Subgroups — #15833446

Order 3328.

614 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	#13125419	C₂	1	1	yes	maximal, characteristic
2	#13123765	C₂	1	2	yes	maximal
3	#13113733	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113732	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113730	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#6107	C₂	1	2	yes	maximal, complemented
8	#13113593	C₂	1	1	yes	maximal, characteristic
9	#13113289	C₂	1	2	yes	maximal
10	C₈²×D₁₃	C₂	1	1	yes	maximal, characteristic
11	C₈×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719283	C₂²	1	2	yes	—
13	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
14	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
15	#11718849	C₄	1	1	yes	characteristic
16	#11718842	C₄	1	1	yes	characteristic
17	C₄×#6107	C₂²	1	2	yes	—
18	#11718685	C₂²	1	2	yes	—
19	#11718548	C₄	1	1	yes	characteristic
20	#11718535	C₄	1	1	yes	characteristic
21	#11718500	—	2	2	no	—
22	#11718498	C₂²	1	2	yes	—
23	#11718492	C₂²	1	2	yes	—
24	#11718489	—	2	2	no	—
25	#11718474	C₄	1	2	yes	—
26	#11718453	C₄	1	2	yes	—
27	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
28	#11718350	—	2	2	no	—
29	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
32	C₈×Dic₂₆	—	2	2	no	—
33	C₈×Dic₂₆	C₂²	1	2	yes	—
34	C₁₃×#335	C₂²	1	1	yes	characteristic
35	C₁₃×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₈²×C₁₃	C₂²	1	1	yes	characteristic
37	Q₈⋊#1055	C₂²	1	2	yes	—
38	#11718076	C₂²	1	2	yes	—
39	#11718071	C₂²	1	2	yes	—
40	#11718050	C₂²	1	1	yes	characteristic
41	#11718048	C₂²	1	2	yes	—
42	C₈×#1055	C₂²	1	1	yes	characteristic
43	C₈×#1055	C₂²	1	1	yes	characteristic
44	C₈×#1055	C₂²	1	1	yes	characteristic
45	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
47	#88200	C₂×C₄	1	2	yes	—
48	#88198	C₂×C₄	1	2	yes	—
49	C₂×#6107	C₂×C₄	1	2	yes	—
50	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
51	C₂×#6094	—	2	2	no	—
52	C₂×C₈×D₁₃	—	2	1	no	—
53	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
56	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
57	#88151	C₂×C₄	1	2	yes	—
58	#88150	—	2	2	no	—
59	#88143	—	2	2	no	—
60	#88142	C₂×C₄	1	2	yes	—
61	#88126	—	2	2	no	—
62	C₄²×D₁₃	C₂³	1	1	yes	characteristic
63	C₄×Dic₂₆	—	2	2	no	—
64	C₄×Dic₂₆	C₂³	1	2	yes	—
65	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
66	C₁₃×C₄⋊C₈	—	2	2	no	—
67	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
68	C₁₃×#103	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₁₃	—	2	2	no	—
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	#88075	C₂×C₄	1	2	yes	—
74	#88058	C₂×C₄	1	1	yes	characteristic
75	#88056	C₂×C₄	1	2	yes	—
76	#88055	—	2	2	no	—
77	#88054	—	2	2	no	—
78	#88054	—	2	2	no	—
79	#88054	—	2	2	no	—
80	C₈×Dic₁₃	—	2	2	no	—
81	C₈×Dic₁₃	—	2	1	no	—
82	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₁₃	D₄	1	1	yes	characteristic
