Subgroups — #16015563

Order 3328.

834 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718568	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718562	C₂	1	1	yes	maximal, characteristic
3	#13123258	C₂	1	1	yes	maximal, characteristic
4	#13116434	C₂	1	2	yes	maximal, complemented
5	#13116433	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116431	C₂	1	2	yes	maximal, complemented
7	#13116429	C₂	1	2	yes	maximal, complemented
8	M₃₂×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₆×D₁₃	C₂	1	2	yes	maximal, complemented
11	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88161	C₄	1	1	yes	characteristic
13	C₂×#88155	C₂²	1	1	yes	characteristic
14	#11719227	C₄	1	1	yes	characteristic
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₁₃×#405	C₂²	1	1	yes	characteristic
18	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×#88052	C₂²	1	1	yes	characteristic
21	C₂²×#6090	C₂²	1	1	yes	characteristic
22	#11718658	C₄	1	1	yes	characteristic
23	#11718568	C₄	1	8	yes	complemented
24	#11718562	C₄	1	8	yes	complemented
25	C₂×#88038	C₂²	1	2	yes	—
26	C₂×#88038	C₂²	1	2	yes	—
27	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
28	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
29	#11718369	C₄	1	2	yes	—
30	#11718368	C₄	1	2	yes	—
31	#11718350	C₂²	1	2	yes	—
32	C₄×C₈×D₁₃	C₂²	1	2	yes	—
33	C₁₃×#346	C₂²	1	2	yes	—
34	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
35	#11718094	C₂²	1	2	yes	—
36	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
37	C₁₆×Dic₁₃	C₂²	1	2	yes	—
38	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
39	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
40	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
41	#11718053	C₂²	1	2	yes	—
42	C₄×#6090	C₂²	1	2	yes	—
43	C₂×#6127	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×#6099	C₂×C₄	1	1	yes	characteristic
49	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
50	#88161	C₈	1	8	yes	—
51	#88155	C₂×C₄	1	4	yes	—
52	#88155	C₂×C₄	1	4	yes	—
53	C₂×#6094	C₂×C₄	1	2	yes	—
54	C₂×#6094	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
56	C₂×C₈×D₁₃	C₂³	1	2	yes	—
57	#88147	C₈	1	2	yes	—
58	#88141	C₈	1	2	yes	—
59	#88135	C₈	1	4	yes	—
60	#88126	C₂×C₄	1	2	yes	—
61	C₄²×D₁₃	C₂×C₄	1	2	yes	—
62	C₁₃×M₃₂	C₂×C₄	1	4	yes	complemented
63	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
64	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
65	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
66	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
67	C₁₃×#97	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₃	C₂³	1	2	yes	—
69	#88055	C₂×C₄	1	2	yes	—
70	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
73	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
74	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
75	#88052	C₂×C₄	1	4	yes	complemented
76	C₂×#6090	C₂×C₄	1	4	yes	complemented
77	C₂×#6090	C₂³	1	1	yes	characteristic
78	C₂×#6090	C₂³	1	1	yes	characteristic
79	C₂×#6090	C₂³	1	2	yes	—
80	#88043	C₂×C₄	1	2	yes	—
81	C₄×#1055	C₂×C₄	1	2	yes	—
82	#88038	C₂×C₄	1	8	yes	complemented
83	#88038	C₂×C₄	1	8	yes	complemented
84	C₁₆×D₁₃	C₂×C₄	1	8	yes	complemented
85	C₁₆×D₁₃	C₂×C₄	1	8	yes	complemented
86	C₄×#504	—	13	1	no	maximal, Sylow
87	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
89	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
90	#6127	C₄²	1	4	yes	—
91	#6127	C₂×C₈	1	4	yes	—
92	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
93	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
94	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
95	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
96	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
97	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
102	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
103	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
104	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
105	#6103	C₂×C₈	1	4	yes	—
106	#6101	C₂×C₈	1	4	yes	—
107	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
