Subgroups — #16019723

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127986	C₂	1	1	yes	maximal, characteristic, complemented
2	#13126618	C₂	1	2	yes	maximal, complemented
3	#13117094	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116961	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116953	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116903	C₂	1	2	yes	maximal, complemented
7	#13116807	C₂	1	2	yes	maximal, complemented
8	#13116797	C₂	1	2	yes	maximal, complemented
9	#13116717	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116671	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3415	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#576	C₂²	1	2	yes	—
13	#11719541	—	2	2	no	—
14	#11719476	C₂²	1	2	yes	complemented
15	C₁₃×#509	—	2	2	no	—
16	C₁₃×#509	C₂²	1	2	yes	—
17	C₁₃×#508	—	2	1	no	—
18	C₁₃×#508	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×D₁₆	—	2	1	no	—
20	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
21	C₁₃×#504	C₂²	1	1	yes	characteristic
22	#11718903	—	2	2	no	—
23	#11718862	—	2	2	no	—
24	#11718861	—	2	2	no	—
25	#11718857	C₂²	1	2	yes	complemented
26	#11718856	C₂²	1	2	yes	complemented
27	#11718821	C₂²	1	2	yes	complemented
28	#11718816	C₂²	1	2	yes	complemented
29	C₂×#88067	C₂²	1	2	yes	complemented
30	#11718805	C₂²	1	1	yes	characteristic
31	#11718804	C₂²	1	2	yes	—
32	D₁₃×Q₃₂	—	2	1	no	—
33	#11718576	—	2	2	no	—
34	#11718573	—	2	1	no	—
35	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic, complemented
36	#11718562	C₂²	1	1	yes	characteristic, complemented
37	#11718159	C₂²	1	2	yes	complemented
38	#11718155	C₂²	1	2	yes	complemented
39	#11718153	C₂²	1	1	yes	characteristic
40	#11718109	C₂²	1	2	yes	complemented
41	#11718107	C₂²	1	1	yes	characteristic, complemented
42	#11718103	C₂²	1	1	yes	characteristic, complemented
43	#11718098	C₂²	1	1	yes	characteristic, complemented
44	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
45	C₁₃×H(5,𝔽₂)	—	2	2	no	—
46	#88257	—	2	2	no	—
47	C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
48	C₁₃×AGL(1,ℤ/8)	—	4	2	no	—
49	C₁₃×#135	—	2	2	no	—
50	C₁₃×#135	C₂³	1	2	yes	—
51	C₂×C₁₃×D₈	—	2	2	no	—
52	C₂×C₁₃×D₈	C₂³	1	2	yes	—
53	C₁₃×#131	C₂³	1	1	yes	characteristic
54	#88205	—	2	2	no	—
55	#88205	—	2	2	no	—
56	#88205	—	4	2	no	—
57	#88204	—	4	2	no	—
58	#88204	D₄	1	1	yes	characteristic
59	#88202	—	2	2	no	—
60	#88190	—	2	2	no	—
61	C₂×#6105	—	2	2	no	—
62	C₂×#6105	—	4	2	no	—
63	#88186	—	2	2	no	—
64	#88186	—	4	2	no	—
65	Q₁₆×D₁₃	—	2	2	no	—
66	#88169	—	4	2	no	—
67	#88166	—	2	2	no	—
68	C₂×Dic₅₂	C₂³	1	2	yes	—
