Subgroups — #16018764

Order 3328.

584 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13124484	C₂	1	1	yes	maximal, characteristic
2	#13117041	C₂	1	2	yes	maximal, complemented
3	#13117027	C₂	1	2	yes	maximal, complemented
4	#13116951	C₂	1	2	yes	maximal, complemented
5	#13116934	C₂	1	2	yes	maximal, complemented
6	#13116764	C₂	1	2	yes	maximal, complemented
7	#13116753	C₂	1	2	yes	maximal, complemented
8	#13116450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3059	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
12	C₁₃×#536	—	2	1	no	—
13	#11719535	—	2	1	no	—
14	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
15	C₁₃×#470	—	2	1	no	—
16	C₁₃×#470	—	2	1	no	—
17	C₁₃×#467	C₂²	1	2	yes	—
18	C₁₃×#467	C₂²	1	2	yes	complemented
19	C₁₃×#427	C₂²	1	1	yes	characteristic, complemented
20	C₁₃×#419	—	2	1	no	—
21	C₂×#88068	C₂²	1	2	yes	—
22	C₂×#88068	C₂²	1	2	yes	—
23	#11718796	—	2	1	no	—
24	#11718796	—	2	1	no	—
25	#11718792	—	2	1	no	—
26	#11718776	C₂²	1	2	yes	—
27	#11718776	C₂²	1	2	yes	—
28	#11718767	C₂²	1	1	yes	characteristic
29	C₂×#88052	C₂²	1	1	yes	characteristic, complemented
30	C₂²×#6090	C₂²	1	1	yes	characteristic, complemented
31	#11718156	C₂²	1	2	yes	complemented
32	#11718156	C₂²	1	2	yes	complemented
33	C₂²⋊#6090	C₂²	1	2	yes	—
34	C₂²⋊#6090	C₂²	1	2	yes	complemented
35	#11718083	C₂²	1	2	yes	—
36	#11718083	C₂²	1	2	yes	—
37	#11718082	C₂²	1	2	yes	complemented
38	#11718081	C₂²	1	2	yes	complemented
39	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
40	C₂²×Q₈×C₁₃	—	2	1	no	—
41	C₂²×D₅₂	—	2	1	no	—
42	C₂×C₁₃×Q₁₆	—	2	1	no	—
43	C₂×C₁₃×Q₁₆	—	4	1	no	—
44	C₂×C₁₃×Q₁₆	—	4	1	no	—
45	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
46	C₁₃×#131	—	4	1	no	—
47	C₁₃×#131	—	4	1	no	—
48	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
49	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
50	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×C₄⋊Q₈	—	4	1	no	—
55	C₁₃×#124	—	4	1	no	—
56	C₁₃×#123	—	4	1	no	—
57	C₁₃×#123	—	4	1	no	—
58	C₁₃×C₂²⋊Q₈	—	2	2	no	—
59	C₁₃×C₂²⋊Q₈	—	2	2	no	—
60	C₁₃×#117	—	2	1	no	—
61	#88201	—	4	1	no	—
62	#88194	—	4	1	no	—
63	#88191	—	4	1	no	—
64	#88191	—	4	1	no	—
65	#88184	—	2	2	no	—
66	#88184	—	2	2	no	—
67	#88180	—	2	1	no	—
68	C₂×D₁₀₄	—	2	1	no	—
69	C₂×D₁₀₄	—	4	1	no	—
70	C₂×D₁₀₄	—	4	1	no	—
71	C₂×D₁₀₄	C₂³	1	2	yes	—
72	C₁₃×#107	C₂³	1	2	yes	—
73	C₁₃×#106	—	2	1	no	—
74	C₁₃×#103	—	2	2	no	—
75	C₁₃×#103	—	2	2	no	—
76	#88068	—	4	2	no	—
77	#88068	—	4	2	no	—
78	#88061	—	2	2	no	—
79	#88061	—	2	2	no	—
80	#88058	C₂³	1	2	yes	—
81	#88057	—	2	1	no	—
82	#88052	—	4	1	no	—
83	C₂×#6090	—	4	1	no	—
84	C₂×#6090	C₂³	1	2	yes	—
85	C₂×#6090	C₂³	1	2	yes	—
86	#33766	—	13	1	no	maximal, Sylow
87	C₂³×D₁₃	—	4	1	no	—
88	C₁₃×#41	—	4	1	no	—
89	C₁₃×#41	—	4	1	no	—
90	C₂×Q₈×C₁₃	—	2	2	no	—
91	C₂×Q₈×C₁₃	—	4	2	no	—
92	C₂×Q₈×C₁₃	—	4	2	no	—
93	C₂×Q₈×C₁₃	—	4	1	no	—
94	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
95	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
96	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
97	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
98	C₂²×C₄×C₁₃	—	4	1	no	—
99	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₂²⋊D₁₃	—	4	2	no	—
103	C₂×C₂²⋊D₁₃	—	4	2	no	—
104	C₂×D₅₂	—	2	2	no	—
105	C₂×D₅₂	—	4	2	no	—
106	C₂×D₅₂	—	4	2	no	—
107	C₂×D₅₂	—	4	1	no	—
108	C₁₃×Q₁₆	—	2	2	no	—
109	C₁₃×Q₁₆	—	2	2	no	—
110	C₁₃×Q₁₆	—	4	2	no	—
111	C₁₃×M₁₆	—	4	1	no	—
112	C₁₃×M₁₆	—	4	1	no	—
113	C₁₃×M₁₆	—	4	1	no	—
114	C₂×C₈×C₁₃	—	2	1	no	—
115	C₂×C₈×C₁₃	—	2	1	no	—
116	C₂×C₈×C₁₃	—	2	1	no	—
117	C₂×C₈×C₁₃	—	2	1	no	—
118	C₂×C₈×C₁₃	—	2	1	no	—
