Subgroups — C₁₀₉×#11233 #42129227

Order 27904.

694 conjugacy classes of subgroups.

The subgroup lattice has 336 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	C₂²×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#341×C₁₀₉	C₂	1	2	yes	maximal
4	C₂×#341×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1728	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#108×C₁₀₉	—	2	4	no	—
12	C₂×#108×C₁₀₉	—	2	2	no	—
13	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
14	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
15	C₂×C₄⋊C₈×C₁₀₉	C₄	1	4	yes	—
16	#341×C₁₀₉	—	2	4	no	—
17	#341×C₁₀₉	—	2	2	no	—
18	#336×C₁₀₉	—	2	4	no	—
19	C₂³×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
20	C₂³×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
30	C₂×M₁₆×C₁₀₉	—	2	2	no	—
31	C₂×M₁₆×C₁₀₉	—	2	2	no	—
32	C₂×M₁₆×C₁₀₉	D₄	1	4	yes	—
33	C₂×M₁₆×C₁₀₉	Q₈	1	4	yes	—
34	C₂²×C₈×C₁₀₉	—	2	2	no	—
35	C₂²×C₈×C₁₀₉	—	2	2	no	—
36	C₂²×C₈×C₁₀₉	—	2	2	no	—
37	C₂²×C₈×C₁₀₉	—	2	1	no	—
38	C₂²×C₈×C₁₀₉	—	2	1	no	—
39	C₂²×C₈×C₁₀₉	—	2	2	no	—
40	C₂²×C₈×C₁₀₉	—	2	2	no	—
41	C₂²×C₈×C₁₀₉	—	2	2	no	—
42	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
45	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
46	C₂×C₄²×C₁₀₉	—	2	1	no	—
47	C₂×C₄²×C₁₀₉	—	2	1	no	—
48	C₂×C₄²×C₁₀₉	—	2	1	no	—
49	C₂×C₄²×C₁₀₉	—	2	1	no	—
50	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
52	#108×C₁₀₉	—	2	8	no	—
53	C₄⋊C₈×C₁₀₉	—	2	8	no	—
54	C₂⁴×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	—	2	1	no	—
56	C₂²×C₄×C₁₀₉	—	2	1	no	—
57	C₂²×C₄×C₁₀₉	—	2	4	no	—
58	C₂²×C₄×C₁₀₉	—	2	1	no	—
59	C₂²×C₄×C₁₀₉	—	2	1	no	—
60	C₂²×C₄×C₁₀₉	—	4	2	no	—
61	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
72	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
73	M₁₆×C₁₀₉	—	2	4	no	—
74	M₁₆×C₁₀₉	—	2	4	no	—
75	M₁₆×C₁₀₉	—	2	4	no	—
76	M₁₆×C₁₀₉	—	2	4	no	—
77	M₁₆×C₁₀₉	—	4	2	no	—
78	M₁₆×C₁₀₉	—	4	2	no	—
79	M₁₆×C₁₀₉	—	4	2	no	—
80	M₁₆×C₁₀₉	—	4	2	no	—
81	C₂×C₈×C₁₀₉	—	2	4	no	—
82	C₂×C₈×C₁₀₉	—	2	2	no	—
83	C₂×C₈×C₁₀₉	—	2	4	no	—
84	C₂×C₈×C₁₀₉	—	2	2	no	—
85	C₂×C₈×C₁₀₉	—	2	2	no	—
86	C₂×C₈×C₁₀₉	—	2	4	no	—
87	C₂×C₈×C₁₀₉	—	2	4	no	—
88	C₂×C₈×C₁₀₉	—	2	4	no	—
89	C₂×C₈×C₁₀₉	—	4	1	no	—
90	C₂×C₈×C₁₀₉	—	4	2	no	—
91	C₂×C₈×C₁₀₉	—	4	2	no	—
92	C₂×C₈×C₁₀₉	—	4	2	no	—
93	C₂×C₈×C₁₀₉	—	4	2	no	—
94	C₂×C₈×C₁₀₉	—	4	2	no	—
95	C₂×C₈×C₁₀₉	—	4	1	no	—
96	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
97	C₄²×C₁₀₉	—	2	2	no	—
98	C₄²×C₁₀₉	—	2	2	no	—
99	C₄²×C₁₀₉	—	2	2	no	—
100	C₄²×C₁₀₉	—	2	2	no	—
101	C₂³×C₁₀₉	—	2	2	no	—
102	C₂³×C₁₀₉	—	2	2	no	—
103	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂³×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₁₀₉	#125	1	2	yes	—
107	C₂³×C₁₀₉	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₁₀₉	—	2	4	no	—
109	C₂×C₄×C₁₀₉	—	2	4	no	—
110	C₂×C₄×C₁₀₉	—	2	2	no	—
111	C₂×C₄×C₁₀₉	—	2	2	no	—
112	C₂×C₄×C₁₀₉	—	2	2	no	—
