Subgroups — #28375567

Order 13952.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295749	C₂	1	1	yes	maximal, characteristic
2	#20295717	C₂	1	1	yes	maximal, characteristic
3	#20295435	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295366	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295365	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295365	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295360	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295333	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295321	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295281	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295264	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295099	C₂	1	1	yes	maximal, characteristic
13	#20295065	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295061	C₂	1	1	yes	maximal, characteristic, complemented
15	#441×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182012	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	#16812699	C₂²	1	1	yes	characteristic
19	#16812694	—	2	1	no	—
20	#16812694	—	2	1	no	—
21	#16812693	—	2	1	no	—
22	#16812691	C₂²	1	1	yes	characteristic
23	#16812689	—	2	1	no	—
24	#16812687	—	2	1	no	—
25	#16812684	—	2	1	no	—
26	#16812675	C₂²	1	1	yes	characteristic
27	#16812675	C₂²	1	1	yes	characteristic
28	#16812675	C₂²	1	1	yes	characteristic
29	#16812675	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₁₈	—	2	1	no	—
31	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
32	#16812657	C₂²	1	1	yes	characteristic
33	#16812657	C₂²	1	1	yes	characteristic
34	#16812657	C₂²	1	1	yes	characteristic
35	#16812656	—	2	1	no	—
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	#16812632	C₂²	1	1	yes	characteristic
42	#16812631	C₂²	1	1	yes	characteristic
43	#16812631	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	#16812623	C₂²	1	1	yes	characteristic
49	#16812623	C₂²	1	1	yes	characteristic
50	#16812611	C₂²	1	1	yes	characteristic
51	#16812611	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#97×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×D₄₃₆	D₄	1	1	yes	characteristic
61	C₂×C₄×D₁₀₉	—	2	1	no	—
62	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
64	#13182012	—	2	2	no	—
65	C₂×#11724784	C₂³	1	1	yes	characteristic
66	C₂×#11724784	C₂³	1	1	yes	characteristic
67	C₈×D₁₀₉	—	2	2	no	—
68	C₄⋊Dic₁₀₉	—	2	1	no	—
69	C₄⋊Dic₁₀₉	D₄	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	#13182008	—	2	1	no	—
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	#13182006	—	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	D₄	1	1	yes	characteristic
89	#13182006	D₄	1	1	yes	characteristic
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
95	Q₁₆×C₁₀₉	—	2	2	no	—
96	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
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	C₄²×C₁₀₉	—	2	1	no	—
103	C₄²×C₁₀₉	—	2	1	no	—
104	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₁₀₉	—	2	1	no	—
