Subgroups — #28372849

Order 13952.

384 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13182028	C₂	1	1	yes	maximal, characteristic
3	#20295336	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295334	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295332	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295330	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295328	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295326	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295324	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295067	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295065	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295048	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295024	C₂	1	1	yes	maximal, characteristic
14	#20295024	C₂	1	1	yes	maximal, characteristic
15	#478×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182028	C₂²	1	1	yes	characteristic
17	C₂×#13182012	C₂²	1	1	yes	characteristic
18	C₂×#13182012	C₂²	1	1	yes	characteristic
19	#16812691	C₂²	1	1	yes	characteristic
20	#16812690	C₂²	1	1	yes	characteristic
21	#16812690	C₂²	1	1	yes	characteristic
22	#16812689	—	2	1	no	—
23	#16812689	—	2	1	no	—
24	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
25	#16812664	C₂²	1	1	yes	characteristic
26	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
28	#16812659	—	2	1	no	—
29	#16812657	C₂²	1	1	yes	characteristic
30	#16812655	—	2	1	no	—
31	C₄⋊C₄×D₁₀₉	—	2	1	no	—
32	C₄⋊C₄×D₁₀₉	—	2	1	no	—
33	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₁₀₉	—	2	1	no	—
36	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
37	#16812633	C₂²	1	1	yes	characteristic
38	#16812632	C₂²	1	1	yes	characteristic
39	#16812632	C₂²	1	1	yes	characteristic
40	#16812631	C₂²	1	1	yes	characteristic
41	#16812630	C₂²	1	1	yes	characteristic
42	#16812630	C₂²	1	1	yes	characteristic
43	C₄⋊#11724784	C₂²	1	1	yes	characteristic
44	#16812628	C₂²	1	1	yes	characteristic
45	#16812628	C₂²	1	1	yes	characteristic
46	#16812624	C₂²	1	1	yes	characteristic
47	#16812624	C₂²	1	1	yes	characteristic
48	#16812622	C₂²	1	1	yes	characteristic
49	#16812622	C₂²	1	1	yes	characteristic
50	#16812620	C₂²	1	1	yes	characteristic
51	#16812620	C₂²	1	1	yes	characteristic
52	#125×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
54	#107×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	#13182028	—	2	2	no	—
60	#13182028	Q₈	1	4	yes	—
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
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	C₂×C₄×D₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
70	#13182012	—	2	2	no	—
71	#13182012	—	2	2	no	—
72	C₂×#11724784	C₂³	1	1	yes	characteristic
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	—	2	1	no	—
75	C₄⋊Dic₁₀₉	—	2	1	no	—
76	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	—	2	1	no	—
86	#13182008	—	2	1	no	—
87	C₂²⋊Dic₁₀₉	—	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	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
93	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₀₉	—	2	1	no	—
