Subgroups — C₁₀₉×#3339 #28360593

Order 13952.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#553×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#499×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#482×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#481×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#478×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#466×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#466×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	#126×C₁₀₉	—	2	2	no	—
16	#125×C₁₀₉	C₂²	1	2	yes	—
17	#123×C₁₀₉	—	2	2	no	—
18	#121×C₁₀₉	—	2	2	no	—
19	#121×C₁₀₉	C₂²	1	2	yes	—
20	#121×C₁₀₉	C₂²	1	2	yes	—
21	C₄×D₄×C₁₀₉	—	2	2	no	—
22	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
23	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
24	#117×C₁₀₉	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
26	#107×C₁₀₉	C₂²	1	1	yes	characteristic
27	#107×C₁₀₉	C₂²	1	1	yes	characteristic
28	#107×C₁₀₉	C₂²	1	1	yes	characteristic
29	#107×C₁₀₉	C₂²	1	2	yes	—
30	#106×C₁₀₉	C₂²	1	2	yes	—
31	#106×C₁₀₉	C₂²	1	2	yes	—
32	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
33	#102×C₁₀₉	C₂²	1	2	yes	—
34	#102×C₁₀₉	C₂²	1	2	yes	—
35	#102×C₁₀₉	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂×D₄×C₁₀₉	—	2	2	no	—
40	C₂×D₄×C₁₀₉	—	2	2	no	—
41	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
42	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₀₉	—	2	2	no	—
44	C₂²×C₄×C₁₀₉	—	2	2	no	—
45	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
47	D₈×C₁₀₉	—	2	2	no	—
48	C₂×C₈×C₁₀₉	—	2	2	no	—
49	C₂×C₈×C₁₀₉	—	2	2	no	—
50	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₁₀₉	—	2	2	no	—
55	C₄⋊C₄×C₁₀₉	—	2	2	no	—
56	C₄⋊C₄×C₁₀₉	—	2	2	no	—
57	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
59	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
60	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
62	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
63	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
64	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
65	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
67	C₄²×C₁₀₉	—	2	2	no	—
68	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂³×C₁₀₉	—	2	2	no	—
70	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
72	D₄×C₁₀₉	—	2	2	no	—
73	D₄×C₁₀₉	—	2	2	no	—
74	D₄×C₁₀₉	—	2	2	no	—
75	D₄×C₁₀₉	—	4	4	no	—
76	D₄×C₁₀₉	—	8	2	no	—
77	C₂×C₄×C₁₀₉	—	2	1	no	—
78	C₂×C₄×C₁₀₉	—	2	1	no	—
79	C₂×C₄×C₁₀₉	—	2	1	no	—
80	C₂×C₄×C₁₀₉	—	2	1	no	—
81	C₂×C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	8	2	no	—
85	C₂×C₄×C₁₀₉	—	8	2	no	—
86	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
91	C₈×C₁₀₉	—	2	1	no	—
92	C₈×C₁₀₉	—	2	1	no	—
93	C₈×C₁₀₉	—	2	1	no	—
94	C₈×C₁₀₉	#41	1	2	yes	—
95	C₂²×C₁₀₉	—	2	1	no	—
96	C₂²×C₁₀₉	—	2	1	no	—
97	C₂²×C₁₀₉	—	2	1	no	—
98	C₂²×C₁₀₉	—	2	1	no	—
99	C₂²×C₁₀₉	—	2	1	no	—
100	C₂²×C₁₀₉	—	4	2	no	—
101	C₂²×C₁₀₉	—	4	2	no	—
102	C₂²×C₁₀₉	—	4	2	no	—
103	C₂²×C₁₀₉	#135	1	2	yes	—
104	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₁₀₉	—	2	1	no	—
106	C₄×C₁₀₉	—	2	1	no	—
107	C₄×C₁₀₉	—	2	1	no	—
108	C₄×C₁₀₉	—	2	1	no	—
109	C₄×C₁₀₉	—	4	2	no	—
110	C₄×C₁₀₉	—	4	2	no	—
111	C₄×C₁₀₉	—	4	2	no	—
112	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
113	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×C₁₀₉	—	2	2	no	—
115	C₂×C₁₀₉	—	4	1	no	—
116	C₂×C₁₀₉	—	8	2	no	—
117	C₂×C₁₀₉	#547	1	1	yes	characteristic
118	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
119	C₂×C₁₀₉	#576	1	1	yes	characteristic
120	#3339	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₁₀₉	#3339	1	1	yes	characteristic, direct product factor, Sylow
122	#553	C₂×C₁₀₉	1	2	yes	complemented
123	#499	C₂×C₁₀₉	1	1	yes	characteristic, complemented
124	#482	C₂×C₁₀₉	1	2	yes	complemented
125	#481	C₂×C₁₀₉	1	2	yes	complemented
126	#478	C₂×C₁₀₉	1	2	yes	complemented
127	#466	C₂×C₁₀₉	1	1	yes	characteristic, complemented
128	#466	C₂×C₁₀₉	1	1	yes	characteristic, complemented
129	C₄×D₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
130	C₈×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
131	#427	C₂×C₁₀₉	1	1	yes	characteristic, complemented
132	C₂×#107	C₂×C₁₀₉	1	1	yes	characteristic, complemented
133	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
134	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
135	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
136	#126	—	2	2	no	—
137	#125	C₂²×C₁₀₉	1	2	yes	—
138	#123	—	2	2	no	—
139	#121	—	2	2	no	—
140	#121	C₂²×C₁₀₉	1	2	yes	—
141	#121	C₂²×C₁₀₉	1	2	yes	—
142	C₄×D₄	—	2	2	no	—
143	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
144	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
145	#117	C₂²×C₁₀₉	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
147	#107	C₂²×C₁₀₉	1	1	yes	characteristic
148	#107	C₂²×C₁₀₉	1	1	yes	characteristic
149	#107	C₂²×C₁₀₉	1	2	yes	—
150	#107	C₂²×C₁₀₉	1	1	yes	characteristic
151	#106	C₂²×C₁₀₉	1	2	yes	—
152	#106	C₂²×C₁₀₉	1	2	yes	—
153	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
154	#102	C₂²×C₁₀₉	1	2	yes	—
155	#102	C₂²×C₁₀₉	1	2	yes	—
156	#102	C₂²×C₁₀₉	1	2	yes	—
157	C₂²⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
158	C₂²⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
159	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
163	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
168	D₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
172	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
174	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
180	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
181	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
187	C₂²⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
188	C₄²	—	2	2	no	—
189	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
190	C₂³	—	2	2	no	—
191	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
192	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
193	D₄	—	2	2	no	—
194	D₄	—	2	2	no	—
195	D₄	—	2	2	no	—
196	D₄	—	4	4	no	—
197	D₄	—	8	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
211	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	#41×C₁₀₉	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	#135×C₁₀₉	1	2	yes	—
225	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
234	C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	—	8	2	no	—
238	C₂	#547×C₁₀₉	1	1	yes	characteristic
239	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
240	C₂	#576×C₁₀₉	1	1	yes	characteristic

