Subgroups — #16796492

Order 3328.

626 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131737	C₂	1	2	yes	maximal
2	#13130476	C₂	1	2	yes	maximal, complemented
3	#13130462	C₂	1	4	yes	maximal, complemented
4	#13130424	C₂	1	2	yes	maximal, complemented
5	#13130397	C₂	1	1	yes	maximal, characteristic
6	#13130179	C₂	1	2	yes	maximal
7	C₈×#6118	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₁₆×D₁₃	C₄	1	1	yes	characteristic
10	C₄×Q₈×D₁₃	C₂²	1	2	yes	—
11	#11719158	C₂²	1	4	yes	—
12	#11719078	—	2	4	no	—
13	#11719074	C₂²	1	4	yes	—
14	#11719055	—	2	2	no	—
15	#11719054	C₂²	1	2	yes	—
16	#11719051	C₂²	1	1	yes	characteristic
17	#11719049	—	2	2	no	—
18	C₄×#6118	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
20	Q₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
21	C₄×#6107	C₄	1	2	yes	—
22	D₁₃×#107	C₄	1	1	yes	characteristic
23	#11718534	C₂²	1	1	yes	characteristic
24	D₁₃×#103	C₄	1	2	yes	—
25	#11718451	C₂²	1	2	yes	—
26	C₄×Dic₅₂	C₄	1	1	yes	characteristic
27	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
28	Q₈⋊#1057	C₂²	1	4	yes	—
29	#11718254	C₂²	1	2	yes	—
30	#11718253	C₂²	1	2	yes	—
31	#1057⋊C₈	C₂²	1	2	yes	—
32	C₈×#1057	C₂²	1	2	yes	—
33	#11718244	C₂²	1	4	yes	—
34	C₂×Q₈×D₁₃	C₂×C₄	1	2	yes	—
35	C₂×#6118	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
37	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
38	Q₈×Dic₁₃	C₂³	1	2	yes	—
39	C₂×#6107	C₂×C₄	1	2	yes	—
40	Q₁₆×D₁₃	—	2	4	no	—
41	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄⋊C₄×D₁₃	—	2	2	no	—
45	C₄⋊C₄×D₁₃	C₂×C₄	1	2	yes	—
46	#88141	—	2	2	no	—
47	#88141	C₂³	1	2	yes	—
48	C₄²×D₁₃	—	2	2	no	—
49	C₄²×D₁₃	C₂³	1	1	yes	characteristic
50	C₄×Dic₂₆	—	2	2	no	—
51	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
52	#88112	—	2	4	no	—
53	#88112	—	2	4	no	—
54	#88111	—	2	2	no	—
55	#88111	D₄	1	1	yes	characteristic
56	#88111	D₄	1	1	yes	characteristic
57	#88110	—	2	4	no	—
58	#88110	—	2	2	no	—
59	#88110	—	2	1	no	—
60	C₄⋊#1057	D₄	1	2	yes	—
61	C₄⋊#1057	C₂³	1	4	yes	—
62	C₄×#1057	—	2	4	no	—
63	C₄×#1057	C₂³	1	2	yes	—
64	C₄×#1057	C₂³	1	2	yes	—
65	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
66	C₁₃×#103	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
68	#88075	C₂×C₄	1	2	yes	—
69	#88058	C₂×C₄	1	1	yes	characteristic
70	#88056	C₂×C₄	1	2	yes	—
71	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
73	#88050	C₂×C₄	1	2	yes	—
74	#88047	C₂×C₄	1	2	yes	—
75	C₄×#1055	C₂×C₄	1	1	yes	characteristic
76	#18058	—	13	1	no	maximal, Sylow
77	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
78	Q₈×D₁₃	—	2	4	no	—
79	Q₈×D₁₃	—	4	2	no	—
80	C₂×C₄×D₁₃	—	2	2	no	—
81	C₂×C₄×D₁₃	—	2	2	no	—
82	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
85	C₂×Dic₂₆	—	2	2	no	—
86	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
87	C₂×#1057	—	2	2	no	—
88	C₂×#1057	—	2	4	no	—
89	C₂×#1057	C₂×D₄	1	2	yes	—
90	C₂×#1057	C₂×D₄	1	2	yes	—
91	#6118	—	2	4	no	—
92	C₁₃×Q₁₆	—	2	2	no	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	#6107	—	2	4	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
102	#6101	—	2	2	no	—
103	#6101	—	2	2	no	—
104	#6101	—	2	2	no	—
