Subgroups — #16020554

Order 3328.

482 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	#13127258	C₂	1	1	yes	maximal, characteristic
2	#13127228	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117046	C₂	1	2	yes	maximal, complemented
4	#13117018	C₂	1	2	yes	maximal, complemented
5	#13116985	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116980	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116950	C₂	1	2	yes	maximal, complemented
8	C₄×#88040	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116669	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116621	C₂	1	2	yes	maximal, complemented
11	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719439	—	2	1	no	—
13	#11719435	—	2	1	no	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	#11718752	—	2	1	no	—
17	#11718745	—	2	1	no	—
18	C₂×#88040	C₂²	1	1	yes	characteristic
19	C₂×#88038	C₂²	1	2	yes	—
20	#11718548	C₂²	1	2	yes	—
21	#11718546	—	2	2	no	—
22	#11718545	—	2	2	no	—
23	#11718544	C₂²	1	2	yes	—
24	#11718541	C₂²	1	2	yes	—
25	D₁₃×#107	C₂²	1	2	yes	—
26	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
27	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
28	C₈×D₅₂	C₂²	1	1	yes	characteristic
29	C₁₃×#366	C₂²	1	2	yes	—
30	C₁₃×#366	C₂²	1	2	yes	—
31	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
32	#11718101	C₂²	1	1	yes	characteristic
33	#11718099	C₂²	1	2	yes	—
34	#11718097	C₂²	1	1	yes	characteristic
35	#11718095	C₂²	1	1	yes	characteristic
36	#11718084	C₂²	1	2	yes	—
37	#11718083	C₂²	1	2	yes	—
38	#11718082	C₂²	1	2	yes	—
39	#11718081	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	1	yes	characteristic
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	#88201	—	4	2	no	—
44	#88200	—	4	2	no	—
45	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
46	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₁₃	C₂³	1	2	yes	—
48	#88151	—	2	2	no	—
49	#88149	—	2	2	no	—
50	#88149	—	4	1	no	—
51	#88147	—	4	1	no	—
52	#88146	—	2	2	no	—
53	#88146	—	4	1	no	—
54	C₄⋊C₄×D₁₃	—	2	2	no	—
55	#88144	—	4	1	no	—
56	C₄×D₅₂	—	2	1	no	—
57	C₄×D₅₂	D₄	1	1	yes	characteristic
58	C₄×Dic₂₆	—	2	1	no	—
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₁₃×#107	—	2	1	no	—
62	C₁₃×#107	—	2	1	no	—
63	C₁₃×#107	C₂³	1	2	yes	—
64	C₁₃×#107	C₂³	1	2	yes	—
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	#88061	—	2	1	no	—
67	#88060	D₄	1	2	yes	—
68	#88058	C₂³	1	1	yes	characteristic
69	#88058	C₂³	1	1	yes	characteristic
70	#88056	—	2	1	no	—
71	C₂×#6090	C₂³	1	2	yes	—
72	#88050	—	2	2	no	—
73	#88049	—	2	2	no	—
74	#88048	—	2	2	no	—
75	#88047	—	2	2	no	—
76	C₄⋊#1055	D₄	1	1	yes	characteristic
77	#88040	—	2	2	no	—
78	#88038	—	2	4	no	—
79	#33924	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	4	1	no	—
81	C₂×Q₈×C₁₃	—	4	1	no	—
82	C₂×D₅₂	—	2	1	no	—
83	C₂×D₅₂	—	4	2	no	—
84	C₂×D₅₂	D₈	1	1	yes	characteristic
85	C₂×C₄×D₁₃	—	4	1	no	—
86	C₂×C₄×D₁₃	—	4	2	no	—
87	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
88	C₂×Dic₂₆	—	2	1	no	—
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	4	1	no	—
94	C₁₃×C₄⋊C₄	—	4	1	no	—
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	#6103	—	4	2	no	—
97	#6103	—	4	2	no	—
98	#6103	—	4	1	no	—
99	#6103	—	4	2	no	—
