Subgroups — C₂×C₁₀₉×#2090 #42141667

Order 27904.

810 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#428×C₁₀₉	C₂	1	2	yes	maximal
2	C₂²×#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#2090	C₂	1	8	yes	maximal, direct product factor
5	C₂×#343×C₁₀₉	C₂	1	2	yes	maximal
6	C₂×#341×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
12	#428×C₁₀₉	—	2	4	no	—
13	#428×C₁₀₉	C₂²	1	8	yes	—
14	C₂×#107×C₁₀₉	—	2	2	no	—
15	C₂×#106×C₁₀₉	—	2	2	no	—
16	C₂×#106×C₁₀₉	—	2	2	no	—
17	C₂×#106×C₁₀₉	—	2	2	no	—
18	C₂×#106×C₁₀₉	C₂²	1	4	yes	—
19	#424×C₁₀₉	—	2	2	no	—
20	#424×C₁₀₉	C₂²	1	4	yes	—
21	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
22	#343×C₁₀₉	C₂²	1	8	yes	—
23	#341×C₁₀₉	C₂²	1	4	yes	—
24	C₂³×C₄×C₁₀₉	—	2	2	no	—
25	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₁₀₉	—	2	2	no	—
27	C₂×M₁₆×C₁₀₉	—	2	4	no	—
28	C₂×M₁₆×C₁₀₉	—	2	1	no	—
29	C₂×M₁₆×C₁₀₉	—	2	4	no	—
30	C₂×M₁₆×C₁₀₉	Q₈	1	8	yes	—
31	C₂×M₁₆×C₁₀₉	C₂³	1	2	yes	—
32	C₂×M₁₆×C₁₀₉	C₂³	1	4	yes	—
33	C₂²×C₈×C₁₀₉	—	2	2	no	—
34	C₂²×C₈×C₁₀₉	—	2	4	no	—
35	C₂²×C₈×C₁₀₉	—	2	1	no	—
36	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
37	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
38	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
39	#117×C₁₀₉	—	2	1	no	—
40	#117×C₁₀₉	—	2	1	no	—
41	#117×C₁₀₉	D₄	1	2	yes	—
42	#117×C₁₀₉	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
50	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
51	C₂×C₄²×C₁₀₉	—	2	1	no	—
52	#107×C₁₀₉	—	2	8	no	—
53	#106×C₁₀₉	—	2	8	no	—
54	#106×C₁₀₉	—	2	8	no	—
55	C₄⋊C₈×C₁₀₉	—	2	8	no	—
56	C₂⁴×C₁₀₉	#41	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	—	2	2	no	—
58	C₂²×C₄×C₁₀₉	—	2	4	no	—
59	C₂²×C₄×C₁₀₉	—	4	2	no	—
60	C₂²×C₄×C₁₀₉	—	4	2	no	—
61	C₂²×C₄×C₁₀₉	—	8	4	no	—
62	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	C₂×D₄	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×Q₈	1	4	yes	—
67	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₉	#41	1	4	yes	—
69	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
70	M₁₆×C₁₀₉	—	2	8	no	—
71	M₁₆×C₁₀₉	—	2	4	no	—
72	M₁₆×C₁₀₉	—	2	4	no	—
73	M₁₆×C₁₀₉	—	4	2	no	—
74	M₁₆×C₁₀₉	C₂×Q₈	1	8	yes	—
75	C₂×C₈×C₁₀₉	—	2	8	no	—
76	C₂×C₈×C₁₀₉	—	2	4	no	—
77	C₂×C₈×C₁₀₉	—	2	4	no	—
78	C₂×C₈×C₁₀₉	—	2	4	no	—
79	C₂×C₈×C₁₀₉	—	2	4	no	—
80	C₂×C₈×C₁₀₉	—	2	2	no	—
81	C₂×C₈×C₁₀₉	—	4	2	no	—
82	C₂×C₈×C₁₀₉	C₂×D₄	1	4	yes	—
83	C₄⋊C₄×C₁₀₉	—	2	2	no	—
84	C₄⋊C₄×C₁₀₉	—	2	4	no	—
85	C₄⋊C₄×C₁₀₉	—	2	4	no	—
86	C₄⋊C₄×C₁₀₉	—	4	2	no	—
87	C₄⋊C₄×C₁₀₉	—	4	2	no	—
88	C₄⋊C₄×C₁₀₉	C₂×D₄	1	4	yes	—
89	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
90	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
91	C₄²×C₁₀₉	—	2	2	no	—
92	C₄²×C₁₀₉	—	2	2	no	—
93	C₂³×C₁₀₉	—	2	4	no	—
94	C₂³×C₁₀₉	#121	1	1	yes	characteristic
95	C₂³×C₁₀₉	#121	1	1	yes	characteristic
96	C₂³×C₁₀₉	#125	1	4	yes	—
