Subgroups — #15770824

Order 3328.

670 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123400	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718125	C₂	1	2	yes	maximal
3	#13114575	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718122	C₂	1	2	yes	maximal
5	C₁₃×#2317	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₁₃×#389	—	2	2	no	—
12	C₁₃×#385	C₄	1	4	yes	—
13	C₁₃×#384	—	2	1	no	—
14	C₁₃×#382	—	2	1	no	—
15	#11718660	—	2	4	no	—
16	#11718585	—	2	2	no	—
17	#11718125	—	2	4	no	—
18	#11718122	—	2	1	no	—
19	#11718122	—	2	1	no	—
20	#11718122	—	2	4	no	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₁₃×#117	—	2	2	no	—
24	C₁₃×#117	—	2	1	no	—
25	C₁₃×#117	—	2	1	no	—
26	C₁₃×#117	—	2	2	no	—
27	C₁₃×#117	D₄	1	2	yes	—
28	C₁₃×#117	Q₈	1	2	yes	—
29	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
35	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
36	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
37	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	—	2	1	no	—
40	C₂×C₄²×C₁₃	D₄	1	2	yes	—
41	C₂×#6099	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	—	2	2	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	—	2	2	no	—
50	C₂×#6099	—	2	2	no	—
51	C₂×#6099	D₄	1	4	yes	—
52	C₂×#6099	Q₈	1	4	yes	—
53	C₂²×#1055	—	2	2	no	—
54	C₂²×#1055	—	2	2	no	—
55	C₁₃×#95	—	2	4	no	—
56	C₄⋊#1055	—	2	8	no	—
57	#11206	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	—	8	2	no	—
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	#41	1	2	yes	—
74	C₂²×C₄×C₁₃	#41	1	2	yes	—
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	4	1	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	4	1	no	—
82	C₁₃×C₄⋊C₄	—	8	4	no	—
83	C₁₃×C₂²⋊C₄	—	2	2	no	—
84	C₁₃×C₂²⋊C₄	—	2	4	no	—
85	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	#6099	—	2	4	no	—
91	#6099	—	2	4	no	—
92	#6099	—	2	4	no	—
93	#6099	—	2	4	no	—
94	#6099	—	4	2	no	—
95	#6099	—	4	2	no	—
96	#6099	—	4	2	no	—
97	#6099	—	4	2	no	—
98	C₂×#1055	—	2	4	no	—
99	C₂×#1055	—	2	4	no	—
100	C₂×#1055	—	4	2	no	—
101	C₂×#1055	—	4	2	no	—
102	C₂×#1055	—	4	2	no	—
103	C₂×#1055	—	4	2	no	—
104	#2905	—	13	1	no	—
105	#2317	D₁₃	1	1	yes	characteristic
106	C₂×#344	—	13	2	no	—
107	#1742	—	13	1	no	—
108	C₂×#343	—	13	2	no	—
109	C₂³×C₁₃	—	4	1	no	—
110	C₂³×C₁₃	—	4	1	no	—
111	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂³×C₁₃	#117	1	1	yes	characteristic
114	C₂³×C₁₃	C₄×D₄	1	2	yes	—
115	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	8	4	no	—
126	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#121	1	2	yes	—
128	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#121	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#123	1	2	yes	—
137	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#125	1	2	yes	—
140	C₂×C₄×C₁₃	#125	1	2	yes	—
141	C₂×C₄×C₁₃	#126	1	2	yes	—
142	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
143	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
144	#1055	—	4	4	no	—
145	#1055	—	4	4	no	—
146	C₂²×M₁₆	—	13	2	no	—
147	C₂²×M₁₆	—	13	2	no	—
148	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
149	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
150	C₂²×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
151	#424	—	26	4	no	—
152	#423	—	26	2	no	—
153	#389	—	2	2	no	—
154	#385	Dic₁₃	1	4	yes	—
155	#384	—	2	1	no	—
156	#382	—	2	1	no	—
157	#344	—	26	4	no	—
158	#343	—	26	4	no	—
159	#343	—	26	1	no	—
160	#343	—	26	1	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	2	1	no	—
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	—	4	1	no	—
166	C₂²×C₁₃	—	4	1	no	—
167	C₂²×C₁₃	—	4	2	no	—
168	C₂²×C₁₃	#380	1	1	yes	characteristic
169	C₂²×C₁₃	#386	1	1	yes	characteristic
170	C₂²×C₁₃	#387	1	2	yes	—
171	C₂²×C₁₃	#389	1	2	yes	—
172	C₂²×C₁₃	#391	1	1	yes	characteristic
173	C₄×C₁₃	—	4	4	no	—
174	C₄×C₁₃	—	4	2	no	—
175	C₄×C₁₃	—	4	2	no	—
176	C₄×C₁₃	—	8	2	no	—
177	C₄×C₁₃	#394	1	1	yes	characteristic
178	C₄×C₁₃	#395	1	1	yes	characteristic
179	C₄×C₁₃	#396	1	2	yes	—
180	C₄×C₁₃	#398	1	1	yes	characteristic
181	C₄×C₁₃	#398	1	2	yes	—
182	C₄×C₁₃	#400	1	1	yes	characteristic
183	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
