Subgroups — #28373263

Order 13952.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295647	C₂	1	1	yes	maximal, characteristic
2	#20295503	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295368	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295351	C₂	1	1	yes	maximal, characteristic
5	#20295322	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295312	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295249	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295249	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295184	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295167	C₂	1	1	yes	maximal, characteristic
11	#20295134	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295128	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13182018	C₂	1	1	yes	maximal, characteristic, complemented
15	#433×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
19	C₂×#13182018	C₂²	1	1	yes	characteristic
20	C₂×#13182018	C₂²	1	1	yes	characteristic
21	#16812699	C₂²	1	1	yes	characteristic
22	#16812694	—	2	1	no	—
23	#16812687	C₂²	1	1	yes	characteristic
24	#16812685	—	2	1	no	—
25	#16812681	—	2	1	no	—
26	#16812678	C₂²	1	1	yes	characteristic
27	#16812677	—	2	1	no	—
28	#16812673	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
34	#16812664	—	2	1	no	—
35	#16812664	—	2	1	no	—
36	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
37	#16812658	C₂²	1	1	yes	characteristic
38	#16812655	—	2	1	no	—
39	#16812655	—	2	1	no	—
40	#16812633	C₂²	1	1	yes	characteristic
41	#16812633	C₂²	1	1	yes	characteristic
42	#16812631	C₂²	1	1	yes	characteristic
43	#16812631	C₂²	1	1	yes	characteristic
44	#16812624	C₂²	1	1	yes	characteristic
45	#16812624	C₂²	1	1	yes	characteristic
46	#16812624	C₂²	1	1	yes	characteristic
47	#16812624	C₂²	1	1	yes	characteristic
48	#16812624	C₂²	1	1	yes	characteristic
49	#16812624	C₂²	1	1	yes	characteristic
50	#16812621	C₂²	1	1	yes	characteristic
51	#16812621	C₂²	1	1	yes	characteristic
52	#117×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#97×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	—	2	2	no	—
61	C₂×Dic₂₁₈	—	2	2	no	—
62	C₂×Dic₂₁₈	—	2	1	no	—
63	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
64	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
65	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
66	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
68	C₂²×Dic₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	Dic₄₃₆	—	2	2	no	—
71	Dic₄₃₆	—	2	2	no	—
72	#13182018	—	2	2	no	—
73	#13182018	—	2	2	no	—
74	C₄⋊Dic₁₀₉	—	2	1	no	—
75	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₀₉	C₂³	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	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	C₂²⋊Dic₁₀₉	—	2	1	no	—
86	C₂²⋊Dic₁₀₉	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	—	2	1	no	—
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	—	2	1	no	—
97	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×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	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
