Subgroups — #15846166

Order 3328.

606 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125786	C₂	1	2	yes	maximal
2	C₂×#11718343	C₂	1	1	yes	maximal, characteristic
3	#13123770	C₂	1	1	yes	maximal, characteristic
4	#13114355	C₂	1	4	yes	maximal, complemented
5	#13114231	C₂	1	4	yes	maximal, complemented
6	#13113520	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	#11718510	C₂²	1	8	yes	—
18	#11718400	—	2	4	no	—
19	#11718360	—	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	#11718074	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	8	no	—
43	#88124	—	2	1	no	—
44	#88123	—	2	2	no	—
45	#88123	—	2	2	no	—
46	#88123	—	2	2	no	—
47	#88123	D₄	1	1	yes	characteristic
48	#88123	D₄	1	1	yes	characteristic
49	#88123	C₂³	1	4	yes	—
50	C₄×Dic₂₆	—	2	1	no	—
51	C₄×Dic₂₆	C₂³	1	2	yes	—
52	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
53	C₁₃×C₂²⋊C₈	—	2	2	no	—
54	C₄×C₈×C₁₃	—	2	2	no	—
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	#88058	—	2	2	no	—
60	#88058	—	2	2	no	—
61	#88058	—	2	2	no	—
62	#88058	—	2	2	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	#19208	—	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	complemented
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	Dic₅₂	—	8	8	no	—
106	#3209	—	13	2	no	—
107	C₂×#500	—	13	1	no	—
108	#2960	—	13	1	no	—
109	#1694	—	13	4	no	—
110	#1674	—	13	4	no	—
111	#1564	D₁₃	1	1	yes	characteristic
112	#1549	—	13	2	no	—
113	C₂³×C₁₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
120	C₂×C₄×C₁₃	#137	1	2	yes	—
121	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₁₃	—	2	2	no	—
125	C₂×Dic₁₃	—	2	2	no	—
126	C₂×Dic₁₃	—	4	4	no	—
127	C₂×Dic₁₃	—	8	4	no	—
128	Dic₂₆	—	2	4	no	—
129	Dic₂₆	—	4	2	no	—
130	Dic₂₆	—	4	8	no	—
131	Dic₂₆	—	8	4	no	—
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	—	2	1	no	—
134	C₈×C₁₃	—	4	4	no	—
135	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
136	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
137	#1055	—	4	2	no	—
138	#533	—	13	1	no	—
139	C₂×C₄⋊Q₈	—	13	2	no	—
140	#500	—	13	2	no	—
141	#500	—	13	2	no	—
142	#500	—	26	1	no	—
143	#500	—	26	1	no	—
144	#494	—	26	2	no	—
145	#487	—	26	2	no	—
146	#462	—	13	8	no	—
147	#451	—	26	4	no	—
148	#446	—	13	2	no	—
149	#446	—	26	1	no	—
150	C₈×Q₈	—	13	2	no	—
151	C₈×Q₈	—	26	1	no	—
152	#431	C₂×D₁₃	1	2	yes	—
153	C₂×#107	—	26	1	no	—
154	C₂×#107	—	26	1	no	—
155	#424	—	26	1	no	—
156	#423	—	13	1	no	—
157	#405	—	26	1	no	—
158	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
159	#335	C₂×D₁₃	1	2	yes	—
160	#334	C₂×D₁₃	1	2	yes	—
161	#332	—	13	8	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	—	2	1	no	—
164	C₂²×C₁₃	#462	1	2	yes	—
165	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	#462	1	2	yes	—
169	C₄×C₁₃	#473	1	2	yes	—
170	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
171	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
172	C₄×C₁₃	#576	1	2	yes	—
173	Dic₁₃	—	4	2	no	—
174	Dic₁₃	—	4	2	no	—
175	Dic₁₃	—	8	4	no	—
