Subgroups — #16018438

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125347	C₂	1	1	yes	maximal, characteristic
2	D₁₃×#500	C₂	1	1	yes	maximal, characteristic
3	#13117120	C₂	1	4	yes	maximal, complemented
4	#13117118	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117116	C₂	1	1	yes	maximal, characteristic, complemented
6	#13117109	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116621	C₂	1	4	yes	maximal
8	C₄×#88038	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719433	—	2	1	no	—
11	D₁₃×C₄⋊Q₈	—	2	1	no	—
12	C₁₃×#500	C₂²	1	1	yes	characteristic
13	C₁₃×#500	C₂²	1	1	yes	characteristic
14	#11718740	—	2	1	no	—
15	#11718739	—	2	1	no	—
16	C₂×#88038	C₂²	1	2	yes	—
17	#11718541	—	2	1	no	—
18	#11718541	C₂²	1	2	yes	—
19	D₁₃×#107	—	2	1	no	—
20	D₁₃×#107	C₂²	1	2	yes	—
21	#11718539	—	2	1	no	—
22	#11718539	C₂²	1	2	yes	—
23	#11718535	—	2	1	no	—
24	#11718535	C₂²	1	2	yes	—
25	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
26	#11718343	C₂²	1	1	yes	characteristic
27	#11718343	C₂²	1	1	yes	characteristic
28	C₁₃×#366	C₂²	1	4	yes	—
29	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
30	#11718097	C₂²	1	4	yes	—
31	#11718094	C₂²	1	2	yes	—
32	#11718082	C₂²	1	4	yes	—
33	#11718081	C₂²	1	4	yes	—
34	C₄×#6090	C₂²	1	1	yes	characteristic
35	C₂×Q₈×D₁₃	—	4	1	no	—
36	C₁₃×C₄⋊Q₈	—	2	1	no	—
37	C₁₃×C₄⋊Q₈	—	2	1	no	—
38	#88200	—	4	1	no	—
39	Q₈×Dic₁₃	—	4	1	no	—
40	#88198	—	4	1	no	—
41	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
43	#88151	—	4	1	no	—
44	#88146	—	2	2	no	—
45	C₄⋊C₄×D₁₃	—	2	2	no	—
46	C₄⋊C₄×D₁₃	—	4	1	no	—
47	#88144	—	2	2	no	—
48	#88142	—	2	2	no	—
49	#88142	—	4	1	no	—
50	#88141	—	4	1	no	—
51	C₄²×D₁₃	D₄	1	1	yes	characteristic
52	#88123	—	2	1	no	—
53	#88123	—	2	1	no	—
54	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
55	C₁₃×#107	—	2	1	no	—
56	C₁₃×#107	—	2	1	no	—
57	C₁₃×#107	C₂³	1	2	yes	—
58	C₁₃×#107	C₂³	1	2	yes	—
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	#88058	—	2	1	no	—
61	#88058	—	2	1	no	—
62	#88058	C₂³	1	2	yes	—
63	#88058	C₂³	1	2	yes	—
64	C₈×Dic₁₃	D₄	1	1	yes	characteristic
65	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
66	C₂×#6090	C₂³	1	2	yes	—
67	#88048	—	2	2	no	—
68	#88048	—	2	2	no	—
69	#88047	—	2	2	no	—
70	#88047	—	2	2	no	—
71	C₄×#1055	D₄	1	1	yes	characteristic
72	#88038	Q₈	1	8	yes	complemented
73	#33731	—	13	1	no	maximal, Sylow
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×Q₈×C₁₃	—	4	1	no	—
76	Q₈×D₁₃	—	8	4	no	—
77	C₂×C₄×D₁₃	—	4	2	no	—
78	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
79	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
80	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×Dic₂₆	—	4	1	no	—
82	C₂×Dic₂₆	—	4	1	no	—
83	C₂×Dic₂₆	—	4	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×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	4	1	no	—
89	C₁₃×C₄⋊C₄	—	4	1	no	—
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	#6103	—	4	2	no	—
92	C₄⋊Dic₁₃	—	2	2	no	—
93	C₄⋊Dic₁₃	—	2	2	no	—
94	C₄⋊Dic₁₃	—	4	1	no	—
95	C₄⋊Dic₁₃	—	4	1	no	—
96	C₄⋊Dic₁₃	—	4	2	no	—
