Subgroups — #28376064

Order 13952.

350 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	#20295754	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295748	C₂	1	1	yes	maximal, characteristic
3	#20295420	C₂	1	1	yes	maximal, characteristic
4	#20295419	C₂	1	1	yes	maximal, characteristic
5	#20295404	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295251	C₂	1	1	yes	maximal, characteristic
7	#20295248	C₂	1	1	yes	maximal, characteristic
8	#20295236	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295235	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295235	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295230	C₂	1	1	yes	maximal, characteristic
12	#20295154	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295132	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295113	C₂	1	1	yes	maximal, characteristic
15	#441×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
18	C₂×#13182019	C₂²	1	1	yes	characteristic
19	C₂×#13182019	C₂²	1	1	yes	characteristic
20	C₂×#13182019	C₂²	1	1	yes	characteristic
21	C₂×#13182018	C₂²	1	1	yes	characteristic
22	C₂×#13182018	C₂²	1	1	yes	characteristic
23	C₂×#13182016	C₂²	1	1	yes	characteristic
24	#16812699	C₂²	1	1	yes	characteristic
25	#16812697	C₂²	1	1	yes	characteristic
26	#16812697	C₂²	1	1	yes	characteristic
27	#16812697	C₂²	1	1	yes	characteristic
28	#16812694	—	2	1	no	—
29	#16812693	—	2	1	no	—
30	#16812691	—	2	1	no	—
31	#16812689	—	2	1	no	—
32	#16812689	—	2	1	no	—
33	#16812688	C₂²	1	1	yes	characteristic
34	#16812687	C₂²	1	1	yes	characteristic
35	#16812677	—	2	1	no	—
36	C₄×Dic₂₁₈	—	2	1	no	—
37	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
38	#16812664	C₂²	1	1	yes	characteristic
39	#16812664	C₂²	1	1	yes	characteristic
40	#16812664	C₂²	1	1	yes	characteristic
41	#16812661	C₂²	1	1	yes	characteristic
42	#16812656	—	2	1	no	—
43	C₄⋊#11724784	C₂²	1	1	yes	characteristic
44	#16812628	C₂²	1	1	yes	characteristic
45	#16812628	C₂²	1	1	yes	characteristic
46	#16812622	C₂²	1	1	yes	characteristic
47	#16812622	C₂²	1	1	yes	characteristic
48	#16812622	C₂²	1	1	yes	characteristic
49	#16812619	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	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	D₄	1	1	yes	characteristic
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
63	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₀₉	—	2	1	no	—
65	Dic₄₃₆	—	2	2	no	—
66	Dic₄₃₆	—	4	1	no	—
67	#13182019	—	4	1	no	—
68	#13182019	—	4	1	no	—
69	#13182019	—	4	1	no	—
70	#13182018	—	2	2	no	—
71	#13182018	—	4	1	no	—
72	#13182016	—	4	1	no	—
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₂×#11724784	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₀₉	—	2	1	no	—
76	C₄⋊Dic₁₀₉	—	2	1	no	—
77	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
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	#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	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	#13182006	—	2	1	no	—
