Subgroups — #16017559

Order 3328.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88158	C₂	1	1	yes	maximal, characteristic
2	#13116997	C₂	1	2	yes	maximal, complemented
3	#13116990	C₂	1	2	yes	maximal, complemented
4	#13116985	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116980	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116975	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116970	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116618	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116541	C₂	1	2	yes	maximal
11	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#533	—	2	1	no	—
13	C₂×#88158	C₂²	1	1	yes	characteristic
14	C₄×#6127	—	2	1	no	—
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₁₃×#500	C₂²	1	1	yes	characteristic
17	C₁₃×#499	—	2	1	no	—
18	C₁₃×#499	C₂²	1	2	yes	—
19	C₁₃×#498	—	2	1	no	—
20	C₁₃×#427	—	2	1	no	—
21	C₁₃×#427	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
23	#11718774	—	2	1	no	—
24	#11718767	C₂²	1	1	yes	characteristic
25	C₂²×#6090	C₂²	1	2	yes	—
26	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
27	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
28	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
29	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
30	C₄×#6095	—	2	1	no	—
31	C₄×#6095	—	2	1	no	—
32	#11718084	C₂²	1	2	yes	—
33	#11718084	C₂²	1	2	yes	—
34	#11718083	C₂²	1	2	yes	—
35	#11718083	C₂²	1	2	yes	—
36	#11718082	C₂²	1	2	yes	—
37	#11718082	C₂²	1	2	yes	—
38	#11718081	C₂²	1	2	yes	—
39	#11718081	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	1	yes	characteristic
41	C₄⋊#6090	C₂²	1	1	yes	characteristic
42	C₄⋊#6090	C₂²	1	2	yes	—
43	C₂×#6127	—	2	1	no	—
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×C₄⋊Q₈	—	2	1	no	—
48	C₁₃×#125	—	2	2	no	—
49	C₁₃×C₂²⋊Q₈	—	4	1	no	—
50	C₁₃×C₂²⋊Q₈	—	4	1	no	—
51	C₄×Q₈×C₁₃	—	4	1	no	—
52	C₄×Q₈×C₁₃	—	4	1	no	—
53	C₁₃×#117	—	2	2	no	—
54	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
55	C₄×C₂²⋊D₁₃	—	4	1	no	—
56	C₄×C₂²⋊D₁₃	—	4	1	no	—
57	#88180	—	2	1	no	—
58	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
59	#88158	—	2	2	no	—
60	#88158	Q₈	1	4	yes	complemented
61	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
62	C₂×#6095	—	2	1	no	—
63	C₄×D₅₂	—	2	1	no	—
64	C₄×D₅₂	—	2	1	no	—
65	#88126	—	4	1	no	—
66	C₄²×D₁₃	—	4	1	no	—
67	C₄×Dic₂₆	—	2	1	no	—
68	C₄×Dic₂₆	—	2	1	no	—
69	C₁₃×#107	—	2	1	no	—
70	C₁₃×#107	—	2	1	no	—
71	C₁₃×#107	C₂³	1	2	yes	—
72	C₁₃×#107	C₂³	1	2	yes	—
73	C₁₃×#106	—	2	2	no	—
74	C₁₃×#106	—	2	2	no	—
75	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
79	#88061	—	2	1	no	—
80	#88061	—	2	1	no	—
81	#88058	C₂³	1	1	yes	characteristic
82	#88058	C₂³	1	1	yes	characteristic
83	#88057	—	2	1	no	—
84	#88056	—	2	1	no	—
85	#88056	—	2	1	no	—
86	C₂×#6090	—	2	4	no	—