	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 ◁ C₄×D₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #553×C₁₀₉	C₂	semidirect
2. #499×C₁₀₉	C₂	semidirect
3. #482×C₁₀₉	C₂	semidirect
4. #481×C₁₀₉	C₂	semidirect
5. #478×C₁₀₉	C₂	semidirect
6. #466×C₁₀₉	C₂	semidirect
7. #466×C₁₀₉	C₂	semidirect
8. C₄×D₈×C₁₀₉	C₂	semidirect
9. C₈×D₄×C₁₀₉	C₂	semidirect
10. #427×C₁₀₉	C₂	semidirect
11. C₂×#107×C₁₀₉	C₂	semidirect
120. #3339	C₁₀₉	direct (N × H)
121. C₁₀₉	#3339	direct (N × H)
122. #553	C₂×C₁₀₉	semidirect
123. #499	C₂×C₁₀₉	semidirect
124. #482	C₂×C₁₀₉	semidirect
125. #481	C₂×C₁₀₉	semidirect
126. #478	C₂×C₁₀₉	semidirect
127. #466	C₂×C₁₀₉	semidirect
128. #466	C₂×C₁₀₉	semidirect
129. C₄×D₈	C₂×C₁₀₉	semidirect
130. C₈×D₄	C₂×C₁₀₉	semidirect
131. #427	C₂×C₁₀₉	semidirect
132. C₂×#107	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#553×C₁₀₉	2
#499×C₁₀₉	2
#482×C₁₀₉	2
#481×C₁₀₉	2
#478×C₁₀₉	2
#466×C₁₀₉	2
C₄×D₈×C₁₀₉	2
C₈×D₄×C₁₀₉	2
#427×C₁₀₉	2
C₂×#107×C₁₀₉	2
#3339	109

Subgroups — C109×#3339 #28360593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3339 #28360593