105	C₄×Dic₁₃	—	2	2	no	—
106	C₄×Dic₁₃	—	2	2	no	—
107	C₄×Dic₁₃	—	2	2	no	—
108	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄×Dic₁₃	#41	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
111	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
112	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
113	Dic₅₂	M₁₆	1	4	yes	complemented
114	C₈×D₁₃	—	2	2	no	—
115	C₈×D₁₃	—	4	1	no	—
116	#2984	—	13	2	no	—
117	C₂×C₄×Q₁₆	—	13	1	no	—
118	#1591	—	13	2	no	—
119	#1587	—	13	4	no	—
120	C₈×Q₁₆	—	13	2	no	—
121	#1568	—	13	1	no	—
122	#1491	—	13	2	no	—
123	C₈×M₁₆	—	13	1	no	—
124	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
125	Q₈×C₁₃	—	2	2	no	—
126	Q₈×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×Dic₁₃	—	2	2	no	—
133	C₂×Dic₁₃	—	2	2	no	—
134	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
135	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
137	C₄×D₁₃	—	2	2	no	—
138	C₄×D₁₃	—	2	1	no	—
139	C₄×D₁₃	—	2	1	no	—
140	C₄×D₁₃	—	2	2	no	—
141	C₄×D₁₃	—	4	4	no	—
142	C₄×D₁₃	—	4	4	no	—
143	Dic₂₆	—	2	2	no	—
144	Dic₂₆	—	4	4	no	—
145	Dic₂₆	C₂×M₁₆	1	4	yes	—
146	#1057	—	2	4	no	—
147	#1057	—	2	2	no	—
148	#1057	—	2	2	no	—
149	#1057	—	4	4	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
152	#1055	—	2	1	no	—
153	#1055	—	2	1	no	—
154	C₂²×Q₁₆	—	13	1	no	—
155	C₂×C₄×Q₈	—	13	2	no	—
156	#446	—	13	4	no	—
157	#446	—	26	4	no	—
158	C₈×Q₈	—	13	4	no	—
159	C₄×Q₁₆	—	13	1	no	—
160	C₄×Q₁₆	—	13	2	no	—
161	C₄×Q₁₆	—	13	1	no	—
162	C₄×Q₁₆	—	13	2	no	—
163	C₄×Q₁₆	—	13	1	no	—
164	C₄×Q₁₆	#261	1	1	yes	characteristic
165	#432	—	13	2	no	—
166	#432	—	26	2	no	—
167	#431	—	13	1	no	—
168	#431	—	26	2	no	—
169	C₂×#107	—	13	1	no	—
170	C₂×#103	—	13	2	no	—
171	C₄×M₁₆	—	13	1	no	—
172	C₂×C₄×C₈	—	13	1	no	—
173	#335	—	13	2	no	—
174	#334	—	13	2	no	—
175	Q₈⋊C₈	—	13	4	no	—
176	Q₈⋊C₈	—	13	4	no	—
177	#322	—	13	2	no	—
178	C₈²	—	13	2	no	—
179	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂×D₁₃	—	2	1	no	—
181	C₂×D₁₃	—	2	1	no	—
182	C₂×D₁₃	#443	1	2	yes	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	4	2	no	—
186	C₄×C₁₃	—	4	2	no	—
187	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
188	C₄×C₁₃	#568	1	1	yes	characteristic
189	Dic₁₃	—	2	4	no	—
190	Dic₁₃	—	2	1	no	—
191	Dic₁₃	—	2	1	no	—
192	Dic₁₃	—	4	2	no	—
193	Dic₁₃	#578	1	2	yes	—
194	C₂²×Q₈	—	13	2	no	—
195	C₂×Q₁₆	—	13	2	no	—
196	C₂×Q₁₆	—	13	1	no	—
197	C₂×Q₁₆	—	26	4	no	—
198	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
199	C₂×M₁₆	—	26	1	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₄×Q₈	—	13	2	no	—
203	C₄×Q₈	—	13	2	no	—
204	C₄×Q₈	—	13	2	no	—
205	C₄×Q₈	—	26	2	no	—
206	C₄×Q₈	—	26	2	no	—
207	C₄×Q₈	C₂×#261	1	2	yes	—
208	C₂×C₄⋊C₄	—	13	2	no	—
209	C₂×C₄⋊C₄	—	26	2	no	—
210	C₂×C₄²	—	13	1	no	—
211	C₂×C₄²	—	26	2	no	—
212	#107	—	13	1	no	—
213	#107	—	13	2	no	—
214	#107	C₂×#261	1	1	yes	characteristic
215	C₄⋊C₈	—	13	2	no	—
216	C₄⋊C₈	—	13	4	no	—
217	C₄⋊C₈	—	26	4	no	—
218	C₄⋊C₈	—	26	4	no	—
219	#103	—	13	2	no	—
220	#103	—	13	2	no	—
221	#103	—	13	2	no	—
222	#103	C₂×#261	1	2	yes	—
223	C₂²⋊C₈	—	13	1	no	—
224	C₂²⋊C₈	—	13	1	no	—
225	C₂²⋊C₈	—	26	2	no	—
226	#97	—	26	1	no	—
227	#97	—	26	2	no	—
