Subgroups — #15833773

Order 3328.

584 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125395	C₂	1	1	yes	maximal, characteristic
2	#13123930	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123911	C₂	1	1	yes	maximal, characteristic
4	#13113716	C₂	1	2	yes	maximal, complemented
5	#13113695	C₂	1	2	yes	maximal, complemented
6	#13113681	C₂	1	2	yes	maximal, complemented
7	#13113674	C₂	1	2	yes	maximal, complemented
8	#13113599	C₂	1	1	yes	maximal, characteristic, complemented
9	#13113310	C₂	1	1	yes	maximal, characteristic, complemented
10	#13113306	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719182	C₂²	1	1	yes	characteristic
13	#11719175	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₁₃×#431	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#11718775	C₄	1	2	yes	—
18	#11718775	C₄	1	2	yes	—
19	#11718587	C₂²	1	1	yes	characteristic
20	#11718587	C₂²	1	1	yes	characteristic
21	#11718497	—	2	2	no	—
22	#11718489	—	2	2	no	—
23	#11718388	—	2	2	no	—
24	#11718381	—	2	2	no	—
25	#11718355	C₄	1	1	yes	characteristic
26	#11718354	C₄	1	1	yes	characteristic
27	C₄×#6095	C₂²	1	2	yes	—
28	C₄×#6095	C₂²	1	2	yes	—
29	#11718349	C₂²	1	2	yes	—
30	C₈×D₅₂	C₂²	1	2	yes	—
31	#11718342	C₄	1	1	yes	characteristic
32	#11718341	C₄	1	1	yes	characteristic
33	#11718340	C₂²	1	2	yes	—
34	C₈×Dic₂₆	C₂²	1	2	yes	—
35	C₁₃×#322	C₂²	1	1	yes	characteristic
36	C₁₃×#322	C₂²	1	1	yes	characteristic
37	C₁₃×#322	C₂²	1	1	yes	characteristic
38	C₁₃×#322	C₂²	1	1	yes	characteristic
39	#11718051	C₂²	1	2	yes	—
40	#11718051	C₂²	1	2	yes	—
41	#11718050	C₂²	1	2	yes	—
42	#11718049	C₂²	1	2	yes	—
43	#11718048	C₂²	1	2	yes	—
44	#11718048	C₂²	1	2	yes	—
45	C₂×#6127	—	2	1	no	—
46	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	#88184	—	2	1	no	—
50	#88184	C₂×C₄	1	2	yes	—
51	#88180	—	2	1	no	—
52	#88178	—	2	1	no	—
53	#88178	C₂×C₄	1	2	yes	—
54	C₂×#6095	C₂×C₄	1	2	yes	—
55	C₂×#6095	C₂×C₄	1	2	yes	—
56	C₂×#6094	—	2	2	no	—
57	C₂×C₈×D₁₃	—	2	2	no	—
58	#88129	—	2	1	no	—
59	#88128	C₂×C₄	1	1	yes	characteristic
60	C₄×D₅₂	—	2	1	no	—
61	C₄×D₅₂	C₂³	1	2	yes	—
62	#88124	—	2	1	no	—
63	#88123	C₂×C₄	1	1	yes	characteristic
64	#88123	C₂×C₄	1	1	yes	characteristic
65	#88123	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₂₆	—	2	1	no	—
67	C₄×Dic₂₆	C₂³	1	2	yes	—
68	C₁₃×C₄⋊C₈	—	2	1	no	—
69	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
70	C₁₃×C₂²⋊C₈	—	2	1	no	—
71	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
72	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
80	C₂²⋊#1055	—	2	1	no	—
81	C₂²⋊#1055	—	2	1	no	—
82	#88061	C₂×C₄	1	2	yes	—
83	#88061	C₂×C₄	1	2	yes	—
84	#88060	—	2	2	no	—
85	#88060	—	2	2	no	—
86	#88057	C₂×C₄	1	2	yes	—
87	#88057	C₂×C₄	1	2	yes	—
88	#88056	C₂×C₄	1	2	yes	—
89	#88056	C₂×C₄	1	2	yes	—
90	#88055	—	2	2	no	—
91	#88054	—	2	2	no	—
92	#88054	—	2	2	no	—
93	C₈×Dic₁₃	—	2	2	no	—
