Subgroups — #16016451

Order 3328.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123808	C₂	1	1	yes	maximal, characteristic
2	#13116741	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₆×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116668	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116664	C₂	1	2	yes	maximal, complemented
6	C₁₆×C₂²⋊D₁₃	C₂	1	2	yes	maximal, complemented
7	#13116640	C₂	1	2	yes	maximal, complemented
8	#13116638	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116432	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719262	C₄	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
14	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₈×C₂²⋊D₁₃	C₂²	1	2	yes	—
16	#11718762	C₂²	1	2	yes	—
17	C₂×#88052	C₂²	1	2	yes	—
18	#11718673	C₄	1	1	yes	characteristic
19	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
21	#11718496	C₄	1	1	yes	characteristic
22	#11718492	C₄	1	1	yes	characteristic
23	#11718389	C₄	1	2	yes	—
24	#11718381	C₄	1	2	yes	—
25	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
26	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
27	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
29	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
30	C₂²⋊#6090	C₂²	1	2	yes	—
31	#11718099	C₂²	1	2	yes	—
32	#11718094	C₂²	1	2	yes	—
33	#11718094	C₂²	1	2	yes	—
34	#11718093	C₂²	1	2	yes	—
35	#11718093	C₂²	1	2	yes	—
36	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
37	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
38	C₁₆×Dic₁₃	C₂²	1	2	yes	—
39	C₄⋊#6090	C₂²	1	1	yes	characteristic
40	C₄×#6090	C₂²	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
42	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
43	#88192	C₈	1	1	yes	characteristic
44	#88190	C₈	1	1	yes	characteristic
45	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
46	#88180	C₂×C₄	1	2	yes	—
47	C₂×#6099	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
50	#88151	C₈	1	1	yes	characteristic
51	#88144	C₈	1	1	yes	characteristic
52	#88137	C₈	1	2	yes	—
53	#88133	C₈	1	2	yes	—
54	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₃×C₁₆	—	2	4	no	—
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²⋊#1055	C₂×C₄	1	2	yes	—
64	#88060	C₂×C₄	1	2	yes	—
65	#88055	C₂×C₄	1	2	yes	—
66	#88055	C₂×C₄	1	2	yes	—
67	#88054	C₂×C₄	1	2	yes	—
68	#88054	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
71	C₈×Dic₁₃	C₂³	1	2	yes	—
72	#88052	—	2	4	no	—
73	C₂×#6090	C₂³	1	2	yes	—
74	C₂×#6090	C₂³	1	1	yes	characteristic
75	C₂×#6090	C₂³	1	1	yes	characteristic
76	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
77	C₄×#1055	C₂×C₄	1	1	yes	characteristic
78	C₁₆×D₁₃	—	2	2	no	—
79	C₁₆×D₁₃	—	2	2	no	—
80	#33480	—	13	1	no	maximal, Sylow
81	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
94	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
95	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
97	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
98	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
99	#6103	C₂×C₈	1	2	yes	—
100	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
101	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
102	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
103	#6101	C₂×C₈	1	2	yes	—
104	#6101	C₂×C₈	1	2	yes	—
105	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
106	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
107	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
108	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
