Subgroups — #15768530

Order 3328.

918 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718142	C₂	1	2	yes	maximal
3	C₂×#11718142	C₂	1	2	yes	maximal
4	C₄×#88071	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
9	C₂³×#1055	C₂²	1	1	yes	characteristic
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	C₂×#88071	C₄	1	2	yes	—
12	C₂×#88071	C₄	1	2	yes	—
13	C₂×C₄×#1055	—	2	4	no	—
14	C₂×C₄×#1055	C₄	1	4	yes	complemented
15	C₁₃×#336	C₄	1	4	yes	complemented
16	#11718142	—	2	4	no	—
17	#11718142	—	2	4	no	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₂³×Dic₁₃	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₃	—	2	2	no	—
30	C₂×C₄²×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₁₃	—	2	2	no	—
34	C₂×C₄×Dic₁₃	—	2	2	no	—
35	C₂×C₄×Dic₁₃	—	2	2	no	—
36	C₂×C₄×Dic₁₃	—	2	2	no	—
37	C₂×C₄×Dic₁₃	—	2	2	no	—
38	C₂×C₄×Dic₁₃	—	2	2	no	—
39	C₂²×#1055	—	2	2	no	—
40	C₂²×#1055	—	2	2	no	—
41	C₂²×#1055	—	2	2	no	—
42	C₂²×#1055	—	2	1	no	—
43	C₂²×#1055	—	2	1	no	—
44	C₂²×#1055	C₂×C₄	1	2	yes	—
45	C₂²×#1055	C₂×C₄	1	2	yes	—
46	C₂²×#1055	D₄	1	4	yes	—
47	C₂²×#1055	D₄	1	2	yes	—
48	C₂²×#1055	Q₈	1	2	yes	—
49	#88071	—	2	4	no	—
50	#88071	C₈	1	8	yes	complemented
51	C₄×#1055	—	2	8	no	—
52	#10797	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
59	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
61	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	#41	1	2	yes	—
68	C₂²×C₄×C₁₃	#41	1	2	yes	—
69	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	#41	1	2	yes	—
71	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
72	C₂²×Dic₁₃	—	2	4	no	—
73	C₂²×Dic₁₃	—	2	4	no	—
74	C₂²×Dic₁₃	—	2	4	no	—
75	C₂²×Dic₁₃	—	2	4	no	—
76	C₂²×Dic₁₃	—	2	4	no	—
77	C₂²×Dic₁₃	—	2	4	no	—
78	C₂²×Dic₁₃	C₂×C₈	1	8	yes	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₂×C₈×C₁₃	—	4	2	no	—
83	C₂×C₈×C₁₃	—	4	2	no	—
84	C₂×C₈×C₁₃	—	4	2	no	—
85	C₂×C₈×C₁₃	—	4	2	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	2	4	no	—
89	C₄×Dic₁₃	—	4	2	no	—
90	C₄×Dic₁₃	—	4	2	no	—
91	C₄×Dic₁₃	—	4	2	no	—
92	C₄×Dic₁₃	—	4	2	no	—
93	C₂×#1055	—	2	2	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	—	2	2	no	—
97	C₂×#1055	—	2	4	no	—
98	C₂×#1055	—	2	4	no	—
99	C₂×#1055	—	2	4	no	—
100	C₂×#1055	—	2	4	no	—
101	C₂×#1055	C₂²⋊C₄	1	4	yes	—
102	C₂×#1055	C₄⋊C₄	1	4	yes	—
103	C₂²×C₄×C₈	—	13	1	no	—
104	C₂×#336	—	13	2	no	—
105	C₂×#336	—	13	2	no	—
106	C₂×#336	D₁₃	1	1	yes	characteristic
107	C₄×#95	—	13	1	no	—
108	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
109	C₂³×C₁₃	#117	1	2	yes	—
110	C₂³×C₁₃	C₄×D₄	1	2	yes	—
111	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
113	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
114	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
115	C₂³×C₁₃	#131	1	2	yes	—
116	C₂³×C₁₃	#131	1	2	yes	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	8	no	—
122	C₂×C₄×C₁₃	—	2	8	no	—
123	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
124	C₂×C₄×C₁₃	#97	1	4	yes	—
125	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#117	1	2	yes	—
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#121	1	2	yes	—
136	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₁₃	#123	1	2	yes	—
138	C₂×C₄×C₁₃	#123	1	2	yes	—
139	C₂×C₄×C₁₃	#124	1	2	yes	—
140	C₂×C₄×C₁₃	#125	1	2	yes	—
141	C₂×C₄×C₁₃	#126	1	2	yes	—
142	C₂×C₄×C₁₃	#126	1	2	yes	—
143	C₂×Dic₁₃	—	2	8	no	—
144	C₂×Dic₁₃	—	2	8	no	—