94	C₄⋊#1055	C₂³	1	2	yes	—
95	C₄⋊#1055	C₂³	1	2	yes	—
96	#88043	—	2	1	no	—
97	#88043	—	2	1	no	—
98	#17990	—	13	1	no	maximal, Sylow
99	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
102	C₂×C₂²⋊D₁₃	—	2	2	no	—
103	#6127	—	8	2	no	—
104	C₂×D₅₂	—	2	1	no	—
105	C₂×D₅₂	—	2	1	no	—
106	C₂×D₅₂	C₂²×C₄	1	2	yes	—
107	C₂×C₄×D₁₃	—	2	2	no	—
108	C₂×Dic₂₆	—	2	1	no	—
109	C₂×Dic₂₆	—	2	1	no	—
110	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₁₃	—	2	1	no	—
112	C₂×C₈×C₁₃	—	2	1	no	—
113	C₂×C₈×C₁₃	—	4	1	no	—
114	C₂×C₈×C₁₃	—	4	1	no	—
115	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
121	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₁₃	#41	1	1	yes	characteristic
124	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
125	C₂²⋊Dic₁₃	—	2	2	no	—
126	#6103	—	2	2	no	—
127	#6103	—	2	2	no	—
128	C₄⋊Dic₁₃	—	2	1	no	—
129	C₄⋊Dic₁₃	—	2	1	no	—
130	C₄⋊Dic₁₃	—	2	1	no	—
131	C₄⋊Dic₁₃	—	2	1	no	—
132	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
133	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
134	#6101	—	2	2	no	—
135	#6101	—	2	2	no	—
136	C₄×Dic₁₃	—	2	2	no	—
137	C₂×#1055	—	2	2	no	—
138	C₂×#1055	—	2	2	no	—
139	#6095	—	4	2	no	—
140	#6095	—	4	2	no	—
141	#6094	—	8	2	no	—
142	C₈×D₁₃	—	8	2	no	—
143	#3167	—	13	1	no	—
144	#2985	—	13	1	no	—
145	#2981	—	13	1	no	—
146	#1588	—	13	2	no	—
147	#1585	—	13	2	no	—
148	#1583	—	13	2	no	—
149	#1582	—	13	2	no	—
150	#1568	—	13	1	no	—
151	#1497	—	13	1	no	—
152	#1496	—	13	1	no	—
153	#1452	D₁₃	1	1	yes	characteristic, complemented
154	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
155	C₂²×D₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	—	2	1	no	—
157	C₂×C₄×C₁₃	—	2	1	no	—
158	C₂×C₄×C₁₃	—	2	2	no	—
159	C₂×C₄×C₁₃	—	4	1	no	—
160	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
166	C₂²⋊D₁₃	—	8	2	no	—
167	C₂×Dic₁₃	—	2	2	no	—
168	C₂×Dic₁₃	—	2	2	no	—
169	C₂×Dic₁₃	—	2	2	no	—
170	D₅₂	—	2	2	no	—
171	D₅₂	—	2	2	no	—
172	D₅₂	—	8	1	no	—
173	D₅₂	—	8	1	no	—
174	C₄×D₁₃	—	4	2	no	—
175	C₄×D₁₃	—	4	2	no	—
176	Dic₂₆	—	2	2	no	—
177	Dic₂₆	—	2	2	no	—
178	Dic₂₆	—	8	1	no	—
179	Dic₂₆	—	8	1	no	—
180	C₈×C₁₃	—	2	1	no	—
181	C₈×C₁₃	—	2	1	no	—
182	C₈×C₁₃	—	2	1	no	—
183	C₈×C₁₃	—	2	1	no	—
184	C₈×C₁₃	—	2	2	no	—
185	C₈×C₁₃	—	2	2	no	—
186	C₈×C₁₃	—	4	1	no	—
187	C₈×C₁₃	—	4	1	no	—
188	#1055	—	4	2	no	—
189	#1055	—	4	2	no	—
190	#533	—	13	1	no	—
191	#532	—	13	1	no	—
192	#498	—	13	1	no	—
193	#492	—	13	1	no	—
194	#487	—	13	1	no	—
195	#486	—	13	1	no	—
196	#465	—	13	2	no	—
197	#465	—	13	2	no	—
198	#446	—	13	2	no	—
199	#446	—	26	2	no	—
200	C₈×Q₈	—	13	2	no	—
201	C₄×SD₁₆	—	13	2	no	—
202	C₄×SD₁₆	—	13	2	no	—
203	#436	—	13	2	no	—
204	#435	—	26	2	no	—
205	C₈×D₄	—	13	2	no	—
206	#433	—	26	2	no	—
207	#432	—	13	1	no	—
208	#432	—	13	1	no	—
209	#431	C₂×D₁₃	1	1	yes	characteristic
210	#431	C₂×D₁₃	1	1	yes	characteristic
211	#408	—	26	2	no	—
