Subgroups — C₁₀₉×#32702 #42150696

Order 27904.

480 conjugacy classes of subgroups.

The subgroup lattice has 280 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×#1937	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×#1837	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×C₄×C₈×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	2	no	—
52	#104×C₁₀₉	—	2	2	no	—
53	#104×C₁₀₉	—	4	2	no	—
54	#103×C₁₀₉	—	2	2	no	—
55	#102×C₁₀₉	—	2	2	no	—
56	#101×C₁₀₉	—	2	2	no	—
57	#100×C₁₀₉	—	2	2	no	—
58	C₂²⋊C₈×C₁₀₉	—	4	2	no	—
59	C₄×C₈×C₁₀₉	—	2	1	no	—
60	C₄×C₈×C₁₀₉	—	2	1	no	—
61	#41×C₁₀₉	—	2	2	no	—
62	#41×C₁₀₉	—	2	2	no	—
63	#41×C₁₀₉	—	2	2	no	—
64	#41×C₁₀₉	—	2	2	no	—
65	C₂×Q₈×C₁₀₉	#41	1	2	yes	—
66	C₂×D₄×C₁₀₉	—	2	2	no	—
67	C₂×D₄×C₁₀₉	#41	1	2	yes	—
68	C₂²×C₄×C₁₀₉	—	2	1	no	—
69	C₂²×C₄×C₁₀₉	—	2	2	no	—
70	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
71	Q₁₆×C₁₀₉	—	4	1	no	—
72	SD₁₆×C₁₀₉	—	4	2	no	—
73	D₈×C₁₀₉	—	4	1	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	—	2	2	no	—
81	M₁₆×C₁₀₉	—	2	2	no	—
82	M₁₆×C₁₀₉	—	4	4	no	—
83	C₂×C₈×C₁₀₉	—	2	2	no	—
84	C₂×C₈×C₁₀₉	—	2	2	no	—
85	C₂×C₈×C₁₀₉	—	2	2	no	—
86	C₂×C₈×C₁₀₉	—	2	2	no	—
87	C₂×C₈×C₁₀₉	—	2	1	no	—
88	C₂×C₈×C₁₀₉	—	2	2	no	—
89	C₂×C₈×C₁₀₉	—	2	2	no	—
90	C₂×C₈×C₁₀₉	—	2	1	no	—
91	C₂×C₈×C₁₀₉	—	4	4	no	—
92	C₄⋊C₄×C₁₀₉	—	2	2	no	—
93	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
94	C₄²×C₁₀₉	—	2	1	no	—
95	C₄²×C₁₀₉	—	2	2	no	—
96	C₄²×C₁₀₉	#41	1	1	yes	characteristic
97	C₄²×C₁₀₉	#41	1	1	yes	characteristic
98	C₂³×C₁₀₉	—	2	2	no	—
99	C₂³×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
100	Q₈×C₁₀₉	—	2	2	no	—
101	Q₈×C₁₀₉	—	2	2	no	—
102	D₄×C₁₀₉	—	2	2	no	—
103	D₄×C₁₀₉	—	2	2	no	—
104	D₄×C₁₀₉	—	4	2	no	—
105	D₄×C₁₀₉	—	4	2	no	—
106	C₂×C₄×C₁₀₉	—	2	1	no	—
107	C₂×C₄×C₁₀₉	—	2	1	no	—
108	C₂×C₄×C₁₀₉	—	2	2	no	—
109	C₂×C₄×C₁₀₉	—	4	2	no	—
110	C₂×C₄×C₁₀₉	—	4	2	no	—
111	C₂×C₄×C₁₀₉	—	4	2	no	—
112	C₂×C₄×C₁₀₉	—	4	2	no	—
113	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
114	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
117	C₈×C₁₀₉	—	4	1	no	—
118	C₈×C₁₀₉	—	4	1	no	—
119	C₈×C₁₀₉	—	4	2	no	—
120	C₈×C₁₀₉	—	4	2	no	—
121	C₈×C₁₀₉	—	4	2	no	—
122	C₈×C₁₀₉	—	4	2	no	—
123	C₂²×C₁₀₉	—	4	1	no	—
124	C₂²×C₁₀₉	—	4	2	no	—
125	C₂²×C₁₀₉	—	8	2	no	—
126	C₂²×C₁₀₉	#539	1	1	yes	characteristic
127	C₂²×C₁₀₉	#539	1	2	yes	—
128	C₄×C₁₀₉	—	2	2	no	—
129	C₄×C₁₀₉	—	4	1	no	—
130	C₄×C₁₀₉	—	4	2	no	—
131	C₄×C₁₀₉	—	8	2	no	—
132	C₄×C₁₀₉	#551	1	1	yes	characteristic
133	C₄×C₁₀₉	#553	1	2	yes	—
134	C₄×C₁₀₉	H(3,𝔽₄)	1	1	yes	characteristic
135	#32702	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₁₀₉	—	2	1	no	—
137	C₂×C₁₀₉	—	2	2	no	—
138	C₂×C₁₀₉	—	8	2	no	—
139	C₂×C₁₀₉	#2782	1	1	yes	characteristic
140	#2103	C₂×C₁₀₉	1	2	yes	complemented
141	#2102	C₂×C₁₀₉	1	1	yes	characteristic, complemented
142	#2080	C₂×C₁₀₉	1	2	yes	complemented
143	#2050	C₂×C₁₀₉	1	1	yes	characteristic
144	#1937	C₂×C₁₀₉	1	2	yes	complemented
145	#1881	C₂×C₁₀₉	1	2	yes	—
146	#1861	C₂×C₁₀₉	1	2	yes	complemented
147	#1858	C₂×C₁₀₉	1	2	yes	complemented
148	#1837	C₂×C₁₀₉	1	1	yes	characteristic, complemented
149	C₁₀₉	#32702	1	1	yes	characteristic, direct product factor, Sylow
150	#575	C₂²×C₁₀₉	1	1	yes	characteristic
151	#568	C₂²×C₁₀₉	1	2	yes	—
152	#430	C₂²×C₁₀₉	1	1	yes	characteristic
153	#430	C₂²×C₁₀₉	1	2	yes	—
154	#424	C₂²×C₁₀₉	1	2	yes	—
155	#423	C₂²×C₁₀₉	1	1	yes	characteristic
156	#423	C₂²×C₁₀₉	1	1	yes	characteristic
