Subgroups — C₁₀₉×#22719 #42140713

Order 27904.

790 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×M₁₆×C₁₀₉	C₂	1	2	yes	maximal
2	C₂×#436×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1950	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#1855	C₂	1	4	yes	maximal, complemented
5	C₁₀₉×#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₉×#1811	C₂	1	2	yes	maximal, complemented
7	C₄×#108×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1738	C₂	1	2	yes	maximal, complemented
9	C₂²×M₁₆×C₁₀₉	C₂²	1	4	yes	—
10	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
12	#436×C₁₀₉	—	2	2	no	—
13	#436×C₁₀₉	C₄	1	4	yes	—
14	#436×C₁₀₉	C₄	1	4	yes	—
15	#435×C₁₀₉	—	2	4	no	—
16	C₈×D₄×C₁₀₉	—	2	2	no	—
17	C₂×#108×C₁₀₉	C₂²	1	2	yes	—
18	C₂×#108×C₁₀₉	C₂²	1	4	yes	—
19	#423×C₁₀₉	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	#407×C₁₀₉	C₂²	1	4	yes	—
22	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
23	C₄×M₁₆×C₁₀₉	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	#341×C₁₀₉	C₂²	1	4	yes	—
26	#341×C₁₀₉	C₂²	1	1	yes	characteristic
27	#341×C₁₀₉	C₂²	1	2	yes	—
28	#341×C₁₀₉	C₂²	1	1	yes	characteristic
29	#341×C₁₀₉	C₂²	1	2	yes	—
30	C₂²×D₄×C₁₀₉	Q₈	1	1	yes	characteristic
31	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
32	C₂×M₁₆×C₁₀₉	—	2	4	no	—
33	C₂×M₁₆×C₁₀₉	—	2	8	no	—
34	C₂×M₁₆×C₁₀₉	—	4	4	no	—
35	C₂×M₁₆×C₁₀₉	—	4	4	no	—
36	C₂×M₁₆×C₁₀₉	C₂×C₄	1	8	yes	—
37	C₂×M₁₆×C₁₀₉	C₂³	1	4	yes	—
38	C₂×M₁₆×C₁₀₉	C₂³	1	2	yes	—
39	C₂×M₁₆×C₁₀₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₀₉	—	2	4	no	—
41	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₄×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
45	C₄×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₁₀₉	D₄	1	2	yes	—
47	C₄×D₄×C₁₀₉	D₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₀₉	Q₈	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₁₀₉	Q₈	1	2	yes	—
50	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
51	#108×C₁₀₉	—	2	4	no	—
52	#108×C₁₀₉	—	4	4	no	—
53	C₄⋊C₈×C₁₀₉	—	2	2	no	—
54	C₄⋊C₈×C₁₀₉	—	2	2	no	—
55	C₄⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
56	C₄⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
58	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
59	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
60	#97×C₁₀₉	—	2	2	no	—
61	C₄×C₈×C₁₀₉	—	2	2	no	—
62	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
64	C₂⁴×C₁₀₉	C₂×Q₈	1	2	yes	—
65	C₂×D₄×C₁₀₉	—	4	2	no	—
66	C₂×D₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
67	C₂×D₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₀₉	—	2	4	no	—
69	C₂²×C₄×C₁₀₉	—	2	4	no	—
70	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
76	M₁₆×C₁₀₉	—	2	2	no	—
77	M₁₆×C₁₀₉	—	2	2	no	—
78	M₁₆×C₁₀₉	—	2	4	no	—
79	M₁₆×C₁₀₉	—	2	2	no	—
80	M₁₆×C₁₀₉	—	2	4	no	—
81	M₁₆×C₁₀₉	—	4	8	no	—
82	M₁₆×C₁₀₉	—	4	8	no	—
83	M₁₆×C₁₀₉	#41	1	4	yes	—
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	2	no	—
88	C₂×C₈×C₁₀₉	—	2	2	no	—
89	C₂×C₈×C₁₀₉	—	2	4	no	—
90	C₂×C₈×C₁₀₉	—	4	8	no	—
91	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
94	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
95	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
96	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
97	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
98	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
102	C₂³×C₁₀₉	—	2	4	no	—
