Subgroups — #16781852

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130691	C₂	1	2	yes	maximal
2	C₂×#11718255	C₂	1	2	yes	maximal, complemented
3	#13129666	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
6	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88111	—	2	2	no	—
9	C₂×C₄⋊#1057	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₄	1	1	yes	characteristic
14	C₄×C₈×D₁₃	—	2	4	no	—
15	#11718290	—	2	2	no	—
16	#11718255	—	2	4	no	—
17	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
20	C₂²×#1057	—	2	4	no	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₁₃	—	2	2	no	—
30	C₂×C₈×D₁₃	—	2	2	no	—
31	C₂×C₈×D₁₃	—	2	2	no	—
32	C₂×C₈×D₁₃	—	2	2	no	—
33	C₂×C₈×D₁₃	—	2	2	no	—
34	C₂×C₈×D₁₃	C₈	1	4	yes	complemented
35	C₄²×D₁₃	—	2	2	no	—
36	C₄²×D₁₃	—	2	2	no	—
37	#88111	—	4	4	no	—
38	C₄⋊#1057	—	2	4	no	—
39	C₄⋊#1057	—	2	4	no	—
40	C₄×C₈×C₁₃	—	2	2	no	—
41	C₈×Dic₁₃	—	2	2	no	—
42	C₈×Dic₁₃	D₄	1	4	yes	—
43	C₄×#1055	—	2	2	no	—
44	#8595	—	13	1	no	maximal, Sylow
45	C₂³×D₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×D₁₃	—	2	2	no	—
53	C₂×C₄×D₁₃	—	2	2	no	—
54	C₂×C₄×D₁₃	—	2	1	no	—
55	C₂×C₄×D₁₃	—	2	1	no	—
56	C₂×C₄×D₁₃	—	2	1	no	—
57	C₂×C₄×D₁₃	—	2	1	no	—
58	C₂×C₄×D₁₃	—	2	2	no	—
59	C₂×C₄×D₁₃	—	2	2	no	—
60	C₂×C₄×D₁₃	—	2	2	no	—
61	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
62	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
63	C₂×C₄×D₁₃	SD₁₆	1	4	yes	—
64	C₂×#1057	—	4	4	no	—
65	C₂×#1057	—	4	4	no	—
66	C₂×#1057	—	4	4	no	—
67	C₂×C₈×C₁₃	—	2	1	no	—
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
73	C₄²×C₁₃	—	2	1	no	—
74	C₄²×C₁₃	—	2	1	no	—
75	C₄×Dic₁₃	—	2	2	no	—
76	C₄×Dic₁₃	—	2	2	no	—
77	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
78	C₄×Dic₁₃	#41	1	2	yes	—
79	C₂×#1055	—	2	1	no	—
80	C₂×#1055	—	2	1	no	—
81	C₂×#1055	C₂²⋊C₄	1	2	yes	—
82	C₂×#1055	C₂×C₈	1	2	yes	—
83	C₂×#1055	M₁₆	1	2	yes	—
84	C₂×#1055	M₁₆	1	2	yes	—
85	C₈×D₁₃	—	2	8	no	—
86	C₈×D₁₃	—	2	8	no	—
87	C₂²×C₄×C₈	—	13	1	no	—
88	#1797	—	13	2	no	—
89	C₂×#334	—	13	2	no	—
90	#1277	—	13	2	no	—
91	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×D₁₃	—	2	1	no	—
93	C₂²×D₁₃	—	2	1	no	—
94	C₂²×D₁₃	—	2	1	no	—
95	C₂²×D₁₃	—	2	1	no	—
96	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
97	C₂²×D₁₃	#106	1	2	yes	—
98	C₂²×D₁₃	#108	1	2	yes	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	#106	1	2	yes	—
104	C₂×C₄×C₁₃	#108	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂×Dic₁₃	—	2	1	no	—
112	C₂×Dic₁₃	—	2	1	no	—
113	C₂×Dic₁₃	—	2	1	no	—
114	C₂×Dic₁₃	—	2	1	no	—
115	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
116	C₂×Dic₁₃	#121	1	2	yes	—
117	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×Dic₁₃	C₂×SD₁₆	1	2	yes	—
119	C₂×Dic₁₃	#135	1	2	yes	—
120	C₄×D₁₃	—	2	8	no	—
121	C₄×D₁₃	—	2	2	no	—
122	C₄×D₁₃	—	2	2	no	—
123	C₄×D₁₃	—	2	2	no	—
124	C₄×D₁₃	—	2	2	no	—
