Subgroups — #15832183

Order 3328.

588 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	#13123502	C₂	1	2	yes	maximal
2	#13113625	C₂	1	2	yes	maximal, complemented
3	#13113621	C₂	1	2	yes	maximal, complemented
4	#13113616	C₂	1	2	yes	maximal, complemented
5	#13113603	C₂	1	2	yes	maximal, complemented
6	#13113599	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113588	C₂	1	2	yes	maximal, complemented
8	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2965	C₂	1	1	yes	maximal, characteristic
10	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
11	C₁₃×#446	C₄	1	1	yes	characteristic
12	C₁₃×#446	C₄	1	1	yes	characteristic
13	C₁₃×#436	C₄	1	1	yes	characteristic
14	C₁₃×#436	C₄	1	1	yes	characteristic
15	C₁₃×#435	C₄	1	2	yes	—
16	C₁₃×#435	C₄	1	2	yes	—
17	C₁₃×#431	C₂²	1	1	yes	characteristic
18	C₁₃×#431	C₂²	1	2	yes	—
19	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	#11718895	—	2	2	no	—
22	#11718661	—	2	2	no	—
23	#11718661	—	2	2	no	—
24	#11718658	—	2	2	no	—
25	#11718587	C₂²	1	2	yes	—
26	#11718587	C₂²	1	2	yes	—
27	#11718586	C₂²	1	2	yes	—
28	#11718586	C₂²	1	2	yes	—
29	#11718585	C₂²	1	2	yes	—
30	C₂×C₄⋊#1055	C₂²	1	2	yes	—
31	#11718050	C₂²	1	2	yes	—
32	#11718050	C₂²	1	2	yes	—
33	#11718050	C₂²	1	2	yes	—
34	#11718050	C₂²	1	2	yes	—
35	#11718049	C₂²	1	1	yes	characteristic
36	#11718049	C₂²	1	2	yes	—
37	#11718049	C₂²	1	2	yes	—
38	#11718049	C₂²	1	1	yes	characteristic
39	#11718049	C₂²	1	1	yes	characteristic
40	#11718049	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
42	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×Q₈×C₁₃	Q₈	1	1	yes	characteristic
48	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄×D₄×C₁₃	Q₈	1	1	yes	characteristic
51	C₄×D₄×C₁₃	Q₈	1	2	yes	—
52	C₁₃×#117	D₄	1	2	yes	—
53	C₁₃×#117	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
56	C₂×#6099	—	2	2	no	—
57	C₂²×#1055	—	2	2	no	—
58	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
59	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
63	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
64	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
65	C₁₃×#97	C₂×C₄	1	2	yes	—
66	C₁₃×#97	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	2	yes	—
72	C₂²⋊#1055	—	2	2	no	—
73	C₂²⋊#1055	—	2	2	no	—
74	C₂²⋊#1055	—	2	2	no	—
75	C₂²⋊#1055	—	2	2	no	—
76	C₄⋊#1055	—	2	2	no	—
77	C₄⋊#1055	—	2	2	no	—
78	C₄⋊#1055	C₂³	1	2	yes	—
79	C₄⋊#1055	C₂³	1	2	yes	—
80	C₄⋊#1055	C₂³	1	2	yes	—
81	C₄⋊#1055	C₂³	1	2	yes	—
82	#88043	—	2	2	no	—
83	#88043	—	2	2	no	—
84	C₄×#1055	—	2	2	no	—
85	C₄×#1055	—	2	2	no	—
86	#17831	—	13	1	no	maximal, Sylow
87	C₁₃×#41	—	4	2	no	—
88	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂×D₄×C₁₃	C₄⋊C₄	1	2	yes	—
90	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
97	C₁₃×M₁₆	—	2	4	no	—
98	C₁₃×M₁₆	—	4	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	—	4	1	no	—
101	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
108	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
109	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
110	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
111	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
112	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
113	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
114	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃	C₂×D₄	1	2	yes	—
118	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
119	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
120	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
121	#6099	—	4	4	no	—
122	C₂×#1055	—	2	2	no	—
123	C₂×#1055	—	2	2	no	—
124	C₂×#1055	—	2	2	no	—
125	C₂×#1055	—	2	2	no	—
126	C₂×#1055	—	4	4	no	—
127	#2965	D₁₃	1	1	yes	characteristic
128	#2925	—	13	2	no	—
