2	1	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	—	2	2	no	—
205	#117	D₄×C₁₀₉	1	4	yes	—
206	#117	Q₈×C₁₀₉	1	4	yes	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	Q₈×C₁₀₉	1	2	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
216	#107	—	2	4	no	—
217	#106	—	2	4	no	—
218	C₄⋊C₈	—	2	4	no	—
219	C₄⋊C₈	—	2	4	no	—
220	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
224	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
228	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
229	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
230	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
231	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
232	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
233	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	1	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
246	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	2	4	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
267	C₂³	#121×C₁₀₉	1	1	yes	characteristic
268	C₂³	#121×C₁₀₉	1	1	yes	characteristic
269	C₂³	#121×C₁₀₉	1	2	yes	—
270	C₂³	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
278	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
279	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
280	C₂×C₄	C₄×Q₈×C₁₀₉	1	2	yes	—
281	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
282	C₂×C₄	#121×C₁₀₉	1	2	yes	—
283	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
284	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
285	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
286	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
287	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
288	C₂×C₄	#123×C₁₀₉	1	2	yes	—
289	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
290	C₂×C₄	#125×C₁₀₉	1	2	yes	—
291	C₂×C₄	#125×C₁₀₉	1	2	yes	—
292	C₂×C₄	#126×C₁₀₉	1	2	yes	—
293	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
294	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	4	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#380×C₁₀₉	1	1	yes	characteristic
304	C₂²	#389×C₁₀₉	1	2	yes	—
305	C₂²	#394×C₁₀₉	1	1	yes	characteristic
306	C₂²	#396×C₁₀₉	1	2	yes	—
307	C₂²	#398×C₁₀₉	1	1	yes	characteristic
308	C₄	—	4	4	no	—
309	C₄	—	8	4	no	—
310	C₄	—	8	4	no	—
311	C₄	#386×C₁₀₉	1	1	yes	characteristic
312	C₄	#387×C₁₀₉	1	2	yes	—
313	C₄	#391×C₁₀₉	1	1	yes	characteristic
314	C₄	#395×C₁₀₉	1	1	yes	characteristic
315	C₄	#398×C₁₀₉	1	2	yes	—
316	C₄	#400×C₁₀₉	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	C₁₀₉×#1441	1	1	yes	characteristic
322	C₂	C₁₀₉×#1812	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#117
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#380
prime radical	C₂⁴×C₁₀₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂²×M₁₆×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₂×C₄⋊C₈×C₁₀₉	C₄	semidirect
155. #11229	C₁₀₉	direct (N × H)
167. C₁₀₉	#11229	direct (N × H)
179. C₂×C₄⋊C₈	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×#427×C₁₀₉	2
C₂×#424×C₁₀₉	2
C₁₀₉×#1740	2
C₁₀₉×#1735	2
C₂×H(3,ℤ/4)×C₁₀₉	2
#11229	109

Subgroups — C109×#11229 #42129223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11229 #42129223