Subgroups — #427×D₁₀₉ #28371608

Order 13952.

512 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#117×D₁₀₉	C₂	1	2	yes	maximal
3	#20295064	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295063	C₂	1	2	yes	maximal, complemented
5	#20295062	C₂	1	2	yes	maximal, complemented
6	#20295061	C₂	1	2	yes	maximal, complemented
7	#107×D₁₀₉	C₂	1	2	yes	maximal, complemented
8	#106×D₁₀₉	C₂	1	2	yes	maximal, complemented
9	#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×#11724784	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₁₀₉	C₄	1	8	yes	complemented
13	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
17	#16812658	C₂²	1	2	yes	—
18	#16812657	C₂²	1	4	yes	—
19	#16812656	—	2	2	no	—
20	#16812655	—	2	2	no	—
21	C₄⋊C₄×D₁₀₉	C₂²	1	4	yes	—
22	C₂²⋊C₄×D₁₀₉	—	2	2	no	—
23	C₄²×D₁₀₉	—	2	2	no	—
24	#16812633	C₂²	1	2	yes	complemented
25	#16812632	C₂²	1	4	yes	complemented
26	#16812631	C₂²	1	2	yes	complemented
27	#16812630	C₂²	1	4	yes	complemented
28	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
29	#117×C₁₀₉	C₂²	1	2	yes	—
30	#107×C₁₀₉	C₂²	1	2	yes	complemented
31	#106×C₁₀₉	C₂²	1	2	yes	complemented
32	C₂³×D₁₀₉	D₄	1	1	yes	characteristic
33	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₄×D₁₀₉	—	2	4	no	—
35	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
36	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
37	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
38	C₂×C₄×D₁₀₉	Q₈	1	4	yes	—
39	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
40	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
41	C₂×#11724784	C₂×C₄	1	4	yes	complemented
42	C₂×#11724784	C₂³	1	1	yes	characteristic
43	C₂×#11724784	C₂³	1	1	yes	characteristic
44	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
45	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
46	C₈×D₁₀₉	C₂×C₄	1	4	yes	complemented
47	C₈×D₁₀₉	C₂×C₄	1	4	yes	complemented
48	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
49	#13182008	—	2	4	no	—
50	C₂²⋊Dic₁₀₉	—	2	2	no	—
51	#13182006	—	2	4	no	—
52	C₄×Dic₁₀₉	—	2	2	no	—
53	C₄×Dic₁₀₉	—	2	4	no	—
54	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
56	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
59	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
60	C₄²×C₁₀₉	—	2	2	no	—
61	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×D₁₀₉	—	2	4	no	—
63	C₂²×D₁₀₉	C₄⋊C₄	1	4	yes	—
64	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₀₉	—	2	4	no	—
67	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
71	C₂×Dic₁₀₉	—	2	4	no	—
72	C₂×Dic₁₀₉	C₄⋊C₄	1	4	yes	—
73	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
74	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
75	C₄×D₁₀₉	—	4	8	no	—
76	C₄×D₁₀₉	C₄⋊C₄	1	4	yes	—
77	C₄×D₁₀₉	C₄⋊C₄	1	4	yes	—
78	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
79	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
80	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
81	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
82	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
83	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
84	#11724784	C₂²×C₄	1	2	yes	—
85	#11724784	C₂²×C₄	1	2	yes	—
86	C₂²×C₁₀₉	—	2	2	no	—
87	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
88	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₁₀₉	—	2	2	no	—
90	C₂×D₁₀₉	—	2	8	no	—
91	C₂×D₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×D₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×D₁₀₉	#135	1	4	yes	—
94	C₄×C₁₀₉	—	4	4	no	—
95	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
96	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
97	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
98	Dic₁₀₉	—	4	4	no	—
99	Dic₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
100	Dic₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₁₀₉	—	2	2	no	—
102	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₁₀₉	C₂×#135	1	2	yes	—
104	D₁₀₉	—	2	2	no	—
105	D₁₀₉	#427	1	4	yes	direct product factor
106	C₂×#427	—	109	1	no	maximal, Sylow
107	C₁₀₉	C₂×#427	1	1	yes	characteristic, complemented, Sylow
108	C₂³×C₈	—	109	1	no	—
