Subgroups — #15834670

Order 3328.

652 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123961	C₂	1	2	yes	maximal, complemented
2	#13113745	C₂	1	2	yes	maximal, complemented
3	#13113673	C₂	1	4	yes	maximal, complemented
4	#13113660	C₂	1	2	yes	maximal, complemented
5	#13113592	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113275	C₂	1	2	yes	maximal, complemented
7	#13113125	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#135	C₄	1	1	yes	characteristic
10	C₄×C₁₃×#41	C₂²	1	2	yes	—
11	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
12	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×SD₁₆	C₄	1	2	yes	—
14	C₄×C₁₃×SD₁₆	C₂²	1	2	yes	—
15	C₄×C₁₃×D₈	C₄	1	1	yes	characteristic
16	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₄	1	1	yes	characteristic
18	C₁₃×#416	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	#11718685	C₂²	1	4	yes	—
21	#11718672	—	2	4	no	—
22	D₄×#1055	C₂²	1	4	yes	—
23	#11718587	—	2	2	no	—
24	#11718586	C₂²	1	1	yes	characteristic
25	#11718585	C₂²	1	2	yes	—
26	C₂×C₄⋊#1055	—	2	2	no	—
27	C₄×#6099	C₂²	1	1	yes	characteristic
28	Q₈⋊#1055	C₂²	1	4	yes	—
29	D₄⋊#1055	C₂²	1	4	yes	—
30	#11718049	C₂²	1	2	yes	—
31	#11718049	C₂²	1	2	yes	—
32	#11718047	C₂²	1	2	yes	—
33	#11718046	C₂²	1	2	yes	—
34	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
35	C₁₃×#135	—	2	4	no	—
36	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
42	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
43	C₄×D₄×C₁₃	—	2	2	no	—
44	C₄×D₄×C₁₃	—	2	2	no	—
45	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₁₃	C₂³	1	2	yes	—
47	C₁₃×#117	—	2	2	no	—
48	C₁₃×#117	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₃	—	2	2	no	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×#6099	—	2	1	no	—
52	C₂×#6099	—	2	2	no	—
53	C₂²×#1055	—	2	4	no	—
54	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
55	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
56	C₁₃×#106	C₂×C₄	1	2	yes	—
57	C₁₃×#103	C₂×C₄	1	2	yes	—
58	C₁₃×#103	C₂×C₄	1	2	yes	—
59	C₁₃×#102	C₂×C₄	1	2	yes	—
60	C₁₃×#102	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₂²⋊#1055	—	2	4	no	—
66	C₂²⋊#1055	D₄	1	2	yes	—
67	C₄⋊#1055	—	2	2	no	—
68	C₄⋊#1055	—	2	4	no	—
69	C₄⋊#1055	D₄	1	2	yes	—
70	C₄⋊#1055	C₂³	1	4	yes	—
71	#88043	—	2	1	no	—
72	#88043	—	2	4	no	—
73	C₄×#1055	C₂³	1	2	yes	—
74	C₄×#1055	C₂³	1	2	yes	—
75	#18065	—	13	1	no	maximal, Sylow
76	C₁₃×#41	—	2	4	no	—
77	C₁₃×#41	—	4	2	no	—
78	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
87	C₁₃×Q₁₆	—	2	2	no	—
88	C₁₃×SD₁₆	—	2	4	no	—
89	C₁₃×D₈	M₁₆	1	4	yes	complemented
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	4	1	no	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
99	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
100	C₁₃×C₂²⋊C₄	—	2	2	no	—
101	C₁₃×C₂²⋊C₄	—	2	2	no	—
102	C₁₃×C₂²⋊C₄	—	2	2	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	2	2	no	—
105	C₄²×C₁₃	—	2	2	no	—
106	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	#41	1	1	yes	characteristic
109	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
110	#6099	—	4	2	no	—
111	#6099	—	8	2	no	—
112	C₂×#1055	—	2	4	no	—
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	—	8	4	no	—
115	C₂×#1055	C₂×D₄	1	4	yes	—
116	#2990	—	13	2	no	—
117	C₄×#135	D₁₃	1	1	yes	characteristic
118	#1593	—	13	2	no	—
119	#1582	—	13	4	no	—
120	#1580	—	13	2	no	—
121	#1566	—	13	1	no	—
122	#1489	—	13	2	no	—
123	#1451	—	13	1	no	—
124	C₂³×C₁₃	—	2	2	no	—
125	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
126	Q₈×C₁₃	—	2	2	no	—
127	Q₈×C₁₃	—	2	2	no	—
128	D₄×C₁₃	—	2	2	no	—
129	D₄×C₁₃	—	4	4	no	—
130	D₄×C₁₃	C₂×M₁₆	1	4	yes	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	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	—	2	2	no	—
138	C₂×C₄×C₁₃	—	4	4	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
152	#1055	—	2	4	no	—
153	#1055	—	4	2	no	—
154	#1055	—	4	4	no	—
155	C₂×#135	Dic₁₃	1	1	yes	characteristic
156	C₄×#41	C₂×D₁₃	1	2	yes	—
157	#446	—	13	4	no	—
158	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
159	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
160	C₄×SD₁₆	Dic₁₃	1	2	yes	—
161	C₄×SD₁₆	C₂×D₁₃	1	2	yes	—
162	C₄×D₈	Dic₁₃	1	1	yes	characteristic
163	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
164	#435	—	26	4	no	—
165	C₈×D₄	—	13	4	no	—
166	#432	—	26	2	no	—
167	#431	—	13	1	no	—
168	#427	Dic₁₃	1	1	yes	characteristic
169	#423	—	13	2	no	—
170	C₂×C₄⋊C₈	—	26	2	no	—
171	#416	Dic₁₃	1	2	yes	—
172	C₄×M₁₆	—	13	1	no	—
173	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
174	#334	—	13	2	no	—
175	#334	—	13	2	no	—
176	Q₈⋊C₈	—	13	4	no	—
177	D₄⋊C₈	—	13	4	no	—