113	C₂×C₄×C₁₀₉	—	2	4	no	—
114	C₂×C₄×C₁₀₉	—	4	2	no	—
115	C₂×C₄×C₁₀₉	—	4	2	no	—
116	C₂×C₄×C₁₀₉	#108	1	4	yes	complemented
117	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
122	C₂×C₄×C₁₀₉	#121	1	2	yes	—
123	C₂×C₄×C₁₀₉	#121	1	2	yes	—
124	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
130	C₂×C₄×C₁₀₉	#123	1	2	yes	—
131	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
132	C₂×C₄×C₁₀₉	#125	1	2	yes	—
133	C₂×C₄×C₁₀₉	#126	1	2	yes	—
134	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
135	C₈×C₁₀₉	—	2	4	no	—
136	C₈×C₁₀₉	—	4	2	no	—
137	C₈×C₁₀₉	—	4	4	no	—
138	C₈×C₁₀₉	—	4	4	no	—
139	C₂²×C₁₀₉	—	2	1	no	—
140	C₂²×C₁₀₉	—	2	1	no	—
141	C₂²×C₁₀₉	—	2	1	no	—
142	C₂²×C₁₀₉	—	2	1	no	—
143	C₂²×C₁₀₉	—	2	4	no	—
144	C₂²×C₁₀₉	—	2	4	no	—
145	C₂²×C₁₀₉	#380	1	1	yes	characteristic
146	C₂²×C₁₀₉	#389	1	2	yes	—
147	C₂²×C₁₀₉	#395	1	1	yes	characteristic
148	C₂²×C₁₀₉	#398	1	2	yes	—
149	C₂²×C₁₀₉	#400	1	1	yes	characteristic
150	C₂²×C₁₀₉	C₂×#108	1	2	yes	—
151	C₂²×C₁₀₉	#430	1	2	yes	—
152	C₄×C₁₀₉	—	2	4	no	—
153	C₄×C₁₀₉	—	4	2	no	—
154	C₄×C₁₀₉	#386	1	1	yes	characteristic
155	C₄×C₁₀₉	#387	1	2	yes	—
156	C₄×C₁₀₉	#391	1	1	yes	characteristic
157	C₄×C₁₀₉	#394	1	1	yes	characteristic
158	C₄×C₁₀₉	#396	1	2	yes	—
159	C₄×C₁₀₉	#398	1	1	yes	characteristic
160	#11233	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₁₀₉	—	2	1	no	—
162	C₂×C₁₀₉	—	2	2	no	—
163	C₂×C₁₀₉	—	2	1	no	—
164	C₂×C₁₀₉	#1441	1	1	yes	characteristic
165	C₂×C₁₀₉	#1810	1	1	yes	characteristic
166	C₂×C₁₀₉	#1811	1	1	yes	characteristic
167	C₂×C₁₀₉	#1950	1	4	yes	—
168	C₂²×#108	C₂×C₁₀₉	1	1	yes	characteristic
169	C₂²×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
170	C₂×#341	C₂×C₁₀₉	1	2	yes	—
171	C₂×#341	C₂×C₁₀₉	1	1	yes	characteristic
172	#1728	C₂×C₁₀₉	1	2	yes	—
173	C₁₀₉	#11233	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
175	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
176	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
177	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
178	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
179	C₂×#108	—	2	4	no	—
180	C₂×#108	—	2	2	no	—
181	C₂×C₄⋊C₈	—	2	2	no	—
182	C₂×C₄⋊C₈	—	2	2	no	—
183	C₂×C₄⋊C₈	C₄×C₁₀₉	1	4	yes	—
184	#341	—	2	4	no	—
185	#341	—	2	2	no	—
186	#336	—	2	4	no	—
187	C₂³×C₄	Q₈×C₁₀₉	1	1	yes	characteristic
188	C₂³×C₄	Q₈×C₁₀₉	1	1	yes	characteristic
189	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	D₄×C₁₀₉	1	4	yes	—
201	C₂×M₁₆	Q₈×C₁₀₉	1	4	yes	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
211	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
212	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
213	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
219	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
220	#108	—	2	8	no	—
221	C₄⋊C₈	—	2	8	no	—
222	C₂⁴	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
