Subgroups — #15833051

Order 3328.

626 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123917	C₂	1	1	yes	maximal, characteristic, complemented
2	#13123912	C₂	1	1	yes	maximal, characteristic
3	#13113715	C₂	1	2	yes	maximal, complemented
4	#13113680	C₂	1	2	yes	maximal, complemented
5	SD₁₆⋊#1055	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#1055	C₂	1	2	yes	maximal, complemented
7	#13113616	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113283	C₂	1	1	yes	maximal, characteristic, complemented
9	#13113253	C₂	1	1	yes	maximal, characteristic
10	#13113131	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3103	C₂	1	1	yes	maximal, characteristic
12	C₁₃×#532	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#481	C₄	1	2	yes	—
15	C₁₃×#475	C₄	1	2	yes	—
16	C₁₃×#460	C₄	1	2	yes	—
17	C₁₃×#448	C₄	1	2	yes	—
18	C₄×C₁₃×SD₁₆	C₂²	1	2	yes	—
19	C₄×C₁₃×SD₁₆	C₂²	1	2	yes	—
20	C₁₃×#431	C₂²	1	1	yes	characteristic
21	#11718895	—	2	2	no	—
22	Q₈×#1055	—	2	2	no	—
23	Q₈×#1055	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	C₂²	1	2	yes	—
28	#11718587	C₂²	1	1	yes	characteristic
29	#11718586	—	2	2	no	—
30	#11718586	C₂²	1	1	yes	characteristic
31	#11718586	C₂²	1	1	yes	characteristic
32	C₄×#6099	C₂²	1	1	yes	characteristic
33	Q₈⋊#1055	C₂²	1	2	yes	—
34	Q₈⋊#1055	C₂²	1	2	yes	—
35	D₄⋊#1055	C₂²	1	2	yes	—
36	D₄⋊#1055	C₂²	1	2	yes	—
37	#11718050	C₂²	1	2	yes	—
38	#11718049	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	C₂×C₁₃×#41	—	2	1	no	—
44	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
47	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
48	C₁₃×#127	C₂×C₄	1	1	yes	characteristic
49	C₁₃×#124	—	2	1	no	—
50	C₁₃×#121	—	2	1	no	—
51	C₁₃×#121	C₂×C₄	1	2	yes	—
52	C₄×Q₈×C₁₃	—	2	1	no	—
53	C₄×Q₈×C₁₃	—	2	1	no	—
54	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
55	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
56	C₄×D₄×C₁₃	—	2	1	no	—
57	C₄×D₄×C₁₃	C₂³	1	2	yes	—
58	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
59	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₁₃	—	2	1	no	—
61	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×#6099	—	2	1	no	—
63	C₂×#6099	—	2	2	no	—
64	C₂²×#1055	—	2	2	no	—
65	C₁₃×#106	C₂×C₄	1	2	yes	—
66	C₁₃×#106	C₂×C₄	1	2	yes	—
67	C₁₃×C₄⋊C₈	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	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₂²⋊#1055	—	2	1	no	—
76	C₂²⋊#1055	—	2	1	no	—
77	C₂²⋊#1055	—	2	2	no	—
78	C₂²⋊#1055	—	2	2	no	—
79	C₂²⋊#1055	D₄	1	2	yes	—
80	C₄⋊#1055	—	2	2	no	—
81	C₄⋊#1055	—	2	2	no	—
82	C₄⋊#1055	D₄	1	2	yes	—
83	C₄⋊#1055	C₂³	1	2	yes	—
84	C₄⋊#1055	C₂³	1	2	yes	—
85	#88043	—	2	1	no	—
86	#88043	—	2	1	no	—
87	C₄×#1055	—	2	1	no	—
88	C₄×#1055	—	2	2	no	—