145	C₂×Dic₁₃	—	2	8	no	—
146	C₂×Dic₁₃	—	4	4	no	—
147	C₂×Dic₁₃	—	4	4	no	—
148	C₂×Dic₁₃	—	4	4	no	—
149	C₂×Dic₁₃	—	4	4	no	—
150	C₈×C₁₃	—	4	4	no	—
151	#1055	—	2	4	no	—
152	#1055	—	2	4	no	—
153	C₂³×C₈	—	13	1	no	—
154	C₂³×C₈	—	13	1	no	—
155	C₂³×C₈	C₂×D₁₃	1	2	yes	—
156	C₂²×C₄²	—	13	2	no	—
157	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
158	C₂×C₄×C₈	—	13	4	no	—
159	C₂×C₄×C₈	—	26	4	no	—
160	C₂×#95	—	13	2	no	—
161	C₂×#95	—	13	2	no	—
162	#336	—	26	4	no	—
163	#336	—	26	4	no	—
164	#336	Dic₁₃	1	4	yes	complemented
165	C₂²×C₁₃	#376	1	1	yes	characteristic
166	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂²×C₁₃	#387	1	2	yes	—
169	C₂²×C₁₃	#388	1	2	yes	—
170	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
171	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
172	C₂²×C₁₃	#405	1	1	yes	characteristic
173	C₂²×C₁₃	#405	1	1	yes	characteristic
174	C₂²×C₁₃	#431	1	4	yes	—
175	C₂²×C₁₃	#433	1	4	yes	—
176	C₂²×C₁₃	C₈×D₄	1	2	yes	—
177	C₂²×C₁₃	C₈×D₄	1	4	yes	—
178	C₂²×C₁₃	#435	1	4	yes	—
179	C₂²×C₁₃	#436	1	2	yes	—
180	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
181	C₂²×C₁₃	#446	1	2	yes	—
182	C₄×C₁₃	—	2	4	no	—
183	C₄×C₁₃	—	2	4	no	—
184	C₄×C₁₃	#380	1	1	yes	characteristic
185	C₄×C₁₃	#381	1	1	yes	characteristic
186	C₄×C₁₃	#382	1	1	yes	characteristic
187	C₄×C₁₃	#383	1	1	yes	characteristic
188	C₄×C₁₃	#387	1	2	yes	—
189	C₄×C₁₃	#388	1	2	yes	—
190	Dic₁₃	—	4	8	no	—
191	C₂³×C₄	—	13	4	no	—
192	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
193	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
194	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	13	2	no	—
202	C₂²×C₈	—	13	2	no	—
203	C₂²×C₈	—	13	2	no	—
204	C₂²×C₈	—	13	4	no	—
205	C₂²×C₈	—	13	2	no	—
206	C₂²×C₈	—	26	2	no	—
207	C₂²×C₈	—	26	2	no	—
208	C₂²×C₈	—	26	2	no	—
209	C₂²×C₈	—	26	1	no	—
210	C₂²×C₈	—	26	1	no	—
211	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	26	2	no	—
215	C₂×C₄²	—	26	2	no	—
216	C₂×C₄²	—	26	2	no	—
217	C₂×C₄²	—	26	2	no	—
218	C₂×C₄²	—	26	2	no	—
219	C₂×C₄²	—	26	2	no	—
220	C₂×C₄²	C₄×D₁₃	1	2	yes	—
221	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
222	C₄×C₈	—	26	8	no	—
223	#95	—	13	8	no	—
224	#95	—	26	4	no	—
225	C₂×C₁₃	#1434	1	1	yes	characteristic
226	C₂×C₁₃	#1744	1	1	yes	characteristic
227	C₂×C₁₃	#1746	1	1	yes	characteristic
228	C₂×C₁₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₁₃	#1752	1	1	yes	characteristic
230	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
231	C₂×C₁₃	#1770	1	1	yes	characteristic
232	C₂×C₁₃	#1916	1	4	yes	—
233	C₂×C₁₃	#1923	1	4	yes	—
234	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	13	8	no	—
240	C₂²×C₄	—	26	4	no	—
241	C₂²×C₄	—	26	4	no	—
242	C₂²×C₄	—	26	4	no	—
243	C₂²×C₄	—	26	4	no	—
244	C₂²×C₄	—	26	4	no	—
245	C₂²×C₄	—	26	4	no	—
246	C₂²×C₄	C₈×D₁₃	1	4	yes	—
247	C₂²×C₄	#6094	1	4	yes	—
248	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
249	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
250	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
251	C₂²×C₄	#6127	1	2	yes	—
252	C₂²×C₄	#6127	1	2	yes	—
253	C₂²×C₄	D₄×D₁₃	1	2	yes	—
254	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	#6129	1	2	yes	—
256	C₂²×C₄	#6129	1	1	yes	characteristic
257	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
258	C₂²×C₄	#6131	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	13	4	no	—
