Subgroups — C₁₀₉×#12871 #42130865

Order 27904.

442 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2112	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1810	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1387	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₁₀₉	—	2	1	no	—
7	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
8	C₂×#108×C₁₀₉	C₂²	1	2	yes	—
9	C₂×#107×C₁₀₉	—	2	1	no	—
10	C₂×#107×C₁₀₉	—	2	1	no	—
11	C₂×#107×C₁₀₉	—	2	1	no	—
12	C₂×#107×C₁₀₉	C₄	1	2	yes	—
13	C₂×#107×C₁₀₉	C₄	1	2	yes	—
14	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
15	#407×C₁₀₉	—	2	1	no	—
16	C₂²⋊C₁₆×C₁₀₉	—	2	1	no	—
17	C₂²⋊C₁₆×C₁₀₉	—	2	1	no	—
18	#341×C₁₀₉	—	2	1	no	—
19	C₂³×C₄×C₁₀₉	—	4	1	no	—
20	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂×M₁₆×C₁₀₉	—	2	2	no	—
22	C₂²×C₈×C₁₀₉	—	2	4	no	—
23	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₀₉	—	4	1	no	—
33	C₂×C₄⋊C₄×C₁₀₉	—	4	1	no	—
34	C₂×C₄⋊C₄×C₁₀₉	—	4	1	no	—
35	M₃₂×C₁₀₉	—	2	4	no	—
36	C₂×C₁₆×C₁₀₉	—	2	2	no	—
37	#108×C₁₀₉	—	2	4	no	—
38	#107×C₁₀₉	—	2	2	no	—
39	#107×C₁₀₉	—	2	2	no	—
40	#107×C₁₀₉	—	2	2	no	—
41	#107×C₁₀₉	C₂×C₄	1	4	yes	—
42	C₂²⋊C₈×C₁₀₉	—	4	1	no	—
43	C₂²⋊C₈×C₁₀₉	—	4	1	no	—
44	C₂⁴×C₁₀₉	SD₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₉	—	2	2	no	—
46	C₂²×C₄×C₁₀₉	—	2	2	no	—
47	C₂²×C₄×C₁₀₉	—	4	2	no	—
48	C₂²×C₄×C₁₀₉	—	4	2	no	—
49	C₂²×C₄×C₁₀₉	—	4	2	no	—
50	C₂²×C₄×C₁₀₉	—	8	4	no	—
51	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	M₁₆×C₁₀₉	—	4	2	no	—
58	M₁₆×C₁₀₉	—	4	2	no	—
59	C₂×C₈×C₁₀₉	—	2	2	no	—
60	C₂×C₈×C₁₀₉	—	2	2	no	—
61	C₂×C₈×C₁₀₉	—	2	2	no	—
62	C₂×C₈×C₁₀₉	—	2	2	no	—
63	C₂×C₈×C₁₀₉	—	2	2	no	—
64	C₂×C₈×C₁₀₉	—	4	2	no	—
65	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₀₉	#41	1	4	yes	—
70	C₄⋊C₄×C₁₀₉	—	2	4	no	—
71	C₄⋊C₄×C₁₀₉	—	4	2	no	—
72	C₄⋊C₄×C₁₀₉	—	4	2	no	—
73	C₄⋊C₄×C₁₀₉	—	4	2	no	—
74	C₁₆×C₁₀₉	—	4	2	no	—
75	C₂³×C₁₀₉	—	2	2	no	—
76	C₂³×C₁₀₉	—	2	2	no	—
77	C₂³×C₁₀₉	—	2	2	no	—
78	C₂³×C₁₀₉	#102	1	1	yes	characteristic
79	C₂³×C₁₀₉	#102	1	1	yes	characteristic
80	C₂³×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₄×C₁₀₉	—	2	1	no	—
83	C₂×C₄×C₁₀₉	—	2	1	no	—
84	C₂×C₄×C₁₀₉	—	2	1	no	—
85	C₂×C₄×C₁₀₉	—	2	1	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	4	4	no	—
89	C₂×C₄×C₁₀₉	—	8	4	no	—
90	C₂×C₄×C₁₀₉	—	8	4	no	—
91	C₂×C₄×C₁₀₉	—	8	4	no	—
92	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	#123	1	2	yes	—
96	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	#137	1	2	yes	—
98	C₈×C₁₀₉	—	2	1	no	—
99	C₈×C₁₀₉	—	2	1	no	—
100	C₈×C₁₀₉	—	8	2	no	—
101	C₈×C₁₀₉	#117	1	2	yes	—
102	C₈×C₁₀₉	#123	1	2	yes	—
103	C₂²×C₁₀₉	—	2	8	no	—
104	C₂²×C₁₀₉	—	2	1	no	—
105	C₂²×C₁₀₉	—	2	1	no	—
106	C₂²×C₁₀₉	—	2	1	no	—
107	C₂²×C₁₀₉	—	2	1	no	—
108	C₂²×C₁₀₉	—	2	1	no	—
109	C₂²×C₁₀₉	—	2	1	no	—
110	C₂²×C₁₀₉	#361	1	2	yes	—
111	C₂²×C₁₀₉	#362	1	2	yes	—
112	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
113	C₂²×C₁₀₉	#483	1	2	yes	—
114	C₄×C₁₀₉	—	2	1	no	—
115	C₄×C₁₀₉	—	2	1	no	—
116	C₄×C₁₀₉	—	8	4	no	—
117	C₄×C₁₀₉	#381	1	1	yes	characteristic
118	C₄×C₁₀₉	#419	1	1	yes	characteristic
119	C₄×C₁₀₉	#480	1	2	yes	—
120	#12871	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₁₀₉	—	2	2	no	—
122	C₂×C₁₀₉	—	2	2	no	—
123	C₂×C₁₀₉	—	2	2	no	—
124	C₂×C₁₀₉	#1823	1	1	yes	characteristic
125	C₂×C₁₀₉	C₂×#361	1	1	yes	characteristic
126	C₂×C₁₀₉	C₂×#362	1	1	yes	characteristic
127	C₂²×#107	C₂×C₁₀₉	1	1	yes	characteristic
128	#2112	C₂×C₁₀₉	1	1	yes	characteristic
129	#1810	C₂×C₁₀₉	1	1	yes	characteristic
130	#1387	C₂×C₁₀₉	1	4	yes	complemented
