Subgroups — #15768647

Order 3328.

894 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718143	C₂	1	2	yes	maximal
2	C₂×#11718143	C₂	1	2	yes	maximal
3	#13113066	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₁₃	—	2	4	no	—
10	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	complemented
11	C₂×#88071	—	2	2	no	—
12	C₂×#88071	—	2	2	no	—
13	#11718143	—	2	4	no	—
14	#11718143	C₄	1	8	yes	complemented
15	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂³×Dic₁₃	—	2	4	no	—
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	D₄	1	4	yes	—
27	C₂×C₄²×C₁₃	—	2	2	no	—
28	C₂×C₄²×C₁₃	—	2	2	no	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
31	C₂×C₄⋊Dic₁₃	—	2	4	no	—
32	C₂×C₄⋊Dic₁₃	—	2	4	no	—
33	C₂×C₄⋊Dic₁₃	—	2	4	no	—
34	C₂×C₄⋊Dic₁₃	—	2	4	no	—
35	C₂×C₄⋊Dic₁₃	—	2	4	no	—
36	C₂×C₄⋊Dic₁₃	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₁₃	—	2	8	no	—
38	#88071	—	4	4	no	—
39	#88071	—	4	4	no	—
40	#10820	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	—	2	2	no	—
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	D₈	1	2	yes	—
49	C₂²×C₄×C₁₃	SD₁₆	1	4	yes	—
50	C₂²×C₄×C₁₃	Q₁₆	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	#41	1	2	yes	—
56	C₂²×C₄×C₁₃	#41	1	2	yes	—
57	C₂²×C₄×C₁₃	#41	1	2	yes	—
58	C₂²×C₄×C₁₃	#41	1	2	yes	—
59	C₂²×Dic₁₃	—	2	8	no	—
60	C₂²×Dic₁₃	—	4	4	no	—
61	C₂²×Dic₁₃	—	4	4	no	—
62	C₂²×Dic₁₃	—	4	4	no	—
63	C₂²×Dic₁₃	—	4	4	no	—
64	C₂²×Dic₁₃	—	4	4	no	—
65	C₂²×Dic₁₃	—	4	4	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	8	no	—
71	C₂×C₈×C₁₃	—	2	8	no	—
72	C₂×C₈×C₁₃	C₄²	1	4	yes	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	—	2	2	no	—
76	C₄²×C₁₃	—	2	4	no	—
77	C₄⋊Dic₁₃	—	2	8	no	—
78	C₄⋊Dic₁₃	—	2	8	no	—
79	C₄⋊Dic₁₃	—	4	4	no	—
80	C₄⋊Dic₁₃	—	4	4	no	—
81	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
82	C₂×#340	—	13	2	no	—
83	C₂×#340	—	13	2	no	—
84	C₄⋊#95	—	13	2	no	—
85	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂³×C₁₃	C₄×D₄	1	2	yes	—
87	C₂³×C₁₃	#121	1	2	yes	—
88	C₂³×C₁₃	#124	1	2	yes	—
89	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
90	C₂³×C₁₃	C₂×SD₁₆	1	2	yes	—
91	C₂³×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
92	C₂³×C₁₃	#135	1	2	yes	—
93	C₂³×C₁₃	#135	1	2	yes	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	8	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	#106	1	4	yes	—
102	C₂×C₄×C₁₃	#107	1	4	yes	—
103	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#117	1	4	yes	—
108	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
111	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₁₃	#123	1	4	yes	—
114	C₂×C₄×C₁₃	#125	1	2	yes	—
115	C₂×C₄×C₁₃	#126	1	4	yes	—
116	C₂×Dic₁₃	—	4	8	no	—
117	C₂×Dic₁₃	—	4	8	no	—
118	C₂×Dic₁₃	—	4	8	no	—
119	C₂×Dic₁₃	—	8	4	no	—
120	C₂×Dic₁₃	—	8	4	no	—
121	C₂×Dic₁₃	—	8	4	no	—
122	C₂×Dic₁₃	—	8	4	no	—
123	C₈×C₁₃	—	2	4	no	—
124	C₈×C₁₃	—	2	4	no	—
125	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
