Subgroups — #16795123

Order 3328.

588 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131536	C₂	1	2	yes	maximal
2	#13130414	C₂	1	2	yes	maximal, complemented
3	#13130410	C₂	1	2	yes	maximal, complemented
4	#13130408	C₂	1	2	yes	maximal, complemented
5	#13130401	C₂	1	2	yes	maximal, complemented
6	#13130397	C₂	1	1	yes	maximal, characteristic, complemented
7	#13130388	C₂	1	2	yes	maximal, complemented
8	#13130386	C₂	1	1	yes	maximal, characteristic, complemented
9	#13123801	C₂	1	1	yes	maximal, characteristic
10	C₄×#6131	C₂²	1	1	yes	characteristic
11	#11719159	—	2	2	no	—
12	#11719077	—	2	2	no	—
13	#11719077	—	2	2	no	—
14	#11719072	—	2	2	no	—
15	#11719054	C₂²	1	2	yes	—
16	#11719054	C₂²	1	2	yes	—
17	#11719052	C₂²	1	2	yes	—
18	#11719051	C₂²	1	2	yes	—
19	#11719051	C₂²	1	2	yes	—
20	D₁₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
21	C₁₃×#446	C₄	1	1	yes	characteristic
22	#11718685	C₄	1	1	yes	characteristic
23	#11718499	C₄	1	1	yes	characteristic
24	#11718497	C₄	1	2	yes	—
25	#11718494	C₂²	1	1	yes	characteristic
26	#11718494	C₂²	1	2	yes	—
27	#11718368	C₂²	1	2	yes	—
28	#11718367	C₄	1	2	yes	—
29	#11718349	C₄	1	1	yes	characteristic
30	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	#11718256	C₂²	1	2	yes	—
32	#11718255	C₂²	1	2	yes	—
33	#11718255	C₂²	1	2	yes	—
34	#11718254	C₂²	1	1	yes	characteristic
35	#11718254	C₂²	1	1	yes	characteristic
36	#11718254	C₂²	1	1	yes	characteristic
37	#11718254	C₂²	1	1	yes	characteristic
38	#11718254	C₂²	1	2	yes	—
39	#11718253	C₂²	1	2	yes	—
40	#11718253	C₂²	1	2	yes	—
41	C₂×#6131	D₄	1	1	yes	characteristic
42	C₂×#6118	—	2	2	no	—
43	C₂×D₁₃⋊C₈	—	2	2	no	—
44	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	Q₈×Dic₁₃	Q₈	1	1	yes	characteristic
46	C₂×#6094	C₂×C₄	1	2	yes	—
47	C₂×#6094	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
50	#88147	Q₈	1	2	yes	—
51	#88147	Q₈	1	1	yes	characteristic
52	#88146	D₄	1	2	yes	—
53	#88146	D₄	1	1	yes	characteristic
54	C₄×D₅₂	C₂×C₄	1	2	yes	—
55	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₁₃	C₂³	1	2	yes	—
57	C₄²×D₁₃	C₂³	1	1	yes	characteristic
58	#88112	—	2	2	no	—
59	#88111	—	2	2	no	—
60	#88111	—	2	2	no	—
61	#88111	—	2	2	no	—
62	#88111	—	2	2	no	—
63	#88110	—	2	2	no	—
64	#88110	—	2	2	no	—
65	C₄⋊#1057	—	2	2	no	—
66	C₄⋊#1057	C₂³	1	2	yes	—
67	C₄⋊#1057	C₂³	1	2	yes	—
68	C₄⋊#1057	C₂³	1	2	yes	—
69	C₄⋊#1057	C₂³	1	2	yes	—
70	C₄×#1057	—	2	2	no	—
71	C₄×#1057	—	2	2	no	—
72	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
73	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
74	C₁₃×#97	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
76	#88060	C₂×C₄	1	2	yes	—
77	#88060	C₂×C₄	1	2	yes	—