89	C₄×#1055	—	2	2	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	#17923	—	13	1	no	maximal, Sylow
95	C₁₃×#41	—	8	2	no	—
96	C₂×Q₈×C₁₃	—	2	4	no	—
97	C₂×Q₈×C₁₃	—	2	1	no	—
98	C₂×Q₈×C₁₃	—	2	1	no	—
99	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
100	C₂×D₄×C₁₃	—	2	2	no	—
101	C₂×D₄×C₁₃	—	2	1	no	—
102	C₂×D₄×C₁₃	—	2	1	no	—
103	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₁₃	—	2	1	no	—
105	C₂²×C₄×C₁₃	—	2	1	no	—
106	C₂²×C₄×C₁₃	—	2	2	no	—
107	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
110	C₁₃×SD₁₆	—	2	4	no	—
111	C₁₃×SD₁₆	—	2	4	no	—
112	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
114	C₁₃×C₄⋊C₄	—	2	2	no	—
115	C₁₃×C₄⋊C₄	—	2	1	no	—
116	C₁₃×C₄⋊C₄	—	2	1	no	—
117	C₁₃×C₄⋊C₄	—	2	2	no	—
118	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
119	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
120	C₁₃×C₂²⋊C₄	—	2	2	no	—
121	C₁₃×C₂²⋊C₄	—	2	2	no	—
122	C₄²×C₁₃	—	2	2	no	—
123	C₄²×C₁₃	—	2	2	no	—
124	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₁₃	#41	1	1	yes	characteristic
127	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
128	#6099	—	4	2	no	—
129	#6099	—	8	2	no	—
130	C₂×#1055	—	2	1	no	—
131	C₂×#1055	—	2	2	no	—
132	C₂×#1055	—	2	2	no	—
133	C₂×#1055	—	2	1	no	—
134	C₂×#1055	—	8	2	no	—
135	C₂×#1055	C₂×D₄	1	2	yes	—
136	C₂×#1055	C₂×D₄	1	2	yes	—
137	#3103	D₁₃	1	1	yes	characteristic
138	#2982	—	13	1	no	—
139	#2981	—	13	1	no	—
140	#1588	—	13	2	no	—
141	#1583	—	13	2	no	—
142	SD₁₆⋊C₈	—	13	2	no	—
143	C₈×SD₁₆	—	13	2	no	—
144	#1572	—	13	1	no	—
145	#1491	—	13	1	no	—
146	#1484	—	13	1	no	—
147	#1453	—	13	1	no	—
148	C₂³×C₁₃	—	2	2	no	—
149	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
150	Q₈×C₁₃	—	2	4	no	—
151	Q₈×C₁₃	—	2	2	no	—
152	Q₈×C₁₃	—	8	1	no	—
153	Q₈×C₁₃	#131	1	4	yes	—
154	D₄×C₁₃	—	2	2	no	—
155	D₄×C₁₃	—	2	2	no	—
156	D₄×C₁₃	—	8	1	no	—
157	D₄×C₁₃	—	8	1	no	—
158	D₄×C₁₃	—	8	2	no	—
159	C₂×C₄×C₁₃	—	2	4	no	—
160	C₂×C₄×C₁₃	—	2	2	no	—
161	C₂×C₄×C₁₃	—	2	2	no	—
162	C₂×C₄×C₁₃	—	2	2	no	—
163	C₂×C₄×C₁₃	—	2	2	no	—
164	C₂×C₄×C₁₃	—	2	2	no	—
165	C₂×C₄×C₁₃	—	4	1	no	—
166	C₂×C₄×C₁₃	—	4	2	no	—
167	C₂×C₄×C₁₃	—	4	2	no	—
168	C₂×C₄×C₁₃	—	8	1	no	—
169	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
170	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
171	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
172	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
175	C₈×C₁₃	—	2	2	no	—
176	C₈×C₁₃	—	2	2	no	—
177	#1055	—	2	2	no	—
178	#1055	—	2	2	no	—
179	#1055	—	2	2	no	—
180	#1055	—	4	1	no	—
181	#1055	—	4	2	no	—
182	#1055	—	4	2	no	—
183	#532	C₂×D₁₃	1	1	yes	characteristic
184	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
185	#481	Dic₁₃	1	2	yes	—
186	#475	Dic₁₃	1	2	yes	—
187	#460	Dic₁₃	1	2	yes	—
188	#448	Dic₁₃	1	2	yes	—
189	C₈×Q₈	—	13	2	no	—
190	C₈×Q₈	—	13	2	no	—
