Subgroups — #16016165

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127937	C₂	1	2	yes	maximal, complemented
2	#13123734	C₂	1	1	yes	maximal, characteristic
3	#13116679	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116675	C₂	1	2	yes	maximal, complemented
5	#13116673	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116672	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116671	C₂	1	2	yes	maximal, complemented
8	#13116600	C₂	1	1	yes	maximal, characteristic, complemented
9	D₁₃×#350	C₂	1	2	yes	maximal, complemented
10	#13116434	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2171	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719473	C₄	1	1	yes	characteristic
13	D₁₃×#131	C₂²	1	2	yes	—
14	C₁₃×#504	C₂²	1	2	yes	complemented
15	C₁₃×#443	C₂²	1	1	yes	characteristic
16	#11718858	C₄	1	1	yes	characteristic
17	#11718805	C₂²	1	2	yes	—
18	#11718801	C₂²	1	2	yes	complemented
19	#11718568	—	2	2	no	—
20	#11718568	C₂²	1	2	yes	complemented
21	D₁₃×M₃₂	—	2	2	no	—
22	D₁₃×M₃₂	C₂²	1	2	yes	—
23	#11718562	—	2	2	no	—
24	#11718562	C₂²	1	2	yes	complemented
25	C₂×#88038	—	2	2	no	—
26	C₂×#88038	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
28	#11718555	C₄	1	1	yes	characteristic
29	#11718554	C₄	1	1	yes	characteristic
30	#11718484	C₄	1	2	yes	—
31	#11718482	C₄	1	2	yes	—
32	#11718361	C₂²	1	1	yes	characteristic
33	#11718350	C₂²	1	1	yes	characteristic
34	C₁₃×#364	C₂²	1	1	yes	characteristic, complemented
35	C₁₃×#350	C₂²	1	2	yes	complemented
36	C₁₃×#346	C₂²	1	1	yes	characteristic, complemented
37	#11718153	C₂²	1	2	yes	complemented
38	#11718107	C₂²	1	2	yes	complemented
39	#11718103	C₂²	1	2	yes	complemented
40	#11718100	C₂²	1	2	yes	complemented
41	#11718094	C₂²	1	1	yes	characteristic, complemented
42	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
43	#11718055	C₂²	1	1	yes	characteristic, complemented
44	C₄×#6090	C₂²	1	1	yes	characteristic, complemented
45	#41×D₁₃	C₂×C₄	1	2	yes	—
46	C₁₃×#135	C₂×C₄	1	1	yes	characteristic
47	C₁₃×#131	C₂³	1	2	yes	—
48	#88204	C₂×C₄	1	2	yes	—
49	#88202	C₂×C₄	1	2	yes	—
50	#88172	C₈	1	2	yes	—
51	#88169	C₈	1	2	yes	—
52	#88168	C₈	1	2	yes	—
53	#88165	C₈	1	2	yes	—
54	#88161	—	2	2	no	—
55	#88161	C₂×C₄	1	2	yes	—
56	M₁₆×D₁₃	—	2	2	no	—
57	M₁₆×D₁₃	C₂×C₄	1	2	yes	—
58	#88158	C₂×C₄	1	1	yes	characteristic
59	#88155	—	2	2	no	—
60	#88155	C₂³	1	2	yes	—
61	C₂×#6094	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×D₁₃	—	2	2	no	—
63	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
64	#88126	C₂×C₄	1	1	yes	characteristic
65	C₁₃×M₃₂	—	2	2	no	—
66	C₁₃×M₃₂	C₂³	1	2	yes	—
