Subgroups — #16807176

Order 3328.

646 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	D₁₃⋊#504	C₂	1	2	yes	maximal, complemented
2	#13131228	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131227	C₂	1	2	yes	maximal, complemented
4	#13131226	C₂	1	1	yes	maximal, characteristic, complemented
5	#13131225	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131224	C₂	1	2	yes	maximal, complemented
7	#13131223	C₂	1	1	yes	maximal, characteristic, complemented
8	#13131178	C₂	1	1	yes	maximal, characteristic, complemented
9	#13131079	C₂	1	2	yes	maximal, complemented
10	#13131046	C₂	1	1	yes	maximal, characteristic, complemented
11	#13123736	C₂	1	1	yes	maximal, characteristic
12	#11719473	C₄	1	1	yes	characteristic
13	D₁₃×#131	C₂²	1	2	yes	—
14	#11719162	C₂²	1	2	yes	complemented
15	#11719162	C₂²	1	2	yes	complemented
16	#11719096	—	2	2	no	—
17	#11719096	—	2	2	no	—
18	#11719096	C₂²	1	2	yes	complemented
19	#11719096	C₂²	1	2	yes	complemented
20	D₁₃⋊M₃₂	—	2	2	no	—
21	D₁₃⋊M₃₂	C₂²	1	2	yes	—
22	C₂×#88098	C₂²	1	1	yes	characteristic
23	C₂×D₁₃⋊C₁₆	—	2	2	no	—
24	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₁₃×#444	C₄	1	1	yes	characteristic
26	#11718857	C₂²	1	1	yes	characteristic
27	#11718806	C₄	1	2	yes	—
28	#11718556	C₂²	1	1	yes	characteristic
29	#11718554	C₄	1	1	yes	characteristic
30	#11718485	C₂²	1	2	yes	—
31	#11718482	C₄	1	2	yes	—
32	#11718378	C₄	1	1	yes	characteristic
33	#11718369	C₂²	1	1	yes	characteristic
34	#11718304	C₂²	1	2	yes	complemented
35	#11718304	C₂²	1	2	yes	complemented
36	#11718270	C₂²	1	2	yes	complemented
37	#11718270	C₂²	1	2	yes	complemented
38	#11718261	C₂²	1	1	yes	characteristic, complemented
39	#11718261	C₂²	1	1	yes	characteristic, complemented
40	#11718258	C₂²	1	1	yes	characteristic, complemented
41	#11718257	C₂²	1	1	yes	characteristic, complemented
42	#11718233	C₂²	1	2	yes	complemented
43	#11718231	C₂²	1	1	yes	characteristic, complemented
44	C₄×#6092	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	C₂×C₄	1	2	yes	—
48	#88204	C₂×C₄	1	2	yes	—
49	#88202	C₂³	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	C₂³	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	C₂×C₄	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	#88106	—	2	2	no	—
66	#88106	—	2	2	no	—
67	#88106	C₂³	1	2	yes	—
68	#88106	C₂³	1	2	yes	—
69	C₂×#6092	—	2	2	no	—
70	C₂×#6092	—	2	2	no	—
71	C₂×#6092	C₂³	1	1	yes	characteristic
72	C₂×#6092	C₂³	1	1	yes	characteristic
73	C₂×#6092	C₂³	1	1	yes	characteristic
74	C₂×#6092	C₂³	1	1	yes	characteristic
75	#88098	—	2	2	no	—
76	#88098	—	2	2	no	—
77	#88098	—	2	2	no	—
78	#88098	—	2	2	no	—
79	D₁₃⋊C₁₆	—	2	2	no	—
80	D₁₃⋊C₁₆	—	2	2	no	—
81	D₁₃⋊C₁₆	—	2	2	no	—
82	D₁₃⋊C₁₆	D₄	1	2	yes	—
83	D₁₃⋊C₁₆	D₄	1	2	yes	—
84	C₁₃×#108	C₂×C₄	1	1	yes	characteristic
85	C₁₃×#104	C₂×C₄	1	2	yes	—
86	C₁₃×#97	C₂×C₄	1	1	yes	characteristic
87	#88077	C₂×C₄	1	2	yes	—
88	#88065	C₂×C₄	1	2	yes	—
89	#88062	C₂×C₄	1	2	yes	—
90	#88059	C₂×C₄	1	1	yes	characteristic
91	#88055	C₂×C₄	1	1	yes	characteristic
92	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
93	#88045	C₂×C₄	1	2	yes	—
94	C₄×#1055	C₂³	1	1	yes	characteristic
95	#33441	—	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	C₂²×C₄	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	C₂⁴	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	#6092	—	2	2	no	—
142	#6092	—	2	2	no	—
143	#6092	—	2	1	no	—
144	#6092	—	2	1	no	—
145	#6092	C₂×D₄	1	1	yes	characteristic
146	#6092	C₂×D₄	1	2	yes	—
147	#6092	C₂×D₄	1	1	yes	characteristic
148	C₂×#504	—	13	2	no	—
149	#2955	—	13	1	no	—
150	#2171	—	13	1	no	—
151	#2171	—	13	2	no	—
152	#2171	—	13	1	no	—
153	#2170	—	13	1	no	—
154	#2170	—	13	2	no	—
155	#2170	—	13	1	no	—
156	#2153	—	13	1	no	—
157	#2117	—	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	C₄×D₄	1	2	yes	—
184	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
185	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
186	#1055	—	2	2	no	—
187	#1055	C₂²×C₈	1	2	yes	—
188	#1055	C₂²×D₄	1	1	yes	characteristic
189	#1055	C₂×#41	1	1	yes	characteristic
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	—	13	2	no	—
196	#504	—	26	2	no	—
