Subgroups — D₁₀₉⋊#568 #28380024

Order 13952.

1460 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296106	C₂	1	2	yes	maximal, complemented
2	#20296105	C₂	1	8	yes	maximal, complemented
3	#20296104	C₂	1	6	yes	maximal, complemented
4	D₁₀₉⋊#131	C₂	1	8	yes	maximal, complemented
5	C₂×D₁₀₉⋊M₁₆	C₂	1	6	yes	maximal, complemented
6	C₂×#41×D₁₀₉	C₂	1	1	yes	maximal, characteristic
7	#16812787	C₂²	1	16	yes	complemented
8	#16812786	C₂²	1	48	yes	complemented
9	C₂×#13182039	C₂²	1	12	yes	complemented
10	D₁₀₉⋊M₁₆	C₂²	1	4	yes	—
11	D₁₀₉⋊M₁₆	C₂²	1	12	yes	complemented
12	D₁₀₉⋊M₁₆	C₂²	1	12	yes	complemented
13	D₁₀₉⋊M₁₆	C₂²	1	12	yes	complemented
14	C₂×#13182038	C₂²	1	12	yes	complemented
15	C₂×D₁₀₉⋊C₈	C₂²	1	12	yes	complemented
16	C₂×#13182030	C₂²	1	1	yes	characteristic
17	C₂×#13182029	C₄	1	3	yes	—
18	C₂×#13182028	C₂²	1	3	yes	—
19	#41×D₁₀₉	C₄	1	4	yes	—
20	#41×D₁₀₉	C₄	1	4	yes	—
21	#41×D₁₀₉	C₂²	1	4	yes	—
22	#41×D₁₀₉	C₂²	1	4	yes	—
23	C₂×Q₈×D₁₀₉	C₄	1	1	yes	characteristic
24	C₂×D₄×D₁₀₉	C₄	1	3	yes	—
25	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	—
26	C₂×#41×C₁₀₉	C₄	1	1	yes	characteristic
27	#13182039	C₂³	1	8	yes	—
28	#13182039	C₂³	1	24	yes	complemented
29	#13182038	C₂³	1	8	yes	—
30	#13182038	C₂³	1	24	yes	complemented
31	D₁₀₉⋊C₈	C₂³	1	8	yes	—
32	D₁₀₉⋊C₈	C₂³	1	24	yes	complemented
33	C₂×#11724786	C₂³	1	24	yes	complemented
34	C₂³×D₁₀₉	C₂×C₄	1	3	yes	—
35	#13182030	C₂×C₄	1	4	yes	—
36	#13182030	C₂³	1	4	yes	—
37	#13182029	C₂×C₄	1	12	yes	—
38	#13182029	C₂×C₄	1	12	yes	—
39	#13182028	C₂×C₄	1	12	yes	—
40	#13182028	C₂³	1	12	yes	—
41	C₂×Dic₂₁₈	C₂×C₄	1	3	yes	—
42	Q₈×D₁₀₉	C₂×C₄	1	4	yes	—
43	Q₈×D₁₀₉	C₂×C₄	1	4	yes	—
44	C₂×D₄₃₆	C₂×C₄	1	3	yes	—
45	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	6	yes	—
46	D₄×D₁₀₉	C₂×C₄	1	12	yes	—
47	D₄×D₁₀₉	C₂×C₄	1	12	yes	—
48	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
49	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
50	C₂×C₄×D₁₀₉	C₂×C₄	1	6	yes	—
51	C₂×C₄×D₁₀₉	C₂×C₄	1	6	yes	—
52	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
53	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
55	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
56	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
57	#41×C₁₀₉	C₂×C₄	1	4	yes	—
58	#41×C₁₀₉	C₂×C₄	1	4	yes	—
59	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
60	C₂×D₄×C₁₀₉	C₂×C₄	1	3	yes	—
61	C₂²×C₄×C₁₀₉	C₂×C₄	1	3	yes	—
62	C₂³×C₁₀₉	C₂²×C₄	1	3	yes	—
63	C₂²×D₁₀₉	—	4	6	no	—
64	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₁₀₉	C₂²×C₄	1	6	yes	—
66	C₂²×D₁₀₉	C₂²×C₄	1	6	yes	—
67	C₂²×D₁₀₉	C₂²×C₄	1	3	yes	—
68	Q₈×C₁₀₉	C₂²×C₄	1	4	yes	—
69	D₄×C₁₀₉	C₂²×C₄	1	12	yes	—
70	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
72	C₂×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
