Subgroups — #28371944

Order 13952.

452 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#20295791	C₂	1	2	yes	maximal, complemented
3	#20295146	C₂	1	2	yes	maximal, complemented
4	#20295145	C₂	1	2	yes	maximal, complemented
5	#20295144	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295143	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295142	C₂	1	1	yes	maximal, characteristic, complemented
8	#108×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295041	C₂	1	2	yes	maximal, complemented
10	#20294980	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₀₉	C₂²	1	2	yes	—
13	#16812722	C₄	1	1	yes	characteristic, complemented
14	#16812721	C₂²	1	1	yes	characteristic
15	#16812720	C₄	1	2	yes	complemented
16	#16812719	C₂²	1	2	yes	—
17	#16812718	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₁₀₉	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₁₀₉	C₄	1	2	yes	complemented
20	D₈×D₁₀₉	C₄	1	1	yes	characteristic, complemented
21	#16812707	—	2	2	no	—
22	#16812707	C₂²	1	2	yes	complemented
23	#16812706	C₂²	1	2	yes	complemented
24	#16812705	—	2	2	no	—
25	#16812705	C₂²	1	2	yes	complemented
26	C₂×#13182012	—	2	2	no	—
27	C₂×#13182012	C₂²	1	1	yes	characteristic
28	M₁₆×D₁₀₉	—	2	2	no	—
29	M₁₆×D₁₀₉	C₂²	1	2	yes	—
30	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
31	#16812656	C₂²	1	1	yes	characteristic
32	#16812635	C₂²	1	1	yes	characteristic, complemented
33	#16812634	C₂²	1	2	yes	complemented
34	#16812627	C₂²	1	2	yes	complemented
35	#16812626	C₂²	1	2	yes	complemented
36	#16812625	C₂²	1	2	yes	complemented
37	#16812610	C₂²	1	1	yes	characteristic, complemented
38	#16812609	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
40	#135×C₁₀₉	C₂²	1	1	yes	characteristic
41	#131×C₁₀₉	C₂²	1	2	yes	complemented
42	#108×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
43	#104×C₁₀₉	C₂²	1	2	yes	complemented
44	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
45	#13182030	C₂×C₄	1	2	yes	—
46	#13182029	—	2	2	no	—
47	#13182029	C₂³	1	2	yes	—
48	#13182028	—	2	2	no	—
49	#13182028	C₂×C₄	1	2	yes	—
50	Q₈×D₁₀₉	C₂×C₄	1	2	yes	—
51	D₄×D₁₀₉	—	2	2	no	—
52	D₄×D₁₀₉	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₀₉	—	2	2	no	—
54	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
55	Dic₄₃₆	C₂×C₄	1	1	yes	characteristic
56	#13182019	C₂×C₄	1	2	yes	—
57	#13182018	C₂×C₄	1	2	yes	—
58	#13182017	C₂×C₄	1	2	yes	—
59	#13182016	C₂×C₄	1	2	yes	—
60	D₈₇₂	C₂×C₄	1	1	yes	characteristic
61	#13182014	C₂×C₄	1	2	yes	—
62	#13182013	—	2	2	no	—
63	#13182013	C₂³	1	2	yes	—
64	#13182012	—	2	2	no	—
65	#13182012	—	2	2	no	—
66	#13182012	—	2	2	no	—
67	#13182012	D₄	1	1	yes	characteristic
68	#13182012	D₄	1	1	yes	characteristic
69	#13182012	D₄	1	1	yes	characteristic
70	#13182012	D₄	1	1	yes	characteristic
71	C₂×#11724784	—	2	2	no	—
72	C₂×#11724784	C₂³	1	1	yes	characteristic
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₈×D₁₀₉	—	2	2	no	—
75	C₈×D₁₀₉	—	2	2	no	—
76	C₈×D₁₀₉	—	2	2	no	—
77	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
78	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
79	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
80	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
81	#13182008	—	2	1	no	—
82	#13182006	—	2	1	no	—
83	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
84	#41×C₁₀₉	C₂³	1	2	yes	—
85	Q₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
86	SD₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
87	D₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
88	M₁₆×C₁₀₉	—	2	2	no	—
89	M₁₆×C₁₀₉	C₂³	1	2	yes	—
90	C₂×C₈×C₁₀₉	—	2	2	no	—
91	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