178	#322	—	13	2	no	—
179	#322	—	13	2	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	2	4	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	—	4	2	no	—
185	C₂²×C₁₃	—	4	2	no	—
186	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	4	2	no	—
191	C₄×C₁₃	—	4	2	no	—
192	C₄×C₁₃	—	4	2	no	—
193	C₄×C₁₃	#444	1	2	yes	—
194	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
195	C₄×C₁₃	#568	1	1	yes	characteristic
196	C₄×C₁₃	#577	1	2	yes	—
197	C₂×#41	C₂×Dic₁₃	1	2	yes	—
198	#135	—	2	4	no	—
199	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
200	C₂×SD₁₆	C₂×Dic₁₃	1	2	yes	—
201	C₂×D₈	C₂×Dic₁₃	1	1	yes	characteristic
202	C₂×M₁₆	—	26	1	no	—
203	C₂×M₁₆	—	26	2	no	—
204	C₂²×C₈	—	26	4	no	—
205	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
206	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
207	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
208	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
209	C₄×D₄	—	2	2	no	—
210	C₄×D₄	—	2	2	no	—
211	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
212	C₄×D₄	C₂²×D₁₃	1	2	yes	—
213	#117	—	2	2	no	—
214	#117	C₂×Dic₁₃	1	2	yes	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
217	#107	C₂×Dic₁₃	1	1	yes	characteristic
218	#107	C₂×Dic₁₃	1	1	yes	characteristic
219	#106	C₂×Dic₁₃	1	2	yes	—
220	C₄⋊C₈	—	13	2	no	—
221	C₄⋊C₈	—	13	4	no	—
222	C₄⋊C₈	—	26	4	no	—
223	C₄⋊C₈	—	26	2	no	—
224	#103	C₂×Dic₁₃	1	2	yes	—
225	#103	C₂×Dic₁₃	1	2	yes	—
226	#102	C₂×Dic₁₃	1	2	yes	—
227	#102	C₂×Dic₁₃	1	2	yes	—
228	C₂²⋊C₈	—	13	2	no	—
229	C₂²⋊C₈	—	26	4	no	—
230	#97	—	26	4	no	—
231	#97	—	26	1	no	—
232	C₄×C₈	—	13	2	no	—
233	C₄×C₈	—	13	2	no	—
234	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
235	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
238	C₂×C₁₃	—	4	1	no	—
239	C₂×C₁₃	—	4	4	no	—
240	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
241	C₂×C₁₃	#2952	1	1	yes	characteristic
242	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
243	#41	—	2	4	no	—
244	#41	—	4	2	no	—
245	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	C₂²×Dic₁₃	1	2	yes	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
252	C₂²×C₄	#6129	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
254	Q₁₆	—	2	2	no	—
255	SD₁₆	—	2	4	no	—
256	D₈	#6099	1	4	yes	complemented
257	M₁₆	—	52	2	no	—
258	M₁₆	—	104	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	13	4	no	—
262	C₂×C₈	—	26	4	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	—	104	4	no	—
265	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
266	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
267	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
268	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
272	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	D₄×D₁₃	1	1	yes	characteristic
280	C₄²	#6129	1	1	yes	characteristic
281	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
282	C₄²	C₂³×D₁₃	1	1	yes	characteristic
283	C₁₃	#18065	1	1	yes	characteristic, complemented, Sylow
284	C₂³	—	2	2	no	—
285	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	4	no	—
290	D₄	C₂×#6099	1	4	yes	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
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	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
303	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
304	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
306	C₂×C₄	C₂×#6129	1	1	yes	characteristic
307	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	26	4	no	—
312	C₈	—	52	2	no	—
313	C₈	—	52	4	no	—
314	C₈	C₂×#6099	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	#11718858	1	2	yes	—
329	C₄	#11719475	1	2	yes	—
330	C₄	C₂²×#6099	1	1	yes	characteristic
331	C₄	#11719524	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	4	4	no	—
334	C₂	D₄×#6099	1	1	yes	characteristic
335	C₂	#13123720	1	1	yes	characteristic
336	C₂	C₂×#11718858	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₄×C₁₃×#135	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×C₁₃×#135	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₄×#135	D₁₃
hypocenter	C₁₃	#18065
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #13123961	C₂	semidirect
2. #13113745	C₂	semidirect
3. #13113673	C₂	semidirect
4. #13113660	C₂	semidirect
5. #13113592	C₂	semidirect
6. #13113275	C₂	semidirect
7. #13113125	C₂	semidirect
89. C₁₃×D₈	M₁₆	semidirect
256. D₈	#6099	semidirect
283. C₁₃	#18065	semidirect

Maximal subgroups

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

subgroup	index
#13123961	2
#13113745	2
#13113673	2
#13113660	2
#13113592	2
#13113275	2
#13113125	2
C₄×C₁₃×#135	2
#18065	13