Subgroups — C₂×C₁₀₉×#1721 #42137608

Order 27904.

1146 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#493×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₂×#492×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#486×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1721	C₂	1	8	yes	maximal, direct product factor
6	C₂×#322×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×SD₁₆×C₁₀₉	—	2	2	no	—
8	C₂²×D₈×C₁₀₉	—	2	4	no	—
9	C₂²×D₈×C₁₀₉	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#127×C₁₀₉	C₂²	1	2	yes	—
12	C₂×#127×C₁₀₉	C₂²	1	1	yes	characteristic
13	#493×C₁₀₉	—	2	4	no	—
14	#493×C₁₀₉	—	2	2	no	—
15	#493×C₁₀₉	C₂²	1	8	yes	complemented
16	#493×C₁₀₉	C₂²	1	4	yes	complemented
17	#492×C₁₀₉	—	2	2	no	—
18	#492×C₁₀₉	C₂²	1	4	yes	complemented
19	#486×C₁₀₉	—	2	4	no	—
20	#486×C₁₀₉	C₂²	1	8	yes	complemented
21	C₂×#102×C₁₀₉	—	2	4	no	—
22	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	#322×C₁₀₉	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₁₀₉	—	2	1	no	—
26	C₂²×D₄×C₁₀₉	—	2	2	no	—
27	C₂²×D₄×C₁₀₉	—	2	2	no	—
28	C₂²×D₄×C₁₀₉	—	2	2	no	—
29	C₂²×D₄×C₁₀₉	—	2	1	no	—
30	C₂²×D₄×C₁₀₉	—	2	2	no	—
31	C₂×SD₁₆×C₁₀₉	—	2	8	no	—
32	C₂×SD₁₆×C₁₀₉	—	4	8	no	—
33	C₂×D₈×C₁₀₉	—	2	16	no	—
34	C₂×D₈×C₁₀₉	—	2	8	no	—
35	C₂×D₈×C₁₀₉	—	4	16	no	—
36	C₂×D₈×C₁₀₉	—	4	8	no	—
37	C₂²×C₈×C₁₀₉	D₄	1	4	yes	—
38	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
39	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
40	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
41	C₄⋊Q₈×C₁₀₉	C₂³	1	2	yes	—
42	#127×C₁₀₉	—	2	2	no	—
43	#127×C₁₀₉	—	2	1	no	—
44	#127×C₁₀₉	—	2	2	no	—
45	#127×C₁₀₉	—	2	2	no	—
46	#127×C₁₀₉	—	2	2	no	—
47	#127×C₁₀₉	C₂³	1	4	yes	—
48	#127×C₁₀₉	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
50	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
51	#102×C₁₀₉	—	2	16	no	—
52	C₄×C₈×C₁₀₉	D₄	1	4	yes	—
53	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
54	C₄×C₈×C₁₀₉	C₂³	1	4	yes	—
55	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
56	C₂⁴×C₁₀₉	—	4	1	no	—
57	C₂⁴×C₁₀₉	—	4	2	no	—
58	C₂×Q₈×C₁₀₉	—	2	2	no	—
59	C₂×Q₈×C₁₀₉	—	4	1	no	—
60	C₂×Q₈×C₁₀₉	—	4	4	no	—
61	C₂×D₄×C₁₀₉	—	2	4	no	—
62	C₂×D₄×C₁₀₉	—	2	4	no	—
63	C₂×D₄×C₁₀₉	—	2	4	no	—
64	C₂×D₄×C₁₀₉	—	2	4	no	—
65	C₂×D₄×C₁₀₉	—	2	2	no	—
66	C₂×D₄×C₁₀₉	—	4	2	no	—
67	C₂×D₄×C₁₀₉	—	4	1	no	—
68	C₂×D₄×C₁₀₉	—	4	2	no	—
69	C₂×D₄×C₁₀₉	—	4	2	no	—
