Subgroups — C₁₀₉×#20669 #42138663

Order 27904.

1322 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#430×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#2879	C₂	1	2	yes	maximal
4	C₁₀₉×#1811	C₂	1	4	yes	maximal, complemented
5	C₁₀₉×#1810	C₂	1	4	yes	maximal, complemented
6	C₂×#341×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×#131×C₁₀₉	—	2	8	no	—
9	C₂×#131×C₁₀₉	C₄	1	8	yes	—
10	C₂²×M₁₆×C₁₀₉	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
12	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₁₀₉	C₂²	1	2	yes	—
15	#430×C₁₀₉	—	2	8	no	—
16	C₂×#108×C₁₀₉	C₂²	1	8	yes	—
17	#408×C₁₀₉	—	2	8	no	—
18	#407×C₁₀₉	C₂²	1	8	yes	—
19	#341×C₁₀₉	C₂²	1	4	yes	—
20	#341×C₁₀₉	C₂²	1	4	yes	—
21	C₂×#41×C₁₀₉	—	2	4	no	—
22	C₂×#41×C₁₀₉	—	2	4	no	—
23	C₂×#41×C₁₀₉	C₂×C₄	1	4	yes	—
24	C₂×#41×C₁₀₉	D₄	1	4	yes	—
25	C₂²×Q₈×C₁₀₉	Q₈	1	1	yes	characteristic
26	C₂²×D₄×C₁₀₉	Q₈	1	2	yes	—
27	C₂²×D₄×C₁₀₉	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
29	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
30	#131×C₁₀₉	—	2	32	no	—
31	C₂×M₁₆×C₁₀₉	—	2	4	no	—
32	C₂×M₁₆×C₁₀₉	—	2	8	no	—
33	C₂×M₁₆×C₁₀₉	—	2	8	no	—
34	C₂×M₁₆×C₁₀₉	—	2	8	no	—
35	C₂×M₁₆×C₁₀₉	C₂×C₄	1	8	yes	—
36	C₂×M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₁₀₉	D₄	1	8	yes	—
38	C₂×M₁₆×C₁₀₉	C₂³	1	8	yes	—
39	C₂²×C₈×C₁₀₉	—	2	4	no	—
40	C₂²×C₈×C₁₀₉	—	2	8	no	—
41	C₂²×C₈×C₁₀₉	—	2	4	no	—
42	C₂²×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₁₀₉	C₂×C₄	1	8	yes	—
44	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
47	#108×C₁₀₉	—	2	16	no	—
48	C₂²⋊C₈×C₁₀₉	—	2	16	no	—
49	C₂⁴×C₁₀₉	C₂×Q₈	1	2	yes	—
50	C₂⁴×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
51	#41×C₁₀₉	—	2	8	no	—
52	#41×C₁₀₉	—	2	8	no	—
53	#41×C₁₀₉	—	2	16	no	—
54	C₂×Q₈×C₁₀₉	—	2	4	no	—
55	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
56	C₂×D₄×C₁₀₉	—	2	8	no	—
57	C₂×D₄×C₁₀₉	—	2	4	no	—
58	C₂×D₄×C₁₀₉	C₄⋊C₄	1	8	yes	—
59	C₂×D₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₁₀₉	—	2	4	no	—
61	C₂²×C₄×C₁₀₉	—	2	4	no	—
62	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	#41	1	4	yes	—
70	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
71	M₁₆×C₁₀₉	—	2	16	no	—
72	M₁₆×C₁₀₉	—	2	8	no	—
73	M₁₆×C₁₀₉	—	2	8	no	—
74	M₁₆×C₁₀₉	—	4	8	no	—
75	M₁₆×C₁₀₉	C₂²⋊C₄	1	16	yes	—
76	C₂×C₈×C₁₀₉	—	2	8	no	—
77	C₂×C₈×C₁₀₉	—	2	2	no	—
78	C₂×C₈×C₁₀₉	—	2	2	no	—
79	C₂×C₈×C₁₀₉	—	2	16	no	—
80	C₂×C₈×C₁₀₉	—	2	8	no	—
81	C₂×C₈×C₁₀₉	—	4	8	no	—
82	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₁₀₉	C₂×D₄	1	8	yes	—
86	C₂³×C₁₀₉	—	2	4	no	—
87	C₂³×C₁₀₉	—	4	2	no	—
88	C₂³×C₁₀₉	—	4	4	no	—
89	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
90	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
92	C₂³×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
93	Q₈×C₁₀₉	—	2	8	no	—
94	D₄×C₁₀₉	—	2	16	no	—
95	D₄×C₁₀₉	—	2	8	no	—
96	C₂×C₄×C₁₀₉	—	2	4	no	—
97	C₂×C₄×C₁₀₉	—	2	4	no	—
98	C₂×C₄×C₁₀₉	—	2	4	no	—
99	C₂×C₄×C₁₀₉	—	2	8	no	—
100	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