100	C₄⋊C₄×C₁₀₉	—	2	1	no	—
101	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
105	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂³×C₁₀₉	—	2	1	no	—
107	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
108	D₄×C₁₀₉	—	2	1	no	—
109	D₄×C₁₀₉	C₂×Q₈	1	2	yes	—
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	—	4	2	no	—
114	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
117	C₂²⋊D₁₀₉	—	2	4	no	—
118	C₂×Dic₁₀₉	—	2	1	no	—
119	C₂×Dic₁₀₉	—	2	4	no	—
120	C₂×Dic₁₀₉	—	2	1	no	—
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	4	2	no	—
124	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₀₉	—	4	1	no	—
128	C₄×D₁₀₉	—	4	1	no	—
129	C₄×D₁₀₉	—	4	2	no	—
130	C₄×D₁₀₉	—	4	2	no	—
131	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
132	C₄×D₁₀₉	#41	1	2	yes	—
133	Dic₂₁₈	—	2	1	no	—
134	Dic₂₁₈	C₂×Q₈	1	2	yes	—
135	C₈×C₁₀₉	—	2	1	no	—
136	C₈×C₁₀₉	—	2	1	no	—
137	#11724784	—	2	1	no	—
138	#11724784	—	2	1	no	—
139	C₂²×C₁₀₉	—	2	2	no	—
140	C₂²×C₁₀₉	—	4	1	no	—
141	C₂²×C₁₀₉	—	4	1	no	—
142	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₀₉	—	2	1	no	—
144	C₂×D₁₀₉	—	2	1	no	—
145	C₂×D₁₀₉	C₂²⋊Q₈	1	2	yes	—
146	C₄×C₁₀₉	—	2	1	no	—
147	C₄×C₁₀₉	—	2	1	no	—
148	C₄×C₁₀₉	—	4	1	no	—
149	C₄×C₁₀₉	—	4	1	no	—
150	C₄×C₁₀₉	—	4	1	no	—
151	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
152	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
153	Dic₁₀₉	—	2	2	no	—
154	Dic₁₀₉	—	2	1	no	—
155	Dic₁₀₉	—	2	1	no	—
156	Dic₁₀₉	—	4	1	no	—
157	Dic₁₀₉	—	4	1	no	—
158	Dic₁₀₉	—	4	1	no	—
159	Dic₁₀₉	C₂²⋊Q₈	1	2	yes	—
160	C₂×C₁₀₉	—	4	2	no	—
161	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₁₀₉	#575	1	1	yes	characteristic
163	C₂×C₁₀₉	#575	1	1	yes	characteristic
164	D₁₀₉	—	2	2	no	—
165	#3080	—	109	1	no	maximal, Sylow
166	C₁₀₉	#3080	1	1	yes	characteristic, complemented, Sylow
167	C₂×#125	—	109	1	no	—
168	C₄×#41	—	109	1	no	—
169	#479	—	109	1	no	—
170	#479	—	109	1	no	—
171	#479	—	109	1	no	—
172	#479	—	109	1	no	—
173	#478	—	109	1	no	—
174	#478	—	109	1	no	—
175	#478	—	109	1	no	—
176	#478	D₁₀₉	1	1	yes	characteristic, complemented
177	#428	—	109	1	no	—
178	#428	—	109	1	no	—
179	#423	—	109	1	no	—
180	#417	—	109	1	no	—
181	#417	—	109	1	no	—
182	C₂×#41	—	109	1	no	—
183	C₂×M₁₆	—	109	1	no	—
184	C₂×M₁₆	—	109	1	no	—
185	#125	—	109	1	no	—
186	#125	—	109	1	no	—
187	#125	—	109	1	no	—
188	#125	—	218	1	no	—
189	#125	—	218	1	no	—
190	#125	C₂×D₁₀₉	1	1	yes	characteristic
191	C₄×Q₈	—	109	1	no	—
192	C₄×Q₈	—	109	1	no	—
193	C₄×D₄	—	109	1	no	—
194	C₄×D₄	—	218	1	no	—
195	C₄×D₄	—	218	1	no	—
196	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
197	#117	—	109	1	no	—
198	#117	—	218	1	no	—
199	C₂×C₄⋊C₄	—	109	1	no	—
200	C₂×C₄⋊C₄	—	109	1	no	—
201	C₂×C₄⋊C₄	—	218	1	no	—
202	C₂×C₄⋊C₄	—	218	1	no	—
203	C₂×C₄²	—	109	1	no	—
204	C₂×C₄²	—	218	1	no	—
205	#107	—	109	1	no	—
206	#107	—	109	1	no	—
207	#107	—	109	1	no	—
208	#107	C₂×D₁₀₉	1	1	yes	characteristic
209	#106	—	109	1	no	—
210	#106	—	109	1	no	—
211	#106	—	109	1	no	—
212	#106	C₂×D₁₀₉	1	1	yes	characteristic
213	C₄⋊C₈	—	109	1	no	—
214	C₄⋊C₈	—	109	1	no	—
215	C₄⋊C₈	—	109	1	no	—
