Subgroups — C₁₀₉×#12929 #42130923

Order 27904.

458 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#365×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#2112	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1810	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1383	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₁₀₉	—	2	1	no	—
8	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
9	C₂×#108×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#108×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#106×C₁₀₉	—	2	1	no	—
12	C₂×#106×C₁₀₉	—	2	4	no	—
13	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
14	#407×C₁₀₉	—	2	1	no	—
15	#365×C₁₀₉	—	2	4	no	—
16	C₂²⋊C₁₆×C₁₀₉	C₄	1	4	yes	complemented
17	#341×C₁₀₉	—	2	1	no	—
18	C₂³×C₄×C₁₀₉	—	4	1	no	—
19	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₀₉	—	2	1	no	—
21	C₂×M₁₆×C₁₀₉	—	2	1	no	—
22	C₂²×C₈×C₁₀₉	—	2	2	no	—
23	C₂²×C₈×C₁₀₉	—	2	2	no	—
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
27	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
29	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₀₉	—	4	2	no	—
31	C₂×C₄⋊C₄×C₁₀₉	—	4	1	no	—
32	M₃₂×C₁₀₉	—	2	2	no	—
33	M₃₂×C₁₀₉	—	2	2	no	—
34	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
35	#108×C₁₀₉	—	2	2	no	—
36	#108×C₁₀₉	—	2	2	no	—
37	#106×C₁₀₉	—	2	4	no	—
38	#106×C₁₀₉	—	2	2	no	—
39	#106×C₁₀₉	D₄	1	4	yes	—
40	C₂²⋊C₈×C₁₀₉	—	4	2	no	—
41	C₂⁴×C₁₀₉	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	—	2	1	no	—
43	C₂²×C₄×C₁₀₉	—	2	1	no	—
44	C₂²×C₄×C₁₀₉	—	2	1	no	—
45	C₂²×C₄×C₁₀₉	—	2	1	no	—
46	C₂²×C₄×C₁₀₉	—	4	4	no	—
47	C₂²×C₄×C₁₀₉	—	4	2	no	—
48	C₂²×C₄×C₁₀₉	—	8	4	no	—
49	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	M₁₆×C₁₀₉	—	4	1	no	—
54	M₁₆×C₁₀₉	—	4	1	no	—
55	M₁₆×C₁₀₉	—	4	1	no	—
56	M₁₆×C₁₀₉	—	4	1	no	—
57	C₂×C₈×C₁₀₉	—	2	1	no	—
58	C₂×C₈×C₁₀₉	—	2	1	no	—
59	C₂×C₈×C₁₀₉	—	2	4	no	—
60	C₂×C₈×C₁₀₉	—	2	4	no	—
61	C₂×C₈×C₁₀₉	—	4	1	no	—
62	C₂×C₈×C₁₀₉	—	4	1	no	—
63	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₁₀₉	#41	1	2	yes	—
68	C₄⋊C₄×C₁₀₉	—	2	4	no	—
69	C₄⋊C₄×C₁₀₉	—	4	4	no	—
70	C₄⋊C₄×C₁₀₉	—	4	2	no	—
71	C₁₆×C₁₀₉	—	2	4	no	—
72	C₂³×C₁₀₉	—	2	4	no	—
73	C₂³×C₁₀₉	—	2	1	no	—
74	C₂³×C₁₀₉	—	2	1	no	—
75	C₂³×C₁₀₉	#106	1	2	yes	—
76	C₂³×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₀₉	—	2	1	no	—
78	C₂×C₄×C₁₀₉	—	2	1	no	—
79	C₂×C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄×C₁₀₉	—	2	2	no	—
81	C₂×C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	—	4	4	no	—
84	C₂×C₄×C₁₀₉	—	8	8	no	—
85	C₂×C₄×C₁₀₉	—	8	4	no	—
86	C₂×C₄×C₁₀₉	#106	1	2	yes	—
87	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	#137	1	1	yes	characteristic
93	C₈×C₁₀₉	—	2	2	no	—
94	C₈×C₁₀₉	—	8	1	no	—
95	C₈×C₁₀₉	—	8	1	no	—
96	C₈×C₁₀₉	C₄×D₄	1	2	yes	—
97	C₈×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
98	C₂²×C₁₀₉	—	2	4	no	—
99	C₂²×C₁₀₉	—	2	4	no	—
100	C₂²×C₁₀₉	—	2	2	no	—
101	C₂²×C₁₀₉	—	2	2	no	—
102	C₂²×C₁₀₉	—	2	2	no	—
103	C₂²×C₁₀₉	#365	1	4	yes	complemented
104	C₂²×C₁₀₉	C₂×#106	1	1	yes	characteristic
105	C₂²×C₁₀₉	#450	1	1	yes	characteristic
106	C₂²×C₁₀₉	#483	1	1	yes	characteristic
107	C₄×C₁₀₉	—	2	2	no	—
108	C₄×C₁₀₉	—	8	4	no	—
109	C₄×C₁₀₉	#385	1	1	yes	characteristic
110	C₄×C₁₀₉	#428	1	1	yes	characteristic
111	C₄×C₁₀₉	#448	1	1	yes	characteristic
112	C₄×C₁₀₉	#481	1	1	yes	characteristic
113	#12929	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂×C₁₀₉	—	2	1	no	—
115	C₂×C₁₀₉	—	2	4	no	—
116	C₂×C₁₀₉	—	2	1	no	—
117	C₂×C₁₀₉	#1853	1	1	yes	characteristic
118	C₂×C₁₀₉	C₂×#365	1	2	yes	—
119	C₂²×#106	C₂×C₁₀₉	1	1	yes	characteristic
120	C₂×#365	C₂×C₁₀₉	1	2	yes	complemented
121	#2112	C₂×C₁₀₉	1	1	yes	characteristic
122	#1810	C₂×C₁₀₉	1	1	yes	characteristic
123	#1383	C₂×C₁₀₉	1	2	yes	complemented
