Subgroups — #16017577

Order 3328.

534 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125347	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
3	#13117003	C₂	1	2	yes	maximal, complemented
4	#13117002	C₂	1	2	yes	maximal, complemented
5	#13116989	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116988	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116975	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116623	C₂	1	2	yes	maximal
9	#13116590	C₂	1	1	yes	maximal, characteristic
10	#13116554	C₂	1	2	yes	maximal
11	C₁₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₁₃	C₂²	1	1	yes	characteristic
13	#11719433	—	2	1	no	—
14	C₄×Q₈×D₁₃	—	2	1	no	—
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	Q₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
18	#11718740	—	2	1	no	—
19	C₄×#6107	—	2	1	no	—
20	C₂×#88038	C₂²	1	2	yes	—
21	#11718541	—	2	1	no	—
22	#11718541	C₂²	1	2	yes	—
23	D₁₃×#107	C₂²	1	1	yes	characteristic
24	#11718539	—	2	1	no	—
25	#11718539	C₂²	1	2	yes	—
26	#11718534	C₂²	1	1	yes	characteristic
27	D₁₃×#103	—	2	1	no	—
28	#11718451	—	2	1	no	—
29	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	#11718343	C₂²	1	1	yes	characteristic
32	C₁₃×#367	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₃×#358	C₂²	1	2	yes	—
35	#11718156	C₂²	1	2	yes	—
36	#11718096	C₂²	1	2	yes	—
37	#11718095	C₂²	1	2	yes	—
38	#11718093	C₂²	1	2	yes	—
39	#11718084	C₂²	1	2	yes	—
40	#11718082	C₂²	1	2	yes	—
41	#11718081	C₂²	1	2	yes	—
42	C₄⋊#6090	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₁₃	—	2	1	no	—
44	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₄×Q₈×C₁₃	—	2	1	no	—
47	#88200	—	4	1	no	—
48	Q₈×Dic₁₃	—	2	1	no	—
49	Q₈×Dic₁₃	—	4	1	no	—
50	C₂×#6107	—	2	1	no	—
51	Q₁₆×D₁₃	—	2	2	no	—
52	Q₁₆×D₁₃	Q₈	1	4	yes	complemented
53	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
54	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
55	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
56	#88151	—	4	1	no	—
57	#88146	—	2	2	no	—
58	C₄⋊C₄×D₁₃	—	2	1	no	—
59	C₄⋊C₄×D₁₃	—	4	1	no	—
60	#88144	—	2	2	no	—
61	#88141	—	2	1	no	—
62	#88141	—	4	1	no	—
63	#88141	—	4	1	no	—
64	C₄²×D₁₃	—	4	1	no	—
65	C₄²×D₁₃	D₄	1	1	yes	characteristic
66	#88123	—	2	1	no	—
67	C₄×Dic₂₆	—	2	1	no	—
68	C₄×Dic₂₆	—	4	1	no	—
69	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
70	C₁₃×#107	—	2	1	no	—
71	C₁₃×#107	C₂³	1	2	yes	—
72	C₁₃×#107	C₂³	1	1	yes	characteristic
73	C₁₃×#103	—	2	1	no	—
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	#88075	—	2	1	no	—
76	#88058	—	2	1	no	—
77	#88058	C₂³	1	2	yes	—
78	#88058	C₂³	1	1	yes	characteristic
79	#88056	—	2	1	no	—
80	C₈×Dic₁₃	D₄	1	1	yes	characteristic
81	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
82	C₂×#6090	C₂³	1	2	yes	—
83	#88050	—	2	1	no	—
84	#88048	—	2	2	no	—
85	#88048	—	2	2	no	—