126	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
127	C₂²×C₄⋊C₄	—	13	4	no	—
128	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
129	C₂×C₄×C₈	—	2	4	no	—
130	C₂×C₄×C₈	Dic₁₃	1	4	yes	complemented
131	C₂×#95	—	26	2	no	—
132	C₂×#95	—	26	2	no	—
133	#340	—	13	8	no	—
134	#340	—	26	4	no	—
135	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₁₃	#380	1	1	yes	characteristic
137	C₂²×C₁₃	#382	1	1	yes	characteristic
138	C₂²×C₁₃	#388	1	4	yes	—
139	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
140	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
141	C₂²×C₁₃	#427	1	1	yes	characteristic
142	C₂²×C₁₃	#427	1	1	yes	characteristic
143	C₂²×C₁₃	C₄×D₈	1	2	yes	—
144	C₂²×C₁₃	C₄×SD₁₆	1	4	yes	—
145	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
146	C₂²×C₁₃	#465	1	4	yes	—
147	C₂²×C₁₃	#466	1	2	yes	—
148	C₂²×C₁₃	#467	1	2	yes	—
149	C₂²×C₁₃	#486	1	2	yes	—
150	C₂²×C₁₃	#487	1	2	yes	—
151	C₂²×C₁₃	#488	1	4	yes	—
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	—	2	4	no	—
154	C₄×C₁₃	#376	1	1	yes	characteristic
155	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₁₃	#381	1	1	yes	characteristic
157	C₄×C₁₃	#383	1	1	yes	characteristic
158	C₄×C₁₃	#387	1	4	yes	—
159	Dic₁₃	—	8	8	no	—
160	C₂³×C₄	—	26	4	no	—
161	C₂³×C₄	D₅₂	1	1	yes	characteristic
162	C₂³×C₄	D₅₂	1	1	yes	characteristic
163	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	C₄×D₁₃	1	4	yes	—
169	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
170	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
171	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
172	C₂×C₄⋊C₄	—	13	8	no	—
173	C₂×C₄⋊C₄	—	26	4	no	—
174	C₂×C₄⋊C₄	—	26	4	no	—
175	C₂×C₄⋊C₄	—	26	4	no	—
176	C₂×C₄⋊C₄	—	26	4	no	—
177	C₂×C₄⋊C₄	—	26	4	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	Dic₂₆	1	2	yes	—
181	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
182	C₄×C₈	—	2	8	no	—
183	#95	—	52	4	no	—
184	#95	—	52	4	no	—
185	C₂×C₁₃	#1434	1	1	yes	characteristic
186	C₂×C₁₃	C₄×#106	1	1	yes	characteristic
187	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
188	C₂×C₁₃	#1807	1	1	yes	characteristic
189	C₂×C₁₃	#1808	1	1	yes	characteristic
190	C₂×C₁₃	#1834	1	1	yes	characteristic
191	C₂×C₁₃	#1835	1	1	yes	characteristic
192	C₂×C₁₃	#1933	1	4	yes	—
193	C₂×C₁₃	#1934	1	4	yes	—
194	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	26	8	no	—
200	C₂²×C₄	—	52	4	no	—
201	C₂²×C₄	—	52	4	no	—
202	C₂²×C₄	—	52	4	no	—
203	C₂²×C₄	—	52	4	no	—
204	C₂²×C₄	—	52	4	no	—
205	C₂²×C₄	—	52	4	no	—
206	C₂²×C₄	#6095	1	4	yes	—
207	C₂²×C₄	D₁₀₄	1	2	yes	—
208	C₂²×C₄	Dic₅₂	1	2	yes	—
209	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
210	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
211	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
213	C₂²×C₄	#6127	1	2	yes	—
214	C₂²×C₄	#6127	1	2	yes	—
215	C₂²×C₄	#6127	1	2	yes	—
216	C₂²×C₄	#6127	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	—	2	8	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
