Subgroups — #15777760

Order 3328.

614 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	C₂²×#88038	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
4	#13112914	C₂	1	1	yes	maximal, characteristic, complemented
5	#13112913	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112913	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#6094	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88126	C₄	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88055	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂²×#6090	C₂²	1	1	yes	characteristic
17	C₂²×#6090	C₂²	1	1	yes	characteristic
18	C₂×#88043	C₄	1	1	yes	characteristic
19	C₂×#88038	—	2	2	no	—
20	C₂×#88038	—	2	2	no	—
21	C₂×#88038	C₄	1	4	yes	complemented
22	C₂×C₁₆×D₁₃	—	2	2	no	—
23	C₂×C₁₆×D₁₃	—	2	2	no	—
24	C₂×C₁₆×D₁₃	C₄	1	4	yes	complemented
25	#11718350	—	2	2	no	—
26	#11718350	—	2	2	no	—
27	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#6103	C₈	1	2	yes	—
32	C₂×#6101	C₈	1	2	yes	—
33	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂×#6094	—	2	2	no	—
37	C₂×#6094	—	2	2	no	—
38	C₂×#6094	C₈	1	4	yes	—
39	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	D₄	1	2	yes	—
42	C₂×C₈×D₁₃	D₄	1	2	yes	—
43	C₂×C₈×D₁₃	D₄	1	2	yes	—
44	C₂×C₈×D₁₃	Q₈	1	2	yes	—
45	#88126	—	2	2	no	—
46	#88126	—	2	2	no	—
47	C₂×C₁₃×C₁₆	—	2	1	no	—
48	C₂×C₁₃×C₁₆	—	2	1	no	—
49	C₂×C₁₃×C₁₆	—	2	1	no	—
50	C₂×C₁₃×C₁₆	—	2	1	no	—
51	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
52	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	C₄×C₈×C₁₃	—	2	1	no	—
55	#88055	—	2	2	no	—
56	C₈×Dic₁₃	—	2	1	no	—
57	C₈×Dic₁₃	—	2	1	no	—
58	C₂×#6090	—	2	1	no	—
59	C₂×#6090	—	2	1	no	—
60	C₂×#6090	—	2	1	no	—
61	C₂×#6090	—	2	1	no	—
62	C₂×#6090	C₂×C₄	1	2	yes	—
63	C₂×#6090	C₂×C₄	1	2	yes	—
64	#88043	—	2	2	no	—
65	#88038	—	2	8	no	—
66	C₁₆×D₁₃	—	2	8	no	—
67	#12545	—	13	1	no	maximal, Sylow
68	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
72	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₁₃	C₄²	1	2	yes	—
74	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
78	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₁₃	—	2	1	no	—
89	C₄²×C₁₃	—	2	1	no	—
90	#6103	—	2	4	no	—
91	#6101	—	2	4	no	—
92	C₄×Dic₁₃	—	2	1	no	—
93	C₄×Dic₁₃	—	2	1	no	—
94	C₂×#1055	—	2	1	no	—
95	C₂×#1055	—	2	1	no	—
96	C₂×#1055	C₄²	1	2	yes	—
97	C₂×#1055	C₂²⋊C₄	1	2	yes	—
98	C₂×#1055	C₄⋊C₄	1	2	yes	—
99	C₂×#1055	C₂×C₈	1	2	yes	—
100	#6094	—	2	8	no	—
101	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
102	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
103	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
104	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
105	C₁₃×C₁₆	—	2	2	no	—
106	C₁₃×C₁₆	—	2	2	no	—
107	#6090	—	2	2	no	—
108	#6090	—	2	2	no	—
109	C₂²×M₃₂	—	13	1	no	—
110	C₂³×C₁₆	—	13	1	no	—
111	C₂×#405	—	13	1	no	—
112	#1374	—	13	1	no	—
113	#1374	—	13	1	no	—
114	#1374	—	13	1	no	—
115	#1374	D₁₃	1	1	yes	characteristic, complemented
116	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
117	C₂²×D₁₃	—	2	2	no	—
118	C₂²×D₁₃	—	2	2	no	—
119	C₂²×D₁₃	C₄×C₈	1	2	yes	—
120	C₂²×D₁₃	C₂²⋊C₈	1	2	yes	—