157	#421	C₂²×C₁₀₉	1	2	yes	—
158	C₂×#104	C₂²×C₁₀₉	1	2	yes	—
159	#416	C₂²×C₁₀₉	1	2	yes	—
160	#413	C₂²×C₁₀₉	1	2	yes	—
161	#408	C₂²×C₁₀₉	1	2	yes	—
162	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
163	#344	C₂²×C₁₀₉	1	2	yes	—
164	#344	C₂²×C₁₀₉	1	2	yes	—
165	#341	C₂²×C₁₀₉	1	2	yes	—
166	#341	C₂²×C₁₀₉	1	2	yes	—
167	#341	C₂²×C₁₀₉	1	2	yes	—
168	#339	C₂²×C₁₀₉	1	2	yes	—
169	#338	C₂²×C₁₀₉	1	1	yes	characteristic
170	H(3,ℤ/4)	C₂²×C₁₀₉	1	1	yes	characteristic
171	C₂×#41	C₂³×C₁₀₉	1	2	yes	—
172	#137	—	2	2	no	—
173	AGL(1,ℤ/8)	—	2	2	no	—
174	#135	—	4	1	no	—
175	#131	—	4	2	no	—
176	#131	—	4	2	no	—
177	C₂×M₁₆	—	4	2	no	—
178	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
179	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
180	C₂×M₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
181	C₂×M₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
182	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
183	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
184	#117	C₂³×C₁₀₉	1	2	yes	—
185	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
186	#108	—	4	2	no	—
187	#108	—	4	1	no	—
188	C₄⋊C₈	—	4	1	no	—
189	C₄⋊C₈	—	4	1	no	—
190	C₄⋊C₈	—	4	2	no	—
191	#104	—	2	2	no	—
192	#104	—	2	2	no	—
193	#104	—	4	2	no	—
194	#103	—	2	2	no	—
195	#102	—	2	2	no	—
196	#101	—	2	2	no	—
197	#100	—	2	2	no	—
198	C₂²⋊C₈	—	4	2	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	#41	—	2	2	no	—
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	#41	—	2	2	no	—
205	C₂×Q₈	#41×C₁₀₉	1	2	yes	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	#41×C₁₀₉	1	2	yes	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
211	Q₁₆	—	4	1	no	—
212	SD₁₆	—	4	2	no	—
213	D₈	—	4	1	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	2	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	4	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	4	4	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	2	no	—
236	C₄²	#41×C₁₀₉	1	1	yes	characteristic
237	C₄²	#41×C₁₀₉	1	1	yes	characteristic
238	C₂³	—	2	2	no	—
239	C₂³	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	2	2	no	—
242	D₄	—	2	2	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
254	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
255	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
256	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	4	2	no	—
262	C₈	—	4	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	#539×C₁₀₉	1	1	yes	characteristic
267	C₂²	#539×C₁₀₉	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	2	no	—
271	C₄	—	8	2	no	—
272	C₄	#551×C₁₀₉	1	1	yes	characteristic
273	C₄	#553×C₁₀₉	1	2	yes	—
274	C₄	H(3,𝔽₄)×C₁₀₉	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	2	no	—
277	C₂	—	8	2	no	—
278	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×#1937	C₂	semidirect
7. C₁₀₉×#1861	C₂	semidirect
8. C₁₀₉×#1858	C₂	semidirect
9. C₁₀₉×#1837	C₂	semidirect
135. #32702	C₁₀₉	direct (N × H)
140. #2103	C₂×C₁₀₉	semidirect
141. #2102	C₂×C₁₀₉	semidirect
142. #2080	C₂×C₁₀₉	semidirect
144. #1937	C₂×C₁₀₉	semidirect
146. #1861	C₂×C₁₀₉	semidirect
147. #1858	C₂×C₁₀₉	semidirect
148. #1837	C₂×C₁₀₉	semidirect
149. C₁₀₉	#32702	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×#1937	2
C₁₀₉×#1881	2
C₁₀₉×#1861	2
C₁₀₉×#1858	2
C₁₀₉×#1837	2
#32702	109

Subgroups — C109×#32702 #42150696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32702 #42150696