93	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₂²×D₁₀₉	—	2	2	no	—
95	C₂²×D₁₀₉	#41	1	1	yes	characteristic
96	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
97	D₄×C₁₀₉	—	2	2	no	—
98	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₀₉	—	2	2	no	—
100	C₂×C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₁₀₉	—	2	2	no	—
103	C₂²⋊D₁₀₉	—	2	2	no	—
104	C₂²⋊D₁₀₉	—	2	2	no	—
105	C₂×Dic₁₀₉	—	2	2	no	—
106	C₂×Dic₁₀₉	—	2	1	no	—
107	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
108	D₄₃₆	—	2	2	no	—
109	D₄₃₆	C₂²×C₄	1	2	yes	—
110	C₄×D₁₀₉	—	2	2	no	—
111	C₄×D₁₀₉	—	2	2	no	—
112	C₄×D₁₀₉	—	2	2	no	—
113	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
114	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
115	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
116	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
117	Dic₂₁₈	—	2	2	no	—
118	Dic₂₁₈	C₂²×C₄	1	2	yes	—
119	C₈×C₁₀₉	—	2	2	no	—
120	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
121	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
122	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	#11724784	—	2	2	no	—
125	#11724784	C₂²×C₄	1	1	yes	characteristic
126	#11724784	C₂²×C₄	1	1	yes	characteristic
127	#11724784	C₂×D₄	1	1	yes	characteristic
128	#11724784	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₁₀₉	—	2	2	no	—
130	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
131	C₂×D₁₀₉	—	2	2	no	—
132	C₂×D₁₀₉	—	4	1	no	—
133	C₂×D₁₀₉	—	8	2	no	—
134	C₂×D₁₀₉	C₄×D₄	1	1	yes	characteristic
135	C₂×D₁₀₉	C₄×D₄	1	1	yes	characteristic
136	C₄×C₁₀₉	—	2	2	no	—
137	C₄×C₁₀₉	—	4	1	no	—
138	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
140	Dic₁₀₉	—	2	2	no	—
141	Dic₁₀₉	—	4	1	no	—
142	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
143	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
144	C₂×C₁₀₉	—	2	1	no	—
145	C₂×C₁₀₉	—	4	2	no	—
146	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
147	D₁₀₉	—	2	1	no	—
148	D₁₀₉	—	2	1	no	—
149	D₁₀₉	—	4	2	no	—
150	#2957	—	109	1	no	maximal, Sylow
151	C₁₀₉	#2957	1	1	yes	characteristic, complemented, Sylow
152	C₂×#135	—	109	1	no	—
153	#568	—	109	2	no	—
154	#444	—	109	2	no	—
155	#444	—	109	2	no	—
156	#443	—	109	1	no	—
157	#443	—	109	1	no	—
158	#443	—	109	1	no	—
159	#443	D₁₀₉	1	1	yes	characteristic, complemented
160	C₂×#108	—	109	1	no	—
161	#421	—	109	2	no	—
162	#405	—	109	1	no	—
163	C₂×#41	—	109	2	no	—
164	#135	—	109	2	no	—
165	#135	—	109	1	no	—
166	#135	—	109	2	no	—
167	#135	—	109	1	no	—
168	#135	—	109	1	no	—
169	#135	C₂×D₁₀₉	1	1	yes	characteristic
170	C₂×Q₁₆	—	109	1	no	—
171	C₂×SD₁₆	—	109	2	no	—
172	C₂×D₈	—	109	1	no	—
173	#131	—	109	2	no	—
174	#131	—	109	2	no	—
175	#131	—	109	2	no	—
176	#131	—	218	2	no	—
177	#131	—	218	2	no	—
178	#131	C₂×D₁₀₉	1	2	yes	complemented
179	C₂×M₁₆	—	109	1	no	—
180	C₂×M₁₆	—	109	2	no	—
181	C₂×M₁₆	—	218	2	no	—
182	C₂×M₁₆	—	218	2	no	—
183	C₂²×C₈	—	109	1	no	—
184	#117	—	109	1	no	—
185	#108	—	109	2	no	—
186	#108	—	109	1	no	—
187	#108	C₂×D₁₀₉	1	1	yes	characteristic, complemented
188	#104	—	109	2	no	—
189	#104	—	109	2	no	—
190	#104	—	109	2	no	—
191	#104	C₂×D₁₀₉	1	2	yes	complemented
192	#97	—	109	1	no	—
193	#97	—	109	1	no	—
194	C₄×C₈	—	109	1	no	—
195	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
196	#41	—	109	2	no	—
197	#41	—	109	2	no	—
198	#41	—	109	2	no	—
199	#41	—	218	2	no	—
200	#41	—	218	2	no	—
201	#41	C₂²×D₁₀₉	1	2	yes	—
202	C₂×Q₈	—	109	2	no	—
203	C₂×D₄	—	109	2	no	—
204	C₂×D₄	—	218	2	no	—
205	C₂²×C₄	—	109	1	no	—
206	C₂²×C₄	—	218	2	no	—
207	Q₁₆	—	109	2	no	—
208	Q₁₆	—	109	1	no	—
209	Q₁₆	C₄×D₁₀₉	1	1	yes	characteristic
210	SD₁₆	—	109	2	no	—
211	SD₁₆	—	109	2	no	—
