Subgroups — #15833495

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123812	C₂	1	2	yes	maximal, complemented
2	#13113750	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113745	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113739	C₂	1	2	yes	maximal, complemented
5	#13113660	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113655	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆⋊#1055	C₂	1	2	yes	maximal, complemented
8	#13113599	C₂	1	1	yes	maximal, characteristic, complemented
9	#13113296	C₂	1	2	yes	maximal, complemented
10	C₂×#11718047	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3168	C₂	1	1	yes	maximal, characteristic
12	C₁₃×#529	C₂²	1	2	yes	—
13	C₁₃×#499	C₄	1	1	yes	characteristic
14	C₁₃×#495	C₄	1	1	yes	characteristic
15	C₁₃×#488	C₄	1	2	yes	—
16	C₁₃×#467	C₄	1	1	yes	characteristic
17	C₁₃×#466	C₄	1	1	yes	characteristic
18	C₁₃×#465	C₄	1	2	yes	—
19	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
20	C₄×C₁₃×SD₁₆	C₂²	1	2	yes	—
21	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
23	#11718685	C₂²	1	2	yes	—
24	Q₈×#1055	C₂²	1	2	yes	—
25	#11718673	C₂²	1	2	yes	—
26	#11718672	—	2	2	no	—
27	D₄×#1055	—	2	2	no	—
28	D₄×#1055	C₂²	1	2	yes	—
29	#11718661	—	2	2	no	—
30	#11718658	—	2	2	no	—
31	#11718587	C₂²	1	2	yes	—
32	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
33	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
34	Q₈⋊#1055	C₂²	1	2	yes	—
35	Q₈⋊#1055	C₂²	1	2	yes	—
36	D₄⋊#1055	C₂²	1	2	yes	—
37	D₄⋊#1055	C₂²	1	2	yes	—
38	#11718050	C₂²	1	2	yes	—
39	#11718049	C₂²	1	1	yes	characteristic
40	#11718049	C₂²	1	1	yes	characteristic
41	#11718047	C₂²	1	1	yes	characteristic
42	#11718047	C₂²	1	1	yes	characteristic
43	#11718047	C₂²	1	1	yes	characteristic
44	#11718047	C₂²	1	1	yes	characteristic
45	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
47	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₁₃×#126	—	2	2	no	—
51	C₁₃×#125	C₂×C₄	1	2	yes	—
52	C₁₃×#124	C₂×C₄	1	2	yes	—
53	C₁₃×#123	—	2	2	no	—
54	C₁₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
55	C₁₃×#121	C₂×C₄	1	2	yes	—
56	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
57	C₄×D₄×C₁₃	—	2	2	no	—
58	C₄×D₄×C₁₃	C₂³	1	2	yes	—
59	C₁₃×#117	—	2	2	no	—
60	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×#6099	—	2	2	no	—
62	C₂²×#1055	—	2	1	no	—
63	C₂²×#1055	—	2	2	no	—
64	C₂²×#1055	D₄	1	2	yes	—
65	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
66	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
67	C₁₃×#106	C₂×C₄	1	2	yes	—
68	C₁₃×#103	C₂×C₄	1	2	yes	—
69	C₁₃×#103	C₂×C₄	1	2	yes	—
70	C₁₃×#102	C₂×C₄	1	2	yes	—
71	C₁₃×#102	C₂×C₄	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	C₂²⋊#1055	—	2	2	no	—
77	C₂²⋊#1055	—	2	2	no	—
78	C₂²⋊#1055	—	2	2	no	—
79	C₄⋊#1055	—	2	2	no	—
80	C₄⋊#1055	—	2	2	no	—
81	C₄⋊#1055	C₂³	1	2	yes	—
82	C₄⋊#1055	C₂³	1	2	yes	—
83	#88043	—	2	2	no	—
84	#88043	—	2	2	no	—
85	C₄×#1055	—	2	1	no	—
86	C₄×#1055	—	2	2	no	—
87	C₄×#1055	D₄	1	2	yes	—
88	C₄×#1055	C₂³	1	1	yes	characteristic
89	C₄×#1055	C₂³	1	1	yes	characteristic
90	C₄×#1055	C₂³	1	1	yes	characteristic
91	C₄×#1055	C₂³	1	1	yes	characteristic
92	#17963	—	13	1	no	maximal, Sylow
93	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×D₄×C₁₃	—	2	2	no	—
95	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₁₃	—	2	2	no	—
97	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
100	C₁₃×Q₁₆	—	2	2	no	—
101	C₁₃×SD₁₆	—	2	4	no	—
102	C₁₃×D₈	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	2	2	no	—
113	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
114	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
115	C₁₃×C₂²⋊C₄	—	2	2	no	—
116	C₁₃×C₂²⋊C₄	—	2	2	no	—