67	C₂×C₁₃×C₁₆	—	2	2	no	—
68	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₁₃×#108	C₂×C₄	1	1	yes	characteristic
71	C₁₃×#104	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	#88077	C₂×C₄	1	2	yes	—
74	#88065	C₂×C₄	1	2	yes	—
75	#88062	C₂×C₄	1	2	yes	—
76	#88059	C₂×C₄	1	1	yes	characteristic
77	#88055	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
79	#88052	—	2	2	no	—
80	#88052	C₂³	1	2	yes	—
81	C₂×#6090	—	2	2	no	—
82	C₂×#6090	C₂³	1	1	yes	characteristic
83	C₂×#6090	C₂³	1	1	yes	characteristic
84	#88045	C₂×C₄	1	2	yes	—
85	#88043	C₂×C₄	1	1	yes	characteristic
86	#88038	—	2	2	no	—
87	#88038	—	2	2	no	—
88	#88038	—	2	2	no	—
89	#88038	D₄	1	2	yes	—
90	#88038	D₄	1	2	yes	—
91	C₁₆×D₁₃	—	2	2	no	—
92	C₁₆×D₁₃	—	2	2	no	—
93	C₁₆×D₁₃	—	2	2	no	—
94	C₁₆×D₁₃	—	2	2	no	—
95	#33442	—	13	1	no	maximal, Sylow
96	C₁₃×#41	C₂²×C₄	1	2	yes	—
97	#6131	C₂×C₈	1	2	yes	—
98	Q₈×D₁₃	C₂×C₈	1	2	yes	—
99	#6129	—	2	2	no	—
100	#6129	C₂×C₈	1	2	yes	—
101	D₄×D₁₃	—	2	2	no	—
102	D₄×D₁₃	C₂×C₈	1	2	yes	—
103	#6127	—	2	2	no	—
104	#6127	C₂²×C₄	1	2	yes	—
105	C₂×C₄×D₁₃	—	2	2	no	—
106	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₁₃×Q₁₆	C₂×C₈	1	1	yes	characteristic
108	C₁₃×SD₁₆	C₂×C₈	1	2	yes	—
109	C₁₃×D₈	C₂×C₈	1	1	yes	characteristic
110	C₁₃×M₁₆	—	2	2	no	—
111	C₁₃×M₁₆	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₁₃	—	2	2	no	—
113	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
115	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
116	#6107	C₂×C₈	1	2	yes	—
117	#6106	C₂×C₈	1	2	yes	—
118	#6105	C₂×C₈	1	2	yes	—
119	#6104	C₂×C₈	1	2	yes	—
120	#6103	—	2	1	no	—
121	#6101	—	2	1	no	—
122	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
123	#6099	—	2	2	no	—
124	#6099	C₂²×C₄	1	2	yes	—
125	C₂×#1055	—	2	2	no	—
126	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
127	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
128	Dic₅₂	C₂×C₈	1	1	yes	characteristic
129	D₁₀₄	C₂×C₈	1	1	yes	characteristic
130	#6095	C₂×C₈	1	2	yes	—
131	#6094	—	2	2	no	—
132	#6094	—	2	2	no	—
133	#6094	—	2	2	no	—
134	#6094	C₂×C₈	1	2	yes	—
135	#6094	C₂×C₈	1	2	yes	—
136	C₈×D₁₃	—	2	2	no	—
137	C₈×D₁₃	—	2	2	no	—
138	C₈×D₁₃	—	2	2	no	—
139	C₈×D₁₃	C₂×D₄	1	2	yes	—
140	C₈×D₁₃	#41	1	2	yes	—
141	C₁₃×C₁₆	—	2	2	no	—
142	C₁₃×C₁₆	—	2	1	no	—
143	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
144	#6090	—	2	2	no	—
145	#6090	—	2	1	no	—
146	#6090	C₂×D₄	1	1	yes	characteristic
147	#6090	C₂×D₄	1	1	yes	characteristic
148	#3407	—	13	2	no	—
149	#2955	—	13	1	no	—
150	#2171	—	13	2	no	—
151	#2171	—	13	1	no	—
152	#2171	D₁₃	1	1	yes	characteristic, complemented