69	#88155	C₂³	1	1	yes	characteristic
70	C₁₃×Q₃₂	D₄	1	1	yes	characteristic
71	C₁₃×SD₃₂	—	2	2	no	—
72	C₁₃×SD₃₂	D₄	1	2	yes	—
73	C₁₃×D₁₆	—	2	2	no	—
74	C₁₃×D₁₆	—	2	1	no	—
75	C₁₃×D₁₆	—	2	1	no	—
76	C₁₃×D₁₆	D₄	1	1	yes	characteristic
77	C₁₃×M₃₂	—	2	1	no	—
78	C₁₃×M₃₂	C₂³	1	1	yes	characteristic
79	C₂×C₁₃×C₁₆	—	2	1	no	—
80	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
81	#88077	—	2	2	no	—
82	#88077	D₄	1	1	yes	characteristic
83	#88073	—	2	2	no	—
84	#88069	—	2	2	no	—
85	#88067	—	2	2	no	—
86	#88067	—	4	2	no	—
87	#88065	D₄	1	1	yes	characteristic
88	#88064	—	2	2	no	—
89	#88062	D₄	1	1	yes	characteristic
90	#88059	C₂³	1	2	yes	—
91	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
92	#88052	C₂³	1	1	yes	characteristic
93	C₂×#6090	C₂³	1	1	yes	characteristic
94	Dic₁₀₄	—	2	1	no	—
95	Dic₁₀₄	—	2	1	no	—
96	#88038	—	2	1	no	—
97	C₁₆×D₁₃	—	2	1	no	—
98	#33859	—	13	1	no	maximal, Sylow
99	C₁₃×#41	—	2	2	no	—
100	C₁₃×#41	—	4	2	no	—
101	C₁₃×#41	—	4	2	no	—
102	C₁₃×#41	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄×C₁₃	—	2	2	no	—
104	C₂×D₄×C₁₃	—	4	2	no	—
105	C₂×D₄×C₁₃	—	4	2	no	—
106	C₂×D₄×C₁₃	—	8	2	no	—
107	Q₈×D₁₃	—	4	2	no	—
108	#6129	—	4	2	no	—
109	#6129	—	8	2	no	—
110	#6127	—	4	2	no	—
111	#6127	C₂×D₄	1	1	yes	characteristic
112	C₂×Dic₂₆	—	2	2	no	—
113	C₂×Dic₂₆	—	4	2	no	—
114	C₁₃×Q₁₆	C₂×D₄	1	2	yes	—
115	C₁₃×SD₁₆	—	2	2	no	—
116	C₁₃×SD₁₆	—	4	2	no	—
117	C₁₃×D₈	—	2	2	no	—
118	C₁₃×D₈	—	2	2	no	—
119	C₁₃×D₈	—	4	2	no	—
120	C₁₃×D₈	C₂×D₄	1	2	yes	—
121	C₁₃×M₁₆	—	2	1	no	—
122	C₁₃×M₁₆	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₁₃	—	2	1	no	—
124	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
125	C₂²⋊Dic₁₃	—	8	2	no	—
126	#6107	—	4	2	no	—
127	#6107	—	4	2	no	—
128	#6107	D₈	1	1	yes	characteristic
129	#6106	D₈	1	1	yes	characteristic
130	#6105	—	4	2	no	—
131	#6105	—	4	2	no	—
132	#6105	—	4	2	no	—
133	#6105	—	4	2	no	—
134	#6105	—	4	2	no	—
135	#6105	D₈	1	1	yes	characteristic
136	#6104	—	4	2	no	—
137	#6104	D₈	1	1	yes	characteristic
138	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
139	#6099	—	2	2	no	—
140	#6099	—	4	2	no	—
141	#6099	C₂×D₄	1	1	yes	characteristic
142	C₂×#1055	—	4	2	no	—
143	C₂×#1055	C₂×D₄	1	1	yes	characteristic
144	Dic₅₂	—	2	2	no	—
145	Dic₅₂	—	2	2	no	—
146	#6094	—	2	1	no	—
147	C₈×D₁₃	—	2	1	no	—
148	C₁₃×C₁₆	—	2	1	no	—
149	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
150	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
151	#6090	—	2	1	no	—
152	#6090	—	2	1	no	—
153	#3417	—	13	1	no	—
154	#3415	D₁₃	1	1	yes	characteristic, complemented
155	#3290	—	13	2	no	—
156	#2249	—	13	1	no	—
157	#2222	—	13	1	no	—