70	C₂×D₄×C₁₀₉	—	4	2	no	—
71	C₂×D₄×C₁₀₉	—	4	8	no	—
72	C₂×D₄×C₁₀₉	—	4	4	no	—
73	C₂×D₄×C₁₀₉	—	4	8	no	—
74	C₂×D₄×C₁₀₉	—	4	8	no	—
75	C₂×D₄×C₁₀₉	—	4	8	no	—
76	C₂²×C₄×C₁₀₉	—	4	1	no	—
77	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
79	SD₁₆×C₁₀₉	—	4	16	no	—
80	D₈×C₁₀₉	—	4	32	no	—
81	D₈×C₁₀₉	—	4	16	no	—
82	C₂×C₈×C₁₀₉	—	2	8	no	—
83	C₂×C₈×C₁₀₉	D₈	1	8	yes	complemented
84	C₂×C₈×C₁₀₉	C₂×D₄	1	8	yes	—
85	C₂×C₈×C₁₀₉	C₂×D₄	1	4	yes	—
86	C₄⋊C₄×C₁₀₉	—	2	4	no	—
87	C₄⋊C₄×C₁₀₉	—	4	2	no	—
88	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
89	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
91	C₂³×C₁₀₉	—	4	4	no	—
92	C₂³×C₁₀₉	—	4	2	no	—
93	C₂³×C₁₀₉	—	8	4	no	—
94	C₂³×C₁₀₉	—	8	2	no	—
95	C₂³×C₁₀₉	—	8	4	no	—
96	C₂³×C₁₀₉	—	8	4	no	—
97	C₂³×C₁₀₉	—	8	4	no	—
98	C₂³×C₁₀₉	#127	1	1	yes	characteristic
99	Q₈×C₁₀₉	—	4	4	no	—
100	Q₈×C₁₀₉	—	8	2	no	—
101	D₄×C₁₀₉	—	4	8	no	—
102	D₄×C₁₀₉	—	4	4	no	—
103	D₄×C₁₀₉	—	4	8	no	—
104	D₄×C₁₀₉	—	4	8	no	—
105	D₄×C₁₀₉	—	4	8	no	—
106	D₄×C₁₀₉	—	8	4	no	—
107	D₄×C₁₀₉	—	8	2	no	—
108	D₄×C₁₀₉	—	8	4	no	—
109	D₄×C₁₀₉	—	8	4	no	—
110	D₄×C₁₀₉	—	8	4	no	—
111	C₂×C₄×C₁₀₉	—	4	2	no	—
112	C₂×C₄×C₁₀₉	—	8	2	no	—
113	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	#127	1	2	yes	—
115	C₂×C₄×C₁₀₉	C₂×D₈	1	4	yes	—
116	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	8	yes	—
117	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
119	C₈×C₁₀₉	—	2	8	no	—
120	C₈×C₁₀₉	C₂×D₈	1	8	yes	complemented
121	C₂²×C₁₀₉	—	8	4	no	—
122	C₂²×C₁₀₉	—	8	2	no	—
123	C₂²×C₁₀₉	—	8	4	no	—
124	C₂²×C₁₀₉	—	8	4	no	—
125	C₂²×C₁₀₉	—	8	4	no	—
126	C₂²×C₁₀₉	—	16	8	no	—
127	C₂²×C₁₀₉	—	16	4	no	—
128	C₂²×C₁₀₉	#493	1	2	yes	—
129	C₂²×C₁₀₉	#496	1	4	yes	—
130	C₂²×C₁₀₉	C₂×#127	1	1	yes	characteristic
131	C₄×C₁₀₉	—	8	2	no	—
132	C₄×C₁₀₉	#493	1	2	yes	—
133	C₄×C₁₀₉	#496	1	4	yes	—
134	C₄×C₁₀₉	C₂²×D₈	1	2	yes	—
135	C₄×C₁₀₉	C₂×AGL(1,ℤ/8)	1	4	yes	—
136	C₂×#1721	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₁₀₉	—	16	4	no	—
138	C₂×C₁₀₉	—	16	2	no	—
139	C₂×C₁₀₉	#1721	1	4	yes	direct product factor
140	C₂×C₁₀₉	C₂×#493	1	1	yes	characteristic
141	C₂×C₁₀₉	C₂×#496	1	2	yes	—
142	C₂×#493	C₂×C₁₀₉	1	1	yes	characteristic, complemented
143	C₂×#493	C₂×C₁₀₉	1	2	yes	complemented
144	C₂×#492	C₂×C₁₀₉	1	1	yes	characteristic, complemented
145	C₂×#486	C₂×C₁₀₉	1	2	yes	complemented
146	#1721	C₂×C₁₀₉	1	8	yes	direct product factor
