Subgroups — #16018510

Order 3328.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125471	C₂	1	2	yes	maximal, complemented
2	#13117110	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117094	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117071	C₂	1	2	yes	maximal, complemented
5	#13116948	C₂	1	2	yes	maximal, complemented
6	#13116928	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116919	C₂	1	1	yes	maximal, characteristic
8	#13116700	C₂	1	2	yes	maximal, complemented
9	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116431	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719266	—	2	2	no	—
13	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×D₈	C₂²	1	2	yes	—
15	#11718818	C₂²	1	2	yes	—
16	#11718816	C₂²	1	2	yes	—
17	C₂×#88067	C₂²	1	2	yes	—
18	#11718679	—	2	2	no	—
19	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
20	C₂×#88038	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
22	#11718548	C₂²	1	2	yes	—
23	#11718535	C₂²	1	2	yes	—
24	#11718441	—	2	2	no	—
25	#11718426	—	2	2	no	—
26	C₄×Dic₅₂	C₂²	1	2	yes	—
27	#11718350	C₂²	1	1	yes	characteristic
28	C₁₃×#366	C₂²	1	1	yes	characteristic
29	C₁₃×#357	C₂²	1	2	yes	—
30	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
31	#11718154	C₂²	1	2	yes	—
32	#11718096	C₂²	1	1	yes	characteristic
33	#11718095	C₂²	1	2	yes	—
34	#11718094	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
36	#11718084	C₂²	1	2	yes	—
37	#11718081	C₂²	1	2	yes	—
38	#11718053	C₂²	1	1	yes	characteristic
39	C₂×C₁₃×D₈	C₂³	1	2	yes	—
40	C₄×D₄×C₁₃	—	2	2	no	—
41	#88192	—	2	2	no	—
42	#88190	—	2	2	no	—
43	C₂×#6105	—	2	2	no	—
44	#88183	—	4	2	no	—
45	#88178	—	4	2	no	—
46	C₂×Dic₅₂	C₂³	1	2	yes	—
47	C₂×#6094	D₄	1	1	yes	characteristic
48	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
49	#88151	—	2	2	no	—
50	#88142	—	2	2	no	—
51	#88139	—	4	2	no	—
52	#88133	—	4	2	no	—
53	#88126	D₄	1	1	yes	characteristic
54	C₄×Dic₂₆	—	2	2	no	—
55	C₁₃×D₁₆	—	2	2	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₁₃×#107	C₂³	1	2	yes	—
59	C₁₃×#102	—	2	2	no	—
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	#88072	—	2	2	no	—
62	#88067	—	2	4	no	—
63	#88058	C₂³	1	2	yes	—
64	#88056	—	2	2	no	—
65	#88055	D₄	1	1	yes	characteristic
66	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
67	C₂×#6090	C₂³	1	1	yes	characteristic
68	C₂×#6090	C₂³	1	1	yes	characteristic
69	#88050	—	2	2	no	—
70	#88047	—	2	2	no	—
71	#88043	D₄	1	1	yes	characteristic
72	Dic₁₀₄	—	2	2	no	—
73	#88038	—	2	2	no	—
74	C₁₆×D₁₃	—	2	2	no	—
75	#33739	—	13	1	no	maximal, Sylow
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	4	2	no	—
78	C₂×C₂²⋊D₁₃	—	4	2	no	—
79	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×Dic₂₆	—	2	2	no	—
81	C₂×Dic₂₆	—	4	2	no	—
82	C₁₃×D₈	—	2	2	no	—
83	C₁₃×D₈	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₂²⋊C₄	—	4	2	no	—
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₃	—	4	2	no	—
90	C₂²⋊Dic₁₃	—	4	2	no	—
91	C₂²⋊Dic₁₃	—	4	2	no	—
92	#6105	—	4	4	no	—
93	#6103	—	4	2	no	—
94	#6103	D₈	1	1	yes	characteristic
95	#6103	D₈	1	1	yes	characteristic
96	C₄⋊Dic₁₃	—	2	2	no	—
97	C₄⋊Dic₁₃	—	4	2	no	—
98	#6101	—	4	2	no	—
99	#6101	—	4	2	no	—
100	#6101	D₈	1	1	yes	characteristic
