Subgroups — #15833799

Order 3328.

576 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125406	C₂	1	1	yes	maximal, characteristic
2	#13123987	C₂	1	2	yes	maximal, complemented
3	#13113709	C₂	1	2	yes	maximal, complemented
4	#13113702	C₂	1	2	yes	maximal, complemented
5	#13113681	C₂	1	2	yes	maximal, complemented
6	#13113674	C₂	1	2	yes	maximal, complemented
7	#13113592	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113302	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719184	C₂²	1	2	yes	—
11	C₁₃×#431	C₂²	1	1	yes	characteristic
12	C₁₃×#431	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718777	C₄	1	1	yes	characteristic
15	#11718776	C₄	1	1	yes	characteristic
16	#11718775	C₄	1	2	yes	—
17	#11718585	C₂²	1	2	yes	—
18	#11718501	—	2	2	no	—
19	#11718500	—	2	2	no	—
20	#11718390	—	2	2	no	—
21	#11718389	—	2	2	no	—
22	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
23	#11718358	C₄	1	2	yes	—
24	#11718357	C₄	1	1	yes	characteristic
25	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
26	C₄×#6095	C₂²	1	2	yes	—
27	#11718349	C₂²	1	2	yes	—
28	C₈×D₅₂	C₂²	1	2	yes	—
29	#11718344	C₄	1	1	yes	characteristic
30	#11718340	C₂²	1	2	yes	—
31	C₈×Dic₂₆	C₂²	1	2	yes	—
32	C₁₃×#322	C₂²	1	2	yes	—
33	C₁₃×#322	C₂²	1	2	yes	—
34	#11718051	C₂²	1	2	yes	—
35	#11718051	C₂²	1	2	yes	—
36	#11718049	C₂²	1	1	yes	characteristic
37	#11718049	C₂²	1	1	yes	characteristic
38	#11718049	C₂²	1	2	yes	—
39	#11718048	C₂²	1	2	yes	—
40	#11718048	C₂²	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	#88184	C₂×C₄	1	2	yes	—
45	#88183	—	2	2	no	—
46	C₄×C₂²⋊D₁₃	—	2	2	no	—
47	#88178	C₂×C₄	1	2	yes	—
48	C₂×#6099	—	2	2	no	—
49	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
50	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
51	C₂×#6095	C₂×C₄	1	2	yes	—
52	C₂×#6094	—	2	2	no	—
53	C₂×C₈×D₁₃	—	2	2	no	—
54	#88130	—	2	2	no	—
55	#88129	C₂×C₄	1	2	yes	—
56	C₄×D₅₂	C₂³	1	2	yes	—
57	#88126	—	2	2	no	—
58	#88124	C₂×C₄	1	2	yes	—
59	C₄×Dic₂₆	C₂³	1	2	yes	—
60	C₁₃×C₄⋊C₈	—	2	1	no	—
61	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
62	C₁₃×C₂²⋊C₈	—	2	1	no	—
63	C₁₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
64	C₁₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	2	yes	—
71	#88061	C₂×C₄	1	2	yes	—
72	#88061	C₂×C₄	1	2	yes	—
73	#88060	—	2	2	no	—
74	#88060	—	2	2	no	—
75	#88058	C₂×C₄	1	1	yes	characteristic
76	#88058	C₂×C₄	1	1	yes	characteristic
77	#88057	C₂×C₄	1	2	yes	—
78	#88056	C₂×C₄	1	2	yes	—
79	#88056	C₂×C₄	1	2	yes	—
80	#88055	—	2	2	no	—
81	#88054	—	2	2	no	—
82	#88054	—	2	2	no	—
83	C₈×Dic₁₃	—	2	2	no	—
84	C₄⋊#1055	—	2	2	no	—
85	C₄⋊#1055	C₂³	1	2	yes	—
86	C₄⋊#1055	C₂³	1	2	yes	—
87	#17993	—	13	1	no	maximal, Sylow
88	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
91	C₂×C₂²⋊D₁₃	—	2	2	no	—
92	C₂×D₅₂	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₁₃	—	2	2	no	—
94	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	4	1	no	—
97	C₂×C₈×C₁₃	—	4	1	no	—
98	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
104	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₃	#41	1	1	yes	characteristic
107	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₁₃	—	2	2	no	—
109	#6103	—	2	2	no	—
110	#6103	—	2	2	no	—
111	#6103	—	2	2	no	—
112	#6103	—	2	2	no	—
113	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
114	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
115	#6101	—	2	2	no	—
116	#6101	—	2	2	no	—