147	C₂×#322	C₂×C₁₀₉	1	1	yes	characteristic, complemented
148	C₁₀₉	C₂×#1721	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×SD₁₆	—	2	2	no	—
150	C₂²×D₈	—	2	4	no	—
151	C₂²×D₈	—	2	2	no	—
152	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
153	C₂×#127	C₂²×C₁₀₉	1	2	yes	—
154	C₂×#127	C₂²×C₁₀₉	1	1	yes	characteristic
155	#493	—	2	4	no	—
156	#493	—	2	2	no	—
157	#493	C₂²×C₁₀₉	1	8	yes	complemented
158	#493	C₂²×C₁₀₉	1	4	yes	complemented
159	#492	—	2	2	no	—
160	#492	C₂²×C₁₀₉	1	4	yes	complemented
161	#486	—	2	4	no	—
162	#486	C₂²×C₁₀₉	1	8	yes	complemented
163	C₂×#102	—	2	4	no	—
164	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
165	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
166	#322	C₂²×C₁₀₉	1	4	yes	complemented
167	C₂²×Q₈	—	2	1	no	—
168	C₂²×D₄	—	2	2	no	—
169	C₂²×D₄	—	2	2	no	—
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	—	2	1	no	—
172	C₂²×D₄	—	2	2	no	—
173	C₂×SD₁₆	—	2	8	no	—
174	C₂×SD₁₆	—	4	8	no	—
175	C₂×D₈	—	2	16	no	—
176	C₂×D₈	—	2	8	no	—
177	C₂×D₈	—	4	16	no	—
178	C₂×D₈	—	4	8	no	—
179	C₂²×C₈	D₄×C₁₀₉	1	4	yes	—
180	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
181	C₄⋊Q₈	—	2	2	no	—
182	C₄⋊Q₈	—	2	1	no	—
183	C₄⋊Q₈	C₂³×C₁₀₉	1	2	yes	—
184	#127	—	2	2	no	—
185	#127	—	2	1	no	—
186	#127	—	2	2	no	—
187	#127	—	2	2	no	—
188	#127	—	2	2	no	—
189	#127	C₂³×C₁₀₉	1	4	yes	—
190	#127	C₂³×C₁₀₉	1	2	yes	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
193	#102	—	2	16	no	—
194	C₄×C₈	D₄×C₁₀₉	1	4	yes	—
195	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
196	C₄×C₈	C₂³×C₁₀₉	1	4	yes	—
197	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
198	C₂⁴	—	4	1	no	—
199	C₂⁴	—	4	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	8	no	—
214	C₂×D₄	—	4	4	no	—
215	C₂×D₄	—	4	8	no	—
216	C₂×D₄	—	4	8	no	—
217	C₂×D₄	—	4	8	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
221	SD₁₆	—	4	16	no	—
222	D₈	—	4	32	no	—
223	D₈	—	4	16	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	D₈×C₁₀₉	1	8	yes	complemented
226	C₂×C₈	C₂×D₄×C₁₀₉	1	8	yes	—
227	C₂×C₈	C₂×D₄×C₁₀₉	1	4	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
231	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
232	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
233	C₂³	—	4	4	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	8	4	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	4	no	—