197	#504	—	26	2	no	—
198	C₂×M₃₂	—	13	1	no	—
199	C₂×M₃₂	—	13	2	no	—
200	C₂×M₃₂	—	26	2	no	—
201	C₂²×C₁₆	—	13	1	no	—
202	C₂²×C₁₆	—	26	2	no	—
203	#444	—	13	2	no	—
204	#444	—	13	1	no	—
205	#444	#261	1	1	yes	characteristic
206	#443	—	13	1	no	—
207	#443	—	13	1	no	—
208	#443	—	13	2	no	—
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	—	13	1	no	—
215	#350	—	13	2	no	—
216	#350	—	13	2	no	—
217	#350	—	13	2	no	—
218	#350	—	13	2	no	—
219	#346	—	13	1	no	—
220	#346	—	13	1	no	—
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	—	4	4	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	C₂×#261	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	C₂×#261	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	—	13	2	no	—
257	M₃₂	—	13	2	no	—
258	M₃₂	—	26	2	no	—
259	M₃₂	—	26	2	no	—
260	M₃₂	—	26	2	no	—
261	M₃₂	—	26	2	no	—
262	M₃₂	—	26	2	no	—
263	M₃₂	—	26	2	no	—
264	C₂×C₁₆	—	13	2	no	—
265	C₂×C₁₆	—	13	2	no	—
266	C₂×C₁₆	—	13	1	no	—
267	C₂×C₁₆	—	13	1	no	—
268	C₂×C₁₆	—	13	1	no	—
269	C₂×C₁₆	—	13	1	no	—
270	C₂×C₁₆	—	26	2	no	—
271	C₂×C₁₆	—	26	2	no	—
272	C₂×C₁₆	—	26	2	no	—
273	C₂×C₁₆	—	26	2	no	—
274	C₂×C₁₆	—	26	2	no	—
275	#108	—	13	2	no	—
276	#108	—	13	1	no	—
277	#108	C₂×#261	1	1	yes	characteristic
278	#104	—	13	2	no	—
279	#104	—	13	2	no	—
280	#104	—	13	2	no	—
281	#104	C₂×#261	1	2	yes	—
282	#97	—	13	1	no	—
283	#97	C₂×#261	1	1	yes	characteristic
284	C₄×C₈	—	13	1	no	—
285	C₄×C₈	—	13	1	no	—
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	C₂²×#261	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	D₁₃⋊C₈	1	1	yes	characteristic
305	SD₁₆	—	13	2	no	—
306	SD₁₆	—	13	2	no	—
307	SD₁₆	—	13	2	no	—
308	SD₁₆	D₁₃⋊C₈	1	2	yes	—
309	D₈	—	13	2	no	—
310	D₈	—	13	1	no	—
311	D₈	D₁₃⋊C₈	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	C₂²×#261	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	C₂²×#261	1	1	yes	characteristic
331	C₂×C₈	C₂²×#261	1	1	yes	characteristic
332	C₄⋊C₄	—	26	1	no	—
333	C₂²⋊C₄	—	26	1	no	—
334	C₄²	—	13	1	no	—
335	C₄²	C₂²×#261	1	1	yes	characteristic
336	C₁₆	—	13	1	no	—
337	C₁₆	—	13	2	no	—
338	C₁₆	—	13	1	no	—
339	C₁₆	—	26	1	no	—
340	C₁₆	—	26	1	no	—
341	C₁₆	—	26	2	no	—
342	C₁₆	—	26	2	no	—
343	C₁₃	#33441	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×D₁₃⋊C₈	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×D₁₃⋊C₈	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	C₂³×#261	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	13	2	no	—
369	C₈	—	13	1	no	—
370	C₈	—	13	1	no	—
371	C₈	—	26	2	no	—
372	C₈	C₂×D₁₃⋊C₈	1	1	yes	characteristic
373	C₈	C₂×D₁₃⋊C₈	1	1	yes	characteristic
374	C₈	D₄×#261	1	2	yes	—
375	C₂²	—	2	2	no	—
376	C₂²	—	13	2	no	—
377	C₂²	—	26	2	no	—
378	C₂²	—	52	1	no	—
379	C₂²	—	52	4	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	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
387	C₄	C₂×D₄×#261	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	D₄×D₁₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#444	C₄
socle	C₂×C₁₃	C₂×C₈×D₄
prime radical	#11719473	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#444	#261
hypocenter	C₁₃	#33441
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
1. D₁₃⋊#504	C₂	semidirect
2. #13131228	C₂	semidirect
3. #13131227	C₂	semidirect
4. #13131226	C₂	semidirect
5. #13131225	C₂	semidirect
6. #13131224	C₂	semidirect
7. #13131223	C₂	semidirect
8. #13131178	C₂	semidirect
9. #13131079	C₂	semidirect
10. #13131046	C₂	semidirect
14. #11719162	C₂²	semidirect
15. #11719162	C₂²	semidirect
18. #11719096	C₂²	semidirect
19. #11719096	C₂²	semidirect
34. #11718304	C₂²	semidirect
35. #11718304	C₂²	semidirect
36. #11718270	C₂²	semidirect
37. #11718270	C₂²	semidirect
38. #11718261	C₂²	semidirect
39. #11718261	C₂²	semidirect
40. #11718258	C₂²	semidirect
41. #11718257	C₂²	semidirect
42. #11718233	C₂²	semidirect
43. #11718231	C₂²	semidirect
44. C₄×#6092	C₂²	semidirect
343. C₁₃	#33441	semidirect

Maximal subgroups

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

subgroup	index
D₁₃⋊#504	2
#13131228	2
#13131227	2
#13131226	2
#13131225	2
#13131224	2
#13131223	2
#13131178	2
#13131079	2
#13131046	2
#13123736	2
#33441	13