109	#6099	—	2	4	no	—
110	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
111	C₂×#1055	C₂²×C₄	1	2	yes	—
112	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
113	C₈×D₁₃	—	2	1	no	—
114	C₈×D₁₃	—	2	1	no	—
115	C₈×D₁₃	—	2	2	no	—
116	C₁₃×C₁₆	—	2	2	no	—
117	C₁₃×C₁₆	—	2	1	no	—
118	C₁₃×C₁₆	#41	1	2	yes	—
119	#6090	—	2	1	no	—
120	#6090	—	2	2	no	—
121	#6090	#41	1	2	yes	—
122	#2966	—	13	1	no	—
123	#2183	—	13	1	no	—
124	Q₈×C₁₆	—	13	1	no	—
125	#2169	—	13	1	no	—
126	#2168	—	13	2	no	—
127	D₄×C₁₆	—	13	2	no	—
128	D₄×C₁₆	D₁₃	1	1	yes	characteristic, complemented
129	#2164	—	13	2	no	—
130	#2163	—	13	1	no	—
131	#2108	—	13	2	no	—
132	C₂×C₄×C₁₆	—	13	1	no	—
133	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
134	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
135	D₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₂²⋊D₁₃	—	2	4	no	—
143	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
144	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
145	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
147	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
148	C₄×D₁₃	—	2	2	no	—
149	C₄×D₁₃	—	2	1	no	—
150	C₄×D₁₃	—	2	1	no	—
151	C₄×D₁₃	—	2	2	no	—
152	Dic₂₆	—	2	2	no	—
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
156	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
157	#1055	—	2	1	no	—
158	#1055	—	2	2	no	—
159	#1055	#131	1	2	yes	—
160	#529	—	13	1	no	—
161	C₂×M₃₂	—	13	2	no	—
162	C₂²×C₁₆	—	13	1	no	—
163	C₂²×C₁₆	—	13	1	no	—
164	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
165	#446	—	13	1	no	—
166	C₈×Q₈	—	13	1	no	—
167	#436	—	13	1	no	—
168	#435	—	13	2	no	—
169	C₈×D₄	—	13	2	no	—
170	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
171	#433	—	13	2	no	—
172	#432	—	13	1	no	—
173	#405	—	13	2	no	—
174	C₂×C₄×C₈	—	13	1	no	—
175	C₄⋊C₁₆	—	13	2	no	—
176	C₄⋊C₁₆	—	13	2	no	—
177	C₄⋊C₁₆	—	13	1	no	—
178	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
179	C₂²⋊C₁₆	—	13	2	no	—
180	C₂²⋊C₁₆	—	13	2	no	—
181	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
182	#346	—	13	2	no	—
183	#346	—	13	2	no	—
184	C₄×C₁₆	—	13	1	no	—
185	C₄×C₁₆	—	13	1	no	—
186	C₄×C₁₆	—	13	2	no	—
187	C₄×C₁₆	—	13	1	no	—
188	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
189	C₂²×C₁₃	—	2	2	no	—
190	C₂²×C₁₃	—	2	2	no	—
191	C₂²×C₁₃	—	2	2	no	—
192	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
193	C₂×D₁₃	—	2	1	no	—
194	C₂×D₁₃	—	2	1	no	—
195	C₂×D₁₃	#504	1	2	yes	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	2	no	—
198	C₄×C₁₃	#504	1	2	yes	—
199	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
200	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
201	Dic₁₃	—	2	2	no	—
202	Dic₁₃	—	2	1	no	—
203	Dic₁₃	—	2	1	no	—
204	Dic₁₃	—	2	2	no	—
205	Dic₁₃	—	2	1	no	—
206	C₂×M₁₆	—	13	2	no	—
207	C₂²×C₈	—	13	1	no	—
208	C₂²×C₈	—	13	1	no	—
209	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
210	#126	—	13	2	no	—
211	#125	—	13	1	no	—
212	#124	—	13	1	no	—
213	#123	—	13	2	no	—
214	C₂²⋊Q₈	—	13	1	no	—
215	#121	—	13	1	no	—
216	C₄×Q₈	—	13	1	no	—
217	C₄×D₄	—	13	2	no	—
218	C₄×D₄	C₄×D₁₃	1	1	yes	characteristic
219	#117	—	13	2	no	—
220	C₂×C₄²	—	13	1	no	—
221	M₃₂	—	26	4	no	—
222	C₂×C₁₆	—	2	4	no	—
223	C₂×C₁₆	—	13	1	no	—
224	C₂×C₁₆	—	13	1	no	—
225	C₂×C₁₆	—	13	2	no	—
226	C₂×C₁₆	—	26	2	no	—
227	C₂×C₁₆	—	26	2	no	—
228	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
230	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
231	C₄⋊C₈	—	13	2	no	—
232	C₄⋊C₈	—	13	2	no	—
233	C₄⋊C₈	—	13	1	no	—
