Subgroups — #15832010

Order 3328.

668 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123469	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718586	C₂	1	1	yes	maximal, characteristic
3	#13113624	C₂	1	4	yes	maximal, complemented
4	#13113619	C₂	1	4	yes	maximal, complemented
5	#13113582	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#431	—	2	1	no	—
9	C₁₃×#431	—	2	1	no	—
10	C₁₃×#431	C₂²	1	2	yes	—
11	C₁₃×#431	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	—	2	1	no	—
15	#11718868	—	2	1	no	—
16	C₂×C₂²⋊#1055	C₄	1	2	yes	—
17	#11718587	—	2	1	no	—
18	#11718587	C₂²	1	2	yes	—
19	#11718586	—	2	1	no	—
20	#11718586	C₄	1	2	yes	—
21	#11718586	C₄	1	2	yes	—
22	#11718586	C₂²	1	2	yes	—
23	#11718586	C₂²	1	2	yes	—
24	#11718586	C₂²	1	2	yes	—
25	#11718585	C₂²	1	2	yes	—
26	C₂×C₄⋊#1055	C₄	1	2	yes	—
27	C₄×#6099	—	2	1	no	—
28	C₂×C₄×#1055	C₂²	1	2	yes	—
29	#11718072	C₂²	1	8	yes	complemented
30	#11718071	C₂²	1	8	yes	complemented
31	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	—	2	2	no	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×#6099	—	2	2	no	—
48	C₂²×#1055	C₂×C₄	1	4	yes	—
49	C₁₃×C₄⋊C₈	—	2	2	no	—
50	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
51	C₁₃×C₂²⋊C₈	—	2	2	no	—
52	C₁₃×C₂²⋊C₈	—	2	2	no	—
53	C₄×C₈×C₁₃	—	2	2	no	—
54	C₄×C₈×C₁₃	—	2	2	no	—
55	C₂²⋊#1055	—	2	2	no	—
56	C₂²⋊#1055	—	2	2	no	—
57	C₂²⋊#1055	C₂×C₄	1	4	yes	—
58	C₂²⋊#1055	C₂×C₄	1	4	yes	—
59	C₄⋊#1055	—	2	2	no	—
60	C₄⋊#1055	C₂×C₄	1	4	yes	—
61	C₄⋊#1055	C₂×C₄	1	4	yes	—
62	C₄⋊#1055	C₂³	1	4	yes	—
63	#88043	—	2	2	no	—
64	C₄×#1055	—	2	2	no	—
65	C₄×#1055	C₂×C₄	1	4	yes	—
66	C₄×#1055	C₂³	1	4	yes	—
67	#17806	—	13	1	no	maximal, Sylow
68	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₃	—	4	2	no	—
72	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₁₃	—	2	4	no	—
91	C₂×C₈×C₁₃	—	4	4	no	—
92	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
106	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
107	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
108	#6099	—	4	4	no	—
109	C₂×#1055	—	2	8	no	—
110	C₂×#1055	—	2	4	no	—
111	C₂×#1055	C₂²×C₄	1	8	yes	—
112	#2920	—	13	1	no	—
113	C₂×#431	—	13	1	no	—
114	C₂×#431	D₁₃	1	1	yes	characteristic
115	#1574	—	13	4	no	—
116	#1573	—	13	4	no	—
117	#1564	—	13	4	no	—
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	—	2	2	no	—
120	C₂³×C₁₃	—	4	1	no	—
121	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	4	1	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	#108	1	4	yes	—
141	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	#131	1	4	yes	—
151	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
154	C₈×C₁₃	—	4	4	no	—
155	#1055	—	2	8	no	—
156	#1055	—	4	4	no	—
157	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
158	#432	—	13	2	no	—
159	#432	—	26	1	no	—
160	#431	—	2	1	no	—
161	#431	—	2	1	no	—
162	#431	—	13	2	no	—
163	#431	—	13	2	no	—
164	#431	—	13	2	no	—
165	#431	—	13	2	no	—
166	#431	—	13	2	no	—
167	#431	—	26	1	no	—
168	#431	C₂×D₁₃	1	2	yes	—
169	#431	C₂×D₁₃	1	2	yes	—
170	#423	—	13	2	no	—
171	C₂×C₄⋊C₈	—	13	2	no	—
172	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
173	#407	—	26	1	no	—
174	C₂×C₂²⋊C₈	—	2	1	no	—
175	C₂×C₂²⋊C₈	—	13	2	no	—
176	C₄×M₁₆	—	26	1	no	—
177	C₂×C₄×C₈	—	2	1	no	—
178	C₂×C₄×C₈	—	13	2	no	—
179	#335	—	13	8	no	—
180	#334	—	13	8	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	—	2	1	no	—
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	—	2	1	no	—
187	C₂²×C₁₃	—	2	1	no	—
188	C₂²×C₁₃	—	4	4	no	—
189	C₂²×C₁₃	#424	1	2	yes	—
190	C₂²×C₁₃	C₂×#108	1	2	yes	—
191	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	C₄×C₁₃	—	2	1	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	2	1	no	—
199	C₄×C₁₃	#424	1	2	yes	—
200	C₄×C₁₃	C₂×#108	1	2	yes	—
201	C₄×C₁₃	C₂×#108	1	2	yes	—
202	C₄×C₁₃	C₂×#131	1	2	yes	—
203	C₄×C₁₃	#575	1	2	yes	—
204	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
205	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
206	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
207	C₂×M₁₆	—	26	2	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	13	4	no	—
210	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
