Subgroups — D₁₀₉⋊#443 #28379798

Order 13952.

472 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₀₉⋊#131	C₂	1	2	yes	maximal, complemented
2	#20296070	C₂	1	2	yes	maximal, complemented
3	#20296069	C₂	1	4	yes	maximal, complemented
4	#20296068	C₂	1	2	yes	maximal, complemented
5	D₁₀₉⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₀₉⋊#104	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#89137	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₀₉	C₂	1	1	yes	maximal, characteristic
9	#16812787	C₂²	1	4	yes	complemented
10	#16812786	—	2	4	no	—
11	#16812786	C₂²	1	4	yes	complemented
12	D₁₀₉⋊M₁₆	—	2	2	no	—
13	D₁₀₉⋊M₁₆	C₂²	1	2	yes	—
14	C₂×D₁₀₉⋊C₈	—	2	2	no	—
15	C₂×D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×#89137	C₂²	1	1	yes	characteristic
17	#16812770	C₂²	1	2	yes	complemented
18	#16812769	C₂²	1	2	yes	complemented
19	#16812764	C₂²	1	4	yes	complemented
20	#16812763	C₂²	1	4	yes	complemented
21	C₈×#89137	C₂²	1	2	yes	complemented
22	C₈×#89137	C₂²	1	2	yes	complemented
23	#41×D₁₀₉	C₂²	1	2	yes	—
24	#16812722	C₂²	1	1	yes	characteristic
25	#16812721	C₄	1	1	yes	characteristic, complemented
26	#16812720	C₂²	1	2	yes	—
27	#16812719	C₄	1	2	yes	complemented
28	#16812718	C₂²	1	1	yes	characteristic
29	Q₁₆×D₁₀₉	C₄	1	1	yes	characteristic, complemented
30	SD₁₆×D₁₀₉	C₄	1	2	yes	complemented
31	D₈×D₁₀₉	C₄	1	1	yes	characteristic, complemented
32	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
33	#135×C₁₀₉	C₄	1	1	yes	characteristic, complemented
34	C₂²×#89137	—	2	1	no	—
35	#13182039	—	2	4	no	—
36	#13182039	C₂³	1	4	yes	—
37	#13182038	—	2	2	no	—
38	#13182038	—	2	4	no	—
39	D₁₀₉⋊C₈	—	2	2	no	—
40	D₁₀₉⋊C₈	—	2	4	no	—
41	D₁₀₉⋊C₈	D₄	1	1	yes	characteristic
42	D₁₀₉⋊C₈	D₄	1	1	yes	characteristic
43	D₁₀₉⋊C₈	C₂³	1	2	yes	—
44	C₂×#11724786	—	2	4	no	—
45	C₂×#11724786	D₄	1	2	yes	—
46	C₄×#89137	—	2	1	no	—
47	C₄×#89137	C₂³	1	2	yes	—
48	#13182030	C₂³	1	2	yes	—
49	#13182029	—	2	2	no	—
50	#13182029	C₂×C₄	1	2	yes	—
51	#13182028	—	2	2	no	—
52	#13182028	C₂³	1	2	yes	—
53	Q₈×D₁₀₉	C₂×C₄	1	2	yes	—
54	D₄×D₁₀₉	—	2	2	no	—
55	D₄×D₁₀₉	C₂×C₄	1	2	yes	—
56	C₂×C₄×D₁₀₉	—	2	2	no	—
57	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
58	Dic₄₃₆	C₂×C₄	1	1	yes	characteristic, complemented
59	#13182019	C₂×C₄	1	2	yes	complemented
60	#13182018	C₂×C₄	1	2	yes	complemented
61	#13182017	C₂×C₄	1	2	yes	complemented
62	#13182016	C₂×C₄	1	2	yes	complemented
63	D₈₇₂	C₂×C₄	1	1	yes	characteristic, complemented
64	#13182014	C₂×C₄	1	2	yes	complemented
65	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
66	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
67	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
68	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
69	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
70	#41×C₁₀₉	C₂×C₄	1	2	yes	—
71	Q₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
72	SD₁₆×C₁₀₉	C₂×C₄	1	2	yes	complemented
73	D₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
74	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
75	C₂²×D₁₀₉	—	2	2	no	—
76	C₂²×D₁₀₉	#41	1	1	yes	characteristic
77	C₂×#89137	—	2	4	no	—
78	C₂×#89137	—	2	2	no	—
79	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
80	D₄×C₁₀₉	—	2	2	no	—
81	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²⋊D₁₀₉	—	2	2	no	—
