Subgroups — #16807183

Order 3328.

648 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131804	C₂	1	2	yes	maximal, complemented
2	#13131227	C₂	1	4	yes	maximal, complemented
3	#13131225	C₂	1	2	yes	maximal, complemented
4	#13131223	C₂	1	2	yes	maximal, complemented
5	D₁₃⋊#364	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131079	C₂	1	2	yes	maximal, complemented
7	#13131043	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₃×#443	C₂	1	1	yes	maximal, characteristic
9	D₁₃×#135	C₄	1	1	yes	characteristic
10	D₁₃×#131	C₂²	1	2	yes	—
11	#11719162	C₂²	1	4	yes	complemented
12	#11719096	—	2	4	no	—
13	#11719096	C₂²	1	4	yes	complemented
14	D₁₃⋊M₃₂	—	2	2	no	—
15	D₁₃⋊M₃₂	C₂²	1	2	yes	—
16	C₂×#88098	—	2	2	no	—
17	C₂×#88098	C₂²	1	1	yes	characteristic
18	C₂×#88098	C₂²	1	1	yes	characteristic
19	C₁₃×#443	C₄	1	1	yes	characteristic
20	#11718857	C₂²	1	1	yes	characteristic
21	#11718805	C₄	1	2	yes	—
22	#11718556	C₂²	1	1	yes	characteristic
23	D₁₃×#108	C₄	1	1	yes	characteristic
24	#11718485	C₂²	1	2	yes	—
25	D₁₃×#104	C₄	1	2	yes	—
26	#11718361	C₄	1	1	yes	characteristic
27	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
28	#11718304	C₂²	1	4	yes	complemented
29	#11718270	C₂²	1	4	yes	complemented
30	#11718261	C₂²	1	2	yes	complemented
31	#11718257	C₂²	1	2	yes	complemented
32	#11718233	C₂²	1	2	yes	complemented
33	#11718231	C₂²	1	2	yes	complemented
34	#41×D₁₃	C₂×C₄	1	2	yes	—
35	C₁₃×#135	C₂×C₄	1	1	yes	characteristic
36	C₁₃×#131	C₂×C₄	1	2	yes	—
37	#88204	C₂×C₄	1	2	yes	—
38	#88202	C₂³	1	2	yes	—
39	#88173	C₈	1	1	yes	characteristic
40	Q₁₆×D₁₃	C₈	1	1	yes	characteristic
41	#88170	C₈	1	2	yes	—
42	SD₁₆×D₁₃	C₈	1	2	yes	—
43	#88166	C₈	1	1	yes	characteristic
44	D₈×D₁₃	C₈	1	1	yes	characteristic
45	#88161	—	2	2	no	—
46	#88161	C₂³	1	2	yes	—
47	M₁₆×D₁₃	—	2	2	no	—
48	M₁₆×D₁₃	C₂×C₄	1	2	yes	—
49	#88158	C₂×C₄	1	1	yes	characteristic
50	#88155	—	2	2	no	—
51	#88155	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₁₃	—	2	2	no	—
53	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
55	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
56	#88106	—	2	4	no	—
57	#88106	C₂³	1	4	yes	—
58	C₂×#6092	—	2	4	no	—
59	C₂×#6092	C₂³	1	2	yes	—
60	C₂×#6092	C₂³	1	2	yes	—
61	#88098	—	2	2	no	—
62	#88098	—	2	1	no	—
63	#88098	—	2	1	no	—
64	#88098	—	2	4	no	—
65	#88098	D₄	1	1	yes	characteristic
66	#88098	D₄	1	1	yes	characteristic
67	#88098	D₄	1	2	yes	—
68	D₁₃⋊C₁₆	—	2	2	no	—
69	D₁₃⋊C₁₆	—	2	4	no	—
70	C₁₃×#108	C₂×C₄	1	1	yes	characteristic
71	C₁₃×#104	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₁₃	C₂×C₄	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	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
79	#88045	C₂×C₄	1	2	yes	—
80	C₄×#1055	C₂³	1	1	yes	characteristic
81	#33443	—	13	1	no	maximal, Sylow
82	C₁₃×#41	C₂²×C₄	1	2	yes	—
83	#6131	C₂×C₈	1	2	yes	—
84	Q₈×D₁₃	C₂×C₈	1	2	yes	—
85	#6129	—	2	2	no	—
86	#6129	C₂×C₈	1	2	yes	—
87	D₄×D₁₃	—	2	2	no	—
88	D₄×D₁₃	C₂×C₈	1	2	yes	—
89	#6127	—	2	2	no	—
