Subgroups — #15833070

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13124876	C₂	1	1	yes	maximal, characteristic
2	#13123933	C₂	1	1	yes	maximal, characteristic
3	#13123916	C₂	1	1	yes	maximal, characteristic
4	#13113706	C₂	1	2	yes	maximal, complemented
5	#13113685	C₂	1	2	yes	maximal, complemented
6	#13113654	C₂	1	2	yes	maximal, complemented
7	#13113642	C₂	1	2	yes	maximal, complemented
8	#13113282	C₂	1	1	yes	maximal, characteristic
9	#13113256	C₂	1	1	yes	maximal, characteristic
10	#13113131	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1572	C₂	1	1	yes	maximal, characteristic
12	#11719175	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
14	C₁₃×#432	C₂²	1	1	yes	characteristic
15	C₁₃×#431	C₂²	1	1	yes	characteristic
16	C₁₃×#431	C₂²	1	1	yes	characteristic
17	#11718586	C₂²	1	1	yes	characteristic
18	C₄×#6099	C₂²	1	1	yes	characteristic
19	#11718511	C₄	1	2	yes	—
20	#11718510	C₄	1	2	yes	—
21	#11718492	C₂²	1	2	yes	—
22	#11718491	C₂²	1	2	yes	—
23	#11718491	C₂²	1	2	yes	—
24	#11718491	C₂²	1	2	yes	—
25	#11718400	C₄	1	2	yes	—
26	#11718383	C₄	1	2	yes	—
27	#11718381	—	2	2	no	—
28	#11718379	—	2	2	no	—
29	#11718362	—	2	2	no	—
30	C₄×Dic₅₂	C₂²	1	2	yes	—
31	C₄×Dic₅₂	C₂²	1	2	yes	—
32	C₈×Dic₂₆	—	2	2	no	—
33	C₁₃×#335	C₂²	1	2	yes	—
34	C₁₃×#334	C₂²	1	2	yes	—
35	#11718076	C₂²	1	2	yes	—
36	#11718076	C₂²	1	2	yes	—
37	#11718076	C₂²	1	2	yes	—
38	#11718076	C₂²	1	2	yes	—
39	#11718047	C₂²	1	1	yes	characteristic
40	#11718046	C₂²	1	1	yes	characteristic
41	#11718046	C₂²	1	1	yes	characteristic
42	C₈×#1055	C₂²	1	1	yes	characteristic
43	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	#88180	—	2	1	no	—
46	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
47	#88178	—	2	1	no	—
48	#88178	C₂×C₄	1	2	yes	—
49	C₂×#6101	—	2	1	no	—
50	C₂×C₄×Dic₁₃	—	2	1	no	—
51	C₂×#6099	—	2	1	no	—
52	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
53	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
54	#88124	—	2	1	no	—
55	#88123	C₂×C₄	1	1	yes	characteristic
56	#88123	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₂₆	—	2	1	no	—
58	C₄×Dic₂₆	—	2	1	no	—
59	C₄×Dic₂₆	—	2	1	no	—
60	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
61	C₄×Dic₂₆	C₂³	1	2	yes	—
62	C₄×Dic₂₆	C₂³	1	2	yes	—
63	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
64	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
65	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
66	C₁₃×C₂²⋊C₈	—	2	1	no	—
67	C₁₃×C₂²⋊C₈	—	2	1	no	—
68	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
69	C₁₃×#97	—	2	1	no	—
70	C₄×C₈×C₁₃	—	2	1	no	—
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₂²⋊#1055	D₄	1	2	yes	—