101	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₁₀₉	#117	1	4	yes	—
104	C₂×C₄×C₁₀₉	#121	1	8	yes	—
105	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
106	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
110	C₈×C₁₀₉	—	2	2	no	—
111	C₈×C₁₀₉	—	2	2	no	—
112	C₈×C₁₀₉	—	4	8	no	—
113	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	8	yes	—
114	C₂²×C₁₀₉	—	2	2	no	—
115	C₂²×C₁₀₉	—	2	4	no	—
116	C₂²×C₁₀₉	—	2	8	no	—
117	C₂²×C₁₀₉	—	2	4	no	—
118	C₂²×C₁₀₉	—	4	4	no	—
119	C₂²×C₁₀₉	—	4	8	no	—
120	C₂²×C₁₀₉	#380	1	4	yes	—
121	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₁₀₉	C₂×C₂²⋊Q₈	1	2	yes	—
123	C₄×C₁₀₉	—	2	4	no	—
124	C₄×C₁₀₉	#380	1	4	yes	—
125	C₄×C₁₀₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₁₀₉	C₂×#117	1	1	yes	characteristic
127	C₄×C₁₀₉	C₂×#121	1	2	yes	—
128	#20669	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₁₀₉	—	2	2	no	—
130	C₂×C₁₀₉	—	2	4	no	—
131	C₂×C₁₀₉	—	4	4	no	—
132	C₂×C₁₀₉	C₂×#380	1	1	yes	characteristic
133	C₂×C₁₀₉	#2916	1	2	yes	—
134	C₂²×#131	C₂×C₁₀₉	1	1	yes	characteristic
135	C₂×#430	C₂×C₁₀₉	1	2	yes	complemented
136	#2879	C₂×C₁₀₉	1	2	yes	—
137	#1811	C₂×C₁₀₉	1	4	yes	complemented
138	#1810	C₂×C₁₀₉	1	4	yes	complemented
139	C₂×#341	C₂×C₁₀₉	1	2	yes	complemented
140	C₁₀₉	#20669	1	1	yes	characteristic, direct product factor, Sylow
141	C₂²×#41	C₂²×C₁₀₉	1	1	yes	characteristic
142	C₂×#131	—	2	8	no	—
143	C₂×#131	C₄×C₁₀₉	1	8	yes	—
144	C₂²×M₁₆	C₂²×C₁₀₉	1	4	yes	—
145	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
146	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
147	C₂³×C₈	C₂²×C₁₀₉	1	2	yes	—
148	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
149	#430	—	2	8	no	—
150	C₂×#108	C₂²×C₁₀₉	1	8	yes	—
151	#408	—	2	8	no	—
152	#407	C₂²×C₁₀₉	1	8	yes	—
153	#341	C₂²×C₁₀₉	1	4	yes	—
154	#341	C₂²×C₁₀₉	1	4	yes	—
155	C₂×#41	—	2	4	no	—
156	C₂×#41	—	2	4	no	—
157	C₂×#41	C₂×C₄×C₁₀₉	1	4	yes	—
158	C₂×#41	D₄×C₁₀₉	1	4	yes	—
159	C₂²×Q₈	Q₈×C₁₀₉	1	1	yes	characteristic
160	C₂²×D₄	Q₈×C₁₀₉	1	2	yes	—
161	C₂²×D₄	Q₈×C₁₀₉	1	1	yes	characteristic
162	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
163	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
164	#131	—	2	32	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	2	8	no	—
167	C₂×M₁₆	—	2	8	no	—
168	C₂×M₁₆	—	2	8	no	—
169	C₂×M₁₆	C₂×C₄×C₁₀₉	1	8	yes	—
170	C₂×M₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
171	C₂×M₁₆	D₄×C₁₀₉	1	8	yes	—
172	C₂×M₁₆	C₂³×C₁₀₉	1	8	yes	—
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	2	8	no	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
177	C₂²×C₈	C₂×C₄×C₁₀₉	1	8	yes	—
178	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
181	#108	—	2	16	no	—
182	C₂²⋊C₈	—	2	16	no	—
183	C₂⁴	C₂×Q₈×C₁₀₉	1	2	yes	—
184	C₂⁴	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
185	#41	—	2	8	no	—
186	#41	—	2	8	no	—
187	#41	—	2	16	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	4	yes	—
190	C₂×D₄	—	2	8	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	C₄⋊C₄×C₁₀₉	1	8	yes	—