234	C₄⋊C₈	C₄×D₁₃	1	1	yes	characteristic
235	C₂²⋊C₈	—	13	2	no	—
236	C₂²⋊C₈	—	13	2	no	—
237	C₂²⋊C₈	C₄×D₁₃	1	2	yes	—
238	#97	—	13	2	no	—
239	#97	—	13	2	no	—
240	C₄×C₈	—	13	1	no	—
241	C₄×C₈	—	13	1	no	—
242	C₄×C₈	—	13	2	no	—
243	C₄×C₈	—	13	1	no	—
244	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
245	C₂×C₁₃	—	4	2	no	—
246	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
247	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
248	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
249	D₁₃	—	2	2	no	—
250	C₂×Q₈	—	13	1	no	—
251	C₂×D₄	—	13	2	no	—
252	C₂×D₄	C₈×D₁₃	1	1	yes	characteristic
253	C₂²×C₄	—	13	1	no	—
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	13	1	no	—
256	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
257	M₁₆	—	26	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	13	1	no	—
261	C₂×C₈	—	13	1	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	—	26	1	no	—
265	C₂×C₈	—	26	1	no	—
266	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
268	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
269	C₄⋊C₄	—	13	2	no	—
270	C₄⋊C₄	—	13	2	no	—
271	C₄⋊C₄	—	13	2	no	—
272	C₄⋊C₄	—	13	1	no	—
273	C₄⋊C₄	—	13	2	no	—
274	C₄⋊C₄	C₈×D₁₃	1	1	yes	characteristic
275	C₂²⋊C₄	—	13	2	no	—
276	C₂²⋊C₄	—	13	2	no	—
277	C₂²⋊C₄	—	13	2	no	—
278	C₂²⋊C₄	—	13	2	no	—
279	C₂²⋊C₄	C₈×D₁₃	1	2	yes	—
280	C₄²	—	13	1	no	—
281	C₄²	—	13	1	no	—
282	C₄²	—	13	1	no	—
283	C₄²	—	13	2	no	—
284	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	13	2	no	—
288	C₁₆	—	26	1	no	—
289	C₁₆	—	26	2	no	—
290	C₁₆	#6129	1	2	yes	—
291	C₁₃	#33480	1	1	yes	characteristic, complemented, Sylow
292	C₂³	—	13	1	no	—
293	C₂³	C₂×C₈×D₁₃	1	2	yes	—
294	Q₈	—	26	2	no	—
295	D₄	—	2	2	no	—
296	D₄	—	26	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	13	1	no	—
300	C₂×C₄	—	13	1	no	—
301	C₂×C₄	—	13	1	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	13	2	no	—
304	C₂×C₄	—	26	2	no	—
305	C₂×C₄	—	26	1	no	—
306	C₂×C₄	—	26	1	no	—
307	C₂×C₄	—	26	2	no	—
308	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
310	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	13	2	no	—
315	C₈	—	26	2	no	—
316	C₈	—	26	1	no	—
317	C₈	C₂×#6129	1	1	yes	characteristic
318	C₈	#41×D₁₃	1	1	yes	characteristic
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	13	2	no	—
323	C₂²	—	26	1	no	—
324	C₂²	—	26	1	no	—
325	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	26	1	no	—
329	C₄	—	26	1	no	—
330	C₄	—	26	2	no	—
331	C₄	—	26	1	no	—
332	C₄	—	26	2	no	—
333	C₄	#11718562	1	2	yes	—
334	C₄	C₄×#6129	1	1	yes	characteristic
335	C₄	D₁₃×#131	1	1	yes	characteristic
336	C₂	—	4	2	no	—
337	C₂	—	26	2	no	—
338	C₂	C₈×#6129	1	1	yes	characteristic
339	C₂	D₁₃×#504	1	1	yes	characteristic
340	C₂	C₂×#11718562	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	D₄×C₁₃×C₁₆	C₂
prime radical	#88192	C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	D₄×C₁₆	D₁₃
hypocenter	C₁₃	#33480
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13116741	C₂	semidirect
3. C₁₆×Dic₂₆	C₂	semidirect
4. #13116668	C₂	semidirect
5. #13116664	C₂	semidirect
6. C₁₆×C₂²⋊D₁₃	C₂	semidirect
7. #13116640	C₂	semidirect
8. #13116638	C₂	semidirect
9. #13116432	C₂	semidirect
10. C₄×C₁₆×D₁₃	C₂	semidirect
11. D₄×C₁₃×C₁₆	C₂	semidirect
128. D₄×C₁₆	D₁₃	semidirect
291. C₁₃	#33480	semidirect

Maximal subgroups

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

subgroup	index
#13123808	2
#13116741	2
C₁₆×Dic₂₆	2
#13116668	2
#13116664	2
C₁₆×C₂²⋊D₁₃	2
#13116640	2
#13116638	2
#13116432	2
C₄×C₁₆×D₁₃	2
D₄×C₁₃×C₁₆	2
#33480	13