Subgroups — #28375597

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	#20295726	C₂	1	1	yes	maximal, characteristic
2	#20295717	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295428	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295365	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295355	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295349	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295343	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295333	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295326	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295269	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295269	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295105	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295065	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295065	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182012	C₂²	1	1	yes	characteristic
17	C₂×#13182012	C₂²	1	1	yes	characteristic
18	#16812693	—	2	1	no	—
19	#16812692	—	2	1	no	—
20	#16812691	C₂²	1	1	yes	characteristic
21	#16812691	C₂²	1	1	yes	characteristic
22	#16812689	—	2	1	no	—
23	#16812684	—	2	1	no	—
24	#16812682	—	2	1	no	—
25	#16812675	C₂²	1	1	yes	characteristic
26	#16812675	C₂²	1	1	yes	characteristic
27	#16812672	—	2	1	no	—
28	#16812670	C₂²	1	1	yes	characteristic
29	#16812670	C₂²	1	1	yes	characteristic
30	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
32	#16812657	C₂²	1	1	yes	characteristic
33	#16812657	C₂²	1	1	yes	characteristic
34	#16812656	—	2	1	no	—
35	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
37	#16812633	C₂²	1	1	yes	characteristic
38	#16812633	C₂²	1	1	yes	characteristic
39	#16812632	C₂²	1	1	yes	characteristic
40	#16812632	C₂²	1	1	yes	characteristic
41	#16812631	C₂²	1	1	yes	characteristic
42	#16812631	C₂²	1	1	yes	characteristic
43	#16812630	C₂²	1	1	yes	characteristic
44	#16812630	C₂²	1	1	yes	characteristic
45	C₄⋊#11724784	C₂²	1	1	yes	characteristic
46	#16812623	C₂²	1	1	yes	characteristic
47	#16812623	C₂²	1	1	yes	characteristic
48	#16812620	C₂²	1	1	yes	characteristic
49	#16812620	C₂²	1	1	yes	characteristic
50	#16812611	C₂²	1	1	yes	characteristic
51	#16812611	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×D₄₃₆	D₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
61	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
62	C₂×C₄×D₁₀₉	—	2	1	no	—
63	C₂×C₄×D₁₀₉	—	2	1	no	—
64	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
66	#13182012	—	2	2	no	—
67	#13182012	—	2	2	no	—
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₂×#11724784	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₀₉	D₄	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	#13182008	—	2	1	no	—
83	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	C₂²⋊Dic₁₀₉	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	D₄	1	1	yes	characteristic
90	#13182006	D₄	1	1	yes	characteristic
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₀₉	—	2	1	no	—
96	SD₁₆×C₁₀₉	—	2	2	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	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
103	C₄²×C₁₀₉	—	2	1	no	—
104	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂³×C₁₀₉	—	2	1	no	—
106	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₁₀₉	—	2	1	no	—
109	Q₈×C₁₀₉	—	2	1	no	—
110	D₄×C₁₀₉	—	2	1	no	—
111	D₄×C₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	8	1	no	—
116	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₁₀₉	—	4	2	no	—
120	C₂²⋊D₁₀₉	—	8	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	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
127	D₄₃₆	—	4	1	no	—
128	C₄×D₁₀₉	—	2	1	no	—
129	C₄×D₁₀₉	—	2	1	no	—
130	C₄×D₁₀₉	—	2	1	no	—
131	C₄×D₁₀₉	—	2	1	no	—
132	C₄×D₁₀₉	—	4	2	no	—
133	C₄×D₁₀₉	—	8	1	no	—
134	C₈×C₁₀₉	—	2	1	no	—
135	C₈×C₁₀₉	#41	1	2	yes	—
136	#11724784	—	2	1	no	—
137	#11724784	—	2	1	no	—
138	C₂²×C₁₀₉	—	4	1	no	—
139	C₂²×C₁₀₉	—	4	1	no	—
140	C₂²×C₁₀₉	—	4	1	no	—
141	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₀₉	—	2	1	no	—
143	C₂×D₁₀₉	—	2	1	no	—
144	C₂×D₁₀₉	—	2	1	no	—
145	C₂×D₁₀₉	—	2	1	no	—
