Subgroups — C₁₀₉×#11240 #42129234

Order 27904.

638 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#344×C₁₀₉	C₂	1	2	yes	maximal
4	C₁₀₉×#1735	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1733	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	2	yes	—
10	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#108×C₁₀₉	—	2	4	no	—
12	C₂×#108×C₁₀₉	—	2	2	no	—
13	#424×C₁₀₉	—	2	2	no	—
14	#424×C₁₀₉	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
16	#344×C₁₀₉	—	2	4	no	—
17	#340×C₁₀₉	—	2	2	no	—
18	#339×C₁₀₉	—	2	4	no	—
19	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₀₉	—	2	1	no	—
21	C₂×M₁₆×C₁₀₉	—	2	2	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	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₁₀₉	D₄	1	4	yes	—
30	C₂²×C₈×C₁₀₉	—	2	2	no	—
31	C₂²×C₈×C₁₀₉	—	2	2	no	—
32	C₂²×C₈×C₁₀₉	—	2	1	no	—
33	C₂²×C₈×C₁₀₉	—	2	2	no	—
34	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
36	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
37	#117×C₁₀₉	—	2	1	no	—
38	#117×C₁₀₉	—	2	1	no	—
39	#117×C₁₀₉	—	2	2	no	—
40	#117×C₁₀₉	—	2	2	no	—
41	#117×C₁₀₉	—	2	2	no	—
42	#117×C₁₀₉	C₂×C₄	1	2	yes	—
43	#117×C₁₀₉	D₄	1	2	yes	—
44	#117×C₁₀₉	Q₈	1	4	yes	—
45	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₀₉	Q₈	1	2	yes	—
47	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
49	C₂×C₄²×C₁₀₉	—	2	1	no	—
50	C₂×C₄²×C₁₀₉	—	2	2	no	—
51	#108×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	—	4	2	no	—
56	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
64	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
65	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
66	M₁₆×C₁₀₉	—	2	4	no	—
67	M₁₆×C₁₀₉	—	2	4	no	—
68	M₁₆×C₁₀₉	—	2	4	no	—
69	M₁₆×C₁₀₉	—	4	2	no	—
70	M₁₆×C₁₀₉	—	4	2	no	—
71	M₁₆×C₁₀₉	—	4	2	no	—
72	C₂×C₈×C₁₀₉	—	2	4	no	—
73	C₂×C₈×C₁₀₉	—	2	2	no	—
74	C₂×C₈×C₁₀₉	—	2	4	no	—
75	C₂×C₈×C₁₀₉	—	2	2	no	—
76	C₂×C₈×C₁₀₉	—	2	2	no	—
77	C₂×C₈×C₁₀₉	—	2	4	no	—
78	C₂×C₈×C₁₀₉	—	4	1	no	—
79	C₂×C₈×C₁₀₉	—	4	2	no	—
80	C₂×C₈×C₁₀₉	—	4	1	no	—
81	C₂×C₈×C₁₀₉	—	4	2	no	—
82	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
83	C₄⋊C₄×C₁₀₉	—	2	2	no	—
84	C₄⋊C₄×C₁₀₉	—	2	4	no	—
85	C₄⋊C₄×C₁₀₉	—	4	2	no	—
86	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
87	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
88	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
89	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
90	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
91	C₄²×C₁₀₉	—	2	2	no	—
92	C₄²×C₁₀₉	—	2	2	no	—
93	C₄²×C₁₀₉	—	2	4	no	—
94	C₄²×C₁₀₉	—	4	2	no	—
95	C₂³×C₁₀₉	—	4	2	no	—
96	C₂³×C₁₀₉	#117	1	1	yes	characteristic
97	C₂³×C₁₀₉	#121	1	1	yes	characteristic
98	C₂³×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
100	C₂³×C₁₀₉	#126	1	2	yes	—
101	C₂×C₄×C₁₀₉	—	2	4	no	—
102	C₂×C₄×C₁₀₉	—	4	4	no	—
103	C₂×C₄×C₁₀₉	—	4	4	no	—
104	C₂×C₄×C₁₀₉	—	8	4	no	—
105	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
109	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	#121	1	2	yes	—
111	C₂×C₄×C₁₀₉	#121	1	2	yes	—
112	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	#123	1	2	yes	—
118	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	#125	1	2	yes	—
120	C₂×C₄×C₁₀₉	#125	1	2	yes	—
121	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
123	C₈×C₁₀₉	—	2	4	no	—
124	C₈×C₁₀₉	—	4	2	no	—
125	C₈×C₁₀₉	—	4	4	no	—
126	C₂²×C₁₀₉	—	2	2	no	—
127	C₂²×C₁₀₉	—	2	2	no	—
128	C₂²×C₁₀₉	—	2	2	no	—
129	C₂²×C₁₀₉	—	4	2	no	—
130	C₂²×C₁₀₉	—	4	2	no	—
131	C₂²×C₁₀₉	#380	1	1	yes	characteristic
132	C₂²×C₁₀₉	#387	1	2	yes	—
133	C₂²×C₁₀₉	#395	1	1	yes	characteristic
134	C₂²×C₁₀₉	#396	1	2	yes	—
135	C₂²×C₁₀₉	#400	1	1	yes	characteristic
136	C₄×C₁₀₉	—	4	4	no	—
137	C₄×C₁₀₉	—	8	4	no	—
138	C₄×C₁₀₉	#386	1	1	yes	characteristic
139	C₄×C₁₀₉	#389	1	2	yes	—
140	C₄×C₁₀₉	#391	1	1	yes	characteristic
141	C₄×C₁₀₉	#394	1	1	yes	characteristic
142	C₄×C₁₀₉	#398	1	2	yes	—
143	C₄×C₁₀₉	#398	1	1	yes	characteristic