119	C₂×C₈×C₁₃	—	2	1	no	—
120	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
124	C₁₃×C₄⋊C₄	—	2	2	no	—
125	C₁₃×C₄⋊C₄	—	4	2	no	—
126	C₁₃×C₄⋊C₄	—	4	2	no	—
127	C₁₃×C₂²⋊C₄	—	4	1	no	—
128	C₁₃×C₂²⋊C₄	—	4	2	no	—
129	C₁₃×C₂²⋊C₄	—	4	2	no	—
130	C₄²×C₁₃	—	4	1	no	—
131	C₂²⋊Dic₁₃	—	4	1	no	—
132	#6103	—	4	2	no	—
133	#6103	—	4	2	no	—
134	C₄⋊Dic₁₃	—	2	2	no	—
135	C₄×Dic₁₃	—	4	1	no	—
136	D₁₀₄	—	2	2	no	—
137	D₁₀₄	—	2	2	no	—
138	D₁₀₄	—	4	2	no	—
139	#6090	—	2	2	no	—
140	#6090	—	2	2	no	—
141	#3059	D₁₃	1	1	yes	characteristic, complemented
142	#3058	—	13	1	no	—
143	#2237	—	13	2	no	—
144	#2234	—	13	2	no	—
145	#2220	—	13	2	no	—
146	#2217	—	13	2	no	—
147	#2187	—	13	2	no	—
148	#2185	—	13	2	no	—
149	#2113	—	13	1	no	—
150	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
153	C₂²×D₁₃	—	4	2	no	—
154	C₂²×D₁₃	—	8	2	no	—
155	C₂²×D₁₃	—	16	2	no	—
156	Q₈×C₁₃	—	4	1	no	—
157	Q₈×C₁₃	—	4	2	no	—
158	Q₈×C₁₃	—	4	2	no	—
159	Q₈×C₁₃	—	8	2	no	—
160	D₄×C₁₃	—	4	1	no	—
161	D₄×C₁₃	—	4	1	no	—
162	D₄×C₁₃	—	4	1	no	—
163	C₂×C₄×C₁₃	—	2	1	no	—
164	C₂×C₄×C₁₃	—	2	1	no	—
165	C₂×C₄×C₁₃	—	2	1	no	—
166	C₂×C₄×C₁₃	—	2	1	no	—
167	C₂×C₄×C₁₃	—	2	1	no	—
168	C₂×C₄×C₁₃	—	2	1	no	—
169	C₂×C₄×C₁₃	—	4	2	no	—
170	C₂×C₄×C₁₃	—	4	2	no	—
171	C₂×C₄×C₁₃	—	8	2	no	—
172	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
176	C₂²⋊D₁₃	—	16	2	no	—
177	C₂²⋊D₁₃	—	16	2	no	—
178	C₂×Dic₁₃	—	4	2	no	—
179	D₅₂	—	4	2	no	—
180	D₅₂	—	4	2	no	—
181	D₅₂	—	8	2	no	—
182	C₈×C₁₃	—	2	1	no	—
183	C₈×C₁₃	—	2	1	no	—
184	C₈×C₁₃	—	2	1	no	—
185	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
186	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
188	C₂²×D₈	—	13	1	no	—
189	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
190	#536	—	2	1	no	—
191	#535	—	26	1	no	—
192	C₂×SD₃₂	—	13	2	no	—
193	C₂×SD₃₂	—	13	2	no	—
194	C₂×M₃₂	—	13	1	no	—
195	C₂²×C₁₆	—	13	1	no	—
196	#470	—	2	1	no	—
197	#470	—	2	1	no	—
198	#469	—	26	1	no	—
199	#469	—	26	1	no	—
200	#467	C₂×D₁₃	1	2	yes	—
201	#467	C₂×D₁₃	1	2	yes	complemented
202	#466	—	13	2	no	—
203	#466	—	13	2	no	—
204	#427	—	13	1	no	—
205	#427	C₂×D₁₃	1	1	yes	characteristic, complemented
206	#419	—	2	1	no	—
207	#418	—	26	1	no	—
208	#367	—	13	2	no	—
209	#366	—	13	2	no	—
210	#358	—	13	2	no	—
211	#358	—	13	2	no	—
212	#357	—	13	2	no	—
213	#357	—	13	2	no	—
214	C₂²⋊C₁₆	—	13	2	no	—
215	C₂²⋊C₁₆	—	13	2	no	—
216	C₂²×C₁₃	—	2	2	no	—
217	C₂²×C₁₃	—	2	2	no	—
218	C₂²×C₁₃	—	2	2	no	—
219	C₂²×C₁₃	—	2	1	no	—
220	C₂²×C₁₃	—	2	1	no	—
221	C₂²×C₁₃	—	2	1	no	—
222	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
223	C₂×D₁₃	—	8	2	no	—
224	C₂×D₁₃	—	8	2	no	—
225	C₂×D₁₃	—	8	2	no	—
226	C₂×D₁₃	—	16	4	no	—
227	C₄×C₁₃	—	2	1	no	—
228	C₄×C₁₃	—	2	1	no	—
229	C₄×C₁₃	—	2	1	no	—
230	C₄×C₁₃	—	8	2	no	—
231	C₄×C₁₃	—	8	2	no	—
232	C₄×C₁₃	#534	1	1	yes	characteristic
233	C₄×C₁₃	#575	1	1	yes	characteristic
234	Dic₁₃	—	8	2	no	—
235	C₂×#41	C₂²⋊D₁₃	1	1	yes	characteristic
236	C₂²×Q₈	—	2	1	no	—
237	C₂²×D₄	—	26	1	no	—
238	C₂×Q₁₆	—	2	1	no	—
239	C₂×Q₁₆	—	4	1	no	—
240	C₂×Q₁₆	—	4	1	no	—
241	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
242	C₂×D₈	—	13	2	no	—
243	C₂×D₈	—	26	1	no	—
244	C₂×D₈	—	52	1	no	—
245	C₂×D₈	—	52	1	no	—
246	#131	—	4	1	no	—
247	#131	—	4	1	no	—
248	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
252	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
253	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
254	C₄⋊Q₈	—	4	1	no	—
255	#127	—	52	1	no	—
256	#124	—	4	1	no	—
257	#124	—	52	1	no	—
258	#123	—	4	1	no	—
259	#123	—	4	1	no	—
260	C₂²⋊Q₈	—	2	2	no	—
261	C₂²⋊Q₈	—	2	2	no	—
262	#121	—	26	2	no	—
263	#121	—	26	2	no	—
