Subgroups — #15846134

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125982	C₂	1	2	yes	maximal
2	C₂×#11718360	C₂	1	1	yes	maximal, characteristic
3	#13123770	C₂	1	1	yes	maximal, characteristic
4	#13114331	C₂	1	4	yes	maximal, complemented
5	#13114207	C₂	1	4	yes	maximal, complemented
6	#13113507	C₂	1	2	yes	maximal
7	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₅₂	—	2	1	no	—
9	C₂²×Dic₅₂	—	2	1	no	—
10	#11719175	C₂²	1	1	yes	characteristic
11	C₂×#88123	C₂²	1	2	yes	—
12	C₁₃×#431	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718763	—	2	1	no	—
15	#11718762	—	2	1	no	—
16	#11718585	C₂²	1	1	yes	characteristic
17	#11718512	C₂²	1	8	yes	—
18	#11718382	—	2	4	no	—
19	#11718360	—	2	1	no	—
20	#11718360	—	2	1	no	—
21	#11718360	C₂²	1	2	yes	—
22	#11718360	C₂²	1	2	yes	—
23	#11718342	—	2	2	no	—
24	#11718341	—	2	2	no	—
25	#11718340	—	2	1	no	—
26	#11718340	C₂²	1	2	yes	—
27	C₈×Dic₂₆	—	2	1	no	—
28	C₈×Dic₂₆	C₂²	1	2	yes	—
29	C₁₃×#335	C₂²	1	2	yes	—
30	C₁₃×#334	C₂²	1	2	yes	—
31	#11718078	C₂²	1	8	yes	—
32	C₂²×Dic₂₆	—	2	2	no	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	#88180	—	2	1	no	—
37	C₂×C₄⋊Dic₁₃	—	2	2	no	—
38	C₂×C₄⋊Dic₁₃	—	2	2	no	—
39	#88178	—	2	1	no	—
40	#88178	D₄	1	2	yes	—
41	C₂×#6099	—	2	1	no	—
42	C₂×Dic₅₂	—	2	2	no	—
43	C₂×Dic₅₂	—	2	2	no	—
44	C₂×Dic₅₂	—	2	2	no	—
45	C₂×Dic₅₂	—	2	2	no	—
46	C₂×Dic₅₂	—	4	4	no	—
47	C₂×Dic₅₂	—	4	4	no	—
48	#88124	—	2	1	no	—
49	#88123	—	2	2	no	—
50	#88123	—	2	2	no	—
51	#88123	—	2	2	no	—
52	#88123	D₄	1	1	yes	characteristic
53	#88123	D₄	1	1	yes	characteristic
54	#88123	C₂³	1	4	yes	—
55	C₄×Dic₂₆	—	2	1	no	—
56	C₄×Dic₂₆	C₂³	1	2	yes	—
57	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
58	C₁₃×C₂²⋊C₈	—	2	2	no	—
59	C₄×C₈×C₁₃	—	2	2	no	—
60	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	#88057	—	2	8	no	—
65	#88056	—	2	8	no	—
66	#88055	—	2	2	no	—
67	#88054	—	2	2	no	—
68	#88054	—	2	2	no	—
69	C₈×Dic₁₃	—	2	2	no	—
70	C₄⋊#1055	—	2	1	no	—
71	C₄⋊#1055	C₂³	1	2	yes	—
72	#19204	—	13	1	no	maximal, Sylow
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
76	C₂²×Dic₁₃	—	4	2	no	—
77	C₂×Dic₂₆	—	2	1	no	—
78	C₂×Dic₂₆	—	2	4	no	—
79	C₂×Dic₂₆	—	4	4	no	—
80	C₂×Dic₂₆	—	4	4	no	—
81	C₂×Dic₂₆	—	4	2	no	—
82	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₃	#41	1	1	yes	characteristic
93	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊Dic₁₃	—	2	2	no	—
95	C₄⋊Dic₁₃	—	2	1	no	—
96	C₄⋊Dic₁₃	—	2	1	no	—
97	C₄⋊Dic₁₃	—	2	1	no	—
98	C₄⋊Dic₁₃	—	2	4	no	—
99	C₄⋊Dic₁₃	—	2	4	no	—
100	C₄⋊Dic₁₃	—	4	2	no	—
101	C₄⋊Dic₁₃	—	4	2	no	—
102	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
103	#6101	—	2	2	no	—
104	#6101	—	2	2	no	—
105	C₄×Dic₁₃	—	2	2	no	—
106	#6099	—	4	2	no	—
107	C₂×#1055	—	2	2	no	—
108	Dic₅₂	—	4	4	no	—
109	Dic₅₂	—	4	4	no	—
110	Dic₅₂	—	4	4	no	—
111	Dic₅₂	—	4	4	no	—
112	#3229	—	13	2	no	—
113	C₂×#494	—	13	1	no	—
114	#2960	—	13	1	no	—
115	#1690	—	13	4	no	—
116	#1670	—	13	4	no	—