117	C₁₃×C₂²⋊C₄	—	2	2	no	—
118	C₁₃×C₂²⋊C₄	—	2	2	no	—
119	C₁₃×C₂²⋊C₄	—	2	2	no	—
120	C₄²×C₁₃	—	2	2	no	—
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	#6099	—	4	4	no	—
126	C₂×#1055	—	2	2	no	—
127	C₂×#1055	—	2	2	no	—
128	C₂×#1055	—	2	1	no	—
129	C₂×#1055	—	2	1	no	—
130	C₂×#1055	—	4	2	no	—
131	C₂×#1055	—	4	4	no	—
132	C₂×#1055	—	4	2	no	—
133	C₂×#1055	C₂×D₄	1	2	yes	—
134	C₂×#1055	C₂×D₄	1	2	yes	—
135	#3168	D₁₃	1	1	yes	characteristic
136	#2966	—	13	2	no	—
137	#1594	—	13	1	no	—
138	#1593	—	13	1	no	—
139	#1592	—	13	2	no	—
140	#1580	—	13	1	no	—
141	#1579	—	13	1	no	—
142	SD₁₆⋊C₈	—	13	2	no	—
143	#1568	—	13	1	no	—
144	#1494	—	13	2	no	—
145	C₂×#322	—	13	1	no	—
146	C₂³×C₁₃	—	2	2	no	—
147	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
148	Q₈×C₁₃	—	2	2	no	—
149	Q₈×C₁₃	—	2	2	no	—
150	D₄×C₁₃	—	2	2	no	—
151	D₄×C₁₃	—	2	2	no	—
152	D₄×C₁₃	—	4	4	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	—	2	1	no	—
155	C₂×C₄×C₁₃	—	2	1	no	—
156	C₂×C₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	—	2	2	no	—
158	C₂×C₄×C₁₃	—	2	2	no	—
159	C₂×C₄×C₁₃	—	2	2	no	—
160	C₂×C₄×C₁₃	—	4	2	no	—
161	C₂×C₄×C₁₃	—	4	2	no	—
162	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
168	C₈×C₁₃	—	2	1	no	—
169	C₈×C₁₃	—	2	1	no	—
170	C₈×C₁₃	#131	1	2	yes	—
171	C₈×C₁₃	#131	1	2	yes	—
172	#1055	—	2	2	no	—
173	#1055	—	2	2	no	—
174	#1055	—	4	1	no	—
175	#1055	—	4	1	no	—
176	#1055	—	4	2	no	—
177	#1055	—	4	2	no	—
178	#529	C₂×D₁₃	1	2	yes	—
179	#499	Dic₁₃	1	1	yes	characteristic
180	#495	Dic₁₃	1	1	yes	characteristic
181	#488	Dic₁₃	1	2	yes	—
182	#467	Dic₁₃	1	1	yes	characteristic
183	#466	Dic₁₃	1	1	yes	characteristic
184	#465	Dic₁₃	1	2	yes	—
185	#446	—	13	2	no	—
186	C₈×Q₈	—	13	2	no	—
187	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
188	C₄×SD₁₆	C₂×D₁₃	1	2	yes	—
189	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
190	#436	—	13	2	no	—
191	#435	—	26	2	no	—
192	C₈×D₄	—	13	2	no	—
193	C₈×D₄	—	26	2	no	—
194	#433	—	26	2	no	—
195	#432	—	13	2	no	—
196	#405	—	26	2	no	—
197	C₂×C₄×C₈	—	13	1	no	—
198	C₂×C₄×C₈	—	13	1	no	—
199	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
200	#335	—	13	2	no	—
201	#334	—	13	1	no	—
202	#334	—	13	1	no	—
203	Q₈⋊C₈	—	13	2	no	—
204	Q₈⋊C₈	—	13	2	no	—
205	D₄⋊C₈	—	13	2	no	—
206	D₄⋊C₈	—	13	2	no	—
207	#322	—	13	1	no	—
208	#322	—	13	1	no	—
209	#322	—	13	1	no	—
210	#322	—	13	1	no	—
211	C₂²×C₁₃	—	2	1	no	—
212	C₂²×C₁₃	—	2	1	no	—
213	C₂²×C₁₃	—	4	2	no	—
214	C₂²×C₁₃	—	4	2	no	—
215	C₂²×C₁₃	—	4	2	no	—
216	C₂²×C₁₃	#444	1	2	yes	—
217	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
218	C₄×C₁₃	—	2	1	no	—
219	C₄×C₁₃	—	2	1	no	—
220	C₄×C₁₃	—	2	1	no	—
221	C₄×C₁₃	—	2	1	no	—
222	C₄×C₁₃	—	4	2	no	—
223	C₄×C₁₃	—	4	2	no	—
224	C₄×C₁₃	—	4	2	no	—
225	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
226	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
227	C₄×C₁₃	#575	1	2	yes	—
228	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
229	C₂×SD₁₆	C₂×Dic₁₃	1	2	yes	—
230	C₂×D₈	C₂×Dic₁₃	1	1	yes	characteristic
231	C₂×M₁₆	—	26	2	no	—
232	C₂²×C₈	—	13	2	no	—
233	C₂²×C₈	—	26	2	no	—
234	C₂²×C₈	—	26	1	no	—
235	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
236	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
237	#126	—	2	2	no	—
238	#125	C₂×Dic₁₃	1	2	yes	—
239	#124	C₂×Dic₁₃	1	2	yes	—
240	#123	—	2	2	no	—
241	C₂²⋊Q₈	C₂×Dic₁₃	1	2	yes	—
242	#121	C₂×Dic₁₃	1	2	yes	—
243	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
244	C₄×D₄	—	2	2	no	—
245	C₄×D₄	C₂²×D₁₃	1	2	yes	—
246	#117	—	2	2	no	—
247	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
248	#107	C₂×Dic₁₃	1	1	yes	characteristic
249	#107	C₂×Dic₁₃	1	1	yes	characteristic
250	#106	C₂×Dic₁₃	1	2	yes	—
251	C₄⋊C₈	—	13	2	no	—
