Subgroups — #15761978

Order 3328.

622 conjugacy classes of subgroups.

The subgroup lattice has 372 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	#13114987	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718139	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×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×#88201	—	2	2	no	—
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×#88054	—	2	2	no	—
14	C₂×C₈×Dic₁₃	—	2	1	no	—
15	#11718649	—	2	1	no	—
16	#11718649	—	2	1	no	—
17	#11718649	C₄	1	4	yes	—
18	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
21	C₁₃×#332	—	2	2	no	—
22	#11718142	—	2	1	no	—
23	#11718139	—	2	2	no	—
24	#11718077	—	2	2	no	—
25	#11718077	—	2	2	no	—
26	C₂²×Q₈×C₁₃	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	#88201	—	4	4	no	—
40	#88201	—	4	4	no	—
41	C₂×#6103	—	2	4	no	—
42	C₂×#6103	—	2	4	no	—
43	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
45	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
46	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
47	C₂²×#1055	—	2	1	no	—
48	C₂²×#1055	—	2	1	no	—
49	#88128	—	2	1	no	—
50	#88128	—	2	1	no	—
51	#88128	—	2	1	no	—
52	#88128	—	2	1	no	—
53	C₁₃×C₄⋊C₈	—	2	1	no	—
54	C₁₃×C₄⋊C₈	—	2	1	no	—
55	C₁₃×C₄⋊C₈	—	2	1	no	—
56	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
57	#88054	—	4	4	no	—
58	C₈×Dic₁₃	—	4	2	no	—
59	C₄⋊#1055	—	2	1	no	—
60	C₄⋊#1055	—	2	1	no	—
61	C₄⋊#1055	—	2	1	no	—
62	C₄⋊#1055	D₄	1	2	yes	—
63	#9698	—	13	1	no	maximal, Sylow
64	C₂³×D₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	—	4	1	no	—
66	C₂×Q₈×C₁₃	—	4	1	no	—
67	C₂×Q₈×C₁₃	—	4	2	no	—
68	C₂×Q₈×C₁₃	—	4	2	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
74	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂×D₅₂	—	2	2	no	—
76	C₂×D₅₂	—	2	2	no	—
77	C₂×D₅₂	—	4	4	no	—
78	C₂×D₅₂	—	4	1	no	—
79	C₂×D₅₂	—	4	1	no	—
80	C₂×D₅₂	—	4	1	no	—
81	C₂×D₅₂	—	4	1	no	—
82	C₂×D₅₂	—	4	2	no	—
83	C₂×D₅₂	—	4	2	no	—
84	C₂×D₅₂	—	4	1	no	—
85	C₂×D₅₂	—	4	1	no	—
86	C₂×D₅₂	—	4	1	no	—
87	C₂×D₅₂	—	4	1	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	4	1	no	—
90	C₂×C₈×C₁₃	—	4	1	no	—
91	C₂×C₈×C₁₃	—	4	1	no	—
92	C₂×C₈×C₁₃	—	4	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	4	1	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃	#41	1	1	yes	characteristic
100	C₄²×C₁₃	#41	1	1	yes	characteristic
101	#6103	—	4	8	no	—
102	#6103	—	4	8	no	—
103	C₄×Dic₁₃	—	2	1	no	—
104	C₄×Dic₁₃	—	2	1	no	—
105	C₄×Dic₁₃	—	2	1	no	—
106	C₄×Dic₁₃	—	2	1	no	—
107	C₄×Dic₁₃	—	4	2	no	—
108	C₄×Dic₁₃	D₈	1	4	yes	—
109	C₄×Dic₁₃	SD₁₆	1	4	yes	—
110	C₂×#1055	—	2	2	no	—
111	C₂×#1055	—	4	1	no	—
112	C₂×#1055	—	4	1	no	—
113	C₂×#1055	—	4	1	no	—
114	C₂×#1055	—	4	2	no	—
115	#2858	—	13	1	no	—
116	#1842	—	13	1	no	—
117	C₄×C₄⋊C₈	—	13	1	no	—
118	C₂×#332	—	13	1	no	—
119	C₂×#332	D₁₃	1	1	yes	characteristic, complemented
120	C₂×#331	—	13	1	no	—
121	C₂×#331	—	13	1	no	—
122	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×D₁₃	—	4	2	no	—
124	C₂²×D₁₃	—	4	4	no	—
125	C₂²×D₁₃	—	4	4	no	—
126	C₂²×D₁₃	—	4	2	no	—
127	C₂²×D₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	8	2	no	—
129	Q₈×C₁₃	—	8	1	no	—
130	Q₈×C₁₃	—	8	1	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	4	no	—
134	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
142	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
144	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