264	#120	—	52	1	no	—
265	#120	—	52	1	no	—
266	#117	—	2	1	no	—
267	#117	—	26	1	no	—
268	SD₃₂	—	52	2	no	—
269	SD₃₂	—	52	2	no	—
270	M₃₂	—	52	1	no	—
271	C₂×C₁₆	—	13	2	no	—
272	C₂×C₁₆	—	13	2	no	—
273	C₂×C₁₆	—	52	1	no	—
274	#107	—	13	2	no	—
275	#107	C₂²×D₁₃	1	2	yes	—
276	#106	—	2	1	no	—
277	#106	—	26	1	no	—
278	#103	—	2	2	no	—
279	#103	—	2	2	no	—
280	#102	—	26	2	no	—
281	#102	—	26	2	no	—
282	C₂×C₁₃	—	4	1	no	—
283	C₂×C₁₃	—	4	1	no	—
284	C₂×C₁₃	—	4	1	no	—
285	C₂×C₁₃	#3186	1	1	yes	characteristic
286	C₂×C₁₃	#3416	1	2	yes	—
287	D₁₃	—	16	2	no	—
288	C₂⁴	—	52	1	no	—
289	#41	—	4	1	no	—
290	#41	—	4	1	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	4	2	no	—
293	C₂×Q₈	—	4	2	no	—
294	C₂×Q₈	—	4	1	no	—
295	C₂×Q₈	#6104	1	1	yes	characteristic
296	C₂×D₄	—	26	2	no	—
297	C₂×D₄	—	52	2	no	—
298	C₂×D₄	—	52	2	no	—
299	C₂×D₄	—	52	2	no	—
300	C₂×D₄	—	52	2	no	—
301	C₂×D₄	—	52	1	no	—
302	C₂×D₄	#6104	1	1	yes	characteristic
303	C₂×D₄	#6104	1	1	yes	characteristic
304	C₂×D₄	#6104	1	1	yes	characteristic
305	C₂²×C₄	—	4	1	no	—
306	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
307	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
308	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
309	Q₁₆	—	2	2	no	—
310	Q₁₆	—	2	2	no	—
311	Q₁₆	—	4	2	no	—
312	D₈	—	26	2	no	—
313	D₈	—	26	2	no	—
314	D₈	—	52	2	no	—
315	M₁₆	—	4	1	no	—
316	M₁₆	—	4	1	no	—
317	M₁₆	—	4	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
325	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
326	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
327	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	4	2	no	—
330	C₄⋊C₄	—	4	2	no	—
331	C₄⋊C₄	—	26	2	no	—
332	C₂²⋊C₄	—	4	1	no	—
333	C₂²⋊C₄	—	4	2	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₂²⋊C₄	—	52	1	no	—
336	C₂²⋊C₄	—	52	2	no	—
337	C₂²⋊C₄	—	52	2	no	—
338	C₄²	—	4	1	no	—
339	C₄²	—	52	1	no	—
340	C₁₆	—	26	2	no	—
341	C₁₆	—	26	2	no	—
342	C₁₃	#33766	1	1	yes	characteristic, complemented, Sylow
343	C₂³	—	52	2	no	—
344	C₂³	—	104	2	no	—
345	C₂³	—	208	2	no	—
346	C₂³	C₂×#6104	1	1	yes	characteristic
347	C₂³	C₂×#6104	1	1	yes	characteristic
348	C₂³	C₂×#6104	1	1	yes	characteristic
349	Q₈	—	4	1	no	—
350	Q₈	—	4	2	no	—
351	Q₈	—	4	2	no	—
352	Q₈	—	8	2	no	—
353	D₄	—	4	1	no	—
354	D₄	—	4	1	no	—
355	D₄	—	4	1	no	—
356	D₄	—	52	2	no	—
357	D₄	—	52	2	no	—
358	D₄	—	104	2	no	—
359	D₄	—	208	2	no	—
360	D₄	—	208	2	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	8	2	no	—
370	C₂×C₄	—	52	2	no	—
371	C₂×C₄	C₂×#6104	1	1	yes	characteristic
372	C₂×C₄	C₂×#6104	1	1	yes	characteristic
373	C₂×C₄	C₂×#6104	1	1	yes	characteristic
374	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
375	C₈	—	2	1	no	—
376	C₈	—	2	1	no	—
377	C₈	—	2	1	no	—
378	C₈	#88251	1	1	yes	characteristic
379	C₈	#88251	1	1	yes	characteristic
380	C₂²	—	2	2	no	—
381	C₂²	—	2	2	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	104	2	no	—
387	C₂²	—	104	2	no	—
388	C₂²	—	104	2	no	—
389	C₂²	—	208	4	no	—
390	C₂²	C₂²×#6104	1	1	yes	characteristic
391	C₄	—	2	1	no	—
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	8	2	no	—
395	C₄	—	8	2	no	—
396	C₄	—	104	2	no	—
397	C₄	#11719509	1	1	yes	characteristic
398	C₄	#11719527	1	1	yes	characteristic
399	C₂	—	4	1	no	—
400	C₂	—	4	1	no	—
401	C₂	—	4	1	no	—
402	C₂	—	208	2	no	—
403	C₂	#13125584	1	1	yes	characteristic
404	C₂	#13127980	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3059	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13124484	C₂
hypercenter	#3059	D₁₃
hypocenter	C₁₃	#33766
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ #3059