216	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
217	#103	—	109	1	no	—
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	C₂×D₁₀₉	1	1	yes	characteristic
224	#102	C₂×D₁₀₉	1	1	yes	characteristic
225	#41	—	109	4	no	—
226	#41	—	218	2	no	—
227	C₂×Q₈	—	109	1	no	—
228	C₂×D₄	—	218	1	no	—
229	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	109	1	no	—
232	C₂²×C₄	—	109	1	no	—
233	C₂²×C₄	—	109	1	no	—
234	C₂²×C₄	—	218	1	no	—
235	C₂²×C₄	—	218	1	no	—
236	C₂²×C₄	—	218	1	no	—
237	C₂²×C₄	—	218	1	no	—
238	M₁₆	—	218	2	no	—
239	M₁₆	—	218	2	no	—
240	C₂×C₈	—	109	1	no	—
241	C₂×C₈	—	109	1	no	—
242	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	109	1	no	—
247	C₄⋊C₄	—	109	1	no	—
248	C₄⋊C₄	—	109	1	no	—
249	C₄⋊C₄	—	218	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	C₂²×D₁₀₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	218	1	no	—
264	C₂²⋊C₄	—	218	1	no	—
265	C₄²	—	109	1	no	—
266	C₄²	—	109	1	no	—
267	C₄²	—	109	1	no	—
268	C₄²	—	218	1	no	—
269	C₄²	—	218	1	no	—
270	C₄²	—	218	1	no	—
271	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	109	1	no	—
274	Q₈	—	109	2	no	—
275	Q₈	—	218	1	no	—
276	D₄	—	2	1	no	—
277	D₄	—	218	4	no	—
278	D₄	Q₈×D₁₀₉	1	2	yes	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	109	1	no	—
284	C₂×C₄	—	109	2	no	—
285	C₂×C₄	—	109	2	no	—
286	C₂×C₄	—	109	1	no	—
287	C₂×C₄	—	109	1	no	—
288	C₂×C₄	—	436	1	no	—
289	C₂×C₄	—	436	1	no	—
290	C₂×C₄	—	436	2	no	—
291	C₂×C₄	—	436	2	no	—
292	C₂×C₄	—	436	2	no	—
293	C₂×C₄	—	218	4	no	—
294	C₂×C₄	—	218	1	no	—
295	C₂×C₄	—	218	1	no	—
296	C₂×C₄	—	218	1	no	—
297	C₂×C₄	—	218	1	no	—
298	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
299	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	218	1	no	—
304	C₈	—	218	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	109	2	no	—
309	C₂²	—	218	1	no	—
310	C₂²	—	218	1	no	—
311	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	109	2	no	—
318	C₄	—	436	1	no	—
319	C₄	—	436	1	no	—
320	C₄	—	436	1	no	—
321	C₄	—	218	2	no	—
322	C₄	—	218	1	no	—
323	C₄	—	218	1	no	—
324	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
325	C₄	#41×D₁₀₉	1	1	yes	characteristic
326	C₂	—	4	2	no	—
327	C₂	—	218	2	no	—
328	C₂	C₂²⋊Q₈×D₁₀₉	1	1	yes	characteristic
329	C₂	#20295822	1	1	yes	characteristic
330	C₂	#20295823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#478×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#13182028	C₂²
hypercenter	#478	D₁₀₉
hypocenter	C₁₀₉	#3080
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #125×D₁₀₉	C₂	semidirect
3. #20295336	C₂	semidirect
4. #20295334	C₂	semidirect
5. #20295332	C₂	semidirect
6. #20295330	C₂	semidirect
7. #20295328	C₂	semidirect
8. #20295326	C₂	semidirect
9. #20295324	C₂	semidirect
10. #20295067	C₂	semidirect
11. #20295065	C₂	semidirect
12. #20295048	C₂	semidirect
15. #478×C₁₀₉	C₂	semidirect
166. C₁₀₉	#3080	semidirect
176. #478	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#125×D₁₀₉	2
C₄×#13182028	2
#20295336	2
#20295334	2
#20295332	2
#20295330	2
#20295328	2
#20295326	2
#20295324	2
#20295067	2
#20295065	2
#20295048	2
#20295024	2
#478×C₁₀₉	2
#3080	109