Subgroups — #16789298

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719089	C₂	1	1	yes	maximal, characteristic
2	D₁₃⋊(C₂²⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718216	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×#261	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×#261)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×#261)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×#88237	C₄	1	1	yes	characteristic
9	C₂×D₁₃⋊M₁₆	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
11	#11719089	—	2	2	no	—
12	#11719089	—	2	2	no	—
13	C₂×#88100	C₄	1	1	yes	characteristic
14	C₂×#88100	C₄	1	1	yes	characteristic
15	C₂×C₈×#261	C₂²	1	1	yes	characteristic
16	C₂×C₈×#261	C₂²	1	1	yes	characteristic
17	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
21	C₂²⋊#6092	C₄	1	2	yes	complemented
22	#11718259	C₄	1	2	yes	complemented
23	#11718216	—	2	2	no	—
24	C₁₆×#261	—	2	2	no	—
25	C₁₃×C₂²⋊C₁₆	C₄	1	1	yes	characteristic, complemented
26	C₂²⋊#6090	C₄	1	1	yes	characteristic, complemented
27	#11718099	C₄	1	1	yes	characteristic, complemented
28	#11718099	C₄	1	1	yes	characteristic, complemented
29	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₂²⋊#261	C₈	1	2	yes	—
31	C₂×#6122	C₈	1	2	yes	—
32	#88237	—	2	2	no	—
33	#88237	—	2	2	no	—
34	C₂×C₄⋊#261	C₈	1	2	yes	—
35	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
37	D₁₃⋊M₁₆	—	2	2	no	—
38	D₁₃⋊M₁₆	—	2	2	no	—
39	C₂×#6118	C₈	1	2	yes	—
40	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₁₃	—	2	1	no	—
45	C₂×C₈×D₁₃	—	2	1	no	—
46	C₂×C₈×D₁₃	—	2	1	no	—
47	C₂×C₈×D₁₃	—	2	1	no	—
48	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
52	C₂×#6092	C₂×C₄	1	2	yes	—
53	C₂×#6092	C₂×C₄	1	2	yes	—
54	#88100	—	2	2	no	—
55	#88100	—	2	2	no	—
56	C₈×#261	—	2	1	no	—
57	C₈×#261	—	2	1	no	—
58	C₈×#261	D₄	1	2	yes	—
59	C₈×#261	D₄	1	2	yes	—
60	D₁₃⋊C₁₆	—	2	1	no	—
61	D₁₃⋊C₁₆	—	2	1	no	—
62	D₁₃⋊C₁₆	—	2	1	no	—
63	D₁₃⋊C₁₆	—	2	1	no	—
64	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
65	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
66	C₂×#6090	C₂×C₄	1	1	yes	characteristic
67	C₂×#6090	C₂×C₄	1	1	yes	characteristic
68	C₁₆×D₁₃	—	2	1	no	—
69	C₁₆×D₁₃	—	2	1	no	—
70	C₁₆×D₁₃	—	2	1	no	—
71	C₁₆×D₁₃	—	2	1	no	—
72	#12600	—	13	1	no	maximal, Sylow
73	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×#261	C₂×C₈	1	2	yes	—
75	C₂²×#261	C₂×C₈	1	2	yes	—
76	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
77	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄×D₁₃	—	2	1	no	—
79	C₂×C₄×D₁₃	—	2	1	no	—
80	C₂×C₄×D₁₃	—	2	1	no	—
81	C₂×C₄×D₁₃	—	2	1	no	—
82	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²⋊#261	—	2	4	no	—
87	#6122	—	2	4	no	—
88	C₂×#1057	C₂×C₈	1	2	yes	—
89	C₂×#1057	C₂×C₈	1	2	yes	—
90	C₄⋊#261	—	2	4	no	—
91	C₄×#261	—	2	1	no	—
92	C₄×#261	—	2	1	no	—
93	C₄×#261	C₂²⋊C₄	1	2	yes	—