108	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
109	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
111	#6099	C₄²	1	4	yes	—
112	C₂×#1055	C₂×C₈	1	4	yes	—
113	C₂×#1055	C₂²×C₄	1	2	yes	—
114	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
115	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
116	#6094	C₄²	1	8	yes	—
117	#6094	C₂×C₈	1	8	yes	—
118	C₈×D₁₃	C₂×C₈	1	8	yes	—
119	C₈×D₁₃	C₂²×C₄	1	4	yes	—
120	C₈×D₁₃	C₂²×C₄	1	4	yes	—
121	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
122	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
123	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
124	#6090	C₂²×C₄	1	2	yes	—
125	#6090	C₂²×C₄	1	4	yes	—
126	#6090	C₂²×C₄	1	2	yes	—
127	C₂×#504	—	13	1	no	—
128	C₂×#504	—	13	1	no	—
129	C₄×#131	—	13	1	no	—
130	#2108	—	13	1	no	—
131	#2108	—	13	2	no	—
132	#2108	—	13	2	no	—
133	#2108	D₁₃	1	1	yes	characteristic, complemented
134	C₄×M₃₂	—	13	1	no	—
135	C₄×M₃₂	—	13	2	no	—
136	C₂×C₄×C₁₆	—	13	1	no	—
137	C₂×C₄×C₁₆	—	13	2	no	—
138	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
139	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
141	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
142	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
143	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
145	C₂²⋊D₁₃	C₄×C₈	1	8	yes	—
146	C₂×Dic₁₃	—	2	4	no	—
147	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
148	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
149	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
150	D₅₂	C₄×C₈	1	4	yes	—
151	C₄×D₁₃	—	2	8	no	—
152	C₄×D₁₃	C₂×C₄²	1	4	yes	—
153	C₄×D₁₃	C₂²×C₈	1	4	yes	—
154	Dic₂₆	C₄×C₈	1	4	yes	—
155	C₈×C₁₃	C₂×C₄²	1	2	yes	—
156	C₈×C₁₃	C₂²×C₈	1	4	yes	—
157	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
158	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
159	#1055	C₂×C₄²	1	4	yes	—
160	#1055	C₂²×C₈	1	2	yes	—
161	#1055	C₂²×C₈	1	2	yes	—
162	C₂×#131	—	13	1	no	—
163	C₂×#131	—	13	1	no	—
164	C₄×#41	—	13	1	no	—
165	#504	—	13	8	no	—
166	#504	—	13	8	no	—
167	C₂×M₃₂	—	13	1	no	—
168	C₂×M₃₂	—	13	2	no	—
169	C₂×M₃₂	—	13	2	no	—
170	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
171	C₂²×C₁₆	—	13	1	no	—
172	C₂²×C₁₆	—	13	2	no	—
173	C₂²×C₁₆	—	13	2	no	—
174	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
175	#405	—	13	1	no	—
176	#405	—	13	2	no	—
177	#405	—	13	2	no	—
178	#405	C₂×D₁₃	1	1	yes	characteristic
179	C₄×M₁₆	—	13	1	no	—
180	C₄×M₁₆	—	13	2	no	—
181	C₂×C₄×C₈	—	13	1	no	—
182	C₂×C₄×C₈	—	13	2	no	—
183	#346	—	13	2	no	—
184	#346	—	13	2	no	—
185	#346	C₂×D₁₃	1	2	yes	—
186	C₄×C₁₆	—	13	1	no	—
187	C₄×C₁₆	—	13	2	no	—
188	C₄×C₁₆	—	13	1	no	—
189	C₄×C₁₆	—	13	1	no	—
190	C₄×C₁₆	—	13	1	no	—
191	C₄×C₁₆	—	13	2	no	—
192	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
193	C₂²×C₁₃	—	2	2	no	—
194	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
195	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
196	C₂×D₁₃	—	2	4	no	—
197	C₂×D₁₃	C₂×C₄×C₈	1	4	yes	—
198	C₄×C₁₃	—	2	4	no	—
199	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
200	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
201	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
202	Dic₁₃	—	2	2	no	—
203	Dic₁₃	C₂×C₄×C₈	1	4	yes	—
204	Dic₁₃	#504	1	4	yes	complemented
205	C₂×#41	—	13	1	no	—
206	#131	—	13	8	no	—
207	#131	—	13	4	no	—
208	#131	—	13	4	no	—
209	C₂×M₁₆	—	13	1	no	—
210	C₂×M₁₆	—	13	2	no	—
211	C₂×M₁₆	—	13	2	no	—
212	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
213	C₂²×C₈	—	13	1	no	—
214	C₂²×C₈	—	13	2	no	—
215	C₂²×C₈	—	13	2	no	—
216	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
217	C₄×Q₈	—	13	2	no	—
218	C₄×D₄	—	13	2	no	—
219	C₄×D₄	—	13	4	no	—
220	#117	—	13	2	no	—
221	#117	C₄×D₁₃	1	1	yes	characteristic
222	C₂×C₄²	—	13	1	no	—
223	C₂×C₄²	—	13	2	no	—
224	M₃₂	—	13	4	no	—
225	M₃₂	—	13	8	no	—
226	M₃₂	—	13	8	no	—
227	M₃₂	C₄×D₁₃	1	4	yes	complemented
228	C₂×C₁₆	—	13	4	no	—
229	C₂×C₁₆	—	13	1	no	—
230	C₂×C₁₆	—	13	1	no	—
231	C₂×C₁₆	—	13	2	no	—
232	C₂×C₁₆	—	13	8	no	—
233	C₂×C₁₆	—	13	8	no	—
