Subgroups — #28376232

Order 13952.

1540 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#20295509	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295507	C₂	1	3	yes	maximal, complemented
4	#20295506	C₂	1	3	yes	maximal, complemented
5	#20295502	C₂	1	2	yes	maximal, complemented
6	#20295498	C₂	1	6	yes	maximal, complemented
7	C₄×#13182030	C₂	1	2	yes	maximal, complemented
8	C₄×#13182028	C₂	1	6	yes	maximal, complemented
9	C₂×#16812657	C₂	1	3	yes	maximal, complemented
10	#117×D₁₀₉	C₂	1	3	yes	maximal, complemented
11	#520×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182030	C₂²	1	1	yes	characteristic
13	C₂×#13182029	C₂²	1	3	yes	—
14	C₂×#13182028	C₂²	1	3	yes	—
15	#41×D₁₀₉	C₄	1	16	yes	complemented
16	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₁₀₉	C₂²	1	3	yes	—
18	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	—
19	Q₈×Dic₁₀₉	C₂²	1	2	yes	complemented
20	C₄×Dic₂₁₈	C₂²	1	6	yes	complemented
21	#16812664	C₂²	1	6	yes	complemented
22	C₄×D₄₃₆	C₂²	1	6	yes	complemented
23	D₄×Dic₁₀₉	C₂²	1	6	yes	complemented
24	#16812661	C₂²	1	6	yes	complemented
25	C₄×C₂²⋊D₁₀₉	C₂²	1	12	yes	complemented
26	#16812659	C₂²	1	12	yes	complemented
27	#16812658	C₂²	1	3	yes	complemented
28	#16812657	C₂²	1	1	yes	characteristic
29	#16812657	C₂²	1	3	yes	—
30	#16812657	C₂²	1	6	yes	—
31	#16812657	C₂²	1	6	yes	complemented
32	#16812657	C₂²	1	6	yes	complemented
33	#16812656	C₂²	1	6	yes	—
34	#16812655	C₂²	1	6	yes	complemented
35	C₂×C₄⋊Dic₁₀₉	C₂²	1	3	yes	complemented
36	C₄⋊C₄×D₁₀₉	C₂²	1	3	yes	complemented
37	C₄⋊C₄×D₁₀₉	C₂²	1	3	yes	complemented
38	C₂×#13182006	C₂²	1	6	yes	—
39	C₂²⋊C₄×D₁₀₉	C₂²	1	6	yes	—
40	C₂×C₄×Dic₁₀₉	C₂²	1	6	yes	complemented
41	C₄²×D₁₀₉	C₂²	1	6	yes	complemented
42	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	complemented
44	C₄×D₄×C₁₀₉	C₂²	1	6	yes	complemented
45	#117×C₁₀₉	C₂²	1	3	yes	complemented
46	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	complemented
47	C₂³×D₁₀₉	C₂³	1	3	yes	—
48	#13182030	C₂×C₄	1	8	yes	complemented
49	#13182029	C₂×C₄	1	24	yes	complemented
50	#13182028	C₂×C₄	1	24	yes	complemented
51	C₂×Dic₂₁₈	C₂³	1	3	yes	—
52	Q₈×D₁₀₉	C₂×C₄	1	8	yes	complemented
53	C₂×D₄₃₆	C₂³	1	3	yes	—
54	C₂×C₂²⋊D₁₀₉	C₂³	1	6	yes	—
55	D₄×D₁₀₉	C₂×C₄	1	24	yes	complemented
56	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
57	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
58	C₂×C₄×D₁₀₉	C₂×C₄	1	24	yes	complemented
59	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
60	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
61	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
62	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
63	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
64	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₁₀₉	C₂³	1	3	yes	—
67	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
