Subgroups — #28373398

Order 13952.

372 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295656	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295601	C₂	1	1	yes	maximal, characteristic
3	#20295393	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295387	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295386	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295383	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295382	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295376	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295375	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295033	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295026	C₂	1	1	yes	maximal, characteristic
12	#20295021	C₂	1	1	yes	maximal, characteristic
13	#103×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#20294983	C₂	1	1	yes	maximal, characteristic, complemented
15	#491×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182030	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×#13182012	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
20	#16812699	C₂²	1	1	yes	characteristic
21	#16812698	—	2	1	no	—
22	#16812698	C₂²	1	1	yes	characteristic
23	#16812697	C₂²	1	1	yes	characteristic
24	#16812690	C₂²	1	1	yes	characteristic
25	#16812689	—	2	1	no	—
26	#16812688	C₂²	1	1	yes	characteristic
27	#16812687	C₂²	1	1	yes	characteristic
28	#16812684	—	2	1	no	—
29	#16812684	—	2	1	no	—
30	#16812679	—	2	1	no	—
31	#16812677	—	2	1	no	—
32	#16812677	—	2	1	no	—
33	#16812677	—	2	1	no	—
34	#16812657	C₂²	1	1	yes	characteristic
35	#16812656	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
37	#16812624	C₂²	1	1	yes	characteristic
38	#16812624	C₂²	1	1	yes	characteristic
39	#16812623	C₂²	1	1	yes	characteristic
40	#16812623	C₂²	1	1	yes	characteristic
41	#16812623	C₂²	1	1	yes	characteristic
42	#16812623	C₂²	1	1	yes	characteristic
43	#16812622	C₂²	1	1	yes	characteristic
44	#16812622	C₂²	1	1	yes	characteristic
45	#16812621	C₂²	1	1	yes	characteristic
46	#16812621	C₂²	1	1	yes	characteristic
47	#16812619	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic
49	#16812609	C₂²	1	1	yes	characteristic
50	C₄×#11724784	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	#125×C₁₀₉	C₂²	1	1	yes	characteristic
54	#103×C₁₀₉	C₂²	1	1	yes	characteristic
55	#103×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	#13182030	—	2	2	no	—
60	#13182030	—	2	2	no	—
61	C₂×Dic₂₁₈	—	2	1	no	—
62	C₂×Dic₂₁₈	—	2	1	no	—
63	C₂×Dic₂₁₈	—	2	1	no	—
64	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
65	Q₈×D₁₀₉	—	2	2	no	—
66	Q₈×D₁₀₉	—	2	2	no	—
67	C₂×D₄₃₆	—	2	1	no	—
68	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	—	2	1	no	—
71	C₂×C₄×D₁₀₉	—	2	1	no	—
72	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
73	#13182012	—	2	2	no	—
74	C₂×#11724784	C₂³	1	1	yes	characteristic
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₈×D₁₀₉	—	2	2	no	—
77	C₄⋊Dic₁₀₉	—	2	1	no	—
78	C₄⋊Dic₁₀₉	—	2	1	no	—
79	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
87	#13182008	D₄	1	1	yes	characteristic
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	#13182006	D₄	1	1	yes	characteristic
94	#13182006	D₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×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	—	2	1	no	—
