Subgroups — #15833634

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125396	C₂	1	1	yes	maximal, characteristic
2	#13123549	C₂	1	1	yes	maximal, characteristic
3	#13113679	C₂	1	4	yes	maximal, complemented
4	#13113670	C₂	1	4	yes	maximal, complemented
5	#13113591	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113291	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113286	C₂	1	1	yes	maximal, characteristic
8	C₂×#11718047	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#533	C₂²	1	1	yes	characteristic
11	C₂×#88155	—	2	1	no	—
12	#11719182	C₂²	1	1	yes	characteristic
13	C₁₃×#446	—	2	1	no	—
14	C₁₃×#446	C₂²	1	2	yes	—
15	C₈×Q₈×C₁₃	—	2	1	no	—
16	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
17	C₁₃×#424	—	2	1	no	—
18	C₁₃×#423	C₂²	1	1	yes	characteristic
19	C₁₃×#405	—	2	1	no	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	#11718771	—	2	1	no	—
22	#11718749	C₄	1	1	yes	characteristic
23	#11718746	C₄	1	1	yes	characteristic
24	#11718742	C₄	1	1	yes	characteristic
25	#11718740	C₄	1	1	yes	characteristic
26	#11718709	C₄	1	4	yes	—
27	C₄×#6106	C₂²	1	4	yes	—
28	#11718585	C₂²	1	1	yes	characteristic
29	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
30	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
31	#11718349	—	2	1	no	—
32	#11718349	C₂²	1	2	yes	—
33	C₈×D₅₂	—	2	1	no	—
34	C₈×D₅₂	C₂²	1	2	yes	—
35	Q₈⋊#1055	C₂²	1	4	yes	—
36	#11718075	C₂²	1	4	yes	—
37	#11718072	C₂²	1	4	yes	—
38	#11718047	C₂²	1	2	yes	—
39	#11718047	C₂²	1	2	yes	—
40	C₂×#6127	—	2	1	no	—
41	C₂×C₁₃×M₁₆	—	2	1	no	—
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
45	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
46	C₁₃×#125	—	2	1	no	—
47	C₁₃×C₂²⋊Q₈	—	2	1	no	—
48	C₁₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
49	C₄×Q₈×C₁₃	—	2	1	no	—
50	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
51	C₁₃×#117	—	2	1	no	—
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×#6106	C₂×C₄	1	4	yes	—
54	#88184	—	2	1	no	—
55	#88184	C₂×C₄	1	2	yes	—
56	C₂×#6099	—	2	1	no	—
57	C₂²×#1055	—	2	1	no	—
58	C₂²×#1055	C₂×C₄	1	2	yes	—
59	#88155	—	4	4	no	—
60	C₂×#6094	—	2	2	no	—
61	C₂×C₈×D₁₃	—	2	2	no	—
62	#88129	—	2	1	no	—
63	#88128	C₂×C₄	1	1	yes	characteristic
64	C₄×D₅₂	—	2	1	no	—
65	C₄×D₅₂	C₂³	1	2	yes	—
66	#88123	C₂×C₄	1	1	yes	characteristic
67	C₁₃×C₄⋊C₈	—	2	2	no	—
68	C₁₃×C₄⋊C₈	—	2	2	no	—
69	C₁₃×C₄⋊C₈	—	2	1	no	—
70	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
71	C₁₃×#97	—	2	2	no	—
72	C₄×C₈×C₁₃	—	2	2	no	—
73	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
74	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
77	#88075	C₂×C₄	1	4	yes	—
78	#88060	—	2	2	no	—
79	#88060	—	2	2	no	—
80	#88049	C₂×C₄	1	4	yes	—
81	#88048	C₂×C₄	1	4	yes	—
82	C₄⋊#1055	—	2	1	no	—
83	C₄⋊#1055	C₂³	1	2	yes	—
84	C₄×#1055	—	2	1	no	—
85	C₄×#1055	C₂×C₄	1	1	yes	characteristic