212	SD₁₆	—	109	2	no	—
213	SD₁₆	C₄×D₁₀₉	1	2	yes	—
214	D₈	—	109	2	no	—
215	D₈	—	109	1	no	—
216	D₈	C₄×D₁₀₉	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	109	1	no	—
219	M₁₆	—	109	1	no	—
220	M₁₆	—	109	1	no	—
221	M₁₆	—	109	1	no	—
222	M₁₆	—	109	2	no	—
223	M₁₆	—	218	2	no	—
224	M₁₆	—	218	2	no	—
225	M₁₆	—	218	2	no	—
226	M₁₆	—	218	2	no	—
227	M₁₆	C₂²×D₁₀₉	1	2	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	109	1	no	—
230	C₂×C₈	—	109	1	no	—
231	C₂×C₈	—	109	1	no	—
232	C₂×C₈	—	109	1	no	—
233	C₂×C₈	—	109	1	no	—
234	C₂×C₈	—	109	1	no	—
235	C₂×C₈	—	218	2	no	—
236	C₂×C₈	—	218	2	no	—
237	C₂×C₈	—	218	2	no	—
238	C₂×C₈	—	218	2	no	—
239	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
241	C₄⋊C₄	—	218	1	no	—
242	C₂²⋊C₄	—	218	1	no	—
243	C₄²	—	109	1	no	—
244	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
245	C₂³	—	109	1	no	—
246	C₂³	—	218	2	no	—
247	Q₈	—	109	2	no	—
248	Q₈	—	218	2	no	—
249	Q₈	C₂×C₄×D₁₀₉	1	2	yes	—
250	D₄	—	2	2	no	—
251	D₄	—	109	2	no	—
252	D₄	—	218	2	no	—
253	D₄	—	218	2	no	—
254	D₄	—	218	2	no	—
255	D₄	—	218	2	no	—
256	D₄	C₂×C₄×D₁₀₉	1	2	yes	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	109	1	no	—
260	C₂×C₄	—	109	1	no	—
261	C₂×C₄	—	109	1	no	—
262	C₂×C₄	—	109	1	no	—
263	C₂×C₄	—	109	1	no	—
264	C₂×C₄	—	218	2	no	—
265	C₂×C₄	—	218	2	no	—
266	C₂×C₄	—	218	2	no	—
267	C₂×C₄	—	218	1	no	—
268	C₂×C₄	—	218	2	no	—
269	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	109	1	no	—
272	C₈	—	109	1	no	—
273	C₈	—	109	1	no	—
274	C₈	—	109	1	no	—
275	C₈	—	218	2	no	—
276	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
277	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
278	C₈	D₄×D₁₀₉	1	1	yes	characteristic
279	C₈	D₄×D₁₀₉	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	872	2	no	—
282	C₂²	—	109	1	no	—
283	C₂²	—	109	1	no	—
284	C₂²	—	436	1	no	—
285	C₂²	—	218	2	no	—
286	C₂²	#41×D₁₀₉	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	109	1	no	—
290	C₄	—	109	1	no	—
291	C₄	—	436	1	no	—
292	C₄	—	218	2	no	—
293	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
294	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	—	436	2	no	—
298	C₂	—	218	1	no	—
299	C₂	—	218	1	no	—
300	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	#443×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×C₄×D₄
prime radical	#135×D₁₀₉	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#443	D₁₀₉
hypocenter	C₁₀₉	#2957
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #20295791	C₂	semidirect
3. #20295146	C₂	semidirect
4. #20295145	C₂	semidirect
5. #20295144	C₂	semidirect
6. #20295143	C₂	semidirect
7. #20295142	C₂	semidirect
8. #108×D₁₀₉	C₂	semidirect
9. #20295041	C₂	semidirect
10. #20294980	C₂	semidirect
11. #443×C₁₀₉	C₂	semidirect
13. #16812722	C₄	semidirect
15. #16812720	C₄	semidirect
17. #16812718	C₄	semidirect
18. Q₁₆×D₁₀₉	C₄	semidirect
19. SD₁₆×D₁₀₉	C₄	semidirect
20. D₈×D₁₀₉	C₄	semidirect
22. #16812707	C₂²	semidirect
23. #16812706	C₂²	semidirect
25. #16812705	C₂²	semidirect
32. #16812635	C₂²	semidirect
33. #16812634	C₂²	semidirect
34. #16812627	C₂²	semidirect
35. #16812626	C₂²	semidirect
36. #16812625	C₂²	semidirect
37. #16812610	C₂²	semidirect
38. #16812609	C₂²	semidirect
39. C₈×Dic₁₀₉	C₂²	semidirect
41. #131×C₁₀₉	C₂²	semidirect
42. #108×C₁₀₉	C₂²	semidirect
43. #104×C₁₀₉	C₂²	semidirect
44. C₄×C₈×C₁₀₉	C₂²	semidirect
151. C₁₀₉	#2957	semidirect
159. #443	D₁₀₉	semidirect
178. #131	C₂×D₁₀₉	semidirect
187. #108	C₂×D₁₀₉	semidirect
191. #104	C₂×D₁₀₉	semidirect
195. C₄×C₈	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#135×D₁₀₉	2
#20295791	2
#20295146	2
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