73	C₂×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
74	C₂²⋊D₁₀₉	C₂²×C₄	1	24	yes	—
75	C₂×Dic₁₀₉	C₂²×C₄	1	3	yes	—
76	C₂×Dic₁₀₉	C₂²×C₄	1	6	yes	—
77	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
79	D₄₃₆	C₂²×C₄	1	12	yes	—
80	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₁₀₉	C₂²×C₄	1	6	yes	—
83	C₄×D₁₀₉	C₂²×C₄	1	6	yes	—
84	C₄×D₁₀₉	C₂²×C₄	1	6	yes	—
85	C₄×D₁₀₉	C₂²×C₄	1	6	yes	—
86	C₄×D₁₀₉	C₂²×C₄	1	6	yes	—
87	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
88	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
89	C₄×D₁₀₉	C₂⁴	1	6	yes	—
90	Dic₂₁₈	C₂²×C₄	1	12	yes	—
91	#11724786	C₂⁴	1	16	yes	—
92	C₂²×C₁₀₉	—	4	3	no	—
93	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
94	C₂²×C₁₀₉	C₂³×C₄	1	6	yes	—
95	C₂×D₁₀₉	—	2	2	no	—
96	C₂×D₁₀₉	—	2	12	no	—
97	C₂×D₁₀₉	—	4	6	no	—
98	C₂×D₁₀₉	—	4	3	no	—
99	C₂×D₁₀₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×D₁₀₉	C₂³×C₄	1	6	yes	—
101	C₂×D₁₀₉	C₂³×C₄	1	1	yes	characteristic
102	C₄×C₁₀₉	C₂³×C₄	1	6	yes	—
103	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
104	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
105	Dic₁₀₉	C₂³×C₄	1	6	yes	—
106	Dic₁₀₉	C₂³×C₄	1	1	yes	characteristic
107	Dic₁₀₉	C₂⁵	1	1	yes	characteristic
108	C₂×C₁₀₉	—	2	1	no	—
109	C₂×C₁₀₉	—	2	6	no	—
110	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
111	D₁₀₉	—	2	1	no	—
112	D₁₀₉	—	2	6	no	—
113	D₁₀₉	#568	1	2	yes	complemented
114	C₂×#568	—	109	1	no	maximal, Sylow
115	C₁₀₉	C₂×#568	1	1	yes	characteristic, complemented, Sylow
116	C₂²×#41	—	109	1	no	—
117	#568	—	109	6	no	—
118	#568	—	109	8	no	—
119	#568	—	109	2	no	—
120	C₂×#131	—	109	8	no	—
121	C₂²×M₁₆	—	109	6	no	—
122	C₂×#41	—	109	4	no	—
123	C₂×#41	—	109	4	no	—
124	C₂×#41	—	109	4	no	—
125	C₂×#41	—	109	4	no	—
126	C₂×#41	—	109	1	no	—
127	C₂×#41	—	109	3	no	—
128	C₂×#41	—	109	3	no	—
129	C₂×#41	#89137	1	1	yes	characteristic
130	C₂²×Q₈	—	109	1	no	—
131	C₂²×D₄	—	109	3	no	—
132	C₂³×C₄	—	109	3	no	—
133	#131	—	109	48	no	—
134	#131	—	109	16	no	—
135	C₂×M₁₆	—	109	4	no	—
136	C₂×M₁₆	—	109	12	no	—
137	C₂×M₁₆	—	109	12	no	—
138	C₂×M₁₆	—	109	12	no	—
139	C₂×M₁₆	—	109	12	no	—
140	C₂×M₁₆	—	109	12	no	—
141	C₂²×C₈	—	109	12	no	—
142	C₂⁴	—	109	3	no	—
143	#41	—	109	4	no	—
144	#41	—	109	4	no	—
145	#41	—	109	12	no	—
146	#41	—	109	12	no	—
147	#41	—	109	12	no	—
148	#41	—	109	12	no	—
149	#41	C₂×#89137	1	4	yes	—
150	#41	C₂×#89137	1	4	yes	—
151	C₂×Q₈	—	109	4	no	—
152	C₂×Q₈	—	109	4	no	—
153	C₂×Q₈	—	109	3	no	—
154	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
155	C₂×D₄	—	109	6	no	—
156	C₂×D₄	—	109	12	no	—
157	C₂×D₄	—	109	12	no	—
158	C₂×D₄	—	109	3	no	—
159	C₂×D₄	C₂×#89137	1	3	yes	—
160	C₂²×C₄	—	109	3	no	—
161	C₂²×C₄	—	109	3	no	—
162	C₂²×C₄	—	109	3	no	—
163	C₂²×C₄	—	109	6	no	—
164	C₂²×C₄	—	109	6	no	—