158	#2221	—	13	1	no	—
159	#2212	—	13	2	no	—
160	#2194	—	13	2	no	—
161	#2193	—	13	2	no	—
162	#2178	—	13	1	no	—
163	#2170	—	13	1	no	—
164	C₂³×C₁₃	—	4	2	no	—
165	C₂³×C₁₃	—	8	2	no	—
166	Q₈×C₁₃	—	2	2	no	—
167	Q₈×C₁₃	C₂×D₈	1	1	yes	characteristic
168	D₄×C₁₃	—	2	1	no	—
169	D₄×C₁₃	—	2	2	no	—
170	D₄×C₁₃	—	4	2	no	—
171	D₄×C₁₃	—	4	2	no	—
172	D₄×C₁₃	—	4	2	no	—
173	D₄×C₁₃	—	8	2	no	—
174	D₄×C₁₃	—	8	2	no	—
175	D₄×C₁₃	C₂×D₈	1	1	yes	characteristic
176	C₂×C₄×C₁₃	—	2	1	no	—
177	C₂×C₄×C₁₃	—	4	2	no	—
178	C₂×C₄×C₁₃	—	8	2	no	—
179	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
180	C₂²⋊D₁₃	—	2	1	no	—
181	C₂²⋊D₁₃	—	8	2	no	—
182	C₂×Dic₁₃	—	2	1	no	—
183	C₂×Dic₁₃	—	4	2	no	—
184	C₂×Dic₁₃	—	8	2	no	—
185	D₅₂	C₂×D₈	1	1	yes	characteristic
186	C₄×D₁₃	—	2	1	no	—
187	C₄×D₁₃	—	8	2	no	—
188	Dic₂₆	—	4	2	no	—
189	Dic₂₆	—	4	2	no	—
190	Dic₂₆	—	8	2	no	—
191	Dic₂₆	C₂×D₈	1	1	yes	characteristic
192	C₈×C₁₃	—	2	1	no	—
193	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
194	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
195	#1055	—	2	1	no	—
196	#1055	—	4	2	no	—
197	#1055	C₂×D₈	1	1	yes	characteristic
198	#1055	C₂×D₈	1	1	yes	characteristic
199	#578	—	13	2	no	—
200	#577	—	26	2	no	—
201	#576	C₂×D₁₃	1	2	yes	—
202	#510	—	13	2	no	—
203	#510	—	26	2	no	—
204	#509	—	2	2	no	—
205	#509	C₂×D₁₃	1	2	yes	—
206	#508	—	2	1	no	—
207	#508	—	26	1	no	—
208	#508	C₂×D₁₃	1	1	yes	characteristic
209	C₂×Q₃₂	—	13	1	no	—
210	C₂×Q₃₂	—	26	1	no	—
211	C₂×SD₃₂	—	13	2	no	—
212	C₂×D₁₆	—	2	1	no	—
213	C₂×D₁₆	C₂×D₁₃	1	1	yes	characteristic
214	#504	—	13	1	no	—
215	#504	C₂×D₁₃	1	1	yes	characteristic
216	#495	—	13	2	no	—
217	#473	—	13	2	no	—
218	#472	—	13	2	no	—
219	#471	—	26	2	no	—
220	#455	—	26	2	no	—
221	#454	—	26	2	no	—
222	#443	—	13	1	no	—
223	#443	—	13	2	no	—
224	#368	—	13	1	no	—
225	#364	—	13	1	no	—
226	#362	—	13	2	no	—
227	#361	—	13	2	no	—
228	#359	—	13	2	no	—
229	#350	—	13	1	no	—
230	#350	—	13	1	no	—
231	C₄×C₁₆	—	13	1	no	—
232	C₂²×C₁₃	—	2	1	no	—
233	C₂²×C₁₃	—	4	2	no	—
234	C₂²×C₁₃	—	8	2	no	—
235	C₂²×C₁₃	—	16	2	no	—
236	C₂²×C₁₃	—	32	2	no	—
237	C₂²×C₁₃	#576	1	1	yes	characteristic
238	C₂×D₁₃	—	2	1	no	—
239	C₄×C₁₃	—	2	1	no	—
240	C₄×C₁₃	—	4	2	no	—
241	C₄×C₁₃	D₄²	1	1	yes	characteristic
242	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
243	Dic₁₃	—	2	1	no	—
244	Dic₁₃	—	4	1	no	—
245	Dic₁₃	—	8	2	no	—
246	2₋¹⁺⁴	—	26	2	no	—
247	H(5,𝔽₂)	—	2	2	no	—
248	#137	—	26	2	no	—
249	#137	—	26	2	no	—
250	#137	—	52	2	no	—
251	#137	—	52	2	no	—
252	AGL(1,ℤ/8)	—	2	2	no	—
253	AGL(1,ℤ/8)	—	4	2	no	—
254	AGL(1,ℤ/8)	—	26	2	no	—
255	AGL(1,ℤ/8)	—	52	2	no	—
256	#135	—	2	2	no	—
