Subgroups — #135×#1055 #15832298

Order 3328.

750 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#1055	C₂	1	2	yes	maximal, complemented
2	Q₁₆×#1055	C₂	1	2	yes	maximal, complemented
3	SD₁₆×#1055	C₂	1	4	yes	maximal, complemented
4	D₈×#1055	C₂	1	2	yes	maximal, complemented
5	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113218	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#1055	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	Q₈×#1055	C₂²	1	4	yes	—
21	D₄×#1055	—	2	4	no	—
22	D₄×#1055	C₂²	1	4	yes	—
23	#11718586	—	2	2	no	—
24	#11718586	C₂²	1	2	yes	—
25	C₂×C₄×#1055	—	2	2	no	—
26	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
27	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
28	Q₈⋊#1055	C₂²	1	4	yes	—
29	D₄⋊#1055	C₂²	1	4	yes	—
30	#11718050	C₂²	1	2	yes	—
31	#11718049	C₂²	1	2	yes	—
32	C₈×#1055	C₂²	1	2	yes	—
33	C₈×#1055	C₂²	1	2	yes	—
34	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
35	C₁₃×#135	C₈	1	8	yes	complemented
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	C₂²×#1055	—	2	1	no	—
52	C₂²×#1055	—	2	4	no	—
53	C₂²×#1055	D₄	1	2	yes	—
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	2	no	—
66	C₂²⋊#1055	—	2	4	no	—
67	C₄⋊#1055	—	2	4	no	—
68	C₄⋊#1055	C₂³	1	4	yes	—
69	C₄×#1055	—	2	2	no	—
70	C₄×#1055	—	2	1	no	—
71	C₄×#1055	—	2	4	no	—
72	C₄×#1055	D₄	1	1	yes	characteristic
73	C₄×#1055	D₄	1	1	yes	characteristic
74	C₄×#1055	C₂³	1	2	yes	—
75	C₄×#1055	C₂³	1	2	yes	—
76	C₈×#135	—	13	1	no	maximal, Sylow
77	C₁₃×#41	—	2	4	no	—
78	C₁₃×#41	C₂×C₈	1	8	yes	complemented
79	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×D₄×C₁₃	—	2	2	no	—
81	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	—	2	2	no	—
85	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
88	C₁₃×Q₁₆	C₂×C₈	1	4	yes	complemented
89	C₁₃×SD₁₆	C₂×C₈	1	8	yes	complemented
90	C₁₃×D₈	C₂×C₈	1	4	yes	complemented
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	complemented
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	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
100	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
101	C₁₃×C₂²⋊C₄	—	2	2	no	—
102	C₁₃×C₂²⋊C₄	—	2	2	no	—
103	C₁₃×C₂²⋊C₄	—	2	2	no	—
104	C₄²×C₁₃	—	2	2	no	—
105	C₄²×C₁₃	—	2	2	no	—
106	C₄²×C₁₃	—	2	2	no	—
107	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	#41	1	1	yes	characteristic
110	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
111	C₂×#1055	—	2	4	no	—
112	C₂×#1055	—	2	2	no	—
113	C₂×#1055	—	2	4	no	—
114	C₂×#1055	—	2	4	no	—
115	C₂×#1055	—	4	8	no	—
116	C₂×#1055	C₂×D₄	1	2	yes	—
117	C₂×#1055	C₂×D₄	1	2	yes	—
118	C₈×#41	—	13	2	no	—
119	C₄×#135	D₁₃	1	1	yes	characteristic
120	C₈×Q₁₆	—	13	2	no	—
121	C₈×SD₁₆	—	13	4	no	—
122	C₈×D₈	—	13	2	no	—
123	#1564	—	13	1	no	—
124	#1476	—	13	2	no	—
125	C₂×C₈²	—	13	1	no	—
126	C₂³×C₁₃	—	2	2	no	—
127	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
128	Q₈×C₁₃	—	2	2	no	—
129	Q₈×C₁₃	C₂²×C₈	1	4	yes	—
130	D₄×C₁₃	—	2	8	no	—
131	D₄×C₁₃	—	2	2	no	—
132	D₄×C₁₃	C₂²×C₈	1	4	yes	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	8	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	4	4	no	—
140	C₂×C₄×C₁₃	—	4	4	no	—
141	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