100	#6103	D₈	1	2	yes	—
101	C₄⋊Dic₁₃	—	2	1	no	—
102	C₄⋊Dic₁₃	—	2	1	no	—
103	C₄⋊Dic₁₃	—	4	2	no	—
104	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
105	#6101	—	4	1	no	—
106	#6101	—	4	2	no	—
107	C₄×Dic₁₃	—	4	1	no	—
108	C₄×Dic₁₃	—	4	2	no	—
109	C₂×#1055	C₂×D₄	1	2	yes	—
110	Dic₅₂	—	2	1	no	—
111	Dic₅₂	—	2	1	no	—
112	D₁₀₄	—	2	1	no	—
113	D₁₀₄	—	2	1	no	—
114	C₈×D₁₃	—	2	2	no	—
115	C₈×D₁₃	—	2	2	no	—
116	C₁₃×C₁₆	—	2	1	no	—
117	C₁₃×C₁₆	#41	1	2	yes	—
118	#6090	—	2	2	no	—
119	#3341	—	13	1	no	—
120	#3338	—	13	1	no	—
121	#2252	D₁₃	1	1	yes	characteristic, complemented
122	#2237	—	13	2	no	—
123	#2232	—	13	2	no	—
124	#2227	—	13	1	no	—
125	#2226	—	13	1	no	—
126	#2220	—	13	2	no	—
127	C₄×SD₃₂	—	13	1	no	—
128	#2169	—	13	1	no	—
129	#2159	—	13	2	no	—
130	C₂²×D₁₃	—	4	1	no	—
131	C₂²×D₁₃	C₂×D₈	1	2	yes	—
132	Q₈×C₁₃	—	16	1	no	—
133	Q₈×C₁₃	—	16	1	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₁₃	—	4	1	no	—
140	C₂×Dic₁₃	—	4	1	no	—
141	C₂×Dic₁₃	—	4	1	no	—
142	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
143	D₅₂	—	2	2	no	—
144	D₅₂	—	4	1	no	—
145	D₅₂	—	4	1	no	—
146	D₅₂	—	16	2	no	—
147	C₄×D₁₃	—	2	2	no	—
148	C₄×D₁₃	—	2	2	no	—
149	C₄×D₁₃	—	8	2	no	—
150	C₄×D₁₃	—	16	2	no	—
151	Dic₂₆	—	4	1	no	—
152	Dic₂₆	—	4	1	no	—
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
155	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
156	#1055	—	2	2	no	—
157	#555	—	26	1	no	—
158	#554	—	26	1	no	—
159	C₂×SD₃₂	—	13	1	no	—
160	C₂×M₃₂	—	13	2	no	—
161	#500	C₂×D₁₃	1	1	yes	characteristic
162	#500	C₂×D₁₃	1	1	yes	characteristic
163	#484	—	26	2	no	—
164	#482	—	26	2	no	—
165	#477	—	26	1	no	—
166	#476	—	26	1	no	—
167	#467	—	13	2	no	—
168	#466	—	13	2	no	—
169	C₄×Q₁₆	—	13	1	no	—
170	C₄×D₈	—	13	1	no	—
171	C₈×D₄	—	13	1	no	—
172	#427	—	13	2	no	—
173	C₂×#107	—	13	2	no	—
174	#367	—	13	1	no	—
175	#367	—	13	2	no	—
176	#366	—	13	2	no	—
177	#366	C₂×D₁₃	1	2	yes	—
178	#366	C₂×D₁₃	1	2	yes	—
179	C₄⋊C₁₆	—	13	1	no	—
180	#358	—	13	1	no	—
181	#358	—	13	2	no	—
182	#357	—	13	1	no	—
183	#357	—	13	2	no	—
184	C₂²⋊C₁₆	—	13	2	no	—
185	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
186	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
187	C₂×D₁₃	—	2	2	no	—
188	C₂×D₁₃	—	2	2	no	—
189	C₂×D₁₃	—	2	2	no	—
190	C₂×D₁₃	—	8	1	no	—
191	C₂×D₁₃	—	8	1	no	—
192	C₂×D₁₃	—	8	1	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	—	8	2	no	—
196	C₄×C₁₃	#510	1	2	yes	—
197	C₄×C₁₃	#547	1	1	yes	characteristic
198	C₄×C₁₃	#578	1	1	yes	characteristic
199	Dic₁₃	—	2	2	no	—
200	Dic₁₃	—	8	1	no	—
201	Dic₁₃	—	8	1	no	—
202	Dic₁₃	—	8	1	no	—
203	C₂×Q₁₆	—	13	1	no	—
204	C₂×D₈	—	13	1	no	—
205	C₂²×C₈	—	13	2	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	#125	—	52	1	no	—
209	#124	—	52	2	no	—
210	C₂²⋊Q₈	—	26	2	no	—
211	C₂²⋊Q₈	—	52	2	no	—
212	#121	—	26	2	no	—
213	#121	—	52	1	no	—
214	C₄×Q₈	—	26	1	no	—
215	C₄×D₄	—	13	1	no	—
216	C₄×D₄	—	26	1	no	—
217	C₄×D₄	—	52	1	no	—
218	#117	—	26	2	no	—
219	#117	—	52	1	no	—
220	C₂×C₄⋊C₄	—	26	2	no	—
221	SD₃₂	—	26	2	no	—
222	M₃₂	—	26	4	no	—
223	C₂×C₁₆	—	13	2	no	—
224	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
226	#107	—	2	1	no	—
227	#107	—	2	1	no	—
228	#107	—	13	1	no	—
229	#107	—	13	1	no	—
230	#107	—	26	2	no	—