252	C₄⋊C₈	—	13	2	no	—
253	C₄⋊C₈	—	26	2	no	—
254	C₄⋊C₈	—	26	2	no	—
255	#103	C₂×Dic₁₃	1	2	yes	—
256	#103	C₂×Dic₁₃	1	2	yes	—
257	#102	C₂×Dic₁₃	1	2	yes	—
258	#102	C₂×Dic₁₃	1	2	yes	—
259	C₂²⋊C₈	—	26	2	no	—
260	C₂²⋊C₈	—	26	2	no	—
261	C₂²⋊C₈	—	26	2	no	—
262	#97	—	26	2	no	—
263	#97	—	26	2	no	—
264	C₄×C₈	—	13	2	no	—
265	C₄×C₈	—	13	1	no	—
266	C₄×C₈	—	13	1	no	—
267	C₄×C₈	—	13	1	no	—
268	C₄×C₈	—	13	1	no	—
269	C₄×C₈	—	26	2	no	—
270	C₄×C₈	—	26	1	no	—
271	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
272	C₄×C₈	C₂×Dic₁₃	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	—	8	2	no	—
277	C₂×C₁₃	#2931	1	1	yes	characteristic
278	C₂×C₁₃	#2943	1	1	yes	characteristic
279	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
280	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	C₂²×Dic₁₃	1	2	yes	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
285	C₂²×C₄	#6129	1	1	yes	characteristic
286	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
287	Q₁₆	—	2	2	no	—
288	SD₁₆	—	2	4	no	—
289	D₈	—	2	2	no	—
290	M₁₆	—	52	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	13	2	no	—
294	C₂×C₈	—	13	2	no	—
295	C₂×C₈	—	26	2	no	—
296	C₂×C₈	—	26	2	no	—
297	C₂×C₈	—	26	1	no	—
298	C₂×C₈	—	26	1	no	—
299	C₂×C₈	—	52	2	no	—
300	C₂×C₈	—	52	4	no	—
301	C₂×C₈	—	52	2	no	—
302	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
303	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
304	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
305	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
311	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
312	C₂²⋊C₄	—	2	2	no	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	D₄×D₁₃	1	1	yes	characteristic
319	C₄²	#6129	1	1	yes	characteristic
320	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
321	C₄²	C₂³×D₁₃	1	1	yes	characteristic
322	C₁₃	#17963	1	1	yes	characteristic, complemented, Sylow
323	C₂³	—	2	2	no	—
324	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
325	Q₈	—	2	2	no	—
326	Q₈	—	2	2	no	—
327	D₄	—	2	2	no	—
328	D₄	—	2	2	no	—
329	D₄	—	4	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
340	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
341	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
343	C₂×C₄	C₂×#6129	1	1	yes	characteristic
344	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	26	2	no	—
348	C₈	—	26	2	no	—
349	C₈	—	52	1	no	—
350	C₈	—	52	1	no	—
351	C₈	—	52	2	no	—
352	C₈	—	52	2	no	—
353	C₈	#88202	1	2	yes	—
354	C₈	#88202	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	#11718858	1	2	yes	—
361	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	2	no	—
369	C₄	#11719473	1	2	yes	—
370	C₄	C₂×#88202	1	1	yes	characteristic
371	C₄	C₂×#88202	1	1	yes	characteristic
372	C₂	—	2	2	no	—
373	C₂	—	8	2	no	—
374	C₂	#13123548	1	1	yes	characteristic
375	C₂	#13123637	1	1	yes	characteristic
376	C₂	C₂×#11718858	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×#3168	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₁₃×#466	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#3168	D₁₃
hypocenter	C₁₃	#17963
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #13123812	C₂	semidirect
2. #13113750	C₂	semidirect
3. #13113745	C₂	semidirect
4. #13113739	C₂	semidirect
5. #13113660	C₂	semidirect
6. #13113655	C₂	semidirect
7. SD₁₆⋊#1055	C₂	semidirect
8. #13113599	C₂	semidirect
9. #13113296	C₂	semidirect
10. C₂×#11718047	C₂	semidirect
322. C₁₃	#17963	semidirect

Maximal subgroups

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

subgroup	index
#13123812	2
#13113750	2
#13113745	2
#13113739	2
#13113660	2
#13113655	2
SD₁₆⋊#1055	2
#13113599	2
#13113296	2
C₂×#11718047	2
C₁₃×#3168	2
#17963	13