Subgroups — #15834893

Order 3328.

652 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125415	C₂	1	1	yes	maximal, characteristic
2	#13123902	C₂	1	2	yes	maximal, complemented
3	#13113722	C₂	1	2	yes	maximal, complemented
4	#13113702	C₂	1	4	yes	maximal, complemented
5	C₈×D₁₀₄	C₂	1	2	yes	maximal, complemented
6	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113254	C₂	1	2	yes	maximal, complemented
8	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#11719182	C₂²	1	2	yes	—
10	C₁₃×#431	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718776	C₄	1	2	yes	—
14	#11718776	C₄	1	2	yes	—
15	#11718586	C₂²	1	2	yes	—
16	#11718499	—	2	2	no	—
17	#11718495	—	2	2	no	—
18	#11718388	—	2	4	no	—
19	#11718356	C₄	1	1	yes	characteristic
20	#11718355	C₄	1	2	yes	—
21	C₄×D₁₀₄	C₂²	1	2	yes	—
22	C₄×D₁₀₄	C₂²	1	2	yes	—
23	#11718349	C₂²	1	4	yes	—
24	C₈×D₅₂	C₂²	1	4	yes	—
25	#11718343	C₄	1	1	yes	characteristic
26	C₁₃×#322	C₂²	1	2	yes	—
27	C₈²×C₁₃	C₂²	1	2	yes	—
28	#11718051	C₂²	1	4	yes	—
29	#11718051	C₂²	1	4	yes	—
30	#11718050	C₂²	1	1	yes	characteristic
31	#11718050	C₂²	1	1	yes	characteristic
32	#11718049	C₂²	1	2	yes	—
33	C₂×#6127	—	2	2	no	—
34	C₂×C₁₃×M₁₆	—	2	1	no	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	#88184	—	2	2	no	—
39	#88184	C₂×C₄	1	4	yes	—
40	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
41	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
42	C₂×#6094	—	2	4	no	—
43	C₂×C₈×D₁₃	—	2	4	no	—
44	#88129	—	2	2	no	—
45	#88128	C₂×C₄	1	2	yes	—
46	C₄×D₅₂	—	2	2	no	—
47	C₄×D₅₂	C₂³	1	4	yes	—
48	#88123	C₂×C₄	1	2	yes	—
49	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
50	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
51	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
52	C₁₃×#97	—	2	1	no	—
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂³	1	2	yes	—
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	2	yes	—
59	C₂²⋊#1055	—	2	2	no	—
60	#88061	C₂×C₄	1	4	yes	—
61	#88061	C₂×C₄	1	4	yes	—
62	#88060	—	2	4	no	—
63	#88060	—	2	4	no	—
64	#88058	C₂×C₄	1	2	yes	—
65	#88058	C₂×C₄	1	2	yes	—
66	C₄⋊#1055	C₂³	1	2	yes	—
67	C₄⋊#1055	C₂³	1	2	yes	—
68	C₄×#1055	—	2	2	no	—
69	#18084	—	13	1	no	maximal, Sylow
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
73	C₂×C₂²⋊D₁₃	—	2	4	no	—
74	#6127	—	8	2	no	—
75	#6127	—	8	2	no	—
76	C₂×D₅₂	—	2	2	no	—
77	C₂×D₅₂	—	2	2	no	—
78	C₂×D₅₂	C₂²×C₄	1	4	yes	—
79	C₂×C₄×D₁₃	—	2	4	no	—
80	C₂×Dic₂₆	—	2	2	no	—
81	C₁₃×M₁₆	—	2	4	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	4	1	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃	#41	1	1	yes	characteristic
94	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
95	#6103	—	2	4	no	—
96	#6103	—	2	4	no	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	2	2	no	—
101	D₁₀₄	—	2	4	no	—
102	D₁₀₄	—	4	2	no	—
103	#6094	—	8	4	no	—
104	C₈×D₁₃	—	8	4	no	—
105	#3169	—	13	1	no	—
106	#2979	—	13	2	no	—
107	#1589	—	13	2	no	—
108	#1586	—	13	4	no	—
109	C₈×D₈	—	13	2	no	—
110	#1564	—	13	1	no	—
111	#1484	—	13	2	no	—
112	C₈×M₁₆	D₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂²×D₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂²⋊D₁₃	—	8	4	no	—
127	C₂×Dic₁₃	—	2	4	no	—
128	D₅₂	—	2	4	no	—
129	D₅₂	—	2	4	no	—
130	D₅₂	—	4	4	no	—
131	D₅₂	—	8	2	no	—
132	C₄×D₁₃	—	4	4	no	—
133	C₄×D₁₃	—	4	4	no	—
134	Dic₂₆	—	8	2	no	—
135	C₈×C₁₃	—	2	4	no	—
136	C₈×C₁₃	—	2	1	no	—
137	C₈×C₁₃	—	2	1	no	—
138	C₈×C₁₃	—	2	1	no	—
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	—	2	2	no	—
141	C₈×C₁₃	#131	1	2	yes	—
142	#1055	—	4	2	no	—
143	#1055	—	4	2	no	—
144	#532	—	13	2	no	—
145	#500	—	13	1	no	—
146	#493	—	13	1	no	—
147	#486	—	13	2	no	—
148	#466	—	13	2	no	—
149	#466	—	13	2	no	—
150	C₄×D₈	—	13	2	no	—
151	C₄×D₈	—	13	2	no	—
152	#436	—	13	4	no	—
153	#436	—	26	2	no	—
154	C₈×D₄	—	13	4	no	—
155	C₈×D₄	—	26	2	no	—
156	#431	—	13	2	no	—
157	#431	C₂×D₁₃	1	1	yes	characteristic
158	#408	—	26	4	no	—
159	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
161	#335	—	13	1	no	—
162	#335	—	13	1	no	—
163	#334	—	13	2	no	—
164	D₄⋊C₈	—	13	4	no	—
165	D₄⋊C₈	—	13	4	no	—
166	#322	C₂×D₁₃	1	2	yes	—
167	C₈²	C₂×D₁₃	1	2	yes	—