106	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₀₉	—	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	C₂×D₄	1	1	yes	characteristic
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	C₂²⋊D₁₀₉	—	8	1	no	—
119	C₂×Dic₁₀₉	—	2	4	no	—
120	C₂×Dic₁₀₉	—	2	1	no	—
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	—	2	1	no	—
123	C₂×Dic₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	D₄₃₆	—	2	1	no	—
128	D₄₃₆	—	2	1	no	—
129	C₄×D₁₀₉	—	8	1	no	—
130	Dic₂₁₈	—	2	4	no	—
131	Dic₂₁₈	—	2	1	no	—
132	Dic₂₁₈	—	2	1	no	—
133	Dic₂₁₈	—	2	1	no	—
134	Dic₂₁₈	—	2	1	no	—
135	Dic₂₁₈	—	8	1	no	—
136	Dic₂₁₈	#41	1	2	yes	—
137	Dic₂₁₈	#41	1	2	yes	—
138	C₈×C₁₀₉	—	2	1	no	—
139	C₈×C₁₀₉	—	2	1	no	—
140	C₈×C₁₀₉	—	2	1	no	—
141	C₈×C₁₀₉	—	2	1	no	—
142	C₂²×C₁₀₉	—	2	1	no	—
143	C₂²×C₁₀₉	—	2	1	no	—
144	C₂²×C₁₀₉	—	2	1	no	—
145	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₀₉	—	4	1	no	—
147	C₂×D₁₀₉	—	4	1	no	—
148	C₂×D₁₀₉	—	4	1	no	—
149	C₄×C₁₀₉	—	2	1	no	—
150	C₄×C₁₀₉	—	2	1	no	—
151	C₄×C₁₀₉	—	2	1	no	—
152	C₄×C₁₀₉	—	2	1	no	—
153	C₄×C₁₀₉	—	2	1	no	—
154	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
155	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
156	Dic₁₀₉	—	2	2	no	—
157	Dic₁₀₉	—	2	2	no	—
158	Dic₁₀₉	—	4	1	no	—
159	Dic₁₀₉	—	4	1	no	—
160	Dic₁₀₉	—	4	1	no	—
161	Dic₁₀₉	—	4	1	no	—
162	Dic₁₀₉	—	4	1	no	—
163	C₂×C₁₀₉	—	4	1	no	—
164	C₂×C₁₀₉	#525	1	1	yes	characteristic
165	C₂×C₁₀₉	#577	1	1	yes	characteristic
166	C₂×C₁₀₉	#578	1	1	yes	characteristic
167	D₁₀₉	—	8	1	no	—
168	#3121	—	109	1	no	maximal, Sylow
169	C₁₀₉	#3121	1	1	yes	characteristic, complemented, Sylow
170	#542	—	109	1	no	—
171	#519	—	109	1	no	—
172	#485	—	109	1	no	—
173	#482	—	109	1	no	—
174	#477	—	109	1	no	—
175	#475	—	109	1	no	—
176	#464	—	109	1	no	—
177	#464	—	109	1	no	—
178	#451	—	109	1	no	—
179	#448	—	109	1	no	—
180	#441	—	109	1	no	—
181	#440	—	109	1	no	—
182	C₄×Q₁₆	—	109	1	no	—
183	C₄×SD₁₆	—	109	1	no	—
184	#433	D₁₀₉	1	1	yes	characteristic, complemented
185	C₂²×Q₈	—	109	1	no	—
186	C₂×Q₁₆	—	109	1	no	—
187	C₂×Q₁₆	—	109	1	no	—
188	C₂×SD₁₆	—	109	1	no	—
189	C₂×SD₁₆	—	109	1	no	—
190	C₄⋊Q₈	—	109	1	no	—
191	#126	—	218	1	no	—
192	#124	—	109	1	no	—
193	#124	—	218	1	no	—
194	#123	—	218	1	no	—
195	C₂²⋊Q₈	—	109	1	no	—
196	C₂²⋊Q₈	—	218	1	no	—
197	#121	—	109	1	no	—
198	C₄×Q₈	—	109	1	no	—
199	C₄×Q₈	—	109	1	no	—
200	C₄×Q₈	—	109	1	no	—
201	C₄×Q₈	—	109	1	no	—
202	C₄×Q₈	—	109	1	no	—
203	C₄×Q₈	—	218	1	no	—
204	C₄×Q₈	—	218	1	no	—
205	C₄×D₄	—	109	1	no	—
206	#117	—	109	1	no	—
207	#117	—	218	1	no	—
208	#117	—	218	1	no	—
209	#117	C₂×D₁₀₉	1	1	yes	characteristic
210	#107	—	109	1	no	—
211	#107	—	109	1	no	—
212	#106	—	109	1	no	—
213	#106	—	109	1	no	—
214	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
215	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
216	#103	—	109	1	no	—
217	#103	—	109	1	no	—
218	#103	—	109	1	no	—
219	#103	—	109	1	no	—
220	#103	—	109	1	no	—
221	#103	—	109	1	no	—
222	#102	—	109	1	no	—
223	#102	—	109	1	no	—
224	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
225	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
226	#97	C₂×D₁₀₉	1	1	yes	characteristic
227	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