97	C₂³×C₁₀₉	C₂×#41	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	—	2	4	no	—
99	C₂×C₄×C₁₀₉	—	4	4	no	—
100	C₂×C₄×C₁₀₉	—	4	4	no	—
101	C₂×C₄×C₁₀₉	—	8	4	no	—
102	C₂×C₄×C₁₀₉	—	8	4	no	—
103	C₂×C₄×C₁₀₉	—	8	4	no	—
104	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
108	C₂×C₄×C₁₀₉	#123	1	2	yes	—
109	C₂×C₄×C₁₀₉	#123	1	2	yes	—
110	C₂×C₄×C₁₀₉	#125	1	4	yes	—
111	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
113	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	C₂²×Q₈	1	2	yes	—
116	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
117	C₈×C₁₀₉	—	2	8	no	—
118	C₈×C₁₀₉	—	2	4	no	—
119	C₈×C₁₀₉	—	4	2	no	—
120	C₂²×C₁₀₉	—	2	8	no	—
121	C₂²×C₁₀₉	—	2	2	no	—
122	C₂²×C₁₀₉	—	2	2	no	—
123	C₂²×C₁₀₉	#398	1	2	yes	—
124	C₂²×C₁₀₉	#398	1	2	yes	—
125	C₂²×C₁₀₉	#471	1	4	yes	—
126	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
127	C₂²×C₁₀₉	C₂×#125	1	2	yes	—
128	C₄×C₁₀₉	—	4	4	no	—
129	C₄×C₁₀₉	—	8	4	no	—
130	C₄×C₁₀₉	#398	1	2	yes	—
131	C₄×C₁₀₉	#398	1	2	yes	—
132	C₄×C₁₀₉	C₂×C₂²⋊Q₈	1	2	yes	—
133	C₄×C₁₀₉	C₂×#123	1	1	yes	characteristic
134	C₄×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
135	C₂×#2090	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₁₀₉	—	2	2	no	—
137	C₂×C₁₀₉	—	2	2	no	—
138	C₂×C₁₀₉	#2090	1	4	yes	direct product factor
139	C₂×C₁₀₉	C₂×#398	1	1	yes	characteristic
140	C₂×C₁₀₉	C₂×#471	1	2	yes	—
141	C₂×#428	C₂×C₁₀₉	1	2	yes	—
142	C₂²×#106	C₂×C₁₀₉	1	1	yes	characteristic
143	C₂×#424	C₂×C₁₀₉	1	1	yes	characteristic
144	#2090	C₂×C₁₀₉	1	8	yes	direct product factor
145	C₂×#343	C₂×C₁₀₉	1	2	yes	—
146	C₂×#341	C₂×C₁₀₉	1	1	yes	characteristic
147	C₁₀₉	C₂×#2090	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
149	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
150	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
151	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
152	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
153	#428	—	2	4	no	—
154	#428	C₂²×C₁₀₉	1	8	yes	—
155	C₂×#107	—	2	2	no	—
156	C₂×#106	—	2	2	no	—
157	C₂×#106	—	2	2	no	—
158	C₂×#106	—	2	2	no	—
159	C₂×#106	C₂²×C₁₀₉	1	4	yes	—
160	#424	—	2	2	no	—
161	#424	C₂²×C₁₀₉	1	4	yes	—
162	C₂×C₄⋊C₈	—	2	2	no	—
163	#343	C₂²×C₁₀₉	1	8	yes	—
164	#341	C₂²×C₁₀₉	1	4	yes	—
165	C₂³×C₄	—	2	2	no	—
166	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	Q₈×C₁₀₉	1	8	yes	—
172	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
173	C₂×M₁₆	C₂³×C₁₀₉	1	4	yes	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
178	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
179	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
180	#117	—	2	1	no	—
181	#117	—	2	1	no	—
182	#117	D₄×C₁₀₉	1	2	yes	—
183	#117	C₂³×C₁₀₉	1	2	yes	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
190	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	1	no	—
