Subgroups — C₁₀₉×#32703 #42150697

Order 27904.

476 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	C₁₀₉×#2103	C₂	1	2	yes	maximal, complemented
2	C₁₀₉×#2102	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#2080	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#2050	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1938	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1881	C₂	1	2	yes	maximal
7	C₁₀₉×#1861	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#1858	C₂	1	2	yes	maximal, complemented
9	C₁₀₉×#1838	C₂	1	1	yes	maximal, characteristic, complemented
10	#575×C₁₀₉	C₂²	1	1	yes	characteristic
11	#568×C₁₀₉	C₂²	1	2	yes	—
12	#430×C₁₀₉	C₂²	1	1	yes	characteristic
13	#430×C₁₀₉	C₂²	1	2	yes	—
14	#424×C₁₀₉	C₂²	1	2	yes	—
15	#423×C₁₀₉	C₂²	1	1	yes	characteristic
16	#423×C₁₀₉	C₂²	1	1	yes	characteristic
17	#421×C₁₀₉	C₂²	1	2	yes	—
18	C₂×#104×C₁₀₉	C₂²	1	2	yes	—
19	#416×C₁₀₉	C₂²	1	2	yes	—
20	#413×C₁₀₉	C₂²	1	2	yes	—
21	#408×C₁₀₉	C₂²	1	2	yes	—
22	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
23	#344×C₁₀₉	C₂²	1	2	yes	—
24	#344×C₁₀₉	C₂²	1	2	yes	—
25	#341×C₁₀₉	C₂²	1	2	yes	—
26	#341×C₁₀₉	C₂²	1	2	yes	—
27	#341×C₁₀₉	C₂²	1	2	yes	—
28	#339×C₁₀₉	C₂²	1	2	yes	—
29	#338×C₁₀₉	C₂²	1	1	yes	characteristic
30	H(3,ℤ/4)×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂×#41×C₁₀₉	C₂³	1	2	yes	—
32	#137×C₁₀₉	—	2	2	no	—
33	AGL(1,ℤ/8)×C₁₀₉	—	2	2	no	—
34	#135×C₁₀₉	—	4	1	no	—
35	#131×C₁₀₉	—	4	2	no	—
36	#131×C₁₀₉	—	4	2	no	—
37	C₂×M₁₆×C₁₀₉	—	4	2	no	—
38	C₂×M₁₆×C₁₀₉	C₂³	1	2	yes	—
39	C₂×M₁₆×C₁₀₉	C₂³	1	2	yes	—
40	C₂×M₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₁₀₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
44	#117×C₁₀₉	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
46	#108×C₁₀₉	—	4	2	no	—
47	#108×C₁₀₉	—	4	1	no	—
48	C₄⋊C₈×C₁₀₉	—	4	1	no	—
49	C₄⋊C₈×C₁₀₉	—	4	1	no	—
50	C₄⋊C₈×C₁₀₉	—	4	2	no	—
51	#104×C₁₀₉	—	2	4	no	—
52	#104×C₁₀₉	—	4	2	no	—
53	#103×C₁₀₉	—	2	2	no	—
54	#102×C₁₀₉	—	2	2	no	—
55	#101×C₁₀₉	—	2	2	no	—
56	#100×C₁₀₉	—	2	2	no	—
57	C₂²⋊C₈×C₁₀₉	—	4	2	no	—
58	#97×C₁₀₉	—	2	2	no	—
59	#41×C₁₀₉	—	2	2	no	—
60	#41×C₁₀₉	—	2	2	no	—
61	#41×C₁₀₉	—	2	2	no	—
62	#41×C₁₀₉	—	2	2	no	—
63	C₂×Q₈×C₁₀₉	#41	1	2	yes	—
64	C₂×D₄×C₁₀₉	—	2	2	no	—
65	C₂×D₄×C₁₀₉	#41	1	2	yes	—
66	C₂²×C₄×C₁₀₉	—	2	1	no	—
67	C₂²×C₄×C₁₀₉	—	2	2	no	—
68	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
69	Q₁₆×C₁₀₉	—	4	1	no	—
70	SD₁₆×C₁₀₉	—	4	2	no	—
71	D₈×C₁₀₉	—	4	1	no	—
72	M₁₆×C₁₀₉	—	2	2	no	—
73	M₁₆×C₁₀₉	—	2	2	no	—
74	M₁₆×C₁₀₉	—	2	2	no	—
75	M₁₆×C₁₀₉	—	2	2	no	—
76	M₁₆×C₁₀₉	—	2	2	no	—
77	M₁₆×C₁₀₉	—	2	2	no	—
78	M₁₆×C₁₀₉	—	2	2	no	—
79	M₁₆×C₁₀₉	—	2	2	no	—
80	M₁₆×C₁₀₉	—	4	4	no	—
81	C₂×C₈×C₁₀₉	—	2	2	no	—
82	C₂×C₈×C₁₀₉	—	2	2	no	—
83	C₂×C₈×C₁₀₉	—	2	2	no	—
84	C₂×C₈×C₁₀₉	—	2	2	no	—
85	C₂×C₈×C₁₀₉	—	2	1	no	—
86	C₂×C₈×C₁₀₉	—	2	2	no	—
87	C₂×C₈×C₁₀₉	—	2	2	no	—
88	C₂×C₈×C₁₀₉	—	2	1	no	—
