Subgroups — #15834752

Order 3328.

592 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123841	C₂	1	2	yes	maximal, complemented
2	#13113739	C₂	1	2	yes	maximal, complemented
3	#13113708	C₂	1	2	yes	maximal, complemented
4	#13113701	C₂	1	2	yes	maximal, complemented
5	SD₁₆⋊#1055	C₂	1	2	yes	maximal, complemented
6	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113244	C₂	1	2	yes	maximal, complemented
8	#13113125	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3168	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#529	C₂²	1	2	yes	—
11	C₁₃×#499	C₄	1	1	yes	characteristic
12	C₁₃×#495	C₄	1	1	yes	characteristic
13	C₁₃×#488	C₄	1	2	yes	—
14	C₁₃×#467	C₄	1	1	yes	characteristic
15	C₁₃×#466	C₄	1	1	yes	characteristic
16	C₁₃×#465	C₄	1	2	yes	—
17	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
18	C₄×C₁₃×SD₁₆	C₂²	1	2	yes	—
19	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	#11718685	C₂²	1	2	yes	—
22	Q₈×#1055	C₂²	1	2	yes	—
23	#11718673	—	2	2	no	—
24	#11718673	C₂²	1	2	yes	—
25	#11718672	—	2	2	no	—
26	D₄×#1055	C₂²	1	2	yes	—
27	#11718661	—	2	2	no	—
28	#11718658	—	2	2	no	—
29	#11718586	C₂²	1	2	yes	—
30	#11718586	C₂²	1	1	yes	characteristic
31	C₄×#6099	C₂²	1	1	yes	characteristic
32	Q₈⋊#1055	C₂²	1	2	yes	—
33	Q₈⋊#1055	C₂²	1	2	yes	—
34	D₄⋊#1055	C₂²	1	2	yes	—
35	D₄⋊#1055	C₂²	1	2	yes	—
36	#11718050	C₂²	1	1	yes	characteristic
37	#11718050	C₂²	1	1	yes	characteristic
38	#11718049	C₂²	1	2	yes	—
39	#11718047	C₂²	1	2	yes	—
40	#11718046	C₂²	1	2	yes	—
41	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₁₃×#126	—	2	2	no	—
47	C₁₃×#125	C₂×C₄	1	2	yes	—
48	C₁₃×#124	C₂×C₄	1	2	yes	—
49	C₁₃×#123	—	2	2	no	—
50	C₁₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
51	C₁₃×#121	C₂×C₄	1	2	yes	—
52	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
53	C₄×D₄×C₁₃	—	2	2	no	—
54	C₄×D₄×C₁₃	C₂³	1	2	yes	—
55	C₁₃×#117	—	2	2	no	—
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×#6099	—	2	2	no	—
58	C₂×#6099	D₄	1	1	yes	characteristic
59	C₂×#6099	D₄	1	1	yes	characteristic
60	C₂²×#1055	—	2	2	no	—
61	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
62	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
63	C₁₃×#106	C₂×C₄	1	2	yes	—
64	C₁₃×#103	C₂×C₄	1	2	yes	—
65	C₁₃×#103	C₂×C₄	1	2	yes	—
66	C₁₃×#102	C₂×C₄	1	2	yes	—
67	C₁₃×#102	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₂²⋊#1055	—	2	2	no	—
73	C₂²⋊#1055	—	2	1	no	—
74	C₂²⋊#1055	—	2	2	no	—
75	C₂²⋊#1055	—	2	2	no	—
76	C₄⋊#1055	—	2	1	no	—
77	C₄⋊#1055	—	2	2	no	—
78	C₄⋊#1055	—	2	2	no	—
79	C₄⋊#1055	C₂³	1	2	yes	—
80	C₄⋊#1055	C₂³	1	2	yes	—
81	#88043	—	2	2	no	—
82	#88043	D₄	1	2	yes	—
83	C₄×#1055	—	2	2	no	—
84	C₄×#1055	—	2	2	no	—
85	C₄×#1055	C₂³	1	2	yes	—
86	C₄×#1055	C₂³	1	2	yes	—
87	#18074	—	13	1	no	maximal, Sylow
88	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×D₄×C₁₃	—	2	2	no	—
90	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₁₃	—	2	2	no	—
92	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
95	C₁₃×Q₁₆	—	4	1	no	—
96	C₁₃×SD₁₆	—	2	4	no	—
97	C₁₃×D₈	—	4	1	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	4	1	no	—
100	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	2	2	no	—
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	2	2	no	—
108	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
109	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
110	C₁₃×C₂²⋊C₄	—	2	2	no	—
111	C₁₃×C₂²⋊C₄	—	2	2	no	—
112	C₁₃×C₂²⋊C₄	—	2	2	no	—