101	#6101	D₈	1	1	yes	characteristic
102	C₄×Dic₁₃	—	4	2	no	—
103	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×#1055	C₂×D₄	1	1	yes	characteristic
105	C₂×#1055	C₂×D₄	1	1	yes	characteristic
106	Dic₅₂	—	2	2	no	—
107	Dic₅₂	—	2	2	no	—
108	#6094	—	2	2	no	—
109	C₈×D₁₃	—	2	1	no	—
110	C₈×D₁₃	—	2	1	no	—
111	C₁₃×C₁₆	—	2	1	no	—
112	C₁₃×C₁₆	#41	1	2	yes	—
113	#6090	—	2	1	no	—
114	#6090	#41	1	2	yes	—
115	#3175	—	13	2	no	—
116	#2252	—	13	1	no	—
117	#2249	—	13	1	no	—
118	#2243	—	13	2	no	—
119	#2220	—	13	2	no	—
120	#2216	—	13	1	no	—
121	#2215	—	13	1	no	—
122	#2175	—	13	2	no	—
123	C₄×Q₃₂	—	13	1	no	—
124	C₄×D₁₆	D₁₃	1	1	yes	characteristic, complemented
125	#2108	—	13	1	no	—
126	C₂³×C₁₃	—	4	2	no	—
127	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
128	D₄×C₁₃	—	4	2	no	—
129	D₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	16	2	no	—
132	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₁₃	—	16	2	no	—
136	C₂×Dic₁₃	—	4	2	no	—
137	C₂×Dic₁₃	—	4	2	no	—
138	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
140	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
141	C₄×D₁₃	—	2	1	no	—
142	C₄×D₁₃	—	2	1	no	—
143	Dic₂₆	—	4	2	no	—
144	Dic₂₆	—	4	2	no	—
145	Dic₂₆	—	16	2	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
148	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
149	#1055	—	2	1	no	—
150	#1055	#135	1	2	yes	—
151	#542	—	26	2	no	—
152	C₂×Q₃₂	—	13	1	no	—
153	C₂×SD₃₂	—	13	2	no	—
154	C₂×D₁₆	C₂×D₁₃	1	1	yes	characteristic
155	C₂×M₃₂	—	13	1	no	—
156	C₂²×C₁₆	—	13	1	no	—
157	#500	—	13	2	no	—
158	#495	—	13	2	no	—
159	#489	—	26	2	no	—
160	#468	—	26	2	no	—
161	#467	—	13	2	no	—
162	#466	—	13	2	no	—
163	#440	—	26	2	no	—
164	C₄×Q₁₆	—	13	2	no	—
165	C₄×D₈	C₂×D₁₃	1	2	yes	—
166	#405	—	13	1	no	—
167	#366	—	13	1	no	—
168	#366	—	13	2	no	—
169	#366	C₂×D₁₃	1	1	yes	characteristic
170	#358	—	13	2	no	—
171	#358	—	13	2	no	—
172	#357	—	13	2	no	—
173	#357	C₂×D₁₃	1	2	yes	—
174	#346	—	13	1	no	—
175	#346	—	13	1	no	—
176	C₄×C₁₆	—	13	1	no	—
177	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
178	C₂²×C₁₃	—	8	2	no	—
179	C₂²×C₁₃	—	8	2	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
182	C₂×D₁₃	—	2	1	no	—
183	C₂×D₁₃	—	2	1	no	—
184	C₂×D₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	#532	1	1	yes	characteristic
189	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
190	Dic₁₃	—	2	1	no	—
191	Dic₁₃	—	2	1	no	—
192	Dic₁₃	—	2	1	no	—
193	Dic₁₃	—	8	2	no	—
194	Dic₁₃	—	8	2	no	—
195	C₂×Q₁₆	—	13	2	no	—
196	C₂×SD₁₆	—	26	2	no	—
197	C₂×D₈	C₂²×D₁₃	1	2	yes	—
198	C₂×M₁₆	—	13	1	no	—
199	C₂²×C₈	—	13	1	no	—
200	C₄⋊Q₈	—	26	2	no	—
201	#126	—	52	2	no	—
202	#124	—	26	2	no	—
203	#124	—	52	2	no	—
204	#123	—	52	2	no	—
205	C₂²⋊Q₈	—	26	2	no	—
206	C₂²⋊Q₈	—	52	2	no	—
207	#121	—	26	2	no	—
208	C₄×Q₈	—	26	2	no	—
209	C₄×D₄	—	2	2	no	—
210	#117	—	13	1	no	—
211	Q₃₂	—	26	2	no	—
212	SD₃₂	—	26	4	no	—
213	D₁₆	—	2	2	no	—
214	M₃₂	—	26	2	no	—
215	C₂×C₁₆	—	13	1	no	—
216	C₂×C₁₆	—	13	1	no	—
217	C₂×C₁₆	—	26	2	no	—
218	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
219	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
220	#107	—	13	2	no	—
221	#107	—	26	2	no	—
222	#107	C₂²×D₁₃	1	2	yes	—
223	#103	—	26	2	no	—