145	C₂×C₄×C₁₃	#131	1	2	yes	—
146	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	C₂²×C₈	1	2	yes	—
152	C₈×C₁₃	C₂²×C₈	1	2	yes	—
153	#1055	—	2	4	no	—
154	#1055	—	2	2	no	—
155	#1055	—	4	4	no	—
156	#1055	#135	1	4	yes	direct product factor
157	C₂×#135	Dic₁₃	1	1	yes	characteristic
158	C₄×#41	C₂×D₁₃	1	2	yes	—
159	C₈×Q₈	—	13	4	no	—
160	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
161	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
162	C₄×SD₁₆	Dic₁₃	1	2	yes	—
163	C₄×SD₁₆	C₂×D₁₃	1	2	yes	—
164	C₄×D₈	Dic₁₃	1	1	yes	characteristic
165	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
166	C₈×D₄	—	13	4	no	—
167	C₈×D₄	—	26	4	no	—
168	#431	—	13	2	no	—
169	#431	—	26	2	no	—
170	#427	Dic₁₃	1	1	yes	characteristic
171	#416	Dic₁₃	1	2	yes	—
172	C₂×C₄×C₈	—	13	1	no	—
173	C₂×C₄×C₈	—	13	1	no	—
174	C₂×C₄×C₈	—	26	2	no	—
175	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
176	#335	—	13	2	no	—
177	#334	—	13	2	no	—
178	Q₈⋊C₈	—	13	4	no	—
179	D₄⋊C₈	—	13	4	no	—
180	C₈²	—	13	2	no	—
181	C₈²	—	13	2	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	—	2	4	no	—
185	C₂²×C₁₃	—	4	2	no	—
186	C₂²×C₁₃	—	4	2	no	—
187	C₂²×C₁₃	C₈×D₄	1	2	yes	—
188	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
189	C₄×C₁₃	—	2	4	no	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	2	1	no	—
192	C₄×C₁₃	—	4	2	no	—
193	C₄×C₁₃	—	4	2	no	—
194	C₄×C₁₃	C₈×D₄	1	2	yes	—
195	C₄×C₁₃	#443	1	2	yes	—
196	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
197	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
198	C₄×C₁₃	C₂×#135	1	2	yes	—
199	C₂×#41	C₂×Dic₁₃	1	2	yes	—
200	#135	#1055	1	8	yes	direct product factor
201	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
202	C₂×SD₁₆	C₂×Dic₁₃	1	2	yes	—
203	C₂×D₈	C₂×Dic₁₃	1	1	yes	characteristic
204	C₂²×C₈	—	13	2	no	—
205	C₂²×C₈	—	26	4	no	—
206	C₂²×C₈	—	26	1	no	—
207	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
208	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
209	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
210	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
214	C₄×D₄	C₂²×D₁₃	1	2	yes	—
215	#117	—	2	2	no	—
216	#117	C₂×Dic₁₃	1	2	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
219	#107	C₂×Dic₁₃	1	1	yes	characteristic
220	#107	C₂×Dic₁₃	1	1	yes	characteristic
221	#106	C₂×Dic₁₃	1	2	yes	—
222	C₄⋊C₈	—	13	4	no	—
223	C₄⋊C₈	—	26	4	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	—	26	4	no	—
229	C₂²⋊C₈	—	26	2	no	—
230	C₄×C₈	—	13	1	no	—
231	C₄×C₈	—	13	1	no	—
232	C₄×C₈	—	13	2	no	—
233	C₄×C₈	—	13	2	no	—
234	C₄×C₈	—	26	4	no	—
235	C₄×C₈	—	26	2	no	—
236	C₄×C₈	—	26	1	no	—
237	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
238	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
241	C₂×C₁₃	—	2	2	no	—
242	C₂×C₁₃	—	4	4	no	—
243	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
244	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
245	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
246	#41	—	2	4	no	—
247	#41	C₂×#1055	1	8	yes	complemented
248	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	C₂²×Dic₁₃	1	2	yes	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	#6129	1	1	yes	characteristic
256	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
