Subgroups — #15761983

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123194	C₂	1	1	yes	maximal, characteristic
2	#13114952	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718138	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718077	C₂	1	1	yes	maximal, characteristic
6	C₂×#11718077	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×#88201	C₄	1	4	yes	—
10	C₂×#88128	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	—	2	1	no	—
14	#11718649	—	2	1	no	—
15	#11718649	—	2	1	no	—
16	#11718649	—	2	2	no	—
17	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
20	C₁₃×#333	—	2	2	no	—
21	#11718142	—	2	1	no	—
22	#11718142	—	2	2	no	—
23	#11718138	—	2	2	no	—
24	#11718077	—	2	2	no	—
25	#11718077	—	2	2	no	—
26	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₂²×D₅₂	—	2	1	no	—
28	C₂²×D₅₂	—	2	1	no	—
29	C₂²×D₅₂	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃	—	2	1	no	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₁₃×C₄⋊Q₈	—	2	1	no	—
33	C₁₃×C₄⋊Q₈	—	2	1	no	—
34	C₁₃×C₄⋊Q₈	—	2	1	no	—
35	C₁₃×C₄⋊Q₈	—	2	1	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	#88201	—	4	8	no	—
40	C₂×#6103	—	2	4	no	—
41	C₂×#6103	—	2	4	no	—
42	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
45	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
46	C₂²×#1055	—	2	1	no	—
47	C₂²×#1055	—	2	1	no	—
48	#88128	—	2	1	no	—
49	#88128	—	2	1	no	—
50	#88128	—	2	1	no	—
51	#88128	—	2	1	no	—
52	C₁₃×C₄⋊C₈	—	2	1	no	—
53	C₁₃×C₄⋊C₈	—	2	1	no	—
54	C₁₃×C₄⋊C₈	—	2	1	no	—
55	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
56	C₈×Dic₁₃	—	4	2	no	—
57	C₄⋊#1055	—	2	1	no	—
58	C₄⋊#1055	—	2	1	no	—
59	C₄⋊#1055	—	2	1	no	—
60	C₄⋊#1055	D₄	1	2	yes	—
61	#9699	—	13	1	no	maximal, Sylow
62	C₂³×D₁₃	—	2	2	no	—
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	—	4	4	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
71	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
72	C₂×D₅₂	—	2	2	no	—
73	C₂×D₅₂	—	2	2	no	—
74	C₂×D₅₂	—	4	4	no	—
75	C₂×D₅₂	—	4	1	no	—
76	C₂×D₅₂	—	4	1	no	—
77	C₂×D₅₂	—	4	1	no	—
78	C₂×D₅₂	—	4	1	no	—
79	C₂×D₅₂	—	4	1	no	—
80	C₂×D₅₂	—	4	1	no	—
81	C₂×D₅₂	—	4	2	no	—
82	C₂×D₅₂	—	4	2	no	—
83	C₂×D₅₂	—	4	1	no	—
84	C₂×D₅₂	—	4	1	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	4	1	no	—
87	C₂×C₈×C₁₃	—	4	1	no	—
88	C₂×C₈×C₁₃	—	4	1	no	—
89	C₂×C₈×C₁₃	—	4	2	no	—
90	C₁₃×C₄⋊C₄	—	4	1	no	—
91	C₁₃×C₄⋊C₄	—	4	1	no	—
92	C₁₃×C₄⋊C₄	—	4	1	no	—
93	C₁₃×C₄⋊C₄	—	4	1	no	—
94	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₃	#41	1	1	yes	characteristic
97	C₄²×C₁₃	#41	1	1	yes	characteristic
98	#6103	—	4	8	no	—
99	#6103	—	4	8	no	—
100	C₄×Dic₁₃	—	2	2	no	—
101	C₄×Dic₁₃	—	2	1	no	—
102	C₄×Dic₁₃	—	2	2	no	—
103	C₄×Dic₁₃	—	2	1	no	—
104	C₄×Dic₁₃	—	2	1	no	—
105	C₄×Dic₁₃	—	2	1	no	—
106	C₄×Dic₁₃	—	2	4	no	—
107	C₂×#1055	—	2	2	no	—
108	C₂×#1055	—	4	1	no	—
109	C₂×#1055	—	4	1	no	—
110	C₂×#1055	—	4	1	no	—
111	C₂×#1055	—	4	2	no	—
112	#2858	—	13	1	no	—
113	#1831	—	13	1	no	—
114	C₄×C₄⋊C₈	—	13	1	no	—
115	C₂×#333	—	13	1	no	—
116	C₂×#333	D₁₃	1	1	yes	characteristic, complemented
117	C₂×#331	—	13	1	no	—
118	C₂×#331	—	13	1	no	—
119	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×D₁₃	—	4	2	no	—
121	C₂²×D₁₃	—	4	4	no	—
122	C₂²×D₁₃	—	4	2	no	—
123	C₂²×D₁₃	—	4	4	no	—
124	C₂²×D₁₃	—	4	2	no	—
125	Q₈×C₁₃	—	8	2	no	—
