Subgroups — #28371932

Order 13952.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295823	C₂	1	1	yes	maximal, characteristic
2	#131×D₁₀₉	C₂	1	2	yes	maximal, complemented
3	#20295149	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295146	C₂	1	2	yes	maximal, complemented
5	#20295143	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295142	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295140	C₂	1	2	yes	maximal, complemented
8	#20295073	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295041	C₂	1	2	yes	maximal, complemented
10	#20294983	C₂	1	1	yes	maximal, characteristic, complemented
11	#444×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₀₉	C₂²	1	2	yes	—
13	#16812731	C₄	1	2	yes	complemented
14	#16812728	C₄	1	2	yes	complemented
15	#16812726	C₄	1	2	yes	complemented
16	#16812723	C₄	1	2	yes	complemented
17	#16812721	C₂²	1	1	yes	characteristic
18	#16812719	C₂²	1	2	yes	—
19	#16812707	—	2	2	no	—
20	#16812707	C₂²	1	2	yes	complemented
21	#16812706	C₂²	1	2	yes	complemented
22	#16812705	—	2	2	no	—
23	#16812705	C₂²	1	2	yes	complemented
24	C₂×#13182012	C₂²	1	1	yes	characteristic
25	M₁₆×D₁₀₉	—	2	2	no	—
26	M₁₆×D₁₀₉	C₂²	1	2	yes	—
27	C₂×C₈×D₁₀₉	—	2	2	no	—
28	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
29	#16812656	C₂²	1	1	yes	characteristic
30	#16812635	C₂²	1	1	yes	characteristic, complemented
31	#16812634	C₂²	1	2	yes	complemented
32	#16812627	C₂²	1	2	yes	complemented
33	#16812626	C₂²	1	2	yes	complemented
34	#16812625	C₂²	1	2	yes	complemented
35	#16812609	C₂²	1	1	yes	characteristic, complemented
36	C₄×#11724784	C₂²	1	1	yes	characteristic, complemented
37	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
38	#135×C₁₀₉	C₂²	1	1	yes	characteristic
39	#131×C₁₀₉	C₂²	1	2	yes	complemented
40	#108×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
41	#104×C₁₀₉	C₂²	1	2	yes	complemented
42	#97×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
43	#13182030	C₂×C₄	1	2	yes	—
44	#13182029	—	2	2	no	—
45	#13182029	C₂³	1	2	yes	—
46	#13182028	—	2	2	no	—
47	#13182028	C₂×C₄	1	2	yes	—
48	Q₈×D₁₀₉	C₂×C₄	1	2	yes	—
49	D₄×D₁₀₉	—	2	2	no	—
50	D₄×D₁₀₉	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₀₉	—	2	2	no	—
52	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
53	Dic₄₃₆	C₂×C₄	1	1	yes	characteristic
54	#13182019	C₂×C₄	1	2	yes	—
55	#13182018	C₂×C₄	1	2	yes	—
56	#13182017	C₂×C₄	1	2	yes	—
57	#13182016	C₂×C₄	1	2	yes	—
58	D₈₇₂	C₂×C₄	1	1	yes	characteristic
59	#13182014	C₂×C₄	1	2	yes	—
60	#13182013	—	2	2	no	—
61	#13182013	C₂³	1	2	yes	—
62	#13182012	—	2	2	no	—
63	#13182012	—	2	2	no	—
64	#13182012	—	2	2	no	—
65	#13182012	C₂×C₄	1	2	yes	—
66	#13182012	C₂×C₄	1	2	yes	—
67	C₂×#11724784	—	2	2	no	—
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₂×#11724784	C₂³	1	1	yes	characteristic
70	C₈×D₁₀₉	—	2	2	no	—
71	C₈×D₁₀₉	—	2	2	no	—
72	C₈×D₁₀₉	—	2	2	no	—
73	C₈×D₁₀₉	D₄	1	2	yes	—
74	C₈×D₁₀₉	D₄	1	2	yes	—
75	#13182008	—	2	1	no	—
76	#13182006	—	2	1	no	—
77	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
78	#41×C₁₀₉	C₂³	1	2	yes	—
79	Q₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
80	SD₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
81	D₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
82	M₁₆×C₁₀₉	—	2	2	no	—
83	M₁₆×C₁₀₉	C₂³	1	2	yes	—
