Subgroups — #28376088

Order 13952.

362 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295758	C₂	1	1	yes	maximal, characteristic
2	#20295754	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295421	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295419	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295397	C₂	1	1	yes	maximal, characteristic
6	#20295251	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295234	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295232	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295229	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295225	C₂	1	1	yes	maximal, characteristic
11	#20295155	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295124	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295123	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295118	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182019	C₂²	1	1	yes	characteristic
17	C₂×#13182019	C₂²	1	1	yes	characteristic
18	C₂×#13182018	C₂²	1	1	yes	characteristic
19	C₂×#13182018	C₂²	1	1	yes	characteristic
20	C₂×#13182017	C₂²	1	1	yes	characteristic
21	C₂×#13182017	C₂²	1	1	yes	characteristic
22	C₂×#13182016	C₂²	1	1	yes	characteristic
23	#16812699	C₂²	1	1	yes	characteristic
24	#16812697	C₂²	1	1	yes	characteristic
25	#16812697	C₂²	1	1	yes	characteristic
26	#16812695	C₂²	1	1	yes	characteristic
27	#16812691	—	2	1	no	—
28	#16812689	—	2	1	no	—
29	#16812689	—	2	1	no	—
30	#16812688	C₂²	1	1	yes	characteristic
31	#16812685	—	2	1	no	—
32	#16812683	—	2	1	no	—
33	#16812681	—	2	1	no	—
34	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×Dic₂₁₈	—	2	1	no	—
36	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
37	#16812664	C₂²	1	1	yes	characteristic
38	#16812664	C₂²	1	1	yes	characteristic
39	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
40	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
41	#16812659	—	2	1	no	—
42	#16812632	C₂²	1	1	yes	characteristic
43	#16812628	C₂²	1	1	yes	characteristic
44	#16812628	C₂²	1	1	yes	characteristic
45	#16812624	C₂²	1	1	yes	characteristic
46	#16812624	C₂²	1	1	yes	characteristic
47	#16812623	C₂²	1	1	yes	characteristic
48	#16812619	C₂²	1	1	yes	characteristic
49	#16812610	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	#124×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
65	C₂²×Dic₁₀₉	—	2	1	no	—
66	C₂×C₄×D₁₀₉	—	2	1	no	—
67	#13182019	—	2	2	no	—
68	#13182019	—	4	1	no	—
69	#13182018	—	4	1	no	—
70	#13182018	—	4	1	no	—
71	#13182017	—	2	2	no	—
72	#13182017	—	4	1	no	—
73	#13182016	—	2	2	no	—
74	C₂×#11724784	C₂³	1	1	yes	characteristic
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₀₉	—	2	1	no	—
77	C₄⋊Dic₁₀₉	—	2	1	no	—
78	C₄⋊Dic₁₀₉	—	2	1	no	—
79	C₄⋊Dic₁₀₉	D₄	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	—	2	1	no	—
87	#13182008	D₄	1	1	yes	characteristic
88	#13182008	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₀₉	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	—	2	1	no	—
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	Q₁₆×C₁₀₉	—	4	1	no	—
102	SD₁₆×C₁₀₉	—	4	1	no	—
103	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	—	2	1	no	—
106	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
108	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