257	#135	—	13	1	no	—
258	#135	—	26	2	no	—
259	#135	—	52	2	no	—
260	#135	C₂²×D₁₃	1	2	yes	—
261	C₂×Q₁₆	—	13	2	no	—
262	C₂×Q₁₆	—	26	2	no	—
263	C₂×SD₁₆	—	26	2	no	—
264	C₂×SD₁₆	—	52	2	no	—
265	C₂×D₈	—	2	2	no	—
266	C₂×D₈	C₂²×D₁₃	1	2	yes	—
267	#131	—	13	1	no	—
268	#131	—	26	2	no	—
269	#131	C₂²×D₁₃	1	1	yes	characteristic
270	#124	—	26	2	no	—
271	Q₃₂	—	26	2	no	—
272	Q₃₂	—	26	1	no	—
273	Q₃₂	—	26	1	no	—
274	Q₃₂	C₂²⋊D₁₃	1	1	yes	characteristic
275	SD₃₂	—	2	2	no	—
276	SD₃₂	—	26	2	no	—
277	SD₃₂	—	52	2	no	—
278	SD₃₂	C₂²⋊D₁₃	1	2	yes	—
279	D₁₆	—	2	2	no	—
280	D₁₆	—	2	1	no	—
281	D₁₆	—	2	1	no	—
282	D₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
283	M₃₂	—	2	1	no	—
284	M₃₂	—	13	1	no	—
285	M₃₂	—	26	1	no	—
286	M₃₂	C₂²×D₁₃	1	1	yes	characteristic
287	C₂×C₁₆	—	2	1	no	—
288	C₂×C₁₆	—	13	1	no	—
289	C₂×C₁₆	—	26	1	no	—
290	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
291	#108	—	13	1	no	—
292	#108	—	13	2	no	—
293	#104	—	13	1	no	—
294	#104	—	13	1	no	—
295	#104	—	26	2	no	—
296	#101	—	26	2	no	—
297	#100	—	26	2	no	—
298	C₄×C₈	—	13	1	no	—
299	C₂×C₁₃	—	2	1	no	—
300	C₂×C₁₃	—	4	1	no	—
301	C₂×C₁₃	—	8	2	no	—
302	C₂×C₁₃	—	16	2	no	—
303	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
304	D₁₃	—	4	1	no	—
305	#41	—	2	2	no	—
306	#41	—	4	2	no	—
307	#41	—	4	2	no	—
308	#41	—	13	1	no	—
309	#41	—	52	2	no	—
310	#41	—	52	2	no	—
311	#41	—	104	2	no	—
312	#41	D₄×D₁₃	1	1	yes	characteristic
313	C₂×Q₈	—	26	2	no	—
314	C₂×Q₈	—	52	2	no	—
315	C₂×Q₈	—	52	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	4	2	no	—
318	C₂×D₄	—	4	2	no	—
319	C₂×D₄	—	8	2	no	—
320	Q₁₆	—	13	1	no	—
321	Q₁₆	—	26	2	no	—
322	Q₁₆	—	26	2	no	—
323	Q₁₆	—	52	2	no	—
324	Q₁₆	—	52	2	no	—
325	Q₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
326	SD₁₆	—	2	2	no	—
327	SD₁₆	—	4	2	no	—
328	SD₁₆	—	13	1	no	—
329	SD₁₆	—	13	1	no	—
330	SD₁₆	—	52	2	no	—
331	SD₁₆	—	52	2	no	—
332	SD₁₆	—	52	2	no	—
333	SD₁₆	—	52	2	no	—
334	SD₁₆	—	52	2	no	—
335	D₈	—	2	2	no	—
336	D₈	—	2	2	no	—
337	D₈	—	4	2	no	—
338	D₈	—	13	1	no	—
339	D₈	—	52	2	no	—
340	D₈	C₂×C₂²⋊D₁₃	1	2	yes	—
341	M₁₆	—	2	1	no	—
342	M₁₆	—	13	1	no	—
343	M₁₆	—	26	1	no	—
344	M₁₆	—	26	2	no	—
345	M₁₆	—	52	2	no	—
346	M₁₆	D₄×D₁₃	1	1	yes	characteristic
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	13	1	no	—
349	C₂×C₈	—	26	1	no	—
350	C₂×C₈	—	52	2	no	—
351	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
352	C₂²⋊C₄	—	104	2	no	—
353	C₄²	—	13	1	no	—
354	C₁₆	—	2	1	no	—
355	C₁₆	—	26	1	no	—
356	C₁₆	—	26	1	no	—
357	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
358	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