129	#1574	—	13	2	no	—
130	#1573	—	13	2	no	—
131	#1572	—	13	2	no	—
132	#1569	—	13	2	no	—
133	#1568	—	13	1	no	—
134	#1565	—	13	2	no	—
135	#1564	—	13	1	no	—
136	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
138	Q₈×C₁₃	—	4	1	no	—
139	D₄×C₁₃	—	4	2	no	—
140	D₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	1	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	4	1	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
159	C₈×C₁₃	—	2	1	no	—
160	C₈×C₁₃	—	2	2	no	—
161	C₈×C₁₃	—	2	1	no	—
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	—	2	2	no	—
164	C₈×C₁₃	#131	1	2	yes	—
165	#1055	—	4	2	no	—
166	#1055	—	4	2	no	—
167	#1055	—	4	2	no	—
168	#1055	—	4	2	no	—
169	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
170	#446	Dic₁₃	1	1	yes	characteristic
171	#446	Dic₁₃	1	1	yes	characteristic
172	#436	Dic₁₃	1	1	yes	characteristic
173	#436	Dic₁₃	1	1	yes	characteristic
174	#435	Dic₁₃	1	2	yes	—
175	#435	Dic₁₃	1	2	yes	—
176	#433	—	26	2	no	—
177	#433	—	26	2	no	—
178	#432	—	13	2	no	—
179	#432	—	13	2	no	—
180	#431	—	13	2	no	—
181	#431	—	13	2	no	—
182	#431	C₂×D₁₃	1	1	yes	characteristic
183	#431	C₂×D₁₃	1	2	yes	—
184	#423	—	13	2	no	—
185	C₂×C₄⋊C₈	—	13	2	no	—
186	#408	—	26	2	no	—
187	#405	—	26	2	no	—
188	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
189	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
190	#335	—	13	2	no	—
191	#335	—	13	2	no	—
192	#335	—	13	2	no	—
193	#335	—	13	2	no	—
194	#334	—	13	1	no	—
195	#334	—	13	1	no	—
196	#334	—	13	2	no	—
197	#334	—	13	1	no	—
198	#334	—	13	1	no	—
199	#334	—	13	2	no	—
200	C₂²×C₁₃	—	2	1	no	—
201	C₂²×C₁₃	—	2	2	no	—
202	C₂²×C₁₃	—	2	2	no	—
203	C₂²×C₁₃	—	2	1	no	—
204	C₂²×C₁₃	—	2	1	no	—
205	C₂²×C₁₃	—	2	2	no	—
206	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
207	C₄×C₁₃	—	2	1	no	—
208	C₄×C₁₃	—	2	2	no	—
209	C₄×C₁₃	—	2	2	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	—	2	1	no	—
213	C₄×C₁₃	—	2	2	no	—
214	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
215	C₄×C₁₃	#568	1	1	yes	characteristic
216	C₄×C₁₃	#575	1	2	yes	—
217	C₂×#41	D₅₂	1	1	yes	characteristic
218	C₂×M₁₆	—	26	2	no	—
219	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
220	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
221	C₂²×C₈	—	26	2	no	—
222	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
223	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
224	C₄×Q₈	Dic₂₆	1	1	yes	characteristic
225	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
226	C₄×D₄	Dic₂₆	1	1	yes	characteristic
227	C₄×D₄	Dic₂₆	1	2	yes	—
228	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
229	C₄×D₄	C₂×Dic₁₃	1	1	yes	characteristic
230	#117	D₅₂	1	2	yes	—
231	#117	D₅₂	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
234	C₄⋊C₈	—	13	2	no	—
235	C₄⋊C₈	—	13	2	no	—
236	C₄⋊C₈	—	13	2	no	—
237	C₄⋊C₈	—	13	2	no	—
238	C₄⋊C₈	—	26	2	no	—
239	C₄⋊C₈	—	26	2	no	—
240	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
241	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
242	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
243	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
244	C₂²⋊C₈	—	26	2	no	—
245	C₂²⋊C₈	—	26	2	no	—
246	C₂²⋊C₈	—	26	2	no	—
247	C₂²⋊C₈	—	26	2	no	—
248	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
249	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
250	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
251	#97	—	26	2	no	—
252	#97	—	26	2	no	—
253	#97	C₂×Dic₁₃	1	2	yes	—
254	#97	C₂×Dic₁₃	1	2	yes	—
255	C₄×C₈	—	26	2	no	—
256	C₄×C₈	—	26	2	no	—
257	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
258	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₁₃	1	2	yes	—
262	C₂×C₁₃	—	4	1	no	—
263	C₂×C₁₃	—	4	2	no	—
264	C₂×C₁₃	#2906	1	1	yes	characteristic