121	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
131	C₂×Dic₁₃	—	2	2	no	—
132	C₂×Dic₁₃	—	2	2	no	—
133	C₂×Dic₁₃	—	2	2	no	—
134	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
135	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
136	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
137	C₄×D₁₃	#95	1	4	yes	—
138	C₄×D₁₃	#97	1	4	yes	—
139	C₄×D₁₃	C₂²⋊C₈	1	4	yes	—
140	C₄×D₁₃	C₄⋊C₈	1	4	yes	—
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	#1055	—	2	2	no	—
147	#1055	#95	1	4	yes	—
148	C₂²×M₁₆	—	13	1	no	—
149	C₂³×C₈	—	13	1	no	—
150	C₂×#117	—	13	1	no	—
151	C₂×M₃₂	—	13	4	no	—
152	C₂×M₃₂	—	26	2	no	—
153	C₂×M₃₂	—	26	2	no	—
154	C₂²×C₁₆	—	13	1	no	—
155	C₂²×C₁₆	—	13	1	no	—
156	C₂²×C₁₆	—	13	4	no	—
157	C₂²×C₁₆	—	26	2	no	—
158	C₂²×C₁₆	—	26	2	no	—
159	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
160	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
161	#405	—	26	2	no	—
162	#405	—	26	2	no	—
163	C₂×#97	—	13	1	no	—
164	C₂×#97	—	13	1	no	—
165	C₂×C₄×C₈	—	13	1	no	—
166	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
167	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
168	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
170	C₂²×C₁₃	#504	1	2	yes	—
171	C₂²×C₁₃	#504	1	2	yes	—
172	C₂×D₁₃	—	2	2	no	—
173	C₂×D₁₃	—	2	2	no	—
174	C₂×D₁₃	—	2	4	no	—
175	C₂×D₁₃	—	2	4	no	—
176	C₂×D₁₃	#336	1	4	yes	—
177	C₄×C₁₃	—	4	2	no	—
178	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
179	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
180	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
181	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
182	Dic₁₃	—	4	2	no	—
183	Dic₁₃	#336	1	4	yes	—
184	C₂³×C₄	—	13	1	no	—
185	C₂×M₁₆	—	13	4	no	—
186	C₂×M₁₆	—	26	2	no	—
187	C₂×M₁₆	—	26	2	no	—
188	C₂²×C₈	—	13	1	no	—
189	C₂²×C₈	—	13	2	no	—
190	C₂²×C₈	—	13	2	no	—
191	C₂²×C₈	—	13	2	no	—
192	C₂²×C₈	—	13	2	no	—
193	C₂²×C₈	—	13	2	no	—
194	C₂²×C₈	—	13	2	no	—
195	C₂²×C₈	—	13	1	no	—
196	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
198	#117	—	26	2	no	—
199	#117	—	26	2	no	—
200	C₂×C₄⋊C₄	—	13	2	no	—
201	C₂×C₂²⋊C₄	—	13	2	no	—
202	C₂×C₄²	—	13	1	no	—
203	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
204	M₃₂	—	26	8	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	13	2	no	—
210	C₂×C₁₆	—	13	2	no	—
211	C₂×C₁₆	—	26	8	no	—
212	C₂×C₁₆	—	26	1	no	—
213	C₂×C₁₆	—	26	1	no	—
214	C₂×C₁₆	—	26	1	no	—
215	C₂×C₁₆	—	26	1	no	—
216	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
217	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
218	#97	—	26	2	no	—
219	#97	—	26	2	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	26	1	no	—
223	C₄×C₈	—	26	1	no	—
224	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
225	C₂×C₁₃	#2108	1	1	yes	characteristic
226	C₂×C₁₃	#2108	1	1	yes	characteristic
227	C₂×C₁₃	#2113	1	2	yes	—
228	C₂×C₁₃	#2151	1	2	yes	—
229	D₁₃	—	2	4	no	—
230	C₂⁴	—	13	1	no	—
231	C₂²×C₄	—	13	1	no	—
232	C₂²×C₄	—	13	2	no	—
233	C₂²×C₄	—	13	2	no	—
234	C₂²×C₄	—	13	2	no	—
235	C₂²×C₄	—	13	2	no	—
236	C₂²×C₄	—	13	2	no	—
237	C₂²×C₄	—	13	2	no	—
238	C₂²×C₄	—	13	2	no	—
239	C₂²×C₄	C₈×D₁₃	1	2	yes	—
