Subgroups — #16020507

Order 3328.

502 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	#13127184	C₂	1	1	yes	maximal, characteristic
2	#13117109	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117036	C₂	1	2	yes	maximal, complemented
4	#13117027	C₂	1	2	yes	maximal, complemented
5	#13116978	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116970	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116951	C₂	1	2	yes	maximal, complemented
8	C₄×#88068	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116668	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116625	C₂	1	2	yes	maximal, complemented
11	C₁₃×#3342	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#556	—	2	1	no	—
13	#11719260	—	2	1	no	—
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₁₃×#485	—	2	2	no	—
16	C₁₃×#475	—	2	1	no	—
17	C₁₃×#467	C₂²	1	2	yes	—
18	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
19	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#427	C₂²	1	2	yes	—
21	C₂×#88068	C₂²	1	1	yes	characteristic
22	#11718776	C₂²	1	2	yes	—
23	C₂×#88058	C₂²	1	2	yes	—
24	C₂×#88052	C₂²	1	2	yes	—
25	#11718507	—	2	1	no	—
26	#11718397	—	2	2	no	—
27	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
28	#11718343	C₂²	1	1	yes	characteristic
29	#11718156	C₂²	1	1	yes	characteristic
30	#11718156	C₂²	1	2	yes	—
31	C₂²⋊#6090	C₂²	1	2	yes	—
32	#11718083	C₂²	1	1	yes	characteristic
33	#11718083	C₂²	1	2	yes	—
34	#11718082	C₂²	1	1	yes	characteristic
35	#11718082	C₂²	1	2	yes	—
36	#11718081	C₂²	1	2	yes	—
37	#11718081	C₂²	1	2	yes	—
38	#11718081	C₂²	1	2	yes	—
39	C₄⋊#6090	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂×#6129	—	4	1	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
44	C₁₃×C₄⋊Q₈	—	2	1	no	—
45	C₁₃×#126	—	4	2	no	—
46	C₁₃×#124	—	4	1	no	—
47	C₁₃×C₂²⋊Q₈	—	2	2	no	—
48	C₄×Q₈×C₁₃	—	2	1	no	—
49	C₄×Q₈×C₁₃	—	4	1	no	—
50	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
51	C₁₃×#117	—	2	2	no	—
52	#88184	—	2	2	no	—
53	C₂×C₄⋊Dic₁₃	—	2	2	no	—
54	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
55	#88151	—	4	1	no	—
56	#88140	—	4	2	no	—
57	C₄×D₅₂	—	2	1	no	—
58	#88123	—	2	1	no	—
59	C₁₃×#107	—	2	1	no	—
60	C₁₃×#107	C₂³	1	1	yes	characteristic
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#106	—	2	2	no	—
63	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₁₃×#103	—	2	1	no	—
65	C₁₃×#103	—	2	2	no	—
66	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	#88068	—	2	2	no	—
69	#88061	—	2	2	no	—
70	#88061	—	2	1	no	—
71	#88058	—	2	2	no	—
72	#88058	—	2	1	no	—
73	#88058	C₂³	1	1	yes	characteristic
74	#88058	C₂³	1	2	yes	—
75	#88052	—	2	4	no	—
76	C₂×#6090	C₂³	1	2	yes	—
77	C₂×#6090	C₂³	1	1	yes	characteristic
78	C₂×#6090	C₂³	1	1	yes	characteristic
79	#33920	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	—	4	1	no	—
82	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₂²⋊D₁₃	—	4	2	no	—
85	C₂²×Dic₁₃	—	4	2	no	—
86	#6129	—	8	4	no	—
87	C₂×D₅₂	—	2	1	no	—
88	C₂×C₄×D₁₃	—	4	1	no	—
89	C₂×Dic₂₆	—	4	1	no	—
90	C₁₃×Q₁₆	—	2	1	no	—
91	C₁₃×Q₁₆	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
103	C₁₃×C₂²⋊C₄	—	4	2	no	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
106	C₄²×C₁₃	—	4	2	no	—
107	C₄²×C₁₃	—	4	1	no	—