265	C₂×C₁₃	#2911	1	1	yes	characteristic
266	C₂×C₁₃	#2916	1	1	yes	characteristic
267	#41	—	4	2	no	—
268	C₂×Q₈	C₄⋊Dic₁₃	1	1	yes	characteristic
269	C₂×D₄	C₄⋊Dic₁₃	1	2	yes	—
270	C₂×D₄	C₄⋊Dic₁₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
272	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
273	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₅₂	1	2	yes	—
275	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
276	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	4	2	no	—
279	M₁₆	—	52	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	26	2	no	—
283	C₂×C₈	—	26	2	no	—
284	C₂×C₈	—	26	2	no	—
285	C₂×C₈	—	26	2	no	—
286	C₂×C₈	—	52	4	no	—
287	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
288	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
289	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
290	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
291	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
292	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
293	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
294	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
295	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
296	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
297	C₂²⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
298	C₂²⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
299	C₂²⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
300	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
301	C₄²	C₂×Dic₂₆	1	2	yes	—
302	C₄²	C₂×D₅₂	1	1	yes	characteristic
303	C₄²	C₂×D₅₂	1	2	yes	—
304	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
305	C₄²	C₂²×Dic₁₃	1	2	yes	—
306	C₄²	C₂³×D₁₃	1	1	yes	characteristic
307	C₁₃	#17831	1	1	yes	characteristic, complemented, Sylow
308	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
309	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
310	Q₈	—	4	1	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
323	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
324	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
326	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
327	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
328	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
329	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
330	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	2	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	2	no	—
336	C₈	—	52	2	no	—
337	C₈	—	52	2	no	—
338	C₈	—	52	2	no	—
339	C₈	—	52	2	no	—
340	C₈	#88202	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	2	no	—
347	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	#11719453	1	2	yes	—
356	C₄	C₂×#88202	1	1	yes	characteristic
357	C₄	#11719524	1	1	yes	characteristic
358	C₂	—	4	1	no	—
359	C₂	—	4	2	no	—
360	C₂	#13123404	1	1	yes	characteristic
361	C₂	#13123426	1	1	yes	characteristic
362	C₂	#13123447	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#2965	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×C₁₃×#41	D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#2965	D₁₃
hypocenter	C₁₃	#17831
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2965

Complements

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

N	H ≅ G/N	type
2. #13113625	C₂	semidirect
3. #13113621	C₂	semidirect
4. #13113616	C₂	semidirect
5. #13113603	C₂	semidirect
6. #13113599	C₂	semidirect
7. #13113588	C₂	semidirect
8. #13113584	C₂	semidirect
307. C₁₃	#17831	semidirect

Maximal subgroups

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

subgroup	index
#13123502	2
#13113625	2
#13113621	2
#13113616	2
#13113603	2
#13113599	2
#13113588	2
#13113584	2
C₁₃×#2965	2
#17831	13