211	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
212	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
214	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
215	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
219	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	13	4	no	—
224	C₄⋊C₈	—	13	4	no	—
225	C₄⋊C₈	—	13	4	no	—
226	C₄⋊C₈	—	26	2	no	—
227	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
228	C₂²⋊C₈	—	2	2	no	—
229	C₂²⋊C₈	—	2	2	no	—
230	C₂²⋊C₈	—	13	4	no	—
231	C₂²⋊C₈	—	13	4	no	—
232	C₂²⋊C₈	—	26	2	no	—
233	C₂²⋊C₈	—	26	2	no	—
234	#97	—	26	2	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	—	13	4	no	—
238	C₄×C₈	—	13	4	no	—
239	C₄×C₈	—	26	2	no	—
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	—	2	2	no	—
242	C₂×C₁₃	—	4	1	no	—
243	C₂×C₁₃	C₂×#424	1	1	yes	characteristic
244	C₂×C₁₃	#2911	1	1	yes	characteristic
245	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
246	C₂⁴	#6101	1	1	yes	characteristic
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	#6101	1	1	yes	characteristic
251	C₂²×C₄	#6101	1	1	yes	characteristic
252	C₂²×C₄	#6101	1	1	yes	characteristic
253	C₂²×C₄	#6101	1	1	yes	characteristic
254	C₂²×C₄	#6101	1	1	yes	characteristic
255	C₂²×C₄	#6101	1	1	yes	characteristic
256	C₂²×C₄	#6101	1	1	yes	characteristic
257	C₂²×C₄	#6101	1	1	yes	characteristic
258	C₂²×C₄	#6101	1	1	yes	characteristic
259	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
260	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
261	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
268	M₁₆	—	52	4	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	4	4	no	—
271	C₂×C₈	—	13	8	no	—
272	C₂×C₈	—	26	8	no	—
273	C₂×C₈	—	26	4	no	—
274	C₄²	#6101	1	1	yes	characteristic
275	C₄²	#6101	1	1	yes	characteristic
276	C₄²	#6101	1	1	yes	characteristic
277	C₄²	#6101	1	1	yes	characteristic
278	C₄²	#6101	1	1	yes	characteristic
279	C₄²	#6101	1	1	yes	characteristic
280	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
281	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
282	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
283	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
284	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
289	C₄²	C₂³×D₁₃	1	1	yes	characteristic
290	C₁₃	#17806	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	—	4	1	no	—
294	C₂³	C₂×#6101	1	1	yes	characteristic
295	C₂³	C₂×#6101	1	1	yes	characteristic
296	C₂³	C₂×#6101	1	1	yes	characteristic
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	#88062	1	4	yes	—
314	C₂×C₄	#88155	1	4	yes	—
315	C₂×C₄	C₂×#6101	1	1	yes	characteristic
316	C₂×C₄	C₂×#6101	1	1	yes	characteristic
317	C₂×C₄	C₂×#6101	1	1	yes	characteristic
318	C₂×C₄	C₂×#6101	1	1	yes	characteristic
319	C₂×C₄	C₂×#6101	1	1	yes	characteristic
320	C₂×C₄	C₂×#6101	1	1	yes	characteristic
321	C₂×C₄	C₂×#6101	1	1	yes	characteristic
322	C₂×C₄	C₂×#6101	1	1	yes	characteristic
323	C₂×C₄	C₂×#6101	1	1	yes	characteristic
324	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
326	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
327	C₈	—	4	4	no	—
328	C₈	—	26	8	no	—
329	C₈	—	52	4	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	4	no	—
338	C₂²	#11718763	1	2	yes	—
339	C₂²	C₂×#88062	1	2	yes	—
340	C₂²	C₂²×#6101	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	#11718763	1	2	yes	—
349	C₄	C₂×#88062	1	2	yes	—
350	C₄	C₂×#88062	1	2	yes	—
351	C₄	C₂×#88155	1	2	yes	—
352	C₄	#11719527	1	2	yes	—
353	C₂	—	2	2	no	—
354	C₂	—	2	2	no	—
355	C₂	—	4	1	no	—
356	C₂	C₂×#11718763	1	1	yes	characteristic
357	C₂	#13123425	1	1	yes	characteristic
358	C₂	C₂²×#88062	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#431	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#431	D₁₃
hypocenter	C₁₃	#17806
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #13113624	C₂	semidirect
4. #13113619	C₂	semidirect
5. #13113582	C₂	semidirect
29. #11718072	C₂²	semidirect
30. #11718071	C₂²	semidirect
290. C₁₃	#17806	semidirect

Maximal subgroups

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

subgroup	index
#13123469	2
C₂×#11718586	2
#13113624	2
#13113619	2
#13113582	2
C₂×C₁₃×#431	2
#17806	13