68	#13182008	C₂³	1	6	yes	—
69	#13182008	C₂³	1	6	yes	—
70	C₂²⋊Dic₁₀₉	C₂³	1	6	yes	—
71	#13182006	C₂³	1	2	yes	—
72	#13182006	C₂³	1	6	yes	—
73	#13182006	C₂³	1	12	yes	—
74	C₄×Dic₁₀₉	C₂³	1	2	yes	—
75	C₄×Dic₁₀₉	C₂³	1	6	yes	—
76	C₄×Dic₁₀₉	C₂³	1	6	yes	—
77	C₄×Dic₁₀₉	C₂³	1	12	yes	—
78	#41×C₁₀₉	C₂×C₄	1	8	yes	complemented
79	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₂×D₄×C₁₀₉	C₂³	1	3	yes	—
81	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
82	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
83	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
84	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
85	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
87	C₄²×C₁₀₉	C₂³	1	6	yes	—
88	C₂³×C₁₀₉	C₂⁴	1	3	yes	—
89	C₂²×D₁₀₉	—	4	3	no	—
90	C₂²×D₁₀₉	—	4	3	no	—
91	C₂²×D₁₀₉	C₂²×C₄	1	12	yes	—
92	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
93	C₂²×D₁₀₉	C₂⁴	1	3	yes	—
94	Q₈×C₁₀₉	C₂²×C₄	1	4	yes	—
95	D₄×C₁₀₉	C₂²×C₄	1	12	yes	—
96	C₂×C₄×C₁₀₉	—	4	6	no	—
97	C₂×C₄×C₁₀₉	C₂²×C₄	1	12	yes	—
98	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
99	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
100	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
101	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₁₀₉	C₂²×C₄	1	24	yes	—
103	C₂×Dic₁₀₉	—	4	6	no	—
104	C₂×Dic₁₀₉	C₂²×C₄	1	12	yes	—
105	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
106	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
107	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
108	C₂×Dic₁₀₉	C₂⁴	1	3	yes	—
109	D₄₃₆	C₂²×C₄	1	12	yes	—
110	C₄×D₁₀₉	—	4	6	no	—
111	C₄×D₁₀₉	—	4	6	no	—
112	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
113	C₄×D₁₀₉	C₂²×C₄	1	12	yes	—
114	C₄×D₁₀₉	C₂²×C₄	1	12	yes	—
115	C₄×D₁₀₉	C₂²×C₄	1	12	yes	—
116	Dic₂₁₈	C₂²×C₄	1	12	yes	—
117	C₂²×C₁₀₉	—	2	3	no	—
118	C₂²×C₁₀₉	—	2	3	no	—
119	C₂²×C₁₀₉	C₂³×C₄	1	6	yes	—
120	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
121	C₂×D₁₀₉	—	2	3	no	—
122	C₂×D₁₀₉	—	2	3	no	—
123	C₂×D₁₀₉	—	2	1	no	—
124	C₂×D₁₀₉	—	2	1	no	—
125	C₂×D₁₀₉	—	4	12	no	—
126	C₂×D₁₀₉	C₂³×C₄	1	2	yes	—
127	C₂×D₁₀₉	C₂³×C₄	1	6	yes	—
128	C₄×C₁₀₉	—	2	2	no	—
129	C₄×C₁₀₉	—	2	6	no	—
130	C₄×C₁₀₉	C₂³×C₄	1	6	yes	—
131	C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
132	Dic₁₀₉	—	2	2	no	—
133	Dic₁₀₉	—	2	6	no	—
134	Dic₁₀₉	C₂³×C₄	1	6	yes	—
135	Dic₁₀₉	C₂³×C₄	1	2	yes	—
136	C₂×C₁₀₉	—	2	6	no	—
137	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
138	C₂×C₁₀₉	#585	1	1	yes	characteristic
139	C₂×C₁₀₉	#585	1	1	yes	characteristic
140	D₁₀₉	—	2	2	no	—
141	D₁₀₉	—	2	6	no	—
142	#3428	—	109	1	no	maximal, Sylow
143	C₁₀₉	#3428	1	1	yes	characteristic, complemented, Sylow
144	C₂²×#41	—	109	1	no	—
145	#520	—	109	1	no	—
146	#520	—	109	3	no	—
147	#520	—	109	3	no	—
148	#520	D₁₀₉	1	1	yes	characteristic, complemented
149	#519	—	109	2	no	—