191	C₈×Q₈	—	26	2	no	—
192	C₄×SD₁₆	C₂×D₁₃	1	2	yes	—
193	C₄×SD₁₆	C₂×D₁₃	1	2	yes	—
194	#436	—	13	2	no	—
195	#435	—	26	2	no	—
196	C₈×D₄	—	13	2	no	—
197	#432	—	13	1	no	—
198	#431	—	13	1	no	—
199	#431	—	13	1	no	—
200	#431	—	26	2	no	—
201	#431	C₂×D₁₃	1	1	yes	characteristic
202	#408	—	26	2	no	—
203	C₄×M₁₆	—	13	1	no	—
204	#335	—	13	2	no	—
205	#334	—	13	2	no	—
206	Q₈⋊C₈	—	13	2	no	—
207	Q₈⋊C₈	—	13	2	no	—
208	D₄⋊C₈	—	13	2	no	—
209	D₄⋊C₈	—	13	2	no	—
210	#322	—	13	1	no	—
211	#322	—	13	1	no	—
212	#322	—	13	1	no	—
213	C₈²	—	13	1	no	—
214	C₂²×C₁₃	—	2	1	no	—
215	C₂²×C₁₃	—	2	1	no	—
216	C₂²×C₁₃	—	2	1	no	—
217	C₂²×C₁₃	—	4	2	no	—
218	C₂²×C₁₃	—	4	2	no	—
219	C₂²×C₁₃	—	4	2	no	—
220	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
221	C₄×C₁₃	—	2	4	no	—
222	C₄×C₁₃	—	2	1	no	—
223	C₄×C₁₃	—	2	1	no	—
224	C₄×C₁₃	—	2	1	no	—
225	C₄×C₁₃	—	2	1	no	—
226	C₄×C₁₃	—	4	2	no	—
227	C₄×C₁₃	—	4	2	no	—
228	C₄×C₁₃	C₂×#131	1	2	yes	—
229	C₄×C₁₃	#577	1	2	yes	—
230	C₂×#41	—	2	1	no	—
231	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
232	C₂×SD₁₆	C₂×Dic₁₃	1	2	yes	—
233	C₂×SD₁₆	C₂×Dic₁₃	1	2	yes	—
234	C₂×M₁₆	—	26	2	no	—
235	C₂×M₁₆	—	26	1	no	—
236	C₂²×C₈	—	26	2	no	—
237	C₄⋊Q₈	C₂×Dic₁₃	1	1	yes	characteristic
238	#127	C₂×Dic₁₃	1	1	yes	characteristic
239	#124	—	2	1	no	—
240	#121	—	2	1	no	—
241	#121	C₂×Dic₁₃	1	2	yes	—
242	C₄×Q₈	—	2	1	no	—
243	C₄×Q₈	—	2	1	no	—
244	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
245	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
246	C₄×D₄	—	2	1	no	—
247	C₄×D₄	C₂²×D₁₃	1	2	yes	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
250	C₂×C₄²	—	2	1	no	—
251	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
252	#106	C₂×Dic₁₃	1	2	yes	—
253	#106	C₂×Dic₁₃	1	2	yes	—
254	C₄⋊C₈	—	13	2	no	—
255	C₄⋊C₈	—	13	2	no	—
256	C₄⋊C₈	—	13	2	no	—
257	C₄⋊C₈	—	26	2	no	—
258	C₄⋊C₈	—	26	2	no	—
259	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
260	#103	C₂×Dic₁₃	1	2	yes	—
261	#103	C₂×Dic₁₃	1	2	yes	—
262	#102	C₂×Dic₁₃	1	2	yes	—
263	#102	C₂×Dic₁₃	1	2	yes	—
264	C₂²⋊C₈	—	13	2	no	—
265	C₂²⋊C₈	—	26	2	no	—
266	C₂²⋊C₈	—	26	2	no	—
267	C₂²⋊C₈	—	26	1	no	—
268	C₂²⋊C₈	—	26	1	no	—
269	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
270	#97	—	26	1	no	—
271	#97	—	26	1	no	—
272	C₄×C₈	—	13	1	no	—
273	C₄×C₈	—	13	1	no	—
274	C₄×C₈	—	13	1	no	—
275	C₄×C₈	—	13	1	no	—
276	C₄×C₈	—	26	2	no	—
277	C₄×C₈	—	26	2	no	—
278	C₄×C₈	—	26	1	no	—
279	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
281	C₂×C₁₃	—	4	1	no	—
282	C₂×C₁₃	—	8	2	no	—
283	C₂×C₁₃	#2929	1	1	yes	characteristic
284	C₂×C₁₃	#2949	1	1	yes	characteristic
285	C₂×C₁₃	#2956	1	1	yes	characteristic
286	#41	—	8	2	no	—
287	C₂×Q₈	—	2	4	no	—
