Subgroups — #28375594

Order 13952.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295722	C₂	1	1	yes	maximal, characteristic
2	#20295716	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295428	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295368	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295357	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295351	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295345	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295334	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295327	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295272	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295272	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13182018	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295068	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295068	C₂	1	1	yes	maximal, characteristic, complemented
15	#436×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182018	C₂²	1	1	yes	characteristic
17	#16812694	—	2	1	no	—
18	#16812692	—	2	1	no	—
19	#16812691	C₂²	1	1	yes	characteristic
20	#16812691	C₂²	1	1	yes	characteristic
21	#16812689	—	2	1	no	—
22	#16812681	—	2	1	no	—
23	#16812680	—	2	1	no	—
24	#16812678	C₂²	1	1	yes	characteristic
25	#16812678	C₂²	1	1	yes	characteristic
26	#16812673	C₂²	1	1	yes	characteristic
27	#16812673	C₂²	1	1	yes	characteristic
28	#16812671	—	2	1	no	—
29	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
30	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
31	#16812659	—	2	1	no	—
32	#16812658	C₂²	1	1	yes	characteristic
33	#16812658	C₂²	1	1	yes	characteristic
34	#16812655	—	2	1	no	—
35	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
37	#16812633	C₂²	1	1	yes	characteristic
38	#16812633	C₂²	1	1	yes	characteristic
39	#16812633	C₂²	1	1	yes	characteristic
40	#16812633	C₂²	1	1	yes	characteristic
41	#16812631	C₂²	1	1	yes	characteristic
42	#16812631	C₂²	1	1	yes	characteristic
43	#16812631	C₂²	1	1	yes	characteristic
44	#16812631	C₂²	1	1	yes	characteristic
45	#16812631	C₂²	1	1	yes	characteristic
46	#16812624	C₂²	1	1	yes	characteristic
47	#16812624	C₂²	1	1	yes	characteristic
48	#16812624	C₂²	1	1	yes	characteristic
49	#16812621	C₂²	1	1	yes	characteristic
50	#16812621	C₂²	1	1	yes	characteristic
51	#16812621	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
60	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
62	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
63	C₂²×Dic₁₀₉	—	2	1	no	—
64	C₂²×Dic₁₀₉	—	2	1	no	—
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	#13182018	—	2	2	no	—
67	C₄⋊Dic₁₀₉	—	2	1	no	—
68	C₄⋊Dic₁₀₉	—	2	1	no	—
69	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
75	#13182008	—	2	1	no	—
76	#13182008	—	2	1	no	—
77	#13182008	—	2	1	no	—
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	C₂²⋊Dic₁₀₉	—	2	1	no	—
83	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	C₂²⋊Dic₁₀₉	—	2	1	no	—
85	C₂²⋊Dic₁₀₉	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	C₄×Dic₁₀₉	—	2	1	no	—
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
95	M₁₆×C₁₀₉	—	2	2	no	—
96	M₁₆×C₁₀₉	—	2	2	no	—
97	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