226	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
227	C₄⋊C₄	—	26	8	no	—
228	C₄⋊C₄	—	26	8	no	—
229	C₄⋊C₄	—	52	4	no	—
230	C₄⋊C₄	—	52	4	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₁₃	#10820	1	1	yes	characteristic, complemented, Sylow
235	C₂³	C₄×D₅₂	1	2	yes	—
236	C₂³	#88129	1	2	yes	—
237	C₂³	C₂×#6095	1	2	yes	—
238	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
239	C₂³	#88158	1	2	yes	—
240	C₂³	#88158	1	2	yes	—
241	C₂³	C₂×Dic₅₂	1	1	yes	characteristic
242	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
243	C₂³	#88184	1	2	yes	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	52	8	no	—
252	C₂×C₄	—	52	8	no	—
253	C₂×C₄	—	52	8	no	—
254	C₂×C₄	—	104	4	no	—
255	C₂×C₄	—	104	4	no	—
256	C₂×C₄	—	104	4	no	—
257	C₂×C₄	—	104	4	no	—
258	C₂×C₄	#88057	1	4	yes	—
259	C₂×C₄	#88058	1	4	yes	—
260	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
261	C₂×C₄	#88124	1	2	yes	—
262	C₂×C₄	#88126	1	4	yes	—
263	C₂×C₄	#88130	1	4	yes	—
264	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
265	C₂×C₄	#88178	1	2	yes	—
266	C₂×C₄	#88180	1	1	yes	characteristic
267	C₂×C₄	#88180	1	1	yes	characteristic
268	C₂×C₄	#88180	1	1	yes	characteristic
269	C₂×C₄	#88180	1	1	yes	characteristic
270	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
271	C₂×C₄	#88183	1	4	yes	—
272	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	2	4	no	—
275	C₂²	#11718342	1	2	yes	—
276	C₂²	C₄×#6095	1	4	yes	—
277	C₂²	C₄×D₁₀₄	1	2	yes	—
278	C₂²	#11718355	1	2	yes	—
279	C₂²	#11718358	1	4	yes	—
280	C₂²	C₄×Dic₅₂	1	2	yes	—
281	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
282	C₂²	#11718597	1	1	yes	characteristic
283	C₂²	#11718601	1	4	yes	—
284	C₂²	C₂×#88057	1	1	yes	characteristic
285	C₂²	C₂×#88058	1	1	yes	characteristic
286	C₂²	#11718767	1	1	yes	characteristic
287	C₂²	#11718767	1	1	yes	characteristic
288	C₂²	#11718775	1	4	yes	—
289	C₂²	#11718776	1	2	yes	—
290	C₂²	#11718777	1	2	yes	—
291	C₂²	#11718873	1	1	yes	characteristic
292	C₄	—	2	4	no	—
293	C₄	—	2	4	no	—
294	C₄	—	104	8	no	—
295	C₄	#11718593	1	1	yes	characteristic
296	C₄	#11718594	1	4	yes	—
297	C₄	#11718598	1	1	yes	characteristic
298	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
299	C₄	#11718872	1	1	yes	characteristic
300	C₂	#13113063	1	1	yes	characteristic
301	C₂	C₄×#88057	1	1	yes	characteristic
302	C₂	C₄×#88058	1	1	yes	characteristic
303	C₂	#13114861	1	1	yes	characteristic
304	C₂	#13114865	1	1	yes	characteristic
305	C₂	#13114964	1	1	yes	characteristic
306	C₂	#13114967	1	1	yes	characteristic
307	C₂	#13115512	1	4	yes	—
308	C₂	#13115519	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#10820
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×C₈×C₁₃	C₄	semidirect
14. #11718143	C₄	semidirect
130. C₂×C₄×C₈	Dic₁₃	semidirect
234. C₁₃	#10820	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718143	2
#13113066	2
C₂²×C₄×C₈×C₁₃	2
#10820	13