86	#88047	—	2	1	no	—
87	C₄×#1055	D₄	1	1	yes	characteristic
88	#88038	—	2	4	no	—
89	#33628	—	13	1	no	maximal, Sylow
90	C₂×Q₈×C₁₃	—	2	1	no	—
91	C₂×Q₈×C₁₃	—	4	1	no	—
92	Q₈×D₁₃	—	2	4	no	—
93	Q₈×D₁₃	—	4	2	no	—
94	C₂×C₄×D₁₃	—	4	1	no	—
95	C₂×C₄×D₁₃	—	4	1	no	—
96	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
97	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
98	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₂₆	—	2	1	no	—
100	C₂×Dic₂₆	—	4	1	no	—
101	C₂×Dic₂₆	—	4	1	no	—
102	C₁₃×Q₁₆	—	2	1	no	—
103	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	—
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
106	C₁₃×C₄⋊C₄	—	2	1	no	—
107	C₁₃×C₄⋊C₄	—	2	2	no	—
108	C₁₃×C₄⋊C₄	—	4	1	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₄²×C₁₃	—	4	1	no	—
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	#6107	—	2	4	no	—
113	#6103	—	4	2	no	—
114	C₄⋊Dic₁₃	—	2	1	no	—
115	C₄⋊Dic₁₃	—	2	2	no	—
116	C₄⋊Dic₁₃	—	4	1	no	—
117	C₄⋊Dic₁₃	—	4	1	no	—
118	C₄⋊Dic₁₃	—	4	2	no	—
119	#6101	—	4	1	no	—
120	#6101	—	4	1	no	—
121	#6101	—	4	1	no	—
122	#6101	—	4	2	no	—
123	C₄×Dic₁₃	—	4	1	no	—
124	C₄×Dic₁₃	—	4	1	no	—
125	C₄×Dic₁₃	—	4	1	no	—
126	C₄×Dic₁₃	—	4	2	no	—
127	C₄×Dic₁₃	D₈	1	1	yes	characteristic
128	C₄×Dic₁₃	D₈	1	1	yes	characteristic
129	C₄×Dic₁₃	C₂×D₄	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	1	no	—
133	Dic₅₂	C₂×Q₈	1	2	yes	—
134	C₈×D₁₃	—	2	2	no	—
135	C₈×D₁₃	C₂×Q₈	1	2	yes	—
136	C₈×D₁₃	#41	1	2	yes	—
137	C₁₃×C₁₆	—	2	2	no	—
138	#6090	—	2	2	no	—
139	#3160	—	13	1	no	—
140	C₂×C₄×Q₁₆	—	13	1	no	—
141	#2229	—	13	2	no	—
142	#2229	—	13	2	no	—
143	#2227	—	13	1	no	—
144	#2227	—	13	1	no	—
145	#2225	—	13	1	no	—
146	#2225	D₁₃	1	1	yes	characteristic, complemented
147	#2159	—	13	2	no	—
148	#2150	—	13	1	no	—
149	#2141	—	13	2	no	—
150	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
151	Q₈×C₁₃	—	2	2	no	—
152	Q₈×C₁₃	—	4	1	no	—
153	Q₈×C₁₃	—	8	2	no	—
154	C₂×C₄×C₁₃	—	4	1	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×Dic₁₃	—	4	1	no	—
161	C₂×Dic₁₃	—	4	1	no	—
162	C₂×Dic₁₃	—	4	2	no	—
163	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
164	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
165	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
166	C₄×D₁₃	—	2	2	no	—
167	C₄×D₁₃	—	4	1	no	—
168	C₄×D₁₃	—	4	4	no	—
169	C₄×D₁₃	—	8	2	no	—
170	C₄×D₁₃	C₂²⋊Q₈	1	2	yes	—
171	C₄×D₁₃	#137	1	2	yes	—
172	Dic₂₆	—	2	2	no	—
173	Dic₂₆	—	4	1	no	—
174	Dic₂₆	—	4	4	no	—
175	Dic₂₆	—	8	2	no	—
176	C₈×C₁₃	—	2	1	no	—
177	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
178	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
179	#1055	—	2	1	no	—
180	#1055	C₂²⋊Q₈	1	2	yes	—
181	C₂²×Q₁₆	—	13	1	no	—
182	#533	—	26	1	no	—
183	C₂×C₄×Q₈	—	26	1	no	—
184	C₂×M₃₂	—	13	2	no	—
185	#500	—	13	1	no	—
186	#500	C₂×D₁₃	1	1	yes	characteristic
187	#499	—	13	2	no	—