85	C₈×Dic₁₃	D₄	1	1	yes	characteristic
86	#88050	C₂×C₄	1	2	yes	—
87	#88047	C₂×C₄	1	2	yes	—
88	C₄⋊#1055	—	2	2	no	—
89	C₄⋊#1055	C₂³	1	2	yes	—
90	#88043	—	2	2	no	—
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	#17958	—	13	1	no	maximal, Sylow
95	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
99	C₂×Dic₂₆	—	2	2	no	—
100	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
101	C₁₃×Q₁₆	—	2	2	no	—
102	C₂×C₈×C₁₃	—	2	1	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₁₃×C₄⋊C₄	—	2	2	no	—
109	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
110	C₄²×C₁₃	—	2	2	no	—
111	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
112	#6107	—	2	4	no	—
113	#6103	—	2	2	no	—
114	#6103	—	2	2	no	—
115	C₄⋊Dic₁₃	—	2	2	no	—
116	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
117	#6101	—	2	2	no	—
118	#6101	—	2	2	no	—
119	#6101	—	2	2	no	—
120	#6101	—	2	2	no	—
121	#6101	—	2	2	no	—
122	C₄×Dic₁₃	—	2	2	no	—
123	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₁₃	#41	1	1	yes	characteristic
126	C₂×#1055	—	2	1	no	—
127	C₂×#1055	—	2	2	no	—
128	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
129	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
130	C₂×#1055	C₂×D₄	1	1	yes	characteristic
131	C₂×#1055	C₂×D₄	1	1	yes	characteristic
132	Dic₅₂	—	2	2	no	—
133	#6094	—	4	4	no	—
134	C₈×D₁₃	—	2	4	no	—
135	C₈×D₁₃	—	2	2	no	—
136	C₈×D₁₃	—	2	2	no	—
137	C₈×D₁₃	—	2	2	no	—
138	C₈×D₁₃	Q₁₆	1	4	yes	complemented
139	#3170	—	13	1	no	—
140	#2960	—	13	2	no	—
141	#1591	—	13	1	no	—
142	#1591	—	13	1	no	—
143	#1591	—	13	2	no	—
144	C₈×Q₁₆	—	13	2	no	—
145	C₈×Q₁₆	—	13	1	no	—
146	C₈×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
147	#1567	—	13	1	no	—
148	#1493	—	13	2	no	—
149	C₂×C₈²	—	13	1	no	—
150	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
151	Q₈×C₁₃	—	2	2	no	—
152	Q₈×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
158	C₂×Dic₁₃	—	2	2	no	—
159	C₂×Dic₁₃	—	2	2	no	—
160	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
161	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
162	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
163	C₄×D₁₃	—	2	1	no	—
164	C₄×D₁₃	—	2	1	no	—
165	C₄×D₁₃	—	2	2	no	—
166	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
167	C₄×D₁₃	#135	1	2	yes	—
168	Dic₂₆	—	2	2	no	—
169	Dic₂₆	—	2	2	no	—
170	Dic₂₆	—	4	4	no	—
171	C₈×C₁₃	—	2	2	no	—
172	C₈×C₁₃	—	2	1	no	—
173	C₈×C₁₃	—	2	1	no	—
174	C₈×C₁₃	—	4	2	no	—
175	C₈×C₁₃	#131	1	2	yes	—
176	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
177	#1055	—	2	2	no	—
178	#1055	—	2	1	no	—
179	#1055	—	2	1	no	—
180	#1055	—	4	2	no	—
181	#1055	#131	1	2	yes	—
182	#1055	C₂×Q₁₆	1	2	yes	—
183	#533	—	13	2	no	—
184	#500	—	13	1	no	—
185	#494	—	13	1	no	—
186	#487	—	13	2	no	—
187	#467	—	13	1	no	—
188	#467	—	13	1	no	—
189	#467	—	13	2	no	—
190	#446	—	13	2	no	—
191	#446	—	13	2	no	—
192	#446	—	26	2	no	—
193	C₈×Q₈	—	13	2	no	—