126	Q₈×C₁₃	—	8	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	4	no	—
130	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#135	1	2	yes	—
139	C₂×C₄×C₁₃	#135	1	2	yes	—
140	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
142	C₂×Dic₁₃	—	2	2	no	—
143	C₂×Dic₁₃	—	2	2	no	—
144	C₂×Dic₁₃	—	2	2	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	—	2	2	no	—
147	C₂×Dic₁₃	#100	1	4	yes	—
148	D₅₂	—	8	2	no	—
149	D₅₂	—	8	2	no	—
150	D₅₂	—	8	1	no	—
151	D₅₂	—	8	1	no	—
152	D₅₂	—	8	1	no	—
153	D₅₂	—	8	1	no	—
154	D₅₂	—	8	2	no	—
155	D₅₂	—	8	1	no	—
156	D₅₂	—	8	1	no	—
157	C₈×C₁₃	—	4	2	no	—
158	C₈×C₁₃	—	8	1	no	—
159	#1055	—	4	2	no	—
160	#1055	—	8	1	no	—
161	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
162	C₂×#127	—	13	1	no	—
163	C₂×#124	—	13	4	no	—
164	C₂×C₄⋊C₈	—	13	1	no	—
165	C₂×C₄⋊C₈	—	13	1	no	—
166	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
167	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
168	C₂×C₄×C₈	—	26	1	no	—
169	#390	—	26	2	no	—
170	#390	—	26	1	no	—
171	#390	—	26	1	no	—
172	C₄³	—	13	1	no	—
173	#336	—	26	2	no	—
174	#336	—	26	1	no	—
175	#333	—	2	2	no	—
176	#333	—	26	2	no	—
177	#331	—	26	2	no	—
178	#331	—	26	2	no	—
179	C₂²×C₁₃	#390	1	1	yes	characteristic
180	C₂²×C₁₃	C₂×#100	1	2	yes	—
181	C₂²×C₁₃	#416	1	1	yes	characteristic
182	C₂²×C₁₃	#416	1	2	yes	—
183	C₂²×C₁₃	#417	1	1	yes	characteristic
184	C₂×D₁₃	—	8	4	no	—
185	C₂×D₁₃	—	8	4	no	—
186	C₂×D₁₃	—	8	2	no	—
187	C₂×D₁₃	—	8	2	no	—
188	C₂×D₁₃	—	8	2	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	#463	1	1	yes	characteristic
193	C₄×C₁₃	#463	1	1	yes	characteristic
194	C₄×C₁₃	#464	1	1	yes	characteristic
195	C₄×C₁₃	#464	1	1	yes	characteristic
196	C₄×C₁₃	#488	1	2	yes	—
197	C₄×C₁₃	#495	1	1	yes	characteristic
198	C₄×C₁₃	#495	1	1	yes	characteristic
199	Dic₁₃	—	2	4	no	—
200	Dic₁₃	—	4	2	no	—
201	C₂²×Q₈	C₄×D₁₃	1	2	yes	—
202	C₂²×D₄	—	13	2	no	—
203	C₂²×D₄	—	26	1	no	—
204	C₂²×D₄	—	26	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	26	1	no	—
208	C₂²×C₈	—	26	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	#127	—	26	1	no	—
214	#127	—	26	1	no	—
215	#127	—	26	1	no	—
216	#127	—	26	1	no	—
217	#124	—	52	8	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₂²⋊C₄	—	26	4	no	—
221	C₂×C₂²⋊C₄	—	26	4	no	—
222	C₂×C₄²	—	13	2	no	—
223	C₂×C₄²	—	13	2	no	—
224	C₂×C₄²	—	13	1	no	—
225	C₂×C₄²	—	13	1	no	—
226	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	13	2	no	—
231	C₄⋊C₈	—	26	1	no	—
232	C₄⋊C₈	—	26	1	no	—
233	C₄⋊C₈	—	26	1	no	—
234	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
235	C₄×C₈	—	52	2	no	—
236	C₂×C₁₃	#1543	1	2	yes	—
237	C₂×C₁₃	#1551	1	2	yes	—
238	C₂×C₁₃	#1958	1	1	yes	characteristic
239	C₂×C₁₃	#1965	1	1	yes	characteristic
240	C₂×C₁₃	#1983	1	1	yes	characteristic
241	D₁₃	—	16	2	no	—
242	C₂⁴	—	26	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	4	4	no	—
246	C₂×D₄	—	26	2	no	—
247	C₂×D₄	—	26	2	no	—
248	C₂×D₄	—	52	4	no	—
249	C₂×D₄	—	52	1	no	—
250	C₂×D₄	—	52	1	no	—
251	C₂×D₄	—	52	1	no	—
252	C₂×D₄	—	52	1	no	—
253	C₂×D₄	—	52	2	no	—
254	C₂×D₄	—	52	1	no	—
255	C₂×D₄	—	52	2	no	—
256	C₂×D₄	—	52	1	no	—
257	C₂×D₄	—	52	1	no	—
258	C₂×D₄	—	52	1	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	13	2	no	—
261	C₂²×C₄	—	13	2	no	—
262	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
