Subgroups — #16020986

Order 3328.

510 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	#13127846	C₂	1	1	yes	maximal, characteristic
2	#13127840	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117104	C₂	1	2	yes	maximal, complemented
4	#13117093	C₂	1	1	yes	maximal, characteristic
5	#13116900	C₂	1	2	yes	maximal, complemented
6	#13116880	C₂	1	2	yes	maximal, complemented
7	#13116877	C₂	1	1	yes	maximal, characteristic
8	#13116741	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116703	C₂	1	2	yes	maximal, complemented
10	C₄×#88069	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719426	—	2	1	no	—
13	#11719290	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
15	C₁₃×#494	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	#11718852	C₂²	1	2	yes	—
18	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
19	#11718843	—	2	2	no	—
20	#11718842	C₂²	1	2	yes	—
21	C₂×#88069	C₂²	1	1	yes	characteristic
22	C₂×#88068	C₂²	1	2	yes	—
23	C₂×#88068	C₂²	1	2	yes	—
24	#11718751	—	2	1	no	—
25	#11718688	—	2	1	no	—
26	C₂×#88040	C₂²	1	2	yes	—
27	#11718533	C₂²	1	2	yes	—
28	#11718480	—	2	2	no	—
29	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
30	#11718356	C₂²	1	1	yes	characteristic
31	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₃×#358	C₂²	1	2	yes	—
34	#11718156	C₂²	1	1	yes	characteristic
35	#11718101	C₂²	1	2	yes	—
36	#11718094	C₂²	1	2	yes	—
37	#11718093	C₂²	1	2	yes	—
38	#11718084	C₂²	1	1	yes	characteristic
39	#11718081	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×Q₁₆	—	2	1	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
44	C₁₃×C₄⋊Q₈	—	2	1	no	—
45	C₄×Q₈×C₁₃	—	2	1	no	—
46	#88201	—	2	2	no	—
47	Q₈×Dic₁₃	—	2	2	no	—
48	#88198	—	2	2	no	—
49	C₂×#6107	—	2	2	no	—
50	C₂×#6106	—	2	2	no	—
51	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
52	C₂×D₁₀₄	—	2	1	no	—
53	C₂×D₁₀₄	C₂³	1	2	yes	—
54	#88148	—	4	2	no	—
55	#88147	—	2	2	no	—
56	#88144	—	4	1	no	—
57	#88143	—	4	2	no	—
58	#88141	—	4	1	no	—
59	#88129	—	4	1	no	—
60	#88128	—	2	1	no	—
61	C₄×D₅₂	—	4	1	no	—
62	C₄×Dic₂₆	—	2	1	no	—
63	C₄×Dic₂₆	D₄	1	1	yes	characteristic
64	C₁₃×Q₃₂	—	4	2	no	—
65	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
66	C₁₃×#107	C₂³	1	1	yes	characteristic
67	C₁₃×#103	—	2	1	no	—
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	#88075	—	2	2	no	—
70	#88069	—	2	2	no	—
71	#88068	—	2	4	no	—
72	#88068	—	4	2	no	—
73	#88058	C₂³	1	1	yes	characteristic
74	#88056	—	2	1	no	—
75	#88054	D₄	1	2	yes	—
76	C₈×Dic₁₃	C₂³	1	2	yes	—
77	C₂×#6090	C₂³	1	1	yes	characteristic
78	C₂×#6090	C₂³	1	1	yes	characteristic
79	#88049	—	2	2	no	—
80	C₄⋊#1055	D₄	1	1	yes	characteristic
81	#88040	—	4	2	no	—
82	#33967	—	13	1	no	maximal, Sylow
83	C₂×Q₈×C₁₃	—	2	2	no	—
84	C₂×Q₈×C₁₃	—	2	1	no	—
85	C₂×D₅₂	—	2	2	no	—
86	C₂×D₅₂	—	4	1	no	—
87	C₂×D₅₂	—	4	1	no	—
88	C₂×C₄×D₁₃	—	4	2	no	—
89	C₂×Dic₂₆	—	2	1	no	—
90	C₂×Dic₂₆	D₈	1	1	yes	characteristic
91	C₁₃×Q₁₆	—	2	2	no	—
92	C₁₃×Q₁₆	—	2	1	no	—
93	C₁₃×Q₁₆	—	2	2	no	—
94	C₁₃×Q₁₆	—	2	1	no	—
95	C₁₃×Q₁₆	—	8	1	no	—
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	—	2	1	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₄²×C₁₃	—	4	1	no	—
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	#6107	—	8	2	no	—
105	#6106	—	8	2	no	—
106	#6103	—	4	2	no	—
107	#6103	—	4	2	no	—
108	#6103	—	4	2	no	—
109	C₄⋊Dic₁₃	—	2	1	no	—
110	C₄⋊Dic₁₃	—	4	2	no	—