146	C₂×Dic₁₃	—	2	2	no	—
147	C₂×Dic₁₃	—	2	2	no	—
148	C₂×Dic₁₃	—	2	2	no	—
149	C₂×Dic₁₃	—	2	2	no	—
150	C₂×Dic₁₃	#102	1	4	yes	—
151	C₂×Dic₁₃	#102	1	4	yes	—
152	D₅₂	—	8	2	no	—
153	D₅₂	—	8	2	no	—
154	D₅₂	—	8	1	no	—
155	D₅₂	—	8	1	no	—
156	D₅₂	—	8	1	no	—
157	D₅₂	—	8	1	no	—
158	D₅₂	—	8	2	no	—
159	D₅₂	—	8	1	no	—
160	D₅₂	—	8	1	no	—
161	C₈×C₁₃	—	4	2	no	—
162	C₈×C₁₃	—	8	1	no	—
163	#1055	—	4	2	no	—
164	#1055	—	8	1	no	—
165	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
166	C₂×#127	—	13	1	no	—
167	C₂×#124	—	26	2	no	—
168	C₂×C₄⋊C₈	—	13	1	no	—
169	C₂×C₄⋊C₈	—	13	1	no	—
170	C₂×C₄⋊C₈	—	26	2	no	—
171	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
172	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
173	C₂×C₄×C₈	—	26	1	no	—
174	#390	—	13	4	no	—
175	#390	—	26	1	no	—
176	#390	—	26	1	no	—
177	C₄³	—	13	1	no	—
178	#336	—	26	1	no	—
179	#332	—	2	2	no	—
180	#332	—	26	2	no	—
181	#331	—	26	2	no	—
182	#331	—	26	2	no	—
183	C₂²×C₁₃	#390	1	1	yes	characteristic
184	C₂²×C₁₃	#413	1	2	yes	—
185	C₂²×C₁₃	C₂×#102	1	2	yes	—
186	C₂²×C₁₃	#416	1	1	yes	characteristic
187	C₂²×C₁₃	#417	1	1	yes	characteristic
188	C₂×D₁₃	—	8	4	no	—
189	C₂×D₁₃	—	8	4	no	—
190	C₂×D₁₃	—	8	2	no	—
191	C₂×D₁₃	—	8	2	no	—
192	C₂×D₁₃	—	8	2	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	8	2	no	—
196	C₄×C₁₃	#463	1	1	yes	characteristic
197	C₄×C₁₃	#463	1	1	yes	characteristic
198	C₄×C₁₃	#464	1	1	yes	characteristic
199	C₄×C₁₃	#464	1	1	yes	characteristic
200	C₄×C₁₃	#486	1	2	yes	—
201	C₄×C₁₃	#492	1	1	yes	characteristic
202	C₄×C₁₃	#493	1	1	yes	characteristic
203	Dic₁₃	—	2	4	no	—
204	Dic₁₃	—	2	4	no	—
205	C₂²×Q₈	—	2	1	no	—
206	C₂²×D₄	—	13	2	no	—
207	C₂²×D₄	—	26	1	no	—
208	C₂²×D₄	—	26	1	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	26	1	no	—
212	C₂²×C₈	—	26	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	#127	—	26	1	no	—
218	#127	—	26	1	no	—
219	#127	—	26	1	no	—
220	#127	—	26	1	no	—
221	#124	—	52	4	no	—
222	#124	—	52	4	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	C₄×D₁₃	1	2	yes	—
225	C₂×C₂²⋊C₄	—	26	4	no	—
226	C₂×C₂²⋊C₄	—	26	4	no	—
227	C₂×C₄²	—	13	2	no	—
228	C₂×C₄²	—	13	1	no	—
229	C₂×C₄²	—	13	1	no	—
230	C₂×C₄²	—	13	2	no	—
231	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
232	C₄⋊C₈	—	2	1	no	—
233	C₄⋊C₈	—	2	1	no	—
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	13	2	no	—
236	C₄⋊C₈	—	26	1	no	—
237	C₄⋊C₈	—	26	1	no	—
238	C₄⋊C₈	—	26	1	no	—
239	C₄⋊C₈	—	52	4	no	—
240	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
241	C₄×C₈	—	52	2	no	—
242	C₂×C₁₃	#1540	1	2	yes	—
243	C₂×C₁₃	#1547	1	2	yes	—
244	C₂×C₁₃	#1957	1	1	yes	characteristic
245	C₂×C₁₃	#1967	1	1	yes	characteristic
246	C₂×C₁₃	#1983	1	1	yes	characteristic
247	D₁₃	—	16	2	no	—
248	C₂⁴	—	26	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×Q₈	—	4	2	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×D₄	—	26	2	no	—
254	C₂×D₄	—	26	2	no	—
255	C₂×D₄	—	52	4	no	—
256	C₂×D₄	—	52	1	no	—
257	C₂×D₄	—	52	1	no	—
258	C₂×D₄	—	52	2	no	—
259	C₂×D₄	—	52	2	no	—
260	C₂×D₄	—	52	1	no	—
261	C₂×D₄	—	52	1	no	—
262	C₂×D₄	—	52	1	no	—
263	C₂×D₄	—	52	1	no	—
264	C₂×D₄	—	52	1	no	—
265	C₂×D₄	—	52	1	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	13	2	no	—
268	C₂²×C₄	—	13	2	no	—
269	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
270	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