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×D₁₃	—	4	4	no	—
173	C₂×D₁₃	—	4	4	no	—
174	C₂×D₁₃	—	4	4	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	#443	1	2	yes	—
179	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
180	C₄×C₁₃	#568	1	1	yes	characteristic
181	C₄×C₁₃	#576	1	1	yes	characteristic
182	C₄×C₁₃	#576	1	1	yes	characteristic
183	Dic₁₃	—	4	4	no	—
184	C₂×#41	—	26	2	no	—
185	C₂×D₈	—	13	2	no	—
186	C₂×D₈	—	13	2	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	26	4	no	—
189	C₂²×C₈	—	26	4	no	—
190	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
192	C₄⋊Q₈	—	13	2	no	—
193	#127	—	13	2	no	—
194	#124	—	26	2	no	—
195	#121	—	13	4	no	—
196	#121	—	26	2	no	—
197	C₄×D₄	—	13	4	no	—
198	C₄×D₄	—	26	2	no	—
199	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
200	#107	—	13	2	no	—
201	#107	—	13	2	no	—
202	C₄⋊C₈	—	13	2	no	—
203	C₄⋊C₈	—	13	2	no	—
204	C₄⋊C₈	D₅₂	1	1	yes	characteristic
205	C₄⋊C₈	D₅₂	1	1	yes	characteristic
206	#102	—	13	4	no	—
207	#102	—	13	4	no	—
208	C₂²⋊C₈	—	26	2	no	—
209	C₂²⋊C₈	—	26	4	no	—
210	C₂²⋊C₈	—	26	4	no	—
211	C₂²⋊C₈	D₅₂	1	2	yes	—
212	#97	—	2	1	no	—
213	C₄×C₈	—	26	2	no	—
214	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
215	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₁₃	1	2	yes	—
217	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₁₃	1	2	yes	—
220	C₂×C₁₃	—	4	1	no	—
221	C₂×C₁₃	#2935	1	1	yes	characteristic
222	C₂×C₁₃	#2948	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
224	D₁₃	—	8	4	no	—
225	#41	—	104	2	no	—
226	#41	—	104	2	no	—
227	C₂×Q₈	—	26	2	no	—
228	C₂×D₄	—	13	4	no	—
229	C₂×D₄	—	26	4	no	—
230	C₂×D₄	—	26	2	no	—
231	C₂×D₄	—	26	2	no	—
232	C₂²×C₄	—	26	4	no	—
233	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
235	C₂²×C₄	#6127	1	1	yes	characteristic
236	D₈	—	26	4	no	—
237	D₈	—	52	2	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	104	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	26	2	no	—
244	C₂×C₈	—	26	2	no	—
245	C₂×C₈	—	104	4	no	—
246	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×D₅₂	1	2	yes	—
251	C₂×C₈	C₂×D₅₂	1	2	yes	—
252	C₄⋊C₄	—	13	4	no	—
253	C₄⋊C₄	—	26	2	no	—
254	C₂²⋊C₄	—	26	4	no	—
255	C₂²⋊C₄	—	26	4	no	—
256	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
257	C₄²	C₂×D₅₂	1	1	yes	characteristic
258	C₄²	#6127	1	1	yes	characteristic
259	C₄²	C₂³×D₁₃	1	1	yes	characteristic
260	C₁₃	#18084	1	1	yes	characteristic, complemented, Sylow
261	C₂³	—	26	4	no	—
262	C₂³	C₄×D₅₂	1	1	yes	characteristic
263	Q₈	—	104	2	no	—
264	D₄	—	26	4	no	—
265	D₄	—	26	4	no	—
266	D₄	—	52	4	no	—
267	D₄	—	104	2	no	—
268	D₄	—	104	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	26	4	no	—
275	C₂×C₄	—	52	4	no	—
276	C₂×C₄	—	52	4	no	—
277	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
278	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
279	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
282	C₂×C₄	C₂×#6127	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	52	2	no	—
290	C₈	—	52	2	no	—
291	C₈	#88161	1	2	yes	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	52	4	no	—
296	C₂²	—	52	4	no	—
297	C₂²	—	52	4	no	—
298	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	52	4	no	—
303	C₄	#11718361	1	2	yes	—
304	C₄	C₂×#88161	1	1	yes	characteristic
305	C₄	#11719450	1	1	yes	characteristic
306	C₄	#11719457	1	1	yes	characteristic
307	C₄	#11719457	1	1	yes	characteristic
308	C₂	—	4	1	no	—
309	C₂	—	104	4	no	—
310	C₂	#13123582	1	1	yes	characteristic
311	C₂	#13123689	1	1	yes	characteristic
312	C₂	C₂×#11718361	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#13125415	C₂
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#18084
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13123902	C₂	semidirect
3. #13113722	C₂	semidirect
4. #13113702	C₂	semidirect
5. C₈×D₁₀₄	C₂	semidirect
6. #13113584	C₂	semidirect
7. #13113254	C₂	semidirect
8. C₈×C₁₃×M₁₆	C₂	semidirect
112. C₈×M₁₆	D₁₃	semidirect
260. C₁₃	#18084	semidirect

Maximal subgroups

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

subgroup	index
#13125415	2
#13123902	2
#13113722	2
#13113702	2
C₈×D₁₀₄	2
#13113584	2
#13113254	2
C₈×C₁₃×M₁₆	2
#18084	13