94	C₄×#261	M₁₆	1	2	yes	—
95	#6118	—	2	4	no	—
96	D₁₃⋊C₈	—	2	1	no	—
97	D₁₃⋊C₈	—	2	1	no	—
98	D₁₃⋊C₈	C₂²⋊C₄	1	2	yes	—
99	D₁₃⋊C₈	M₁₆	1	2	yes	—
100	C₂×C₈×C₁₃	—	2	1	no	—
101	C₂×C₈×C₁₃	—	2	1	no	—
102	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×#1055	—	2	1	no	—
105	C₂×#1055	—	2	1	no	—
106	C₂×#1055	C₄²	1	1	yes	characteristic
107	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
108	C₈×D₁₃	—	2	1	no	—
109	C₈×D₁₃	—	2	1	no	—
110	C₈×D₁₃	—	2	1	no	—
111	C₈×D₁₃	—	2	1	no	—
112	C₈×D₁₃	—	2	1	no	—
113	C₈×D₁₃	—	2	1	no	—
114	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
115	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
116	C₈×D₁₃	#41	1	1	yes	characteristic
117	C₈×D₁₃	#41	1	1	yes	characteristic
118	#6092	—	2	2	no	—
119	#6092	—	2	2	no	—
120	C₁₃×C₁₆	—	2	1	no	—
121	C₁₃×C₁₆	—	2	1	no	—
122	#6090	—	2	1	no	—
123	#6090	—	2	1	no	—
124	C₂×#405	—	13	1	no	—
125	C₂×C₂²⋊C₁₆	—	13	1	no	—
126	C₂×C₂²⋊C₁₆	—	13	1	no	—
127	C₂×#346	—	13	1	no	—
128	C₂×C₄×C₁₆	—	13	1	no	—
129	#1374	—	13	1	no	—
130	#1374	—	13	1	no	—
131	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
132	C₂²×D₁₃	—	2	1	no	—
133	C₂²×D₁₃	—	2	1	no	—
134	C₂²×D₁₃	—	2	2	no	—
135	C₂²×D₁₃	—	2	2	no	—
136	C₂²×D₁₃	C₄×C₈	1	1	yes	characteristic
137	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
138	C₂×#261	—	2	2	no	—
139	C₂×#261	—	2	2	no	—
140	C₂×#261	—	2	2	no	—
141	C₂×#261	—	2	2	no	—
142	C₂×#261	—	2	2	no	—
143	C₂×#261	C₂²⋊C₈	1	4	yes	—
144	C₂×C₄×C₁₃	—	2	1	no	—
145	C₂×C₄×C₁₃	—	2	1	no	—
146	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
148	C₂×Dic₁₃	—	2	1	no	—
149	C₂×Dic₁₃	—	2	1	no	—
150	C₂×Dic₁₃	C₄×C₈	1	1	yes	characteristic
151	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
152	C₄×D₁₃	—	2	1	no	—
153	C₄×D₁₃	—	2	1	no	—
154	C₄×D₁₃	—	2	1	no	—
155	C₄×D₁₃	—	2	1	no	—
156	C₄×D₁₃	—	2	1	no	—
157	C₄×D₁₃	—	2	1	no	—
158	C₄×D₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₄×D₁₃	#117	1	1	yes	characteristic
160	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
161	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
162	#1057	—	2	2	no	—
163	#1057	C₂²⋊C₈	1	4	yes	—
164	C₈×C₁₃	—	2	1	no	—
165	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
166	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
167	#1055	—	2	1	no	—
168	#1055	C₄×D₄	1	1	yes	characteristic
169	#1055	C₄×D₄	1	1	yes	characteristic
170	C₂²×M₁₆	—	13	1	no	—
171	C₂³×C₈	—	13	1	no	—
172	C₂×#117	—	13	1	no	—
173	C₂²×C₁₆	—	13	1	no	—
174	C₂²×C₁₆	—	13	1	no	—
175	C₂²×C₁₆	—	13	1	no	—
176	C₂²×C₁₆	—	13	1	no	—
177	#405	—	26	2	no	—
178	#405	—	26	2	no	—
179	C₂×#97	—	13	1	no	—
180	C₂×#97	—	13	1	no	—
181	C₂×C₄×C₈	—	13	1	no	—
182	C₂×C₄×C₈	—	13	1	no	—
183	C₂²⋊C₁₆	—	13	2	no	—
184	C₂²⋊C₁₆	—	13	2	no	—
185	C₂²⋊C₁₆	—	13	1	no	—
186	C₂²⋊C₁₆	—	13	1	no	—
187	C₂²⋊C₁₆	—	13	1	no	—
188	C₂²⋊C₁₆	#261	1	1	yes	characteristic, complemented
189	#346	—	26	2	no	—