89	C₂×C₈×C₁₀₉	—	4	4	no	—
90	C₄⋊C₄×C₁₀₉	—	2	2	no	—
91	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
92	C₄²×C₁₀₉	—	2	1	no	—
93	C₄²×C₁₀₉	—	2	2	no	—
94	C₄²×C₁₀₉	#41	1	2	yes	—
95	C₂³×C₁₀₉	—	2	2	no	—
96	C₂³×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
97	Q₈×C₁₀₉	—	2	2	no	—
98	Q₈×C₁₀₉	—	2	2	no	—
99	D₄×C₁₀₉	—	2	2	no	—
100	D₄×C₁₀₉	—	2	2	no	—
101	D₄×C₁₀₉	—	4	2	no	—
102	D₄×C₁₀₉	—	4	2	no	—
103	C₂×C₄×C₁₀₉	—	2	2	no	—
104	C₂×C₄×C₁₀₉	—	2	2	no	—
105	C₂×C₄×C₁₀₉	—	4	2	no	—
106	C₂×C₄×C₁₀₉	—	4	2	no	—
107	C₂×C₄×C₁₀₉	—	4	2	no	—
108	C₂×C₄×C₁₀₉	—	8	1	no	—
109	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
110	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
113	C₈×C₁₀₉	—	4	1	no	—
114	C₈×C₁₀₉	—	4	1	no	—
115	C₈×C₁₀₉	—	4	2	no	—
116	C₈×C₁₀₉	—	4	2	no	—
117	C₈×C₁₀₉	—	4	2	no	—
118	C₈×C₁₀₉	—	4	2	no	—
119	C₂²×C₁₀₉	—	4	1	no	—
120	C₂²×C₁₀₉	—	4	2	no	—
121	C₂²×C₁₀₉	—	8	2	no	—
122	C₂²×C₁₀₉	#539	1	1	yes	characteristic
123	C₂²×C₁₀₉	#539	1	2	yes	—
124	C₄×C₁₀₉	—	4	2	no	—
125	C₄×C₁₀₉	—	4	2	no	—
126	C₄×C₁₀₉	—	8	2	no	—
127	C₄×C₁₀₉	#551	1	1	yes	characteristic
128	C₄×C₁₀₉	#553	1	2	yes	—
129	C₄×C₁₀₉	H(3,𝔽₄)	1	1	yes	characteristic
130	#32703	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₁₀₉	—	2	1	no	—
132	C₂×C₁₀₉	—	2	2	no	—
133	C₂×C₁₀₉	—	8	2	no	—
134	C₂×C₁₀₉	#2782	1	1	yes	characteristic
135	#2103	C₂×C₁₀₉	1	2	yes	complemented
136	#2102	C₂×C₁₀₉	1	1	yes	characteristic, complemented
137	#2080	C₂×C₁₀₉	1	2	yes	complemented
138	#2050	C₂×C₁₀₉	1	1	yes	characteristic
139	#1938	C₂×C₁₀₉	1	2	yes	complemented
140	#1881	C₂×C₁₀₉	1	2	yes	—
141	#1861	C₂×C₁₀₉	1	2	yes	complemented
142	#1858	C₂×C₁₀₉	1	2	yes	complemented
143	#1838	C₂×C₁₀₉	1	1	yes	characteristic, complemented
144	C₁₀₉	#32703	1	1	yes	characteristic, direct product factor, Sylow
145	#575	C₂²×C₁₀₉	1	1	yes	characteristic
146	#568	C₂²×C₁₀₉	1	2	yes	—
147	#430	C₂²×C₁₀₉	1	1	yes	characteristic
148	#430	C₂²×C₁₀₉	1	2	yes	—
149	#424	C₂²×C₁₀₉	1	2	yes	—
150	#423	C₂²×C₁₀₉	1	1	yes	characteristic
151	#423	C₂²×C₁₀₉	1	1	yes	characteristic
152	#421	C₂²×C₁₀₉	1	2	yes	—
153	C₂×#104	C₂²×C₁₀₉	1	2	yes	—
154	#416	C₂²×C₁₀₉	1	2	yes	—
155	#413	C₂²×C₁₀₉	1	2	yes	—
156	#408	C₂²×C₁₀₉	1	2	yes	—
157	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
158	#344	C₂²×C₁₀₉	1	2	yes	—
159	#344	C₂²×C₁₀₉	1	2	yes	—
160	#341	C₂²×C₁₀₉	1	2	yes	—
161	#341	C₂²×C₁₀₉	1	2	yes	—
162	#341	C₂²×C₁₀₉	1	2	yes	—
163	#339	C₂²×C₁₀₉	1	2	yes	—
164	#338	C₂²×C₁₀₉	1	1	yes	characteristic
165	H(3,ℤ/4)	C₂²×C₁₀₉	1	1	yes	characteristic
166	C₂×#41	C₂³×C₁₀₉	1	2	yes	—
167	#137	—	2	2	no	—
168	AGL(1,ℤ/8)	—	2	2	no	—
169	#135	—	4	1	no	—
170	#131	—	4	2	no	—
171	#131	—	4	2	no	—
172	C₂×M₁₆	—	4	2	no	—
173	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
174	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
175	C₂×M₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