78	#88060	C₂×C₄	1	2	yes	—
79	C₈×Dic₁₃	C₂³	1	2	yes	—
80	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
81	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
82	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
83	C₄⋊#1055	C₂×C₄	1	2	yes	—
84	#88043	C₂×C₄	1	2	yes	—
85	C₄×#1055	C₂×C₄	1	1	yes	characteristic
86	#17833	—	13	1	no	maximal, Sylow
87	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
88	#6131	—	4	1	no	—
89	#6131	—	4	1	no	—
90	C₂×D₅₂	C₄⋊C₄	1	2	yes	—
91	C₂×D₅₂	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
97	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
98	C₂×#1057	—	2	2	no	—
99	C₂×#1057	—	2	2	no	—
100	C₂×#1057	—	2	2	no	—
101	C₂×#1057	—	2	2	no	—
102	#6118	—	4	4	no	—
103	D₁₃⋊C₈	—	4	4	no	—
104	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
108	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
111	#6103	C₄⋊C₄	1	2	yes	—
112	#6103	C₄⋊C₄	1	2	yes	—
113	#6103	C₄⋊C₄	1	2	yes	—
114	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
115	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
116	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
118	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
119	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
120	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
121	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
122	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
123	C₂×#1055	C₂²×C₄	1	2	yes	—
124	#6094	—	2	4	no	—
125	#6094	—	4	2	no	—
126	C₈×D₁₃	—	2	2	no	—
127	C₈×D₁₃	—	4	1	no	—
128	#2965	—	13	1	no	—
129	#2925	—	13	2	no	—
130	#1574	—	13	2	no	—
131	#1573	—	13	2	no	—
132	#1572	—	13	2	no	—
133	#1569	—	13	2	no	—
134	#1568	—	13	1	no	—
135	#1565	—	13	2	no	—
136	#1564	—	13	1	no	—
137	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
139	Q₈×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
149	D₅₂	—	4	2	no	—
150	D₅₂	—	4	1	no	—
151	C₄×D₁₃	—	2	1	no	—
152	C₄×D₁₃	—	2	2	no	—
153	C₄×D₁₃	—	2	1	no	—
154	C₄×D₁₃	—	2	2	no	—
155	C₄×D₁₃	—	2	2	no	—
156	C₄×D₁₃	—	2	2	no	—
157	C₄×D₁₃	—	2	1	no	—
158	C₄×D₁₃	—	2	1	no	—
159	C₄×D₁₃	—	4	2	no	—
160	C₄×D₁₃	—	4	1	no	—
161	#1057	—	4	2	no	—
162	#1057	—	4	2	no	—
163	#1057	—	4	2	no	—
164	#1057	—	4	2	no	—
165	C₈×C₁₃	—	2	1	no	—
166	C₈×C₁₃	—	2	2	no	—
167	C₈×C₁₃	#131	1	2	yes	—
168	#1055	—	2	1	no	—
169	#1055	—	2	2	no	—
170	#1055	—	2	1	no	—
171	C₄×#41	—	13	1	no	—
172	#446	—	13	1	no	—
173	#446	#261	1	1	yes	characteristic
174	#436	—	13	1	no	—
175	#436	—	13	1	no	—
176	#435	—	13	2	no	—
177	#435	—	13	2	no	—
178	#433	—	26	2	no	—
179	#433	—	26	2	no	—
180	#432	—	13	2	no	—
181	#432	—	13	2	no	—
182	#431	—	13	1	no	—
183	#431	—	13	2	no	—