124	C₁₀₉	#12929	1	1	yes	characteristic, direct product factor, Sylow
125	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
126	C₂²×C₄⋊C₄	—	2	1	no	—
127	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
128	C₂×#108	C₂²×C₁₀₉	1	1	yes	characteristic
129	C₂×#108	C₂²×C₁₀₉	1	1	yes	characteristic
130	C₂×#106	—	2	1	no	—
131	C₂×#106	—	2	4	no	—
132	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
133	#407	—	2	1	no	—
134	#365	—	2	4	no	—
135	C₂²⋊C₁₆	C₄×C₁₀₉	1	4	yes	complemented
136	#341	—	2	1	no	—
137	C₂³×C₄	—	4	1	no	—
138	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
139	C₂×M₁₆	—	2	1	no	—
140	C₂×M₁₆	—	2	1	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
144	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
145	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
146	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	4	2	no	—
150	C₂×C₄⋊C₄	—	4	1	no	—
151	M₃₂	—	2	2	no	—
152	M₃₂	—	2	2	no	—
153	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	complemented
154	#108	—	2	2	no	—
155	#108	—	2	2	no	—
156	#106	—	2	4	no	—
157	#106	—	2	2	no	—
158	#106	D₄×C₁₀₉	1	4	yes	—
159	C₂²⋊C₈	—	4	2	no	—
160	C₂⁴	SD₁₆×C₁₀₉	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	—	8	4	no	—
168	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
169	C₂²×C₄	SD₁₆×C₁₀₉	1	1	yes	characteristic
170	C₂²×C₄	SD₁₆×C₁₀₉	1	2	yes	—
171	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
172	M₁₆	—	4	1	no	—
173	M₁₆	—	4	1	no	—
174	M₁₆	—	4	1	no	—
175	M₁₆	—	4	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
183	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
184	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
185	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
186	C₂×C₈	#41×C₁₀₉	1	2	yes	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₁₆	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	1	no	—
194	C₂³	#106×C₁₀₉	1	2	yes	—
195	C₂³	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	8	8	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	#106×C₁₀₉	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
207	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
208	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
209	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
210	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
211	C₂×C₄	#137×C₁₀₉	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	8	1	no	—
214	C₈	—	8	1	no	—
215	C₈	C₄×D₄×C₁₀₉	1	2	yes	—
216	C₈	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	#365×C₁₀₉	1	4	yes	complemented
223	C₂²	C₂×#106×C₁₀₉	1	1	yes	characteristic
224	C₂²	#450×C₁₀₉	1	1	yes	characteristic
225	C₂²	#483×C₁₀₉	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	8	4	no	—
228	C₄	#385×C₁₀₉	1	1	yes	characteristic
229	C₄	#428×C₁₀₉	1	1	yes	characteristic
230	C₄	#448×C₁₀₉	1	1	yes	characteristic
231	C₄	#481×C₁₀₉	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	4	no	—
234	C₂	—	2	1	no	—
235	C₂	C₁₀₉×#1853	1	1	yes	characteristic
236	C₂	C₂×#365×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#106
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
2. C₂×#365×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1383	C₂	semidirect
16. C₂²⋊C₁₆×C₁₀₉	C₄	semidirect
34. C₂×C₁₆×C₁₀₉	C₂×C₄	semidirect
103. C₂²×C₁₀₉	#365	semidirect
113. #12929	C₁₀₉	direct (N × H)
120. C₂×#365	C₂×C₁₀₉	semidirect
123. #1383	C₂×C₁₀₉	semidirect
124. C₁₀₉	#12929	direct (N × H)
135. C₂²⋊C₁₆	C₄×C₁₀₉	semidirect
153. C₂×C₁₆	C₂×C₄×C₁₀₉	semidirect
222. C₂²	#365×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#106×C₁₀₉	2
C₂×#365×C₁₀₉	2
C₁₀₉×#2112	2
C₁₀₉×#1810	2
C₁₀₉×#1383	2
#12929	109

Subgroups — C109×#12929 #42130923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12929 #42130923