125	C₄×D₁₃	—	2	8	no	—
126	#1057	—	8	4	no	—
127	C₈×C₁₃	—	2	2	no	—
128	C₈×C₁₃	—	2	2	no	—
129	#1055	—	2	2	no	—
130	#1055	C₂²⋊C₈	1	4	yes	complemented
131	C₂³×C₈	—	13	1	no	—
132	C₂³×C₈	—	13	1	no	—
133	C₂²×C₄²	—	13	1	no	—
134	C₂×C₄⋊C₈	—	13	4	no	—
135	C₂×C₂²⋊C₈	—	26	2	no	—
136	C₂×C₄×C₈	—	13	1	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	C₂×C₄×C₈	—	13	1	no	—
139	C₂×C₄×C₈	—	26	4	no	—
140	C₂×C₄×C₈	#261	1	1	yes	characteristic
141	#336	—	26	2	no	—
142	#334	—	26	4	no	—
143	C₂²×C₁₃	#380	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
145	C₂²×C₁₃	#423	1	1	yes	characteristic
146	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
147	C₂²×C₁₃	#427	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
150	C₂×D₁₃	—	2	2	no	—
151	C₂×D₁₃	—	2	2	no	—
152	C₂×D₁₃	—	2	2	no	—
153	C₂×D₁₃	—	2	2	no	—
154	C₂×D₁₃	—	2	2	no	—
155	C₂×D₁₃	—	2	2	no	—
156	C₂×D₁₃	#334	1	4	yes	complemented
157	C₄×C₁₃	—	2	2	no	—
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
160	C₄×C₁₃	#407	1	1	yes	characteristic
161	C₄×C₁₃	#431	1	1	yes	characteristic
162	C₄×C₁₃	#432	1	1	yes	characteristic
163	Dic₁₃	—	2	2	no	—
164	Dic₁₃	—	2	2	no	—
165	Dic₁₃	#465	1	4	yes	—
166	C₂³×C₄	—	13	1	no	—
167	C₂³×C₄	—	13	1	no	—
168	C₂³×C₄	—	13	1	no	—
169	C₂²×C₈	—	13	1	no	—
170	C₂²×C₈	—	13	1	no	—
171	C₂²×C₈	—	13	4	no	—
172	C₂²×C₈	—	26	2	no	—
173	C₂²×C₈	—	26	2	no	—
174	C₂²×C₈	—	26	2	no	—
175	C₂²×C₈	—	26	2	no	—
176	C₂²×C₈	—	26	2	no	—
177	C₂²×C₈	—	26	4	no	—
178	C₂²×C₈	C₂×#261	1	1	yes	characteristic
179	C₂²×C₈	C₂×#261	1	1	yes	characteristic
180	C₂×C₄²	—	13	1	no	—
181	C₂×C₄²	—	13	1	no	—
182	C₂×C₄²	—	13	1	no	—
183	C₂×C₄²	—	26	2	no	—
184	C₂×C₄²	—	26	2	no	—
185	C₂×C₄²	C₂×#261	1	1	yes	characteristic
186	C₄⋊C₈	—	26	4	no	—
187	C₄⋊C₈	—	26	4	no	—
188	C₂²⋊C₈	—	52	4	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	13	4	no	—
191	C₄×C₈	—	26	2	no	—
192	C₄×C₈	—	26	2	no	—
193	C₂×C₁₃	C₂×#334	1	1	yes	characteristic
194	C₂×C₁₃	#1564	1	1	yes	characteristic
195	C₂×C₁₃	#1566	1	1	yes	characteristic
196	C₂×C₁₃	#1568	1	1	yes	characteristic
197	C₂×C₁₃	#1797	1	1	yes	characteristic
198	C₂×C₁₃	#1807	1	1	yes	characteristic
199	C₂×C₁₃	#1810	1	1	yes	characteristic
200	D₁₃	—	2	4	no	—
201	C₂⁴	—	13	1	no	—
202	C₂²×C₄	—	13	1	no	—
203	C₂²×C₄	—	13	1	no	—
204	C₂²×C₄	—	13	1	no	—
205	C₂²×C₄	—	13	4	no	—
206	C₂²×C₄	—	13	2	no	—
207	C₂²×C₄	—	13	2	no	—
208	C₂²×C₄	—	26	2	no	—
209	C₂²×C₄	—	26	2	no	—
210	C₂²×C₄	—	26	2	no	—
211	C₂²×C₄	—	26	2	no	—
212	C₂²×C₄	—	26	2	no	—
213	C₂²×C₄	—	26	1	no	—
214	C₂²×C₄	—	26	1	no	—
215	C₂²×C₄	—	26	1	no	—
216	C₂²×C₄	—	26	1	no	—
217	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
218	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
219	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	13	2	no	—
225	C₂×C₈	—	13	2	no	—
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	—	13	2	no	—
228	C₂×C₈	—	26	8	no	—