90	#6127	C₂²×C₄	1	2	yes	—
91	C₂×C₄×D₁₃	—	2	2	no	—
92	C₂×C₄×D₁₃	—	2	1	no	—
93	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₁₃×Q₁₆	C₂×C₈	1	1	yes	characteristic
95	C₁₃×SD₁₆	C₂×C₈	1	2	yes	—
96	C₁₃×D₈	C₂×C₈	1	1	yes	characteristic
97	C₁₃×M₁₆	—	2	2	no	—
98	C₁₃×M₁₆	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	#6107	C₂×C₈	1	2	yes	—
104	#6106	C₂×C₈	1	2	yes	—
105	#6105	C₂×C₈	1	2	yes	—
106	#6104	C₂×C₈	1	2	yes	—
107	C₄×Dic₁₃	—	2	1	no	—
108	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
109	#6099	—	2	2	no	—
110	#6099	C₂²×C₄	1	2	yes	—
111	C₂×#1055	—	2	2	no	—
112	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
113	C₂×#1055	C₂⁴	1	1	yes	characteristic
114	Dic₅₂	C₂×C₈	1	1	yes	characteristic
115	D₁₀₄	C₂×C₈	1	1	yes	characteristic
116	#6095	C₂×C₈	1	2	yes	—
117	#6094	—	2	2	no	—
118	#6094	—	2	2	no	—
119	#6094	—	2	2	no	—
120	C₈×D₁₃	—	2	2	no	—
121	C₈×D₁₃	—	2	2	no	—
122	C₈×D₁₃	—	2	2	no	—
123	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
124	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
125	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
126	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
127	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
128	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
129	C₈×D₁₃	#41	1	1	yes	characteristic
130	C₈×D₁₃	#41	1	1	yes	characteristic
131	#6092	—	2	4	no	—
132	#6092	—	2	2	no	—
133	#6092	C₂×D₄	1	2	yes	—
134	#6092	C₂×D₄	1	2	yes	—
135	#3407	—	13	2	no	—
136	C₂×#443	—	13	1	no	—
137	#2171	—	13	4	no	—
138	#2170	—	13	2	no	—
139	#2170	—	13	2	no	—
140	C₂×#364	—	13	1	no	—
141	#2117	—	13	2	no	—
142	C₄×M₃₂	—	13	1	no	—
143	C₂²×D₁₃	—	2	2	no	—
144	C₂²×D₁₃	#131	1	1	yes	characteristic
145	Q₈×C₁₃	C₂²×C₈	1	2	yes	—
146	D₄×C₁₃	—	2	2	no	—
147	D₄×C₁₃	C₂²×C₈	1	2	yes	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	1	no	—
150	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
151	C₂²⋊D₁₃	—	2	2	no	—
152	C₂²⋊D₁₃	—	2	2	no	—
153	C₂²⋊D₁₃	—	2	2	no	—
154	C₂×Dic₁₃	—	2	1	no	—
155	C₂×Dic₁₃	—	2	2	no	—
156	C₂×Dic₁₃	#131	1	1	yes	characteristic
157	D₅₂	—	2	2	no	—
158	D₅₂	C₂²×C₈	1	2	yes	—
159	C₄×D₁₃	—	2	2	no	—
160	C₄×D₁₃	—	2	2	no	—
161	C₄×D₁₃	—	2	2	no	—
162	C₄×D₁₃	—	4	2	no	—
163	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
164	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
165	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
166	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
167	Dic₂₆	—	2	2	no	—
168	Dic₂₆	C₂²×C₈	1	2	yes	—
169	C₈×C₁₃	—	2	2	no	—
170	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
171	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
172	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
173	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
174	#1055	—	2	2	no	—
175	#1055	C₂²×C₈	1	1	yes	characteristic
176	#1055	C₂²×C₈	1	1	yes	characteristic
177	#1055	C₂²×D₄	1	1	yes	characteristic
178	#1055	C₂×#41	1	1	yes	characteristic
179	C₂×#135	—	13	1	no	—
