Subgroups — #15846173

Order 3328.

582 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125989	C₂	1	2	yes	maximal
2	C₂×#11718343	C₂	1	1	yes	maximal, characteristic
3	#13123770	C₂	1	1	yes	maximal, characteristic
4	#13114365	C₂	1	4	yes	maximal, complemented
5	#13114241	C₂	1	4	yes	maximal, complemented
6	#13113516	C₂	1	2	yes	maximal
7	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic
8	#11719175	C₂²	1	1	yes	characteristic
9	C₂×#88123	C₂²	1	2	yes	—
10	C₁₃×#431	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88058	—	2	1	no	—
13	C₂×#88058	—	2	1	no	—
14	#11718763	—	2	1	no	—
15	#11718762	—	2	1	no	—
16	#11718585	C₂²	1	1	yes	characteristic
17	#11718511	C₂²	1	8	yes	—
18	#11718383	—	2	4	no	—
19	#11718343	—	2	2	no	—
20	#11718343	—	2	1	no	—
21	#11718343	—	2	1	no	—
22	#11718343	C₂²	1	2	yes	—
23	#11718343	C₂²	1	2	yes	—
24	#11718342	—	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	#11718073	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	#88124	—	2	1	no	—
43	#88123	—	2	2	no	—
44	#88123	—	2	2	no	—
45	#88123	—	2	2	no	—
46	#88123	D₄	1	1	yes	characteristic
47	#88123	D₄	1	1	yes	characteristic
48	#88123	C₂³	1	4	yes	—
49	C₄×Dic₂₆	—	2	1	no	—
50	C₄×Dic₂₆	C₂³	1	2	yes	—
51	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
52	C₁₃×C₂²⋊C₈	—	2	2	no	—
53	C₄×C₈×C₁₃	—	2	2	no	—
54	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	#88058	—	2	2	no	—
59	#88058	—	2	2	no	—
60	#88058	—	2	2	no	—
61	#88058	—	2	2	no	—
62	#88058	—	2	8	no	—
63	#88056	—	2	8	no	—
64	#88055	—	2	2	no	—
65	#88054	—	2	2	no	—
66	#88054	—	2	2	no	—
67	C₈×Dic₁₃	—	2	2	no	—
68	C₄⋊#1055	—	2	1	no	—
69	C₄⋊#1055	C₂³	1	2	yes	—
70	#19209	—	13	1	no	maximal, Sylow
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
74	C₂²×Dic₁₃	—	4	2	no	—
75	C₂×Dic₂₆	—	2	1	no	—
76	C₂×Dic₂₆	—	2	4	no	—
77	C₂×Dic₂₆	—	4	2	no	—
78	C₂×Dic₂₆	—	8	4	no	—
79	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃	#41	1	1	yes	characteristic
90	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₂²⋊Dic₁₃	—	2	2	no	—
92	C₄⋊Dic₁₃	—	2	1	no	—
93	C₄⋊Dic₁₃	—	2	1	no	—
94	C₄⋊Dic₁₃	—	2	1	no	—
95	C₄⋊Dic₁₃	—	2	4	no	—
96	C₄⋊Dic₁₃	—	2	4	no	—
97	C₄⋊Dic₁₃	—	4	2	no	—
98	C₄⋊Dic₁₃	—	4	2	no	—
99	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
100	#6101	—	2	2	no	—
101	#6101	—	2	2	no	—
102	C₄×Dic₁₃	—	2	2	no	—
103	#6099	—	4	2	no	—
104	C₂×#1055	—	2	2	no	—
105	#3230	—	13	2	no	—
106	C₂×#500	—	13	1	no	—
107	#2960	—	13	1	no	—
108	#1696	—	13	4	no	—
109	#1676	—	13	4	no	—
110	#1564	D₁₃	1	1	yes	characteristic
111	#1548	—	13	2	no	—
112	C₂³×C₁₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
119	C₂×C₄×C₁₃	#137	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₁₃	—	2	2	no	—
124	C₂×Dic₁₃	—	2	2	no	—
125	C₂×Dic₁₃	—	4	4	no	—
126	C₂×Dic₁₃	—	8	4	no	—
127	Dic₂₆	—	2	4	no	—
128	Dic₂₆	—	4	2	no	—
129	Dic₂₆	—	8	4	no	—
130	Dic₂₆	—	8	4	no	—
131	C₈×C₁₃	—	2	1	no	—
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	—	4	4	no	—
134	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
135	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
136	#1055	—	4	2	no	—
137	#533	—	13	1	no	—
138	C₂×C₄⋊Q₈	—	13	2	no	—
139	#500	—	13	2	no	—
140	#500	—	13	2	no	—
141	#500	—	26	2	no	—
142	#500	—	26	1	no	—
143	#500	—	26	1	no	—
144	#487	—	26	2	no	—
145	#484	—	26	4	no	—
146	#477	—	13	8	no	—
147	#446	—	13	2	no	—
148	#446	—	26	1	no	—
149	C₈×Q₈	—	13	2	no	—
150	C₈×Q₈	—	26	1	no	—
151	#431	C₂×D₁₃	1	2	yes	—
152	C₂×#107	—	26	1	no	—
153	C₂×#107	—	26	1	no	—
154	#424	—	26	1	no	—
155	#423	—	13	1	no	—
156	#405	—	26	1	no	—
157	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
158	#335	C₂×D₁₃	1	2	yes	—
159	#334	C₂×D₁₃	1	2	yes	—
160	#333	—	13	8	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	#462	1	2	yes	—
164	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	#462	1	2	yes	—
168	C₄×C₁₃	#472	1	2	yes	—
169	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
170	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