176	C₂×M₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
177	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
178	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
179	#117	C₂³×C₁₀₉	1	2	yes	—
180	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
181	#108	—	4	2	no	—
182	#108	—	4	1	no	—
183	C₄⋊C₈	—	4	1	no	—
184	C₄⋊C₈	—	4	1	no	—
185	C₄⋊C₈	—	4	2	no	—
186	#104	—	2	4	no	—
187	#104	—	4	2	no	—
188	#103	—	2	2	no	—
189	#102	—	2	2	no	—
190	#101	—	2	2	no	—
191	#100	—	2	2	no	—
192	C₂²⋊C₈	—	4	2	no	—
193	#97	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	C₂×Q₈	#41×C₁₀₉	1	2	yes	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	#41×C₁₀₉	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
204	Q₁₆	—	4	1	no	—
205	SD₁₆	—	4	2	no	—
206	D₈	—	4	1	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	2	2	no	—
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	4	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	4	4	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	2	no	—
229	C₄²	#41×C₁₀₉	1	2	yes	—
230	C₂³	—	2	2	no	—
231	C₂³	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	Q₈	—	2	2	no	—
234	D₄	—	2	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
245	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
246	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
247	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
248	C₈	—	4	1	no	—
249	C₈	—	4	1	no	—
250	C₈	—	4	2	no	—
251	C₈	—	4	2	no	—
252	C₈	—	4	2	no	—
253	C₈	—	4	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	2	no	—
257	C₂²	#539×C₁₀₉	1	1	yes	characteristic
258	C₂²	#539×C₁₀₉	1	2	yes	—
259	C₄	—	4	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	8	2	no	—
262	C₄	#551×C₁₀₉	1	1	yes	characteristic
263	C₄	#553×C₁₀₉	1	2	yes	—
264	C₄	H(3,𝔽₄)×C₁₀₉	1	1	yes	characteristic
265	C₂	—	2	1	no	—
266	C₂	—	2	2	no	—
267	C₂	—	8	2	no	—
268	C₂	C₁₀₉×#2782	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₀₉	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle	C₂×C₁₀₉	#2782
prime radical	#575×C₁₀₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₉×#2103	C₂	semidirect
2. C₁₀₉×#2102	C₂	semidirect
3. C₁₀₉×#2080	C₂	semidirect
5. C₁₀₉×#1938	C₂	semidirect
7. C₁₀₉×#1861	C₂	semidirect
8. C₁₀₉×#1858	C₂	semidirect
9. C₁₀₉×#1838	C₂	semidirect
130. #32703	C₁₀₉	direct (N × H)
135. #2103	C₂×C₁₀₉	semidirect
136. #2102	C₂×C₁₀₉	semidirect
137. #2080	C₂×C₁₀₉	semidirect
139. #1938	C₂×C₁₀₉	semidirect
141. #1861	C₂×C₁₀₉	semidirect
142. #1858	C₂×C₁₀₉	semidirect
143. #1838	C₂×C₁₀₉	semidirect
144. C₁₀₉	#32703	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#2103	2
C₁₀₉×#2102	2
C₁₀₉×#2080	2
C₁₀₉×#2050	2
C₁₀₉×#1938	2
C₁₀₉×#1881	2
C₁₀₉×#1861	2
C₁₀₉×#1858	2
C₁₀₉×#1838	2
#32703	109

Subgroups — C109×#32703 #42150697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32703 #42150697