190	C₄×C₁₆	—	26	2	no	—
191	C₂²×C₁₃	—	2	1	no	—
192	C₂²×C₁₃	—	2	1	no	—
193	C₂²×C₁₃	—	2	1	no	—
194	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
195	C₂×D₁₃	—	2	1	no	—
196	C₂×D₁₃	—	2	1	no	—
197	C₂×D₁₃	—	2	1	no	—
198	C₂×D₁₃	—	4	2	no	—
199	C₂×D₁₃	—	4	2	no	—
200	C₂×D₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
201	C₂×D₁₃	#431	1	1	yes	characteristic
202	C₂×D₁₃	#504	1	2	yes	—
203	C₂×D₁₃	#504	1	2	yes	—
204	#261	—	2	4	no	—
205	#261	—	4	2	no	—
206	C₄×C₁₃	—	2	1	no	—
207	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
208	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
209	Dic₁₃	—	2	1	no	—
210	Dic₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
211	Dic₁₃	#431	1	1	yes	characteristic
212	C₂³×C₄	—	13	1	no	—
213	C₂×M₁₆	—	13	2	no	—
214	C₂×M₁₆	—	13	2	no	—
215	C₂×M₁₆	—	26	2	no	—
216	C₂×M₁₆	—	26	2	no	—
217	C₂²×C₈	—	13	1	no	—
218	C₂²×C₈	—	13	1	no	—
219	C₂²×C₈	—	13	1	no	—
220	C₂²×C₈	—	13	1	no	—
221	C₂²×C₈	—	13	1	no	—
222	C₂²×C₈	—	13	1	no	—
223	C₂²×C₈	—	13	1	no	—
224	C₂²×C₈	—	26	1	no	—
225	C₂²×C₈	—	26	1	no	—
226	C₂²×C₈	—	26	1	no	—
227	C₂²×C₈	—	26	1	no	—
228	C₂²×C₈	C₂×#261	1	1	yes	characteristic
229	#117	—	26	2	no	—
230	#117	—	26	2	no	—
231	C₂×C₄⋊C₄	—	13	2	no	—
232	C₂×C₂²⋊C₄	—	13	2	no	—
233	C₂×C₄²	—	13	1	no	—
234	C₂×C₄²	—	13	1	no	—
235	C₂×C₁₆	—	13	2	no	—
236	C₂×C₁₆	—	13	2	no	—
237	C₂×C₁₆	—	13	1	no	—
238	C₂×C₁₆	—	13	1	no	—
239	C₂×C₁₆	—	26	1	no	—
240	C₂×C₁₆	—	26	1	no	—
241	C₂×C₁₆	—	26	1	no	—
242	C₂×C₁₆	—	26	1	no	—
243	C₂×C₁₆	—	26	1	no	—
244	C₂×C₁₆	—	26	1	no	—
245	C₂×C₁₆	—	26	1	no	—
246	C₂×C₁₆	—	26	1	no	—
247	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
248	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
249	#97	—	26	2	no	—
250	#97	—	26	2	no	—
251	C₄×C₈	—	13	2	no	—
252	C₄×C₈	—	13	2	no	—
253	C₄×C₈	—	26	1	no	—
254	C₄×C₈	—	26	1	no	—
255	C₂×C₁₃	—	4	1	no	—
256	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
257	C₂×C₁₃	#2108	1	1	yes	characteristic
258	C₂×C₁₃	#2108	1	1	yes	characteristic
259	D₁₃	—	4	1	no	—
260	D₁₃	#2113	1	2	yes	—
261	D₁₃	#2164	1	2	yes	—
262	C₂⁴	—	13	1	no	—
263	C₂²×C₄	—	13	1	no	—
264	C₂²×C₄	—	13	2	no	—
265	C₂²×C₄	—	13	1	no	—
266	C₂²×C₄	—	13	1	no	—
267	C₂²×C₄	—	13	1	no	—
268	C₂²×C₄	—	13	1	no	—
269	C₂²×C₄	—	13	2	no	—
270	C₂²×C₄	—	26	1	no	—
271	C₂²×C₄	—	26	1	no	—
272	C₂²×C₄	—	26	1	no	—
273	C₂²×C₄	—	26	1	no	—
274	C₂²×C₄	C₄×#261	1	1	yes	characteristic
275	M₁₆	—	26	4	no	—
276	M₁₆	—	26	4	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	13	2	no	—
280	C₂×C₈	—	13	2	no	—
281	C₂×C₈	—	13	2	no	—
282	C₂×C₈	—	13	1	no	—
283	C₂×C₈	—	13	1	no	—
284	C₂×C₈	—	13	1	no	—
285	C₂×C₈	—	13	1	no	—
286	C₂×C₈	—	13	1	no	—
287	C₂×C₈	—	13	2	no	—
288	C₂×C₈	—	13	1	no	—
289	C₂×C₈	—	26	1	no	—
290	C₂×C₈	—	26	1	no	—
291	C₂×C₈	—	26	1	no	—
292	C₂×C₈	—	26	1	no	—
293	C₂×C₈	—	26	1	no	—
294	C₂×C₈	—	26	1	no	—
295	C₂×C₈	—	26	1	no	—
