Subgroups — #16792758

Order 3328.

798 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131644	C₂	1	1	yes	maximal, characteristic
2	C₄×#88240	C₂	1	1	yes	maximal, characteristic
3	#13130060	C₂	1	2	yes	maximal, complemented
4	#13130054	C₂	1	2	yes	maximal, complemented
5	#13130052	C₂	1	1	yes	maximal, characteristic, complemented
6	#13130044	C₂	1	2	yes	maximal, complemented
7	#13130038	C₂	1	1	yes	maximal, characteristic, complemented
8	#13130038	C₂	1	2	yes	maximal, complemented
9	C₈×#6118	C₂	1	1	yes	maximal, characteristic, complemented
10	#13130032	C₂	1	1	yes	maximal, characteristic, complemented
11	#13123801	C₂	1	1	yes	maximal, characteristic
12	C₂×#88240	C₄	1	2	yes	—
13	C₄×#6131	C₂²	1	1	yes	characteristic
14	#11719158	C₄	1	2	yes	—
15	#11719076	C₄	1	2	yes	—
16	#11719075	C₄	1	4	yes	—
17	#11719072	C₄	1	4	yes	—
18	#11719072	C₄	1	2	yes	—
19	#11719051	C₂²	1	1	yes	characteristic
20	#11719049	C₂²	1	2	yes	—
21	C₄×#6118	C₂²	1	1	yes	characteristic
22	C₄×#6118	C₂²	1	2	yes	—
23	C₄×#6118	C₂²	1	2	yes	—
24	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₄×D₁₃⋊C₈	C₂²	1	2	yes	—
27	C₁₃×#446	C₄	1	1	yes	characteristic
28	#11718685	C₄	1	1	yes	characteristic
29	#11718499	C₄	1	1	yes	characteristic
30	#11718497	C₄	1	2	yes	—
31	#11718494	C₂²	1	1	yes	characteristic
32	#11718494	C₂²	1	2	yes	—
33	#11718368	C₂²	1	2	yes	—
34	#11718367	C₄	1	2	yes	—
35	#11718349	C₄	1	1	yes	characteristic
36	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
37	#1057⋊C₈	C₂²	1	1	yes	characteristic
38	#1057⋊C₈	C₂²	1	1	yes	characteristic
39	#1057⋊C₈	C₂²	1	1	yes	characteristic
40	#1057⋊C₈	C₂²	1	1	yes	characteristic
41	#1057⋊C₈	C₂²	1	2	yes	—
42	#1057⋊C₈	C₂²	1	2	yes	—
43	#1057⋊C₈	C₂²	1	2	yes	—
44	#11718251	C₂²	1	2	yes	—
45	C₈×#1057	C₂²	1	2	yes	—
46	C₈×#1057	C₂²	1	2	yes	—
47	C₂×#6131	C₂×C₄	1	1	yes	characteristic
48	#88240	—	2	8	no	—
49	C₂×#6118	C₂×C₄	1	4	yes	—
50	C₂×#6118	C₂×C₄	1	4	yes	—
51	C₂×#6118	C₂×C₄	1	2	yes	—
52	C₂×D₁₃⋊C₈	C₂×C₄	1	2	yes	—
53	C₂×D₁₃⋊C₈	C₂×C₄	1	4	yes	—
54	C₂×D₁₃⋊C₈	C₂×C₄	1	2	yes	—
55	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
56	Q₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
57	C₂×#6094	C₂×C₄	1	2	yes	—
58	C₂×#6094	C₂×C₄	1	2	yes	—
59	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
61	#88147	C₂×C₄	1	2	yes	—
62	#88147	C₂×C₄	1	1	yes	characteristic
63	#88146	C₂×C₄	1	2	yes	—
64	#88146	C₂×C₄	1	1	yes	characteristic
65	C₄×D₅₂	C₂×C₄	1	2	yes	—
66	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
67	C₄²×D₁₃	C₂³	1	1	yes	characteristic