231	#107	C₂²×D₁₃	1	2	yes	—
232	#107	C₂²×D₁₃	1	2	yes	—
233	#106	—	26	2	no	—
234	C₄⋊C₈	—	13	1	no	—
235	#103	—	26	2	no	—
236	#103	—	26	1	no	—
237	#102	—	26	2	no	—
238	#102	—	26	1	no	—
239	C₂²⋊C₈	—	13	2	no	—
240	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
241	C₂×C₁₃	#3334	1	1	yes	characteristic
242	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
243	C₂×C₁₃	#3415	1	1	yes	characteristic
244	D₁₃	—	4	2	no	—
245	D₁₃	—	16	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	26	1	no	—
249	C₂×D₄	—	13	1	no	—
250	C₂×D₄	—	26	1	no	—
251	C₂×D₄	—	52	2	no	—
252	C₂²×C₄	—	13	2	no	—
253	C₂²×C₄	—	52	2	no	—
254	C₂²×C₄	—	52	1	no	—
255	Q₁₆	—	26	1	no	—
256	Q₁₆	—	26	1	no	—
257	D₈	—	26	1	no	—
258	D₈	—	26	1	no	—
259	C₂×C₈	—	13	2	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	—	26	2	no	—
262	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	13	1	no	—
269	C₄⋊C₄	—	26	1	no	—
270	C₄⋊C₄	—	26	1	no	—
271	C₄⋊C₄	—	52	2	no	—
272	C₄⋊C₄	—	52	2	no	—
273	C₄⋊C₄	—	52	1	no	—
274	C₂²⋊C₄	—	13	2	no	—
275	C₂²⋊C₄	—	52	2	no	—
276	C₂²⋊C₄	—	52	2	no	—
277	C₂²⋊C₄	—	52	2	no	—
278	C₂²⋊C₄	—	52	1	no	—
279	C₄²	—	52	2	no	—
280	C₄²	—	52	1	no	—
281	C₄²	D₄×D₁₃	1	1	yes	characteristic
282	C₁₆	—	2	1	no	—
283	C₁₆	—	26	2	no	—
284	C₁₆	#6131	1	2	yes	—
285	C₁₃	#33924	1	1	yes	characteristic, complemented, Sylow
286	C₂³	—	13	2	no	—
287	C₂³	—	52	1	no	—
288	Q₈	—	16	1	no	—
289	Q₈	—	16	1	no	—
290	Q₈	—	52	1	no	—
291	Q₈	—	52	1	no	—
292	D₄	—	26	2	no	—
293	D₄	—	52	1	no	—
294	D₄	—	52	1	no	—
295	D₄	—	208	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	26	2	no	—
300	C₂×C₄	—	26	2	no	—
301	C₂×C₄	—	52	1	no	—
302	C₂×C₄	—	52	1	no	—
303	C₂×C₄	—	52	1	no	—
304	C₂×C₄	—	104	2	no	—
305	C₂×C₄	—	208	2	no	—
306	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
307	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	26	2	no	—
311	C₈	C₂×#6131	1	1	yes	characteristic
312	C₈	#88254	1	1	yes	characteristic
313	C₂²	—	26	2	no	—
314	C₂²	—	26	2	no	—
315	C₂²	—	26	2	no	—
316	C₂²	—	104	1	no	—
317	C₂²	—	104	1	no	—
318	C₂²	—	104	1	no	—
319	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	26	2	no	—
324	C₄	—	104	1	no	—
325	C₄	—	104	1	no	—
326	C₄	—	104	1	no	—
327	C₄	#11718579	1	2	yes	—
328	C₄	#11719431	1	1	yes	characteristic
329	C₄	#11719471	1	1	yes	characteristic
330	C₂	—	52	2	no	—
331	C₂	—	208	1	no	—
332	C₂	#13127183	1	1	yes	characteristic
333	C₂	C₂×#11718579	1	1	yes	characteristic
334	C₂	#13127973	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2252	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13127258	C₂
hypercenter	#2252	D₁₃
hypocenter	C₁₃	#33924
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. #13127228	C₂	semidirect
3. #13117046	C₂	semidirect
4. #13117018	C₂	semidirect
5. #13116985	C₂	semidirect
6. #13116980	C₂	semidirect
7. #13116950	C₂	semidirect
8. C₄×#88040	C₂	semidirect
9. #13116669	C₂	semidirect
10. #13116621	C₂	semidirect
11. C₁₃×#2252	C₂	semidirect
121. #2252	D₁₃	semidirect
285. C₁₃	#33924	semidirect

Maximal subgroups

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

subgroup	index
#13127258	2
#13127228	2
#13117046	2
#13117018	2
#13116985	2
#13116980	2
#13116950	2
C₄×#88040	2
#13116669	2
#13116621	2
C₁₃×#2252	2
#33924	13