74	#88058	C₂×C₄	1	2	yes	—
75	#88058	C₂×C₄	1	2	yes	—
76	#88056	C₂×C₄	1	2	yes	—
77	#88056	C₂×C₄	1	2	yes	—
78	#88056	C₂×C₄	1	2	yes	—
79	#88056	C₂×C₄	1	2	yes	—
80	#88055	—	2	2	no	—
81	#88054	—	2	2	no	—
82	#88054	—	2	2	no	—
83	#88054	—	2	2	no	—
84	#88054	—	2	2	no	—
85	C₈×Dic₁₃	—	2	2	no	—
86	C₈×Dic₁₃	—	2	2	no	—
87	C₈×Dic₁₃	—	2	2	no	—
88	C₄⋊#1055	D₄	1	2	yes	—
89	#88043	—	2	1	no	—
90	C₄×#1055	C₂³	1	1	yes	characteristic
91	C₄×#1055	C₂³	1	1	yes	characteristic
92	C₄×#1055	C₂³	1	1	yes	characteristic
93	C₄×#1055	C₂³	1	1	yes	characteristic
94	#17925	—	13	1	no	maximal, Sylow
95	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
98	C₂²×Dic₁₃	—	2	1	no	—
99	C₂²×Dic₁₃	—	2	1	no	—
100	C₂×Dic₂₆	—	2	1	no	—
101	C₂×Dic₂₆	—	2	4	no	—
102	C₂×Dic₂₆	—	2	1	no	—
103	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
104	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	—	2	2	no	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₃	#41	1	1	yes	characteristic
112	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₂²⋊Dic₁₃	—	2	2	no	—
114	C₄⋊Dic₁₃	—	2	1	no	—
115	C₄⋊Dic₁₃	—	2	1	no	—
116	C₄⋊Dic₁₃	—	2	1	no	—
117	C₄⋊Dic₁₃	—	2	1	no	—
118	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
119	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
120	#6101	—	2	2	no	—
121	#6101	—	2	2	no	—
122	#6101	—	2	2	no	—
123	#6101	—	2	2	no	—
124	C₄×Dic₁₃	—	2	2	no	—
125	C₄×Dic₁₃	—	2	2	no	—
126	C₄×Dic₁₃	—	2	2	no	—
127	#6099	—	4	2	no	—
128	C₂×#1055	—	2	1	no	—
129	C₂×#1055	—	2	1	no	—
130	C₂×#1055	C₂×D₄	1	2	yes	—
131	C₂×#1055	C₂×D₄	1	2	yes	—
132	Dic₅₂	—	2	4	no	—
133	Dic₅₂	—	2	4	no	—
134	#3104	—	13	1	no	—
135	#2985	—	13	1	no	—
136	#2982	—	13	1	no	—
137	#1587	—	13	2	no	—
138	#1584	—	13	2	no	—
139	#1579	—	13	2	no	—
140	C₈×Q₁₆	—	13	2	no	—
141	#1572	D₁₃	1	1	yes	characteristic
142	#1491	—	13	1	no	—
143	#1485	—	13	1	no	—
144	#1453	—	13	1	no	—
145	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	4	1	no	—
149	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₁₃	—	2	4	no	—
156	C₂×Dic₁₃	—	2	2	no	—
157	C₂×Dic₁₃	—	2	2	no	—
158	C₂×Dic₁₃	—	2	2	no	—
159	C₂×Dic₁₃	—	2	2	no	—
160	C₂×Dic₁₃	—	8	1	no	—
161	Dic₂₆	—	2	4	no	—
162	Dic₂₆	—	2	2	no	—
163	Dic₂₆	—	2	2	no	—
164	Dic₂₆	—	2	2	no	—
165	Dic₂₆	—	8	1	no	—
166	Dic₂₆	#131	1	4	yes	—
167	C₈×C₁₃	—	2	2	no	—
168	C₈×C₁₃	—	2	2	no	—
169	C₈×C₁₃	—	4	2	no	—
170	C₈×C₁₃	—	4	2	no	—
171	#1055	—	2	2	no	—
172	#1055	—	2	2	no	—
173	#1055	—	2	2	no	—
174	#1055	—	4	1	no	—
175	#533	—	13	1	no	—
176	C₂×C₄×Q₈	—	13	1	no	—
177	#484	—	13	2	no	—
178	#477	—	13	2	no	—
179	#462	—	13	2	no	—
180	#451	—	13	2	no	—
181	#446	—	13	2	no	—
182	#446	—	26	2	no	—