240	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
241	M₁₆	—	26	8	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	13	2	no	—
245	C₂×C₈	—	13	2	no	—
246	C₂×C₈	—	13	2	no	—
247	C₂×C₈	—	13	4	no	—
248	C₂×C₈	—	13	4	no	—
249	C₂×C₈	—	13	4	no	—
250	C₂×C₈	—	13	4	no	—
251	C₂×C₈	—	13	2	no	—
252	C₂×C₈	—	26	1	no	—
253	C₂×C₈	—	26	1	no	—
254	C₂×C₈	C₈×D₁₃	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
257	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
258	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
259	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
260	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
261	C₄⋊C₄	—	26	4	no	—
262	C₂²⋊C₄	—	26	4	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	26	1	no	—
266	C₄²	—	26	1	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	26	2	no	—
270	C₁₆	—	26	2	no	—
271	C₁₃	#12545	1	1	yes	characteristic, complemented, Sylow
272	C₂³	—	13	2	no	—
273	C₂³	—	13	2	no	—
274	C₂³	—	13	2	no	—
275	C₂³	—	26	2	no	—
276	C₂³	—	26	2	no	—
277	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	13	2	no	—
282	C₂×C₄	—	13	2	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	13	4	no	—
285	C₂×C₄	—	13	4	no	—
286	C₂×C₄	—	13	4	no	—
287	C₂×C₄	—	13	4	no	—
288	C₂×C₄	—	26	2	no	—
289	C₂×C₄	—	26	2	no	—
290	C₂×C₄	—	26	2	no	—
291	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
293	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
295	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
296	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	13	4	no	—
299	C₈	—	26	2	no	—
300	C₈	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
301	C₈	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
302	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
303	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
304	C₂²	—	13	4	no	—
305	C₂²	—	26	2	no	—
306	C₂²	—	26	2	no	—
307	C₂²	—	26	4	no	—
308	C₂²	—	26	4	no	—
309	C₂²	C₄×C₈×D₁₃	1	1	yes	characteristic
310	C₂²	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
311	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
312	C₂²	#11718562	1	2	yes	—
313	C₂²	#11718568	1	2	yes	—
314	C₄	—	4	2	no	—
315	C₄	—	13	4	no	—
316	C₄	—	52	2	no	—
317	C₄	D₁₃×#95	1	1	yes	characteristic
318	C₄	D₁₃×#97	1	1	yes	characteristic
319	C₄	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
320	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
321	C₂	—	26	4	no	—
322	C₂	D₁₃×#336	1	1	yes	characteristic
323	C₂	#13116431	1	1	yes	characteristic
324	C₂	#13116434	1	1	yes	characteristic
325	C₂	#13116448	1	2	yes	—
326	C₂	#13116593	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₁₃×#1374	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1374	D₁₃
hypocenter	C₁₃	#12545
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
4. #13112914	C₂	semidirect
5. #13112913	C₂	semidirect
6. #13112913	C₂	semidirect
7. C₁₃×#1374	C₂	semidirect
21. C₂×#88038	C₄	semidirect
24. C₂×C₁₆×D₁₃	C₄	semidirect
115. #1374	D₁₃	semidirect
271. C₁₃	#12545	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88038	2
C₂²×C₁₆×D₁₃	2
C₂×#11718350	2
#13112914	2
#13112913	2
C₁₃×#1374	2
#12545	13