228	C₂×Q₈	—	109	1	no	—
229	C₂×Q₈	—	109	1	no	—
230	C₂×Q₈	—	109	1	no	—
231	C₂×Q₈	—	218	2	no	—
232	C₂×Q₈	—	218	2	no	—
233	C₂×Q₈	—	218	1	no	—
234	C₂×Q₈	—	218	1	no	—
235	C₂×D₄	—	109	1	no	—
236	C₂×D₄	—	218	1	no	—
237	C₂²×C₄	—	218	1	no	—
238	C₂²×C₄	—	218	1	no	—
239	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
240	Q₁₆	—	218	2	no	—
241	Q₁₆	—	218	2	no	—
242	SD₁₆	—	218	2	no	—
243	SD₁₆	—	218	2	no	—
244	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
245	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
248	C₄⋊C₄	—	109	1	no	—
249	C₄⋊C₄	—	109	1	no	—
250	C₄⋊C₄	—	109	1	no	—
251	C₄⋊C₄	—	109	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	—	218	1	no	—
257	C₄⋊C₄	—	218	1	no	—
258	C₄⋊C₄	—	218	1	no	—
259	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
260	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
261	C₂²⋊C₄	—	218	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	D₄₃₆	1	1	yes	characteristic
268	C₂²⋊C₄	D₄₃₆	1	1	yes	characteristic
269	C₄²	—	218	1	no	—
270	C₄²	—	218	1	no	—
271	C₄²	—	218	1	no	—
272	C₄²	—	218	1	no	—
273	C₄²	—	218	1	no	—
274	C₄²	—	218	1	no	—
275	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
276	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
277	C₂³	—	218	1	no	—
278	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
279	Q₈	—	872	1	no	—
280	Q₈	—	109	2	no	—
281	Q₈	—	109	2	no	—
282	Q₈	—	218	4	no	—
283	Q₈	—	218	1	no	—
284	Q₈	—	218	1	no	—
285	Q₈	—	218	1	no	—
286	Q₈	—	218	1	no	—
287	D₄	—	872	1	no	—
288	D₄	—	218	1	no	—
289	D₄	—	218	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	872	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	4	no	—
299	C₂×C₄	—	218	1	no	—
300	C₂×C₄	—	218	1	no	—
301	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
302	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	436	1	no	—
315	C₂²	—	436	1	no	—
316	C₂²	—	436	1	no	—
317	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	436	1	no	—
324	C₄	—	436	1	no	—
325	C₄	—	436	1	no	—
326	C₄	—	436	1	no	—
327	C₄	—	436	1	no	—
328	C₄	—	218	2	no	—
329	C₄	—	218	2	no	—
330	C₄	#41×D₁₀₉	1	1	yes	characteristic
331	C₄	#41×D₁₀₉	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	872	1	no	—
334	C₂	#20295538	1	1	yes	characteristic
335	C₂	#20295838	1	1	yes	characteristic
336	C₂	#20295843	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#433×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#16812673	C₂²
hypercenter	#433	D₁₀₉
hypocenter	C₁₀₉	#3121
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
2. #20295503	C₂	semidirect
3. #20295368	C₂	semidirect
5. #20295322	C₂	semidirect
6. #20295312	C₂	semidirect
7. #20295249	C₂	semidirect
8. #20295249	C₂	semidirect
9. #20295184	C₂	semidirect
11. #20295134	C₂	semidirect
12. #20295128	C₂	semidirect
13. C₄×Dic₄₃₆	C₂	semidirect
14. C₄×#13182018	C₂	semidirect
15. #433×C₁₀₉	C₂	semidirect
169. C₁₀₉	#3121	semidirect
184. #433	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295647	2
#20295503	2
#20295368	2
#20295351	2
#20295322	2
#20295312	2
#20295249	2
#20295184	2
#20295167	2
#20295134	2
#20295128	2
C₄×Dic₄₃₆	2
C₄×#13182018	2
#433×C₁₀₉	2
#3121	109