288	C₂×Q₈	—	2	1	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	2	1	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	C₂²×Dic₁₃	1	2	yes	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
299	C₂²×C₄	#6129	1	1	yes	characteristic
300	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
301	SD₁₆	—	2	4	no	—
302	SD₁₆	—	2	4	no	—
303	M₁₆	—	52	2	no	—
304	M₁₆	—	104	2	no	—
305	C₂×C₈	—	13	2	no	—
306	C₂×C₈	—	13	2	no	—
307	C₂×C₈	—	26	1	no	—
308	C₂×C₈	—	26	2	no	—
309	C₂×C₈	—	26	2	no	—
310	C₂×C₈	—	26	1	no	—
311	C₂×C₈	—	104	2	no	—
312	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
313	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
319	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	D₄×D₁₃	1	1	yes	characteristic
325	C₄²	#6129	1	1	yes	characteristic
326	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
327	C₄²	C₂³×D₁₃	1	1	yes	characteristic
328	C₁₃	#17923	1	1	yes	characteristic, complemented, Sylow
329	C₂³	—	2	2	no	—
330	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
331	Q₈	—	2	4	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	8	1	no	—
334	Q₈	#88202	1	4	yes	—
335	D₄	—	2	2	no	—
336	D₄	—	2	2	no	—
337	D₄	—	8	1	no	—
338	D₄	—	8	1	no	—
339	D₄	—	8	2	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	1	no	—
350	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
351	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
352	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
354	C₂×C₄	C₂×#6129	1	1	yes	characteristic
355	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	2	2	no	—
358	C₈	—	26	2	no	—
359	C₈	—	26	2	no	—
360	C₈	—	26	2	no	—
361	C₈	—	52	1	no	—
362	C₈	—	52	2	no	—
363	C₈	—	52	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	4	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	#11719467	1	2	yes	—
379	C₄	C₂×#88202	1	2	yes	—
380	C₂	—	4	1	no	—
381	C₂	—	8	2	no	—
382	C₂	#13123531	1	1	yes	characteristic
383	C₂	#13123696	1	1	yes	characteristic
384	C₂	#13123746	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×#3103	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₁₃×#532	C₂²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#3103	D₁₃
hypocenter	C₁₃	#17923
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3103

Complements

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

N	H ≅ G/N	type
1. #13123917	C₂	semidirect
3. #13113715	C₂	semidirect
4. #13113680	C₂	semidirect
5. SD₁₆⋊#1055	C₂	semidirect
6. SD₁₆×#1055	C₂	semidirect
7. #13113616	C₂	semidirect
8. #13113283	C₂	semidirect
10. #13113131	C₂	semidirect
328. C₁₃	#17923	semidirect

Maximal subgroups

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

subgroup	index
#13123917	2
#13123912	2
#13113715	2
#13113680	2
SD₁₆⋊#1055	2
SD₁₆×#1055	2
#13113616	2
#13113283	2
#13113253	2
#13113131	2
C₁₃×#3103	2
#17923	13