102	C₄⋊C₄×C₁₀₉	—	2	1	no	—
103	C₄⋊C₄×C₁₀₉	—	2	1	no	—
104	C₄⋊C₄×C₁₀₉	—	2	1	no	—
105	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂²×D₁₀₉	—	2	1	no	—
109	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₀₉	—	2	1	no	—
111	Q₈×C₁₀₉	—	2	1	no	—
112	Q₈×C₁₀₉	—	2	1	no	—
113	Q₈×C₁₀₉	—	2	1	no	—
114	C₂×C₄×C₁₀₉	—	2	1	no	—
115	C₂×C₄×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	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	D₄₃₆	—	2	1	no	—
128	D₄₃₆	—	2	1	no	—
129	D₄₃₆	—	4	2	no	—
130	C₄×D₁₀₉	—	4	2	no	—
131	C₄×D₁₀₉	—	4	2	no	—
132	C₄×D₁₀₉	—	8	1	no	—
133	C₄×D₁₀₉	#41	1	2	yes	—
134	C₄×D₁₀₉	#41	1	2	yes	—
135	Dic₂₁₈	—	2	1	no	—
136	Dic₂₁₈	—	2	1	no	—
137	Dic₂₁₈	—	4	2	no	—
138	Dic₂₁₈	—	8	1	no	—
139	Dic₂₁₈	—	8	1	no	—
140	C₈×C₁₀₉	—	2	1	no	—
141	C₈×C₁₀₉	—	2	1	no	—
142	#11724784	—	2	1	no	—
143	#11724784	—	2	1	no	—
144	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₀₉	—	2	1	no	—
146	C₂×D₁₀₉	—	2	1	no	—
147	C₂×D₁₀₉	—	4	1	no	—
148	C₂×D₁₀₉	—	4	1	no	—
149	C₂×D₁₀₉	—	4	1	no	—
150	C₂×D₁₀₉	#124	1	2	yes	—
151	C₄×C₁₀₉	—	2	1	no	—
152	C₄×C₁₀₉	—	2	1	no	—
153	C₄×C₁₀₉	—	4	1	no	—
154	C₄×C₁₀₉	—	4	1	no	—
155	C₄×C₁₀₉	—	4	1	no	—
156	C₄×C₁₀₉	—	4	1	no	—
157	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
159	Dic₁₀₉	—	2	1	no	—
160	Dic₁₀₉	—	2	1	no	—
161	Dic₁₀₉	—	4	1	no	—
162	Dic₁₀₉	—	4	1	no	—
163	Dic₁₀₉	—	4	1	no	—
164	Dic₁₀₉	#124	1	2	yes	—
165	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
166	C₂×C₁₀₉	#577	1	1	yes	characteristic
167	C₂×C₁₀₉	#578	1	1	yes	characteristic
168	D₁₀₉	—	2	2	no	—
169	D₁₀₉	—	8	1	no	—
170	#3137	—	109	1	no	maximal, Sylow
171	C₁₀₉	#3137	1	1	yes	characteristic, complemented, Sylow
172	#544	—	109	1	no	—
173	#536	—	109	1	no	—
174	#491	—	109	1	no	—
175	#491	D₁₀₉	1	1	yes	characteristic, complemented
176	#489	—	109	1	no	—
177	#489	—	109	1	no	—
178	#488	—	109	1	no	—
179	#488	—	109	1	no	—
180	#487	—	109	1	no	—
181	#487	—	109	1	no	—
182	#419	—	109	1	no	—
183	#417	—	109	1	no	—
184	#416	—	109	1	no	—
185	C₂×#103	—	109	1	no	—
186	#405	—	109	1	no	—
187	C₂×#41	—	109	1	no	—
188	C₂²×Q₈	—	109	1	no	—
189	C₂×M₁₆	—	109	1	no	—
190	C₂²×C₈	—	109	1	no	—
191	C₄⋊Q₈	—	109	1	no	—
192	C₄⋊Q₈	—	109	1	no	—
193	C₄⋊Q₈	—	109	1	no	—
194	C₄⋊Q₈	—	218	1	no	—
195	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
196	#125	—	109	1	no	—
197	#125	—	218	1	no	—
198	#125	C₂×D₁₀₉	1	1	yes	characteristic
199	#124	—	109	1	no	—
200	#124	—	109	1	no	—
201	#123	—	218	1	no	—
202	#123	—	218	1	no	—
203	C₂²⋊Q₈	—	218	1	no	—
204	C₂²⋊Q₈	—	218	1	no	—
205	C₂²⋊Q₈	—	218	1	no	—
206	C₂²⋊Q₈	—	218	1	no	—
207	#117	—	109	1	no	—
208	#117	—	109	1	no	—
209	C₂×C₄⋊C₄	—	109	1	no	—
210	#103	—	109	1	no	—
211	#103	—	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	—	109	1	no	—
217	#103	—	109	1	no	—
218	#103	C₂×D₁₀₉	1	1	yes	characteristic
219	#103	C₂×D₁₀₉	1	1	yes	characteristic
220	#103	C₂×D₁₀₉	1	1	yes	characteristic
221	#103	C₂×D₁₀₉	1	1	yes	characteristic
222	#102	—	109	1	no	—
223	#102	—	109	1	no	—
224	#102	—	109	1	no	—
225	#102	—	109	1	no	—
226	#97	—	109	1	no	—
227	#97	C₂×D₁₀₉	1	1	yes	characteristic
228	C₄×C₈	—	109	1	no	—
229	C₄×C₈	—	109	1	no	—
230	#41	—	218	2	no	—
231	#41	—	218	2	no	—
232	C₂×Q₈	—	109	1	no	—