188	#499	—	26	1	no	—
189	#498	—	26	1	no	—
190	C₄×Q₁₆	—	13	1	no	—
191	C₄×Q₁₆	—	13	1	no	—
192	C₄×Q₁₆	—	13	1	no	—
193	C₄×Q₁₆	—	26	1	no	—
194	C₄×Q₁₆	—	26	1	no	—
195	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
196	#427	—	13	2	no	—
197	#427	—	26	1	no	—
198	C₂×#107	—	13	1	no	—
199	C₂×#103	—	26	1	no	—
200	C₂×C₄×C₈	—	13	1	no	—
201	#367	—	13	2	no	—
202	#367	C₂×D₁₃	1	2	yes	—
203	#366	—	13	2	no	—
204	#366	—	13	2	no	—
205	C₄⋊C₁₆	—	13	2	no	—
206	C₄⋊C₁₆	—	13	1	no	—
207	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
208	#358	—	13	2	no	—
209	#358	—	13	2	no	—
210	#358	—	13	2	no	—
211	#358	C₂×D₁₃	1	2	yes	—
212	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
213	C₂×D₁₃	—	2	1	no	—
214	C₂×D₁₃	—	2	1	no	—
215	C₂×D₁₃	#474	1	2	yes	—
216	C₄×C₁₃	—	2	1	no	—
217	C₄×C₁₃	—	4	2	no	—
218	C₄×C₁₃	—	8	2	no	—
219	C₄×C₁₃	—	8	1	no	—
220	C₄×C₁₃	#510	1	2	yes	—
221	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
222	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
223	Dic₁₃	—	2	1	no	—
224	Dic₁₃	—	2	1	no	—
225	Dic₁₃	—	4	2	no	—
226	Dic₁₃	—	8	2	no	—
227	Dic₁₃	—	8	1	no	—
228	Dic₁₃	#474	1	2	yes	—
229	C₂²×Q₈	—	26	1	no	—
230	C₂×Q₁₆	—	13	4	no	—
231	C₂×Q₁₆	—	13	1	no	—
232	C₂×Q₁₆	—	26	2	no	—
233	C₂×Q₁₆	—	26	1	no	—
234	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
235	C₂²×C₈	—	13	1	no	—
236	C₂²×C₈	—	13	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	26	1	no	—
239	#125	—	26	2	no	—
240	C₂²⋊Q₈	—	52	1	no	—
241	C₂²⋊Q₈	—	52	1	no	—
242	C₄×Q₈	—	2	1	no	—
243	C₄×Q₈	—	26	1	no	—
244	C₄×Q₈	—	26	1	no	—
245	C₄×Q₈	—	26	1	no	—
246	C₄×Q₈	—	52	1	no	—
247	C₄×Q₈	—	52	1	no	—
248	C₄×Q₈	—	52	1	no	—
249	C₄×Q₈	—	52	1	no	—
250	#117	—	26	2	no	—
251	C₂×C₄⋊C₄	—	26	1	no	—
252	C₂×C₄⋊C₄	—	52	1	no	—
253	C₂×C₄²	—	13	1	no	—
254	C₂×C₄²	—	52	1	no	—
255	M₃₂	—	26	4	no	—
256	C₂×C₁₆	—	13	2	no	—
257	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
258	#107	—	2	1	no	—
259	#107	—	13	2	no	—
260	#107	—	13	1	no	—
261	#107	—	26	1	no	—
262	#107	—	26	1	no	—
263	#107	C₂²×D₁₃	1	2	yes	—
264	#107	C₂²×D₁₃	1	1	yes	characteristic
265	#106	—	26	2	no	—
266	#106	—	26	2	no	—
267	#103	—	2	1	no	—
268	#103	—	26	1	no	—
269	#103	—	26	1	no	—
270	#103	—	26	1	no	—
271	C₄×C₈	—	13	1	no	—
272	C₄×C₈	—	13	1	no	—
273	C₄×C₈	—	13	1	no	—
274	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
275	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
276	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
277	C₂×C₁₃	#3414	1	1	yes	characteristic
278	D₁₃	—	2	2	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	4	1	no	—
281	C₂×Q₈	—	26	4	no	—
282	C₂×Q₈	—	26	1	no	—
283	C₂×Q₈	—	52	1	no	—
284	C₂×Q₈	—	52	2	no	—
285	C₂×Q₈	—	52	1	no	—
286	C₂²×C₄	—	13	1	no	—
287	C₂²×C₄	—	13	1	no	—
288	C₂²×C₄	—	13	1	no	—
289	C₂²×C₄	—	52	1	no	—
290	C₂²×C₄	—	52	1	no	—
291	Q₁₆	—	2	1	no	—
292	Q₁₆	—	13	2	no	—