171	C₄×C₁₃	#578	1	2	yes	—
172	Dic₁₃	—	4	2	no	—
173	Dic₁₃	—	4	2	no	—
174	Dic₁₃	—	8	4	no	—
175	C₂²×Q₈	—	26	2	no	—
176	C₂×M₁₆	—	26	1	no	—
177	C₂²×C₈	D₅₂	1	1	yes	characteristic
178	C₂²×C₈	D₅₂	1	1	yes	characteristic
179	C₄⋊Q₈	—	13	1	no	—
180	C₄⋊Q₈	—	13	1	no	—
181	C₄⋊Q₈	—	13	4	no	—
182	C₄⋊Q₈	—	26	2	no	—
183	C₄⋊Q₈	—	26	2	no	—
184	C₄⋊Q₈	—	26	2	no	—
185	#125	—	26	1	no	—
186	C₂²⋊Q₈	—	13	2	no	—
187	C₂²⋊Q₈	—	26	1	no	—
188	C₄×Q₈	—	13	2	no	—
189	C₄×Q₈	—	26	1	no	—
190	#117	—	26	1	no	—
191	C₂×C₄⋊C₄	—	26	2	no	—
192	C₂×C₄⋊C₄	—	26	2	no	—
193	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
194	#107	—	26	8	no	—
195	#107	—	26	2	no	—
196	#107	—	26	2	no	—
197	#107	—	26	2	no	—
198	#107	—	26	2	no	—
199	C₄⋊C₈	—	13	2	no	—
200	C₄⋊C₈	—	26	2	no	—
201	C₄⋊C₈	—	26	2	no	—
202	C₄⋊C₈	—	26	1	no	—
203	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
204	#103	—	26	8	no	—
205	C₂²⋊C₈	—	2	2	no	—
206	#97	—	26	2	no	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	—	26	2	no	—
209	C₄×C₈	D₅₂	1	1	yes	characteristic
210	C₄×C₈	D₅₂	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
213	C₂×C₁₃	—	2	2	no	—
214	C₂×C₁₃	C₂×#462	1	1	yes	characteristic
215	C₂×C₁₃	#3059	1	1	yes	characteristic
216	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
217	C₂×Q₈	—	13	2	no	—
218	C₂×Q₈	—	26	1	no	—
219	C₂×Q₈	—	26	4	no	—
220	C₂×Q₈	—	52	2	no	—
221	C₂×Q₈	—	104	4	no	—
222	C₂²×C₄	—	52	2	no	—
223	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
225	C₂²×C₄	#6131	1	1	yes	characteristic
226	M₁₆	—	52	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	26	2	no	—
230	C₂×C₈	Dic₅₂	1	4	yes	—
231	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
232	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
233	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
234	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
235	C₄⋊C₄	—	13	2	no	—
236	C₄⋊C₄	—	26	1	no	—
237	C₄⋊C₄	—	26	1	no	—
238	C₄⋊C₄	—	26	2	no	—
239	C₄⋊C₄	—	26	1	no	—
240	C₄⋊C₄	—	26	2	no	—
241	C₄⋊C₄	—	26	4	no	—
242	C₄⋊C₄	—	26	4	no	—
243	C₄⋊C₄	—	52	2	no	—
244	C₄⋊C₄	—	52	2	no	—
245	C₂²⋊C₄	—	26	2	no	—
246	C₄²	—	26	2	no	—
247	C₄²	C₂×D₅₂	1	1	yes	characteristic
248	C₄²	D₄×D₁₃	1	1	yes	characteristic
249	C₄²	#6131	1	1	yes	characteristic
250	C₄²	C₂³×D₁₃	1	1	yes	characteristic
251	C₁₃	#19209	1	1	yes	characteristic, complemented, Sylow
252	C₂³	#88149	1	1	yes	characteristic
253	Q₈	—	26	4	no	—
254	Q₈	—	52	2	no	—
255	Q₈	—	104	4	no	—
256	Q₈	—	104	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	26	2	no	—
260	C₂×C₄	—	26	2	no	—
261	C₂×C₄	—	52	4	no	—
262	C₂×C₄	—	104	4	no	—
263	C₂×C₄	#88149	1	1	yes	characteristic
264	C₂×C₄	#88149	1	1	yes	characteristic
265	C₂×C₄	#88149	1	1	yes	characteristic
266	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
267	C₂×C₄	#88163	1	2	yes	—
268	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
269	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
270	C₂×C₄	C₂×#6131	1	1	yes	characteristic
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	4	no	—
274	C₈	—	52	2	no	—
275	C₈	C₂×Dic₅₂	1	2	yes	—
276	C₈	C₂×Dic₅₂	1	2	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	#11718510	1	2	yes	—
280	C₂²	C₂×#88149	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	52	2	no	—
284	C₄	—	52	2	no	—
285	C₄	—	104	4	no	—
286	C₄	#11718510	1	2	yes	—
287	C₄	#11718558	1	2	yes	—
288	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
289	C₄	C₂×#88163	1	1	yes	characteristic
290	C₄	#11719476	1	2	yes	—
291	C₂	—	2	2	no	—
292	C₂	C₂×#11718510	1	1	yes	characteristic
293	C₂	#13124492	1	1	yes	characteristic
294	C₂	C₂×#11718558	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	#19209
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. #13114365	C₂	semidirect
5. #13114241	C₂	semidirect
251. C₁₃	#19209	semidirect

Maximal subgroups

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

subgroup	index
#13125989	2
C₂×#11718343	2
#13123770	2
#13114365	2
#13114241	2
#13113516	2
C₁₃×#1564	2
#19209	13