117	#6101	—	2	2	no	—
118	#6101	—	2	2	no	—
119	C₄×Dic₁₃	—	2	2	no	—
120	#6099	—	8	2	no	—
121	C₂×#1055	—	2	2	no	—
122	C₂×#1055	—	2	2	no	—
123	Dic₅₂	—	4	1	no	—
124	D₁₀₄	—	4	1	no	—
125	#6095	—	4	2	no	—
126	#6094	—	8	2	no	—
127	C₈×D₁₃	—	8	2	no	—
128	#3168	—	13	1	no	—
129	#2993	—	13	2	no	—
130	#1587	—	13	2	no	—
131	#1586	—	13	2	no	—
132	#1583	—	13	2	no	—
133	#1582	—	13	2	no	—
134	#1566	—	13	1	no	—
135	#1495	—	13	2	no	—
136	#1452	D₁₃	1	1	yes	characteristic, complemented
137	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
138	C₂²×D₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	4	1	no	—
143	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
149	C₂²⋊D₁₃	—	8	2	no	—
150	C₂×Dic₁₃	—	2	2	no	—
151	C₂×Dic₁₃	—	2	2	no	—
152	C₂×Dic₁₃	—	2	2	no	—
153	D₅₂	—	2	2	no	—
154	D₅₂	—	2	2	no	—
155	C₄×D₁₃	—	4	2	no	—
156	C₄×D₁₃	—	4	2	no	—
157	Dic₂₆	—	2	2	no	—
158	Dic₂₆	—	2	2	no	—
159	C₈×C₁₃	—	2	1	no	—
160	C₈×C₁₃	—	2	1	no	—
161	C₈×C₁₃	—	2	1	no	—
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	—	2	2	no	—
164	C₈×C₁₃	—	2	2	no	—
165	C₈×C₁₃	—	4	2	no	—
166	#1055	—	4	2	no	—
167	#1055	—	4	2	no	—
168	#529	—	13	2	no	—
169	#499	—	13	1	no	—
170	#495	—	13	1	no	—
171	#488	—	13	2	no	—
172	#467	—	13	1	no	—
173	#466	—	13	1	no	—
174	#465	—	13	2	no	—
175	#446	—	13	2	no	—
176	C₈×Q₈	—	13	2	no	—
177	C₄×Q₁₆	—	13	1	no	—
178	C₄×SD₁₆	—	13	2	no	—
179	C₄×D₈	—	13	1	no	—
180	#436	—	13	2	no	—
181	#436	—	26	2	no	—
182	#435	—	26	2	no	—
183	C₈×D₄	—	13	2	no	—
184	#433	—	26	2	no	—
185	#431	C₂×D₁₃	1	1	yes	characteristic
186	#431	C₂×D₁₃	1	1	yes	characteristic
187	#424	—	26	2	no	—
188	#423	—	13	2	no	—
189	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
190	#334	—	13	1	no	—
191	#334	—	13	1	no	—
192	#334	—	13	2	no	—
193	Q₈⋊C₈	—	13	2	no	—
194	Q₈⋊C₈	—	13	2	no	—
195	D₄⋊C₈	—	13	2	no	—
196	D₄⋊C₈	—	13	2	no	—
197	#322	C₂×D₁₃	1	2	yes	—
198	#322	C₂×D₁₃	1	2	yes	—
199	C₂²×C₁₃	—	2	1	no	—
200	C₂²×C₁₃	—	2	1	no	—
201	C₂²×C₁₃	—	2	1	no	—
202	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
203	C₂×D₁₃	—	4	2	no	—
204	C₂×D₁₃	—	4	2	no	—
205	C₂×D₁₃	—	4	2	no	—
206	C₄×C₁₃	—	2	1	no	—
207	C₄×C₁₃	—	2	1	no	—
208	C₄×C₁₃	—	2	1	no	—
209	C₄×C₁₃	—	2	1	no	—
210	C₄×C₁₃	#568	1	1	yes	characteristic
211	C₄×C₁₃	#568	1	1	yes	characteristic
212	C₄×C₁₃	#577	1	2	yes	—
213	Dic₁₃	—	4	2	no	—
214	Dic₁₃	—	4	2	no	—
215	Dic₁₃	—	4	2	no	—
216	C₂×Q₁₆	—	13	1	no	—
217	C₂×SD₁₆	—	13	2	no	—
218	C₂×D₈	—	13	1	no	—
219	C₂×M₁₆	—	26	2	no	—
220	C₂×M₁₆	—	26	2	no	—
221	C₂²×C₈	—	26	2	no	—
222	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
223	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
224	#126	—	26	2	no	—
225	#125	—	13	2	no	—
226	#124	—	13	2	no	—
227	#123	—	26	2	no	—
228	C₂²⋊Q₈	—	13	2	no	—
229	#121	—	13	2	no	—
230	C₄×Q₈	—	13	2	no	—
231	C₄×D₄	—	13	2	no	—
232	C₄×D₄	—	26	2	no	—
233	#117	—	26	2	no	—
234	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
235	#107	—	13	1	no	—
236	#107	—	13	1	no	—
237	#106	—	13	2	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	13	2	no	—
240	C₄⋊C₈	—	13	2	no	—
241	C₄⋊C₈	—	26	2	no	—
242	C₄⋊C₈	—	26	2	no	—
243	C₄⋊C₈	—	26	2	no	—
244	C₄⋊C₈	D₅₂	1	2	yes	—
245	#103	—	13	2	no	—
246	#103	—	13	2	no	—
247	#102	—	13	2	no	—
248	#102	—	13	2	no	—
249	C₂²⋊C₈	—	2	1	no	—
250	C₂²⋊C₈	—	26	2	no	—
251	C₂²⋊C₈	—	26	2	no	—
252	C₂²⋊C₈	D₅₂	1	1	yes	characteristic
253	C₂²⋊C₈	D₅₂	1	1	yes	characteristic
254	#97	—	26	2	no	—
255	C₄×C₈	—	26	2	no	—