257	Q₁₆	C₂×#1055	1	4	yes	complemented
258	SD₁₆	C₂×#1055	1	8	yes	complemented
259	D₈	C₂×#1055	1	4	yes	complemented
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	13	2	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	—	26	4	no	—
264	C₂×C₈	—	26	4	no	—
265	C₂×C₈	—	26	2	no	—
266	C₂×C₈	—	26	4	no	—
267	C₂×C₈	—	52	8	no	—
268	C₂×C₈	C₂×#1055	1	4	yes	complemented
269	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
271	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
272	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
276	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	D₄×D₁₃	1	1	yes	characteristic
284	C₄²	#6129	1	1	yes	characteristic
285	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
286	C₄²	C₂³×D₁₃	1	1	yes	characteristic
287	C₁₃	C₈×#135	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	2	2	no	—
289	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
290	Q₈	—	2	2	no	—
291	Q₈	C₂²×#1055	1	4	yes	—
292	D₄	—	2	8	no	—
293	D₄	—	2	2	no	—
294	D₄	C₂²×#1055	1	4	yes	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	8	no	—
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	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	C₂²×#1055	1	2	yes	—
304	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
305	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
306	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
307	C₂×C₄	#88202	1	2	yes	—
308	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂×#6129	1	1	yes	characteristic
310	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	13	4	no	—
314	C₈	—	26	4	no	—
315	C₈	—	26	2	no	—
316	C₈	—	52	4	no	—
317	C₈	C₂²×#1055	1	2	yes	—
318	C₈	C₂²×#1055	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	D₄×#1055	1	2	yes	—
325	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	4	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	2	no	—
331	C₄	D₄×#1055	1	2	yes	—
332	C₄	#11718857	1	2	yes	—
333	C₄	D₁₃×#135	1	2	yes	—
334	C₄	C₂³×#1055	1	1	yes	characteristic
335	C₄	C₂×#88202	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	4	4	no	—
338	C₂	C₂×D₄×#1055	1	1	yes	characteristic
339	C₂	#135×Dic₁₃	1	1	yes	characteristic
340	C₂	C₂×#11718857	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₈
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	C₈×#135
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #41×#1055	C₂	semidirect
2. Q₁₆×#1055	C₂	semidirect
3. SD₁₆×#1055	C₂	semidirect
4. D₈×#1055	C₂	semidirect
5. #13113584	C₂	semidirect
6. #13113218	C₂	semidirect
7. C₂×C₈×#1055	C₂	semidirect
35. C₁₃×#135	C₈	semidirect
78. C₁₃×#41	C₂×C₈	semidirect
88. C₁₃×Q₁₆	C₂×C₈	semidirect
89. C₁₃×SD₁₆	C₂×C₈	semidirect
90. C₁₃×D₈	C₂×C₈	semidirect
92. C₂×C₈×C₁₃	C₂×C₈	semidirect
156. #1055	#135	direct (N × H)
200. #135	#1055	direct (N × H)
247. #41	C₂×#1055	semidirect
257. Q₁₆	C₂×#1055	semidirect
258. SD₁₆	C₂×#1055	semidirect
259. D₈	C₂×#1055	semidirect
268. C₂×C₈	C₂×#1055	semidirect
287. C₁₃	C₈×#135	semidirect

Maximal subgroups

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

subgroup	index
#41×#1055	2
Q₁₆×#1055	2
SD₁₆×#1055	2
D₈×#1055	2
#13113584	2
#13113218	2
C₂×C₈×#1055	2
C₄×C₁₃×#135	2
C₈×#135	13