Subgroups — C₁₀₉×#3041 #28360295

Order 13952.

526 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×AGL(1,ℤ/8)×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#535×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#518×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#463×C₁₀₉	C₂	1	4	yes	maximal, complemented
6	#459×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#418×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#416×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	AGL(1,ℤ/8)×C₁₀₉	—	2	4	no	—
14	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	complemented
15	C₂×D₈×C₁₀₉	—	2	4	no	—
16	C₂×D₈×C₁₀₉	C₂²	1	4	yes	—
17	C₂×D₈×C₁₀₉	C₂²	1	2	yes	complemented
18	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	#127×C₁₀₉	C₂²	1	2	yes	complemented
21	#124×C₁₀₉	C₂²	1	2	yes	complemented
22	#121×C₁₀₉	—	2	2	no	—
23	#120×C₁₀₉	—	2	2	no	—
24	C₄×D₄×C₁₀₉	—	2	2	no	—
25	C₄×D₄×C₁₀₉	C₂²	1	4	yes	complemented
26	#117×C₁₀₉	C₂²	1	1	yes	characteristic
27	#117×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
29	C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	complemented
30	#103×C₁₀₉	C₂²	1	2	yes	complemented
31	#102×C₁₀₉	C₂²	1	2	yes	complemented
32	#102×C₁₀₉	C₂²	1	4	yes	complemented
33	C₂⁴×C₁₀₉	—	2	1	no	—
34	C₂⁴×C₁₀₉	—	2	1	no	—
35	#41×C₁₀₉	—	2	2	no	—
36	#41×C₁₀₉	—	2	2	no	—
37	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₁₀₉	—	2	1	no	—
39	C₂×D₄×C₁₀₉	—	2	1	no	—
40	C₂×D₄×C₁₀₉	—	2	2	no	—
41	C₂×D₄×C₁₀₉	—	2	1	no	—
42	C₂×D₄×C₁₀₉	—	2	4	no	—
43	C₂×D₄×C₁₀₉	—	2	2	no	—
44	C₂×D₄×C₁₀₉	—	2	2	no	—
45	C₂×D₄×C₁₀₉	D₄	1	4	yes	complemented
46	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
47	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₀₉	—	2	1	no	—
50	C₂²×C₄×C₁₀₉	—	2	4	no	—
51	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
52	SD₁₆×C₁₀₉	—	2	4	no	—
53	D₈×C₁₀₉	—	2	8	no	—
54	D₈×C₁₀₉	—	2	4	no	—
55	M₁₆×C₁₀₉	—	2	2	no	—
56	C₂×C₈×C₁₀₉	—	2	2	no	—
57	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
58	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
59	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
60	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
62	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
63	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
64	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
65	C₄²×C₁₀₉	—	2	1	no	—
66	C₄²×C₁₀₉	C₂³	1	2	yes	—
67	C₂³×C₁₀₉	—	2	2	no	—
68	C₂³×C₁₀₉	—	2	4	no	—
69	C₂³×C₁₀₉	—	2	2	no	—
70	C₂³×C₁₀₉	—	2	1	no	—
71	C₂³×C₁₀₉	—	4	2	no	—
72	C₂³×C₁₀₉	—	8	2	no	—
73	C₂³×C₁₀₉	—	8	2	no	—
74	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₁₀₉	—	2	1	no	—
76	Q₈×C₁₀₉	—	2	1	no	—
77	D₄×C₁₀₉	—	2	4	no	—
78	D₄×C₁₀₉	—	2	2	no	—
79	D₄×C₁₀₉	—	2	1	no	—
80	D₄×C₁₀₉	—	2	2	no	—
81	D₄×C₁₀₉	—	2	2	no	—
82	D₄×C₁₀₉	—	2	1	no	—
83	D₄×C₁₀₉	—	4	2	no	—
84	D₄×C₁₀₉	—	4	2	no	—
85	D₄×C₁₀₉	—	8	4	no	—
86	D₄×C₁₀₉	C₂×D₄	1	4	yes	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	2	1	no	—
89	C₂×C₄×C₁₀₉	—	4	2	no	—
90	C₂×C₄×C₁₀₉	—	8	4	no	—