85	C₂²⋊D₁₀₉	—	2	2	no	—
86	C₂²⋊D₁₀₉	—	2	2	no	—
87	C₂×Dic₁₀₉	—	2	2	no	—
88	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
89	D₄₃₆	—	2	2	no	—
90	D₄₃₆	C₂²×C₄	1	2	yes	—
91	C₄×D₁₀₉	—	2	2	no	—
92	C₄×D₁₀₉	—	2	2	no	—
93	C₄×D₁₀₉	—	2	2	no	—
94	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
95	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
96	C₄×D₁₀₉	#41	1	1	yes	characteristic
97	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
98	Dic₂₁₈	—	2	2	no	—
99	Dic₂₁₈	C₂²×C₄	1	2	yes	—
100	#11724786	—	2	4	no	—
101	#11724786	C₂×D₄	1	2	yes	—
102	#11724786	C₂×D₄	1	2	yes	—
103	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
104	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
105	#11724784	C₂²×C₄	1	1	yes	characteristic
106	#11724784	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₁₀₉	—	2	2	no	—
108	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
109	C₂×D₁₀₉	—	2	2	no	—
110	C₂×D₁₀₉	—	2	2	no	—
111	C₂×D₁₀₉	—	4	4	no	—
112	C₂×D₁₀₉	C₄×D₄	1	1	yes	characteristic
113	C₂×D₁₀₉	C₂×#41	1	1	yes	characteristic
114	#89137	—	2	4	no	—
115	C₄×C₁₀₉	—	2	2	no	—
116	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
117	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
118	Dic₁₀₉	—	2	2	no	—
119	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
120	Dic₁₀₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₁₀₉	—	2	1	no	—
122	C₂×C₁₀₉	—	4	2	no	—
123	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
124	D₁₀₉	—	2	1	no	—
125	D₁₀₉	—	4	2	no	—
126	D₁₀₉	#443	1	2	yes	complemented
127	C₂×#443	—	109	1	no	maximal, Sylow
128	C₁₀₉	C₂×#443	1	1	yes	characteristic, complemented, Sylow
129	C₂×#135	—	109	1	no	—
130	C₂×#131	—	109	2	no	—
131	#443	—	109	2	no	—
132	#443	—	109	4	no	—
133	#443	—	109	2	no	—
134	C₂×#108	—	109	1	no	—
135	C₂×#104	—	109	2	no	—
136	C₂×C₄×C₈	—	109	1	no	—
137	C₂×#41	—	109	2	no	—
138	#135	—	109	2	no	—
139	#135	—	109	1	no	—
140	#135	—	109	2	no	—
141	#135	—	109	1	no	—
142	#135	—	109	1	no	—
143	#135	#89137	1	1	yes	characteristic, complemented
144	C₂×Q₁₆	—	109	1	no	—
145	C₂×SD₁₆	—	109	2	no	—
146	C₂×D₈	—	109	1	no	—
147	#131	—	109	4	no	—
148	#131	—	109	4	no	—
149	#131	—	218	4	no	—
150	C₂×M₁₆	—	109	2	no	—
151	C₂×M₁₆	—	218	2	no	—
152	C₂²×C₈	—	109	1	no	—
153	C₂²×C₈	—	109	1	no	—
154	C₂²×C₈	—	218	2	no	—
155	C₂×C₄²	—	109	1	no	—
156	#108	—	109	2	no	—
157	#108	—	109	2	no	—
158	#104	—	109	4	no	—
159	#104	—	109	4	no	—
160	C₄×C₈	—	109	2	no	—
161	C₄×C₈	—	109	2	no	—
162	#41	—	109	2	no	—
163	#41	—	109	2	no	—
164	#41	—	109	2	no	—
165	#41	—	218	2	no	—
166	#41	—	218	2	no	—
167	#41	C₂×#89137	1	2	yes	—
168	C₂×Q₈	—	109	2	no	—
169	C₂×D₄	—	109	2	no	—
170	C₂×D₄	—	218	2	no	—
171	C₂²×C₄	—	109	1	no	—
172	C₂²×C₄	—	218	2	no	—
173	C₂²×C₄	—	218	1	no	—
174	Q₁₆	—	109	2	no	—
175	Q₁₆	—	109	1	no	—
176	Q₁₆	C₂×#89137	1	1	yes	characteristic, complemented
177	SD₁₆	—	109	2	no	—
178	SD₁₆	—	109	2	no	—
179	SD₁₆	—	109	2	no	—
180	SD₁₆	C₂×#89137	1	2	yes	complemented
181	D₈	—	109	2	no	—
182	D₈	—	109	1	no	—
183	D₈	C₂×#89137	1	1	yes	characteristic, complemented
184	M₁₆	—	109	4	no	—
185	M₁₆	—	218	4	no	—
186	M₁₆	—	218	4	no	—
187	M₁₆	—	218	2	no	—
188	C₂×C₈	—	109	2	no	—
189	C₂×C₈	—	109	1	no	—
190	C₂×C₈	—	109	1	no	—
191	C₂×C₈	—	109	2	no	—
192	C₂×C₈	—	109	1	no	—