193	C₂×D₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
199	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
200	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
201	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
202	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
203	C₂²×C₄	#41×C₁₀₉	1	4	yes	—
204	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
205	M₁₆	—	2	16	no	—
206	M₁₆	—	2	8	no	—
207	M₁₆	—	2	8	no	—
208	M₁₆	—	4	8	no	—
209	M₁₆	C₂²⋊C₄×C₁₀₉	1	16	yes	—
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	16	no	—
214	C₂×C₈	—	2	8	no	—
215	C₂×C₈	—	4	8	no	—
216	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
217	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
218	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
219	C₂×C₈	C₂×D₄×C₁₀₉	1	8	yes	—
220	C₂³	—	2	4	no	—
221	C₂³	—	4	2	no	—
222	C₂³	—	4	4	no	—
223	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
224	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
225	C₂³	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
226	C₂³	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
227	Q₈	—	2	8	no	—
228	D₄	—	2	16	no	—
229	D₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	4	yes	—
235	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
236	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
237	C₂×C₄	#117×C₁₀₉	1	4	yes	—
238	C₂×C₄	#121×C₁₀₉	1	8	yes	—
239	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
240	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
242	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
243	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	4	8	no	—
247	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	8	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	8	no	—
254	C₂²	#380×C₁₀₉	1	4	yes	—
255	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
256	C₂²	C₂×C₂²⋊Q₈×C₁₀₉	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	#380×C₁₀₉	1	4	yes	—
259	C₄	C₂²×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
260	C₄	C₂×#117×C₁₀₉	1	1	yes	characteristic
261	C₄	C₂×#121×C₁₀₉	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	2	4	no	—
264	C₂	—	4	4	no	—
265	C₂	C₂×#380×C₁₀₉	1	1	yes	characteristic
266	C₂	C₁₀₉×#2916	1	2	yes	—

	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×#41×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂²×#41×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#430×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1811	C₂	semidirect
5. C₁₀₉×#1810	C₂	semidirect
6. C₂×#341×C₁₀₉	C₂	semidirect
128. #20669	C₁₀₉	direct (N × H)
135. C₂×#430	C₂×C₁₀₉	semidirect
137. #1811	C₂×C₁₀₉	semidirect
138. #1810	C₂×C₁₀₉	semidirect
139. C₂×#341	C₂×C₁₀₉	semidirect
140. C₁₀₉	#20669	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×#131×C₁₀₉	2
C₂×#430×C₁₀₉	2
C₁₀₉×#2879	2
C₁₀₉×#1811	2
C₁₀₉×#1810	2
C₂×#341×C₁₀₉	2
#20669	109

Subgroups — C109×#20669 #42138663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#20669 #42138663