229	C₂×C₈	—	26	8	no	—
230	C₂×C₈	—	26	1	no	—
231	C₂×C₈	—	26	1	no	—
232	C₂×C₈	—	52	4	no	—
233	C₂×C₈	—	52	4	no	—
234	C₂×C₈	—	52	4	no	—
235	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
236	C₂×C₈	C₂²⋊#261	1	2	yes	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	13	2	no	—
240	C₄²	—	13	2	no	—
241	C₄²	—	26	2	no	—
242	C₄²	—	26	2	no	—
243	C₁₃	#8595	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	13	2	no	—
245	C₂³	—	13	2	no	—
246	C₂³	—	13	2	no	—
247	C₂³	—	26	1	no	—
248	C₂³	—	26	1	no	—
249	C₂³	—	26	1	no	—
250	C₂³	—	26	1	no	—
251	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	13	2	no	—
257	C₂×C₄	—	13	2	no	—
258	C₂×C₄	—	13	2	no	—
259	C₂×C₄	—	13	2	no	—
260	C₂×C₄	—	13	2	no	—
261	C₂×C₄	—	26	8	no	—
262	C₂×C₄	—	26	8	no	—
263	C₂×C₄	—	26	1	no	—
264	C₂×C₄	—	26	1	no	—
265	C₂×C₄	—	26	1	no	—
266	C₂×C₄	—	26	1	no	—
267	C₂×C₄	—	26	2	no	—
268	C₂×C₄	—	26	2	no	—
269	C₂×C₄	—	26	2	no	—
270	C₂×C₄	—	26	2	no	—
271	C₂×C₄	#88101	1	2	yes	—
272	C₂×C₄	#88104	1	2	yes	—
273	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
274	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
275	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
276	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
277	C₂×C₄	#88237	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	13	4	no	—
282	C₈	—	26	2	no	—
283	C₈	—	104	4	no	—
284	C₂²	—	13	4	no	—
285	C₂²	—	26	2	no	—
286	C₂²	—	26	2	no	—
287	C₂²	—	26	2	no	—
288	C₂²	—	26	2	no	—
289	C₂²	—	26	2	no	—
290	C₂²	—	26	2	no	—
291	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
292	C₂²	#11719053	1	1	yes	characteristic
293	C₂²	C₂×#88101	1	1	yes	characteristic
294	C₂²	#11719092	1	1	yes	characteristic
295	C₂²	C₂×#88104	1	1	yes	characteristic
296	C₂²	D₁₃⋊#108	1	1	yes	characteristic
297	C₂²	#11719136	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	13	4	no	—
301	C₄	—	26	2	no	—
302	C₄	—	26	2	no	—
303	C₄	#11719049	1	1	yes	characteristic
304	C₄	#11719055	1	1	yes	characteristic
305	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
306	C₄	#11719126	1	1	yes	characteristic
307	C₂	—	26	4	no	—
308	C₂	D₁₃⋊#334	1	1	yes	characteristic
309	C₂	#13130386	1	1	yes	characteristic
310	C₂	#13130392	1	1	yes	characteristic
311	C₂	#13130399	1	1	yes	characteristic
312	C₂	#13130693	1	1	yes	characteristic
313	C₂	#13130712	1	1	yes	characteristic
314	C₂	#13130716	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃	C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#8595
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718255	C₂	semidirect
3. #13129666	C₂	semidirect
34. C₂×C₈×D₁₃	C₈	semidirect
130. #1055	C₂²⋊C₈	semidirect
156. C₂×D₁₃	#334	semidirect
243. C₁₃	#8595	semidirect

Maximal subgroups

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

subgroup	index
#13130691	2
C₂×#11718255	2
#13129666	2
C₂×C₄×C₈×D₁₃	2
#8595	13