184	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
185	C₂×M₁₆	—	13	4	no	—
186	C₂×M₁₆	—	13	4	no	—
187	C₂×M₁₆	—	26	2	no	—
188	C₂×M₁₆	—	26	2	no	—
189	C₂×M₁₆	—	26	2	no	—
190	C₂×M₁₆	—	26	2	no	—
191	C₂×M₁₆	—	26	2	no	—
192	C₂×M₁₆	—	26	2	no	—
193	C₂×M₁₆	—	26	2	no	—
194	C₂×M₁₆	—	26	2	no	—
195	C₂×M₁₆	—	26	2	no	—
196	C₂×M₁₆	—	26	2	no	—
197	C₂²×C₈	—	26	2	no	—
198	C₂²×C₈	—	26	2	no	—
199	#117	—	2	1	no	—
200	#117	—	2	1	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	Dic₂₆	1	2	yes	—
204	#117	D₅₂	1	2	yes	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
214	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
217	C₄⋊C₈	—	26	8	no	—
218	#95	—	2	4	no	—
219	C₂×C₁₃	—	2	1	no	—
220	C₂×C₁₃	—	2	2	no	—
221	C₂×C₁₃	—	2	2	no	—
222	C₂×C₁₃	—	2	1	no	—
223	C₂×C₁₃	#1441	1	1	yes	characteristic
224	C₂×C₁₃	#1914	1	1	yes	characteristic
225	C₂×C₁₃	#1915	1	1	yes	characteristic
226	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	8	2	no	—
231	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
232	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
233	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
235	C₂²×C₄	#6127	1	1	yes	characteristic
236	C₂²×C₄	#6127	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₁₃	1	2	yes	—
238	C₂²×C₄	#6129	1	2	yes	—
239	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
240	C₂²×C₄	#6131	1	2	yes	—
241	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
243	M₁₆	—	26	4	no	—
244	M₁₆	—	26	4	no	—
245	M₁₆	—	26	4	no	—
246	M₁₆	—	26	4	no	—
247	M₁₆	—	52	2	no	—
248	M₁₆	—	52	2	no	—
249	M₁₆	—	52	2	no	—
250	M₁₆	—	52	2	no	—
251	C₂×C₈	—	26	4	no	—
252	C₂×C₈	—	26	4	no	—
253	C₂×C₈	—	52	2	no	—
254	C₂×C₈	—	52	2	no	—
255	C₂×C₈	—	52	2	no	—
256	C₂×C₈	—	52	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	8	4	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₂²⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₁₃	#11206	1	1	yes	characteristic, complemented, Sylow
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
276	C₂³	#88180	1	1	yes	characteristic
277	C₂³	D₄×Dic₁₃	1	2	yes	—
278	C₂³	Q₈×Dic₁₃	1	2	yes	—
279	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	#88142	1	2	yes	—
291	C₂×C₄	#88143	1	2	yes	—
292	C₂×C₄	#88144	1	2	yes	—
293	C₂×C₄	#88148	1	2	yes	—
294	C₂×C₄	#88149	1	2	yes	—
295	C₂×C₄	#88150	1	2	yes	—
296	C₂×C₄	#88151	1	2	yes	—
297	C₂×C₄	#88152	1	2	yes	—
298	C₂×C₄	#88178	1	1	yes	characteristic
299	C₂×C₄	#88178	1	1	yes	characteristic
300	C₂×C₄	#88184	1	1	yes	characteristic
301	C₂×C₄	#88184	1	1	yes	characteristic
302	C₂×C₄	#88189	1	1	yes	characteristic
303	C₂×C₄	#88191	1	1	yes	characteristic
304	C₂×C₄	#88193	1	2	yes	—
305	C₂×C₄	#88198	1	1	yes	characteristic
306	C₂×C₄	#88200	1	2	yes	—
307	C₂×C₄	#88201	1	1	yes	characteristic
308	C₈	—	52	4	no	—
309	C₈	—	52	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	#11718642	1	2	yes	—
318	C₂²	#11718646	1	2	yes	—
319	C₂²	#11718873	1	1	yes	characteristic
320	C₂²	#11718881	1	1	yes	characteristic
321	C₂²	#11718891	1	1	yes	characteristic
322	C₄	—	4	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	2	no	—
326	C₄	#11718640	1	1	yes	characteristic
327	C₄	#11718644	1	1	yes	characteristic
328	C₄	#11718645	1	2	yes	—
329	C₄	#11718647	1	1	yes	characteristic
330	C₄	#11718652	1	1	yes	characteristic
331	C₄	#11718655	1	2	yes	—
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	2	2	no	—
335	C₂	—	2	2	no	—
336	C₂	#13113089	1	1	yes	characteristic
337	C₂	#13115384	1	1	yes	characteristic
338	C₂	#13115389	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718881
commutator, prime radical	C₂⁴×C₁₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₁₃×#2317	C₂
socle	C₂²×C₁₃	#386
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	#2317	D₁₃
hypocenter	C₁₃	#11206
LCS	G ▷ C₂⁴×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2317

Complements

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

N	H ≅ G/N	type
272. C₁₃	#11206	semidirect

Maximal subgroups

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

subgroup	index
#13123400	2
C₂×#11718125	2
#13114575	2
C₂×#11718122	2
C₁₃×#2317	2
#11206	13