228	#97	—	26	4	no	—
229	C₄×C₈	—	13	2	no	—
230	C₄×C₈	—	13	1	no	—
231	C₄×C₈	—	13	1	no	—
232	C₄×C₈	—	13	2	no	—
233	C₄×C₈	—	13	1	no	—
234	C₄×C₈	—	26	4	no	—
235	C₄×C₈	C₂×#261	1	1	yes	characteristic
236	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
237	C₂×C₁₃	#2950	1	1	yes	characteristic
238	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
239	D₁₃	—	2	2	no	—
240	C₂×Q₈	—	13	2	no	—
241	C₂×Q₈	—	26	2	no	—
242	C₂×Q₈	—	26	4	no	—
243	C₂×Q₈	—	52	2	no	—
244	C₂×Q₈	C₂²×#261	1	2	yes	—
245	C₂²×C₄	—	13	1	no	—
246	C₂²×C₄	—	13	1	no	—
247	C₂²×C₄	—	13	1	no	—
248	C₂²×C₄	—	26	2	no	—
249	C₂²×C₄	—	26	2	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	13	4	no	—
252	Q₁₆	—	26	4	no	—
253	M₁₆	—	26	4	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	1	no	—
257	C₂×C₈	—	13	1	no	—
258	C₂×C₈	—	26	2	no	—
259	C₂×C₈	—	26	2	no	—
260	C₂×C₈	—	26	4	no	—
261	C₂×C₈	—	52	1	no	—
262	C₂×C₈	C₂²×#261	1	1	yes	characteristic
263	C₂×C₈	C₂²×#261	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	13	2	no	—
266	C₄⋊C₄	—	26	2	no	—
267	C₄⋊C₄	—	26	2	no	—
268	C₄⋊C₄	—	26	2	no	—
269	C₄⋊C₄	—	26	2	no	—
270	C₄⋊C₄	C₂²×#261	1	2	yes	—
271	C₄²	—	2	2	no	—
272	C₄²	—	13	1	no	—
273	C₄²	—	13	1	no	—
274	C₄²	—	13	1	no	—
275	C₄²	—	26	2	no	—
276	C₄²	—	26	2	no	—
277	C₄²	—	26	2	no	—
278	C₄²	C₂²×#261	1	1	yes	characteristic
279	C₁₃	#18058	1	1	yes	characteristic, complemented, Sylow
280	C₂³	—	13	1	no	—
281	Q₈	—	2	2	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	13	4	no	—
284	Q₈	—	26	2	no	—
285	Q₈	—	52	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	13	1	no	—
289	C₂×C₄	—	13	1	no	—
290	C₂×C₄	—	13	1	no	—
291	C₂×C₄	—	26	2	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	26	1	no	—
295	C₂×C₄	—	26	1	no	—
296	C₂×C₄	—	26	2	no	—
297	C₂×C₄	—	52	4	no	—
298	C₂×C₄	—	52	4	no	—
299	C₂×C₄	D₄×#261	1	1	yes	characteristic
300	C₂×C₄	D₄×#261	1	1	yes	characteristic
301	C₂×C₄	C₂³×#261	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	26	2	no	—
304	C₈	—	26	2	no	—
305	C₈	—	26	4	no	—
306	C₈	—	26	1	no	—
307	C₈	—	26	1	no	—
308	C₈	—	52	4	no	—
309	C₈	D₁₃⋊M₁₆	1	2	yes	—
310	C₂²	—	13	2	no	—
311	C₂²	—	26	1	no	—
312	C₂²	—	26	1	no	—
313	C₂²	C₂×D₄×#261	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	13	2	no	—
319	C₄	—	26	1	no	—
320	C₄	—	26	4	no	—
321	C₄	—	26	1	no	—
322	C₄	—	52	2	no	—
323	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
324	C₄	#11719629	1	1	yes	characteristic
325	C₂	—	26	2	no	—
326	C₂	#13131577	1	1	yes	characteristic
327	C₂	#13131597	1	1	yes	characteristic
328	C₂	D₁₃⋊#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₄×C₁₃×Q₁₆	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₃	C₄×D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₄×Q₁₆	#261
hypocenter	C₁₃	#18058
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #13130476	C₂	semidirect
3. #13130462	C₂	semidirect
4. #13130424	C₂	semidirect
113. Dic₅₂	M₁₆	semidirect
279. C₁₃	#18058	semidirect

Maximal subgroups

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

subgroup	index
#13131737	2
#13130476	2
#13130462	2
#13130424	2
#13130397	2
#13130179	2
C₈×#6118	2
C₄×Q₁₆×D₁₃	2
#18058	13