103	C₂³×C₁₀₉	—	4	2	no	—
104	C₂³×C₁₀₉	—	4	2	no	—
105	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
106	C₂³×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
107	D₄×C₁₀₉	—	2	4	no	—
108	D₄×C₁₀₉	—	4	2	no	—
109	C₂×C₄×C₁₀₉	—	2	4	no	—
110	C₂×C₄×C₁₀₉	—	2	4	no	—
111	C₂×C₄×C₁₀₉	—	2	4	no	—
112	C₂×C₄×C₁₀₉	—	2	4	no	—
113	C₂×C₄×C₁₀₉	—	2	4	no	—
114	C₂×C₄×C₁₀₉	—	2	2	no	—
115	C₂×C₄×C₁₀₉	—	4	1	no	—
116	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
124	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
125	C₈×C₁₀₉	—	2	2	no	—
126	C₈×C₁₀₉	—	2	4	no	—
127	C₈×C₁₀₉	—	2	2	no	—
128	C₈×C₁₀₉	—	2	4	no	—
129	C₈×C₁₀₉	—	4	4	no	—
130	C₈×C₁₀₉	—	4	2	no	—
131	C₂²×C₁₀₉	—	2	1	no	—
132	C₂²×C₁₀₉	—	2	1	no	—
133	C₂²×C₁₀₉	—	2	4	no	—
134	C₂²×C₁₀₉	—	2	4	no	—
135	C₂²×C₁₀₉	—	2	4	no	—
136	C₂²×C₁₀₉	—	8	4	no	—
137	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₁₀₉	#554	1	2	yes	—
139	C₄×C₁₀₉	—	2	4	no	—
140	C₄×C₁₀₉	—	2	1	no	—
141	C₄×C₁₀₉	—	2	1	no	—
142	C₄×C₁₀₉	—	2	1	no	—
143	C₄×C₁₀₉	#430	1	2	yes	—
144	C₄×C₁₀₉	C₄×#41	1	1	yes	characteristic
145	C₄×C₁₀₉	#518	1	1	yes	characteristic
146	C₄×C₁₀₉	#546	1	2	yes	—
147	#22719	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂×C₁₀₉	—	2	2	no	—
149	C₂×C₁₀₉	—	4	4	no	—
150	C₂×C₁₀₉	#2349	1	1	yes	characteristic
151	C₂×C₁₀₉	C₂×#430	1	1	yes	characteristic
152	C₂×C₁₀₉	#2916	1	1	yes	characteristic
153	D₄×M₁₆	C₂×C₁₀₉	1	2	yes	—
154	C₂×#436	C₂×C₁₀₉	1	1	yes	characteristic
155	#1950	C₂×C₁₀₉	1	2	yes	complemented
156	#1855	C₂×C₁₀₉	1	4	yes	complemented
157	#1837	C₂×C₁₀₉	1	1	yes	characteristic, complemented
158	#1811	C₂×C₁₀₉	1	2	yes	complemented
159	C₄×#108	C₂×C₁₀₉	1	1	yes	characteristic, complemented
160	#1738	C₂×C₁₀₉	1	2	yes	complemented
161	C₁₀₉	#22719	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×M₁₆	C₂²×C₁₀₉	1	4	yes	—
163	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
164	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
165	#436	—	2	2	no	—
166	#436	C₄×C₁₀₉	1	4	yes	—
167	#436	C₄×C₁₀₉	1	4	yes	—
168	#435	—	2	4	no	—
169	C₈×D₄	—	2	2	no	—
170	C₂×#108	C₂²×C₁₀₉	1	2	yes	—
171	C₂×#108	C₂²×C₁₀₉	1	4	yes	—
172	#423	C₂²×C₁₀₉	1	2	yes	—
173	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
174	#407	C₂²×C₁₀₉	1	4	yes	—
175	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
176	C₄×M₁₆	C₂²×C₁₀₉	1	2	yes	—
177	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
178	#341	C₂²×C₁₀₉	1	1	yes	characteristic
179	#341	C₂²×C₁₀₉	1	2	yes	—
180	#341	C₂²×C₁₀₉	1	2	yes	—
181	#341	C₂²×C₁₀₉	1	4	yes	—
182	#341	C₂²×C₁₀₉	1	1	yes	characteristic
183	C₂²×D₄	Q₈×C₁₀₉	1	1	yes	characteristic
184	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
185	C₂×M₁₆	—	2	4	no	—
186	C₂×M₁₆	—	2	8	no	—
187	C₂×M₁₆	—	4	4	no	—
188	C₂×M₁₆	—	4	4	no	—
189	C₂×M₁₆	C₂×C₄×C₁₀₉	1	8	yes	—
190	C₂×M₁₆	C₂³×C₁₀₉	1	4	yes	—
191	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
192	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
196	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
197	C₄×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
198	C₄×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
199	C₄×D₄	D₄×C₁₀₉	1	2	yes	—
200	C₄×D₄	D₄×C₁₀₉	1	2	yes	—
201	C₂×C₄⋊C₄	Q₈×C₁₀₉	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	Q₈×C₁₀₉	1	2	yes	—
203	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
204	#108	—	2	4	no	—