111	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
112	#6101	—	4	1	no	—
113	#6101	—	4	2	no	—
114	#6101	—	4	2	no	—
115	#6101	D₈	1	2	yes	—
116	C₄×Dic₁₃	—	4	1	no	—
117	C₄×Dic₁₃	—	4	2	no	—
118	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
119	C₂×#1055	C₂×D₄	1	2	yes	—
120	Dic₅₂	—	2	1	no	—
121	Dic₅₂	—	2	1	no	—
122	D₁₀₄	—	2	2	no	—
123	D₁₀₄	—	2	2	no	—
124	D₁₀₄	—	8	1	no	—
125	C₁₃×C₁₆	—	2	2	no	—
126	#6090	—	2	1	no	—
127	#6090	#41	1	2	yes	—
128	#3396	—	13	1	no	—
129	#3395	—	13	1	no	—
130	#2251	—	13	2	no	—
131	#2248	—	13	1	no	—
132	#2211	—	13	2	no	—
133	#2209	D₁₃	1	1	yes	characteristic, complemented
134	#2208	—	13	2	no	—
135	#2207	—	13	1	no	—
136	#2183	—	13	1	no	—
137	#2175	—	13	2	no	—
138	C₄×Q₃₂	—	13	1	no	—
139	C₂²×D₁₃	—	4	2	no	—
140	Q₈×C₁₃	—	4	2	no	—
141	Q₈×C₁₃	—	4	1	no	—
142	Q₈×C₁₃	—	4	1	no	—
143	Q₈×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	1	no	—
146	C₂×C₄×C₁₃	—	4	1	no	—
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₁₃	—	4	1	no	—
151	C₂×Dic₁₃	—	4	1	no	—
152	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
153	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
154	D₅₂	—	4	2	no	—
155	D₅₂	—	4	2	no	—
156	D₅₂	—	8	2	no	—
157	D₅₂	—	16	1	no	—
158	C₄×D₁₃	—	16	2	no	—
159	Dic₂₆	—	2	2	no	—
160	Dic₂₆	—	4	1	no	—
161	Dic₂₆	—	4	1	no	—
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
164	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
165	#1055	—	2	2	no	—
166	#559	—	26	1	no	—
167	#557	—	26	1	no	—
168	C₂×Q₃₂	—	13	1	no	—
169	C₂×Q₃₂	C₂×D₁₃	1	2	yes	—
170	C₂×SD₃₂	—	13	2	no	—
171	C₂×SD₃₂	—	13	2	no	—
172	C₂×SD₃₂	—	13	2	no	—
173	#495	—	13	2	no	—
174	#494	—	13	2	no	—
175	#494	C₂×D₁₃	1	1	yes	characteristic
176	#493	—	13	1	no	—
177	#463	—	26	2	no	—
178	#462	—	26	1	no	—
179	#462	—	26	2	no	—
180	#460	—	26	1	no	—
181	C₈×Q₈	—	13	1	no	—
182	C₄×Q₁₆	—	13	2	no	—
183	C₄×Q₁₆	—	13	1	no	—
184	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
185	C₄×D₈	—	13	2	no	—
186	#366	—	13	1	no	—
187	C₄⋊C₁₆	—	13	2	no	—
188	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
189	#358	—	13	1	no	—
190	#358	—	13	1	no	—
191	#358	C₂×D₁₃	1	2	yes	—
192	#357	—	13	2	no	—
193	#346	—	13	2	no	—
194	C₄×C₁₆	—	13	1	no	—
195	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
196	C₂×D₁₃	—	8	2	no	—
197	C₂×D₁₃	—	8	2	no	—
198	C₂×D₁₃	—	8	2	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	8	2	no	—
201	C₄×C₁₃	—	8	1	no	—
202	C₄×C₁₃	—	8	1	no	—
203	C₄×C₁₃	#510	1	2	yes	—
204	C₄×C₁₃	#550	1	1	yes	characteristic
205	C₄×C₁₃	#576	1	1	yes	characteristic
206	Dic₁₃	—	2	2	no	—
207	Dic₁₃	—	2	2	no	—
208	Dic₁₃	—	8	1	no	—
209	Dic₁₃	—	8	1	no	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	—	13	1	no	—
212	C₂×Q₁₆	—	26	2	no	—
213	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
214	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
215	C₂×SD₁₆	—	26	2	no	—
216	C₂×D₈	—	13	2	no	—
217	C₂×D₈	—	26	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	26	2	no	—
220	#127	—	26	1	no	—
221	#125	—	52	2	no	—
222	#125	—	52	1	no	—
223	#124	—	26	2	no	—
224	#124	—	52	1	no	—
225	#123	—	52	2	no	—
226	C₄×Q₈	—	2	1	no	—
227	C₄×Q₈	—	13	1	no	—
228	C₄×Q₈	—	26	2	no	—
229	C₄×Q₈	—	26	1	no	—
230	C₄×Q₈	—	52	1	no	—
231	C₄×D₄	—	26	2	no	—
232	C₄×D₄	—	52	1	no	—
233	Q₃₂	—	4	2	no	—
234	Q₃₂	—	26	2	no	—
235	SD₃₂	—	26	4	no	—
236	SD₃₂	—	52	2	no	—
237	SD₃₂	—	52	2	no	—
238	C₂×C₁₆	—	13	1	no	—
239	C₂×C₁₆	—	13	1	no	—