180	C₂×#131	—	13	2	no	—
181	#504	—	13	4	no	—
182	#504	—	13	4	no	—
183	#504	—	26	4	no	—
184	C₂×M₃₂	—	13	1	no	—
185	C₂×M₃₂	—	13	1	no	—
186	C₂×M₃₂	—	13	2	no	—
187	C₂×M₃₂	—	26	2	no	—
188	C₂×M₃₂	—	26	2	no	—
189	#443	—	13	1	no	—
190	#443	—	13	2	no	—
191	#443	—	13	1	no	—
192	#443	—	13	2	no	—
193	#443	—	13	1	no	—
194	#443	#261	1	1	yes	characteristic
195	C₂×#108	—	13	1	no	—
196	C₂×#104	—	13	2	no	—
197	C₂×C₄×C₈	—	13	1	no	—
198	#364	—	13	2	no	—
199	#364	—	13	2	no	—
200	#350	—	13	4	no	—
201	#350	—	13	4	no	—
202	#346	—	13	2	no	—
203	C₄×C₁₆	—	13	2	no	—
204	C₂²×C₁₃	—	2	2	no	—
205	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
206	C₂×D₁₃	—	2	2	no	—
207	C₂×D₁₃	—	4	1	no	—
208	C₂×D₁₃	—	8	2	no	—
209	C₂×D₁₃	C₈×D₄	1	1	yes	characteristic
210	C₂×D₁₃	C₈×D₄	1	1	yes	characteristic
211	C₄×C₁₃	—	2	2	no	—
212	C₄×C₁₃	—	4	1	no	—
213	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
214	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
215	Dic₁₃	—	2	2	no	—
216	Dic₁₃	—	2	2	no	—
217	Dic₁₃	C₈×D₄	1	1	yes	characteristic
218	Dic₁₃	C₈×D₄	1	1	yes	characteristic
219	C₂×#41	—	13	2	no	—
220	#135	—	13	2	no	—
221	#135	—	13	1	no	—
222	#135	—	13	2	no	—
223	#135	—	13	1	no	—
224	#135	—	13	1	no	—
225	#135	C₂×#261	1	1	yes	characteristic
226	C₂×Q₁₆	—	13	1	no	—
227	C₂×SD₁₆	—	13	2	no	—
228	C₂×D₈	—	13	1	no	—
229	#131	—	13	2	no	—
230	#131	—	13	2	no	—
231	#131	—	13	2	no	—
232	#131	—	26	2	no	—
233	#131	—	26	2	no	—
234	#131	C₂×#261	1	2	yes	—
235	C₂×M₁₆	—	13	2	no	—
236	C₂×M₁₆	—	26	2	no	—
237	C₂²×C₈	—	13	1	no	—
238	C₂²×C₈	—	13	1	no	—
239	C₂²×C₈	—	26	2	no	—
240	C₂×C₄²	—	13	1	no	—
241	M₃₂	—	13	1	no	—
242	M₃₂	—	13	1	no	—
243	M₃₂	—	13	2	no	—
244	M₃₂	—	13	4	no	—
245	M₃₂	—	26	4	no	—
246	M₃₂	—	26	4	no	—
247	M₃₂	—	26	1	no	—
248	M₃₂	—	26	1	no	—
249	M₃₂	—	26	2	no	—
250	C₂×C₁₆	—	13	2	no	—
251	C₂×C₁₆	—	13	2	no	—
252	C₂×C₁₆	—	26	4	no	—
253	C₂×C₁₆	—	26	4	no	—
254	C₂×C₁₆	—	26	2	no	—
255	#108	—	13	2	no	—
256	#108	—	13	1	no	—
257	#108	C₂×#261	1	1	yes	characteristic
258	#104	—	13	2	no	—
259	#104	—	13	2	no	—
260	#104	—	13	2	no	—
261	#104	C₂×#261	1	2	yes	—
262	C₄×C₈	—	13	1	no	—
263	C₄×C₈	—	13	1	no	—
264	C₄×C₈	—	13	1	no	—
265	C₄×C₈	C₂×#261	1	1	yes	characteristic
266	C₂×C₁₃	—	2	1	no	—
267	C₂×C₁₃	—	4	2	no	—
268	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
269	D₁₃	—	2	1	no	—
270	D₁₃	—	2	1	no	—
271	D₁₃	—	4	2	no	—
272	#41	—	13	2	no	—
273	#41	—	13	2	no	—
274	#41	—	13	2	no	—
275	#41	—	26	2	no	—
276	#41	—	26	2	no	—
277	#41	C₂²×#261	1	2	yes	—
278	C₂×Q₈	—	13	2	no	—
279	C₂×D₄	—	13	2	no	—
280	C₂×D₄	—	26	2	no	—
281	C₂²×C₄	—	13	1	no	—
282	C₂²×C₄	—	26	2	no	—
283	C₂²×C₄	—	26	1	no	—
284	Q₁₆	—	13	2	no	—
285	Q₁₆	—	13	1	no	—
286	Q₁₆	D₁₃⋊C₈	1	1	yes	characteristic
287	SD₁₆	—	13	2	no	—
288	SD₁₆	—	13	2	no	—
289	SD₁₆	—	13	2	no	—
290	SD₁₆	D₁₃⋊C₈	1	2	yes	—
291	D₈	—	13	2	no	—
292	D₈	—	13	1	no	—
293	D₈	D₁₃⋊C₈	1	1	yes	characteristic
294	M₁₆	—	2	2	no	—
295	M₁₆	—	13	2	no	—
296	M₁₆	—	26	2	no	—
297	M₁₆	—	26	2	no	—