238	C₂³	—	8	4	no	—
239	C₂³	—	8	4	no	—
240	C₂³	#127×C₁₀₉	1	1	yes	characteristic
241	Q₈	—	4	4	no	—
242	Q₈	—	8	2	no	—
243	D₄	—	4	8	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	8	no	—
246	D₄	—	4	8	no	—
247	D₄	—	4	8	no	—
248	D₄	—	8	4	no	—
249	D₄	—	8	2	no	—
250	D₄	—	8	4	no	—
251	D₄	—	8	4	no	—
252	D₄	—	8	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
256	C₂×C₄	#127×C₁₀₉	1	2	yes	—
257	C₂×C₄	C₂×D₈×C₁₀₉	1	4	yes	—
258	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	8	yes	—
259	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
261	C₈	—	2	8	no	—
262	C₈	C₂×D₈×C₁₀₉	1	8	yes	complemented
263	C₂²	—	8	4	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	16	8	no	—
269	C₂²	—	16	4	no	—
270	C₂²	#493×C₁₀₉	1	2	yes	—
271	C₂²	#496×C₁₀₉	1	4	yes	—
272	C₂²	C₂×#127×C₁₀₉	1	1	yes	characteristic
273	C₄	—	8	2	no	—
274	C₄	#493×C₁₀₉	1	2	yes	—
275	C₄	#496×C₁₀₉	1	4	yes	—
276	C₄	C₂²×D₈×C₁₀₉	1	2	yes	—
277	C₄	C₂×AGL(1,ℤ/8)×C₁₀₉	1	4	yes	—
278	C₂	—	16	4	no	—
279	C₂	—	16	2	no	—
280	C₂	C₁₀₉×#1721	1	4	yes	direct product factor
281	C₂	C₂×#493×C₁₀₉	1	1	yes	characteristic
282	C₂	C₂×#496×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#493×C₁₀₉	C₂	semidirect
2. C₂×#493×C₁₀₉	C₂	semidirect
3. C₂×#492×C₁₀₉	C₂	semidirect
4. C₂×#486×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1721	C₂	direct (N × H)
6. C₂×#322×C₁₀₉	C₂	semidirect
15. #493×C₁₀₉	C₂²	semidirect
16. #493×C₁₀₉	C₂²	semidirect
18. #492×C₁₀₉	C₂²	semidirect
20. #486×C₁₀₉	C₂²	semidirect
24. #322×C₁₀₉	C₂²	semidirect
83. C₂×C₈×C₁₀₉	D₈	semidirect
120. C₈×C₁₀₉	C₂×D₈	semidirect
136. C₂×#1721	C₁₀₉	direct (N × H)
139. C₂×C₁₀₉	#1721	direct (N × H)
142. C₂×#493	C₂×C₁₀₉	semidirect
143. C₂×#493	C₂×C₁₀₉	semidirect
144. C₂×#492	C₂×C₁₀₉	semidirect
145. C₂×#486	C₂×C₁₀₉	semidirect
146. #1721	C₂×C₁₀₉	direct (N × H)
147. C₂×#322	C₂×C₁₀₉	semidirect
148. C₁₀₉	C₂×#1721	direct (N × H)
157. #493	C₂²×C₁₀₉	semidirect
158. #493	C₂²×C₁₀₉	semidirect
160. #492	C₂²×C₁₀₉	semidirect
162. #486	C₂²×C₁₀₉	semidirect
166. #322	C₂²×C₁₀₉	semidirect
225. C₂×C₈	D₈×C₁₀₉	semidirect
262. C₈	C₂×D₈×C₁₀₉	semidirect
280. C₂	C₁₀₉×#1721	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#493×C₁₀₉	2
C₂×#492×C₁₀₉	2
C₂×#486×C₁₀₉	2
C₁₀₉×#1721	2
C₂×#322×C₁₀₉	2
C₂×#1721	109

Subgroups — C2×C109×#1721 #42137608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#1721 #42137608