Subgroups — #430×D₁₀₉ #28371636

Order 13952.

460 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₀₉	C₂	1	2	yes	maximal
2	C₂×M₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20295075	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295074	C₂	1	2	yes	maximal, complemented
5	#20295073	C₂	1	4	yes	maximal, complemented
6	#108×D₁₀₉	C₂	1	4	yes	maximal, complemented
7	#430×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#13182013	C₂²	1	2	yes	—
10	C₂×#13182013	C₂²	1	1	yes	characteristic
11	C₂×#13182012	—	2	2	no	—
12	C₂×#13182012	C₂²	1	2	yes	—
13	M₁₆×D₁₀₉	—	2	2	no	—
14	M₁₆×D₁₀₉	—	2	2	no	—
15	M₁₆×D₁₀₉	C₄	1	4	yes	—
16	M₁₆×D₁₀₉	C₄	1	4	yes	—
17	M₁₆×D₁₀₉	C₂²	1	4	yes	—
18	M₁₆×D₁₀₉	C₂²	1	4	yes	—
19	C₂×C₈×D₁₀₉	—	2	2	no	—
20	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
21	#16812635	C₂²	1	4	yes	complemented
22	#16812634	C₂²	1	8	yes	complemented
23	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
24	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
25	#108×C₁₀₉	C₂²	1	4	yes	complemented
26	C₂³×D₁₀₉	Q₈	1	1	yes	characteristic
27	C₂²×Dic₁₀₉	Q₈	1	1	yes	characteristic
28	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
31	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
32	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
33	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
34	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
35	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
36	#13182013	—	2	2	no	—
37	#13182013	C₂×C₄	1	4	yes	—
38	#13182013	C₂³	1	4	yes	—
39	#13182012	—	2	4	no	—
40	#13182012	—	2	4	no	—
41	#13182012	C₂×C₄	1	8	yes	—
42	C₂×#11724784	—	2	2	no	—
43	C₂×#11724784	C₂³	1	2	yes	—
44	C₈×D₁₀₉	—	2	4	no	—
45	C₈×D₁₀₉	—	2	4	no	—
46	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
47	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
48	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
49	M₁₆×C₁₀₉	—	2	2	no	—
50	M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
51	M₁₆×C₁₀₉	C₂³	1	4	yes	—
52	C₂×C₈×C₁₀₉	—	2	2	no	—
53	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
54	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×D₁₀₉	—	4	2	no	—
56	C₂²×D₁₀₉	C₄⋊C₄	1	2	yes	—
57	C₂²×D₁₀₉	C₄⋊C₄	1	2	yes	—
58	C₂²×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
59	C₂²×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
64	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
65	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
66	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
68	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
69	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
70	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
71	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
72	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
73	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
74	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
75	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
76	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
77	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
78	C₈×C₁₀₉	—	2	4	no	—
79	C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
80	#11724784	—	2	4	no	—
81	#11724784	C₂²×C₄	1	4	yes	—
82	C₂²×C₁₀₉	—	4	1	no	—
83	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
84	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
85	C₂×D₁₀₉	—	2	4	no	—
86	C₂×D₁₀₉	—	2	2	no	—
87	C₂×D₁₀₉	—	4	2	no	—
88	C₂×D₁₀₉	—	4	1	no	—
89	C₂×D₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×D₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×D₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
93	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
94	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
95	Dic₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	Dic₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	Dic₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₁₀₉	—	2	2	no	—
