Subgroups — #15755813

Order 3328.

550 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114861	C₂	1	2	yes	maximal, complemented
2	#13114824	C₂	1	2	yes	maximal
3	#13112704	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₁₃	—	2	2	no	—
9	C₈×D₄×C₁₃	—	2	2	no	—
10	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic, complemented
11	C₂×C₁₃×C₂²⋊C₈	C₄	1	1	yes	characteristic, complemented
12	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
14	C₂×#88074	—	2	2	no	—
15	#11718873	C₂²	1	4	yes	—
16	C₂×#88057	—	2	2	no	—
17	#11718143	—	2	2	no	—
18	#11718118	—	2	2	no	—
19	C₂²×D₄×C₁₃	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	—	2	1	no	—
23	C₂²×C₈×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	—	4	2	no	—
25	C₂²×C₈×C₁₃	—	4	2	no	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	D₄	1	2	yes	—
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₄×D₄×C₁₃	—	2	2	no	—
33	C₄×D₄×C₁₃	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	Q₈	1	1	yes	characteristic
35	C₂×C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
36	C₂×C₁₃×C₂²⋊C₄	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
39	C₂×C₄⋊Dic₁₃	—	2	4	no	—
40	C₁₃×C₄⋊C₈	—	2	2	no	—
41	C₁₃×C₂²⋊C₈	—	2	2	no	—
42	C₁₃×C₂²⋊C₈	—	2	2	no	—
43	C₄×C₈×C₁₃	—	2	1	no	—
44	C₄×C₈×C₁₃	—	2	1	no	—
45	#88074	—	4	4	no	—
46	#88071	—	2	4	no	—
47	#88057	—	4	4	no	—
48	#8608	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	—	2	2	no	—
53	C₂×D₄×C₁₃	SD₁₆	1	4	yes	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	—	4	1	no	—
57	C₂²×C₄×C₁₃	—	4	1	no	—
58	C₂²×C₄×C₁₃	—	4	1	no	—
59	C₂²×C₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×Dic₁₃	—	4	4	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	4	1	no	—
73	C₂×C₈×C₁₃	—	4	2	no	—
74	C₂×C₈×C₁₃	—	4	2	no	—
75	C₂×C₈×C₁₃	—	4	2	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₂×C₈×C₁₃	—	4	1	no	—
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₂²⋊C₄	—	2	2	no	—
83	C₁₃×C₂²⋊C₄	—	2	2	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₂²⋊Dic₁₃	—	4	4	no	—
87	C₂²⋊Dic₁₃	—	4	4	no	—
88	C₄⋊Dic₁₃	—	4	4	no	—
89	C₄⋊Dic₁₃	—	4	4	no	—
90	C₂×C₈×D₄	D₁₃	1	1	yes	characteristic
91	#1807	—	13	2	no	—
92	#1799	—	13	2	no	—
93	#1289	—	13	2	no	—
94	C₂³×C₁₃	—	2	1	no	—
95	C₂³×C₁₃	—	2	1	no	—
96	C₂³×C₁₃	—	4	1	no	—
97	C₂³×C₁₃	—	4	1	no	—
98	C₂³×C₁₃	—	4	1	no	—
99	C₂³×C₁₃	—	4	1	no	—
100	C₂³×C₁₃	#106	1	2	yes	complemented
101	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₃	#121	1	2	yes	—
103	C₂³×C₁₃	C₂×SD₁₆	1	2	yes	—
104	C₂³×C₁₃	#135	1	2	yes	—
105	D₄×C₁₃	—	2	8	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	—	4	1	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	#106	1	2	yes	complemented
122	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
125	C₂×Dic₁₃	—	8	4	no	—
126	C₂×Dic₁₃	—	8	4	no	—
127	C₂×Dic₁₃	—	8	4	no	—
128	C₈×C₁₃	—	2	1	no	—
129	C₈×C₁₃	—	2	2	no	—
130	C₈×C₁₃	—	4	1	no	—
131	C₈×C₁₃	—	4	1	no	—
132	C₈×C₁₃	#99	1	2	yes	complemented
133	C₂³×C₈	Dic₁₃	1	1	yes	characteristic, complemented
134	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
135	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
136	C₈×D₄	—	2	2	no	—
137	C₈×D₄	—	2	2	no	—
138	C₂×#106	—	26	2	no	—
139	C₂×C₄⋊C₈	Dic₁₃	1	1	yes	characteristic, complemented
140	C₂×#99	—	26	2	no	—
141	C₂×C₂²⋊C₈	Dic₁₃	1	1	yes	characteristic, complemented
142	C₂×C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
143	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic, complemented
144	#380	—	13	4	no	—
145	#342	—	26	2	no	—
146	#340	—	26	2	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	1	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	4	2	no	—
153	C₂²×C₁₃	—	4	2	no	—
154	C₂²×C₁₃	—	4	2	no	—
155	C₂²×C₁₃	—	4	2	no	—
156	C₂²×C₁₃	—	4	1	no	—
157	C₂²×C₁₃	—	4	1	no	—
158	C₂²×C₁₃	#380	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
160	C₂²×C₁₃	#427	1	1	yes	characteristic
161	C₂²×C₁₃	#465	1	4	yes	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	4	1	no	—