113	C₁₃×C₂²⋊C₄	—	2	2	no	—
114	C₁₃×C₂²⋊C₄	—	2	2	no	—
115	C₄²×C₁₃	—	2	2	no	—
116	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃	#41	1	1	yes	characteristic
119	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
120	#6099	—	4	1	no	—
121	#6099	—	4	1	no	—
122	#6099	—	4	4	no	—
123	C₂×#1055	—	2	2	no	—
124	C₂×#1055	—	2	2	no	—
125	C₂×#1055	—	2	2	no	—
126	C₂×#1055	—	4	4	no	—
127	C₂×#1055	C₂×D₄	1	2	yes	—
128	C₂×#1055	C₂×D₄	1	2	yes	—
129	#3168	D₁₃	1	1	yes	characteristic
130	#2969	—	13	2	no	—
131	#1592	—	13	2	no	—
132	#1587	—	13	2	no	—
133	#1586	—	13	2	no	—
134	SD₁₆⋊C₈	—	13	2	no	—
135	#1564	—	13	1	no	—
136	#1482	—	13	2	no	—
137	#1451	—	13	1	no	—
138	C₂³×C₁₃	—	2	2	no	—
139	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
140	Q₈×C₁₃	—	2	2	no	—
141	Q₈×C₁₃	—	2	2	no	—
142	D₄×C₁₃	—	2	2	no	—
143	D₄×C₁₃	—	2	2	no	—
144	D₄×C₁₃	—	8	2	no	—
145	C₂×C₄×C₁₃	—	2	1	no	—
146	C₂×C₄×C₁₃	—	2	1	no	—
147	C₂×C₄×C₁₃	—	2	1	no	—
148	C₂×C₄×C₁₃	—	2	1	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
161	C₈×C₁₃	—	2	1	no	—
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	#131	1	2	yes	—
165	#1055	—	2	2	no	—
166	#1055	—	2	2	no	—
167	#1055	—	4	2	no	—
168	#1055	—	4	2	no	—
169	#1055	—	4	2	no	—
170	#529	C₂×D₁₃	1	2	yes	—
171	#499	Dic₁₃	1	1	yes	characteristic
172	#495	Dic₁₃	1	1	yes	characteristic
173	#488	Dic₁₃	1	2	yes	—
174	#467	Dic₁₃	1	1	yes	characteristic
175	#466	Dic₁₃	1	1	yes	characteristic
176	#465	Dic₁₃	1	2	yes	—
177	#446	—	13	2	no	—
178	C₈×Q₈	—	13	2	no	—
179	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
180	C₄×SD₁₆	C₂×D₁₃	1	2	yes	—
181	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
182	#436	—	13	2	no	—
183	#436	—	26	2	no	—
184	#435	—	26	2	no	—
185	C₈×D₄	—	13	2	no	—
186	#433	—	26	2	no	—
187	#431	—	13	1	no	—
188	#431	—	13	2	no	—
189	#405	—	26	2	no	—
190	C₄×M₁₆	—	13	1	no	—
191	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
192	#335	—	13	1	no	—
193	#335	—	13	1	no	—
194	#334	—	13	2	no	—
195	Q₈⋊C₈	—	13	2	no	—
196	Q₈⋊C₈	—	13	2	no	—
197	D₄⋊C₈	—	13	2	no	—
198	D₄⋊C₈	—	13	2	no	—
199	#322	—	13	2	no	—
200	#322	—	13	2	no	—
201	C₂²×C₁₃	—	2	1	no	—
202	C₂²×C₁₃	—	2	1	no	—
203	C₂²×C₁₃	—	2	1	no	—
204	C₂²×C₁₃	—	4	2	no	—
205	C₂²×C₁₃	—	4	2	no	—
206	C₂²×C₁₃	—	4	2	no	—
207	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
208	C₄×C₁₃	—	2	1	no	—
209	C₄×C₁₃	—	2	1	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	—	4	2	no	—
213	C₄×C₁₃	—	4	2	no	—
214	C₄×C₁₃	—	4	2	no	—
215	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
216	C₄×C₁₃	#568	1	1	yes	characteristic
217	C₄×C₁₃	#575	1	1	yes	characteristic
218	C₄×C₁₃	#575	1	1	yes	characteristic
219	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
220	C₂×SD₁₆	C₂×Dic₁₃	1	2	yes	—
221	C₂×D₈	C₂×Dic₁₃	1	1	yes	characteristic
222	C₂×M₁₆	—	13	1	no	—
223	C₂×M₁₆	—	13	1	no	—
224	C₂×M₁₆	—	26	2	no	—
225	C₂²×C₈	—	26	2	no	—
226	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
227	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
228	#126	—	2	2	no	—
229	#125	C₂×Dic₁₃	1	2	yes	—
230	#124	C₂×Dic₁₃	1	2	yes	—
231	#123	—	2	2	no	—
232	C₂²⋊Q₈	C₂×Dic₁₃	1	2	yes	—
233	#121	C₂×Dic₁₃	1	2	yes	—
234	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
235	C₄×D₄	—	2	2	no	—
236	C₄×D₄	C₂²×D₁₃	1	2	yes	—
237	#117	—	2	2	no	—
238	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
239	#107	C₂×Dic₁₃	1	1	yes	characteristic
240	#107	C₂×Dic₁₃	1	1	yes	characteristic