194	C₈×Q₈	—	26	2	no	—
195	C₈×Q₈	C₂×D₁₃	1	2	yes	—
196	C₄×Q₁₆	—	13	2	no	—
197	C₄×Q₁₆	—	13	1	no	—
198	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
199	#424	—	26	2	no	—
200	#423	—	13	2	no	—
201	#405	—	26	2	no	—
202	C₂×C₄×C₈	—	13	1	no	—
203	C₂×C₄×C₈	—	13	1	no	—
204	C₂×C₄×C₈	—	13	1	no	—
205	#335	—	13	2	no	—
206	#335	—	13	1	no	—
207	#335	C₂×D₁₃	1	1	yes	characteristic
208	Q₈⋊C₈	—	13	2	no	—
209	Q₈⋊C₈	—	13	2	no	—
210	Q₈⋊C₈	—	13	2	no	—
211	Q₈⋊C₈	C₂×D₁₃	1	2	yes	—
212	C₈²	—	13	1	no	—
213	C₈²	—	13	1	no	—
214	C₈²	—	13	1	no	—
215	C₈²	C₂×D₁₃	1	1	yes	characteristic
216	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
217	C₂×D₁₃	—	2	1	no	—
218	C₂×D₁₃	—	2	1	no	—
219	C₂×D₁₃	C₄×Q₁₆	1	2	yes	—
220	C₄×C₁₃	—	2	1	no	—
221	C₄×C₁₃	—	4	2	no	—
222	C₄×C₁₃	—	4	2	no	—
223	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
224	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
225	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
226	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
227	Dic₁₃	—	2	1	no	—
228	Dic₁₃	—	4	2	no	—
229	Dic₁₃	—	4	2	no	—
230	Dic₁₃	C₄×Q₁₆	1	2	yes	—
231	Dic₁₃	#443	1	2	yes	—
232	C₂×Q₁₆	—	13	2	no	—
233	C₂×Q₁₆	—	13	1	no	—
234	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic
235	C₂×M₁₆	—	26	2	no	—
236	C₂²×C₈	—	13	1	no	—
237	C₂²×C₈	—	13	1	no	—
238	C₂²×C₈	—	13	1	no	—
239	C₂²×C₈	—	13	1	no	—
240	C₂²×C₈	—	26	1	no	—
241	C₄⋊Q₈	—	13	2	no	—
242	C₄⋊Q₈	—	13	2	no	—
243	#125	—	26	2	no	—
244	C₂²⋊Q₈	—	13	2	no	—
245	C₂²⋊Q₈	—	13	2	no	—
246	C₂²⋊Q₈	—	26	2	no	—
247	C₄×Q₈	—	13	2	no	—
248	C₄×Q₈	—	26	2	no	—
249	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
250	#117	—	26	2	no	—
251	C₂×C₄²	—	13	1	no	—
252	#107	—	13	1	no	—
253	#107	—	13	2	no	—
254	#107	C₄×D₁₃	1	1	yes	characteristic
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	13	2	no	—
257	C₄⋊C₈	—	26	2	no	—
258	C₄⋊C₈	—	26	2	no	—
259	C₄⋊C₈	—	26	2	no	—
260	C₄⋊C₈	—	26	2	no	—
261	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
262	#103	—	13	2	no	—
263	#103	—	13	2	no	—
264	#103	—	13	2	no	—
265	#103	C₄×D₁₃	1	2	yes	—
266	#97	—	26	2	no	—
267	#97	—	26	2	no	—
268	C₄×C₈	—	2	2	no	—
269	C₄×C₈	—	13	1	no	—
270	C₄×C₈	—	13	1	no	—
271	C₄×C₈	—	13	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	1	no	—
277	C₄×C₈	—	26	2	no	—
278	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
281	C₂×C₁₃	#2931	1	1	yes	characteristic
282	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
283	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
284	D₁₃	—	2	2	no	—
285	C₂×Q₈	—	13	2	no	—
286	C₂×Q₈	—	26	2	no	—
287	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
288	C₂²×C₄	—	13	1	no	—
289	C₂²×C₄	—	13	1	no	—
290	C₂²×C₄	—	13	1	no	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	26	4	no	—
293	Q₁₆	—	26	2	no	—
294	M₁₆	—	52	4	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	13	1	no	—
298	C₂×C₈	—	13	1	no	—