263	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
264	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	2	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	26	2	no	—
271	C₂×C₈	—	52	1	no	—
272	C₂×C₈	—	52	1	no	—
273	C₂×C₈	—	52	2	no	—
274	C₂×C₈	—	52	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	52	8	no	—
280	C₂²⋊C₄	—	52	8	no	—
281	C₄²	—	26	4	no	—
282	C₄²	—	26	1	no	—
283	C₄²	—	26	2	no	—
284	C₄²	—	26	1	no	—
285	C₄²	—	26	1	no	—
286	C₄²	—	26	2	no	—
287	C₄²	—	26	1	no	—
288	C₄²	#6127	1	1	yes	characteristic
289	C₄²	D₄×D₁₃	1	1	yes	characteristic
290	C₄²	#6129	1	1	yes	characteristic
291	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
292	C₁₃	#9699	1	1	yes	characteristic, complemented, Sylow
293	C₂³	—	52	4	no	—
294	C₂³	—	52	2	no	—
295	C₂³	—	52	2	no	—
296	C₂³	—	52	4	no	—
297	C₂³	—	52	2	no	—
298	C₂³	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
299	Q₈	—	8	2	no	—
300	Q₈	—	8	2	no	—
301	D₄	—	104	2	no	—
302	D₄	—	104	2	no	—
303	D₄	—	104	1	no	—
304	D₄	—	104	2	no	—
305	D₄	—	104	1	no	—
306	D₄	—	104	1	no	—
307	D₄	—	104	1	no	—
308	D₄	—	104	1	no	—
309	D₄	—	104	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	13	4	no	—
314	C₂×C₄	—	26	2	no	—
315	C₂×C₄	—	26	2	no	—
316	C₂×C₄	—	26	2	no	—
317	C₂×C₄	—	26	2	no	—
318	C₂×C₄	—	26	2	no	—
319	C₂×C₄	#88135	1	1	yes	characteristic
320	C₂×C₄	#88138	1	1	yes	characteristic
321	C₂×C₄	#88139	1	1	yes	characteristic
322	C₂×C₄	#88168	1	1	yes	characteristic
323	C₂×C₄	#88170	1	1	yes	characteristic
324	C₂×C₄	#88170	1	2	yes	—
325	C₂×C₄	#88172	1	1	yes	characteristic
326	C₂×C₄	#88173	1	2	yes	—
327	C₂×C₄	#88173	1	1	yes	characteristic
328	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
329	C₂×C₄	#88193	1	1	yes	characteristic
330	C₂×C₄	#88194	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	8	1	no	—
333	C₈	—	52	2	no	—
334	C₈	—	104	1	no	—
335	C₂²	—	104	2	no	—
336	C₂²	—	104	2	no	—
337	C₂²	—	104	4	no	—
338	C₂²	—	104	4	no	—
339	C₂²	—	104	2	no	—
340	C₂²	D₁₃×#100	1	2	yes	—
341	C₂²	#11718464	1	1	yes	characteristic
342	C₂²	#11718465	1	1	yes	characteristic
343	C₂²	#11718465	1	2	yes	—
344	C₂²	#11718618	1	1	yes	characteristic
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	26	4	no	—
349	C₄	—	52	2	no	—
350	C₄	#11718459	1	1	yes	characteristic
351	C₄	#11718461	1	2	yes	—
352	C₄	#11718480	1	1	yes	characteristic
353	C₄	#11718826	1	1	yes	characteristic
354	C₄	#11718829	1	1	yes	characteristic
355	C₄	#11718846	1	1	yes	characteristic
356	C₄	#11718852	1	1	yes	characteristic
357	C₂	—	208	2	no	—
358	C₂	#13113497	1	2	yes	—
359	C₂	#13113530	1	2	yes	—
360	C₂	#13115644	1	1	yes	characteristic
361	C₂	#13115674	1	1	yes	characteristic
362	C₂	#13115765	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#333	C₂
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂×#88128	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#333	D₁₃
hypocenter	C₁₃	#9699
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
2. #13114952	C₂	semidirect
3. C₄⋊C₈×Dic₁₃	C₂	semidirect
4. C₂×#11718138	C₂	semidirect
7. C₂×C₁₃×#333	C₂	semidirect
116. C₂×#333	D₁₃	semidirect
292. C₁₃	#9699	semidirect

Maximal subgroups

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

subgroup	index
#13123194	2
#13114952	2
C₄⋊C₈×Dic₁₃	2
C₂×#11718138	2
C₂×#11718077	2
C₂×C₁₃×#333	2
#9699	13