296	C₂×C₈	—	26	1	no	—
297	C₂×C₈	—	26	1	no	—
298	C₂×C₈	—	26	1	no	—
299	C₂×C₈	C₄×#261	1	1	yes	characteristic
300	C₂×C₈	C₂²×#261	1	1	yes	characteristic
301	C₄⋊C₄	—	26	4	no	—
302	C₂²⋊C₄	—	26	4	no	—
303	C₄²	—	13	2	no	—
304	C₄²	—	13	2	no	—
305	C₄²	—	26	1	no	—
306	C₄²	—	26	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	1	no	—
309	C₁₆	—	26	2	no	—
310	C₁₆	—	26	2	no	—
311	C₁₆	—	26	1	no	—
312	C₁₆	—	26	1	no	—
313	C₁₃	#12600	1	1	yes	characteristic, complemented, Sylow
314	C₂³	—	13	1	no	—
315	C₂³	—	13	1	no	—
316	C₂³	—	26	1	no	—
317	C₂³	—	26	1	no	—
318	C₂³	—	26	2	no	—
319	C₂³	—	26	2	no	—
320	C₂³	C₈×#261	1	1	yes	characteristic
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	13	1	no	—
324	C₂×C₄	—	13	1	no	—
325	C₂×C₄	—	13	1	no	—
326	C₂×C₄	—	13	1	no	—
327	C₂×C₄	—	13	1	no	—
328	C₂×C₄	—	13	4	no	—
329	C₂×C₄	—	13	1	no	—
330	C₂×C₄	—	26	1	no	—
331	C₂×C₄	—	26	1	no	—
332	C₂×C₄	—	26	1	no	—
333	C₂×C₄	—	26	1	no	—
334	C₂×C₄	—	26	1	no	—
335	C₂×C₄	—	26	1	no	—
336	C₂×C₄	—	26	2	no	—
337	C₂×C₄	—	26	2	no	—
338	C₂×C₄	—	26	1	no	—
339	C₂×C₄	—	26	1	no	—
340	C₂×C₄	—	26	2	no	—
341	C₂×C₄	—	26	2	no	—
342	C₂×C₄	—	26	2	no	—
343	C₂×C₄	C₈×#261	1	1	yes	characteristic
344	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	13	4	no	—
347	C₈	—	13	1	no	—
348	C₈	—	13	1	no	—
349	C₈	—	26	2	no	—
350	C₈	—	26	1	no	—
351	C₈	D₄×#261	1	1	yes	characteristic
352	C₈	D₄×#261	1	1	yes	characteristic
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	13	2	no	—
357	C₂²	—	13	2	no	—
358	C₂²	—	13	1	no	—
359	C₂²	—	13	1	no	—
360	C₂²	—	26	1	no	—
361	C₂²	—	26	1	no	—
362	C₂²	—	26	1	no	—
363	C₂²	—	52	2	no	—
364	C₂²	—	52	2	no	—
365	C₂²	C₂×C₈×#261	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	13	1	no	—
368	C₄	—	13	1	no	—
369	C₄	—	26	4	no	—
370	C₄	—	26	1	no	—
371	C₄	—	52	2	no	—
372	C₄	C₂²⋊C₄×#261	1	1	yes	characteristic
373	C₄	M₁₆×#261	1	1	yes	characteristic
374	C₂	—	4	1	no	—
375	C₂	—	13	2	no	—
376	C₂	—	13	2	no	—
377	C₂	—	52	1	no	—
378	C₂	C₂²⋊C₈×#261	1	1	yes	characteristic
379	C₂	#13131044	1	1	yes	characteristic
380	C₂	#13131048	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#261
commutator, socle	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₁₃×C₂²⋊C₁₆	C₄
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	C₂²⋊C₁₆	#261
hypocenter	C₁₃	#12600
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#11718216	C₂	semidirect
4. C₂×C₁₆×#261	C₂	semidirect
5. (C₂×#261)⋊C₁₆	C₂	semidirect
6. (C₂×#261)⋊C₁₆	C₂	semidirect
21. C₂²⋊#6092	C₄	semidirect
22. #11718259	C₄	semidirect
25. C₁₃×C₂²⋊C₁₆	C₄	semidirect
26. C₂²⋊#6090	C₄	semidirect
27. #11718099	C₄	semidirect
28. #11718099	C₄	semidirect
188. C₂²⋊C₁₆	#261	semidirect
313. C₁₃	#12600	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719089	2
D₁₃⋊(C₂²⋊C₁₆)	2
C₂×#11718216	2
C₂×C₁₆×#261	2
(C₂×#261)⋊C₁₆	2
D₁₃×C₂²⋊C₁₆	2
#12600	13