109	C₄²×C₁₀₉	—	2	1	no	—
110	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
111	C₂³×C₁₀₉	—	2	1	no	—
112	C₂²×D₁₀₉	—	2	1	no	—
113	Q₈×C₁₀₉	—	2	1	no	—
114	Q₈×C₁₀₉	—	2	1	no	—
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	—	2	1	no	—
117	D₄×C₁₀₉	—	2	1	no	—
118	D₄×C₁₀₉	—	2	1	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	2	1	no	—
122	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₁₀₉	—	4	2	no	—
126	C₂²⋊D₁₀₉	—	8	1	no	—
127	C₂×Dic₁₀₉	—	2	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	8	1	no	—
131	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
135	D₄₃₆	—	2	1	no	—
136	D₄₃₆	—	2	1	no	—
137	C₄×D₁₀₉	—	8	1	no	—
138	Dic₂₁₈	—	2	2	no	—
139	Dic₂₁₈	—	2	1	no	—
140	Dic₂₁₈	—	2	1	no	—
141	Dic₂₁₈	—	2	1	no	—
142	Dic₂₁₈	—	2	1	no	—
143	C₈×C₁₀₉	—	2	1	no	—
144	C₈×C₁₀₉	—	2	1	no	—
145	#11724784	—	2	1	no	—
146	#11724784	#41	1	2	yes	—
147	C₂²×C₁₀₉	—	4	1	no	—
148	C₂²×C₁₀₉	—	4	1	no	—
149	C₂²×C₁₀₉	—	4	1	no	—
150	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₀₉	—	4	1	no	—
152	C₂×D₁₀₉	—	4	1	no	—
153	C₂×D₁₀₉	—	4	1	no	—
154	C₄×C₁₀₉	—	2	1	no	—
155	C₄×C₁₀₉	—	2	1	no	—
156	C₄×C₁₀₉	—	4	1	no	—
157	C₄×C₁₀₉	—	4	1	no	—
158	C₄×C₁₀₉	—	4	1	no	—
159	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
160	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
161	Dic₁₀₉	—	2	1	no	—
162	Dic₁₀₉	—	2	1	no	—
163	Dic₁₀₉	—	2	1	no	—
164	Dic₁₀₉	—	4	1	no	—
165	Dic₁₀₉	—	4	1	no	—
166	Dic₁₀₉	—	4	1	no	—
167	Dic₁₀₉	#137	1	2	yes	—
168	C₂×C₁₀₉	—	8	1	no	—
169	C₂×C₁₀₉	#550	1	1	yes	characteristic
170	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
171	C₂×C₁₀₉	#577	1	1	yes	characteristic
172	D₁₀₉	—	8	1	no	—
173	#3400	—	109	1	no	maximal, Sylow
174	C₁₀₉	#3400	1	1	yes	characteristic, complemented, Sylow
175	#559	—	109	1	no	—
176	#557	—	109	1	no	—
177	#497	—	109	1	no	—
178	#497	—	109	1	no	—
179	#492	—	109	1	no	—
180	#464	—	109	1	no	—
181	#464	D₁₀₉	1	1	yes	characteristic, complemented
182	#462	—	109	1	no	—
183	#461	—	109	1	no	—
184	#461	—	109	1	no	—
185	#460	—	109	1	no	—
186	#446	—	109	1	no	—
187	#440	—	109	1	no	—
188	#440	—	109	1	no	—
189	C₄×Q₁₆	—	109	1	no	—
190	C₂×Q₁₆	—	109	1	no	—
191	C₂×Q₁₆	—	109	1	no	—
192	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
193	C₂×SD₁₆	—	109	1	no	—
194	C₂×SD₁₆	—	109	1	no	—
195	C₂×SD₁₆	—	109	1	no	—
196	C₂×SD₁₆	—	109	1	no	—
197	C₂×SD₁₆	—	109	1	no	—
198	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
199	C₄⋊Q₈	—	109	1	no	—
200	C₄⋊Q₈	—	109	1	no	—
201	C₄⋊Q₈	—	109	1	no	—
202	#127	—	109	1	no	—
203	#125	—	218	1	no	—
204	#125	—	218	1	no	—
205	#125	—	218	1	no	—
206	#124	—	109	1	no	—
207	#124	—	218	1	no	—
208	#124	C₂×D₁₀₉	1	1	yes	characteristic
209	#123	—	218	1	no	—
210	#123	—	218	1	no	—
211	C₄×Q₈	—	109	1	no	—
212	C₄×Q₈	—	109	1	no	—
213	C₄×Q₈	—	109	1	no	—
214	C₄×Q₈	—	109	1	no	—
215	C₄×Q₈	—	218	1	no	—
216	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
217	C₄×D₄	—	109	1	no	—
218	C₄×D₄	—	109	1	no	—
219	C₄×D₄	—	218	1	no	—
220	#107	—	109	1	no	—
221	C₄⋊C₈	—	109	1	no	—
222	C₄⋊C₈	—	109	1	no	—
223	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
224	#103	—	109	1	no	—
225	#103	—	109	1	no	—
226	#103	—	109	1	no	—
227	#103	C₂×D₁₀₉	1	1	yes	characteristic
228	#102	—	109	1	no	—
229	#102	C₂×D₁₀₉	1	1	yes	characteristic
230	#97	—	109	1	no	—
231	#97	—	109	1	no	—