183	C₈×Q₈	—	13	2	no	—
184	C₈×Q₈	—	13	2	no	—
185	C₈×Q₈	—	13	2	no	—
186	C₈×Q₈	—	26	2	no	—
187	C₄×Q₁₆	—	13	2	no	—
188	C₄×Q₁₆	—	13	2	no	—
189	#433	—	26	2	no	—
190	#432	C₂×D₁₃	1	1	yes	characteristic
191	#431	—	13	1	no	—
192	#431	—	26	2	no	—
193	#431	C₂×D₁₃	1	1	yes	characteristic
194	#431	C₂×D₁₃	1	1	yes	characteristic
195	C₄×M₁₆	—	13	1	no	—
196	#335	C₂×D₁₃	1	2	yes	—
197	#334	C₂×D₁₃	1	2	yes	—
198	Q₈⋊C₈	—	13	2	no	—
199	Q₈⋊C₈	—	13	2	no	—
200	Q₈⋊C₈	—	13	2	no	—
201	Q₈⋊C₈	—	13	2	no	—
202	#322	—	13	1	no	—
203	#322	—	13	1	no	—
204	#322	—	13	1	no	—
205	C₈²	—	13	1	no	—
206	C₂²×C₁₃	—	2	1	no	—
207	C₂²×C₁₃	—	2	1	no	—
208	C₂²×C₁₃	—	2	1	no	—
209	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	—	2	1	no	—
213	C₄×C₁₃	—	2	1	no	—
214	C₄×C₁₃	C₂×#131	1	2	yes	—
215	C₄×C₁₃	#578	1	2	yes	—
216	Dic₁₃	—	2	4	no	—
217	Dic₁₃	—	4	2	no	—
218	Dic₁₃	—	4	2	no	—
219	Dic₁₃	—	4	2	no	—
220	C₂²×Q₈	—	13	1	no	—
221	C₂×Q₁₆	—	13	2	no	—
222	C₂×Q₁₆	—	13	2	no	—
223	C₂×M₁₆	—	26	1	no	—
224	C₄⋊Q₈	—	13	1	no	—
225	C₄⋊Q₈	—	13	1	no	—
226	#125	—	26	1	no	—
227	C₂²⋊Q₈	—	13	2	no	—
228	C₂²⋊Q₈	—	26	1	no	—
229	C₄×Q₈	—	13	2	no	—
230	C₄×Q₈	—	13	2	no	—
231	C₄×Q₈	—	13	2	no	—
232	C₄×Q₈	—	26	1	no	—
233	C₄×Q₈	—	26	1	no	—
234	C₄×Q₈	—	26	1	no	—
235	#117	—	26	1	no	—
236	C₂×C₄⋊C₄	—	13	1	no	—
237	C₂×C₄⋊C₄	—	26	1	no	—
238	C₂×C₄²	—	26	1	no	—
239	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
240	#107	—	13	2	no	—
241	#107	—	13	2	no	—
242	C₄⋊C₈	—	13	2	no	—
243	C₄⋊C₈	—	26	2	no	—
244	C₄⋊C₈	—	26	2	no	—
245	C₄⋊C₈	—	26	2	no	—
246	C₄⋊C₈	—	26	2	no	—
247	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
248	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
249	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
250	#103	—	13	2	no	—
251	#103	—	13	2	no	—
252	#103	—	13	2	no	—
253	#103	—	13	2	no	—
254	C₂²⋊C₈	—	2	1	no	—
255	C₂²⋊C₈	—	2	1	no	—
256	C₂²⋊C₈	—	13	2	no	—
257	C₂²⋊C₈	C₄×D₁₃	1	2	yes	—
258	#97	—	2	1	no	—
259	#97	—	26	2	no	—
260	#97	—	26	1	no	—
261	C₄×C₈	—	2	1	no	—
262	C₄×C₈	—	13	1	no	—
263	C₄×C₈	—	13	1	no	—
264	C₄×C₈	—	13	1	no	—
265	C₄×C₈	—	13	1	no	—
266	C₄×C₈	—	26	2	no	—
267	C₄×C₈	—	26	2	no	—
268	C₄×C₈	—	26	2	no	—
269	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
271	C₂×C₁₃	—	4	1	no	—
272	C₂×C₁₃	#2929	1	1	yes	characteristic
273	C₂×C₁₃	#2950	1	1	yes	characteristic
274	C₂×C₁₃	#2956	1	1	yes	characteristic
275	C₂×Q₈	—	13	2	no	—
276	C₂×Q₈	—	13	2	no	—
277	C₂×Q₈	—	26	4	no	—
278	C₂×Q₈	—	26	1	no	—
279	C₂×Q₈	—	26	1	no	—
280	C₂²×C₄	—	26	1	no	—
281	C₂²×C₄	—	26	1	no	—
282	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
283	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