94	C₄×Dic₁₀₉	—	2	1	no	—
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	—	2	1	no	—
97	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
98	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	Q₁₆×C₁₀₉	—	2	2	no	—
102	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₀₉	—	2	1	no	—
105	C₄⋊C₄×C₁₀₉	—	2	1	no	—
106	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₄²×C₁₀₉	—	2	1	no	—
109	C₄²×C₁₀₉	—	2	1	no	—
110	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
111	C₂²×D₁₀₉	—	2	1	no	—
112	Q₈×C₁₀₉	—	2	1	no	—
113	Q₈×C₁₀₉	—	2	1	no	—
114	Q₈×C₁₀₉	—	2	1	no	—
115	Q₈×C₁₀₉	—	2	1	no	—
116	C₂×C₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	—	2	1	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
130	D₄₃₆	—	2	1	no	—
131	D₄₃₆	—	2	1	no	—
132	C₄×D₁₀₉	—	4	2	no	—
133	Dic₂₁₈	—	2	2	no	—
134	Dic₂₁₈	—	2	1	no	—
135	Dic₂₁₈	—	2	1	no	—
136	Dic₂₁₈	—	2	1	no	—
137	Dic₂₁₈	—	2	1	no	—
138	Dic₂₁₈	—	2	1	no	—
139	Dic₂₁₈	—	2	1	no	—
140	C₈×C₁₀₉	—	2	1	no	—
141	C₈×C₁₀₉	#41	1	2	yes	—
142	#11724784	—	2	1	no	—
143	#11724784	—	2	1	no	—
144	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
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	H(5,𝔽₂)	1	1	yes	characteristic
156	Dic₁₀₉	—	2	1	no	—
157	Dic₁₀₉	—	2	1	no	—
158	Dic₁₀₉	—	2	1	no	—
159	Dic₁₀₉	—	4	1	no	—
160	Dic₁₀₉	—	4	1	no	—
161	Dic₁₀₉	—	4	1	no	—
162	Dic₁₀₉	#137	1	2	yes	—
163	C₂×C₁₀₉	#550	1	1	yes	characteristic
164	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
165	C₂×C₁₀₉	#578	1	1	yes	characteristic
166	D₁₀₉	—	8	1	no	—
167	#3398	—	109	1	no	maximal, Sylow
168	C₁₀₉	#3398	1	1	yes	characteristic, complemented, Sylow
169	#557	—	109	1	no	—
170	#556	—	109	1	no	—
171	#497	—	109	1	no	—
172	#497	—	109	1	no	—
173	#494	—	109	1	no	—
174	#464	—	109	1	no	—
175	#464	—	109	1	no	—
176	#462	—	109	1	no	—
177	#462	—	109	1	no	—
178	#462	—	109	1	no	—
179	#461	—	109	1	no	—
180	#446	—	109	1	no	—
181	#441	—	109	1	no	—
182	#441	D₁₀₉	1	1	yes	characteristic, complemented
183	C₄×SD₁₆	—	109	1	no	—
184	C₂×Q₁₆	—	109	1	no	—
185	C₂×Q₁₆	—	109	1	no	—
186	C₂×Q₁₆	—	109	1	no	—
187	C₂×Q₁₆	—	109	1	no	—
188	C₂×Q₁₆	—	109	1	no	—
189	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
190	C₂×SD₁₆	—	109	1	no	—
191	C₂×SD₁₆	—	109	1	no	—
192	C₂×SD₁₆	—	109	1	no	—
193	C₄⋊Q₈	—	109	1	no	—
194	C₄⋊Q₈	—	109	1	no	—
195	C₄⋊Q₈	—	109	1	no	—
196	C₄⋊Q₈	—	109	1	no	—
197	#126	—	218	1	no	—
198	#126	—	218	1	no	—
199	#125	—	218	1	no	—
200	#125	—	218	1	no	—
201	#125	—	218	1	no	—
202	#124	—	109	1	no	—
203	#124	—	109	1	no	—
204	C₂²⋊Q₈	—	218	1	no	—
205	C₄×Q₈	—	109	1	no	—
206	C₄×Q₈	—	109	1	no	—
207	C₄×Q₈	—	109	1	no	—
208	C₄×Q₈	—	109	1	no	—
209	C₄×Q₈	—	218	1	no	—
210	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
211	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
212	C₄×D₄	—	109	1	no	—
213	#117	—	218	1	no	—
214	#106	C₂×D₁₀₉	1	1	yes	characteristic
215	C₄⋊C₈	—	109	1	no	—
216	C₄⋊C₈	—	109	1	no	—
217	C₄⋊C₈	—	109	1	no	—
218	#103	—	109	1	no	—
219	#103	—	109	1	no	—
220	#103	—	109	1	no	—