299	C₂×C₈	—	13	4	no	—
300	C₂×C₈	—	13	1	no	—
301	C₂×C₈	—	13	1	no	—
302	C₂×C₈	—	26	2	no	—
303	C₂×C₈	—	26	2	no	—
304	C₂×C₈	—	26	4	no	—
305	C₂×C₈	—	26	1	no	—
306	C₂×C₈	—	26	2	no	—
307	C₂×C₈	—	26	2	no	—
308	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
309	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
310	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
311	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	13	2	no	—
314	C₄⋊C₄	—	26	2	no	—
315	C₄⋊C₄	—	26	2	no	—
316	C₄⋊C₄	—	26	2	no	—
317	C₄⋊C₄	—	26	2	no	—
318	C₄⋊C₄	—	26	2	no	—
319	C₄⋊C₄	—	26	2	no	—
320	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
321	C₂²⋊C₄	—	26	2	no	—
322	C₂²⋊C₄	—	26	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	13	1	no	—
325	C₄²	—	13	1	no	—
326	C₄²	—	13	1	no	—
327	C₄²	—	26	2	no	—
328	C₄²	C₂³×D₁₃	1	1	yes	characteristic
329	C₁₃	#17958	1	1	yes	characteristic, complemented, Sylow
330	C₂³	—	13	1	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	26	2	no	—
334	Q₈	—	26	2	no	—
335	Q₈	—	52	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	13	1	no	—
339	C₂×C₄	—	13	1	no	—
340	C₂×C₄	—	13	2	no	—
341	C₂×C₄	—	13	2	no	—
342	C₂×C₄	—	13	1	no	—
343	C₂×C₄	—	26	2	no	—
344	C₂×C₄	—	26	1	no	—
345	C₂×C₄	—	26	1	no	—
346	C₂×C₄	—	26	2	no	—
347	C₂×C₄	—	26	2	no	—
348	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
349	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
350	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
351	C₈	—	2	2	no	—
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	4	2	no	—
355	C₈	—	13	2	no	—
356	C₈	—	13	2	no	—
357	C₈	—	26	2	no	—
358	C₈	—	26	1	no	—
359	C₈	—	26	1	no	—
360	C₈	—	52	2	no	—
361	C₈	#88155	1	2	yes	—
362	C₈	Q₁₆×D₁₃	1	2	yes	—
363	C₂²	—	13	2	no	—
364	C₂²	—	26	1	no	—
365	C₂²	—	26	1	no	—
366	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	2	no	—
370	C₄	—	13	2	no	—
371	C₄	—	13	2	no	—
372	C₄	—	26	1	no	—
373	C₄	—	52	2	no	—
374	C₄	—	52	2	no	—
375	C₄	C₂×#88155	1	1	yes	characteristic
376	C₄	D₁₃×#131	1	1	yes	characteristic
377	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
378	C₄	D₁₃×#135	1	1	yes	characteristic
379	C₂	—	26	2	no	—
380	C₂	#13123543	1	1	yes	characteristic
381	C₂	C₄×Q₁₆×D₁₃	1	1	yes	characteristic
382	C₂	D₁₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×Q₁₆	D₁₃
hypocenter	C₁₃	#17958
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #13113733	C₂	semidirect
4. #13113732	C₂	semidirect
5. #13113730	C₂	semidirect
6. C₈×Dic₅₂	C₂	semidirect
7. C₈×#6107	C₂	semidirect
11. C₈×C₁₃×Q₁₆	C₂	semidirect
138. C₈×D₁₃	Q₁₆	semidirect
146. C₈×Q₁₆	D₁₃	semidirect
329. C₁₃	#17958	semidirect

Maximal subgroups

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

subgroup	index
#13125419	2
#13123765	2
#13113733	2
#13113732	2
#13113730	2
C₈×Dic₅₂	2
C₈×#6107	2
#13113593	2
#13113289	2
C₈²×D₁₃	2
C₈×C₁₃×Q₁₆	2
#17958	13