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	#6103	—	4	1	no	—
110	#6103	—	4	2	no	—
111	C₄⋊Dic₁₃	—	2	2	no	—
112	C₄⋊Dic₁₃	—	2	1	no	—
113	C₄⋊Dic₁₃	—	4	1	no	—
114	C₄⋊Dic₁₃	—	4	2	no	—
115	D₁₀₄	—	2	1	no	—
116	D₁₀₄	—	2	1	no	—
117	#6090	—	2	2	no	—
118	#6090	—	2	1	no	—
119	#6090	#41	1	2	yes	complemented
120	#3342	D₁₃	1	1	yes	characteristic, complemented
121	#3334	—	13	1	no	—
122	#2252	—	13	1	no	—
123	#2236	—	13	2	no	—
124	#2234	—	13	2	no	—
125	#2225	—	13	1	no	—
126	#2224	—	13	1	no	—
127	#2220	—	13	2	no	—
128	C₄×SD₃₂	—	13	1	no	—
129	#2169	—	13	1	no	—
130	#2159	—	13	2	no	—
131	C₂³×C₁₃	C₂×D₈	1	2	yes	—
132	C₂²×D₁₃	—	4	1	no	—
133	Q₈×C₁₃	—	4	1	no	—
134	Q₈×C₁₃	—	4	1	no	—
135	Q₈×C₁₃	—	8	2	no	—
136	D₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	1	no	—
142	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₁₃	—	8	4	no	—
147	C₂×Dic₁₃	—	4	2	no	—
148	C₂×Dic₁₃	—	4	1	no	—
149	C₂×Dic₁₃	—	8	4	no	—
150	D₅₂	—	4	1	no	—
151	D₅₂	—	4	1	no	—
152	C₄×D₁₃	—	8	2	no	—
153	Dic₂₆	—	8	2	no	—
154	C₈×C₁₃	—	2	2	no	—
155	C₈×C₁₃	—	2	1	no	—
156	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
158	#556	—	2	1	no	—
159	#547	—	26	1	no	—
160	C₂×SD₃₂	—	13	1	no	—
161	C₂×M₃₂	—	13	2	no	—
162	#500	—	13	1	no	—
163	#500	C₂×D₁₃	1	1	yes	characteristic
164	#485	—	2	2	no	—
165	#480	—	26	2	no	—
166	#475	—	2	1	no	—
167	#474	—	26	1	no	—
168	#467	C₂×D₁₃	1	2	yes	—
169	#466	—	13	2	no	—
170	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
171	C₄×D₈	—	13	1	no	—
172	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
173	#427	C₂×D₁₃	1	2	yes	—
174	C₂×#107	—	13	2	no	—
175	#367	—	13	1	no	—
176	#367	—	13	2	no	—
177	#366	—	13	2	no	—
178	#366	—	13	2	no	—
179	#366	—	13	2	no	—
180	C₄⋊C₁₆	—	13	1	no	—
181	#358	—	13	1	no	—
182	#358	—	13	2	no	—
183	#357	—	13	1	no	—
184	#357	—	13	2	no	—
185	C₂²⋊C₁₆	—	13	2	no	—
186	C₄×C₁₆	—	13	1	no	—
187	C₂²×C₁₃	—	2	2	no	—
188	C₂²×C₁₃	—	2	2	no	—
189	C₂²×C₁₃	—	2	2	no	—
190	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
191	C₂×D₁₃	—	8	1	no	—
192	C₂×D₁₃	—	8	1	no	—
193	C₂×D₁₃	—	8	1	no	—
194	C₄×C₁₃	—	2	2	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	8	1	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	—	8	1	no	—
199	C₄×C₁₃	#509	1	2	yes	—
200	C₄×C₁₃	#547	1	1	yes	characteristic
201	C₄×C₁₃	#578	1	1	yes	characteristic
202	Dic₁₃	—	8	2	no	—
203	Dic₁₃	—	8	1	no	—
204	C₂×#41	—	52	1	no	—
205	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
206	C₂×D₈	—	13	1	no	—
207	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	26	1	no	—
210	#126	—	4	2	no	—
211	#124	—	4	1	no	—
212	#123	—	52	2	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	52	1	no	—
215	#121	—	26	2	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₄×Q₈	—	4	1	no	—
218	C₄×D₄	—	26	1	no	—
219	C₄×D₄	C₂²⋊D₁₃	1	1	yes	characteristic
220	#117	—	2	2	no	—
221	C₂×C₄⋊C₄	—	26	2	no	—
222	SD₃₂	—	26	2	no	—
223	M₃₂	—	26	4	no	—
224	C₂×C₁₆	—	13	1	no	—
225	C₂×C₁₆	—	13	1	no	—
226	C₂×C₁₆	—	13	2	no	—
227	#107	—	2	1	no	—
228	#107	—	13	1	no	—
229	#107	—	13	2	no	—
230	#107	—	26	2	no	—
231	#107	—	26	1	no	—
232	#107	C₂²×D₁₃	1	2	yes	—
233	#107	C₂²×D₁₃	1	1	yes	characteristic
234	#106	—	2	2	no	—
235	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