166	C₄×C₁₃	—	4	1	no	—
167	C₄×C₁₃	C₂×#99	1	1	yes	characteristic
168	C₄×C₁₃	#410	1	1	yes	characteristic
169	Dic₁₃	—	16	4	no	—
170	C₂²×D₄	D₅₂	1	1	yes	characteristic
171	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
172	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	4	2	no	—
176	C₂²×C₈	—	4	2	no	—
177	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
179	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
182	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
183	C₄×D₄	—	2	2	no	—
184	C₄×D₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	26	4	no	—
186	C₂×C₄⋊C₄	Dic₂₆	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	—	26	4	no	—
188	C₂×C₂²⋊C₄	Dic₂₆	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	D₅₂	1	1	yes	characteristic
190	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
191	#106	—	52	4	no	—
192	C₄⋊C₈	—	2	2	no	—
193	#99	—	52	4	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	#95	—	26	4	no	—
199	C₂×C₁₃	—	2	4	no	—
200	C₂×C₁₃	—	2	1	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	4	2	no	—
203	C₂×C₁₃	#1799	1	1	yes	characteristic
204	C₂×C₁₃	#1807	1	1	yes	characteristic
205	C₂×C₁₃	#1812	1	1	yes	characteristic
206	C₂⁴	C₄⋊Dic₁₃	1	1	yes	characteristic
207	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	#6095	1	4	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	52	4	no	—
218	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
222	C₂²×C₄	#6127	1	2	yes	—
223	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
237	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	52	4	no	—
240	C₄⋊C₄	—	52	4	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	52	4	no	—
244	C₂²⋊C₄	—	52	4	no	—
245	C₄²	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₁₃	#8608	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	4	1	no	—
254	C₂³	#88057	1	2	yes	complemented
255	C₂³	C₂×#6095	1	2	yes	—
256	C₂³	#88158	1	2	yes	—
257	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
258	C₂³	#88184	1	2	yes	—
259	D₄	—	2	8	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	104	4	no	—
276	C₂×C₄	—	104	4	no	—
277	C₂×C₄	—	104	4	no	—
278	C₂×C₄	#88057	1	2	yes	complemented
279	C₂×C₄	#88178	1	2	yes	—
280	C₂×C₄	#88180	1	1	yes	characteristic
281	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	2	no	—
284	C₈	—	4	1	no	—
285	C₈	—	4	1	no	—
286	C₈	#88074	1	2	yes	complemented
287	C₂²	—	2	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	C₂×#88057	1	1	yes	characteristic
299	C₂²	#11718767	1	1	yes	characteristic
300	C₂²	#11718775	1	4	yes	—
301	C₂²	#11718873	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	208	4	no	—
308	C₄	C₂×#88074	1	1	yes	characteristic
309	C₄	#11718901	1	1	yes	characteristic
310	C₂	—	2	4	no	—
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	4	2	no	—
314	C₂	#13114824	1	1	yes	characteristic
315	C₂	#13114861	1	1	yes	characteristic
316	C₂	#13114882	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718873
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₈×D₄×C₁₃	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂²×D₄×C₁₃	D₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×C₈×D₄	D₁₃
hypocenter	C₁₃	#8608
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #13114861	C₂	semidirect
3. #13112704	C₂	semidirect
5. C₂³×C₈×C₁₃	C₄	semidirect
10. C₂×C₁₃×C₄⋊C₈	C₄	semidirect
11. C₂×C₁₃×C₂²⋊C₈	C₄	semidirect
13. C₂×C₄×C₈×C₁₃	C₄	semidirect
100. C₂³×C₁₃	#106	semidirect
121. C₂×C₄×C₁₃	#106	semidirect
132. C₈×C₁₃	#99	semidirect
133. C₂³×C₈	Dic₁₃	semidirect
139. C₂×C₄⋊C₈	Dic₁₃	semidirect
141. C₂×C₂²⋊C₈	Dic₁₃	semidirect
143. C₂×C₄×C₈	Dic₁₃	semidirect
247. C₁₃	#8608	semidirect
254. C₂³	#88057	semidirect
278. C₂×C₄	#88057	semidirect
286. C₈	#88074	semidirect

Maximal subgroups

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

subgroup	index
#13114861	2
#13114824	2
#13112704	2
C₂×C₈×D₄×C₁₃	2
#8608	13