298	M₁₆	—	26	2	no	—
299	M₁₆	—	26	2	no	—
300	M₁₆	C₂²×#261	1	2	yes	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	13	1	no	—
303	C₂×C₈	—	13	1	no	—
304	C₂×C₈	—	13	1	no	—
305	C₂×C₈	—	13	1	no	—
306	C₂×C₈	—	13	1	no	—
307	C₂×C₈	—	13	1	no	—
308	C₂×C₈	—	13	1	no	—
309	C₂×C₈	—	13	1	no	—
310	C₂×C₈	—	13	1	no	—
311	C₂×C₈	—	13	1	no	—
312	C₂×C₈	—	26	2	no	—
313	C₂×C₈	—	26	2	no	—
314	C₂×C₈	—	26	2	no	—
315	C₂×C₈	—	26	2	no	—
316	C₂×C₈	C₂²×#261	1	1	yes	characteristic
317	C₂×C₈	C₂²×#261	1	1	yes	characteristic
318	C₄²	—	13	1	no	—
319	C₄²	—	26	1	no	—
320	C₄²	C₂²×#261	1	1	yes	characteristic
321	C₁₆	—	13	2	no	—
322	C₁₆	—	13	2	no	—
323	C₁₆	—	26	2	no	—
324	C₁₆	—	26	4	no	—
325	C₁₃	#33443	1	1	yes	characteristic, complemented, Sylow
326	C₂³	—	13	1	no	—
327	C₂³	—	26	2	no	—
328	Q₈	—	13	2	no	—
329	Q₈	—	26	2	no	—
330	Q₈	C₂×D₁₃⋊C₈	1	2	yes	—
331	D₄	—	2	2	no	—
332	D₄	—	13	2	no	—
333	D₄	—	26	2	no	—
334	D₄	—	26	2	no	—
335	D₄	—	26	2	no	—
336	D₄	—	26	2	no	—
337	D₄	C₂×D₁₃⋊C₈	1	2	yes	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	13	1	no	—
341	C₂×C₄	—	13	1	no	—
342	C₂×C₄	—	13	1	no	—
343	C₂×C₄	—	13	1	no	—
344	C₂×C₄	—	13	1	no	—
345	C₂×C₄	—	26	2	no	—
346	C₂×C₄	—	26	2	no	—
347	C₂×C₄	—	26	2	no	—
348	C₂×C₄	—	26	1	no	—
349	C₂×C₄	—	26	2	no	—
350	C₂×C₄	—	52	2	no	—
351	C₂×C₄	C₂³×#261	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	13	1	no	—
354	C₈	—	13	1	no	—
355	C₈	—	13	1	no	—
356	C₈	—	13	1	no	—
357	C₈	—	26	2	no	—
358	C₈	C₂×D₁₃⋊C₈	1	1	yes	characteristic
359	C₈	C₂×D₁₃⋊C₈	1	1	yes	characteristic
360	C₈	D₄×#261	1	1	yes	characteristic
361	C₈	D₄×#261	1	1	yes	characteristic
362	C₂²	—	2	2	no	—
363	C₂²	—	13	1	no	—
364	C₂²	—	13	1	no	—
365	C₂²	—	26	2	no	—
366	C₂²	—	52	1	no	—
367	C₂²	—	104	2	no	—
368	C₂²	D₁₃⋊#131	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	13	1	no	—
372	C₄	—	13	1	no	—
373	C₄	—	26	2	no	—
374	C₄	—	26	2	no	—
375	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
376	C₄	C₂×D₄×#261	1	1	yes	characteristic
377	C₂	—	2	1	no	—
378	C₂	—	4	2	no	—
379	C₂	—	26	1	no	—
380	C₂	—	26	1	no	—
381	C₂	—	52	2	no	—
382	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×#443	C₄
socle	C₂×C₁₃	C₂×C₈×D₄
prime radical	D₁₃×#135	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#443	#261
hypocenter	C₁₃	#33443
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #13131804	C₂	semidirect
2. #13131227	C₂	semidirect
3. #13131225	C₂	semidirect
4. #13131223	C₂	semidirect
5. D₁₃⋊#364	C₂	semidirect
6. #13131079	C₂	semidirect
7. #13131043	C₂	semidirect
11. #11719162	C₂²	semidirect
13. #11719096	C₂²	semidirect
28. #11718304	C₂²	semidirect
29. #11718270	C₂²	semidirect
30. #11718261	C₂²	semidirect
31. #11718257	C₂²	semidirect
32. #11718233	C₂²	semidirect
33. #11718231	C₂²	semidirect
325. C₁₃	#33443	semidirect

Maximal subgroups

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

subgroup	index
#13131804	2
#13131227	2
#13131225	2
#13131223	2
D₁₃⋊#364	2
#13131079	2
#13131043	2
D₁₃×#443	2
#33443	13