241	#106	C₂×Dic₁₃	1	2	yes	—
242	C₄⋊C₈	—	13	2	no	—
243	C₄⋊C₈	—	13	2	no	—
244	C₄⋊C₈	—	26	2	no	—
245	C₄⋊C₈	—	26	2	no	—
246	C₄⋊C₈	—	26	1	no	—
247	#103	C₂×Dic₁₃	1	2	yes	—
248	#103	C₂×Dic₁₃	1	2	yes	—
249	#102	C₂×Dic₁₃	1	2	yes	—
250	#102	C₂×Dic₁₃	1	2	yes	—
251	C₂²⋊C₈	—	26	2	no	—
252	C₂²⋊C₈	—	26	2	no	—
253	C₂²⋊C₈	—	26	2	no	—
254	C₂²⋊C₈	—	26	1	no	—
255	#97	—	13	2	no	—
256	#97	—	26	2	no	—
257	C₄×C₈	—	13	2	no	—
258	C₄×C₈	—	13	2	no	—
259	C₄×C₈	—	26	2	no	—
260	C₄×C₈	—	26	2	no	—
261	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
262	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
265	C₂×C₁₃	—	4	1	no	—
266	C₂×C₁₃	—	8	2	no	—
267	C₂×C₁₃	#2935	1	1	yes	characteristic
268	C₂×C₁₃	#2943	1	1	yes	characteristic
269	C₂×C₁₃	#2957	1	1	yes	characteristic
270	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	C₂²×Dic₁₃	1	2	yes	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
275	C₂²×C₄	#6129	1	1	yes	characteristic
276	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
277	Q₁₆	—	4	1	no	—
278	SD₁₆	—	2	4	no	—
279	D₈	—	4	1	no	—
280	M₁₆	—	52	1	no	—
281	M₁₆	—	52	1	no	—
282	M₁₆	—	52	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	13	2	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	—	52	4	no	—
291	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
292	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
293	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
294	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
300	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	D₄×D₁₃	1	1	yes	characteristic
308	C₄²	#6129	1	1	yes	characteristic
309	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
310	C₄²	C₂³×D₁₃	1	1	yes	characteristic
311	C₁₃	#18074	1	1	yes	characteristic, complemented, Sylow
312	C₂³	—	2	2	no	—
313	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	Q₈	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	8	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
330	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
331	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
333	C₂×C₄	C₂×#6129	1	1	yes	characteristic
334	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	26	2	no	—
339	C₈	—	26	2	no	—
340	C₈	—	52	2	no	—
341	C₈	—	52	2	no	—
342	C₈	—	52	2	no	—
343	C₈	#88202	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	4	2	no	—
350	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	#11719473	1	1	yes	characteristic
359	C₄	#11719473	1	1	yes	characteristic
360	C₄	C₂×#88202	1	1	yes	characteristic
361	C₄	#11719524	1	1	yes	characteristic
362	C₂	—	4	1	no	—
363	C₂	—	8	2	no	—
364	C₂	#13123580	1	1	yes	characteristic
365	C₂	#13123637	1	1	yes	characteristic
366	C₂	#13123753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3168	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₁₃×#466	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#3168	D₁₃
hypocenter	C₁₃	#18074
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #13123841	C₂	semidirect
2. #13113739	C₂	semidirect
3. #13113708	C₂	semidirect
4. #13113701	C₂	semidirect
5. SD₁₆⋊#1055	C₂	semidirect
6. #13113584	C₂	semidirect
7. #13113244	C₂	semidirect
8. #13113125	C₂	semidirect
311. C₁₃	#18074	semidirect

Maximal subgroups

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

subgroup	index
#13123841	2
#13113739	2
#13113708	2
#13113701	2
SD₁₆⋊#1055	2
#13113584	2
#13113244	2
#13113125	2
C₁₃×#3168	2
#18074	13