284	C₂²×C₄	#6131	1	1	yes	characteristic
285	Q₁₆	—	26	4	no	—
286	Q₁₆	—	26	4	no	—
287	M₁₆	—	52	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	13	2	no	—
291	C₂×C₈	—	13	2	no	—
292	C₂×C₈	—	26	1	no	—
293	C₂×C₈	—	26	1	no	—
294	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
295	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
296	C₄⋊C₄	—	13	2	no	—
297	C₄⋊C₄	—	13	2	no	—
298	C₄⋊C₄	—	26	1	no	—
299	C₄⋊C₄	—	26	2	no	—
300	C₄⋊C₄	—	26	1	no	—
301	C₄⋊C₄	—	26	1	no	—
302	C₄⋊C₄	—	26	2	no	—
303	C₄⋊C₄	—	26	1	no	—
304	C₄⋊C₄	—	26	2	no	—
305	C₄⋊C₄	—	26	2	no	—
306	C₂²⋊C₄	—	26	2	no	—
307	C₄²	—	26	2	no	—
308	C₄²	—	26	2	no	—
309	C₄²	—	26	2	no	—
310	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
311	C₄²	D₄×D₁₃	1	1	yes	characteristic
312	C₄²	#6131	1	1	yes	characteristic
313	C₄²	C₂³×D₁₃	1	1	yes	characteristic
314	C₁₃	#17925	1	1	yes	characteristic, complemented, Sylow
315	C₂³	#88147	1	1	yes	characteristic
316	Q₈	—	13	4	no	—
317	Q₈	—	26	4	no	—
318	Q₈	—	26	2	no	—
319	Q₈	—	26	2	no	—
320	Q₈	—	26	2	no	—
321	Q₈	—	104	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	26	4	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	2	no	—
329	C₂×C₄	—	26	2	no	—
330	C₂×C₄	—	104	1	no	—
331	C₂×C₄	#88147	1	1	yes	characteristic
332	C₂×C₄	#88147	1	1	yes	characteristic
333	C₂×C₄	#88147	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
335	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
336	C₂×C₄	C₂×#6131	1	1	yes	characteristic
337	C₈	—	2	2	no	—
338	C₈	—	2	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	26	2	no	—
342	C₈	—	26	2	no	—
343	C₈	—	26	2	no	—
344	C₈	—	52	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	C₂×#88147	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	26	4	no	—
354	C₄	—	52	2	no	—
355	C₄	—	52	2	no	—
356	C₄	—	52	2	no	—
357	C₄	D₁₃×#131	1	2	yes	—
358	C₄	#11719476	1	2	yes	—
359	C₂	—	4	1	no	—
360	C₂	#13123529	1	1	yes	characteristic
361	C₂	#13123703	1	1	yes	characteristic
362	C₂	#13123745	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1572	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1572	D₁₃
hypocenter	C₁₃	#17925
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1572

Complements

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

N	H ≅ G/N	type
4. #13113706	C₂	semidirect
5. #13113685	C₂	semidirect
6. #13113654	C₂	semidirect
7. #13113642	C₂	semidirect
314. C₁₃	#17925	semidirect

Maximal subgroups

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

subgroup	index
#13124876	2
#13123933	2
#13123916	2
#13113706	2
#13113685	2
#13113654	2
#13113642	2
#13113282	2
#13113256	2
#13113131	2
C₁₃×#1572	2
#17925	13