Subgroups — #15846129

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	#13125980	C₂	1	2	yes	maximal
2	#13123770	C₂	1	1	yes	maximal, characteristic
3	#13114327	C₂	1	4	yes	maximal, complemented
4	#13114203	C₂	1	4	yes	maximal, complemented
5	#13113584	C₂	1	1	yes	maximal, characteristic
6	#13113508	C₂	1	2	yes	maximal
7	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×Q₁₆	—	2	1	no	—
9	C₂²×C₁₃×Q₁₆	—	2	1	no	—
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	#11719175	C₂²	1	1	yes	characteristic
12	C₁₃×#494	—	2	1	no	—
13	C₁₃×#494	—	2	1	no	—
14	C₁₃×#494	C₂²	1	2	yes	—
15	C₁₃×#494	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#11718763	—	2	1	no	—
18	#11718762	—	2	1	no	—
19	#11718755	C₂²	1	8	yes	—
20	#11718754	—	2	2	no	—
21	#11718740	—	2	2	no	—
22	#11718632	—	2	4	no	—
23	#11718586	C₂²	1	2	yes	—
24	#11718585	C₂²	1	1	yes	characteristic
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	#11718139	C₂²	1	8	yes	—
30	#11718050	C₂²	1	2	yes	—
31	#11718049	C₂²	1	2	yes	—
32	C₂²×Q₈×C₁₃	—	2	2	no	—
33	C₂×C₁₃×Q₁₆	—	2	2	no	—
34	C₂×C₁₃×Q₁₆	—	2	2	no	—
35	C₂×C₁₃×Q₁₆	—	2	2	no	—
36	C₂×C₁₃×Q₁₆	—	2	2	no	—
37	C₂×C₁₃×Q₁₆	—	4	4	no	—
38	C₂×C₁₃×Q₁₆	—	4	4	no	—
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₁₃×C₄⋊Q₈	—	2	2	no	—
42	C₁₃×C₄⋊Q₈	—	2	2	no	—
43	C₁₃×C₄⋊Q₈	—	2	2	no	—
44	C₁₃×C₄⋊Q₈	C₂³	1	4	yes	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	#88180	—	2	1	no	—
49	#88178	—	2	1	no	—
50	#88178	D₄	1	2	yes	—
51	C₂×#6099	—	2	1	no	—
52	#88124	—	2	1	no	—
53	#88123	D₄	1	1	yes	characteristic
54	#88123	D₄	1	1	yes	characteristic
55	C₄×Dic₂₆	—	2	1	no	—
56	C₄×Dic₂₆	C₂³	1	2	yes	—
57	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²⋊#1055	—	2	2	no	—
62	#88055	—	2	2	no	—
63	#88054	—	2	2	no	—
64	#88054	—	2	2	no	—
65	C₈×Dic₁₃	—	2	2	no	—
66	#88050	—	2	8	no	—
67	#88048	—	2	8	no	—
68	C₄⋊#1055	—	2	1	no	—
69	C₄⋊#1055	C₂³	1	4	yes	—
70	C₄⋊#1055	C₂³	1	2	yes	—
71	C₄×#1055	—	2	2	no	—
72	#19204	—	13	1	no	maximal, Sylow
73	C₂×Q₈×C₁₃	—	2	4	no	—
74	C₂×Q₈×C₁₃	—	4	2	no	—
75	C₂×Q₈×C₁₃	—	4	4	no	—
76	C₂×Q₈×C₁₃	—	4	4	no	—
77	C₂²×C₄×C₁₃	—	4	2	no	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
81	C₂×Dic₂₆	—	2	1	no	—
82	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
83	C₁₃×Q₁₆	—	4	4	no	—
84	C₁₃×Q₁₆	—	4	4	no	—
85	C₁₃×Q₁₆	—	4	4	no	—
86	C₁₃×Q₁₆	—	4	4	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	—	2	4	no	—
95	C₁₃×C₄⋊C₄	—	4	2	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃	#41	1	1	yes	characteristic
100	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
101	C₂²⋊Dic₁₃	—	2	2	no	—
102	C₄⋊Dic₁₃	—	2	1	no	—
103	C₄⋊Dic₁₃	—	2	1	no	—
104	C₄⋊Dic₁₃	—	2	1	no	—
105	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
106	#6101	—	2	2	no	—
107	#6101	—	2	2	no	—
108	C₄×Dic₁₃	—	2	2	no	—
109	#6099	—	4	2	no	—
110	C₂×#1055	—	2	4	no	—
111	C₂×#1055	—	2	2	no	—
112	#3229	—	13	2	no	—
113	C₂×#494	D₁₃	1	1	yes	characteristic
114	#2960	—	13	1	no	—
115	#1690	—	13	4	no	—
116	#1670	—	13	4	no	—
117	#1564	—	13	1	no	—
118	#1546	—	13	2	no	—
119	C₂³×C₁₃	#121	1	1	yes	characteristic
120	Q₈×C₁₃	—	4	4	no	—
121	Q₈×C₁₃	—	4	4	no	—
122	Q₈×C₁₃	—	8	4	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	8	4	no	—
127	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
131	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₁₃	—	2	2	no	—
136	C₂×Dic₁₃	—	2	2	no	—
137	Dic₂₆	—	2	4	no	—
138	Dic₂₆	—	4	2	no	—
139	C₈×C₁₃	—	2	1	no	—
140	C₈×C₁₃	—	2	1	no	—
141	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
142	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
143	#1055	—	4	2	no	—
144	#1055	—	4	4	no	—
145	C₂²×Q₁₆	—	2	1	no	—
146	C₂²×Q₁₆	—	2	1	no	—
147	#533	—	13	1	no	—
148	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
149	#498	—	26	2	no	—
150	#494	—	2	1	no	—
151	#494	—	2	1	no	—
152	#494	C₂×D₁₃	1	2	yes	—