87	C₂×#6090	C₂³	1	2	yes	—
88	C₂×#6090	C₂³	1	2	yes	—
89	#33626	—	13	1	no	maximal, Sylow
90	C₂×Q₈×C₁₃	—	4	1	no	—
91	C₂×Q₈×C₁₃	—	4	1	no	—
92	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₂²⋊D₁₃	—	4	1	no	—
96	#6127	—	2	4	no	—
97	#6127	—	4	2	no	—
98	C₂×D₅₂	—	2	1	no	—
99	C₂×C₄×D₁₃	—	4	1	no	—
100	C₂×C₄×D₁₃	—	4	1	no	—
101	C₂×C₄×D₁₃	—	4	1	no	—
102	C₂×Dic₂₆	—	2	1	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
108	C₂×C₈×C₁₃	#41	1	2	yes	—
109	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	4	2	no	—
113	C₁₃×C₄⋊C₄	—	4	2	no	—
114	C₁₃×C₄⋊C₄	—	4	1	no	—
115	C₁₃×C₄⋊C₄	—	4	1	no	—
116	C₁₃×C₂²⋊C₄	—	4	2	no	—
117	C₄²×C₁₃	—	4	2	no	—
118	C₄²×C₁₃	D₈	1	1	yes	characteristic
119	C₄²×C₁₃	D₈	1	1	yes	characteristic
120	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
122	C₂²⋊Dic₁₃	—	4	1	no	—
123	#6103	—	4	1	no	—
124	#6103	—	4	1	no	—
125	C₄⋊Dic₁₃	—	2	1	no	—
126	C₄⋊Dic₁₃	—	2	1	no	—
127	#6101	—	4	1	no	—
128	#6101	—	4	1	no	—
129	C₄×Dic₁₃	—	4	1	no	—
130	C₄×Dic₁₃	—	4	1	no	—
131	C₄×Dic₁₃	—	4	1	no	—
132	Dic₅₂	—	2	1	no	—
133	Dic₅₂	C₂×Q₈	1	2	yes	—
134	D₁₀₄	—	2	1	no	—
135	D₁₀₄	C₂×Q₈	1	2	yes	—
136	#6095	—	2	4	no	—
137	#6090	—	2	2	no	—
138	#6090	—	2	2	no	—
139	#3160	D₁₃	1	1	yes	characteristic, complemented
140	C₄×#135	—	13	1	no	—
141	#2229	—	13	2	no	—
142	#2228	—	13	2	no	—
143	#2227	—	13	1	no	—
144	#2226	—	13	1	no	—
145	#2225	—	13	1	no	—
146	#2224	—	13	1	no	—
147	#2158	—	13	2	no	—
148	C₂×C₄⋊C₁₆	—	13	1	no	—
149	#2138	—	13	2	no	—
150	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂²×D₁₃	—	4	1	no	—
152	Q₈×C₁₃	—	8	2	no	—
153	Q₈×C₁₃	—	8	2	no	—
154	C₂×C₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	—	4	2	no	—
158	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
159	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₁₃	#137	1	2	yes	—
165	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
166	C₂²⋊D₁₃	—	4	4	no	—
167	C₂×Dic₁₃	—	4	1	no	—
168	C₂×Dic₁₃	—	4	1	no	—
169	C₂×Dic₁₃	—	4	1	no	—
170	D₅₂	—	2	2	no	—
171	D₅₂	—	4	1	no	—
172	C₄×D₁₃	—	4	4	no	—
173	C₄×D₁₃	—	8	2	no	—
174	C₄×D₁₃	—	8	2	no	—
175	Dic₂₆	—	2	2	no	—
176	Dic₂₆	—	4	1	no	—
177	C₈×C₁₃	—	2	1	no	—
178	C₈×C₁₃	—	2	1	no	—
179	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
180	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
181	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
182	C₂×#135	—	13	1	no	—
183	#533	—	2	1	no	—
184	C₄×#41	—	26	1	no	—
185	C₂²×C₁₆	—	13	2	no	—
186	#500	C₂×D₁₃	1	1	yes	characteristic
187	#500	C₂×D₁₃	1	1	yes	characteristic
188	#499	—	2	1	no	—
189	#499	C₂×D₁₃	1	2	yes	—
190	#498	—	2	1	no	—
191	C₄×Q₁₆	—	13	1	no	—
192	C₄×Q₁₆	—	13	1	no	—
193	C₄×SD₁₆	—	26	1	no	—
194	C₄×SD₁₆	—	26	1	no	—
195	C₄×D₈	—	13	1	no	—
196	C₄×D₈	—	13	1	no	—
197	#427	—	2	1	no	—
198	#427	—	13	1	no	—
199	#427	C₂×D₁₃	1	2	yes	—
200	#416	—	26	1	no	—