268	C₂×C₈	—	13	4	no	—
269	C₂×C₈	—	26	2	no	—
270	C₂×C₈	—	26	2	no	—
271	C₂×C₈	—	26	2	no	—
272	C₂×C₈	—	26	4	no	—
273	C₂×C₈	—	26	4	no	—
274	C₂×C₈	—	26	2	no	—
275	C₂×C₈	—	26	4	no	—
276	C₂×C₈	—	26	4	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	4	no	—
280	C₄²	—	52	2	no	—
281	C₄²	—	52	2	no	—
282	C₄²	—	52	2	no	—
283	C₄²	—	52	2	no	—
284	C₁₃	#10797	1	1	yes	characteristic, complemented, Sylow
285	C₂³	#88126	1	2	yes	—
286	C₂³	#88135	1	2	yes	—
287	C₂³	#88141	1	1	yes	characteristic
288	C₂³	#88147	1	1	yes	characteristic
289	C₂³	C₂×C₈×D₁₃	1	2	yes	—
290	C₂³	C₂×#6094	1	2	yes	—
291	C₂³	#88161	1	2	yes	—
292	C₂³	#88161	1	2	yes	—
293	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	26	8	no	—
301	C₂×C₄	—	26	8	no	—
302	C₂×C₄	—	26	8	no	—
303	C₂×C₄	—	52	4	no	—
304	C₂×C₄	—	52	4	no	—
305	C₂×C₄	—	52	4	no	—
306	C₂×C₄	—	52	4	no	—
307	C₂×C₄	C₈×Dic₁₃	1	4	yes	—
308	C₂×C₄	#88055	1	4	yes	—
309	C₂×C₄	#88126	1	1	yes	characteristic
310	C₂×C₄	#88126	1	1	yes	characteristic
311	C₂×C₄	#88131	1	1	yes	characteristic
312	C₂×C₄	#88133	1	2	yes	—
313	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
314	C₂×C₄	#88137	1	2	yes	—
315	C₂×C₄	#88138	1	2	yes	—
316	C₂×C₄	#88139	1	2	yes	—
317	C₂×C₄	#88143	1	2	yes	—
318	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
319	C₂×C₄	#88146	1	1	yes	characteristic
320	C₂×C₄	#88148	1	2	yes	—
321	C₂×C₄	#88150	1	2	yes	—
322	C₂×C₄	#88152	1	2	yes	—
323	C₂×C₄	#88180	1	2	yes	—
324	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
325	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
326	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
327	C₈	—	4	4	no	—
328	C₈	—	26	4	no	—
329	C₈	—	26	4	no	—
330	C₂²	#11718313	1	2	yes	—
331	C₂²	#11718330	1	2	yes	—
332	C₂²	#11718346	1	4	yes	—
333	C₂²	#11718370	1	4	yes	—
334	C₂²	#11718386	1	4	yes	—
335	C₂²	#11718391	1	4	yes	—
336	C₂²	#11718491	1	2	yes	—
337	C₂²	#11718492	1	2	yes	—
338	C₂²	#11718495	1	2	yes	—
339	C₂²	#11718499	1	2	yes	—
340	C₂²	#11718593	1	1	yes	characteristic
341	C₂²	C₂²⋊C₄×Dic₁₃	1	1	yes	characteristic
342	C₂²	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
343	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
344	C₂²	C₂×#88055	1	1	yes	characteristic
345	C₂²	#11718783	1	1	yes	characteristic
346	C₂²	#11718783	1	1	yes	characteristic
347	C₄	—	2	4	no	—
348	C₄	—	2	4	no	—
349	C₄	—	52	8	no	—
350	C₄	#11718311	1	1	yes	characteristic
351	C₄	#11718312	1	1	yes	characteristic
352	C₄	#11718325	1	1	yes	characteristic
353	C₄	#11718328	1	1	yes	characteristic
354	C₄	#11718612	1	2	yes	—
355	C₄	#11718642	1	2	yes	—
356	C₂	#13113061	1	1	yes	characteristic
357	C₂	#13114581	1	1	yes	characteristic
358	C₂	#13114589	1	1	yes	characteristic
359	C₂	#13114610	1	1	yes	characteristic
360	C₂	#13114615	1	1	yes	characteristic
361	C₂	#13114701	1	1	yes	characteristic
362	C₂	#13114706	1	1	yes	characteristic
363	C₂	#13115391	1	4	yes	—
364	C₂	#13115440	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₁₃×#336	C₂
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#336	D₁₃
hypocenter	C₁₃	#10797
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
14. C₂×C₄×#1055	C₄	semidirect
15. C₁₃×#336	C₄	semidirect
50. #88071	C₈	semidirect
164. #336	Dic₁₃	semidirect
284. C₁₃	#10797	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×#1055	2
C₂×#11718142	2
C₄×#88071	2
C₂×C₁₃×#336	2
#10797	13