193	C₂×C₈	—	109	1	no	—
194	C₂×C₈	—	109	1	no	—
195	C₂×C₈	—	109	1	no	—
196	C₂×C₈	—	109	1	no	—
197	C₂×C₈	—	218	4	no	—
198	C₂×C₈	—	218	4	no	—
199	C₂×C₈	—	218	2	no	—
200	C₂×C₈	C₂×#89137	1	1	yes	characteristic
201	C₄²	—	109	2	no	—
202	C₄²	—	218	1	no	—
203	C₂³	—	109	1	no	—
204	C₂³	—	218	2	no	—
205	Q₈	—	109	2	no	—
206	Q₈	—	218	2	no	—
207	Q₈	C₂²×#89137	1	2	yes	—
208	D₄	—	2	2	no	—
209	D₄	—	109	2	no	—
210	D₄	—	218	2	no	—
211	D₄	—	218	2	no	—
212	D₄	—	218	2	no	—
213	D₄	—	218	2	no	—
214	D₄	C₂²×#89137	1	2	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	109	1	no	—
217	C₂×C₄	—	109	1	no	—
218	C₂×C₄	—	109	1	no	—
219	C₂×C₄	—	109	1	no	—
220	C₂×C₄	—	109	1	no	—
221	C₂×C₄	—	218	2	no	—
222	C₂×C₄	—	218	2	no	—
223	C₂×C₄	—	218	2	no	—
224	C₂×C₄	—	218	2	no	—
225	C₂×C₄	—	218	4	no	—
226	C₂×C₄	—	218	2	no	—
227	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
228	C₈	—	109	2	no	—
229	C₈	—	109	2	no	—
230	C₈	—	109	1	no	—
231	C₈	—	109	1	no	—
232	C₈	—	218	4	no	—
233	C₈	C₂²×#89137	1	1	yes	characteristic
234	C₈	C₂²×#89137	1	1	yes	characteristic
235	C₂²	—	2	2	no	—
236	C₂²	—	109	1	no	—
237	C₂²	—	109	1	no	—
238	C₂²	—	436	4	no	—
239	C₂²	—	218	2	no	—
240	C₂²	—	218	2	no	—
241	C₂²	D₄×#89137	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	109	1	no	—
244	C₄	—	109	1	no	—
245	C₄	—	218	4	no	—
246	C₄	—	218	2	no	—
247	C₄	D₄×#89137	1	1	yes	characteristic
248	C₄	C₂³×#89137	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	—	109	2	no	—
252	C₂	—	436	2	no	—
253	C₂	—	218	1	no	—
254	C₂	C₂×D₄×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#89137
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₄	C₂³×#89137
Fitting	#135×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×C₄×D₄
prime radical	#135×D₁₀₉	C₂
cosocle	C₄×D₁₀₉	C₂⁴
hypercenter	#135	#89137
hypocenter	C₁₀₉	C₂×#443
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. D₁₀₉⋊#131	C₂	semidirect
2. #20296070	C₂	semidirect
3. #20296069	C₂	semidirect
4. #20296068	C₂	semidirect
5. D₁₀₉⋊#108	C₂	semidirect
6. D₁₀₉⋊#104	C₂	semidirect
7. C₂×C₈×#89137	C₂	semidirect
9. #16812787	C₂²	semidirect
11. #16812786	C₂²	semidirect
17. #16812770	C₂²	semidirect
18. #16812769	C₂²	semidirect
19. #16812764	C₂²	semidirect
20. #16812763	C₂²	semidirect
21. C₈×#89137	C₂²	semidirect
22. C₈×#89137	C₂²	semidirect
25. #16812721	C₄	semidirect
27. #16812719	C₄	semidirect
29. Q₁₆×D₁₀₉	C₄	semidirect
30. SD₁₆×D₁₀₉	C₄	semidirect
31. D₈×D₁₀₉	C₄	semidirect
33. #135×C₁₀₉	C₄	semidirect
58. Dic₄₃₆	C₂×C₄	semidirect
59. #13182019	C₂×C₄	semidirect
60. #13182018	C₂×C₄	semidirect
61. #13182017	C₂×C₄	semidirect
62. #13182016	C₂×C₄	semidirect
63. D₈₇₂	C₂×C₄	semidirect
64. #13182014	C₂×C₄	semidirect
71. Q₁₆×C₁₀₉	C₂×C₄	semidirect
72. SD₁₆×C₁₀₉	C₂×C₄	semidirect
73. D₈×C₁₀₉	C₂×C₄	semidirect
126. D₁₀₉	#443	semidirect
128. C₁₀₉	C₂×#443	semidirect
143. #135	#89137	semidirect
176. Q₁₆	C₂×#89137	semidirect
180. SD₁₆	C₂×#89137	semidirect
183. D₈	C₂×#89137	semidirect

Maximal subgroups

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

subgroup	index
D₁₀₉⋊#131	2
#20296070	2
#20296069	2
#20296068	2
D₁₀₉⋊#108	2
D₁₀₉⋊#104	2
C₂×C₈×#89137	2
#135×D₁₀₉	2
C₂×#443	109

Subgroups — D109⋊#443 #28379798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₀₉⋊#443 #28379798