Subgroups — C₂×C₁₀₉×#1385 #42130559

Order 27904.

554 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₁₀₉	C₂	1	2	yes	maximal
3	C₁₀₉×#1385	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₁₀₉	—	2	2	no	—
5	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂×M₃₂×C₁₀₉	—	2	2	no	—
7	C₂×M₃₂×C₁₀₉	—	2	2	no	—
8	C₂×M₃₂×C₁₀₉	C₄	1	2	yes	—
9	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₁₀₉	C₄	1	2	yes	—
11	C₂×#108×C₁₀₉	—	2	2	no	—
12	C₂×#108×C₁₀₉	—	2	2	no	—
13	C₂×#108×C₁₀₉	—	2	2	no	—
14	C₂×#108×C₁₀₉	C₄	1	4	yes	—
15	C₂×#108×C₁₀₉	C₄	1	4	yes	—
16	C₂×#108×C₁₀₉	C₂²	1	4	yes	—
17	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
18	C₂×M₁₆×C₁₀₉	—	2	4	no	—
19	C₂×M₁₆×C₁₀₉	—	2	4	no	—
20	C₂×M₁₆×C₁₀₉	—	2	4	no	—
21	C₂×M₁₆×C₁₀₉	—	4	2	no	—
22	C₂×M₁₆×C₁₀₉	—	4	2	no	—
23	C₂×M₁₆×C₁₀₉	—	4	2	no	—
24	C₂²×C₈×C₁₀₉	—	4	2	no	—
25	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
31	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
32	C₂²×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
33	C₂²×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
34	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
35	M₃₂×C₁₀₉	—	2	4	no	—
36	M₃₂×C₁₀₉	—	2	4	no	—
37	C₂×C₁₆×C₁₀₉	—	2	2	no	—
38	C₂×C₁₆×C₁₀₉	—	2	2	no	—
39	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
40	#108×C₁₀₉	—	2	4	no	—
41	#108×C₁₀₉	—	2	4	no	—
42	#108×C₁₀₉	—	2	4	no	—
43	#108×C₁₀₉	C₂×C₄	1	8	yes	—
44	C₂⁴×C₁₀₉	Q₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₁₀₉	SD₁₆	1	2	yes	—
53	C₂²×C₄×C₁₀₉	Q₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
55	M₁₆×C₁₀₉	—	2	8	no	—
56	M₁₆×C₁₀₉	—	4	4	no	—
57	M₁₆×C₁₀₉	—	4	4	no	—
58	M₁₆×C₁₀₉	—	4	4	no	—
59	C₂×C₈×C₁₀₉	—	4	4	no	—
60	C₂×C₈×C₁₀₉	—	4	4	no	—
61	C₂×C₈×C₁₀₉	—	4	4	no	—
62	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
63	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
64	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
65	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
80	C₁₆×C₁₀₉	—	2	4	no	—
81	C₂³×C₁₀₉	—	4	2	no	—
82	C₂³×C₁₀₉	#103	1	2	yes	—
83	C₂³×C₁₀₉	#103	1	2	yes	—
84	C₂³×C₁₀₉	#107	1	1	yes	characteristic
85	C₂³×C₁₀₉	#107	1	1	yes	characteristic
86	C₂³×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	#95	1	2	yes	—
88	C₂×C₄×C₁₀₉	#95	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	#95	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	#102	1	2	yes	—
91	C₂×C₄×C₁₀₉	#102	1	2	yes	—
92	C₂×C₄×C₁₀₉	#102	1	2	yes	—
93	C₂×C₄×C₁₀₉	#102	1	2	yes	—
94	C₂×C₄×C₁₀₉	#103	1	2	yes	—
95	C₂×C₄×C₁₀₉	#103	1	2	yes	—
96	C₂×C₄×C₁₀₉	#106	1	2	yes	—
97	C₂×C₄×C₁₀₉	#106	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	#106	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	#107	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	#107	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
105	C₈×C₁₀₉	—	4	8	no	—
106	C₈×C₁₀₉	#95	1	1	yes	characteristic
107	C₈×C₁₀₉	#95	1	1	yes	characteristic
108	C₈×C₁₀₉	#95	1	2	yes	—
109	C₈×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
110	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₈×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
112	C₂²×C₁₀₉	—	2	4	no	—
113	C₂²×C₁₀₉	—	2	2	no	—
114	C₂²×C₁₀₉	—	4	2	no	—
115	C₂²×C₁₀₉	—	4	1	no	—
116	C₂²×C₁₀₉	—	4	1	no	—
117	C₂²×C₁₀₉	#340	1	2	yes	—
118	C₂²×C₁₀₉	#340	1	1	yes	characteristic
119	C₂²×C₁₀₉	#340	1	1	yes	characteristic
120	C₂²×C₁₀₉	C₂×#103	1	2	yes	—
121	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
122	C₄×C₁₀₉	#340	1	2	yes	—
123	C₄×C₁₀₉	#340	1	1	yes	characteristic
124	C₄×C₁₀₉	#340	1	1	yes	characteristic
125	C₄×C₁₀₉	C₂×#95	1	1	yes	characteristic
126	C₄×C₁₀₉	C₂×#102	1	2	yes	—
127	C₄×C₁₀₉	C₂×#106	1	1	yes	characteristic
128	C₂×#1385	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₁₀₉	—	2	2	no	—
130	C₂×C₁₀₉	—	2	1	no	—
131	C₂×C₁₀₉	—	2	2	no	—
132	C₂×C₁₀₉	—	2	1	no	—
133	C₂×C₁₀₉	#1385	1	2	yes	direct product factor
134	C₂×C₁₀₉	C₂×#340	1	1	yes	characteristic
135	C₂²×M₃₂	C₂×C₁₀₉	1	1	yes	characteristic
136	C₂²×#108	C₂×C₁₀₉	1	2	yes	—
137	#1385	C₂×C₁₀₉	1	4	yes	direct product factor
138	C₁₀₉	C₂×#1385	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²×M₁₆	—	2	2	no	—
140	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
141	C₂×M₃₂	—	2	2	no	—
142	C₂×M₃₂	—	2	2	no	—
143	C₂×M₃₂	C₄×C₁₀₉	1	2	yes	—
144	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
145	C₂²×C₁₆	C₄×C₁₀₉	1	2	yes	—
146	C₂×#108	—	2	2	no	—
147	C₂×#108	—	2	2	no	—
148	C₂×#108	—	2	2	no	—
149	C₂×#108	C₄×C₁₀₉	1	4	yes	—
150	C₂×#108	C₄×C₁₀₉	1	4	yes	—
151	C₂×#108	C₂²×C₁₀₉	1	4	yes	—
152	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	—	4	2	no	—