153	#2170	—	13	2	no	—
154	#2170	—	13	1	no	—
155	#2170	—	13	1	no	—
156	#2153	—	13	1	no	—
157	C₂×#350	—	13	2	no	—
158	#2108	—	13	1	no	—
159	C₂²×D₁₃	—	2	2	no	—
160	C₂²×D₁₃	#131	1	1	yes	characteristic
161	Q₈×C₁₃	C₂²×C₈	1	2	yes	—
162	D₄×C₁₃	—	2	2	no	—
163	D₄×C₁₃	C₂²×C₈	1	2	yes	—
164	C₂×C₄×C₁₃	—	2	2	no	—
165	C₂×C₄×C₁₃	—	2	1	no	—
166	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
167	C₂²⋊D₁₃	—	2	2	no	—
168	C₂²⋊D₁₃	—	2	2	no	—
169	C₂²⋊D₁₃	—	2	2	no	—
170	C₂×Dic₁₃	—	2	2	no	—
171	C₂×Dic₁₃	—	2	1	no	—
172	C₂×Dic₁₃	#131	1	1	yes	characteristic
173	D₅₂	—	2	2	no	—
174	D₅₂	C₂²×C₈	1	2	yes	—
175	C₄×D₁₃	—	2	2	no	—
176	C₄×D₁₃	—	2	2	no	—
177	C₄×D₁₃	—	2	2	no	—
178	C₄×D₁₃	C₄×D₄	1	2	yes	—
179	C₄×D₁₃	C₂²×C₈	1	2	yes	—
180	Dic₂₆	—	2	2	no	—
181	Dic₂₆	C₂²×C₈	1	2	yes	—
182	C₈×C₁₃	—	2	2	no	—
183	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
184	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
185	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
186	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
187	#1055	—	2	2	no	—
188	#1055	C₄×D₄	1	2	yes	—
189	#1055	C₂²×C₈	1	2	yes	—
190	#575	—	13	1	no	—
191	C₂×#131	—	13	2	no	—
192	#504	—	13	2	no	—
193	#504	—	13	2	no	—
194	#504	—	13	2	no	—
195	#504	—	26	2	no	—
196	#504	—	26	2	no	—
197	#504	C₂×D₁₃	1	2	yes	complemented
198	C₂×M₃₂	—	13	1	no	—
199	C₂×M₃₂	—	13	2	no	—
200	C₂×M₃₂	—	26	2	no	—
201	C₂×M₃₂	—	26	2	no	—
202	C₂²×C₁₆	—	13	1	no	—
203	#444	—	13	1	no	—
204	#444	—	13	1	no	—
205	#444	—	13	2	no	—
206	#443	—	13	2	no	—
207	#443	—	13	1	no	—
208	#443	C₂×D₁₃	1	1	yes	characteristic
209	#430	—	13	1	no	—
210	#421	—	13	2	no	—
211	#405	—	13	1	no	—
212	#364	—	13	2	no	—
213	#364	—	13	1	no	—
214	#364	C₂×D₁₃	1	1	yes	characteristic, complemented
215	#350	—	13	2	no	—
216	#350	—	13	2	no	—
217	#350	—	13	2	no	—
218	#350	C₂×D₁₃	1	2	yes	complemented
219	#346	—	13	1	no	—
220	#346	C₂×D₁₃	1	1	yes	characteristic, complemented
221	C₄×C₁₆	—	13	1	no	—
222	C₄×C₁₆	—	13	1	no	—
223	C₂²×C₁₃	—	2	2	no	—
224	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
225	C₂×D₁₃	—	2	2	no	—
226	C₂×D₁₃	—	4	1	no	—
227	C₂×D₁₃	—	8	2	no	—
228	C₂×D₁₃	C₈×D₄	1	2	yes	—
229	C₄×C₁₃	—	2	2	no	—
230	C₄×C₁₃	—	4	1	no	—
231	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
232	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
233	Dic₁₃	—	2	2	no	—
234	Dic₁₃	—	4	1	no	—
235	Dic₁₃	C₈×D₄	1	2	yes	—
236	C₂×#41	—	13	2	no	—
237	#137	—	13	2	no	—
238	#137	—	13	2	no	—
239	AGL(1,ℤ/8)	—	13	2	no	—
240	AGL(1,ℤ/8)	—	13	2	no	—
241	#135	—	13	2	no	—
242	#135	—	13	1	no	—
243	#135	C₄×D₁₃	1	1	yes	characteristic
244	#131	—	13	2	no	—