150	#518	—	109	6	no	—
151	C₄×#41	—	109	2	no	—
152	C₄×#41	—	109	6	no	—
153	C₂×#117	—	109	3	no	—
154	C₂×#117	—	109	3	no	—
155	C₂×#41	—	109	16	no	—
156	C₂×#41	—	109	1	no	—
157	C₂×#41	—	109	3	no	—
158	C₂×#41	—	109	3	no	—
159	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
160	C₂²×Q₈	—	109	1	no	—
161	C₂²×D₄	—	109	3	no	—
162	C₂³×C₄	—	109	3	no	—
163	C₄×Q₈	—	109	2	no	—
164	C₄×Q₈	—	109	6	no	—
165	C₄×Q₈	—	109	6	no	—
166	C₄×Q₈	C₂×D₁₀₉	1	2	yes	complemented
167	C₄×D₄	—	109	6	no	—
168	C₄×D₄	—	109	12	no	—
169	C₄×D₄	—	109	6	no	—
170	C₄×D₄	—	109	12	no	—
171	C₄×D₄	—	109	6	no	—
172	C₄×D₄	C₂×D₁₀₉	1	6	yes	complemented
173	#117	—	109	3	no	—
174	#117	—	109	1	no	—
175	#117	—	109	3	no	—
176	#117	—	109	6	no	—
177	#117	—	109	6	no	—
178	#117	—	109	6	no	—
179	#117	—	109	6	no	—
180	#117	—	109	6	no	—
181	#117	C₂×D₁₀₉	1	3	yes	complemented
182	C₂×C₄⋊C₄	—	109	3	no	—
183	C₂×C₄⋊C₄	—	109	3	no	—
184	C₂×C₄⋊C₄	—	109	3	no	—
185	C₂×C₄⋊C₄	C₂×D₁₀₉	1	3	yes	complemented
186	C₂×C₂²⋊C₄	—	109	6	no	—
187	C₂×C₂²⋊C₄	—	109	6	no	—
188	C₂×C₄²	—	109	6	no	—
189	C₂×C₄²	—	109	6	no	—
190	C₂⁴	—	109	3	no	—
191	#41	—	109	8	no	—
192	#41	—	109	24	no	—
193	#41	—	109	24	no	—
194	#41	C₄×D₁₀₉	1	8	yes	complemented
195	C₂×Q₈	—	109	8	no	—
196	C₂×Q₈	—	109	3	no	—
197	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
198	C₂×D₄	—	109	6	no	—
199	C₂×D₄	—	109	24	no	—
200	C₂×D₄	—	109	3	no	—
201	C₂×D₄	C₂²×D₁₀₉	1	3	yes	—
202	C₂²×C₄	—	109	3	no	—
203	C₂²×C₄	—	109	24	no	—
204	C₂²×C₄	—	109	3	no	—
205	C₂²×C₄	—	109	3	no	—
206	C₂²×C₄	—	109	1	no	—
207	C₂²×C₄	—	109	3	no	—
208	C₂²×C₄	—	109	6	no	—
209	C₂²×C₄	—	109	6	no	—
210	C₂²×C₄	—	109	6	no	—
211	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
212	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
213	C₄⋊C₄	—	109	6	no	—
214	C₄⋊C₄	—	109	6	no	—
215	C₄⋊C₄	—	109	1	no	—
216	C₄⋊C₄	—	109	3	no	—
217	C₄⋊C₄	—	109	6	no	—
218	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
219	C₄⋊C₄	C₂²×D₁₀₉	1	3	yes	—
220	C₄⋊C₄	C₂²×D₁₀₉	1	6	yes	—
221	C₂²⋊C₄	—	109	6	no	—
222	C₂²⋊C₄	—	109	6	no	—
223	C₂²⋊C₄	—	109	2	no	—
224	C₂²⋊C₄	—	109	12	no	—
225	C₂²⋊C₄	C₂²×D₁₀₉	1	6	yes	—
226	C₄²	—	109	6	no	—
227	C₄²	—	109	2	no	—
228	C₄²	—	109	12	no	—
229	C₄²	—	109	6	no	—
230	C₄²	C₂²×D₁₀₉	1	6	yes	—
231	C₂³	—	109	12	no	—
232	C₂³	—	109	1	no	—
233	C₂³	—	109	3	no	—
234	C₂³	—	436	3	no	—
235	C₂³	—	436	3	no	—
236	C₂³	C₂³×D₁₀₉	1	3	yes	—
237	Q₈	—	109	12	no	—
238	Q₈	C₂×C₄×D₁₀₉	1	4	yes	—
239	D₄	—	109	24	no	—
240	D₄	—	109	12	no	—
241	D₄	C₂×C₄×D₁₀₉	1	12	yes	—
242	C₂×C₄	—	4	6	no	—
243	C₂×C₄	—	109	12	no	—
244	C₂×C₄	—	109	1	no	—
245	C₂×C₄	—	109	3	no	—
246	C₂×C₄	—	109	4	no	—
247	C₂×C₄	—	109	12	no	—
248	C₂×C₄	—	109	12	no	—
249	C₂×C₄	—	109	12	no	—
250	C₂×C₄	—	109	2	no	—
251	C₂×C₄	—	109	6	no	—
252	C₂×C₄	—	436	6	no	—
253	C₂×C₄	—	436	6	no	—
254	C₂×C₄	—	436	6	no	—
255	C₂×C₄	C₂×C₄×D₁₀₉	1	12	yes	—