236	#103	—	2	1	no	—
237	#103	—	2	2	no	—
238	#102	—	26	1	no	—
239	#102	—	26	2	no	—
240	C₂²⋊C₈	C₂²⋊D₁₃	1	2	yes	—
241	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₂×C₁₃	—	4	2	no	—
243	C₂×C₁₃	#3334	1	1	yes	characteristic
244	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
245	C₂×C₁₃	#3417	1	1	yes	characteristic
246	D₁₃	—	16	1	no	—
247	#41	—	104	4	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	52	1	no	—
251	C₂×D₄	—	26	1	no	—
252	C₂×D₄	—	52	2	no	—
253	C₂×D₄	#6104	1	1	yes	characteristic
254	C₂²×C₄	—	52	2	no	—
255	C₂²×C₄	—	52	1	no	—
256	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
257	Q₁₆	—	2	1	no	—
258	Q₁₆	—	2	1	no	—
259	D₈	—	26	1	no	—
260	D₈	—	26	1	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
265	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	26	1	no	—
272	C₄⋊C₄	—	26	2	no	—
273	C₄⋊C₄	—	52	2	no	—
274	C₄⋊C₄	—	52	1	no	—
275	C₄⋊C₄	#6104	1	1	yes	characteristic
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	52	1	no	—
279	C₂²⋊C₄	—	52	2	no	—
280	C₂²⋊C₄	#6104	1	2	yes	—
281	C₄²	—	4	2	no	—
282	C₄²	—	4	1	no	—
283	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
284	C₁₆	—	13	2	no	—
285	C₁₆	—	26	2	no	—
286	C₁₆	—	26	1	no	—
287	C₁₃	#33920	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	52	1	no	—
289	C₂³	C₂×#6104	1	2	yes	—
290	Q₈	—	4	1	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	8	2	no	—
293	Q₈	—	104	2	no	—
294	D₄	—	2	2	no	—
295	D₄	—	52	1	no	—
296	D₄	—	52	1	no	—
297	D₄	—	104	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	52	1	no	—
304	C₂×C₄	—	52	2	no	—
305	C₂×C₄	—	104	4	no	—
306	C₂×C₄	—	104	2	no	—
307	C₂×C₄	C₂×#6104	1	2	yes	—
308	C₂×C₄	C₂×#6104	1	1	yes	characteristic
309	C₂×C₄	C₂×#6104	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	#41×D₁₃	1	1	yes	characteristic
314	C₈	#88257	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	104	1	no	—
319	C₂²	—	104	1	no	—
320	C₂²	—	104	1	no	—
321	C₂²	C₂²×#6104	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	1	no	—
327	C₄	—	104	1	no	—
328	C₄	—	104	2	no	—
329	C₄	#11718854	1	2	yes	—
330	C₄	#11719529	1	1	yes	characteristic
331	C₄	#11719543	1	1	yes	characteristic
332	C₂	—	4	2	no	—
333	C₂	—	208	1	no	—
334	C₂	#13127179	1	1	yes	characteristic
335	C₂	C₂×#11718854	1	1	yes	characteristic
336	C₂	#13127987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3342	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13127184	C₂
hypercenter	#3342	D₁₃
hypocenter	C₁₃	#33920
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3342

Complements

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

N	H ≅ G/N	type
2. #13117109	C₂	semidirect
3. #13117036	C₂	semidirect
4. #13117027	C₂	semidirect
5. #13116978	C₂	semidirect
6. #13116970	C₂	semidirect
7. #13116951	C₂	semidirect
8. C₄×#88068	C₂	semidirect
9. #13116668	C₂	semidirect
10. #13116625	C₂	semidirect
11. C₁₃×#3342	C₂	semidirect
119. #6090	#41	semidirect
120. #3342	D₁₃	semidirect
287. C₁₃	#33920	semidirect

Maximal subgroups

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

subgroup	index
#13127184	2
#13117109	2
#13117036	2
#13117027	2
#13116978	2
#13116970	2
#13116951	2
C₄×#88068	2
#13116668	2
#13116625	2
C₁₃×#3342	2
#33920	13