146	C₂×D₁₀₉	—	2	1	no	—
147	C₂×D₁₀₉	—	2	1	no	—
148	C₄×C₁₀₉	—	2	1	no	—
149	C₄×C₁₀₉	—	2	1	no	—
150	C₄×C₁₀₉	—	4	1	no	—
151	C₄×C₁₀₉	—	4	1	no	—
152	C₄×C₁₀₉	—	4	1	no	—
153	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
154	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
155	Dic₁₀₉	—	2	1	no	—
156	Dic₁₀₉	—	2	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	—	8	1	no	—
162	C₂×C₁₀₉	#547	1	1	yes	characteristic
163	C₂×C₁₀₉	#575	1	1	yes	characteristic
164	C₂×C₁₀₉	#577	1	1	yes	characteristic
165	D₁₀₉	—	4	1	no	—
166	D₁₀₉	—	4	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	D₁₀₉	1	1	yes	characteristic, complemented
181	#436	—	109	1	no	—
182	#428	—	109	1	no	—
183	#428	—	109	1	no	—
184	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
185	C₂×M₁₆	—	109	1	no	—
186	C₂×M₁₆	—	109	1	no	—
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	C₂×D₁₀₉	1	1	yes	characteristic
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	C₂×D₁₀₉	1	1	yes	characteristic
217	C₄⋊C₈	—	109	1	no	—
218	#103	—	109	1	no	—
219	#103	—	109	1	no	—
220	#103	C₂×D₁₀₉	1	1	yes	characteristic
221	#102	—	109	1	no	—
222	#102	—	109	1	no	—
223	#102	C₂×D₁₀₉	1	1	yes	characteristic
224	C₂²⋊C₈	—	109	1	no	—
225	C₂²⋊C₈	—	109	1	no	—
226	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
227	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
228	C₂×D₄	—	109	1	no	—
229	C₂×D₄	—	218	1	no	—
230	C₂×D₄	—	218	1	no	—
231	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	109	1	no	—
234	C₂²×C₄	—	109	1	no	—
235	C₂²×C₄	—	218	1	no	—
236	C₂²×C₄	—	218	1	no	—
237	SD₁₆	—	2	2	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	—	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	C₂²×D₁₀₉	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	109	1	no	—
262	C₂²⋊C₄	—	109	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	—	218	1	no	—
268	C₂²⋊C₄	—	218	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	218	1	no	—
271	C₄²	—	218	1	no	—
272	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
273	C₂³	—	2	1	no	—
274	C₂³	—	109	1	no	—
275	C₂³	—	109	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	872	1	no	—
281	D₄	—	436	1	no	—
282	D₄	—	436	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	872	1	no	—
288	C₂×C₄	—	109	1	no	—
289	C₂×C₄	—	109	1	no	—
290	C₂×C₄	—	436	2	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	—	218	1	no	—
298	C₂×C₄	—	218	1	no	—
299	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
300	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
301	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	218	1	no	—
304	C₈	—	218	1	no	—
305	C₈	#13182029	1	2	yes	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	218	1	no	—
310	C₂²	—	218	1	no	—
311	C₂²	—	218	1	no	—
312	C₂²	—	218	1	no	—
313	C₂²	—	218	1	no	—
314	C₂²	—	218	1	no	—
315	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	436	1	no	—
322	C₄	—	436	1	no	—
323	C₄	—	436	1	no	—
324	C₄	—	436	1	no	—
325	C₄	—	218	1	no	—
326	C₄	—	218	1	no	—
327	C₄	C₂×#13182029	1	1	yes	characteristic
328	C₄	#16812746	1	1	yes	characteristic
329	C₂	—	8	1	no	—
330	C₂	—	436	1	no	—
331	C₂	—	436	1	no	—
332	C₂	#20295681	1	1	yes	characteristic
333	C₂	#20295823	1	1	yes	characteristic
334	C₂	#20295832	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₄×SD₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295726	C₂
hypercenter	C₄×SD₁₆	D₁₀₉
hypocenter	C₁₀₉	#3351
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
2. #20295717	C₂	semidirect
3. #20295428	C₂	semidirect
4. #20295365	C₂	semidirect
5. #20295355	C₂	semidirect
6. #20295349	C₂	semidirect
7. #20295343	C₂	semidirect
8. #20295333	C₂	semidirect
9. #20295326	C₂	semidirect
10. #20295269	C₂	semidirect
11. #20295269	C₂	semidirect
12. #20295105	C₂	semidirect
13. #20295065	C₂	semidirect
14. #20295065	C₂	semidirect
15. C₄×SD₁₆×C₁₀₉	C₂	semidirect
168. C₁₀₉	#3351	semidirect
180. C₄×SD₁₆	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295726	2
#20295717	2
#20295428	2
#20295365	2
#20295355	2
#20295349	2
#20295343	2
#20295333	2
#20295326	2
#20295269	2
#20295105	2
#20295065	2
C₄×SD₁₆×C₁₀₉	2
#3351	109