131	C₁₀₉	#12871	1	1	yes	characteristic, direct product factor, Sylow
132	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
133	C₂²×C₄⋊C₄	—	2	1	no	—
134	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
135	C₂×#108	C₂²×C₁₀₉	1	2	yes	—
136	C₂×#107	—	2	1	no	—
137	C₂×#107	—	2	1	no	—
138	C₂×#107	—	2	1	no	—
139	C₂×#107	C₄×C₁₀₉	1	2	yes	—
140	C₂×#107	C₄×C₁₀₉	1	2	yes	—
141	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
142	#407	—	2	1	no	—
143	C₂²⋊C₁₆	—	2	1	no	—
144	C₂²⋊C₁₆	—	2	1	no	—
145	#341	—	2	1	no	—
146	C₂³×C₄	—	4	1	no	—
147	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
148	C₂×M₁₆	—	2	2	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
151	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
152	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
153	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
154	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	4	1	no	—
160	C₂×C₄⋊C₄	—	4	1	no	—
161	C₂×C₄⋊C₄	—	4	1	no	—
162	M₃₂	—	2	4	no	—
163	C₂×C₁₆	—	2	2	no	—
164	#108	—	2	4	no	—
165	#107	—	2	2	no	—
166	#107	—	2	2	no	—
167	#107	—	2	2	no	—
168	#107	C₂×C₄×C₁₀₉	1	4	yes	—
169	C₂²⋊C₈	—	4	1	no	—
170	C₂²⋊C₈	—	4	1	no	—
171	C₂⁴	SD₁₆×C₁₀₉	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	8	4	no	—
178	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
180	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
181	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
182	C₂²×C₄	SD₁₆×C₁₀₉	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
184	M₁₆	—	4	2	no	—
185	M₁₆	—	4	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
193	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
194	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
195	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
196	C₂×C₈	#41×C₁₀₉	1	4	yes	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₁₆	—	4	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	#102×C₁₀₉	1	1	yes	characteristic
206	C₂³	#102×C₁₀₉	1	1	yes	characteristic
207	C₂³	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	8	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
220	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
221	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
222	C₂×C₄	#123×C₁₀₉	1	2	yes	—
223	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
224	C₂×C₄	#137×C₁₀₉	1	2	yes	—
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	8	2	no	—
228	C₈	#117×C₁₀₉	1	2	yes	—
229	C₈	#123×C₁₀₉	1	2	yes	—
230	C₂²	—	2	8	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	#361×C₁₀₉	1	2	yes	—
238	C₂²	#362×C₁₀₉	1	2	yes	—
239	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
240	C₂²	#483×C₁₀₉	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	8	4	no	—
244	C₄	#381×C₁₀₉	1	1	yes	characteristic
245	C₄	#419×C₁₀₉	1	1	yes	characteristic
246	C₄	#480×C₁₀₉	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	2	2	no	—
250	C₂	C₁₀₉×#1823	1	1	yes	characteristic
251	C₂	C₂×#361×C₁₀₉	1	1	yes	characteristic
252	C₂	C₂×#362×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
prime radical	C₂⁴×C₁₀₉	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#1387	C₂	semidirect
120. #12871	C₁₀₉	direct (N × H)
130. #1387	C₂×C₁₀₉	semidirect
131. C₁₀₉	#12871	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×#107×C₁₀₉	2
C₁₀₉×#2112	2
C₁₀₉×#1810	2
C₁₀₉×#1387	2
#12871	109

Subgroups — C109×#12871 #42130865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12871 #42130865