359	C₁₃	#33859	1	1	yes	characteristic, complemented, Sylow
360	C₂³	—	4	2	no	—
361	C₂³	—	8	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	13	1	no	—
364	Q₈	—	52	2	no	—
365	Q₈	—	52	2	no	—
366	Q₈	—	104	2	no	—
367	Q₈	D₈×D₁₃	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	2	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	8	2	no	—
374	D₄	—	8	2	no	—
375	D₄	—	13	1	no	—
376	D₄	—	26	1	no	—
377	D₄	—	104	2	no	—
378	D₄	D₈×D₁₃	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	8	2	no	—
382	C₂×C₄	—	26	1	no	—
383	C₂×C₄	—	26	1	no	—
384	C₂×C₄	—	52	2	no	—
385	C₂×C₄	—	104	2	no	—
386	C₂×C₄	—	104	2	no	—
387	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	13	1	no	—
390	C₈	—	13	1	no	—
391	C₈	—	26	1	no	—
392	C₈	—	52	2	no	—
393	C₈	#88251	1	1	yes	characteristic
394	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	16	2	no	—
399	C₂²	—	26	1	no	—
400	C₂²	—	32	2	no	—
401	C₂²	#11719462	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	26	1	no	—
405	C₄	—	52	1	no	—
406	C₄	—	104	2	no	—
407	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
408	C₄	D₄×C₂²⋊D₁₃	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	8	2	no	—
412	C₂	—	16	2	no	—
413	C₂	—	52	1	no	—
414	C₂	D₈×C₂²⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×C₂²⋊D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3415	C₂
socle	C₂×C₁₃	D₄×D₈
hypercenter	#3415	D₁₃
hypocenter	C₁₃	#33859
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #3415

Complements

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

N	H ≅ G/N	type
1. #13127986	C₂	semidirect
2. #13126618	C₂	semidirect
3. #13117094	C₂	semidirect
4. #13116961	C₂	semidirect
5. #13116953	C₂	semidirect
6. #13116903	C₂	semidirect
7. #13116807	C₂	semidirect
8. #13116797	C₂	semidirect
9. #13116717	C₂	semidirect
10. #13116671	C₂	semidirect
11. C₁₃×#3415	C₂	semidirect
14. #11719476	C₂²	semidirect
25. #11718857	C₂²	semidirect
26. #11718856	C₂²	semidirect
27. #11718821	C₂²	semidirect
28. #11718816	C₂²	semidirect
29. C₂×#88067	C₂²	semidirect
35. C₂×Dic₁₀₄	C₂²	semidirect
36. #11718562	C₂²	semidirect
37. #11718159	C₂²	semidirect
38. #11718155	C₂²	semidirect
40. #11718109	C₂²	semidirect
41. #11718107	C₂²	semidirect
42. #11718103	C₂²	semidirect
43. #11718098	C₂²	semidirect
44. C₁₆×Dic₁₃	C₂²	semidirect
154. #3415	D₁₃	semidirect
359. C₁₃	#33859	semidirect

Maximal subgroups

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

subgroup	index
#13127986	2
#13126618	2
#13117094	2
#13116961	2
#13116953	2
#13116903	2
#13116807	2
#13116797	2
#13116717	2
#13116671	2
C₁₃×#3415	2
#33859	13