212	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
213	#335	—	13	2	no	—
214	#334	—	13	2	no	—
215	Q₈⋊C₈	—	13	2	no	—
216	Q₈⋊C₈	—	13	2	no	—
217	D₄⋊C₈	—	13	2	no	—
218	D₄⋊C₈	—	13	2	no	—
219	#322	C₂×D₁₃	1	1	yes	characteristic
220	#322	C₂×D₁₃	1	1	yes	characteristic
221	#322	C₂×D₁₃	1	1	yes	characteristic
222	#322	C₂×D₁₃	1	1	yes	characteristic
223	C₂²×C₁₃	—	2	1	no	—
224	C₂²×C₁₃	—	2	1	no	—
225	C₂²×C₁₃	—	2	1	no	—
226	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
227	C₂×D₁₃	—	4	2	no	—
228	C₂×D₁₃	—	4	2	no	—
229	C₂×D₁₃	—	4	2	no	—
230	C₄×C₁₃	—	2	1	no	—
231	C₄×C₁₃	—	2	1	no	—
232	C₄×C₁₃	—	2	1	no	—
233	C₄×C₁₃	—	2	1	no	—
234	C₄×C₁₃	#568	1	1	yes	characteristic
235	C₄×C₁₃	#568	1	1	yes	characteristic
236	C₄×C₁₃	#577	1	2	yes	—
237	Dic₁₃	—	4	2	no	—
238	Dic₁₃	—	4	2	no	—
239	Dic₁₃	—	4	2	no	—
240	C₂×#41	—	26	1	no	—
241	C₂×SD₁₆	—	13	2	no	—
242	C₂×SD₁₆	—	13	2	no	—
243	C₂×M₁₆	—	26	2	no	—
244	C₂²×C₈	—	26	2	no	—
245	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
246	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
247	C₄⋊Q₈	—	13	1	no	—
248	C₄⋊Q₈	—	13	1	no	—
249	C₄⋊Q₈	—	13	1	no	—
250	#127	—	13	1	no	—
251	#125	—	26	1	no	—
252	#124	—	26	1	no	—
253	C₂²⋊Q₈	—	13	2	no	—
254	C₂²⋊Q₈	—	26	1	no	—
255	#121	—	13	2	no	—
256	#121	—	26	1	no	—
257	C₄×Q₈	—	13	2	no	—
258	C₄×Q₈	—	26	1	no	—
259	C₄×D₄	—	13	2	no	—
260	C₄×D₄	—	26	1	no	—
261	#117	—	26	1	no	—
262	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
263	#106	—	13	2	no	—
264	#106	—	13	2	no	—
265	C₄⋊C₈	—	2	1	no	—
266	C₄⋊C₈	—	13	2	no	—
267	C₄⋊C₈	—	13	2	no	—
268	C₄⋊C₈	—	26	2	no	—
269	C₄⋊C₈	—	26	2	no	—
270	C₄⋊C₈	D₅₂	1	2	yes	—
271	#103	—	13	2	no	—
272	#103	—	13	2	no	—
273	#102	—	13	2	no	—
274	#102	—	13	2	no	—
275	C₂²⋊C₈	—	2	1	no	—
276	C₂²⋊C₈	—	26	1	no	—
277	C₂²⋊C₈	—	26	1	no	—
278	C₂²⋊C₈	—	26	2	no	—
279	C₂²⋊C₈	—	26	2	no	—
280	C₂²⋊C₈	D₅₂	1	2	yes	—
281	#97	—	26	2	no	—
282	#97	—	26	1	no	—
283	#97	—	26	1	no	—
284	C₄×C₈	—	26	2	no	—
285	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
286	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
291	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
293	C₂×C₁₃	—	4	1	no	—
294	C₂×C₁₃	#2932	1	1	yes	characteristic
295	C₂×C₁₃	#2949	1	1	yes	characteristic
296	C₂×C₁₃	#2957	1	1	yes	characteristic
297	D₁₃	—	8	2	no	—
298	#41	—	104	2	no	—
299	C₂×Q₈	—	13	2	no	—
300	C₂×Q₈	—	26	1	no	—
301	C₂×Q₈	—	26	1	no	—
302	C₂×D₄	—	13	2	no	—
303	C₂×D₄	—	26	2	no	—
304	C₂×D₄	—	26	1	no	—
305	C₂×D₄	—	26	1	no	—
306	C₂²×C₄	—	26	2	no	—
307	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
309	C₂²×C₄	#6127	1	1	yes	characteristic
310	SD₁₆	—	52	2	no	—
311	SD₁₆	—	52	2	no	—
312	M₁₆	—	104	2	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	4	1	no	—
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	—	26	2	no	—
318	C₂×C₈	—	26	2	no	—
319	C₂×C₈	—	104	2	no	—