99	C₂×C₁₀₉	—	2	1	no	—
100	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	D₁₀₉	—	2	2	no	—
102	D₁₀₉	—	2	1	no	—
103	D₁₀₉	#430	1	2	yes	direct product factor
104	C₂×#430	—	109	1	no	maximal, Sylow
105	C₁₀₉	C₂×#430	1	1	yes	characteristic, complemented, Sylow
106	C₂²×M₁₆	—	109	1	no	—
107	C₂²×M₁₆	—	109	2	no	—
108	#430	—	109	2	no	—
109	#430	—	109	1	no	—
110	#430	—	109	4	no	—
111	#430	D₁₀₉	1	1	yes	characteristic, direct product factor
112	C₂×#108	—	109	4	no	—
113	C₂³×C₄	—	109	1	no	—
114	C₂×M₁₆	—	109	1	no	—
115	C₂×M₁₆	—	109	4	no	—
116	C₂×M₁₆	—	109	4	no	—
117	C₂×M₁₆	—	109	2	no	—
118	C₂×M₁₆	—	109	2	no	—
119	C₂×M₁₆	—	109	4	no	—
120	C₂×M₁₆	—	109	4	no	—
121	C₂×M₁₆	—	218	2	no	—
122	C₂×M₁₆	—	218	2	no	—
123	C₂×M₁₆	—	218	2	no	—
124	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
125	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
126	C₂²×C₈	—	109	2	no	—
127	C₂²×C₈	—	218	2	no	—
128	#108	—	109	8	no	—
129	#108	—	109	4	no	—
130	#108	C₂×D₁₀₉	1	4	yes	complemented
131	C₂⁴	—	109	1	no	—
132	C₂²×C₄	—	109	1	no	—
133	C₂²×C₄	—	109	2	no	—
134	C₂²×C₄	—	109	2	no	—
135	C₂²×C₄	—	109	2	no	—
136	C₂²×C₄	—	109	2	no	—
137	C₂²×C₄	—	109	1	no	—
138	C₂²×C₄	—	109	1	no	—
139	C₂²×C₄	—	109	1	no	—
140	C₂²×C₄	—	109	1	no	—
141	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	109	4	no	—
144	M₁₆	—	109	8	no	—
145	M₁₆	—	109	4	no	—
146	M₁₆	—	218	4	no	—
147	M₁₆	—	218	4	no	—
148	M₁₆	—	218	2	no	—
149	M₁₆	C₄×D₁₀₉	1	4	yes	—
150	M₁₆	C₂²×D₁₀₉	1	4	yes	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	109	2	no	—
153	C₂×C₈	—	109	4	no	—
154	C₂×C₈	—	109	4	no	—
155	C₂×C₈	—	218	4	no	—
156	C₂×C₈	—	218	4	no	—
157	C₂×C₈	—	218	2	no	—
158	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
159	C₂³	—	109	2	no	—
160	C₂³	—	109	1	no	—
161	C₂³	—	109	2	no	—
162	C₂³	—	109	1	no	—
163	C₂³	—	436	2	no	—
164	C₂³	Q₈×D₁₀₉	1	1	yes	characteristic
165	C₂×C₄	—	109	2	no	—
166	C₂×C₄	—	109	2	no	—
167	C₂×C₄	—	109	1	no	—
168	C₂×C₄	—	109	1	no	—
169	C₂×C₄	—	109	2	no	—
170	C₂×C₄	—	109	2	no	—
171	C₂×C₄	—	109	2	no	—
172	C₂×C₄	—	109	2	no	—
173	C₂×C₄	—	109	1	no	—
174	C₂×C₄	—	109	1	no	—
175	C₂×C₄	—	109	2	no	—
176	C₂×C₄	—	109	1	no	—
177	C₂×C₄	—	109	2	no	—
178	C₂×C₄	—	109	1	no	—
179	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
180	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
181	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
182	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
183	C₈	—	2	4	no	—
184	C₈	—	109	4	no	—
185	C₈	—	218	4	no	—
186	C₈	C₂×C₄×D₁₀₉	1	4	yes	—
187	C₂²	—	4	1	no	—
188	C₂²	—	109	1	no	—
189	C₂²	—	109	1	no	—
190	C₂²	—	109	2	no	—
191	C₂²	—	436	2	no	—
192	C₂²	—	436	1	no	—
193	C₂²	—	218	4	no	—
194	C₂²	—	218	2	no	—
195	C₂²	C₄⋊C₄×D₁₀₉	1	2	yes	—
196	C₂²	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
197	C₄	—	109	1	no	—
198	C₄	—	109	1	no	—
199	C₄	—	109	2	no	—
200	C₄	C₄⋊C₄×D₁₀₉	1	2	yes	—
201	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
202	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	2	1	no	—
205	C₂	—	109	2	no	—
206	C₂	—	218	1	no	—
207	C₂	—	218	2	no	—
208	C₂	C₂×C₄⋊C₄×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#430×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₀₉	Q₈
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#430	D₁₀₉
hypocenter	C₁₀₉	C₂×#430
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

Complements

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

N	H ≅ G/N	type
3. #20295075	C₂	semidirect
4. #20295074	C₂	semidirect
5. #20295073	C₂	semidirect
6. #108×D₁₀₉	C₂	semidirect
7. #430×C₁₀₉	C₂	semidirect
21. #16812635	C₂²	semidirect
22. #16812634	C₂²	semidirect
25. #108×C₁₀₉	C₂²	semidirect
103. D₁₀₉	#430	direct (N × H)
105. C₁₀₉	C₂×#430	semidirect
111. #430	D₁₀₉	direct (N × H)
130. #108	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₀₉	2
#20295075	2
#20295074	2
#20295073	2
#108×D₁₀₉	2
#430×C₁₀₉	2
C₂×#430	109

Subgroups — #430×D109 #28371636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #430×D₁₀₉ #28371636