97	#6101	—	4	2	no	—
98	#6101	—	4	2	no	—
99	#6101	—	4	2	no	—
100	C₄×Dic₁₃	—	4	2	no	—
101	C₄×Dic₁₃	D₈	1	1	yes	characteristic
102	C₄×Dic₁₃	D₈	1	1	yes	characteristic
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	C₈×D₁₃	Q₁₆	1	4	yes	—
107	C₈×D₁₃	C₂×Q₈	1	4	yes	—
108	C₁₃×C₁₆	C₂×Q₈	1	4	yes	complemented
109	#6090	C₂×Q₈	1	4	yes	complemented
110	#3160	—	13	1	no	—
111	C₂×#500	—	13	1	no	—
112	#2255	—	13	4	no	—
113	#2254	—	13	1	no	—
114	#2254	—	13	1	no	—
115	#2252	—	13	1	no	—
116	#2252	D₁₃	1	1	yes	characteristic, complemented
117	#2159	—	13	4	no	—
118	C₄×M₃₂	—	13	1	no	—
119	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₁₃	—	8	2	no	—
121	Q₈×C₁₃	—	8	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₁₃	—	4	2	no	—
128	C₂×Dic₁₃	—	4	2	no	—
129	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
132	C₄×D₁₃	—	2	2	no	—
133	C₄×D₁₃	—	2	2	no	—
134	C₄×D₁₃	—	8	4	no	—
135	C₄×D₁₃	C₄⋊Q₈	1	2	yes	—
136	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
137	Dic₂₆	—	8	4	no	—
138	Dic₂₆	—	8	2	no	—
139	Dic₂₆	—	8	2	no	—
140	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
141	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
142	#1055	C₄⋊Q₈	1	2	yes	—
143	#1055	C₂×Q₁₆	1	2	yes	—
144	#533	—	26	1	no	—
145	C₂×C₄⋊Q₈	—	26	1	no	—
146	C₂×M₃₂	—	13	2	no	—
147	#500	—	13	2	no	—
148	#500	—	13	1	no	—
149	#500	—	13	1	no	—
150	#500	—	26	1	no	—
151	#500	—	26	1	no	—
152	#500	C₂×D₁₃	1	1	yes	characteristic
153	#500	C₂×D₁₃	1	1	yes	characteristic
154	#499	—	13	2	no	—
155	#499	—	26	1	no	—
156	#498	—	26	1	no	—
157	#427	—	13	2	no	—
158	#427	—	26	1	no	—
159	C₂×#107	—	13	2	no	—
160	C₂×#107	—	26	1	no	—
161	C₂×C₄×C₈	—	13	1	no	—
162	#367	—	13	4	no	—
163	#367	—	13	4	no	—
164	#366	—	13	4	no	—
165	#366	C₂×D₁₃	1	4	yes	—
166	#346	—	13	2	no	—
167	C₄×C₁₆	—	13	1	no	—
168	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
169	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
170	C₂×D₁₃	—	2	1	no	—
171	C₂×D₁₃	—	2	1	no	—
172	C₂×D₁₃	#500	1	2	yes	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	—	8	2	no	—
175	C₄×C₁₃	#509	1	2	yes	—
176	C₄×C₁₃	#510	1	2	yes	—
177	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
178	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
179	Dic₁₃	—	2	1	no	—
180	Dic₁₃	—	2	1	no	—
181	Dic₁₃	—	8	2	no	—
182	Dic₁₃	—	8	2	no	—
183	Dic₁₃	#500	1	2	yes	—
184	C₂²×Q₈	—	52	1	no	—
185	C₂²×C₈	—	13	1	no	—
186	C₂²×C₈	—	13	1	no	—
187	C₄⋊Q₈	—	2	1	no	—
188	C₄⋊Q₈	—	2	1	no	—
189	C₄⋊Q₈	—	26	2	no	—
190	C₄⋊Q₈	—	26	1	no	—
191	C₄⋊Q₈	—	26	1	no	—
192	C₄⋊Q₈	—	52	1	no	—
193	C₄⋊Q₈	—	52	1	no	—
194	#125	—	26	2	no	—
195	C₂²⋊Q₈	—	52	1	no	—
196	C₂²⋊Q₈	—	52	1	no	—
197	C₄×Q₈	—	52	1	no	—
198	C₄×Q₈	—	52	1	no	—
199	#117	—	26	2	no	—
200	C₂×C₄⋊C₄	—	26	2	no	—
201	C₂×C₄⋊C₄	—	52	1	no	—
202	C₂×C₄²	—	13	1	no	—
203	M₃₂	—	13	8	no	—
204	C₂×C₁₆	—	13	2	no	—
205	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
206	#107	—	2	1	no	—
207	#107	—	2	1	no	—
208	#107	—	13	2	no	—
209	#107	—	13	2	no	—
210	#107	—	26	2	no	—
211	#107	—	26	2	no	—
212	#107	—	26	1	no	—
213	#107	—	26	1	no	—
214	#107	C₂²×D₁₃	1	2	yes	—