256	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
257	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
258	C₄×C₈	C₂²×D₁₃	1	2	yes	—
259	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₁₃	1	2	yes	—
262	C₂×C₁₃	—	4	1	no	—
263	C₂×C₁₃	#2932	1	1	yes	characteristic
264	C₂×C₁₃	#2952	1	1	yes	characteristic
265	C₂×C₁₃	#2957	1	1	yes	characteristic
266	D₁₃	—	8	2	no	—
267	C₂×Q₈	—	13	2	no	—
268	C₂×D₄	—	13	2	no	—
269	C₂×D₄	—	26	2	no	—
270	C₂²×C₄	—	26	2	no	—
271	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
273	C₂²×C₄	#6127	1	1	yes	characteristic
274	Q₁₆	—	52	1	no	—
275	SD₁₆	—	52	2	no	—
276	D₈	—	52	1	no	—
277	M₁₆	—	104	2	no	—
278	M₁₆	—	104	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	26	2	no	—
283	C₂×C₈	—	26	2	no	—
284	C₂×C₈	—	104	2	no	—
285	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
286	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
287	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
289	C₂×C₈	C₂×D₅₂	1	2	yes	—
290	C₂×C₈	C₂×D₅₂	1	2	yes	—
291	C₄⋊C₄	—	13	2	no	—
292	C₄⋊C₄	—	13	2	no	—
293	C₄⋊C₄	—	26	2	no	—
294	C₄⋊C₄	—	26	2	no	—
295	C₄⋊C₄	—	26	2	no	—
296	C₄⋊C₄	—	26	2	no	—
297	C₂²⋊C₄	—	26	2	no	—
298	C₂²⋊C₄	—	26	2	no	—
299	C₂²⋊C₄	—	26	2	no	—
300	C₂²⋊C₄	—	26	2	no	—
301	C₂²⋊C₄	—	26	2	no	—
302	C₄²	—	26	2	no	—
303	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
304	C₄²	C₂×D₅₂	1	1	yes	characteristic
305	C₄²	#6127	1	1	yes	characteristic
306	C₄²	C₂³×D₁₃	1	1	yes	characteristic
307	C₁₃	#17993	1	1	yes	characteristic, complemented, Sylow
308	C₂³	—	26	2	no	—
309	C₂³	C₄×D₅₂	1	1	yes	characteristic
310	Q₈	—	26	2	no	—
311	Q₈	—	26	2	no	—
312	D₄	—	26	2	no	—
313	D₄	—	26	2	no	—
314	D₄	—	104	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	26	2	no	—
320	C₂×C₄	—	26	2	no	—
321	C₂×C₄	—	26	2	no	—
322	C₂×C₄	—	52	2	no	—
323	C₂×C₄	—	52	2	no	—
324	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
325	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
326	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
327	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
329	C₂×C₄	C₂×#6127	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	2	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	52	2	no	—
338	C₈	—	52	2	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	52	2	no	—
343	C₂²	—	52	2	no	—
344	C₂²	—	52	2	no	—
345	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	52	2	no	—
351	C₄	—	52	2	no	—
352	C₄	—	52	2	no	—
353	C₄	#11719450	1	1	yes	characteristic
354	C₄	#11719450	1	1	yes	characteristic
355	C₄	#11719456	1	2	yes	—
356	C₂	—	4	1	no	—
357	C₂	—	104	2	no	—
358	C₂	#13123555	1	1	yes	characteristic
359	C₂	#13123721	1	1	yes	characteristic
360	C₂	#13123754	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1452	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11718776	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1452	D₁₃
hypocenter	C₁₃	#17993
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #13123987	C₂	semidirect
3. #13113709	C₂	semidirect
4. #13113702	C₂	semidirect
5. #13113681	C₂	semidirect
6. #13113674	C₂	semidirect
7. #13113592	C₂	semidirect
8. #13113302	C₂	semidirect
9. C₁₃×#1452	C₂	semidirect
136. #1452	D₁₃	semidirect
307. C₁₃	#17993	semidirect

Maximal subgroups

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

subgroup	index
#13125406	2
#13123987	2
#13113709	2
#13113702	2
#13113681	2
#13113674	2
#13113592	2
#13113302	2
C₁₃×#1452	2
#17993	13