201	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
202	#367	—	13	2	no	—
203	#367	—	13	2	no	—
204	#366	—	13	2	no	—
205	#366	—	13	2	no	—
206	C₄⋊C₁₆	—	13	1	no	—
207	C₄⋊C₁₆	—	13	1	no	—
208	C₄⋊C₁₆	—	13	2	no	—
209	#358	—	13	2	no	—
210	#358	—	13	2	no	—
211	#357	—	13	2	no	—
212	#357	—	13	2	no	—
213	C₂²×C₁₃	—	2	1	no	—
214	C₂²×C₁₃	—	2	1	no	—
215	C₂²×C₁₃	#474	1	2	yes	—
216	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
217	C₂×D₁₃	—	4	2	no	—
218	C₂×D₁₃	—	8	1	no	—
219	C₂×D₁₃	—	8	1	no	—
220	C₄×C₁₃	—	2	1	no	—
221	C₄×C₁₃	—	2	1	no	—
222	C₄×C₁₃	—	2	1	no	—
223	C₄×C₁₃	—	8	2	no	—
224	C₄×C₁₃	—	8	2	no	—
225	C₄×C₁₃	#474	1	2	yes	—
226	C₄×C₁₃	#508	1	2	yes	—
227	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
228	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
229	Dic₁₃	—	4	2	no	—
230	Dic₁₃	—	8	1	no	—
231	Dic₁₃	—	8	1	no	—
232	C₂×#41	—	26	1	no	—
233	#135	—	13	4	no	—
234	#135	—	26	2	no	—
235	C₂×Q₁₆	—	13	1	no	—
236	C₂×SD₁₆	—	26	1	no	—
237	C₂×D₈	—	13	1	no	—
238	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
239	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
240	C₄⋊Q₈	—	2	1	no	—
241	C₄⋊Q₈	—	2	1	no	—
242	#125	—	2	2	no	—
243	C₂²⋊Q₈	—	4	1	no	—
244	C₂²⋊Q₈	—	4	1	no	—
245	C₄×Q₈	—	4	1	no	—
246	C₄×Q₈	—	4	1	no	—
247	C₄×Q₈	—	26	1	no	—
248	C₄×Q₈	—	26	1	no	—
249	C₄×D₄	—	26	1	no	—
250	C₄×D₄	—	26	1	no	—
251	C₄×D₄	—	52	1	no	—
252	C₄×D₄	—	52	1	no	—
253	#117	—	2	2	no	—
254	#117	—	26	1	no	—
255	#117	—	52	1	no	—
256	C₂×C₄²	—	52	1	no	—
257	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
258	C₂×C₁₆	—	13	2	no	—
259	C₂×C₁₆	—	13	2	no	—
260	C₂×C₁₆	—	26	4	no	—
261	#107	—	2	1	no	—
262	#107	—	2	1	no	—
263	#107	—	13	1	no	—
264	#107	—	13	1	no	—
265	#107	C₂²×D₁₃	1	2	yes	—
266	#107	C₂²×D₁₃	1	2	yes	—
267	#106	—	2	2	no	—
268	#106	—	2	2	no	—
269	#106	—	26	1	no	—
270	#103	—	26	1	no	—
271	#103	—	26	1	no	—
272	#102	—	26	1	no	—
273	#102	—	26	1	no	—
274	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
275	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
278	C₂×C₁₃	—	2	2	no	—
279	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
280	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
281	C₂×C₁₃	#3414	1	1	yes	characteristic
282	D₁₃	—	8	2	no	—
283	#41	—	26	4	no	—
284	#41	—	52	2	no	—
285	C₂×Q₈	—	4	1	no	—
286	C₂×Q₈	—	4	1	no	—
287	C₂×Q₈	—	26	1	no	—
288	C₂×D₄	—	26	1	no	—
289	C₂×D₄	—	52	1	no	—
290	C₂²×C₄	—	52	1	no	—
291	C₂²×C₄	—	52	1	no	—
292	C₂²×C₄	—	52	1	no	—
293	C₂²×C₄	#6104	1	1	yes	characteristic
294	C₂²×C₄	#6104	1	1	yes	characteristic
295	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
296	Q₁₆	—	13	2	no	—
297	Q₁₆	—	26	1	no	—
298	SD₁₆	—	26	4	no	—
299	D₈	—	13	2	no	—
300	D₈	—	26	1	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	Q₈×D₁₃	1	2	yes	—
303	C₂×C₈	#6131	1	2	yes	—