68	C₄²×D₁₃	C₂³	1	2	yes	—
69	#88112	C₂×C₄	1	4	yes	—
70	#88111	C₂×C₄	1	4	yes	—
71	#88111	C₂×C₄	1	2	yes	—
72	#88110	C₂×C₄	1	4	yes	—
73	#88110	C₂×C₄	1	2	yes	—
74	#88110	C₂×C₄	1	4	yes	—
75	C₄⋊#1057	C₂×C₄	1	2	yes	—
76	C₄×#1057	C₂×C₄	1	2	yes	—
77	C₄×#1057	C₂×C₄	1	4	yes	—
78	C₄×#1057	C₂×C₄	1	2	yes	—
79	C₄×#1057	C₂³	1	2	yes	—
80	C₄×#1057	C₂³	1	1	yes	characteristic
81	C₄×#1057	C₂³	1	2	yes	—
82	C₄×#1057	C₂³	1	1	yes	characteristic
83	C₄×#1057	C₂³	1	1	yes	characteristic
84	C₄×#1057	C₂³	1	1	yes	characteristic
85	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
86	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
87	C₁₃×#97	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
89	#88060	C₂×C₄	1	2	yes	—
90	#88060	C₂×C₄	1	2	yes	—
91	#88060	C₂×C₄	1	2	yes	—
92	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
93	C₈×Dic₁₃	C₂³	1	2	yes	—
94	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
95	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
96	C₄⋊#1055	C₂×C₄	1	2	yes	—
97	#88043	C₂×C₄	1	2	yes	—
98	C₄×#1055	C₂×C₄	1	1	yes	characteristic
99	#17337	—	13	1	no	maximal, Sylow
100	C₂×Q₈×C₁₃	C₄²	1	1	yes	characteristic
101	#6131	—	2	4	no	—
102	C₂×D₅₂	C₄²	1	2	yes	—
103	C₂×D₅₂	C₄²	1	1	yes	characteristic
104	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
106	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
108	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
109	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×#1057	C₂²×C₄	1	2	yes	—
111	C₂×#1057	C₂²×C₄	1	2	yes	—
112	C₂×#1057	C₂²×C₄	1	2	yes	—
113	C₂×#1057	C₂²×C₄	1	4	yes	—
114	C₂×#1057	C₂²×C₄	1	4	yes	—
115	C₂×#1057	C₂²×C₄	1	2	yes	—
116	#6118	—	2	8	no	—
117	#6118	—	2	8	no	—
118	#6118	M₁₆	1	8	yes	complemented
119	D₁₃⋊C₈	—	2	4	no	—
120	D₁₃⋊C₈	—	2	8	no	—
121	D₁₃⋊C₈	—	2	4	no	—
122	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
125	C₁₃×C₄⋊C₄	C₄²	1	2	yes	—
126	C₁₃×C₄⋊C₄	C₄²	1	1	yes	characteristic
127	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
129	#6103	C₄²	1	2	yes	—
130	#6103	C₄²	1	2	yes	—
131	#6103	C₄²	1	2	yes	—
132	C₄⋊Dic₁₃	C₄²	1	2	yes	—
133	C₄⋊Dic₁₃	C₄²	1	1	yes	characteristic
134	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
135	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
136	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
137	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
138	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
139	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
140	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