109	C₂×#117	—	109	2	no	—
110	#427	—	109	1	no	—
111	#427	—	109	2	no	—
112	#427	—	109	2	no	—
113	#427	—	109	2	no	—
114	#427	D₁₀₉	1	1	yes	characteristic, direct product factor
115	C₂×#107	—	109	2	no	—
116	C₂×#106	—	109	2	no	—
117	C₂³×C₄	—	109	1	no	—
118	C₂²×C₈	—	109	1	no	—
119	C₂²×C₈	—	109	8	no	—
120	C₂²×C₈	—	109	1	no	—
121	C₂²×C₈	—	109	1	no	—
122	C₂²×C₈	—	109	1	no	—
123	C₂²×C₈	—	109	1	no	—
124	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
125	#117	—	109	2	no	—
126	#117	—	109	4	no	—
127	#117	—	218	2	no	—
128	#117	—	218	2	no	—
129	#117	C₂×D₁₀₉	1	2	yes	—
130	C₂×C₄⋊C₄	—	109	4	no	—
131	C₂×C₂²⋊C₄	—	218	2	no	—
132	C₂×C₄²	—	218	2	no	—
133	#107	—	109	2	no	—
134	#107	—	109	4	no	—
135	#107	C₂×D₁₀₉	1	2	yes	complemented
136	#106	—	109	2	no	—
137	#106	—	109	4	no	—
138	#106	C₂×D₁₀₉	1	2	yes	complemented
139	C₂⁴	—	109	1	no	—
140	C₂²×C₄	—	109	1	no	—
141	C₂²×C₄	—	109	4	no	—
142	C₂²×C₄	—	109	1	no	—
143	C₂²×C₄	—	109	1	no	—
144	C₂²×C₄	—	109	4	no	—
145	C₂²×C₄	—	109	1	no	—
146	C₂²×C₄	—	109	1	no	—
147	C₂²×C₄	—	218	4	no	—
148	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
149	C₂×C₈	—	109	4	no	—
150	C₂×C₈	—	109	1	no	—
151	C₂×C₈	—	109	1	no	—
152	C₂×C₈	—	109	4	no	—
153	C₂×C₈	—	109	4	no	—
154	C₂×C₈	—	109	4	no	—
155	C₂×C₈	—	109	4	no	—
156	C₂×C₈	C₄×D₁₀₉	1	4	yes	complemented
157	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
159	C₄⋊C₄	—	109	4	no	—
160	C₄⋊C₄	—	218	4	no	—
161	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	218	4	no	—
164	C₂²⋊C₄	—	218	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	218	4	no	—
167	C₄²	—	218	2	no	—
168	C₂³	—	109	1	no	—
169	C₂³	—	109	4	no	—
170	C₂³	—	109	1	no	—
171	C₂³	—	218	4	no	—
172	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	109	4	no	—
175	C₂×C₄	—	109	1	no	—
176	C₂×C₄	—	109	1	no	—
177	C₂×C₄	—	109	4	no	—
178	C₂×C₄	—	109	4	no	—
179	C₂×C₄	—	109	2	no	—
180	C₂×C₄	—	109	2	no	—
181	C₂×C₄	—	109	2	no	—
182	C₂×C₄	—	109	2	no	—
183	C₂×C₄	—	436	8	no	—
184	C₂×C₄	—	218	4	no	—
185	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
186	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
187	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
188	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
189	C₈	—	109	2	no	—
190	C₈	—	109	2	no	—
191	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
192	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	109	2	no	—
195	C₂²	—	109	4	no	—
196	C₂²	—	109	2	no	—
197	C₂²	—	218	8	no	—
198	C₂²	—	218	2	no	—
199	C₂²	C₄⋊C₄×D₁₀₉	1	2	yes	—
200	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
201	C₄	—	4	4	no	—
202	C₄	—	109	2	no	—
203	C₄	—	109	2	no	—
204	C₄	—	436	4	no	—
205	C₄	C₄⋊C₄×D₁₀₉	1	2	yes	—
206	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
207	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	—	109	4	no	—
210	C₂	—	218	2	no	—
211	C₂	C₂×C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
212	C₂	#135×D₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#427×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂³×D₁₀₉	D₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#427	D₁₀₉
hypocenter	C₁₀₉	C₂×#427
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
3. #20295064	C₂	semidirect
4. #20295063	C₂	semidirect
5. #20295062	C₂	semidirect
6. #20295061	C₂	semidirect
7. #107×D₁₀₉	C₂	semidirect
8. #106×D₁₀₉	C₂	semidirect
9. #427×C₁₀₉	C₂	semidirect
12. C₂×C₈×D₁₀₉	C₄	semidirect
24. #16812633	C₂²	semidirect
25. #16812632	C₂²	semidirect
26. #16812631	C₂²	semidirect
27. #16812630	C₂²	semidirect
30. #107×C₁₀₉	C₂²	semidirect
31. #106×C₁₀₉	C₂²	semidirect
41. C₂×#11724784	C₂×C₄	semidirect
46. C₈×D₁₀₉	C₂×C₄	semidirect
47. C₈×D₁₀₉	C₂×C₄	semidirect
55. C₂×C₈×C₁₀₉	C₂×C₄	semidirect
105. D₁₀₉	#427	direct (N × H)
107. C₁₀₉	C₂×#427	semidirect
114. #427	D₁₀₉	direct (N × H)
135. #107	C₂×D₁₀₉	semidirect
138. #106	C₂×D₁₀₉	semidirect
156. C₂×C₈	C₄×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×D₁₀₉	2
#117×D₁₀₉	2
#20295064	2
#20295063	2
#20295062	2
#20295061	2
#107×D₁₀₉	2
#106×D₁₀₉	2
#427×C₁₀₉	2
C₂×#427	109

Subgroups — #427×D109 #28371608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #427×D₁₀₉ #28371608