103	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
104	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₁₀₉	—	2	1	no	—
108	D₄×C₁₀₉	—	4	1	no	—
109	C₂×C₄×C₁₀₉	—	2	1	no	—
110	C₂×C₄×C₁₀₉	—	2	1	no	—
111	C₂×C₄×C₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₀₉	—	4	2	no	—
119	C₂²⋊D₁₀₉	—	8	1	no	—
120	C₂×Dic₁₀₉	—	2	1	no	—
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	4	2	no	—
128	C₂×Dic₁₀₉	—	8	1	no	—
129	D₄₃₆	—	2	1	no	—
130	D₄₃₆	—	2	1	no	—
131	C₄×D₁₀₉	—	8	1	no	—
132	Dic₂₁₈	—	2	1	no	—
133	Dic₂₁₈	—	2	1	no	—
134	C₈×C₁₀₉	—	2	1	no	—
135	C₈×C₁₀₉	—	2	1	no	—
136	C₈×C₁₀₉	—	2	1	no	—
137	C₈×C₁₀₉	#41	1	2	yes	—
138	C₂²×C₁₀₉	—	2	1	no	—
139	C₂²×C₁₀₉	—	2	1	no	—
140	C₂²×C₁₀₉	—	2	1	no	—
141	C₂²×C₁₀₉	—	2	1	no	—
142	C₂²×C₁₀₉	—	2	1	no	—
143	C₂²×C₁₀₉	—	2	1	no	—
144	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₀₉	—	4	1	no	—
146	C₂×D₁₀₉	—	4	1	no	—
147	C₂×D₁₀₉	—	4	1	no	—
148	C₄×C₁₀₉	—	2	1	no	—
149	C₄×C₁₀₉	—	2	1	no	—
150	C₄×C₁₀₉	—	2	1	no	—
151	C₄×C₁₀₉	—	2	1	no	—
152	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
153	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₁₀₉	—	4	1	no	—
155	Dic₁₀₉	—	4	1	no	—
156	Dic₁₀₉	—	4	1	no	—
157	Dic₁₀₉	—	4	1	no	—
158	Dic₁₀₉	—	4	1	no	—
159	Dic₁₀₉	—	4	1	no	—
160	Dic₁₀₉	—	4	1	no	—
161	C₂×C₁₀₉	—	4	1	no	—
162	C₂×C₁₀₉	—	4	1	no	—
163	C₂×C₁₀₉	#547	1	1	yes	characteristic
164	C₂×C₁₀₉	#575	1	1	yes	characteristic
165	C₂×C₁₀₉	#577	1	1	yes	characteristic
166	D₁₀₉	—	8	1	no	—
167	#3351	—	109	1	no	maximal, Sylow
168	C₁₀₉	#3351	1	1	yes	characteristic, complemented, Sylow
169	#553	—	109	1	no	—
170	#552	—	109	1	no	—
171	#499	—	109	1	no	—
172	#485	—	109	1	no	—
173	#483	—	109	1	no	—
174	#482	—	109	1	no	—
175	#481	—	109	1	no	—
176	#479	—	109	1	no	—
177	#478	—	109	1	no	—
178	#468	—	109	1	no	—
179	#468	—	109	1	no	—
180	C₄×SD₁₆	—	109	1	no	—
181	#436	D₁₀₉	1	1	yes	characteristic, complemented
182	#428	—	109	1	no	—
183	#428	—	109	1	no	—
184	C₂×SD₁₆	—	109	1	no	—
185	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
186	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
187	#126	—	218	1	no	—
188	#126	—	218	1	no	—
189	#125	—	109	1	no	—
190	#125	—	109	1	no	—
191	#125	—	218	1	no	—
192	#123	—	218	1	no	—
193	#123	—	218	1	no	—
194	C₂²⋊Q₈	—	109	1	no	—
195	C₂²⋊Q₈	—	109	1	no	—
196	#121	—	109	1	no	—
197	#121	—	109	1	no	—
198	#121	—	218	1	no	—
199	C₄×Q₈	—	109	1	no	—
200	C₄×D₄	—	109	1	no	—
201	C₄×D₄	—	218	1	no	—
202	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
203	#117	—	109	1	no	—
204	#117	—	109	1	no	—
205	#117	—	218	1	no	—
206	C₂×C₄⋊C₄	—	109	1	no	—
207	C₂×C₄⋊C₄	—	109	1	no	—
208	#107	—	109	1	no	—
209	#107	—	109	1	no	—
210	#107	—	109	1	no	—
211	#107	—	109	1	no	—
212	#106	—	109	1	no	—
213	#106	—	109	1	no	—
214	#106	—	109	1	no	—
215	#106	—	109	1	no	—
216	#106	—	109	1	no	—
217	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
218	#103	—	109	1	no	—
219	#103	—	109	1	no	—
220	#103	—	109	1	no	—
221	#102	—	109	1	no	—
222	#102	—	109	1	no	—
223	#102	—	109	1	no	—
224	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
225	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
226	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
227	C₂×Q₈	—	109	1	no	—
228	C₂×D₄	—	109	1	no	—
229	C₂×D₄	—	218	1	no	—