224	#103	—	26	2	no	—
225	#102	—	2	2	no	—
226	#102	—	26	2	no	—
227	#97	—	13	1	no	—
228	#97	—	13	1	no	—
229	C₄×C₈	—	13	1	no	—
230	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
231	C₂×C₁₃	—	16	2	no	—
232	C₂×C₁₃	#3169	1	1	yes	characteristic
233	C₂×C₁₃	#3415	1	1	yes	characteristic
234	C₂×C₁₃	#3417	1	1	yes	characteristic
235	D₁₃	—	4	1	no	—
236	C₂×Q₈	—	26	2	no	—
237	C₂×Q₈	—	52	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	52	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	13	1	no	—
242	Q₁₆	—	26	2	no	—
243	Q₁₆	—	26	2	no	—
244	SD₁₆	—	52	4	no	—
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	M₁₆	—	26	2	no	—
248	C₂×C₈	—	13	1	no	—
249	C₂×C₈	—	13	1	no	—
250	C₂×C₈	—	26	1	no	—
251	C₂×C₈	—	26	1	no	—
252	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	13	1	no	—
256	C₄⋊C₄	—	13	1	no	—
257	C₄⋊C₄	—	26	2	no	—
258	C₄⋊C₄	—	52	2	no	—
259	C₄⋊C₄	—	52	2	no	—
260	C₄⋊C₄	—	52	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	13	1	no	—
263	C₂²⋊C₄	—	13	1	no	—
264	C₂²⋊C₄	—	52	2	no	—
265	C₂²⋊C₄	—	52	2	no	—
266	C₂²⋊C₄	—	52	2	no	—
267	C₂²⋊C₄	—	52	2	no	—
268	C₄²	—	13	1	no	—
269	C₄²	—	52	2	no	—
270	C₄²	D₄×D₁₃	1	1	yes	characteristic
271	C₁₆	—	2	1	no	—
272	C₁₆	—	13	2	no	—
273	C₁₆	—	26	1	no	—
274	C₁₆	#6127	1	2	yes	—
275	C₁₃	#33739	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	4	2	no	—
277	C₂³	—	13	1	no	—
278	Q₈	—	52	2	no	—
279	Q₈	—	52	2	no	—
280	Q₈	—	208	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	208	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	13	1	no	—
286	C₂×C₄	—	13	1	no	—
287	C₂×C₄	—	13	1	no	—
288	C₂×C₄	—	16	2	no	—
289	C₂×C₄	—	26	1	no	—
290	C₂×C₄	—	26	1	no	—
291	C₂×C₄	—	52	2	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
294	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	13	2	no	—
298	C₈	—	26	1	no	—
299	C₈	C₂×#6127	1	1	yes	characteristic
300	C₈	#41×D₁₃	1	1	yes	characteristic
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	26	1	no	—
305	C₂²	—	26	1	no	—
306	C₂²	—	26	1	no	—
307	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	8	2	no	—
311	C₄	—	26	1	no	—
312	C₄	—	26	1	no	—
313	C₄	—	26	1	no	—
314	C₄	—	104	2	no	—
315	C₄	—	104	2	no	—
316	C₄	#11719253	1	1	yes	characteristic
317	C₄	D₁₃×#135	1	1	yes	characteristic
318	C₂	—	16	2	no	—
319	C₂	—	52	1	no	—
320	C₂	#13125412	1	1	yes	characteristic
321	C₂	#13127973	1	1	yes	characteristic
322	C₂	#13127986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×D₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13116919	C₂
hypercenter	C₄×D₁₆	D₁₃
hypocenter	C₁₃	#33739
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #13125471	C₂	semidirect
2. #13117110	C₂	semidirect
3. #13117094	C₂	semidirect
4. #13117071	C₂	semidirect
5. #13116948	C₂	semidirect
6. #13116928	C₂	semidirect
8. #13116700	C₂	semidirect
9. C₄×Dic₁₀₄	C₂	semidirect
10. #13116431	C₂	semidirect
11. C₄×C₁₃×D₁₆	C₂	semidirect
124. C₄×D₁₆	D₁₃	semidirect
275. C₁₃	#33739	semidirect

Maximal subgroups

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

subgroup	index
#13125471	2
#13117110	2
#13117094	2
#13117071	2
#13116948	2
#13116928	2
#13116919	2
#13116700	2
C₄×Dic₁₀₄	2
#13116431	2
C₄×C₁₃×D₁₆	2
#33739	13