Subgroups — C₁₀₉×#11238 #42129232

Order 27904.

682 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#343×C₁₀₉	C₂	1	2	yes	maximal
4	C₂×#341×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1736	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
11	C₂×#106×C₁₀₉	—	2	2	no	—
12	C₂×#106×C₁₀₉	—	2	4	no	—
13	#424×C₁₀₉	—	2	2	no	—
14	#424×C₁₀₉	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
16	#343×C₁₀₉	—	2	4	no	—
17	#341×C₁₀₉	—	2	2	no	—
18	#340×C₁₀₉	—	2	4	no	—
19	C₂³×C₄×C₁₀₉	—	2	2	no	—
20	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₀₉	—	2	1	no	—
22	C₂×M₁₆×C₁₀₉	—	2	2	no	—
23	C₂×M₁₆×C₁₀₉	—	2	2	no	—
24	C₂×M₁₆×C₁₀₉	—	2	2	no	—
25	C₂×M₁₆×C₁₀₉	—	2	2	no	—
26	C₂×M₁₆×C₁₀₉	—	2	2	no	—
27	C₂×M₁₆×C₁₀₉	—	2	2	no	—
28	C₂×M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₁₀₉	Q₈	1	4	yes	—
31	C₂²×C₈×C₁₀₉	—	2	2	no	—
32	C₂²×C₈×C₁₀₉	—	2	2	no	—
33	C₂²×C₈×C₁₀₉	—	2	2	no	—
34	C₂²×C₈×C₁₀₉	—	2	1	no	—
35	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
37	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
38	#117×C₁₀₉	—	2	1	no	—
39	#117×C₁₀₉	—	2	1	no	—
40	#117×C₁₀₉	C₂×C₄	1	2	yes	—
41	#117×C₁₀₉	Q₈	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₀₉	D₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
50	C₂×C₄²×C₁₀₉	—	2	1	no	—
51	#106×C₁₀₉	—	2	8	no	—
52	C₄⋊C₈×C₁₀₉	—	2	8	no	—
53	C₂⁴×C₁₀₉	#41	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	—	2	2	no	—
55	C₂²×C₄×C₁₀₉	—	2	4	no	—
56	C₂²×C₄×C₁₀₉	—	4	2	no	—
57	C₂²×C₄×C₁₀₉	—	4	2	no	—
58	C₂²×C₄×C₁₀₉	—	8	4	no	—
59	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
67	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
68	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
69	M₁₆×C₁₀₉	—	2	4	no	—
70	M₁₆×C₁₀₉	—	2	4	no	—
71	M₁₆×C₁₀₉	—	2	4	no	—
72	M₁₆×C₁₀₉	—	4	2	no	—
73	M₁₆×C₁₀₉	—	4	2	no	—
74	M₁₆×C₁₀₉	—	4	2	no	—
75	C₂×C₈×C₁₀₉	—	2	4	no	—
76	C₂×C₈×C₁₀₉	—	2	2	no	—
77	C₂×C₈×C₁₀₉	—	2	4	no	—
78	C₂×C₈×C₁₀₉	—	2	2	no	—
79	C₂×C₈×C₁₀₉	—	2	2	no	—
80	C₂×C₈×C₁₀₉	—	2	4	no	—
81	C₂×C₈×C₁₀₉	—	4	1	no	—
82	C₂×C₈×C₁₀₉	—	4	2	no	—
83	C₂×C₈×C₁₀₉	—	4	2	no	—
84	C₂×C₈×C₁₀₉	—	4	1	no	—
85	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
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	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
92	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
93	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
94	C₄²×C₁₀₉	—	2	2	no	—
95	C₄²×C₁₀₉	—	2	2	no	—
96	C₂³×C₁₀₉	—	2	4	no	—
97	C₂³×C₁₀₉	#117	1	1	yes	characteristic
98	C₂³×C₁₀₉	#121	1	1	yes	characteristic
99	C₂³×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₁₀₉	#123	1	2	yes	—
101	C₂³×C₁₀₉	#125	1	2	yes	—
102	C₂×C₄×C₁₀₉	—	2	4	no	—
103	C₂×C₄×C₁₀₉	—	4	4	no	—
104	C₂×C₄×C₁₀₉	—	4	4	no	—
105	C₂×C₄×C₁₀₉	—	8	4	no	—
106	C₂×C₄×C₁₀₉	—	8	4	no	—
107	C₂×C₄×C₁₀₉	—	8	4	no	—
108	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	#121	1	2	yes	—
114	C₂×C₄×C₁₀₉	#121	1	2	yes	—
115	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
121	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₀₉	#125	1	2	yes	—
123	C₂×C₄×C₁₀₉	#126	1	2	yes	—
124	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
126	C₈×C₁₀₉	—	2	4	no	—
127	C₈×C₁₀₉	—	4	2	no	—
128	C₈×C₁₀₉	—	4	4	no	—
129	C₂²×C₁₀₉	—	2	2	no	—
130	C₂²×C₁₀₉	—	2	2	no	—
131	C₂²×C₁₀₉	—	2	4	no	—
132	C₂²×C₁₀₉	—	2	2	no	—
133	C₂²×C₁₀₉	—	2	2	no	—
134	C₂²×C₁₀₉	#380	1	1	yes	characteristic
135	C₂²×C₁₀₉	#387	1	2	yes	—
136	C₂²×C₁₀₉	#394	1	1	yes	characteristic
137	C₂²×C₁₀₉	#398	1	2	yes	—
138	C₂²×C₁₀₉	#398	1	1	yes	characteristic
139	C₂²×C₁₀₉	#471	1	4	yes	—
140	C₄×C₁₀₉	—	4	4	no	—
141	C₄×C₁₀₉	—	8	4	no	—
142	C₄×C₁₀₉	#386	1	1	yes	characteristic
143	C₄×C₁₀₉	#389	1	2	yes	—
144	C₄×C₁₀₉	#391	1	1	yes	characteristic
145	C₄×C₁₀₉	#395	1	1	yes	characteristic
146	C₄×C₁₀₉	#396	1	2	yes	—
147	C₄×C₁₀₉	#400	1	1	yes	characteristic
148	#11238	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂×C₁₀₉	—	2	2	no	—
150	C₂×C₁₀₉	—	2	2	no	—
151	C₂×C₁₀₉	#1441	1	1	yes	characteristic