91	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
92	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₀₉	#41	1	2	yes	—
95	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
96	C₈×C₁₀₉	—	2	2	no	—
97	C₈×C₁₀₉	—	2	2	no	—
98	C₂²×C₁₀₉	—	2	2	no	—
99	C₂²×C₁₀₉	—	2	4	no	—
100	C₂²×C₁₀₉	—	4	4	no	—
101	C₂²×C₁₀₉	—	4	1	no	—
102	C₂²×C₁₀₉	—	4	2	no	—
103	C₂²×C₁₀₉	—	4	2	no	—
104	C₂²×C₁₀₉	—	4	4	no	—
105	C₂²×C₁₀₉	—	8	4	no	—
106	C₂²×C₁₀₉	—	8	4	no	—
107	C₂²×C₁₀₉	#121	1	2	yes	—
108	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₁₀₉	—	2	4	no	—
110	C₄×C₁₀₉	—	4	1	no	—
111	C₄×C₁₀₉	—	4	2	no	—
112	C₄×C₁₀₉	#121	1	2	yes	—
113	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
115	C₂×C₁₀₉	—	2	2	no	—
116	C₂×C₁₀₉	—	4	4	no	—
117	C₂×C₁₀₉	—	8	1	no	—
118	C₂×C₁₀₉	—	8	2	no	—
119	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
120	C₂×C₁₀₉	#576	1	2	yes	—
121	#3041	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₁₀₉	#3041	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×AGL(1,ℤ/8)	C₂×C₁₀₉	1	1	yes	characteristic, complemented
124	C₂²×D₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
125	#535	C₂×C₁₀₉	1	1	yes	characteristic, complemented
126	#518	C₂×C₁₀₉	1	1	yes	characteristic, complemented
127	#463	C₂×C₁₀₉	1	4	yes	complemented
128	#459	C₂×C₁₀₉	1	4	yes	complemented
129	#424	C₂×C₁₀₉	1	1	yes	characteristic, complemented
130	#418	C₂×C₁₀₉	1	1	yes	characteristic, complemented
131	#416	C₂×C₁₀₉	1	1	yes	characteristic, complemented
132	C₂×#41	C₂²×C₁₀₉	1	1	yes	characteristic
133	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
134	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
135	AGL(1,ℤ/8)	—	2	4	no	—
136	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	complemented
137	C₂×D₈	—	2	4	no	—
138	C₂×D₈	C₂²×C₁₀₉	1	4	yes	—
139	C₂×D₈	C₂²×C₁₀₉	1	2	yes	complemented
140	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
141	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
142	#127	C₂²×C₁₀₉	1	2	yes	complemented
143	#124	C₂²×C₁₀₉	1	2	yes	complemented
144	#121	—	2	2	no	—
145	#120	—	2	2	no	—
146	C₄×D₄	—	2	2	no	—
147	C₄×D₄	C₂²×C₁₀₉	1	4	yes	complemented
148	#117	C₂²×C₁₀₉	1	1	yes	characteristic
149	#117	C₂²×C₁₀₉	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	—	2	2	no	—
151	C₄⋊C₈	C₂²×C₁₀₉	1	4	yes	complemented
152	#103	C₂²×C₁₀₉	1	2	yes	complemented
153	#102	C₂²×C₁₀₉	1	2	yes	complemented
154	#102	C₂²×C₁₀₉	1	4	yes	complemented
155	C₂⁴	—	2	1	no	—
156	C₂⁴	—	2	1	no	—
157	#41	—	2	2	no	—
158	#41	—	2	2	no	—
159	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	—	2	1	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	D₄×C₁₀₉	1	4	yes	complemented
168	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
169	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
170	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
174	SD₁₆	—	2	4	no	—
175	D₈	—	2	8	no	—
176	D₈	—	2	4	no	—
177	M₁₆	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
180	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