256	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
257	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
258	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
259	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
260	C₂²	—	2	3	no	—
261	C₂²	—	2	3	no	—
262	C₂²	—	109	2	no	—
263	C₂²	—	109	6	no	—
264	C₂²	—	436	12	no	—
265	C₂²	—	218	3	no	—
266	C₂²	—	218	1	no	—
267	C₂²	—	218	3	no	—
268	C₂²	—	218	1	no	—
269	C₂²	C₂²×C₄×D₁₀₉	1	6	yes	—
270	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
271	C₄	—	2	6	no	—
272	C₄	—	2	2	no	—
273	C₄	—	109	2	no	—
274	C₄	—	109	6	no	—
275	C₄	—	218	2	no	—
276	C₄	—	218	6	no	—
277	C₄	C₂²×C₄×D₁₀₉	1	2	yes	—
278	C₄	C₂²×C₄×D₁₀₉	1	6	yes	—
279	C₂	—	2	6	no	—
280	C₂	—	218	2	no	—
281	C₂	—	218	6	no	—
282	C₂	C₂³×C₄×D₁₀₉	1	1	yes	characteristic
283	C₂	#20295861	1	1	yes	characteristic
284	C₂	#20295863	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Fitting	#520×C₁₀₉	C₂
socle, cosocle	C₂²×C₁₀₉	C₂⁵
prime radical	C₂×#41×D₁₀₉	C₂
hypercenter	#520	D₁₀₉
hypocenter	C₁₀₉	#3428
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #520

Complements

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

N	H ≅ G/N	type
2. #20295509	C₂	semidirect
3. #20295507	C₂	semidirect
4. #20295506	C₂	semidirect
5. #20295502	C₂	semidirect
6. #20295498	C₂	semidirect
7. C₄×#13182030	C₂	semidirect
8. C₄×#13182028	C₂	semidirect
9. C₂×#16812657	C₂	semidirect
10. #117×D₁₀₉	C₂	semidirect
11. #520×C₁₀₉	C₂	semidirect
15. #41×D₁₀₉	C₄	semidirect
19. Q₈×Dic₁₀₉	C₂²	semidirect
20. C₄×Dic₂₁₈	C₂²	semidirect
21. #16812664	C₂²	semidirect
22. C₄×D₄₃₆	C₂²	semidirect
23. D₄×Dic₁₀₉	C₂²	semidirect
24. #16812661	C₂²	semidirect
25. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
26. #16812659	C₂²	semidirect
27. #16812658	C₂²	semidirect
31. #16812657	C₂²	semidirect
32. #16812657	C₂²	semidirect
34. #16812655	C₂²	semidirect
35. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
36. C₄⋊C₄×D₁₀₉	C₂²	semidirect
37. C₄⋊C₄×D₁₀₉	C₂²	semidirect
40. C₂×C₄×Dic₁₀₉	C₂²	semidirect
41. C₄²×D₁₀₉	C₂²	semidirect
43. C₄×Q₈×C₁₀₉	C₂²	semidirect
44. C₄×D₄×C₁₀₉	C₂²	semidirect
45. #117×C₁₀₉	C₂²	semidirect
46. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
48. #13182030	C₂×C₄	semidirect
49. #13182029	C₂×C₄	semidirect
50. #13182028	C₂×C₄	semidirect
52. Q₈×D₁₀₉	C₂×C₄	semidirect
55. D₄×D₁₀₉	C₂×C₄	semidirect
58. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
78. #41×C₁₀₉	C₂×C₄	semidirect
143. C₁₀₉	#3428	semidirect
148. #520	D₁₀₉	semidirect
166. C₄×Q₈	C₂×D₁₀₉	semidirect
172. C₄×D₄	C₂×D₁₀₉	semidirect
181. #117	C₂×D₁₀₉	semidirect
185. C₂×C₄⋊C₄	C₂×D₁₀₉	semidirect
194. #41	C₄×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₁₀₉	2
#20295509	2
#20295507	2
#20295506	2
#20295502	2
#20295498	2
C₄×#13182030	2
C₄×#13182028	2
C₂×#16812657	2
#117×D₁₀₉	2
#520×C₁₀₉	2
#3428	109