144	#11240	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₁₀₉	—	2	2	no	—
146	C₂×C₁₀₉	—	2	2	no	—
147	C₂×C₁₀₉	—	2	1	no	—
148	C₂×C₁₀₉	—	2	1	no	—
149	C₂×C₁₀₉	#1441	1	1	yes	characteristic
150	C₂×C₁₀₉	#1814	1	2	yes	—
151	C₂²×#108	C₂×C₁₀₉	1	1	yes	characteristic
152	C₂×#424	C₂×C₁₀₉	1	1	yes	characteristic
153	C₂×#344	C₂×C₁₀₉	1	2	yes	—
154	#1735	C₂×C₁₀₉	1	1	yes	characteristic
155	#1733	C₂×C₁₀₉	1	2	yes	—
156	C₁₀₉	#11240	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
158	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
159	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
160	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
161	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
162	C₂×#108	—	2	4	no	—
163	C₂×#108	—	2	2	no	—
164	#424	—	2	2	no	—
165	#424	C₄×C₁₀₉	1	4	yes	complemented
166	C₂×C₄⋊C₈	—	2	2	no	—
167	#344	—	2	4	no	—
168	#340	—	2	2	no	—
169	#339	—	2	4	no	—
170	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
180	C₂×M₁₆	D₄×C₁₀₉	1	4	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
186	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
187	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	—	2	1	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	C₂×C₄×C₁₀₉	1	2	yes	—
194	#117	D₄×C₁₀₉	1	2	yes	—
195	#117	Q₈×C₁₀₉	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	Q₈×C₁₀₉	1	2	yes	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	2	no	—
202	#108	—	2	8	no	—
203	C₄⋊C₈	—	2	8	no	—
204	C₂⁴	#41×C₁₀₉	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
212	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
214	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
215	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
216	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	4	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	#117×C₁₀₉	1	1	yes	characteristic
248	C₂³	#121×C₁₀₉	1	1	yes	characteristic
249	C₂³	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
250	C₂³	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
251	C₂³	#126×C₁₀₉	1	2	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
258	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
259	C₂×C₄	C₄×Q₈×C₁₀₉	1	2	yes	—
260	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
261	C₂×C₄	#121×C₁₀₉	1	2	yes	—
262	C₂×C₄	#121×C₁₀₉	1	2	yes	—
263	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
264	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
265	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
266	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
267	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
268	C₂×C₄	#123×C₁₀₉	1	2	yes	—
269	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
270	C₂×C₄	#125×C₁₀₉	1	2	yes	—
271	C₂×C₄	#125×C₁₀₉	1	2	yes	—
272	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
273	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#380×C₁₀₉	1	1	yes	characteristic
283	C₂²	#387×C₁₀₉	1	2	yes	—
284	C₂²	#395×C₁₀₉	1	1	yes	characteristic
285	C₂²	#396×C₁₀₉	1	2	yes	—
286	C₂²	#400×C₁₀₉	1	1	yes	characteristic
287	C₄	—	4	4	no	—
288	C₄	—	8	4	no	—
289	C₄	#386×C₁₀₉	1	1	yes	characteristic
290	C₄	#389×C₁₀₉	1	2	yes	—
291	C₄	#391×C₁₀₉	1	1	yes	characteristic
292	C₄	#394×C₁₀₉	1	1	yes	characteristic
293	C₄	#398×C₁₀₉	1	2	yes	—
294	C₄	#398×C₁₀₉	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	C₁₀₉×#1441	1	1	yes	characteristic
300	C₂	C₁₀₉×#1814	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#121
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#380
prime radical	C₂⁴×C₁₀₉	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
144. #11240	C₁₀₉	direct (N × H)
156. C₁₀₉	#11240	direct (N × H)
165. #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×#108×C₁₀₉	2
C₂×#424×C₁₀₉	2
C₂×#344×C₁₀₉	2
C₁₀₉×#1735	2
C₁₀₉×#1733	2
#11240	109

Subgroups — C109×#11240 #42129234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11240 #42129234