181	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
182	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
183	C₂²⋊C₄	—	2	1	no	—
184	C₂²⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
187	C₄²	—	2	1	no	—
188	C₄²	C₂³×C₁₀₉	1	2	yes	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	4	2	no	—
194	C₂³	—	8	2	no	—
195	C₂³	—	8	2	no	—
196	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
197	Q₈	—	2	1	no	—
198	Q₈	—	2	1	no	—
199	D₄	—	2	4	no	—
200	D₄	—	2	2	no	—
201	D₄	—	2	1	no	—
202	D₄	—	2	2	no	—
203	D₄	—	2	2	no	—
204	D₄	—	2	1	no	—
205	D₄	—	4	2	no	—
206	D₄	—	4	2	no	—
207	D₄	—	8	4	no	—
208	D₄	C₂×D₄×C₁₀₉	1	4	yes	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
214	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
215	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
216	C₂×C₄	#41×C₁₀₉	1	2	yes	—
217	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	4	4	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	4	no	—
227	C₂²	—	8	4	no	—
228	C₂²	—	8	4	no	—
229	C₂²	#121×C₁₀₉	1	2	yes	—
230	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
231	C₄	—	2	4	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	2	no	—
234	C₄	#121×C₁₀₉	1	2	yes	—
235	C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
236	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	—	4	4	no	—
239	C₂	—	8	2	no	—
240	C₂	—	8	1	no	—
241	C₂	C₂×#121×C₁₀₉	1	1	yes	characteristic
242	C₂	#576×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ #117×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×AGL(1,ℤ/8)×C₁₀₉	C₂	semidirect
2. C₂²×D₈×C₁₀₉	C₂	semidirect
3. #535×C₁₀₉	C₂	semidirect
4. #518×C₁₀₉	C₂	semidirect
5. #463×C₁₀₉	C₂	semidirect
6. #459×C₁₀₉	C₂	semidirect
7. #424×C₁₀₉	C₂	semidirect
8. #418×C₁₀₉	C₂	semidirect
9. #416×C₁₀₉	C₂	semidirect
14. C₂×SD₁₆×C₁₀₉	C₂²	semidirect
17. C₂×D₈×C₁₀₉	C₂²	semidirect
20. #127×C₁₀₉	C₂²	semidirect
21. #124×C₁₀₉	C₂²	semidirect
25. C₄×D₄×C₁₀₉	C₂²	semidirect
29. C₄⋊C₈×C₁₀₉	C₂²	semidirect
30. #103×C₁₀₉	C₂²	semidirect
31. #102×C₁₀₉	C₂²	semidirect
32. #102×C₁₀₉	C₂²	semidirect
45. C₂×D₄×C₁₀₉	D₄	semidirect
121. #3041	C₁₀₉	direct (N × H)
122. C₁₀₉	#3041	direct (N × H)
123. C₂×AGL(1,ℤ/8)	C₂×C₁₀₉	semidirect
124. C₂²×D₈	C₂×C₁₀₉	semidirect
125. #535	C₂×C₁₀₉	semidirect
126. #518	C₂×C₁₀₉	semidirect
127. #463	C₂×C₁₀₉	semidirect
128. #459	C₂×C₁₀₉	semidirect
129. #424	C₂×C₁₀₉	semidirect
130. #418	C₂×C₁₀₉	semidirect
131. #416	C₂×C₁₀₉	semidirect
136. C₂×SD₁₆	C₂²×C₁₀₉	semidirect
139. C₂×D₈	C₂²×C₁₀₉	semidirect
142. #127	C₂²×C₁₀₉	semidirect
143. #124	C₂²×C₁₀₉	semidirect
147. C₄×D₄	C₂²×C₁₀₉	semidirect
151. C₄⋊C₈	C₂²×C₁₀₉	semidirect
152. #103	C₂²×C₁₀₉	semidirect
153. #102	C₂²×C₁₀₉	semidirect
154. #102	C₂²×C₁₀₉	semidirect
167. C₂×D₄	D₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×AGL(1,ℤ/8)×C₁₀₉	2
C₂²×D₈×C₁₀₉	2
#535×C₁₀₉	2
#518×C₁₀₉	2
#463×C₁₀₉	2
#459×C₁₀₉	2
#424×C₁₀₉	2
#418×C₁₀₉	2
#416×C₁₀₉	2
#3041	109

Subgroups — C109×#3041 #28360295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3041 #28360295