Subgroups — #28375591

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	#20295729	C₂	1	1	yes	maximal, characteristic
2	#20295714	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295370	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295358	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295352	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295346	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295330	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295323	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295273	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295273	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13182018	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295105	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295067	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295067	C₂	1	1	yes	maximal, characteristic, complemented
15	#499×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182018	C₂²	1	1	yes	characteristic
17	C₂×#13182012	C₂²	1	1	yes	characteristic
18	C₂×#13182012	C₂²	1	1	yes	characteristic
19	#16812694	—	2	1	no	—
20	#16812694	—	2	1	no	—
21	#16812690	—	2	1	no	—
22	#16812684	—	2	1	no	—
23	#16812684	—	2	1	no	—
24	#16812679	C₂²	1	1	yes	characteristic
25	#16812679	C₂²	1	1	yes	characteristic
26	#16812674	—	2	1	no	—
27	#16812674	C₂²	1	1	yes	characteristic
28	#16812674	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
30	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
31	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
32	#16812661	—	2	1	no	—
33	#16812657	—	2	1	no	—
34	#16812657	C₂²	1	1	yes	characteristic
35	#16812657	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₀₉	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	#16812624	C₂²	1	1	yes	characteristic
45	#16812622	C₂²	1	1	yes	characteristic
46	#16812622	C₂²	1	1	yes	characteristic
47	#16812621	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic
49	#16812619	C₂²	1	1	yes	characteristic
50	#16812611	C₂²	1	1	yes	characteristic
51	#16812611	C₂²	1	1	yes	characteristic
52	#125×C₁₀₉	C₂²	1	1	yes	characteristic
53	#125×C₁₀₉	C₂²	1	1	yes	characteristic
54	#107×C₁₀₉	C₂²	1	1	yes	characteristic
55	#107×C₁₀₉	C₂²	1	1	yes	characteristic
56	#106×C₁₀₉	C₂²	1	1	yes	characteristic
57	#106×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	—	2	1	no	—
61	C₂×D₄₃₆	—	2	1	no	—
62	C₂×D₄₃₆	D₄	1	1	yes	characteristic
63	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₀₉	—	2	1	no	—
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	C₂×C₄×D₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
69	#13182018	—	2	2	no	—
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	D₄	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	#13182006	—	2	1	no	—
83	#13182006	—	2	1	no	—
84	#13182006	—	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	C₄×Dic₁₀₉	—	2	1	no	—
94	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₀₉	—	2	1	no	—
97	C₄⋊C₄×C₁₀₉	—	2	1	no	—
98	C₄⋊C₄×C₁₀₉	—	2	1	no	—
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	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂²×D₁₀₉	—	2	1	no	—
106	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	—	2	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	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₁₀₉	—	2	1	no	—
116	C₂×Dic₁₀₉	—	2	1	no	—
117	C₂×Dic₁₀₉	—	2	1	no	—
118	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
120	D₄₃₆	—	2	1	no	—
121	D₄₃₆	—	2	1	no	—
122	D₄₃₆	—	4	1	no	—
123	D₄₃₆	—	4	2	no	—
124	D₄₃₆	—	8	1	no	—
125	C₄×D₁₀₉	—	2	1	no	—
126	C₄×D₁₀₉	—	2	1	no	—
127	C₄×D₁₀₉	—	2	1	no	—
128	C₄×D₁₀₉	—	2	1	no	—
129	C₄×D₁₀₉	—	4	2	no	—
130	C₄×D₁₀₉	—	8	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	#41	1	2	yes	—
136	#11724784	—	2	1	no	—
137	#11724784	—	2	1	no	—
138	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₀₉	—	2	1	no	—