165	C₂²×C₄	—	109	6	no	—
166	C₂²×C₄	—	109	1	no	—
167	C₂²×C₄	—	109	6	no	—
168	C₂²×C₄	—	109	3	no	—
169	C₂²×C₄	C₂×#89137	1	3	yes	—
170	M₁₆	—	109	8	no	—
171	M₁₆	—	109	24	no	—
172	M₁₆	—	109	8	no	—
173	M₁₆	—	109	24	no	—
174	C₂×C₈	—	109	24	no	—
175	C₂×C₈	—	109	8	no	—
176	C₂×C₈	—	109	24	no	—
177	C₂³	—	109	6	no	—
178	C₂³	—	109	3	no	—
179	C₂³	—	109	1	no	—
180	C₂³	—	109	6	no	—
181	C₂³	—	436	6	no	—
182	C₂³	C₂²×#89137	1	3	yes	—
183	Q₈	—	109	12	no	—
184	Q₈	C₂²×#89137	1	4	yes	—
185	D₄	—	109	24	no	—
186	D₄	—	109	12	no	—
187	D₄	C₂²×#89137	1	12	yes	—
188	C₂×C₄	—	109	6	no	—
189	C₂×C₄	—	109	1	no	—
190	C₂×C₄	—	109	3	no	—
191	C₂×C₄	—	109	6	no	—
192	C₂×C₄	—	109	6	no	—
193	C₂×C₄	—	109	1	no	—
194	C₂×C₄	—	109	1	no	—
195	C₂×C₄	—	109	6	no	—
196	C₂×C₄	—	109	6	no	—
197	C₂×C₄	—	109	6	no	—
198	C₂×C₄	—	109	6	no	—
199	C₂×C₄	—	109	1	no	—
200	C₂×C₄	—	109	1	no	—
201	C₂×C₄	—	109	6	no	—
202	C₂×C₄	C₂²×#89137	1	3	yes	—
203	C₂×C₄	C₂²×#89137	1	6	yes	—
204	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
205	C₂×C₄	C₂²×#89137	1	6	yes	—
206	C₈	—	109	16	no	—
207	C₂²	—	4	3	no	—
208	C₂²	—	109	1	no	—
209	C₂²	—	109	1	no	—
210	C₂²	—	109	6	no	—
211	C₂²	—	436	3	no	—
212	C₂²	—	436	6	no	—
213	C₂²	—	218	2	no	—
214	C₂²	—	218	12	no	—
215	C₂²	C₂³×#89137	1	1	yes	characteristic
216	C₂²	C₂³×#89137	1	6	yes	—
217	C₄	—	109	1	no	—
218	C₄	—	109	1	no	—
219	C₄	—	109	6	no	—
220	C₄	C₂³×#89137	1	1	yes	characteristic
221	C₄	C₂³×#89137	1	6	yes	—
222	C₄	C₂³×#89137	1	1	yes	characteristic
223	C₂	—	2	1	no	—
224	C₂	—	2	6	no	—
225	C₂	—	109	2	no	—
226	C₂	—	218	1	no	—
227	C₂	—	218	6	no	—
228	C₂	C₂⁴×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂³×#89137
commutator, socle	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂	C₂⁴×#89137
Fitting	C₂×#41×C₁₀₉	C₄
prime radical	C₂×#41×D₁₀₉	C₂
cosocle	Dic₁₀₉	C₂⁵
hypercenter	C₂×#41	#89137
hypocenter	C₁₀₉	C₂×#568
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #20296106	C₂	semidirect
2. #20296105	C₂	semidirect
3. #20296104	C₂	semidirect
4. D₁₀₉⋊#131	C₂	semidirect
5. C₂×D₁₀₉⋊M₁₆	C₂	semidirect
7. #16812787	C₂²	semidirect
8. #16812786	C₂²	semidirect
9. C₂×#13182039	C₂²	semidirect
11. D₁₀₉⋊M₁₆	C₂²	semidirect
12. D₁₀₉⋊M₁₆	C₂²	semidirect
13. D₁₀₉⋊M₁₆	C₂²	semidirect
14. C₂×#13182038	C₂²	semidirect
15. C₂×D₁₀₉⋊C₈	C₂²	semidirect
28. #13182039	C₂³	semidirect
30. #13182038	C₂³	semidirect
32. D₁₀₉⋊C₈	C₂³	semidirect
33. C₂×#11724786	C₂³	semidirect
113. D₁₀₉	#568	semidirect
115. C₁₀₉	C₂×#568	semidirect

Maximal subgroups

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

subgroup	index
#20296106	2
#20296105	2
#20296104	2
D₁₀₉⋊#131	2
C₂×D₁₀₉⋊M₁₆	2
C₂×#41×D₁₀₉	2
C₂×#568	109

Subgroups — D109⋊#568 #28380024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₀₉⋊#568 #28380024