184	#431	—	13	2	no	—
185	#431	—	13	2	no	—
186	#423	—	13	2	no	—
187	C₂×C₄⋊C₈	—	13	2	no	—
188	#408	—	26	2	no	—
189	#405	—	26	2	no	—
190	C₄×M₁₆	—	13	2	no	—
191	C₂×C₄×C₈	—	13	1	no	—
192	#335	—	13	1	no	—
193	#335	—	13	1	no	—
194	#335	—	13	2	no	—
195	#335	—	13	1	no	—
196	#335	—	13	1	no	—
197	#335	—	13	2	no	—
198	#334	—	13	2	no	—
199	#334	—	13	2	no	—
200	#334	—	13	2	no	—
201	#334	—	13	2	no	—
202	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₂×D₁₃	—	2	1	no	—
204	C₂×D₁₃	—	2	2	no	—
205	C₂×D₁₃	—	2	2	no	—
206	C₂×D₁₃	—	2	1	no	—
207	C₂×D₁₃	—	2	1	no	—
208	C₂×D₁₃	—	2	2	no	—
209	C₄×C₁₃	—	2	2	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	—	2	2	no	—
213	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
214	C₄×C₁₃	#568	1	1	yes	characteristic
215	Dic₁₃	—	2	1	no	—
216	Dic₁₃	—	2	2	no	—
217	Dic₁₃	—	2	1	no	—
218	Dic₁₃	#575	1	1	yes	characteristic
219	Dic₁₃	#575	1	1	yes	characteristic
220	C₂×#41	—	13	1	no	—
221	C₂×M₁₆	—	13	2	no	—
222	C₂×M₁₆	—	13	2	no	—
223	C₂×M₁₆	—	26	2	no	—
224	C₂²×C₈	—	13	1	no	—
225	C₂²×C₈	—	13	1	no	—
226	C₂²×C₈	—	26	2	no	—
227	C₄×Q₈	—	13	1	no	—
228	C₄×Q₈	C₂×#261	1	1	yes	characteristic
229	C₄×D₄	—	13	1	no	—
230	C₄×D₄	—	13	2	no	—
231	C₄×D₄	—	13	2	no	—
232	C₄×D₄	—	13	1	no	—
233	#117	—	13	2	no	—
234	#117	—	13	1	no	—
235	C₂×C₄²	—	13	1	no	—
236	C₂×C₄²	—	13	2	no	—
237	C₄⋊C₈	—	13	2	no	—
238	C₄⋊C₈	—	13	2	no	—
239	C₄⋊C₈	—	13	2	no	—
240	C₄⋊C₈	—	13	2	no	—
241	C₄⋊C₈	—	13	2	no	—
242	C₄⋊C₈	—	13	1	no	—
243	C₄⋊C₈	—	26	2	no	—
244	C₄⋊C₈	—	26	2	no	—
245	C₄⋊C₈	C₂×#261	1	2	yes	—
246	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
247	C₂²⋊C₈	—	13	2	no	—
248	C₂²⋊C₈	—	13	2	no	—
249	C₂²⋊C₈	—	13	2	no	—
250	C₂²⋊C₈	—	26	2	no	—
251	C₂²⋊C₈	—	26	2	no	—
252	C₂²⋊C₈	—	26	2	no	—
253	C₂²⋊C₈	—	26	2	no	—
254	#97	—	13	2	no	—
255	#97	—	26	2	no	—
256	#97	—	26	2	no	—
257	#97	C₂×#261	1	2	yes	—
258	C₄×C₈	—	13	1	no	—
259	C₄×C₈	—	13	1	no	—
260	C₄×C₈	—	13	2	no	—
261	C₄×C₈	—	13	1	no	—
262	C₄×C₈	—	26	2	no	—
263	C₄×C₈	—	26	2	no	—
264	C₄×C₈	C₂×#261	1	1	yes	characteristic
265	C₂×C₁₃	#2906	1	1	yes	characteristic
266	C₂×C₁₃	#2911	1	1	yes	characteristic
267	C₂×C₁₃	#2916	1	1	yes	characteristic
268	D₁₃	—	4	1	no	—
269	D₁₃	—	4	2	no	—
270	#41	—	52	1	no	—
271	#41	—	52	1	no	—
272	C₂×Q₈	C₄⋊#261	1	1	yes	characteristic
273	C₂×D₄	—	13	2	no	—
274	C₂×D₄	—	13	1	no	—
275	C₂²×C₄	—	13	1	no	—
276	C₂²×C₄	—	13	2	no	—
277	C₂²×C₄	—	13	1	no	—
278	C₂²×C₄	—	13	2	no	—
279	C₂²×C₄	—	13	1	no	—
280	C₂²×C₄	—	13	2	no	—
281	M₁₆	—	26	4	no	—
282	M₁₆	—	52	2	no	—
283	M₁₆	—	52	4	no	—
284	C₂×C₈	—	13	1	no	—
285	C₂×C₈	—	13	1	no	—
286	C₂×C₈	—	13	2	no	—
287	C₂×C₈	—	26	2	no	—
288	C₂×C₈	—	26	2	no	—
289	C₂×C₈	—	26	2	no	—