304	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
307	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	4	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	26	1	no	—
315	C₄⋊C₄	—	26	1	no	—
316	C₄⋊C₄	—	52	1	no	—
317	C₄⋊C₄	—	52	1	no	—
318	C₂²⋊C₄	—	4	2	no	—
319	C₂²⋊C₄	—	52	1	no	—
320	C₂²⋊C₄	—	52	1	no	—
321	C₂²⋊C₄	—	52	1	no	—
322	C₄²	—	4	2	no	—
323	C₄²	—	52	1	no	—
324	C₄²	—	52	1	no	—
325	C₄²	—	52	1	no	—
326	C₄²	#6104	1	1	yes	characteristic
327	C₄²	#6104	1	1	yes	characteristic
328	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
329	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
330	C₁₆	—	26	2	no	—
331	C₁₆	—	26	2	no	—
332	C₁₃	#33626	1	1	yes	characteristic, complemented, Sylow
333	C₂³	—	52	1	no	—
334	C₂³	C₂×#6104	1	1	yes	characteristic
335	Q₈	—	8	2	no	—
336	Q₈	—	8	2	no	—
337	Q₈	—	26	2	no	—
338	Q₈	—	52	1	no	—
339	D₄	—	26	2	no	—
340	D₄	—	52	4	no	—
341	D₄	—	52	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	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	—	104	2	no	—
351	C₂×C₄	—	104	2	no	—
352	C₂×C₄	C₂×#6104	1	1	yes	characteristic
353	C₂×C₄	C₂×#6104	1	1	yes	characteristic
354	C₂×C₄	C₂×#6104	1	1	yes	characteristic
355	C₂×C₄	C₂×#6104	1	1	yes	characteristic
356	C₂×C₄	C₂×#6104	1	1	yes	characteristic
357	C₂×C₄	#88196	1	2	yes	—
358	C₂×C₄	#88200	1	2	yes	—
359	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	#88200	1	2	yes	—
363	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
364	C₈	C₂×#6131	1	1	yes	characteristic
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	52	2	no	—
368	C₂²	—	104	1	no	—
369	C₂²	—	104	1	no	—
370	C₂²	#11718748	1	2	yes	—
371	C₂²	C₂²×#6104	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	8	2	no	—
376	C₄	—	8	2	no	—
377	C₄	—	52	2	no	—
378	C₄	—	104	1	no	—
379	C₄	—	104	1	no	—
380	C₄	#11718748	1	2	yes	—
381	C₄	#11718855	1	2	yes	—
382	C₄	C₂×#88196	1	1	yes	characteristic
383	C₄	C₂×#88200	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	104	2	no	—
386	C₂	C₂×#11718748	1	1	yes	characteristic
387	C₂	C₂×#11718855	1	1	yes	characteristic
388	C₂	#13127970	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3160	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#88158	C₂²
hypercenter	#3160	D₁₃
hypocenter	C₁₃	#33626
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #13116997	C₂	semidirect
3. #13116990	C₂	semidirect
4. #13116985	C₂	semidirect
5. #13116980	C₂	semidirect
6. #13116975	C₂	semidirect
7. #13116970	C₂	semidirect
8. #13116618	C₂	semidirect
9. C₂×C₄⋊#6090	C₂	semidirect
11. C₁₃×#3160	C₂	semidirect
60. #88158	Q₈	semidirect
139. #3160	D₁₃	semidirect
332. C₁₃	#33626	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88158	2
#13116997	2
#13116990	2
#13116985	2
#13116980	2
#13116975	2
#13116970	2
#13116618	2
C₂×C₄⋊#6090	2
#13116541	2
C₁₃×#3160	2
#33626	13