117	#1564	D₁₃	1	1	yes	characteristic
118	#1546	—	13	2	no	—
119	C₂³×C₁₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
126	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
130	C₂×Dic₁₃	—	2	2	no	—
131	C₂×Dic₁₃	—	2	2	no	—
132	C₂×Dic₁₃	—	4	4	no	—
133	C₂×Dic₁₃	—	8	4	no	—
134	Dic₂₆	—	2	4	no	—
135	Dic₂₆	—	4	2	no	—
136	Dic₂₆	—	4	4	no	—
137	Dic₂₆	—	4	4	no	—
138	Dic₂₆	—	8	4	no	—
139	C₈×C₁₃	—	2	1	no	—
140	C₈×C₁₃	—	2	1	no	—
141	C₈×C₁₃	—	4	4	no	—
142	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
143	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
144	#1055	—	4	2	no	—
145	C₂²×Q₁₆	—	26	1	no	—
146	C₂²×Q₁₆	—	26	1	no	—
147	#533	—	13	1	no	—
148	C₂×C₄⋊Q₈	—	13	2	no	—
149	#498	—	26	2	no	—
150	#494	—	13	2	no	—
151	#494	—	13	2	no	—
152	#494	—	26	1	no	—
153	#494	—	26	1	no	—
154	#487	—	26	2	no	—
155	#483	—	26	4	no	—
156	#475	—	13	8	no	—
157	#446	—	13	2	no	—
158	#446	—	26	1	no	—
159	C₈×Q₈	—	13	2	no	—
160	C₈×Q₈	—	26	1	no	—
161	#431	C₂×D₁₃	1	2	yes	—
162	#424	—	26	1	no	—
163	#423	—	13	1	no	—
164	#405	—	26	1	no	—
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	#335	C₂×D₁₃	1	2	yes	—
167	#334	C₂×D₁₃	1	2	yes	—
168	#332	—	13	8	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	#460	1	2	yes	—
172	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	#460	1	2	yes	—
176	C₄×C₁₃	#473	1	2	yes	—
177	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
178	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₄×C₁₃	#578	1	2	yes	—
180	Dic₁₃	—	4	2	no	—
181	Dic₁₃	—	4	2	no	—
182	Dic₁₃	—	8	4	no	—
183	C₂²×Q₈	—	26	2	no	—
184	C₂×Q₁₆	—	26	2	no	—
185	C₂×Q₁₆	—	26	2	no	—
186	C₂×Q₁₆	—	26	2	no	—
187	C₂×Q₁₆	—	26	2	no	—
188	C₂×Q₁₆	—	52	4	no	—
189	C₂×Q₁₆	—	52	4	no	—
190	C₂×M₁₆	—	26	1	no	—
191	C₂²×C₈	D₅₂	1	1	yes	characteristic
192	C₂²×C₈	D₅₂	1	1	yes	characteristic
193	C₄⋊Q₈	—	13	1	no	—
194	C₄⋊Q₈	—	13	1	no	—
195	C₄⋊Q₈	—	13	4	no	—
196	C₄⋊Q₈	—	26	2	no	—
197	C₄⋊Q₈	—	26	2	no	—
198	C₄⋊Q₈	—	26	2	no	—
199	#125	—	26	1	no	—
200	C₂²⋊Q₈	—	13	2	no	—
201	C₂²⋊Q₈	—	26	1	no	—
202	C₄×Q₈	—	13	2	no	—
203	C₄×Q₈	—	26	1	no	—
204	#117	—	26	1	no	—
205	C₂×C₄⋊C₄	—	26	2	no	—
206	C₂×C₄⋊C₄	—	26	2	no	—
207	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
208	#106	—	26	8	no	—
209	C₄⋊C₈	—	13	2	no	—
210	C₄⋊C₈	—	26	2	no	—
211	C₄⋊C₈	—	26	2	no	—
212	C₄⋊C₈	—	26	1	no	—
213	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
214	#103	—	26	8	no	—
215	C₂²⋊C₈	—	2	2	no	—
216	#97	—	26	2	no	—
217	C₄×C₈	—	2	2	no	—
218	C₄×C₈	—	26	2	no	—
219	C₄×C₈	D₅₂	1	1	yes	characteristic
220	C₄×C₈	D₅₂	1	1	yes	characteristic
221	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
223	C₂×C₁₃	—	2	2	no	—
224	C₂×C₁₃	C₂×#460	1	1	yes	characteristic
225	C₂×C₁₃	#3059	1	1	yes	characteristic
226	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
227	C₂×Q₈	—	13	2	no	—
228	C₂×Q₈	—	26	1	no	—
229	C₂×Q₈	—	26	4	no	—
230	C₂×Q₈	—	52	2	no	—
231	C₂×Q₈	—	52	4	no	—