141	C₂×#1055	C₂²×C₄	1	2	yes	—
142	#6094	—	4	2	no	—
143	#6094	—	4	2	no	—
144	C₈×D₁₃	—	4	1	no	—
145	C₈×D₁₃	—	4	1	no	—
146	#2965	—	13	1	no	—
147	#2965	—	13	1	no	—
148	C₄×#131	—	13	1	no	—
149	#1455	—	13	2	no	—
150	#1453	—	13	2	no	—
151	#1452	—	13	1	no	—
152	#1451	—	13	2	no	—
153	C₈×M₁₆	—	13	1	no	—
154	C₈×M₁₆	—	13	2	no	—
155	C₈×M₁₆	—	13	1	no	—
156	C₂×#322	—	13	1	no	—
157	C₂²×D₁₃	C₂×C₄²	1	2	yes	—
158	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
159	Q₈×C₁₃	—	2	2	no	—
160	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
161	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
163	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
164	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
165	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
166	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
167	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
168	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
169	D₅₂	—	2	4	no	—
170	D₅₂	—	2	2	no	—
171	C₄×D₁₃	—	2	1	no	—
172	C₄×D₁₃	—	2	1	no	—
173	C₄×D₁₃	—	2	2	no	—
174	C₄×D₁₃	—	2	2	no	—
175	C₄×D₁₃	—	2	4	no	—
176	C₄×D₁₃	—	2	2	no	—
177	C₄×D₁₃	—	2	4	no	—
178	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
179	#1057	—	2	2	no	—
180	#1057	—	2	2	no	—
181	#1057	—	2	2	no	—
182	#1057	—	2	4	no	—
183	#1057	C₂×M₁₆	1	8	yes	—
184	#1057	#131	1	4	yes	complemented
185	C₈×C₁₃	—	2	1	no	—
186	C₈×C₁₃	—	2	1	no	—
187	C₈×C₁₃	—	2	2	no	—
188	#1055	—	2	1	no	—
189	#1055	—	2	1	no	—
190	#1055	—	2	2	no	—
191	C₂×#131	—	13	2	no	—
192	C₄×#41	—	13	1	no	—
193	#446	—	13	2	no	—
194	#446	—	13	1	no	—
195	#446	#261	1	1	yes	characteristic
196	#436	—	13	2	no	—
197	#436	—	13	1	no	—
198	#436	—	13	1	no	—
199	#435	—	13	4	no	—
200	#435	—	13	2	no	—
201	#435	—	13	2	no	—
202	#431	—	13	2	no	—
203	#431	—	13	1	no	—
204	#431	—	13	1	no	—
205	#431	—	13	2	no	—
206	#405	—	13	2	no	—
207	#405	—	13	4	no	—
208	C₄×M₁₆	—	13	1	no	—
209	C₄×M₁₆	—	13	2	no	—
210	C₄×M₁₆	—	13	2	no	—
211	C₄×M₁₆	—	13	2	no	—
212	C₂×C₄×C₈	—	13	1	no	—
213	C₂×C₄×C₈	—	13	2	no	—
214	C₂×C₄×C₈	—	13	1	no	—
215	C₂×C₄×C₈	—	13	1	no	—
216	#322	—	13	2	no	—
217	#322	—	13	1	no	—
218	#322	—	13	1	no	—
219	#322	—	13	1	no	—
220	#322	—	13	1	no	—
221	#322	—	13	2	no	—
222	#322	—	13	2	no	—
223	#322	—	13	2	no	—
224	C₈²	—	13	2	no	—
225	C₈²	—	13	2	no	—
226	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
227	C₂×D₁₃	—	2	1	no	—
228	C₂×D₁₃	—	2	1	no	—
229	C₂×D₁₃	—	2	2	no	—
230	C₂×D₁₃	—	2	2	no	—
231	C₂×D₁₃	—	2	2	no	—
232	C₂×D₁₃	C₄×M₁₆	1	2	yes	—
233	C₄×C₁₃	—	2	2	no	—
234	C₄×C₁₃	—	2	1	no	—