245	#131	—	13	2	no	—
246	#131	—	13	2	no	—
247	#131	—	26	2	no	—
248	#131	—	26	2	no	—
249	#131	C₂²×D₁₃	1	2	yes	—
250	C₂×M₁₆	—	13	1	no	—
251	C₂×M₁₆	—	13	2	no	—
252	C₂×M₁₆	—	26	2	no	—
253	C₂²×C₈	—	13	1	no	—
254	C₂²×C₈	—	26	2	no	—
255	#117	—	13	1	no	—
256	M₃₂	—	2	2	no	—
257	M₃₂	—	13	2	no	—
258	M₃₂	—	13	2	no	—
259	M₃₂	—	13	2	no	—
260	M₃₂	—	26	2	no	—
261	M₃₂	—	26	2	no	—
262	M₃₂	—	26	2	no	—
263	M₃₂	—	26	2	no	—
264	M₃₂	C₂²×D₁₃	1	2	yes	—
265	C₂×C₁₆	—	2	2	no	—
266	C₂×C₁₆	—	13	1	no	—
267	C₂×C₁₆	—	13	1	no	—
268	C₂×C₁₆	—	26	2	no	—
269	C₂×C₁₆	—	26	2	no	—
270	C₂×C₁₆	—	26	2	no	—
271	C₂×C₁₆	—	26	2	no	—
272	C₂×C₁₆	—	26	2	no	—
273	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
274	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
275	#108	—	13	2	no	—
276	#108	—	13	1	no	—
277	#108	C₄×D₁₃	1	1	yes	characteristic
278	#104	—	13	2	no	—
279	#104	—	13	2	no	—
280	#104	—	13	2	no	—
281	#104	C₄×D₁₃	1	2	yes	—
282	#97	—	13	1	no	—
283	#97	—	13	1	no	—
284	C₄×C₈	—	13	1	no	—
285	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
286	C₂×C₁₃	—	2	1	no	—
287	C₂×C₁₃	—	4	2	no	—
288	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
289	D₁₃	—	2	2	no	—
290	D₁₃	—	4	2	no	—
291	#41	—	13	2	no	—
292	#41	—	13	2	no	—
293	#41	—	13	2	no	—
294	#41	—	26	2	no	—
295	#41	—	26	2	no	—
296	#41	C₂×C₄×D₁₃	1	2	yes	—
297	C₂×Q₈	—	13	2	no	—
298	C₂×D₄	—	13	2	no	—
299	C₂×D₄	—	26	2	no	—
300	C₂²×C₄	—	13	1	no	—
301	C₂²×C₄	—	26	2	no	—
302	Q₁₆	—	13	2	no	—
303	Q₁₆	—	13	1	no	—
304	Q₁₆	C₈×D₁₃	1	1	yes	characteristic
305	SD₁₆	—	13	2	no	—
306	SD₁₆	—	13	2	no	—
307	SD₁₆	—	13	2	no	—
308	SD₁₆	C₈×D₁₃	1	2	yes	—
309	D₈	—	13	2	no	—
310	D₈	—	13	1	no	—
311	D₈	C₈×D₁₃	1	1	yes	characteristic
312	M₁₆	—	2	2	no	—
313	M₁₆	—	13	2	no	—
314	M₁₆	—	13	2	no	—
315	M₁₆	—	13	2	no	—
316	M₁₆	—	26	2	no	—
317	M₁₆	—	26	2	no	—
318	M₁₆	—	26	2	no	—
319	M₁₆	—	26	2	no	—
320	M₁₆	C₂×C₄×D₁₃	1	2	yes	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	13	1	no	—
323	C₂×C₈	—	13	1	no	—
324	C₂×C₈	—	13	2	no	—
325	C₂×C₈	—	13	2	no	—
326	C₂×C₈	—	26	2	no	—
327	C₂×C₈	—	26	2	no	—
328	C₂×C₈	—	26	2	no	—
329	C₂×C₈	—	26	2	no	—
330	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
331	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
332	C₄⋊C₄	—	26	1	no	—
333	C₂²⋊C₄	—	26	1	no	—
334	C₄²	—	13	1	no	—
335	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
336	C₁₆	—	2	2	no	—
337	C₁₆	—	2	1	no	—
338	C₁₆	—	13	1	no	—
339	C₁₆	—	13	1	no	—
340	C₁₆	—	26	2	no	—
341	C₁₆	—	26	1	no	—
342	C₁₆	D₄×D₁₃	1	2	yes	—
343	C₁₃	#33442	1	1	yes	characteristic, complemented, Sylow
344	C₂³	—	13	1	no	—
345	C₂³	—	26	2	no	—
346	Q₈	—	13	2	no	—