192	C₂×C₄²	—	2	1	no	—
193	#107	—	2	8	no	—
194	#106	—	2	8	no	—
195	#106	—	2	8	no	—
196	C₄⋊C₈	—	2	8	no	—
197	C₂⁴	#41×C₁₀₉	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	8	4	no	—
203	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
207	C₂²×C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
208	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
209	C₂²×C₄	#41×C₁₀₉	1	4	yes	—
210	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
211	M₁₆	—	2	8	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	C₂×Q₈×C₁₀₉	1	8	yes	—
216	C₂×C₈	—	2	8	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₂×D₄×C₁₀₉	1	4	yes	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	C₂×D₄×C₁₀₉	1	4	yes	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₂³	—	2	4	no	—
235	C₂³	#121×C₁₀₉	1	1	yes	characteristic
236	C₂³	#121×C₁₀₉	1	1	yes	characteristic
237	C₂³	#125×C₁₀₉	1	4	yes	—
238	C₂³	C₂×#41×C₁₀₉	1	1	yes	characteristic
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	8	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
246	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
247	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
248	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
249	C₂×C₄	#123×C₁₀₉	1	2	yes	—
250	C₂×C₄	#123×C₁₀₉	1	2	yes	—
251	C₂×C₄	#125×C₁₀₉	1	4	yes	—
252	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
253	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
254	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
256	C₂×C₄	C₂²×Q₈×C₁₀₉	1	2	yes	—
257	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
258	C₈	—	2	8	no	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	8	no	—
264	C₂²	#398×C₁₀₉	1	2	yes	—
265	C₂²	#398×C₁₀₉	1	2	yes	—
266	C₂²	#471×C₁₀₉	1	4	yes	—
267	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
268	C₂²	C₂×#125×C₁₀₉	1	2	yes	—
269	C₄	—	4	4	no	—
270	C₄	—	8	4	no	—
271	C₄	#398×C₁₀₉	1	2	yes	—
272	C₄	#398×C₁₀₉	1	2	yes	—
273	C₄	C₂×C₂²⋊Q₈×C₁₀₉	1	2	yes	—
274	C₄	C₂×#123×C₁₀₉	1	1	yes	characteristic
275	C₄	C₂×C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	—	2	2	no	—
278	C₂	C₁₀₉×#2090	1	4	yes	direct product factor
279	C₂	C₂×#398×C₁₀₉	1	1	yes	characteristic
280	C₂	C₂×#471×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂⁴×C₁₀₉	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#2090	C₂	direct (N × H)
135. C₂×#2090	C₁₀₉	direct (N × H)
138. C₂×C₁₀₉	#2090	direct (N × H)
144. #2090	C₂×C₁₀₉	direct (N × H)
147. C₁₀₉	C₂×#2090	direct (N × H)
278. C₂	C₁₀₉×#2090	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×#428×C₁₀₉	2
C₂²×#106×C₁₀₉	2
C₂×#424×C₁₀₉	2
C₁₀₉×#2090	2
C₂×#343×C₁₀₉	2
C₂×#341×C₁₀₉	2
C₂×#2090	109

Subgroups — C2×C109×#2090 #42141667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#2090 #42141667