234	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
235	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
237	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
238	#97	—	13	2	no	—
239	#97	—	13	2	no	—
240	#97	C₄×D₁₃	1	2	yes	—
241	C₄×C₈	—	13	1	no	—
242	C₄×C₈	—	13	2	no	—
243	C₄×C₈	—	13	1	no	—
244	C₄×C₈	—	13	1	no	—
245	C₄×C₈	—	13	1	no	—
246	C₄×C₈	—	13	2	no	—
247	C₄×C₈	C₂²×D₁₃	1	2	yes	—
248	C₂×C₁₃	—	2	2	no	—
249	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
250	C₂×C₁₃	C₂×#504	1	2	yes	—
251	D₁₃	—	2	4	no	—
252	#41	—	13	4	no	—
253	#41	—	13	4	no	—
254	C₂×Q₈	—	13	1	no	—
255	C₂×D₄	—	13	2	no	—
256	C₂×D₄	—	13	1	no	—
257	C₂²×C₄	—	13	2	no	—
258	C₂²×C₄	—	13	2	no	—
259	C₂²×C₄	—	13	4	no	—
260	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
261	M₁₆	—	13	4	no	—
262	M₁₆	—	13	8	no	—
263	M₁₆	—	13	8	no	—
264	M₁₆	C₈×D₁₃	1	4	yes	—
265	C₂×C₈	—	13	4	no	—
266	C₂×C₈	—	13	1	no	—
267	C₂×C₈	—	13	2	no	—
268	C₂×C₈	—	13	1	no	—
269	C₂×C₈	—	13	8	no	—
270	C₂×C₈	—	13	4	no	—
271	C₂×C₈	—	13	4	no	—
272	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
273	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
274	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
275	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
276	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
277	C₄⋊C₄	—	13	4	no	—
278	C₄⋊C₄	C₈×D₁₃	1	2	yes	—
279	C₂²⋊C₄	—	13	4	no	—
280	C₂²⋊C₄	C₈×D₁₃	1	2	yes	—
281	C₄²	—	13	4	no	—
282	C₄²	—	13	2	no	—
283	C₄²	—	13	1	no	—
284	C₄²	—	13	1	no	—
285	C₄²	C₂×C₄×D₁₃	1	2	yes	—
286	C₁₆	—	13	2	no	—
287	C₁₆	—	13	2	no	—
288	C₁₆	—	13	4	no	—
289	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
290	C₁₆	C₂×C₄×D₁₃	1	4	yes	—
291	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
292	C₁₃	C₄×#504	1	1	yes	characteristic, complemented, Sylow
293	C₂³	—	13	2	no	—
294	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
295	Q₈	—	13	4	no	—
296	D₄	—	13	8	no	—
297	D₄	—	13	4	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	13	2	no	—
300	C₂×C₄	—	13	4	no	—
301	C₂×C₄	—	13	4	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	26	8	no	—
304	C₂×C₄	—	26	4	no	—
305	C₂×C₄	C₄²×D₁₃	1	2	yes	—
306	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
307	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
308	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
310	C₈	—	13	4	no	—
311	C₈	—	13	2	no	—
312	C₈	—	13	2	no	—
313	C₈	C₄²×D₁₃	1	2	yes	—
314	C₈	C₂×C₈×D₁₃	1	4	yes	—
315	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
316	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	13	4	no	—
319	C₂²	—	26	4	no	—
320	C₂²	C₄×C₈×D₁₃	1	2	yes	—
321	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
322	C₄	—	2	4	no	—
323	C₄	—	13	4	no	—
324	C₄	—	13	4	no	—
325	C₄	—	26	2	no	—
326	C₄	C₄×C₈×D₁₃	1	2	yes	—
327	C₄	C₂×C₄²×D₁₃	1	1	yes	characteristic
328	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	26	4	no	—
331	C₂	C₂×C₄×C₈×D₁₃	1	1	yes	characteristic
332	C₂	D₁₃×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2108	C₂
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2108	D₁₃
hypocenter	C₁₃	C₄×#504
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
4. #13116434	C₂	semidirect
5. #13116433	C₂	semidirect
6. #13116431	C₂	semidirect
7. #13116429	C₂	semidirect
8. M₃₂×Dic₁₃	C₂	semidirect
9. C₂×C₁₆×Dic₁₃	C₂	semidirect
10. C₄×C₁₆×D₁₃	C₂	semidirect
11. C₁₃×#2108	C₂	semidirect
23. #11718568	C₄	semidirect
24. #11718562	C₄	semidirect
62. C₁₃×M₃₂	C₂×C₄	semidirect
63. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
75. #88052	C₂×C₄	semidirect
76. C₂×#6090	C₂×C₄	semidirect
82. #88038	C₂×C₄	semidirect
83. #88038	C₂×C₄	semidirect
84. C₁₆×D₁₃	C₂×C₄	semidirect
85. C₁₆×D₁₃	C₂×C₄	semidirect
133. #2108	D₁₃	semidirect
204. Dic₁₃	#504	semidirect
227. M₃₂	C₄×D₁₃	semidirect
234. C₂×C₁₆	C₄×D₁₃	semidirect
292. C₁₃	C₄×#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718568	2
C₂×#11718562	2
#13123258	2
#13116434	2
#13116433	2
#13116431	2
#13116429	2
M₃₂×Dic₁₃	2
C₂×C₁₆×Dic₁₃	2
C₄×C₁₆×D₁₃	2
C₁₃×#2108	2
C₄×#504	13