84	C₂×C₈×C₁₀₉	—	2	2	no	—
85	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
88	C₂²×D₁₀₉	—	2	2	no	—
89	C₂²×D₁₀₉	#41	1	1	yes	characteristic
90	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
91	D₄×C₁₀₉	—	2	2	no	—
92	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₁₀₉	—	2	2	no	—
97	C₂²⋊D₁₀₉	—	2	2	no	—
98	C₂²⋊D₁₀₉	—	2	2	no	—
99	C₂×Dic₁₀₉	—	2	2	no	—
100	C₂×Dic₁₀₉	—	2	1	no	—
101	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
102	D₄₃₆	—	2	2	no	—
103	D₄₃₆	C₂²×C₄	1	2	yes	—
104	C₄×D₁₀₉	—	2	2	no	—
105	C₄×D₁₀₉	—	2	2	no	—
106	C₄×D₁₀₉	—	2	2	no	—
107	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
108	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
109	Dic₂₁₈	—	2	2	no	—
110	Dic₂₁₈	C₂²×C₄	1	2	yes	—
111	C₈×C₁₀₉	—	2	2	no	—
112	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
113	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
115	#11724784	—	2	2	no	—
116	#11724784	C₂²×C₄	1	2	yes	—
117	#11724784	C₂×D₄	1	1	yes	characteristic
118	#11724784	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₁₀₉	—	2	2	no	—
120	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
121	C₂×D₁₀₉	—	2	2	no	—
122	C₂×D₁₀₉	—	4	1	no	—
123	C₂×D₁₀₉	—	4	4	no	—
124	C₂×D₁₀₉	C₄×D₄	1	2	yes	—
125	C₄×C₁₀₉	—	2	2	no	—
126	C₄×C₁₀₉	—	4	1	no	—
127	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
129	Dic₁₀₉	—	2	2	no	—
130	Dic₁₀₉	—	4	1	no	—
131	Dic₁₀₉	C₄×D₄	1	2	yes	—
132	C₂×C₁₀₉	—	2	1	no	—
133	C₂×C₁₀₉	—	4	2	no	—
134	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
135	D₁₀₉	—	2	2	no	—
136	D₁₀₉	—	4	2	no	—
137	#2955	—	109	1	no	maximal, Sylow
138	C₁₀₉	#2955	1	1	yes	characteristic, complemented, Sylow
139	#575	—	109	1	no	—
140	C₂×#131	—	109	2	no	—
141	#444	—	109	2	no	—
142	#444	—	109	1	no	—
143	#444	D₁₀₉	1	1	yes	characteristic, complemented
144	#443	—	109	1	no	—
145	#443	—	109	1	no	—
146	#443	—	109	2	no	—
147	#430	—	109	1	no	—
148	#421	—	109	2	no	—
149	#405	—	109	1	no	—
150	C₂×#41	—	109	2	no	—
151	#137	—	109	2	no	—
152	#137	—	109	2	no	—
153	AGL(1,ℤ/8)	—	109	2	no	—
154	AGL(1,ℤ/8)	—	109	2	no	—
155	#135	—	109	2	no	—
156	#135	—	109	1	no	—
157	#135	C₂×D₁₀₉	1	1	yes	characteristic
158	#131	—	109	2	no	—
159	#131	—	109	2	no	—
160	#131	—	109	2	no	—
161	#131	—	218	2	no	—
162	#131	—	218	2	no	—
163	#131	C₂×D₁₀₉	1	2	yes	complemented
164	C₂×M₁₆	—	109	1	no	—
165	C₂×M₁₆	—	109	2	no	—
166	C₂×M₁₆	—	218	2	no	—
167	C₂²×C₈	—	109	1	no	—
168	C₂²×C₈	—	218	2	no	—
169	#117	—	109	1	no	—
170	#108	—	109	2	no	—
171	#108	—	109	1	no	—
172	#108	C₂×D₁₀₉	1	1	yes	characteristic, complemented
173	#104	—	109	2	no	—
174	#104	—	109	2	no	—
175	#104	—	109	2	no	—
176	#104	C₂×D₁₀₉	1	2	yes	complemented
177	#97	—	109	1	no	—
178	#97	C₂×D₁₀₉	1	1	yes	characteristic, complemented
179	C₄×C₈	—	109	1	no	—
180	C₄×C₈	—	109	1	no	—
181	#41	—	109	2	no	—
182	#41	—	109	2	no	—
183	#41	—	109	2	no	—
184	#41	—	218	2	no	—
185	#41	—	218	2	no	—
186	#41	C₂²×D₁₀₉	1	2	yes	—
187	C₂×Q₈	—	109	2	no	—
188	C₂×D₄	—	109	2	no	—
189	C₂×D₄	—	218	2	no	—
190	C₂²×C₄	—	109	1	no	—
191	C₂²×C₄	—	218	2	no	—
192	Q₁₆	—	109	2	no	—
193	Q₁₆	—	109	1	no	—
194	Q₁₆	C₄×D₁₀₉	1	1	yes	characteristic
195	SD₁₆	—	109	2	no	—
196	SD₁₆	—	109	2	no	—
197	SD₁₆	—	109	2	no	—
198	SD₁₆	C₄×D₁₀₉	1	2	yes	—
199	D₈	—	109	2	no	—
200	D₈	—	109	1	no	—
