Subgroups — #16016034

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116643	C₂	1	4	yes	maximal, complemented
2	#13116639	C₂	1	2	yes	maximal, complemented
3	#13116635	C₂	1	2	yes	maximal, complemented
4	#13116592	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116450	C₂	1	2	yes	maximal
7	#13116433	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
10	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
12	C₁₃×#405	C₄	1	1	yes	characteristic
13	C₁₃×#405	C₄	1	2	yes	—
14	C₁₃×#405	C₂²	1	1	yes	characteristic
15	C₁₃×#405	C₂²	1	2	yes	—
16	C₄×C₁₃×M₁₆	C₄	1	2	yes	—
17	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
20	C₂×#88052	C₂²	1	2	yes	—
21	C₂²×#6090	C₂²	1	2	yes	—
22	C₂²⋊#6090	C₂²	1	4	yes	—
23	C₂²⋊#6090	C₂²	1	4	yes	—
24	C₄⋊#6090	C₂²	1	1	yes	characteristic
25	C₄⋊#6090	C₂²	1	1	yes	characteristic
26	C₄⋊#6090	C₂²	1	2	yes	—
27	C₄⋊#6090	C₂²	1	2	yes	—
28	C₄⋊#6090	C₂²	1	2	yes	—
29	#11718053	C₂²	1	4	yes	—
30	C₄×#6090	C₂²	1	4	yes	—
31	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
32	C₁₃×#131	—	2	4	no	—
33	C₁₃×#131	—	2	2	no	—
34	C₁₃×#131	—	2	2	no	—
35	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
44	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
45	C₄×D₄×C₁₃	C₈	1	2	yes	—
46	C₄×D₄×C₁₃	C₈	1	1	yes	characteristic
47	C₄×D₄×C₁₃	C₈	1	2	yes	—
48	C₄×D₄×C₁₃	C₈	1	1	yes	characteristic
49	C₁₃×#117	C₂×C₄	1	2	yes	—
50	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₁₃×#97	C₂×C₄	1	2	yes	—
54	C₁₃×#97	C₂×C₄	1	2	yes	—
55	C₁₃×#97	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	2	yes	—
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	#88052	—	2	4	no	—
64	C₂×#6090	—	2	4	no	—
65	C₂×#6090	C₂³	1	2	yes	—
66	C₂×#6090	C₂³	1	2	yes	—
67	C₂×#6090	C₂³	1	4	yes	—
68	#33423	—	13	1	no	maximal, Sylow
69	C₁₃×#41	—	2	2	no	—
70	C₁₃×#41	—	2	2	no	—
71	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
72	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
73	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₁₃×M₁₆	—	2	2	no	—
81	C₁₃×M₁₆	—	2	4	no	—
82	C₁₃×M₁₆	—	2	4	no	—
83	C₁₃×M₁₆	M₁₆	1	4	yes	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₃	#41	1	2	yes	—
94	C₂×C₈×C₁₃	#41	1	2	yes	—
95	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
97	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
98	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
100	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
101	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
102	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
103	C₄²×C₁₃	C₂×C₈	1	2	yes	—
104	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×C₈	1	2	yes	—
106	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
107	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	#6090	—	2	2	no	—
111	#6090	—	2	2	no	—
112	#6090	—	2	4	no	—
113	C₄×#131	D₁₃	1	1	yes	characteristic
114	#2164	—	13	4	no	—
115	#2163	—	13	2	no	—
116	#2162	—	13	2	no	—
117	#2150	—	13	1	no	—
118	C₂×C₄⋊C₁₆	—	13	1	no	—
119	#2113	—	13	2	no	—
120	#2108	—	13	2	no	—
121	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
122	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₁₃	—	2	2	no	—
124	D₄×C₁₃	—	2	4	no	—
125	D₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	#117	1	2	yes	—
133	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
142	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
143	C₈×C₁₃	—	2	2	no	—
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	#117	1	2	yes	—
147	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
148	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
149	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
150	C₂×#131	Dic₁₃	1	1	yes	characteristic
151	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
152	C₄×#41	Dic₁₃	1	1	yes	characteristic
153	C₂×M₃₂	—	13	2	no	—
154	C₂²×C₁₆	—	13	2	no	—
155	#405	Dic₁₃	1	1	yes	characteristic
156	#405	Dic₁₃	1	2	yes	—
157	#405	C₂×D₁₃	1	1	yes	characteristic
158	#405	C₂×D₁₃	1	2	yes	—
159	C₄×M₁₆	Dic₁₃	1	2	yes	—
160	C₄×M₁₆	Dic₁₃	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
163	C₄⋊C₁₆	—	13	1	no	—
164	C₄⋊C₁₆	—	13	1	no	—
165	C₄⋊C₁₆	—	13	2	no	—
166	C₄⋊C₁₆	—	13	2	no	—
167	C₄⋊C₁₆	—	13	2	no	—
168	C₂²⋊C₁₆	—	13	4	no	—
169	C₂²⋊C₁₆	—	13	4	no	—
170	#346	—	13	4	no	—
171	C₄×C₁₆	—	13	4	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	—	2	2	no	—
176	C₂²×C₁₃	—	2	2	no	—
177	C₂²×C₁₃	#431	1	2	yes	—
178	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	2	2	no	—
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	#431	1	2	yes	—
186	C₄×C₁₃	#504	1	2	yes	—
187	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
188	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
189	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
190	#131	—	2	4	no	—