215	#107	C₂²×D₁₃	1	2	yes	—
216	#106	—	26	2	no	—
217	#106	—	26	2	no	—
218	C₄×C₈	—	13	1	no	—
219	C₄×C₈	—	13	1	no	—
220	C₄×C₈	—	13	1	no	—
221	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
222	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
224	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
225	D₁₃	—	2	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	52	2	no	—
229	C₂×Q₈	—	52	1	no	—
230	C₂×Q₈	—	52	1	no	—
231	C₂×Q₈	—	104	4	no	—
232	C₂²×C₄	—	13	1	no	—
233	C₂²×C₄	—	13	1	no	—
234	C₂²×C₄	—	13	1	no	—
235	C₂²×C₄	—	52	2	no	—
236	C₂×C₈	—	13	1	no	—
237	C₂×C₈	—	13	1	no	—
238	C₂×C₈	—	13	4	no	—
239	C₂×C₈	—	13	4	no	—
240	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
241	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	26	2	no	—
247	C₄⋊C₄	—	26	2	no	—
248	C₄⋊C₄	—	52	2	no	—
249	C₄⋊C₄	—	52	2	no	—
250	C₄⋊C₄	—	52	2	no	—
251	C₄⋊C₄	—	52	2	no	—
252	C₄⋊C₄	—	52	1	no	—
253	C₄⋊C₄	—	52	1	no	—
254	C₂²⋊C₄	—	52	2	no	—
255	C₄²	—	13	1	no	—
256	C₄²	—	13	1	no	—
257	C₄²	—	13	1	no	—
258	C₄²	—	52	2	no	—
259	C₄²	D₄×D₁₃	1	1	yes	characteristic
260	C₁₆	—	13	4	no	—
261	C₁₆	Q₈×D₁₃	1	4	yes	complemented
262	C₁₃	#33731	1	1	yes	characteristic, complemented, Sylow
263	C₂³	—	13	1	no	—
264	Q₈	—	8	2	no	—
265	Q₈	—	8	2	no	—
266	Q₈	—	104	4	no	—
267	Q₈	—	104	2	no	—
268	Q₈	—	104	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	13	1	no	—
272	C₂×C₄	—	13	2	no	—
273	C₂×C₄	—	13	2	no	—
274	C₂×C₄	—	13	1	no	—
275	C₂×C₄	—	13	1	no	—
276	C₂×C₄	—	26	2	no	—
277	C₂×C₄	—	26	2	no	—
278	C₂×C₄	—	52	2	no	—
279	C₂×C₄	—	52	2	no	—
280	C₂×C₄	—	104	4	no	—
281	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
282	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
283	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
284	C₈	—	13	2	no	—
285	C₈	—	13	2	no	—
286	C₈	Q₁₆×D₁₃	1	2	yes	—
287	C₈	C₂×Q₈×D₁₃	1	2	yes	—
288	C₂²	—	13	2	no	—
289	C₂²	—	26	1	no	—
290	C₂²	—	26	1	no	—
291	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	—	13	2	no	—
295	C₄	—	26	1	no	—
296	C₄	—	26	1	no	—
297	C₄	—	104	2	no	—
298	C₄	—	104	2	no	—
299	C₄	#11718572	1	2	yes	—
300	C₄	#11718579	1	2	yes	—
301	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
302	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
303	C₂	—	26	2	no	—
304	C₂	D₁₃×#500	1	1	yes	characteristic
305	C₂	C₂×#11718572	1	1	yes	characteristic
306	C₂	C₂×#11718579	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	C₂²×D₁₃	C₂×D₈
hypercenter	#2252	D₁₃
hypocenter	C₁₃	#33731
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
3. #13117120	C₂	semidirect
4. #13117118	C₂	semidirect
5. #13117116	C₂	semidirect
6. #13117109	C₂	semidirect
9. C₁₃×#2252	C₂	semidirect
72. #88038	Q₈	semidirect
108. C₁₃×C₁₆	C₂×Q₈	semidirect
109. #6090	C₂×Q₈	semidirect
116. #2252	D₁₃	semidirect
261. C₁₆	Q₈×D₁₃	semidirect
262. C₁₃	#33731	semidirect

Maximal subgroups

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

subgroup	index
#13125347	2
D₁₃×#500	2
#13117120	2
#13117118	2
#13117116	2
#13117109	2
#13116621	2
C₄×#88038	2
C₁₃×#2252	2
#33731	13