235	C₄×C₁₃	—	2	2	no	—
236	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
237	C₄×C₁₃	#568	1	1	yes	characteristic
238	C₄×C₁₃	#568	1	1	yes	characteristic
239	Dic₁₃	—	2	1	no	—
240	Dic₁₃	—	2	2	no	—
241	Dic₁₃	C₂²×M₁₆	1	2	yes	—
242	Dic₁₃	C₂×#131	1	2	yes	—
243	C₂×#41	—	13	1	no	—
244	#131	—	26	8	no	—
245	C₂×M₁₆	—	13	2	no	—
246	C₂×M₁₆	—	13	4	no	—
247	C₂×M₁₆	—	13	4	no	—
248	C₂×M₁₆	—	13	2	no	—
249	C₂×M₁₆	—	13	2	no	—
250	C₂²×C₈	—	13	4	no	—
251	C₂²×C₈	—	13	2	no	—
252	C₂²×C₈	—	13	2	no	—
253	C₂²×C₈	—	13	1	no	—
254	C₂²×C₈	—	13	1	no	—
255	C₄×Q₈	—	13	1	no	—
256	C₄×Q₈	C₂×#261	1	1	yes	characteristic
257	C₄×D₄	—	13	2	no	—
258	C₄×D₄	—	13	1	no	—
259	C₄×D₄	—	13	2	no	—
260	C₄×D₄	—	13	1	no	—
261	#117	—	13	1	no	—
262	#117	—	13	2	no	—
263	C₂×C₄²	—	13	1	no	—
264	C₂×C₄²	—	13	2	no	—
265	C₄⋊C₈	—	13	2	no	—
266	C₄⋊C₈	—	13	4	no	—
267	C₄⋊C₈	—	13	2	no	—
268	C₄⋊C₈	—	13	1	no	—
269	C₄⋊C₈	C₂×#261	1	2	yes	—
270	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
271	C₂²⋊C₈	—	13	2	no	—
272	C₂²⋊C₈	—	13	4	no	—
273	C₂²⋊C₈	—	13	2	no	—
274	C₂²⋊C₈	—	13	2	no	—
275	C₂²⋊C₈	—	13	2	no	—
276	#97	—	13	2	no	—
277	#97	—	13	4	no	—
278	#97	—	13	4	no	—
279	#97	—	13	2	no	—
280	#97	C₂×#261	1	2	yes	—
281	C₄×C₈	—	13	4	no	—
282	C₄×C₈	—	13	2	no	—
283	C₄×C₈	—	13	2	no	—
284	C₄×C₈	—	13	2	no	—
285	C₄×C₈	—	13	1	no	—
286	C₄×C₈	—	13	2	no	—
287	C₄×C₈	—	13	1	no	—
288	C₄×C₈	—	13	1	no	—
289	C₄×C₈	—	13	1	no	—
290	C₄×C₈	—	13	1	no	—
291	C₄×C₈	—	13	2	no	—
292	C₄×C₈	—	13	1	no	—
293	C₄×C₈	—	13	1	no	—
294	C₄×C₈	C₂×#261	1	1	yes	characteristic
295	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
296	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
297	C₂×C₁₃	#2875	1	1	yes	characteristic
298	D₁₃	—	2	2	no	—
299	D₁₃	—	4	2	no	—
300	#41	—	26	4	no	—
301	C₂×Q₈	C₄×#261	1	1	yes	characteristic
302	C₂×D₄	—	13	2	no	—
303	C₂×D₄	—	13	1	no	—
304	C₂²×C₄	—	13	1	no	—
305	C₂²×C₄	—	13	1	no	—
306	C₂²×C₄	—	13	2	no	—
307	C₂²×C₄	—	13	2	no	—
308	C₂²×C₄	—	13	1	no	—
309	C₂²×C₄	—	13	2	no	—
310	M₁₆	—	13	8	no	—
311	M₁₆	—	26	8	no	—
312	M₁₆	—	26	8	no	—
313	M₁₆	—	52	2	no	—
314	M₁₆	—	52	2	no	—
315	C₂×C₈	—	13	4	no	—
316	C₂×C₈	—	13	2	no	—
317	C₂×C₈	—	13	2	no	—
318	C₂×C₈	—	13	2	no	—
319	C₂×C₈	—	13	2	no	—
320	C₂×C₈	—	13	4	no	—
321	C₂×C₈	—	13	1	no	—
322	C₂×C₈	—	13	1	no	—
323	C₂×C₈	—	13	2	no	—
324	C₂×C₈	—	26	8	no	—
325	C₂×C₈	—	26	4	no	—
326	C₂×C₈	—	26	4	no	—
327	C₂×C₈	—	52	1	no	—
328	C₂×C₈	—	52	1	no	—
329	C₂×C₈	C₂²×#261	1	1	yes	characteristic
330	C₂×C₈	C₂²×#261	1	1	yes	characteristic
331	C₂×C₈	C₂²×#261	1	2	yes	—
332	C₄⋊C₄	—	13	2	no	—