271	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	4	2	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	26	2	no	—
278	C₂×C₈	—	52	1	no	—
279	C₂×C₈	—	52	1	no	—
280	C₂×C₈	—	52	2	no	—
281	C₂×C₈	—	52	1	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	52	8	no	—
287	C₂²⋊C₄	—	52	8	no	—
288	C₄²	—	13	4	no	—
289	C₄²	—	13	4	no	—
290	C₄²	—	26	1	no	—
291	C₄²	—	26	1	no	—
292	C₄²	—	26	1	no	—
293	C₄²	—	26	1	no	—
294	C₄²	—	52	2	no	—
295	C₄²	#6127	1	1	yes	characteristic
296	C₄²	D₄×D₁₃	1	1	yes	characteristic
297	C₄²	#6129	1	1	yes	characteristic
298	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
299	C₁₃	#9698	1	1	yes	characteristic, complemented, Sylow
300	C₂³	—	52	4	no	—
301	C₂³	—	52	4	no	—
302	C₂³	—	52	2	no	—
303	C₂³	—	52	2	no	—
304	C₂³	—	52	2	no	—
305	C₂³	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
306	Q₈	—	8	2	no	—
307	Q₈	—	8	1	no	—
308	Q₈	—	8	1	no	—
309	D₄	—	104	2	no	—
310	D₄	—	104	2	no	—
311	D₄	—	104	2	no	—
312	D₄	—	104	1	no	—
313	D₄	—	104	1	no	—
314	D₄	—	104	1	no	—
315	D₄	—	104	1	no	—
316	D₄	—	104	1	no	—
317	D₄	—	104	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	13	4	no	—
322	C₂×C₄	—	13	4	no	—
323	C₂×C₄	—	26	2	no	—
324	C₂×C₄	—	26	2	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	#88135	1	1	yes	characteristic
328	C₂×C₄	#88138	1	1	yes	characteristic
329	C₂×C₄	#88139	1	1	yes	characteristic
330	C₂×C₄	D₈×D₁₃	1	2	yes	—
331	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
332	C₂×C₄	#88168	1	1	yes	characteristic
333	C₂×C₄	#88170	1	1	yes	characteristic
334	C₂×C₄	#88172	1	1	yes	characteristic
335	C₂×C₄	#88173	1	1	yes	characteristic
336	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
337	C₂×C₄	#88193	1	1	yes	characteristic
338	C₂×C₄	#88194	1	1	yes	characteristic
339	C₈	—	4	2	no	—
340	C₈	—	8	1	no	—
341	C₈	—	52	2	no	—
342	C₈	—	104	1	no	—
343	C₂²	—	104	2	no	—
344	C₂²	—	104	2	no	—
345	C₂²	—	104	4	no	—
346	C₂²	—	104	4	no	—
347	C₂²	—	104	2	no	—
348	C₂²	#11718412	1	2	yes	—
349	C₂²	D₁₃×#102	1	2	yes	—
350	C₂²	#11718464	1	1	yes	characteristic
351	C₂²	#11718465	1	1	yes	characteristic
352	C₂²	#11718618	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	8	2	no	—
356	C₄	—	26	4	no	—
357	C₄	—	26	4	no	—
358	C₄	#11718421	1	2	yes	—
359	C₄	#11718459	1	1	yes	characteristic
360	C₄	#11718480	1	1	yes	characteristic
361	C₄	#11718812	1	1	yes	characteristic
362	C₄	#11718826	1	1	yes	characteristic
363	C₄	#11718836	1	1	yes	characteristic
364	C₄	#11718846	1	1	yes	characteristic
365	C₂	—	208	2	no	—
366	C₂	#13113484	1	2	yes	—
367	C₂	#13113512	1	2	yes	—
368	C₂	#13115640	1	1	yes	characteristic
369	C₂	#13115684	1	1	yes	characteristic
370	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×#332	C₂
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂×#88128	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#332	D₁₃
hypocenter	C₁₃	#9698
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

Complements

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

N	H ≅ G/N	type
2. #13114987	C₂	semidirect
3. C₄⋊C₈×Dic₁₃	C₂	semidirect
4. C₂×#11718139	C₂	semidirect
7. C₂×C₁₃×#332	C₂	semidirect
119. C₂×#332	D₁₃	semidirect
299. C₁₃	#9698	semidirect

Maximal subgroups

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

subgroup	index
#13123194	2
#13114987	2
C₄⋊C₈×Dic₁₃	2
C₂×#11718139	2
C₂×#11718077	2
C₂×C₁₃×#332	2
#9698	13