140	C₂×D₁₀₉	—	2	1	no	—
141	C₂×D₁₀₉	—	2	1	no	—
142	C₂×D₁₀₉	—	2	1	no	—
143	C₂×D₁₀₉	—	2	1	no	—
144	C₂×D₁₀₉	—	2	1	no	—
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	—	4	1	no	—
151	C₄×C₁₀₉	—	4	1	no	—
152	C₄×C₁₀₉	—	4	1	no	—
153	C₄×C₁₀₉	—	4	1	no	—
154	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
155	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₁₀₉	—	2	1	no	—
157	Dic₁₀₉	—	2	1	no	—
158	Dic₁₀₉	—	4	1	no	—
159	Dic₁₀₉	—	4	1	no	—
160	Dic₁₀₉	—	4	1	no	—
161	C₂×C₁₀₉	#547	1	1	yes	characteristic
162	C₂×C₁₀₉	#575	1	1	yes	characteristic
163	C₂×C₁₀₉	#577	1	1	yes	characteristic
164	D₁₀₉	—	4	1	no	—
165	D₁₀₉	—	4	1	no	—
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	D₁₀₉	1	1	yes	characteristic, complemented
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	—	109	1	no	—
182	#428	—	109	1	no	—
183	#428	—	109	1	no	—
184	C₂×SD₁₆	—	109	1	no	—
185	C₂×M₁₆	—	109	1	no	—
186	C₂×M₁₆	—	109	1	no	—
187	#126	—	218	1	no	—
188	#126	—	218	1	no	—
189	#125	—	218	1	no	—
190	#125	C₂×D₁₀₉	1	1	yes	characteristic
191	#125	C₂×D₁₀₉	1	1	yes	characteristic
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	—	109	1	no	—
202	C₄×D₄	—	218	1	no	—
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	C₂×D₁₀₉	1	1	yes	characteristic
211	#107	C₂×D₁₀₉	1	1	yes	characteristic
212	#106	—	109	1	no	—
213	#106	—	109	1	no	—
214	#106	—	109	1	no	—
215	#106	C₂×D₁₀₉	1	1	yes	characteristic
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	—	109	1	no	—
221	#102	—	109	1	no	—
222	#102	—	109	1	no	—
223	#102	—	109	1	no	—
224	C₂²⋊C₈	—	109	1	no	—
225	C₂²⋊C₈	—	109	1	no	—
226	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
227	C₂×Q₈	—	109	1	no	—
228	C₂×D₄	—	109	1	no	—
229	C₂×D₄	—	109	1	no	—
230	C₂×D₄	—	218	1	no	—
231	C₂×D₄	—	218	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	SD₁₆	—	218	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	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	109	1	no	—
249	C₄⋊C₄	—	109	1	no	—
250	C₄⋊C₄	—	109	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	C₂²×D₁₀₉	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
260	C₂²⋊C₄	—	109	1	no	—
261	C₂²⋊C₄	—	109	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	—	218	1	no	—
268	C₂²⋊C₄	—	218	1	no	—
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	—	109	1	no	—
274	C₂³	—	109	1	no	—
275	C₂³	—	218	1	no	—
276	Q₈	—	218	1	no	—
277	Q₈	—	218	1	no	—
278	D₄	—	872	1	no	—
279	D₄	—	436	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	—	109	1	no	—
290	C₂×C₄	—	109	1	no	—
291	C₂×C₄	—	436	2	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	#13182030	1	2	yes	—
306	C₂²	—	436	1	no	—
307	C₂²	—	436	1	no	—
308	C₂²	—	436	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	—	4	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×#13182030	1	1	yes	characteristic
328	C₄	#16812747	1	1	yes	characteristic
329	C₂	—	872	1	no	—
330	C₂	—	436	1	no	—
331	C₂	—	436	1	no	—
332	C₂	#20295679	1	1	yes	characteristic
333	C₂	#20295823	1	1	yes	characteristic
334	C₂	#20295833	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#499×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295729	C₂
hypercenter	#499	D₁₀₉
hypocenter	C₁₀₉	#3351
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
2. #20295714	C₂	semidirect
3. #20295370	C₂	semidirect
4. #20295358	C₂	semidirect
5. #20295352	C₂	semidirect
6. #20295346	C₂	semidirect
7. #20295330	C₂	semidirect
8. #20295323	C₂	semidirect
9. #20295273	C₂	semidirect
10. #20295273	C₂	semidirect
11. C₄×#13182018	C₂	semidirect
12. #20295105	C₂	semidirect
13. #20295067	C₂	semidirect
14. #20295067	C₂	semidirect
15. #499×C₁₀₉	C₂	semidirect
168. C₁₀₉	#3351	semidirect
171. #499	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295729	2
#20295714	2
#20295370	2
#20295358	2
#20295352	2
#20295346	2
#20295330	2
#20295323	2
#20295273	2
C₄×#13182018	2
#20295105	2
#20295067	2
#499×C₁₀₉	2
#3351	109