108	Q₈×C₁₀₉	—	2	1	no	—
109	Q₈×C₁₀₉	—	2	1	no	—
110	Q₈×C₁₀₉	—	2	1	no	—
111	C₂×C₄×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	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	C₂×Dic₁₀₉	—	2	1	no	—
119	C₂×Dic₁₀₉	—	2	1	no	—
120	C₂×Dic₁₀₉	—	2	1	no	—
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
124	D₄₃₆	—	4	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	Dic₂₁₈	—	4	2	no	—
131	C₈×C₁₀₉	—	2	1	no	—
132	C₈×C₁₀₉	#41	1	2	yes	—
133	#11724784	—	2	1	no	—
134	#11724784	—	2	1	no	—
135	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₁₀₉	—	2	1	no	—
137	C₂×D₁₀₉	—	2	1	no	—
138	C₂×D₁₀₉	—	2	1	no	—
139	C₂×D₁₀₉	—	2	1	no	—
140	C₂×D₁₀₉	—	2	1	no	—
141	C₂×D₁₀₉	—	2	1	no	—
142	C₄×C₁₀₉	—	2	1	no	—
143	C₄×C₁₀₉	—	2	1	no	—
144	C₄×C₁₀₉	—	4	1	no	—
145	C₄×C₁₀₉	—	4	1	no	—
146	C₄×C₁₀₉	—	4	1	no	—
147	C₄×C₁₀₉	—	4	1	no	—
148	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
149	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
150	Dic₁₀₉	—	2	1	no	—
151	Dic₁₀₉	—	2	1	no	—
152	Dic₁₀₉	—	4	1	no	—
153	Dic₁₀₉	—	4	1	no	—
154	Dic₁₀₉	—	4	1	no	—
155	Dic₁₀₉	—	4	1	no	—
156	C₂×C₁₀₉	#547	1	1	yes	characteristic
157	C₂×C₁₀₉	#575	1	1	yes	characteristic
158	C₂×C₁₀₉	#578	1	1	yes	characteristic
159	D₁₀₉	—	4	1	no	—
160	D₁₀₉	—	4	1	no	—
161	#3349	—	109	1	no	maximal, Sylow
162	C₁₀₉	#3349	1	1	yes	characteristic, complemented, Sylow
163	#556	—	109	1	no	—
164	#552	—	109	1	no	—
165	#501	—	109	1	no	—
166	#485	—	109	1	no	—
167	#485	—	109	1	no	—
168	#485	—	109	1	no	—
169	#484	—	109	1	no	—
170	#479	—	109	1	no	—
171	#477	—	109	1	no	—
172	#470	—	109	1	no	—
173	#467	—	109	1	no	—
174	#441	D₁₀₉	1	1	yes	characteristic, complemented
175	#435	—	109	1	no	—
176	#428	—	109	1	no	—
177	#427	—	109	1	no	—
178	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
179	C₂×M₁₆	—	109	1	no	—
180	C₂²×C₈	—	109	1	no	—
181	C₄⋊Q₈	—	109	1	no	—
182	#126	—	218	1	no	—
183	#126	—	218	1	no	—
184	#126	—	218	1	no	—
185	#125	—	109	1	no	—
186	#125	—	218	1	no	—
187	#124	—	218	1	no	—
188	#123	—	218	1	no	—
189	C₂²⋊Q₈	—	109	1	no	—
190	C₂²⋊Q₈	—	109	1	no	—
191	C₂²⋊Q₈	—	109	1	no	—
192	C₂²⋊Q₈	—	109	1	no	—
193	C₄×Q₈	—	218	1	no	—
194	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
195	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
196	C₄×D₄	—	109	1	no	—
197	#117	—	109	1	no	—
198	#117	—	109	1	no	—
199	#117	—	109	1	no	—
200	#117	—	218	1	no	—
201	C₂×C₄⋊C₄	—	109	1	no	—
202	#107	—	109	1	no	—
203	#107	—	109	1	no	—
204	#107	—	109	1	no	—
205	#107	—	109	1	no	—
206	#107	—	109	1	no	—
207	#106	—	109	1	no	—
208	#106	—	109	1	no	—
209	#106	—	109	1	no	—
210	#106	C₂×D₁₀₉	1	1	yes	characteristic
211	C₄⋊C₈	—	109	1	no	—
212	#103	—	109	1	no	—
213	#103	—	109	1	no	—
214	#103	—	109	1	no	—
215	#103	—	109	1	no	—
216	#103	C₂×D₁₀₉	1	1	yes	characteristic
217	#103	C₂×D₁₀₉	1	1	yes	characteristic
218	C₂²⋊C₈	—	109	1	no	—
219	C₂²⋊C₈	—	109	1	no	—
220	#97	C₂×D₁₀₉	1	1	yes	characteristic
221	C₂×Q₈	—	218	1	no	—
222	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