221	#103	C₂×D₁₀₉	1	1	yes	characteristic
222	#103	C₂×D₁₀₉	1	1	yes	characteristic
223	#102	—	109	1	no	—
224	#97	—	109	1	no	—
225	#97	C₂×D₁₀₉	1	1	yes	characteristic
226	C₄×C₈	—	109	1	no	—
227	C₂×Q₈	—	109	1	no	—
228	C₂×Q₈	—	109	1	no	—
229	C₂×Q₈	—	109	1	no	—
230	C₂×Q₈	—	109	1	no	—
231	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
233	C₂×D₄	—	109	1	no	—
234	C₂²×C₄	—	218	1	no	—
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	436	1	no	—
237	Q₁₆	—	436	1	no	—
238	Q₁₆	—	436	1	no	—
239	Q₁₆	—	436	1	no	—
240	Q₁₆	—	218	2	no	—
241	SD₁₆	—	436	1	no	—
242	SD₁₆	—	436	1	no	—
243	SD₁₆	—	218	2	no	—
244	C₂×C₈	—	109	1	no	—
245	C₂×C₈	—	109	1	no	—
246	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	109	1	no	—
251	C₄⋊C₄	—	109	1	no	—
252	C₄⋊C₄	—	109	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	—	218	1	no	—
259	C₄⋊C₄	—	218	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	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
266	C₂²⋊C₄	—	218	1	no	—
267	C₂²⋊C₄	—	218	1	no	—
268	C₂²⋊C₄	—	218	1	no	—
269	C₂²⋊C₄	—	218	1	no	—
270	C₂²⋊C₄	—	218	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	109	1	no	—
274	C₄²	—	109	1	no	—
275	C₄²	—	218	1	no	—
276	C₄²	—	218	1	no	—
277	C₄²	—	218	1	no	—
278	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
279	C₂³	—	218	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	218	2	no	—
285	Q₈	—	218	1	no	—
286	Q₈	—	218	1	no	—
287	Q₈	—	218	1	no	—
288	Q₈	—	218	1	no	—
289	Q₈	—	218	1	no	—
290	Q₈	—	218	1	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	—	109	1	no	—
298	C₂×C₄	—	109	1	no	—
299	C₂×C₄	—	109	1	no	—
300	C₂×C₄	—	109	1	no	—
301	C₂×C₄	—	436	2	no	—
302	C₂×C₄	—	218	1	no	—
303	C₂×C₄	—	218	1	no	—
304	C₂×C₄	—	218	1	no	—
305	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
306	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	218	1	no	—
310	C₈	—	218	1	no	—
311	C₈	#13182029	1	2	yes	—
312	C₂²	—	436	1	no	—
313	C₂²	—	436	1	no	—
314	C₂²	—	436	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	—	109	2	no	—
323	C₄	—	436	1	no	—
324	C₄	—	436	1	no	—
325	C₄	—	436	1	no	—
326	C₄	—	218	1	no	—
327	C₄	—	218	1	no	—
328	C₄	—	218	1	no	—
329	C₄	C₂×#13182029	1	1	yes	characteristic
330	C₄	#16812745	1	1	yes	characteristic
331	C₂	—	872	1	no	—
332	C₂	#20295697	1	1	yes	characteristic
333	C₂	#137×D₁₀₉	1	1	yes	characteristic
334	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	C₂³
hypercenter	#441	D₁₀₉
hypocenter	C₁₀₉	#3398
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
1. #20295754	C₂	semidirect
5. #20295404	C₂	semidirect
8. #20295236	C₂	semidirect
9. #20295235	C₂	semidirect
10. #20295235	C₂	semidirect
12. #20295154	C₂	semidirect
13. #20295132	C₂	semidirect
15. #441×C₁₀₉	C₂	semidirect
168. C₁₀₉	#3398	semidirect
182. #441	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295754	2
#20295748	2
#20295420	2
#20295419	2
#20295404	2
#20295251	2
#20295248	2
#20295236	2
#20295235	2
#20295230	2
#20295154	2
#20295132	2
#20295113	2
#441×C₁₀₉	2
#3398	109