232	C₄×C₈	—	109	1	no	—
233	C₂×Q₈	—	109	1	no	—
234	C₂×Q₈	—	109	1	no	—
235	C₂×Q₈	—	109	1	no	—
236	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
237	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
238	C₂×D₄	—	109	1	no	—
239	C₂×D₄	—	218	1	no	—
240	C₂×D₄	—	218	1	no	—
241	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
242	C₂²×C₄	—	218	1	no	—
243	C₂²×C₄	—	218	1	no	—
244	Q₁₆	—	4	1	no	—
245	Q₁₆	—	436	1	no	—
246	Q₁₆	—	218	2	no	—
247	SD₁₆	—	4	1	no	—
248	SD₁₆	—	436	1	no	—
249	SD₁₆	—	436	1	no	—
250	SD₁₆	—	436	1	no	—
251	SD₁₆	—	218	2	no	—
252	SD₁₆	—	218	2	no	—
253	C₂×C₈	—	109	1	no	—
254	C₂×C₈	—	109	1	no	—
255	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	109	1	no	—
259	C₄⋊C₄	—	109	1	no	—
260	C₄⋊C₄	—	109	1	no	—
261	C₄⋊C₄	—	109	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	—	218	1	no	—
268	C₄⋊C₄	—	218	1	no	—
269	C₄⋊C₄	—	218	1	no	—
270	C₄⋊C₄	—	218	1	no	—
271	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	218	1	no	—
275	C₂²⋊C₄	—	218	1	no	—
276	C₂²⋊C₄	—	218	1	no	—
277	C₂²⋊C₄	—	218	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	109	1	no	—
280	C₄²	—	109	1	no	—
281	C₄²	—	218	1	no	—
282	C₄²	—	218	1	no	—
283	C₄²	—	218	1	no	—
284	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	—	218	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	218	2	no	—
292	Q₈	—	218	1	no	—
293	Q₈	—	218	1	no	—
294	Q₈	—	218	1	no	—
295	Q₈	—	218	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	872	1	no	—
299	D₄	—	436	2	no	—
300	D₄	—	218	1	no	—
301	D₄	—	218	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	872	1	no	—
306	C₂×C₄	—	872	1	no	—
307	C₂×C₄	—	109	1	no	—
308	C₂×C₄	—	109	1	no	—
309	C₂×C₄	—	109	1	no	—
310	C₂×C₄	—	109	1	no	—
311	C₂×C₄	—	218	1	no	—
312	C₂×C₄	—	218	1	no	—
313	C₂×C₄	—	218	1	no	—
314	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
315	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	109	2	no	—
320	C₈	—	218	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	436	1	no	—
325	C₂²	—	436	1	no	—
326	C₂²	—	436	1	no	—
327	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	109	2	no	—
334	C₄	—	436	1	no	—
335	C₄	—	436	1	no	—
336	C₄	—	436	1	no	—
337	C₄	—	218	1	no	—
338	C₄	—	218	1	no	—
339	C₄	—	218	1	no	—
340	C₄	#41×D₁₀₉	1	1	yes	characteristic
341	C₄	#16812744	1	1	yes	characteristic
342	C₂	—	8	1	no	—
343	C₂	—	872	1	no	—
344	C₂	#20295694	1	1	yes	characteristic
345	C₂	#137×D₁₀₉	1	1	yes	characteristic
346	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	#464×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#16812695	C₂²
hypercenter	#464	D₁₀₉
hypocenter	C₁₀₉	#3400
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
2. #20295754	C₂	semidirect
3. #20295421	C₂	semidirect
4. #20295419	C₂	semidirect
6. #20295251	C₂	semidirect
7. #20295234	C₂	semidirect
8. #20295232	C₂	semidirect
9. #20295229	C₂	semidirect
11. #20295155	C₂	semidirect
12. #20295124	C₂	semidirect
13. #20295123	C₂	semidirect
14. #20295118	C₂	semidirect
15. #464×C₁₀₉	C₂	semidirect
174. C₁₀₉	#3400	semidirect
181. #464	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295758	2
#20295754	2
#20295421	2
#20295419	2
#20295397	2
#20295251	2
#20295234	2
#20295232	2
#20295229	2
#20295225	2
#20295155	2
#20295124	2
#20295123	2
#20295118	2
#464×C₁₀₉	2
#3400	109