333	C₄⋊C₄	—	13	1	no	—
334	C₄⋊C₄	C₄×#261	1	2	yes	—
335	C₄⋊C₄	C₄×#261	1	1	yes	characteristic
336	C₂²⋊C₄	—	13	2	no	—
337	C₂²⋊C₄	—	13	2	no	—
338	C₂²⋊C₄	—	13	2	no	—
339	C₄²	—	13	1	no	—
340	C₄²	—	13	2	no	—
341	C₄²	—	13	1	no	—
342	C₄²	—	13	2	no	—
343	C₄²	—	13	1	no	—
344	C₄²	C₂²×#261	1	1	yes	characteristic
345	C₄²	C₂²×#261	1	2	yes	—
346	C₁₃	#17337	1	1	yes	characteristic, complemented, Sylow
347	C₂³	—	13	1	no	—
348	C₂³	—	13	2	no	—
349	Q₈	—	2	2	no	—
350	D₄	—	26	4	no	—
351	D₄	—	26	2	no	—
352	C₂×C₄	—	13	1	no	—
353	C₂×C₄	—	13	1	no	—
354	C₂×C₄	—	13	4	no	—
355	C₂×C₄	—	13	1	no	—
356	C₂×C₄	—	13	2	no	—
357	C₂×C₄	—	26	4	no	—
358	C₂×C₄	—	26	2	no	—
359	C₂×C₄	—	26	4	no	—
360	C₂×C₄	—	26	1	no	—
361	C₂×C₄	—	26	2	no	—
362	C₂×C₄	—	26	1	no	—
363	C₂×C₄	—	26	2	no	—
364	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
365	C₂×C₄	C₂×C₄×#261	1	2	yes	—
366	C₂×C₄	C₂×C₄×#261	1	2	yes	—
367	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
368	C₂×C₄	C₂³×#261	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	2	no	—
372	C₈	—	13	8	no	—
373	C₈	—	13	4	no	—
374	C₈	—	26	2	no	—
375	C₈	—	26	2	no	—
376	C₈	—	26	2	no	—
377	C₈	—	26	4	no	—
378	C₈	—	26	1	no	—
379	C₈	—	26	2	no	—
380	C₈	—	26	1	no	—
381	C₂²	—	13	2	no	—
382	C₂²	—	26	1	no	—
383	C₂²	—	26	1	no	—
384	C₂²	—	26	2	no	—
385	C₂²	—	26	2	no	—
386	C₂²	—	26	2	no	—
387	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
388	C₄	—	2	2	no	—
389	C₄	—	2	1	no	—
390	C₄	—	2	2	no	—
391	C₄	—	13	2	no	—
392	C₄	—	13	2	no	—
393	C₄	—	26	1	no	—
394	C₄	—	26	2	no	—
395	C₄	M₁₆×#261	1	2	yes	—
396	C₄	#11719625	1	1	yes	characteristic
397	C₄	#11719629	1	1	yes	characteristic
398	C₂	—	26	2	no	—
399	C₂	—	52	2	no	—
400	C₂	C₂×M₁₆×#261	1	1	yes	characteristic
401	C₂	#131×#261	1	1	yes	characteristic
402	C₂	#13131408	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂²×C₄²
Fitting	C₁₃×#446	C₄
prime radical	C₂×#6131	C₂×C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#446	#261
hypocenter	C₁₃	#17337
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
3. #13130060	C₂	semidirect
4. #13130054	C₂	semidirect
5. #13130052	C₂	semidirect
6. #13130044	C₂	semidirect
7. #13130038	C₂	semidirect
8. #13130038	C₂	semidirect
9. C₈×#6118	C₂	semidirect
10. #13130032	C₂	semidirect
118. #6118	M₁₆	semidirect
184. #1057	#131	semidirect
346. C₁₃	#17337	semidirect

Maximal subgroups

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

subgroup	index
#13131644	2
C₄×#88240	2
#13130060	2
#13130054	2
#13130052	2
#13130044	2
#13130038	2
C₈×#6118	2
#13130032	2
#13123801	2
#17337	13