232	C₂×Q₈	—	52	4	no	—
233	C₂²×C₄	—	52	2	no	—
234	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
235	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
236	C₂²×C₄	#6131	1	1	yes	characteristic
237	Q₁₆	—	52	4	no	—
238	Q₁₆	—	52	4	no	—
239	Q₁₆	—	52	4	no	—
240	Q₁₆	—	52	4	no	—
241	M₁₆	—	52	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	26	2	no	—
245	C₂×C₈	#6095	1	4	yes	—
246	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
247	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
248	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
249	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
250	C₄⋊C₄	—	13	2	no	—
251	C₄⋊C₄	—	26	1	no	—
252	C₄⋊C₄	—	26	1	no	—
253	C₄⋊C₄	—	26	2	no	—
254	C₄⋊C₄	—	26	1	no	—
255	C₄⋊C₄	—	26	2	no	—
256	C₄⋊C₄	—	26	4	no	—
257	C₄⋊C₄	—	26	4	no	—
258	C₄⋊C₄	—	52	2	no	—
259	C₄⋊C₄	—	52	2	no	—
260	C₂²⋊C₄	—	26	2	no	—
261	C₄²	—	26	2	no	—
262	C₄²	C₂×D₅₂	1	1	yes	characteristic
263	C₄²	D₄×D₁₃	1	1	yes	characteristic
264	C₄²	#6131	1	1	yes	characteristic
265	C₄²	C₂³×D₁₃	1	1	yes	characteristic
266	C₁₃	#19204	1	1	yes	characteristic, complemented, Sylow
267	C₂³	#88149	1	1	yes	characteristic
268	Q₈	—	26	4	no	—
269	Q₈	—	52	2	no	—
270	Q₈	—	52	4	no	—
271	Q₈	—	52	4	no	—
272	Q₈	—	104	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	26	2	no	—
276	C₂×C₄	—	26	2	no	—
277	C₂×C₄	—	52	4	no	—
278	C₂×C₄	—	104	4	no	—
279	C₂×C₄	#88149	1	1	yes	characteristic
280	C₂×C₄	#88149	1	1	yes	characteristic
281	C₂×C₄	#88149	1	1	yes	characteristic
282	C₂×C₄	C₂×#6095	1	2	yes	—
283	C₂×C₄	#88162	1	2	yes	—
284	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
285	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
286	C₂×C₄	C₂×#6131	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	—	4	4	no	—
290	C₈	—	52	2	no	—
291	C₈	C₂×#6095	1	2	yes	—
292	C₈	C₂×#6095	1	2	yes	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	#11718509	1	2	yes	—
296	C₂²	C₂×#88149	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	52	2	no	—
300	C₄	—	52	2	no	—
301	C₄	—	104	4	no	—
302	C₄	#11718509	1	2	yes	—
303	C₄	#11718557	1	2	yes	—
304	C₄	C₂²×#6095	1	1	yes	characteristic
305	C₄	C₂×#88162	1	1	yes	characteristic
306	C₄	#11719476	1	2	yes	—
307	C₂	—	2	2	no	—
308	C₂	C₂×#11718509	1	1	yes	characteristic
309	C₂	#13124492	1	1	yes	characteristic
310	C₂	C₂×#11718557	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88149
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1564	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂³×C₁₃	#121
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#19204
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
4. #13114331	C₂	semidirect
5. #13114207	C₂	semidirect
266. C₁₃	#19204	semidirect

Maximal subgroups

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

subgroup	index
#13125982	2
C₂×#11718360	2
#13123770	2
#13114331	2
#13114207	2
#13113507	2
C₁₃×#1564	2
#19204	13