230	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
237	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
238	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
239	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
240	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
272	C₂³	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
273	C₂³	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
274	C₂³	#125×C₁₀₉	1	2	yes	—
275	C₂³	C₄⋊Q₈×C₁₀₉	1	2	yes	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	#108×C₁₀₉	1	4	yes	complemented
285	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
286	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
287	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
288	C₂×C₄	C₄×Q₈×C₁₀₉	1	2	yes	—
289	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
290	C₂×C₄	#121×C₁₀₉	1	2	yes	—
291	C₂×C₄	#121×C₁₀₉	1	2	yes	—
292	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
293	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
294	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
295	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
296	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
297	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
298	C₂×C₄	#123×C₁₀₉	1	2	yes	—
299	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
300	C₂×C₄	#125×C₁₀₉	1	2	yes	—
301	C₂×C₄	#126×C₁₀₉	1	2	yes	—
302	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
303	C₈	—	2	4	no	—
304	C₈	—	4	2	no	—
305	C₈	—	4	4	no	—
306	C₈	—	4	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	#380×C₁₀₉	1	1	yes	characteristic
314	C₂²	#389×C₁₀₉	1	2	yes	—
315	C₂²	#395×C₁₀₉	1	1	yes	characteristic
316	C₂²	#398×C₁₀₉	1	2	yes	—
317	C₂²	#400×C₁₀₉	1	1	yes	characteristic
318	C₂²	C₂×#108×C₁₀₉	1	2	yes	—
319	C₂²	#430×C₁₀₉	1	2	yes	—
320	C₄	—	2	4	no	—
321	C₄	—	4	2	no	—
322	C₄	#386×C₁₀₉	1	1	yes	characteristic
323	C₄	#387×C₁₀₉	1	2	yes	—
324	C₄	#391×C₁₀₉	1	1	yes	characteristic
325	C₄	#394×C₁₀₉	1	1	yes	characteristic
326	C₄	#396×C₁₀₉	1	2	yes	—
327	C₄	#398×C₁₀₉	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	2	no	—
330	C₂	—	2	1	no	—
331	C₂	C₁₀₉×#1441	1	1	yes	characteristic
332	C₂	C₁₀₉×#1810	1	1	yes	characteristic
333	C₂	C₁₀₉×#1811	1	1	yes	characteristic
334	C₂	C₁₀₉×#1950	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
socle	C₂³×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₀₉	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ C₂²×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
116. C₂×C₄×C₁₀₉	#108	semidirect
160. #11233	C₁₀₉	direct (N × H)
173. C₁₀₉	#11233	direct (N × H)
284. C₂×C₄	#108×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
C₂²×C₄⋊C₈×C₁₀₉	2
C₂×#341×C₁₀₉	2
C₁₀₉×#1728	2
#11233	109

Subgroups — C109×#11233 #42129227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11233 #42129227