233	C₂×Q₈	—	218	1	no	—
234	C₂×Q₈	—	218	1	no	—
235	C₂×Q₈	—	218	2	no	—
236	C₂×Q₈	—	218	2	no	—
237	C₂×Q₈	—	218	1	no	—
238	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
239	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
240	C₂×D₄	—	109	1	no	—
241	C₂×D₄	—	218	1	no	—
242	C₂²×C₄	—	109	1	no	—
243	C₂²×C₄	—	218	1	no	—
244	C₂²×C₄	—	218	1	no	—
245	C₂²×C₄	—	218	1	no	—
246	M₁₆	—	218	2	no	—
247	C₂×C₈	—	109	1	no	—
248	C₂×C₈	—	109	1	no	—
249	C₂×C₈	—	218	2	no	—
250	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	109	1	no	—
257	C₄⋊C₄	—	109	1	no	—
258	C₄⋊C₄	—	109	1	no	—
259	C₄⋊C₄	—	109	1	no	—
260	C₄⋊C₄	—	218	1	no	—
261	C₄⋊C₄	—	218	1	no	—
262	C₄⋊C₄	—	218	1	no	—
263	C₄⋊C₄	—	218	1	no	—
264	C₄⋊C₄	—	218	1	no	—
265	C₄⋊C₄	—	218	1	no	—
266	C₄⋊C₄	—	218	1	no	—
267	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
269	C₂²⋊C₄	—	109	1	no	—
270	C₂²⋊C₄	—	109	1	no	—
271	C₂²⋊C₄	—	218	1	no	—
272	C₂²⋊C₄	—	218	1	no	—
273	C₂²⋊C₄	—	218	1	no	—
274	C₂²⋊C₄	—	218	1	no	—
275	C₂²⋊C₄	—	218	1	no	—
276	C₄²	—	109	1	no	—
277	C₄²	—	218	1	no	—
278	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
279	C₂³	—	109	1	no	—
280	C₂³	—	218	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	872	1	no	—
286	Q₈	—	872	1	no	—
287	Q₈	—	436	2	no	—
288	Q₈	—	218	1	no	—
289	Q₈	—	218	1	no	—
290	D₄	—	436	2	no	—
291	D₄	—	218	1	no	—
292	D₄	—	218	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	872	1	no	—
298	C₂×C₄	—	109	1	no	—
299	C₂×C₄	—	109	2	no	—
300	C₂×C₄	—	109	2	no	—
301	C₂×C₄	—	109	1	no	—
302	C₂×C₄	—	109	1	no	—
303	C₂×C₄	—	436	2	no	—
304	C₂×C₄	—	436	2	no	—
305	C₂×C₄	—	218	1	no	—
306	C₂×C₄	—	218	1	no	—
307	C₂×C₄	—	218	1	no	—
308	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
309	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	218	1	no	—
314	C₈	—	218	1	no	—
315	C₂²	—	109	2	no	—
316	C₂²	—	436	1	no	—
317	C₂²	—	436	1	no	—
318	C₂²	—	436	1	no	—
319	C₂²	—	218	1	no	—
320	C₂²	—	218	1	no	—
321	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	109	2	no	—
329	C₄	—	436	1	no	—
330	C₄	—	436	1	no	—
331	C₄	—	436	1	no	—
332	C₄	—	218	1	no	—
333	C₄	—	218	1	no	—
334	C₄	#41×D₁₀₉	1	1	yes	characteristic
335	C₄	#41×D₁₀₉	1	1	yes	characteristic
336	C₂	—	872	1	no	—
337	C₂	—	218	2	no	—
338	C₂	#124×D₁₀₉	1	1	yes	characteristic
339	C₂	#20295835	1	1	yes	characteristic
340	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	#491×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#13182030	C₂²
hypercenter	#491	D₁₀₉
hypocenter	C₁₀₉	#3137
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

Complements

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

N	H ≅ G/N	type
1. #20295656	C₂	semidirect
3. #20295393	C₂	semidirect
4. #20295387	C₂	semidirect
5. #20295386	C₂	semidirect
6. #20295383	C₂	semidirect
7. #20295382	C₂	semidirect
8. #20295376	C₂	semidirect
9. #20295375	C₂	semidirect
10. #20295033	C₂	semidirect
13. #103×D₁₀₉	C₂	semidirect
14. #20294983	C₂	semidirect
15. #491×C₁₀₉	C₂	semidirect
171. C₁₀₉	#3137	semidirect
175. #491	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295656	2
#20295601	2
#20295393	2
#20295387	2
#20295386	2
#20295383	2
#20295382	2
#20295376	2
#20295375	2
#20295033	2
#20295026	2
#20295021	2
#103×D₁₀₉	2
#20294983	2
#491×C₁₀₉	2
#3137	109