86	C₄×#1055	C₂×C₄	1	1	yes	characteristic
87	C₄×#1055	C₂³	1	2	yes	—
88	C₄×#1055	C₂³	1	2	yes	—
89	#17976	—	13	1	no	maximal, Sylow
90	C₂×Q₈×C₁₃	—	2	1	no	—
91	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
95	C₂×C₂²⋊D₁₃	—	2	2	no	—
96	#6127	—	4	2	no	—
97	#6127	—	4	2	no	—
98	C₂×D₅₂	—	2	1	no	—
99	C₂×D₅₂	—	2	1	no	—
100	C₂×D₅₂	C₂²×C₄	1	2	yes	—
101	C₂×C₄×D₁₃	—	2	2	no	—
102	C₂×Dic₂₆	—	2	1	no	—
103	C₁₃×M₁₆	—	4	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₁₃×C₄⋊C₄	—	2	1	no	—
112	C₁₃×C₄⋊C₄	—	2	1	no	—
113	C₁₃×C₄⋊C₄	—	2	1	no	—
114	C₁₃×C₄⋊C₄	—	2	2	no	—
115	C₁₃×C₄⋊C₄	—	2	2	no	—
116	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
117	C₁₃×C₂²⋊C₄	—	2	2	no	—
118	C₄²×C₁₃	—	2	2	no	—
119	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₃	#41	1	1	yes	characteristic
122	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
123	#6106	—	4	4	no	—
124	#6103	—	2	2	no	—
125	#6103	—	2	2	no	—
126	C₄⋊Dic₁₃	—	2	1	no	—
127	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
128	#6099	—	4	2	no	—
129	C₂×#1055	—	2	2	no	—
130	C₂×#1055	—	2	2	no	—
131	C₂×#1055	—	4	2	no	—
132	C₂×#1055	—	4	2	no	—
133	C₂×#1055	C₂²×C₄	1	2	yes	—
134	C₂×#1055	C₂²×C₄	1	2	yes	—
135	#6094	—	4	4	no	—
136	C₈×D₁₃	—	4	4	no	—
137	#3167	—	13	1	no	—
138	#2960	D₁₃	1	1	yes	characteristic, complemented
139	#2931	—	13	1	no	—
140	#1583	—	13	4	no	—
141	#1582	—	13	4	no	—
142	#1566	—	13	1	no	—
143	#1493	—	13	1	no	—
144	#1492	—	13	1	no	—
145	C₂×#322	—	13	1	no	—
146	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
147	C₂²×D₁₃	—	2	2	no	—
148	Q₈×C₁₃	—	2	4	no	—
149	Q₈×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
158	C₂×C₄×C₁₃	#135	1	2	yes	—
159	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
162	C₂²⋊D₁₃	—	4	4	no	—
163	C₂×Dic₁₃	—	2	2	no	—
164	D₅₂	—	2	4	no	—
165	D₅₂	—	4	2	no	—
166	D₅₂	—	4	2	no	—
167	C₄×D₁₃	—	4	2	no	—
168	C₄×D₁₃	—	4	2	no	—
169	Dic₂₆	—	4	2	no	—
170	C₈×C₁₃	—	2	1	no	—
171	C₈×C₁₃	—	2	1	no	—
172	C₈×C₁₃	—	4	2	no	—
173	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
174	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
175	#1055	—	2	2	no	—
176	#1055	—	2	2	no	—
177	#1055	—	4	2	no	—
178	#1055	—	4	2	no	—
179	C₂×#131	—	26	1	no	—
180	#533	C₂×D₁₃	1	1	yes	characteristic
181	#532	—	13	1	no	—
182	#498	—	13	1	no	—
183	#492	—	13	1	no	—
184	#487	—	13	1	no	—
185	#486	—	13	1	no	—
186	#465	—	13	4	no	—
187	#446	—	2	1	no	—
188	#446	C₂×D₁₃	1	2	yes	—
189	C₈×Q₈	—	2	1	no	—
190	C₈×Q₈	C₂×D₁₃	1	2	yes	—
191	C₄×SD₁₆	—	13	4	no	—
192	#436	—	13	2	no	—
193	#436	—	26	1	no	—
194	C₈×D₄	—	13	2	no	—
195	C₈×D₄	—	26	1	no	—
196	#424	—	2	1	no	—
197	#423	—	13	1	no	—
198	#423	C₂×D₁₃	1	1	yes	characteristic
199	#408	—	26	1	no	—
200	#405	—	2	1	no	—
201	C₂×C₄×C₈	—	13	1	no	—
202	C₂×C₄×C₈	—	13	1	no	—
203	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
204	#334	—	13	4	no	—
205	Q₈⋊C₈	—	13	4	no	—