205	#108	—	4	4	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
209	C₄⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
210	C₂²⋊C₈	—	2	4	no	—
211	C₂²⋊C₈	—	2	4	no	—
212	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
213	#97	—	2	2	no	—
214	C₄×C₈	—	2	2	no	—
215	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
216	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
217	C₂⁴	C₂×Q₈×C₁₀₉	1	2	yes	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
220	C₂×D₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
224	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
225	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
227	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
228	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
229	M₁₆	—	2	2	no	—
230	M₁₆	—	2	2	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	8	no	—
235	M₁₆	—	4	8	no	—
236	M₁₆	#41×C₁₀₉	1	4	yes	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	8	no	—
244	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
245	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
246	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
247	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
248	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
249	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
250	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
251	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
252	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
254	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
255	C₂³	—	2	4	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
259	C₂³	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
260	D₄	—	2	4	no	—
261	D₄	—	4	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
273	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
274	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
276	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
277	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
278	C₈	—	2	2	no	—
279	C₈	—	2	2	no	—
280	C₈	—	2	4	no	—
281	C₈	—	2	4	no	—
282	C₈	—	4	4	no	—
283	C₈	—	4	2	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	8	4	no	—
290	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
291	C₂²	#554×C₁₀₉	1	2	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	#430×C₁₀₉	1	2	yes	—
297	C₄	C₄×#41×C₁₀₉	1	1	yes	characteristic
298	C₄	#518×C₁₀₉	1	1	yes	characteristic
299	C₄	#546×C₁₀₉	1	2	yes	—
300	C₂	—	2	2	no	—
301	C₂	—	4	4	no	—
302	C₂	C₁₀₉×#2349	1	1	yes	characteristic
303	C₂	C₂×#430×C₁₀₉	1	1	yes	characteristic
304	C₂	C₁₀₉×#2916	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1950	C₂	semidirect
4. C₁₀₉×#1855	C₂	semidirect
5. C₁₀₉×#1837	C₂	semidirect
6. C₁₀₉×#1811	C₂	semidirect
7. C₄×#108×C₁₀₉	C₂	semidirect
8. C₁₀₉×#1738	C₂	semidirect
147. #22719	C₁₀₉	direct (N × H)
155. #1950	C₂×C₁₀₉	semidirect
156. #1855	C₂×C₁₀₉	semidirect
157. #1837	C₂×C₁₀₉	semidirect
158. #1811	C₂×C₁₀₉	semidirect
159. C₄×#108	C₂×C₁₀₉	semidirect
160. #1738	C₂×C₁₀₉	semidirect
161. C₁₀₉	#22719	direct (N × H)

Maximal subgroups

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

subgroup	index
D₄×M₁₆×C₁₀₉	2
C₂×#436×C₁₀₉	2
C₁₀₉×#1950	2
C₁₀₉×#1855	2
C₁₀₉×#1837	2
C₁₀₉×#1811	2
C₄×#108×C₁₀₉	2
C₁₀₉×#1738	2
#22719	109

Subgroups — C109×#22719 #42140713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#22719 #42140713