230	C₂×D₄	—	218	1	no	—
231	C₂×D₄	D₄₃₆	1	1	yes	characteristic
232	C₂²×C₄	—	218	1	no	—
233	C₂²×C₄	—	218	1	no	—
234	C₂²×C₄	—	218	1	no	—
235	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
236	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
237	SD₁₆	—	218	2	no	—
238	M₁₆	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
244	C₄⋊C₄	—	109	1	no	—
245	C₄⋊C₄	—	109	1	no	—
246	C₄⋊C₄	—	109	1	no	—
247	C₄⋊C₄	—	109	1	no	—
248	C₄⋊C₄	—	109	1	no	—
249	C₄⋊C₄	—	109	1	no	—
250	C₄⋊C₄	—	218	1	no	—
251	C₄⋊C₄	—	218	1	no	—
252	C₄⋊C₄	—	218	1	no	—
253	C₄⋊C₄	—	218	1	no	—
254	C₄⋊C₄	—	218	1	no	—
255	C₄⋊C₄	—	218	1	no	—
256	C₄⋊C₄	—	218	1	no	—
257	C₄⋊C₄	—	218	1	no	—
258	C₄⋊C₄	—	218	1	no	—
259	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
260	C₂²⋊C₄	—	218	1	no	—
261	C₂²⋊C₄	—	218	1	no	—
262	C₂²⋊C₄	—	218	1	no	—
263	C₂²⋊C₄	—	218	1	no	—
264	C₂²⋊C₄	—	218	1	no	—
265	C₂²⋊C₄	—	218	1	no	—
266	C₂²⋊C₄	—	218	1	no	—
267	C₂²⋊C₄	D₄₃₆	1	1	yes	characteristic
268	C₂²⋊C₄	D₄₃₆	1	1	yes	characteristic
269	C₄²	—	218	1	no	—
270	C₄²	—	218	1	no	—
271	C₄²	—	218	1	no	—
272	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
273	C₂³	—	218	1	no	—
274	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
275	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
276	Q₈	—	218	1	no	—
277	Q₈	—	218	1	no	—
278	D₄	—	4	1	no	—
279	D₄	—	872	1	no	—
280	D₄	—	436	2	no	—
281	D₄	—	218	1	no	—
282	D₄	—	218	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	872	1	no	—
288	C₂×C₄	—	872	1	no	—
289	C₂×C₄	—	436	2	no	—
290	C₂×C₄	—	218	1	no	—
291	C₂×C₄	—	218	1	no	—
292	C₂×C₄	—	218	1	no	—
293	C₂×C₄	—	218	1	no	—
294	C₂×C₄	—	218	1	no	—
295	C₂×C₄	—	218	1	no	—
296	C₂×C₄	—	218	1	no	—
297	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
301	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	#13182028	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	436	1	no	—
313	C₂²	—	436	1	no	—
314	C₂²	—	436	1	no	—
315	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	436	1	no	—
321	C₄	—	436	1	no	—
322	C₄	—	436	1	no	—
323	C₄	—	436	1	no	—
324	C₄	—	436	1	no	—
325	C₄	—	436	1	no	—
326	C₄	—	436	1	no	—
327	C₄	C₂×#13182028	1	1	yes	characteristic
328	C₄	#16812746	1	1	yes	characteristic
329	C₂	—	4	1	no	—
330	C₂	—	4	1	no	—
331	C₂	—	872	1	no	—
332	C₂	#20295680	1	1	yes	characteristic
333	C₂	#20295826	1	1	yes	characteristic
334	C₂	#20295838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#436×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295722	C₂
hypercenter	#436	D₁₀₉
hypocenter	C₁₀₉	#3351
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. #20295716	C₂	semidirect
3. #20295428	C₂	semidirect
4. #20295368	C₂	semidirect
5. #20295357	C₂	semidirect
6. #20295351	C₂	semidirect
7. #20295345	C₂	semidirect
8. #20295334	C₂	semidirect
9. #20295327	C₂	semidirect
10. #20295272	C₂	semidirect
11. #20295272	C₂	semidirect
12. C₄×#13182018	C₂	semidirect
13. #20295068	C₂	semidirect
14. #20295068	C₂	semidirect
15. #436×C₁₀₉	C₂	semidirect
168. C₁₀₉	#3351	semidirect
181. #436	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295722	2
#20295716	2
#20295428	2
#20295368	2
#20295357	2
#20295351	2
#20295345	2
#20295334	2
#20295327	2
#20295272	2
C₄×#13182018	2
#20295068	2
#436×C₁₀₉	2
#3351	109