Complements

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

N	H ≅ G/N	type
2. #13117041	C₂	semidirect
3. #13117027	C₂	semidirect
4. #13116951	C₂	semidirect
5. #13116934	C₂	semidirect
6. #13116764	C₂	semidirect
7. #13116753	C₂	semidirect
8. #13116450	C₂	semidirect
9. C₁₃×#3059	C₂	semidirect
10. C₂²×C₁₃×Q₁₆	C₂²	semidirect
18. C₁₃×#467	C₂²	semidirect
19. C₁₃×#427	C₂²	semidirect
29. C₂×#88052	C₂²	semidirect
30. C₂²×#6090	C₂²	semidirect
31. #11718156	C₂²	semidirect
32. #11718156	C₂²	semidirect
34. C₂²⋊#6090	C₂²	semidirect
37. #11718082	C₂²	semidirect
38. #11718081	C₂²	semidirect
141. #3059	D₁₃	semidirect
187. C₂²×Q₁₆	C₂×D₁₃	semidirect
201. #467	C₂×D₁₃	semidirect
205. #427	C₂×D₁₃	semidirect
342. C₁₃	#33766	semidirect

Maximal subgroups

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

subgroup	index
#13124484	2
#13117041	2
#13117027	2
#13116951	2
#13116934	2
#13116764	2
#13116753	2
#13116450	2
C₁₃×#3059	2
#33766	13