347	Q₈	—	26	2	no	—
348	Q₈	C₂×C₈×D₁₃	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	13	2	no	—
351	D₄	—	26	2	no	—
352	D₄	—	26	2	no	—
353	D₄	—	26	2	no	—
354	D₄	—	26	2	no	—
355	D₄	C₂×C₈×D₁₃	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	13	1	no	—
359	C₂×C₄	—	13	2	no	—
360	C₂×C₄	—	13	2	no	—
361	C₂×C₄	—	26	2	no	—
362	C₂×C₄	—	26	2	no	—
363	C₂×C₄	—	26	2	no	—
364	C₂×C₄	—	26	2	no	—
365	C₂×C₄	—	26	1	no	—
366	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	13	2	no	—
369	C₈	—	13	2	no	—
370	C₈	—	26	2	no	—
371	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
372	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
373	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
374	C₈	#41×D₁₃	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	13	2	no	—
377	C₂²	—	26	2	no	—
378	C₂²	—	52	1	no	—
379	C₂²	—	104	2	no	—
380	C₂²	D₁₃×#131	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	13	2	no	—
384	C₄	—	26	2	no	—
385	C₄	—	52	1	no	—
386	C₄	C₄×D₄×D₁₃	1	1	yes	characteristic
387	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	26	2	no	—
391	C₂	—	52	2	no	—
392	C₂	C₈×D₄×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2171	C₂
socle	C₂×C₁₃	C₂×C₈×D₄
prime radical	#11719473	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2171	D₁₃
hypocenter	C₁₃	#33442
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #2171

Complements

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

N	H ≅ G/N	type
1. #13127937	C₂	semidirect
3. #13116679	C₂	semidirect
4. #13116675	C₂	semidirect
5. #13116673	C₂	semidirect
6. #13116672	C₂	semidirect
7. #13116671	C₂	semidirect
8. #13116600	C₂	semidirect
9. D₁₃×#350	C₂	semidirect
10. #13116434	C₂	semidirect
11. C₁₃×#2171	C₂	semidirect
14. C₁₃×#504	C₂²	semidirect
18. #11718801	C₂²	semidirect
20. #11718568	C₂²	semidirect
24. #11718562	C₂²	semidirect
34. C₁₃×#364	C₂²	semidirect
35. C₁₃×#350	C₂²	semidirect
36. C₁₃×#346	C₂²	semidirect
37. #11718153	C₂²	semidirect
38. #11718107	C₂²	semidirect
39. #11718103	C₂²	semidirect
40. #11718100	C₂²	semidirect
41. #11718094	C₂²	semidirect
42. C₁₆×Dic₁₃	C₂²	semidirect
43. #11718055	C₂²	semidirect
44. C₄×#6090	C₂²	semidirect
152. #2171	D₁₃	semidirect
197. #504	C₂×D₁₃	semidirect
214. #364	C₂×D₁₃	semidirect
218. #350	C₂×D₁₃	semidirect
220. #346	C₂×D₁₃	semidirect
343. C₁₃	#33442	semidirect

Maximal subgroups

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

subgroup	index
#13127937	2
#13123734	2
#13116679	2
#13116675	2
#13116673	2
#13116672	2
#13116671	2
#13116600	2
D₁₃×#350	2
#13116434	2
C₁₃×#2171	2
#33442	13