201	D₈	C₄×D₁₀₉	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	109	2	no	—
204	M₁₆	—	109	2	no	—
205	M₁₆	—	109	2	no	—
206	M₁₆	—	218	2	no	—
207	M₁₆	—	218	2	no	—
208	M₁₆	—	218	2	no	—
209	M₁₆	—	218	2	no	—
210	M₁₆	C₂²×D₁₀₉	1	2	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	109	1	no	—
213	C₂×C₈	—	109	1	no	—
214	C₂×C₈	—	109	2	no	—
215	C₂×C₈	—	109	2	no	—
216	C₂×C₈	—	218	2	no	—
217	C₂×C₈	—	218	2	no	—
218	C₂×C₈	—	218	2	no	—
219	C₂×C₈	—	218	2	no	—
220	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
222	C₄⋊C₄	—	218	1	no	—
223	C₂²⋊C₄	—	218	1	no	—
224	C₄²	—	109	1	no	—
225	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
226	C₂³	—	109	1	no	—
227	C₂³	—	218	2	no	—
228	Q₈	—	109	2	no	—
229	Q₈	—	218	2	no	—
230	Q₈	C₂×C₄×D₁₀₉	1	2	yes	—
231	D₄	—	2	2	no	—
232	D₄	—	109	2	no	—
233	D₄	—	218	2	no	—
234	D₄	—	218	2	no	—
235	D₄	—	218	2	no	—
236	D₄	—	218	2	no	—
237	D₄	C₂×C₄×D₁₀₉	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	109	1	no	—
241	C₂×C₄	—	109	2	no	—
242	C₂×C₄	—	109	2	no	—
243	C₂×C₄	—	218	2	no	—
244	C₂×C₄	—	218	2	no	—
245	C₂×C₄	—	218	2	no	—
246	C₂×C₄	—	218	1	no	—
247	C₂×C₄	—	218	2	no	—
248	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	109	2	no	—
251	C₈	—	109	1	no	—
252	C₈	—	109	1	no	—
253	C₈	—	218	2	no	—
254	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
255	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
256	C₈	D₄×D₁₀₉	1	2	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	109	2	no	—
259	C₂²	—	436	1	no	—
260	C₂²	—	436	4	no	—
261	C₂²	—	218	2	no	—
262	C₂²	#41×D₁₀₉	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	109	2	no	—
266	C₄	—	436	1	no	—
267	C₄	—	218	2	no	—
268	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
269	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	436	2	no	—
273	C₂	—	218	2	no	—
274	C₂	C₄×D₄×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#444×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×C₄×D₄
prime radical	#20295823	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#444	D₁₀₉
hypocenter	C₁₀₉	#2955
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #131×D₁₀₉	C₂	semidirect
3. #20295149	C₂	semidirect
4. #20295146	C₂	semidirect
5. #20295143	C₂	semidirect
6. #20295142	C₂	semidirect
7. #20295140	C₂	semidirect
8. #20295073	C₂	semidirect
9. #20295041	C₂	semidirect
10. #20294983	C₂	semidirect
11. #444×C₁₀₉	C₂	semidirect
13. #16812731	C₄	semidirect
14. #16812728	C₄	semidirect
15. #16812726	C₄	semidirect
16. #16812723	C₄	semidirect
20. #16812707	C₂²	semidirect
21. #16812706	C₂²	semidirect
23. #16812705	C₂²	semidirect
30. #16812635	C₂²	semidirect
31. #16812634	C₂²	semidirect
32. #16812627	C₂²	semidirect
33. #16812626	C₂²	semidirect
34. #16812625	C₂²	semidirect
35. #16812609	C₂²	semidirect
36. C₄×#11724784	C₂²	semidirect
37. C₈×Dic₁₀₉	C₂²	semidirect
39. #131×C₁₀₉	C₂²	semidirect
40. #108×C₁₀₉	C₂²	semidirect
41. #104×C₁₀₉	C₂²	semidirect
42. #97×C₁₀₉	C₂²	semidirect
138. C₁₀₉	#2955	semidirect
143. #444	D₁₀₉	semidirect
163. #131	C₂×D₁₀₉	semidirect
172. #108	C₂×D₁₀₉	semidirect
176. #104	C₂×D₁₀₉	semidirect
178. #97	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295823	2
#131×D₁₀₉	2
#20295149	2
#20295146	2
#20295143	2
#20295142	2
#20295140	2
#20295073	2
#20295041	2
#20294983	2
#444×C₁₀₉	2
#2955	109