206	D₄⋊C₈	—	13	4	no	—
207	#322	—	13	2	no	—
208	#322	—	13	2	no	—
209	C₂²×C₁₃	—	2	1	no	—
210	C₂²×C₁₃	—	2	1	no	—
211	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
212	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
213	C₂×D₁₃	—	4	2	no	—
214	C₂×D₁₃	—	4	2	no	—
215	C₂×D₁₃	—	4	2	no	—
216	C₄×C₁₃	—	2	1	no	—
217	C₄×C₁₃	—	2	1	no	—
218	C₄×C₁₃	—	4	2	no	—
219	C₄×C₁₃	—	4	2	no	—
220	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
221	C₄×C₁₃	#444	1	2	yes	—
222	C₄×C₁₃	#568	1	2	yes	—
223	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
224	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
225	Dic₁₃	—	4	2	no	—
226	C₂×#41	—	26	1	no	—
227	C₂×SD₁₆	—	13	4	no	—
228	#131	—	52	4	no	—
229	C₂×M₁₆	—	2	1	no	—
230	C₂×M₁₆	—	26	1	no	—
231	C₂×M₁₆	—	26	2	no	—
232	C₂²×C₈	—	13	2	no	—
233	C₂²×C₈	—	26	1	no	—
234	C₂²×C₈	—	26	2	no	—
235	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
236	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
237	C₄⋊Q₈	—	13	1	no	—
238	C₄⋊Q₈	C₄×D₁₃	1	1	yes	characteristic
239	C₄⋊Q₈	C₄×D₁₃	1	1	yes	characteristic
240	#127	—	13	1	no	—
241	#125	—	2	1	no	—
242	#124	—	26	1	no	—
243	C₂²⋊Q₈	—	2	1	no	—
244	C₂²⋊Q₈	C₄×D₁₃	1	2	yes	—
245	#121	—	13	2	no	—
246	#121	—	26	1	no	—
247	C₄×Q₈	—	2	1	no	—
248	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
249	C₄×D₄	—	13	2	no	—
250	C₄×D₄	—	26	1	no	—
251	#117	—	2	1	no	—
252	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
253	#106	—	13	4	no	—
254	C₄⋊C₈	—	2	2	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	13	2	no	—
258	C₄⋊C₈	—	26	1	no	—
259	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
260	#103	—	13	4	no	—
261	#102	—	13	4	no	—
262	C₂²⋊C₈	—	26	2	no	—
263	C₂²⋊C₈	—	26	2	no	—
264	#97	—	2	2	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	13	1	no	—
267	C₄×C₈	—	13	1	no	—
268	C₄×C₈	—	13	2	no	—
269	C₄×C₈	—	13	2	no	—
270	C₄×C₈	—	26	1	no	—
271	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
272	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
275	C₂×C₁₃	—	2	2	no	—
276	C₂×C₁₃	#2932	1	1	yes	characteristic
277	C₂×C₁₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
278	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
279	D₁₃	—	8	2	no	—
280	#41	—	52	2	no	—
281	#41	—	52	2	no	—
282	C₂×Q₈	—	2	1	no	—
283	C₂×Q₈	—	26	1	no	—
284	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
285	C₂×D₄	—	13	2	no	—
286	C₂×D₄	—	26	2	no	—
287	C₂×D₄	—	26	1	no	—
288	C₂×D₄	—	26	1	no	—
289	C₂²×C₄	—	26	2	no	—
290	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
291	C₂²×C₄	#6127	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
293	SD₁₆	—	52	4	no	—
294	M₁₆	—	4	2	no	—
295	M₁₆	—	52	2	no	—
296	M₁₆	—	52	4	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	13	2	no	—
300	C₂×C₈	—	13	2	no	—
301	C₂×C₈	—	26	2	no	—
302	C₂×C₈	—	26	2	no	—
303	C₂×C₈	—	52	2	no	—