293	Q₁₆	—	26	4	no	—
294	Q₁₆	—	26	1	no	—
295	Q₁₆	Q₈×D₁₃	1	2	yes	—
296	C₂×C₈	—	13	1	no	—
297	C₂×C₈	—	13	2	no	—
298	C₂×C₈	—	13	2	no	—
299	C₂×C₈	—	13	1	no	—
300	C₂×C₈	—	26	2	no	—
301	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
302	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
303	C₄⋊C₄	—	2	1	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	26	1	no	—
308	C₄⋊C₄	—	26	2	no	—
309	C₄⋊C₄	—	52	1	no	—
310	C₄⋊C₄	—	52	1	no	—
311	C₄⋊C₄	—	52	1	no	—
312	C₄⋊C₄	—	52	2	no	—
313	C₄⋊C₄	—	52	2	no	—
314	C₄⋊C₄	—	52	1	no	—
315	C₄⋊C₄	—	52	1	no	—
316	C₂²⋊C₄	—	52	2	no	—
317	C₄²	—	4	1	no	—
318	C₄²	—	13	1	no	—
319	C₄²	—	13	1	no	—
320	C₄²	—	13	1	no	—
321	C₄²	—	52	1	no	—
322	C₄²	—	52	1	no	—
323	C₄²	—	52	2	no	—
324	C₄²	—	52	1	no	—
325	C₄²	D₄×D₁₃	1	1	yes	characteristic
326	C₁₆	—	2	2	no	—
327	C₁₆	—	26	2	no	—
328	C₁₃	#33628	1	1	yes	characteristic, complemented, Sylow
329	C₂³	—	13	1	no	—
330	Q₈	—	2	2	no	—
331	Q₈	—	4	1	no	—
332	Q₈	—	8	2	no	—
333	Q₈	—	26	2	no	—
334	Q₈	—	52	4	no	—
335	Q₈	—	52	1	no	—
336	Q₈	—	104	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	13	1	no	—
341	C₂×C₄	—	13	2	no	—
342	C₂×C₄	—	13	2	no	—
343	C₂×C₄	—	13	1	no	—
344	C₂×C₄	—	13	1	no	—
345	C₂×C₄	—	26	2	no	—
346	C₂×C₄	—	52	1	no	—
347	C₂×C₄	—	52	4	no	—
348	C₂×C₄	—	52	1	no	—
349	C₂×C₄	—	52	1	no	—
350	C₂×C₄	—	52	2	no	—
351	C₂×C₄	—	104	2	no	—
352	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
353	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	13	2	no	—
357	C₈	—	26	1	no	—
358	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
359	C₈	#41×D₁₃	1	1	yes	characteristic
360	C₂²	—	13	2	no	—
361	C₂²	—	26	1	no	—
362	C₂²	—	26	1	no	—
363	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	4	2	no	—
366	C₄	—	8	2	no	—
367	C₄	—	8	1	no	—
368	C₄	—	13	2	no	—
369	C₄	—	26	1	no	—
370	C₄	—	26	1	no	—
371	C₄	—	52	2	no	—
372	C₄	—	104	1	no	—
373	C₄	—	104	2	no	—
374	C₄	#11718576	1	2	yes	—
375	C₄	D₁₃×C₂²⋊Q₈	1	1	yes	characteristic
376	C₄	D₁₃×#137	1	1	yes	characteristic
377	C₂	—	26	2	no	—
378	C₂	D₁₃×#474	1	1	yes	characteristic
379	C₂	C₂×#11718576	1	1	yes	characteristic
380	C₂	#13127971	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2225	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₃	C₂×D₈
hypercenter	#2225	D₁₃
hypocenter	C₁₃	#33628
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
3. #13117003	C₂	semidirect
4. #13117002	C₂	semidirect
5. #13116989	C₂	semidirect
6. #13116988	C₂	semidirect
7. #13116975	C₂	semidirect
11. C₁₃×#2225	C₂	semidirect
52. Q₁₆×D₁₃	Q₈	semidirect
146. #2225	D₁₃	semidirect
328. C₁₃	#33628	semidirect

Maximal subgroups

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

subgroup	index
#13125347	2
C₄×Q₁₆×D₁₃	2
#13117003	2
#13117002	2
#13116989	2
#13116988	2
#13116975	2
#13116623	2
#13116590	2
#13116554	2
C₁₃×#2225	2
#33628	13