290	C₂×C₈	—	26	2	no	—
291	C₂×C₈	—	26	2	no	—
292	C₂×C₈	—	52	1	no	—
293	C₂×C₈	—	52	4	no	—
294	C₂×C₈	C₂²×#261	1	1	yes	characteristic
295	C₂×C₈	C₂²×#261	1	2	yes	—
296	C₂×C₈	C₂²×#261	1	1	yes	characteristic
297	C₄⋊C₄	—	13	2	no	—
298	C₄⋊C₄	—	13	1	no	—
299	C₄⋊C₄	C₄⋊#261	1	2	yes	—
300	C₄⋊C₄	C₄⋊#261	1	1	yes	characteristic
301	C₂²⋊C₄	—	13	2	no	—
302	C₂²⋊C₄	—	13	2	no	—
303	C₂²⋊C₄	—	13	2	no	—
304	C₄²	—	13	1	no	—
305	C₄²	—	13	2	no	—
306	C₄²	—	13	1	no	—
307	C₄²	—	13	2	no	—
308	C₄²	—	13	1	no	—
309	C₄²	C₂²×#261	1	1	yes	characteristic
310	C₄²	C₂²×#261	1	2	yes	—
311	C₁₃	#17833	1	1	yes	characteristic, complemented, Sylow
312	C₂³	—	13	1	no	—
313	C₂³	—	13	2	no	—
314	Q₈	—	4	1	no	—
315	D₄	—	52	2	no	—
316	D₄	—	52	1	no	—
317	C₂×C₄	—	13	1	no	—
318	C₂×C₄	—	13	1	no	—
319	C₂×C₄	—	13	1	no	—
320	C₂×C₄	—	13	2	no	—
321	C₂×C₄	—	26	2	no	—
322	C₂×C₄	—	26	2	no	—
323	C₂×C₄	—	26	1	no	—
324	C₂×C₄	—	26	1	no	—
325	C₂×C₄	—	26	1	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	26	1	no	—
328	C₂×C₄	—	26	2	no	—
329	C₂×C₄	—	52	1	no	—
330	C₂×C₄	—	52	2	no	—
331	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
333	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
334	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
335	C₂×C₄	C₂³×#261	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	26	1	no	—
339	C₈	—	26	2	no	—
340	C₈	—	26	1	no	—
341	C₈	—	52	2	no	—
342	C₈	—	52	2	no	—
343	C₈	—	52	2	no	—
344	C₈	—	52	2	no	—
345	C₈	#88240	1	2	yes	—
346	C₂²	—	26	1	no	—
347	C₂²	—	26	2	no	—
348	C₂²	—	26	2	no	—
349	C₂²	—	26	1	no	—
350	C₂²	—	26	1	no	—
351	C₂²	—	26	2	no	—
352	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	2	no	—
357	C₄	—	13	1	no	—
358	C₄	—	13	1	no	—
359	C₄	—	26	1	no	—
360	C₄	—	26	2	no	—
361	C₄	—	26	1	no	—
362	C₄	C₂×#88240	1	1	yes	characteristic
363	C₄	#11719629	1	1	yes	characteristic
364	C₂	—	52	1	no	—
365	C₂	—	52	2	no	—
366	C₂	#13131482	1	1	yes	characteristic
367	C₂	#13131489	1	1	yes	characteristic
368	C₂	#13131498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#446	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×#6131	D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#446	#261
hypocenter	C₁₃	#17833
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #13130414	C₂	semidirect
3. #13130410	C₂	semidirect
4. #13130408	C₂	semidirect
5. #13130401	C₂	semidirect
6. #13130397	C₂	semidirect
7. #13130388	C₂	semidirect
8. #13130386	C₂	semidirect
311. C₁₃	#17833	semidirect

Maximal subgroups

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

subgroup	index
#13131536	2
#13130414	2
#13130410	2
#13130408	2
#13130401	2
#13130397	2
#13130388	2
#13130386	2
#13123801	2
#17833	13