240	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
241	#107	—	13	1	no	—
242	#107	C₂²×D₁₃	1	1	yes	characteristic
243	C₄⋊C₈	—	13	2	no	—
244	C₄⋊C₈	—	13	1	no	—
245	#103	—	2	1	no	—
246	#103	—	26	2	no	—
247	#103	—	26	1	no	—
248	#102	—	26	2	no	—
249	C₄×C₈	—	13	2	no	—
250	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
251	C₂×C₁₃	#3391	1	1	yes	characteristic
252	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
253	C₂×C₁₃	#3416	1	1	yes	characteristic
254	D₁₃	—	16	2	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	13	1	no	—
258	C₂×Q₈	—	26	1	no	—
259	C₂×D₄	—	26	2	no	—
260	C₂×D₄	—	52	1	no	—
261	C₂×D₄	—	52	1	no	—
262	C₂²×C₄	—	52	2	no	—
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	—	2	2	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	8	1	no	—
268	Q₁₆	—	26	1	no	—
269	Q₁₆	—	26	1	no	—
270	Q₁₆	—	104	2	no	—
271	SD₁₆	—	104	2	no	—
272	D₈	—	26	2	no	—
273	D₈	—	26	2	no	—
274	D₈	—	104	1	no	—
275	C₂×C₈	—	13	2	no	—
276	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
277	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	13	1	no	—
283	C₄⋊C₄	—	13	2	no	—
284	C₄⋊C₄	—	26	1	no	—
285	C₄⋊C₄	—	52	2	no	—
286	C₄⋊C₄	—	52	2	no	—
287	C₄⋊C₄	—	52	2	no	—
288	C₄⋊C₄	—	52	1	no	—
289	C₂²⋊C₄	—	52	2	no	—
290	C₂²⋊C₄	—	52	2	no	—
291	C₂²⋊C₄	—	52	2	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	13	2	no	—
294	C₄²	—	52	2	no	—
295	C₄²	—	52	1	no	—
296	C₄²	D₄×D₁₃	1	1	yes	characteristic
297	C₁₆	—	2	2	no	—
298	C₁₆	—	13	2	no	—
299	C₁₆	—	26	1	no	—
300	C₁₃	#33967	1	1	yes	characteristic, complemented, Sylow
301	C₂³	—	52	2	no	—
302	Q₈	—	4	1	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	26	2	no	—
307	Q₈	—	52	1	no	—
308	Q₈	—	52	1	no	—
309	D₄	—	52	2	no	—
310	D₄	—	52	2	no	—
311	D₄	—	104	2	no	—
312	D₄	—	208	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	13	2	no	—
317	C₂×C₄	—	13	2	no	—
318	C₂×C₄	—	52	1	no	—
319	C₂×C₄	—	52	1	no	—
320	C₂×C₄	—	208	2	no	—
321	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
322	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	26	2	no	—
326	C₈	#88251	1	1	yes	characteristic
327	C₈	#41×D₁₃	1	1	yes	characteristic
328	C₂²	—	104	2	no	—
329	C₂²	—	104	2	no	—
330	C₂²	—	104	2	no	—
331	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	26	2	no	—
337	C₄	—	26	2	no	—
338	C₄	—	104	1	no	—
339	C₄	—	104	1	no	—
340	C₄	#11718579	1	2	yes	—
341	C₄	#11719304	1	1	yes	characteristic
342	C₄	#11719479	1	1	yes	characteristic
343	C₂	—	208	2	no	—
344	C₂	#13127807	1	1	yes	characteristic
345	C₂	C₂×#11718579	1	1	yes	characteristic
346	C₂	#13127981	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×#2209	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#11718356	C₂²
hypercenter	#2209	D₁₃
hypocenter	C₁₃	#33967
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

Complements

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

N	H ≅ G/N	type
2. #13127840	C₂	semidirect
3. #13117104	C₂	semidirect
5. #13116900	C₂	semidirect
6. #13116880	C₂	semidirect
8. #13116741	C₂	semidirect
9. #13116703	C₂	semidirect
10. C₄×#88069	C₂	semidirect
11. C₁₃×#2209	C₂	semidirect
133. #2209	D₁₃	semidirect
300. C₁₃	#33967	semidirect

Maximal subgroups

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

subgroup	index
#13127846	2
#13127840	2
#13117104	2
#13117093	2
#13116900	2
#13116880	2
#13116877	2
#13116741	2
#13116703	2
C₄×#88069	2
C₁₃×#2209	2
#33967	13