320	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
321	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
322	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
323	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
324	C₂×C₈	C₂×D₅₂	1	2	yes	—
325	C₂×C₈	C₂×D₅₂	1	2	yes	—
326	C₄⋊C₄	—	13	2	no	—
327	C₄⋊C₄	—	13	2	no	—
328	C₄⋊C₄	—	26	1	no	—
329	C₄⋊C₄	—	26	1	no	—
330	C₄⋊C₄	—	26	1	no	—
331	C₄⋊C₄	—	26	2	no	—
332	C₄⋊C₄	—	26	1	no	—
333	C₄⋊C₄	—	26	2	no	—
334	C₂²⋊C₄	—	26	2	no	—
335	C₂²⋊C₄	—	26	2	no	—
336	C₂²⋊C₄	—	26	2	no	—
337	C₄²	—	26	2	no	—
338	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
339	C₄²	C₂×D₅₂	1	1	yes	characteristic
340	C₄²	#6127	1	1	yes	characteristic
341	C₄²	C₂³×D₁₃	1	1	yes	characteristic
342	C₁₃	#17990	1	1	yes	characteristic, complemented, Sylow
343	C₂³	—	26	2	no	—
344	C₂³	C₄×D₅₂	1	1	yes	characteristic
345	Q₈	—	26	2	no	—
346	Q₈	—	26	2	no	—
347	Q₈	—	104	1	no	—
348	Q₈	—	104	1	no	—
349	D₄	—	26	2	no	—
350	D₄	—	26	2	no	—
351	D₄	—	104	1	no	—
352	D₄	—	104	1	no	—
353	D₄	—	104	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	26	2	no	—
359	C₂×C₄	—	26	2	no	—
360	C₂×C₄	—	26	2	no	—
361	C₂×C₄	—	52	2	no	—
362	C₂×C₄	—	52	2	no	—
363	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
364	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
365	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
367	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
368	C₂×C₄	C₂×#6127	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	2	no	—
374	C₈	—	2	2	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	1	no	—
377	C₈	—	52	2	no	—
378	C₈	—	52	2	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	52	2	no	—
383	C₂²	—	52	2	no	—
384	C₂²	—	52	2	no	—
385	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	52	2	no	—
391	C₄	—	52	2	no	—
392	C₄	—	52	2	no	—
393	C₄	#11719450	1	1	yes	characteristic
394	C₄	#11719450	1	1	yes	characteristic
395	C₄	#11719456	1	2	yes	—
396	C₂	—	4	1	no	—
397	C₂	—	104	2	no	—
398	C₂	#13123555	1	1	yes	characteristic
399	C₂	#13123697	1	1	yes	characteristic
400	C₂	#13123754	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1452	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11719182	C₂²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1452	D₁₃
hypocenter	C₁₃	#17990
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #13123930	C₂	semidirect
4. #13113716	C₂	semidirect
5. #13113695	C₂	semidirect
6. #13113681	C₂	semidirect
7. #13113674	C₂	semidirect
8. #13113599	C₂	semidirect
9. #13113310	C₂	semidirect
11. C₁₃×#1452	C₂	semidirect
153. #1452	D₁₃	semidirect
342. C₁₃	#17990	semidirect

Maximal subgroups

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

subgroup	index
#13125395	2
#13123930	2
#13123911	2
#13113716	2
#13113695	2
#13113681	2
#13113674	2
#13113599	2
#13113310	2
#13113306	2
C₁₃×#1452	2
#17990	13