153	#494	C₂×D₁₃	1	2	yes	—
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	—	13	2	no	—
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	—	13	2	no	—
167	#334	—	13	2	no	—
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	—	8	4	no	—
176	C₄×C₁₃	#460	1	2	yes	—
177	C₄×C₁₃	#473	1	2	yes	—
178	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
179	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₄×C₁₃	#578	1	2	yes	—
181	Dic₁₃	—	4	2	no	—
182	Dic₁₃	—	4	2	no	—
183	C₂²×Q₈	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	2	2	no	—
186	C₂×Q₁₆	—	2	2	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×Q₁₆	—	4	4	no	—
189	C₂×Q₁₆	—	4	4	no	—
190	C₂×M₁₆	—	26	1	no	—
191	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	13	1	no	—
197	C₄⋊Q₈	—	13	1	no	—
198	C₄⋊Q₈	C₂²×D₁₃	1	4	yes	—
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	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	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	—	13	4	no	—
211	C₄⋊C₈	—	26	2	no	—
212	C₄⋊C₈	—	26	2	no	—
213	C₄⋊C₈	—	26	1	no	—
214	#103	—	26	8	no	—
215	C₂²⋊C₈	—	26	2	no	—
216	#97	—	26	2	no	—
217	C₄×C₈	—	26	2	no	—
218	C₄×C₈	—	26	2	no	—
219	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
220	C₄×C₈	C₂²⋊D₁₃	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	—	2	4	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	—	4	4	no	—
230	C₂×Q₈	—	4	4	no	—
231	C₂×Q₈	—	13	2	no	—
232	C₂×Q₈	—	26	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
235	C₂²×C₄	#6129	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
237	Q₁₆	—	4	4	no	—
238	Q₁₆	—	4	4	no	—
239	Q₁₆	—	4	4	no	—
240	Q₁₆	—	4	4	no	—
241	M₁₆	—	52	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	26	4	no	—
244	C₂×C₈	—	26	2	no	—
245	C₂×C₈	#6105	1	4	yes	—
246	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	13	2	no	—
255	C₄⋊C₄	—	26	1	no	—
256	C₄⋊C₄	—	26	1	no	—
257	C₄⋊C₄	—	26	2	no	—
258	C₄⋊C₄	—	26	1	no	—
259	C₄⋊C₄	—	26	2	no	—
260	C₂²⋊C₄	—	26	2	no	—
261	C₄²	—	26	2	no	—
262	C₄²	D₄×D₁₃	1	1	yes	characteristic
263	C₄²	#6129	1	1	yes	characteristic
264	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
265	C₄²	C₂³×D₁₃	1	1	yes	characteristic
266	C₁₃	#19204	1	1	yes	characteristic, complemented, Sylow
267	C₂³	#88192	1	1	yes	characteristic
268	Q₈	—	4	4	no	—
269	Q₈	—	4	4	no	—
270	Q₈	—	8	4	no	—
271	Q₈	—	26	4	no	—
272	Q₈	—	52	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	—	26	2	no	—
278	C₂×C₄	—	26	2	no	—
279	C₂×C₄	#88186	1	2	yes	—
280	C₂×C₄	C₂×#6105	1	2	yes	—
281	C₂×C₄	#88192	1	1	yes	characteristic
282	C₂×C₄	#88192	1	1	yes	characteristic
283	C₂×C₄	#88192	1	1	yes	characteristic
284	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
285	C₂×C₄	C₂×#6129	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	—	52	4	no	—
290	C₈	—	52	2	no	—
291	C₈	C₂×#6105	1	2	yes	—
292	C₈	C₂×#6105	1	2	yes	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	#11718736	1	2	yes	—
296	C₂²	C₂×#88192	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	8	4	no	—
300	C₄	—	52	2	no	—
301	C₄	—	52	2	no	—
302	C₄	#11718736	1	2	yes	—
303	C₄	#11718853	1	2	yes	—
304	C₄	#11719471	1	2	yes	—
305	C₄	C₂×#88186	1	1	yes	characteristic
306	C₄	C₂²×#6105	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	C₂×#11718736	1	1	yes	characteristic
309	C₂	#13124487	1	1	yes	characteristic
310	C₂	C₂×#11718853	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88192
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#494	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂³×C₁₃	#121
hypercenter	C₂×#494	D₁₃
hypocenter	C₁₃	#19204
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #13114327	C₂	semidirect
4. #13114203	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
#13125980	2
#13123770	2
#13114327	2
#13114203	2
#13113584	2
#13113508	2
C₂×C₁₃×#494	2
#19204	13