304	C₂×C₈	—	52	2	no	—
305	C₂×C₈	—	52	4	no	—
306	C₂×C₈	#6106	1	4	yes	complemented
307	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
308	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
309	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
310	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	13	2	no	—
317	C₄⋊C₄	—	26	1	no	—
318	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₂²⋊C₄	—	26	2	no	—
321	C₂²⋊C₄	—	26	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
324	C₄²	#6127	1	1	yes	characteristic
325	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
326	C₄²	C₂³×D₁₃	1	1	yes	characteristic
327	C₁₃	#17976	1	1	yes	characteristic, complemented, Sylow
328	C₂³	—	26	2	no	—
329	C₂³	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
330	Q₈	—	2	4	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	52	2	no	—
333	D₄	—	26	4	no	—
334	D₄	—	52	4	no	—
335	D₄	—	52	2	no	—
336	D₄	—	52	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	26	2	no	—
342	C₂×C₄	—	52	2	no	—
343	C₂×C₄	—	52	2	no	—
344	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
345	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
346	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
347	C₂×C₄	C₂×#6106	1	2	yes	—
348	C₂×C₄	#88204	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
350	C₂×C₄	C₂×#6127	1	1	yes	characteristic
351	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	4	2	no	—
355	C₈	—	26	2	no	—
356	C₈	—	26	2	no	—
357	C₈	—	52	2	no	—
358	C₈	—	52	2	no	—
359	C₈	C₂×#6106	1	2	yes	—
360	C₈	C₂×#6106	1	2	yes	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	52	2	no	—
364	C₂²	—	52	2	no	—
365	C₂²	—	52	2	no	—
366	C₂²	C₄×#6106	1	2	yes	—
367	C₂²	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	52	2	no	—
373	C₄	C₄×#6106	1	2	yes	—
374	C₄	#11718806	1	2	yes	—
375	C₄	#11719455	1	2	yes	—
376	C₄	C₂²×#6106	1	1	yes	characteristic
377	C₄	C₂×#88204	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	104	2	no	—
380	C₂	#13123552	1	1	yes	characteristic
381	C₂	C₂×C₄×#6106	1	1	yes	characteristic
382	C₂	C₂×#11718806	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#2960	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11719182	C₂²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#2960	D₁₃
hypocenter	C₁₃	#17976
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #13113679	C₂	semidirect
4. #13113670	C₂	semidirect
5. #13113591	C₂	semidirect
6. #13113291	C₂	semidirect
8. C₂×#11718047	C₂	semidirect
9. C₁₃×#2960	C₂	semidirect
106. C₂×C₈×C₁₃	SD₁₆	semidirect
138. #2960	D₁₃	semidirect
306. C₂×C₈	#6106	semidirect
327. C₁₃	#17976	semidirect

Maximal subgroups

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

subgroup	index
#13125396	2
#13123549	2
#13113679	2
#13113670	2
#13113591	2
#13113291	2
#13113286	2
C₂×#11718047	2
C₁₃×#2960	2
#17976	13