152	C₂×C₁₀₉	#1815	1	2	yes	—
153	C₂×C₁₀₉	#2046	1	2	yes	—
154	C₂×C₁₀₉	#2090	1	2	yes	—
155	C₂²×#106	C₂×C₁₀₉	1	1	yes	characteristic
156	C₂×#424	C₂×C₁₀₉	1	1	yes	characteristic
157	C₂×#343	C₂×C₁₀₉	1	2	yes	—
158	C₂×#341	C₂×C₁₀₉	1	1	yes	characteristic
159	#1736	C₂×C₁₀₉	1	2	yes	—
160	C₁₀₉	#11238	1	1	yes	characteristic, direct product factor, Sylow
161	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
162	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
163	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
164	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
165	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
166	C₂×#106	—	2	2	no	—
167	C₂×#106	—	2	4	no	—
168	#424	—	2	2	no	—
169	#424	C₄×C₁₀₉	1	4	yes	complemented
170	C₂×C₄⋊C₈	—	2	2	no	—
171	#343	—	2	4	no	—
172	#341	—	2	2	no	—
173	#340	—	2	4	no	—
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
185	C₂×M₁₆	Q₈×C₁₀₉	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
191	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
192	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
193	#117	—	2	1	no	—
194	#117	—	2	1	no	—
195	#117	C₂×C₄×C₁₀₉	1	2	yes	—
196	#117	Q₈×C₁₀₉	1	2	yes	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	D₄×C₁₀₉	1	4	yes	—
203	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
204	C₂×C₂²⋊C₄	—	2	1	no	—
205	C₂×C₄²	—	2	1	no	—
206	#106	—	2	8	no	—
207	C₄⋊C₈	—	2	8	no	—
208	C₂⁴	#41×C₁₀₉	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	8	4	no	—
214	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
219	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
221	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
222	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
223	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	#117×C₁₀₉	1	1	yes	characteristic
253	C₂³	#121×C₁₀₉	1	1	yes	characteristic
254	C₂³	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
255	C₂³	#123×C₁₀₉	1	2	yes	—
256	C₂³	#125×C₁₀₉	1	2	yes	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
266	C₂×C₄	C₄×Q₈×C₁₀₉	1	2	yes	—
267	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
268	C₂×C₄	#121×C₁₀₉	1	2	yes	—
269	C₂×C₄	#121×C₁₀₉	1	2	yes	—
270	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
271	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
272	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
273	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
274	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
275	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
276	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
277	C₂×C₄	#125×C₁₀₉	1	2	yes	—
278	C₂×C₄	#126×C₁₀₉	1	2	yes	—
279	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
280	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	#380×C₁₀₉	1	1	yes	characteristic
290	C₂²	#387×C₁₀₉	1	2	yes	—
291	C₂²	#394×C₁₀₉	1	1	yes	characteristic
292	C₂²	#398×C₁₀₉	1	2	yes	—
293	C₂²	#398×C₁₀₉	1	1	yes	characteristic
294	C₂²	#471×C₁₀₉	1	4	yes	—
295	C₄	—	4	4	no	—
296	C₄	—	8	4	no	—
297	C₄	#386×C₁₀₉	1	1	yes	characteristic
298	C₄	#389×C₁₀₉	1	2	yes	—
299	C₄	#391×C₁₀₉	1	1	yes	characteristic
300	C₄	#395×C₁₀₉	1	1	yes	characteristic
301	C₄	#396×C₁₀₉	1	2	yes	—
302	C₄	#400×C₁₀₉	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	C₁₀₉×#1441	1	1	yes	characteristic
306	C₂	C₁₀₉×#1815	1	2	yes	—
307	C₂	C₁₀₉×#2046	1	2	yes	—
308	C₂	C₁₀₉×#2090	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	#121
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
prime radical	C₂⁴×C₁₀₉	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
14. #424×C₁₀₉	C₄	semidirect
148. #11238	C₁₀₉	direct (N × H)
160. C₁₀₉	#11238	direct (N × H)
169. #424	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#106×C₁₀₉	2
C₂×#424×C₁₀₉	2
C₂×#343×C₁₀₉	2
C₂×#341×C₁₀₉	2
C₁₀₉×#1736	2
#11238	109

Subgroups — C109×#11238 #42129232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11238 #42129232