157	C₂×M₁₆	—	4	2	no	—
158	C₂×M₁₆	—	4	2	no	—
159	C₂²×C₈	—	4	2	no	—
160	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
161	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
162	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
163	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
164	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
165	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
166	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
167	C₂²×C₈	Q₈×C₁₀₉	1	1	yes	characteristic
168	C₂²×C₈	Q₈×C₁₀₉	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
170	M₃₂	—	2	4	no	—
171	M₃₂	—	2	4	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
175	#108	—	2	4	no	—
176	#108	—	2	4	no	—
177	#108	—	2	4	no	—
178	#108	C₂×C₄×C₁₀₉	1	8	yes	—
179	C₂⁴	Q₁₆×C₁₀₉	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
181	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
182	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
184	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
185	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
186	C₂²×C₄	SD₁₆×C₁₀₉	1	2	yes	—
187	C₂²×C₄	SD₁₆×C₁₀₉	1	2	yes	—
188	C₂²×C₄	Q₁₆×C₁₀₉	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
190	M₁₆	—	2	8	no	—
191	M₁₆	—	4	4	no	—
192	M₁₆	—	4	4	no	—
193	M₁₆	—	4	4	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
198	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
199	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
200	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
201	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
202	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
205	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
206	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
207	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
208	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
209	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
210	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
211	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
214	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
215	C₁₆	—	2	4	no	—
216	C₂³	—	4	2	no	—
217	C₂³	#103×C₁₀₉	1	2	yes	—
218	C₂³	#103×C₁₀₉	1	2	yes	—
219	C₂³	#107×C₁₀₉	1	1	yes	characteristic
220	C₂³	#107×C₁₀₉	1	1	yes	characteristic
221	C₂³	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
222	C₂×C₄	#95×C₁₀₉	1	1	yes	characteristic
223	C₂×C₄	#95×C₁₀₉	1	2	yes	—
224	C₂×C₄	#95×C₁₀₉	1	1	yes	characteristic
225	C₂×C₄	#102×C₁₀₉	1	2	yes	—
226	C₂×C₄	#102×C₁₀₉	1	2	yes	—
227	C₂×C₄	#102×C₁₀₉	1	2	yes	—
228	C₂×C₄	#102×C₁₀₉	1	2	yes	—
229	C₂×C₄	#103×C₁₀₉	1	2	yes	—
230	C₂×C₄	#103×C₁₀₉	1	2	yes	—
231	C₂×C₄	#106×C₁₀₉	1	2	yes	—
232	C₂×C₄	#106×C₁₀₉	1	1	yes	characteristic
233	C₂×C₄	#106×C₁₀₉	1	1	yes	characteristic
234	C₂×C₄	#107×C₁₀₉	1	1	yes	characteristic
235	C₂×C₄	#107×C₁₀₉	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	2	yes	—
237	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
238	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
239	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
240	C₈	—	4	8	no	—
241	C₈	#95×C₁₀₉	1	1	yes	characteristic
242	C₈	#95×C₁₀₉	1	1	yes	characteristic
243	C₈	#95×C₁₀₉	1	2	yes	—
244	C₈	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
245	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
246	C₈	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	#340×C₁₀₉	1	2	yes	—
253	C₂²	#340×C₁₀₉	1	1	yes	characteristic
254	C₂²	#340×C₁₀₉	1	1	yes	characteristic
255	C₂²	C₂×#103×C₁₀₉	1	2	yes	—
256	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
257	C₄	#340×C₁₀₉	1	1	yes	characteristic
258	C₄	#340×C₁₀₉	1	2	yes	—
259	C₄	#340×C₁₀₉	1	1	yes	characteristic
260	C₄	C₂×#95×C₁₀₉	1	1	yes	characteristic
261	C₄	C₂×#102×C₁₀₉	1	2	yes	—
262	C₄	C₂×#106×C₁₀₉	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	2	no	—
266	C₂	—	2	1	no	—
267	C₂	C₁₀₉×#1385	1	2	yes	direct product factor
268	C₂	C₂×#340×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#106
commutator	C₈	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#340
prime radical	C₂⁴×C₁₀₉	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1385	C₂	direct (N × H)
128. C₂×#1385	C₁₀₉	direct (N × H)
133. C₂×C₁₀₉	#1385	direct (N × H)
137. #1385	C₂×C₁₀₉	direct (N × H)
138. C₁₀₉	C₂×#1385	direct (N × H)
267. C₂	C₁₀₉×#1385	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₁₀₉	2
C₂²×#108×C₁₀₉	2
C₁₀₉×#1385	2
C₂×#1385	109

Subgroups — C2×C109×#1385 #42130559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#1385 #42130559