191	#131	—	2	2	no	—
192	#131	—	2	2	no	—
193	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
194	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
195	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
196	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
197	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
198	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
199	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
200	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
201	C₄×Q₈	#1055	1	1	yes	characteristic
202	C₄×Q₈	#1055	1	1	yes	characteristic
203	C₄×D₄	#1055	1	2	yes	—
204	C₄×D₄	#1055	1	1	yes	characteristic
205	C₄×D₄	#1055	1	2	yes	—
206	C₄×D₄	#1055	1	1	yes	characteristic
207	#117	C₂×Dic₁₃	1	2	yes	—
208	#117	C₂×Dic₁₃	1	1	yes	characteristic
209	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
210	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
211	M₃₂	—	26	4	no	—
212	C₂×C₁₆	—	13	2	no	—
213	C₂×C₁₆	—	13	4	no	—
214	C₂×C₁₆	—	13	2	no	—
215	C₂×C₁₆	—	26	4	no	—
216	#97	C₂×Dic₁₃	1	2	yes	—
217	#97	C₂×Dic₁₃	1	2	yes	—
218	#97	C₂×Dic₁₃	1	2	yes	—
219	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
220	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
221	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
222	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₁₃	1	2	yes	—
224	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
226	C₂×C₁₃	—	2	2	no	—
227	C₂×C₁₃	—	4	2	no	—
228	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
229	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
230	C₂×C₁₃	#3407	1	1	yes	characteristic
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	C₂×Q₈	C₂×#1055	1	1	yes	characteristic
234	C₂×D₄	C₂×#1055	1	2	yes	—
235	C₂×D₄	C₂×#1055	1	1	yes	characteristic
236	C₂²×C₄	C₂×#1055	1	2	yes	—
237	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
238	C₂²×C₄	C₂×#1055	1	2	yes	—
239	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	#6099	1	4	yes	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	#6127	1	2	yes	—
251	C₂×C₈	#6127	1	2	yes	—
252	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
253	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
254	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
255	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
256	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
257	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
258	C₄⋊C₄	C₂×#1055	1	2	yes	—
259	C₄⋊C₄	C₂×#1055	1	2	yes	—
260	C₄⋊C₄	C₂×#1055	1	1	yes	characteristic
261	C₄⋊C₄	C₂×#1055	1	1	yes	characteristic
262	C₂²⋊C₄	C₂×#1055	1	2	yes	—
263	C₂²⋊C₄	C₂×#1055	1	2	yes	—
264	C₂²⋊C₄	C₂×#1055	1	2	yes	—
265	C₄²	C₂×#1055	1	2	yes	—
266	C₄²	C₂×#1055	1	1	yes	characteristic
267	C₄²	C₂×#1055	1	2	yes	—
268	C₄²	C₂×#1055	1	1	yes	characteristic
269	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
270	C₄²	C₂²×Dic₁₃	1	2	yes	—
271	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
272	C₁₆	—	26	2	no	—
273	C₁₆	—	26	2	no	—
274	C₁₆	—	26	4	no	—
275	C₁₃	#33423	1	1	yes	characteristic, complemented, Sylow
276	C₂³	C₂²×#1055	1	2	yes	—
277	C₂³	C₂²×#1055	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
288	C₂×C₄	C₂²×#1055	1	2	yes	—
289	C₂×C₄	C₂²×#1055	1	2	yes	—
290	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
291	C₂×C₄	C₂²×#1055	1	2	yes	—
292	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
293	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
294	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
295	C₂×C₄	C₂×#6099	1	2	yes	—
296	C₂×C₄	#88180	1	2	yes	—
297	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	C₂×#6099	1	4	yes	—
302	C₈	#88180	1	2	yes	—
303	C₈	C₂×#6127	1	1	yes	characteristic
304	C₈	C₂×#6127	1	1	yes	characteristic
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#11718586	1	2	yes	—
311	C₂²	C₂³×#1055	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	#11718586	1	2	yes	—
319	C₄	#11718801	1	2	yes	—
320	C₄	C₂²×#6099	1	1	yes	characteristic
321	C₄	C₂×#88180	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×#11718586	1	1	yes	characteristic
325	C₂	C₂×#11718801	1	1	yes	characteristic
326	C₂	#13127939	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₄×C₁₃×#131	C₂
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	#33423
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #13116643	C₂	semidirect
2. #13116639	C₂	semidirect
3. #13116635	C₂	semidirect
4. #13116592	C₂	semidirect
5. C₂×C₄⋊#6090	C₂	semidirect
7. #13116433	C₂	semidirect
275. C₁₃	#33423	semidirect

Maximal subgroups

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

subgroup	index
#13116643	2
#13116639	2
#13116635	2
#13116592	2
C₂×C₄⋊#6090	2
#13116450	2
#13116433	2
C₄×C₁₃×#131	2
#33423	13