223	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
224	C₂×D₄	—	109	1	no	—
225	C₂²×C₄	—	109	1	no	—
226	C₂²×C₄	—	109	1	no	—
227	C₂²×C₄	—	218	1	no	—
228	Q₁₆	—	2	2	no	—
229	M₁₆	—	218	2	no	—
230	C₂×C₈	—	109	1	no	—
231	C₂×C₈	—	109	1	no	—
232	C₂×C₈	—	218	2	no	—
233	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	109	1	no	—
238	C₄⋊C₄	—	109	1	no	—
239	C₄⋊C₄	—	109	1	no	—
240	C₄⋊C₄	—	109	1	no	—
241	C₄⋊C₄	—	109	1	no	—
242	C₄⋊C₄	—	218	1	no	—
243	C₄⋊C₄	—	218	1	no	—
244	C₄⋊C₄	—	218	1	no	—
245	C₄⋊C₄	—	218	1	no	—
246	C₄⋊C₄	—	218	1	no	—
247	C₄⋊C₄	—	218	1	no	—
248	C₄⋊C₄	—	218	1	no	—
249	C₄⋊C₄	—	218	1	no	—
250	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₂²⋊C₄	—	109	1	no	—
253	C₂²⋊C₄	—	109	1	no	—
254	C₂²⋊C₄	—	218	1	no	—
255	C₂²⋊C₄	—	218	1	no	—
256	C₂²⋊C₄	—	218	1	no	—
257	C₂²⋊C₄	—	218	1	no	—
258	C₂²⋊C₄	—	218	1	no	—
259	C₂²⋊C₄	—	218	1	no	—
260	C₂²⋊C₄	—	218	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	218	1	no	—
264	C₄²	—	218	1	no	—
265	C₄²	—	218	1	no	—
266	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
267	C₂³	—	109	1	no	—
268	C₂³	—	109	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	436	2	no	—
274	D₄	—	436	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	109	1	no	—
280	C₂×C₄	—	109	1	no	—
281	C₂×C₄	—	436	2	no	—
282	C₂×C₄	—	218	1	no	—
283	C₂×C₄	—	218	1	no	—
284	C₂×C₄	—	218	1	no	—
285	C₂×C₄	—	218	1	no	—
286	C₂×C₄	—	218	1	no	—
287	C₂×C₄	—	218	1	no	—
288	C₂×C₄	—	218	1	no	—
289	C₂×C₄	—	218	1	no	—
290	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
291	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	218	1	no	—
295	C₈	—	218	1	no	—
296	C₈	#13182029	1	2	yes	—
297	C₂²	—	218	1	no	—
298	C₂²	—	218	1	no	—
299	C₂²	—	218	1	no	—
300	C₂²	—	218	1	no	—
301	C₂²	—	218	1	no	—
302	C₂²	—	218	1	no	—
303	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	436	1	no	—
311	C₄	—	436	1	no	—
312	C₄	—	436	1	no	—
313	C₄	—	436	1	no	—
314	C₄	—	218	1	no	—
315	C₄	—	218	1	no	—
316	C₄	C₂×#13182029	1	1	yes	characteristic
317	C₄	#16812746	1	1	yes	characteristic
318	C₂	—	436	1	no	—
319	C₂	—	436	1	no	—
320	C₂	#20295681	1	1	yes	characteristic
321	C₂	#20295825	1	1	yes	characteristic
322	C₂	#20295842	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#441×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄₃₆	D₄
hypercenter	#441	D₁₀₉
hypocenter	C₁₀₉	#3349
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
3. #20295435	C₂	semidirect
4. #20295366	C₂	semidirect
5. #20295365	C₂	semidirect
6. #20295365	C₂	semidirect
7. #20295360	C₂	semidirect
8. #20295333	C₂	semidirect
9. #20295321	C₂	semidirect
10. #20295281	C₂	semidirect
11. #20295264	C₂	semidirect
13. #20295065	C₂	semidirect
14. #20295061	C₂	semidirect
15. #441×C₁₀₉	C₂	semidirect
162. C₁₀₉	#3349	semidirect
174. #441	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295749	2
#20295717	2
#20295435	2
#20295366	2
#20295365	2
#20295360	2
#20295333	2
#20295321	2
#20295281	2
#20295264	2
#20295099	2
#20295065	2
#20295061	2
#441×C₁₀₉	2
#3349	109