176	C₂²×Q₈	—	26	2	no	—
177	C₂×Q₁₆	—	26	8	no	—
178	C₂×M₁₆	—	26	1	no	—
179	C₂²×C₈	D₅₂	1	1	yes	characteristic
180	C₂²×C₈	D₅₂	1	1	yes	characteristic
181	C₄⋊Q₈	—	13	1	no	—
182	C₄⋊Q₈	—	13	1	no	—
183	C₄⋊Q₈	—	13	4	no	—
184	C₄⋊Q₈	—	26	2	no	—
185	C₄⋊Q₈	—	26	2	no	—
186	C₄⋊Q₈	—	26	2	no	—
187	#125	—	26	1	no	—
188	C₂²⋊Q₈	—	13	2	no	—
189	C₂²⋊Q₈	—	26	1	no	—
190	C₄×Q₈	—	13	2	no	—
191	C₄×Q₈	—	26	1	no	—
192	#117	—	26	1	no	—
193	C₂×C₄⋊C₄	—	26	2	no	—
194	C₂×C₄⋊C₄	—	26	2	no	—
195	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
196	#107	—	26	2	no	—
197	#107	—	26	2	no	—
198	#107	—	26	2	no	—
199	#107	—	26	2	no	—
200	C₄⋊C₈	—	13	2	no	—
201	C₄⋊C₈	—	26	2	no	—
202	C₄⋊C₈	—	26	2	no	—
203	C₄⋊C₈	—	26	1	no	—
204	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
205	#103	—	26	8	no	—
206	C₂²⋊C₈	—	2	2	no	—
207	#97	—	26	2	no	—
208	C₄×C₈	—	2	2	no	—
209	C₄×C₈	—	26	2	no	—
210	C₄×C₈	D₅₂	1	1	yes	characteristic
211	C₄×C₈	D₅₂	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₂×C₁₃	—	2	2	no	—
215	C₂×C₁₃	C₂×#462	1	1	yes	characteristic
216	C₂×C₁₃	#3058	1	1	yes	characteristic
217	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
218	C₂×Q₈	—	13	2	no	—
219	C₂×Q₈	—	26	1	no	—
220	C₂×Q₈	—	26	4	no	—
221	C₂×Q₈	—	52	2	no	—
222	C₂×Q₈	—	104	4	no	—
223	C₂²×C₄	—	52	2	no	—
224	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
225	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
226	C₂²×C₄	#6131	1	1	yes	characteristic
227	Q₁₆	—	104	8	no	—
228	M₁₆	—	52	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	26	2	no	—
232	C₂×C₈	Dic₅₂	1	4	yes	complemented
233	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
234	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
235	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
236	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
237	C₄⋊C₄	—	13	2	no	—
238	C₄⋊C₄	—	26	1	no	—
239	C₄⋊C₄	—	26	1	no	—
240	C₄⋊C₄	—	26	2	no	—
241	C₄⋊C₄	—	26	1	no	—
242	C₄⋊C₄	—	26	2	no	—
243	C₄⋊C₄	—	26	4	no	—
244	C₄⋊C₄	—	26	4	no	—
245	C₄⋊C₄	—	52	2	no	—
246	C₄⋊C₄	—	52	2	no	—
247	C₂²⋊C₄	—	26	2	no	—
248	C₄²	—	26	2	no	—
249	C₄²	C₂×D₅₂	1	1	yes	characteristic
250	C₄²	D₄×D₁₃	1	1	yes	characteristic
251	C₄²	#6131	1	1	yes	characteristic
252	C₄²	C₂³×D₁₃	1	1	yes	characteristic
253	C₁₃	#19208	1	1	yes	characteristic, complemented, Sylow
254	C₂³	#88149	1	1	yes	characteristic
255	Q₈	—	26	4	no	—
256	Q₈	—	52	2	no	—
257	Q₈	—	52	8	no	—
258	Q₈	—	104	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	26	2	no	—
262	C₂×C₄	—	26	2	no	—
263	C₂×C₄	—	52	4	no	—
264	C₂×C₄	—	104	4	no	—
265	C₂×C₄	#88149	1	1	yes	characteristic
266	C₂×C₄	#88149	1	1	yes	characteristic
267	C₂×C₄	#88149	1	1	yes	characteristic
268	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
269	C₂×C₄	#88163	1	2	yes	—
270	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
271	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
272	C₂×C₄	C₂×#6131	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	4	4	no	—
276	C₈	—	52	2	no	—
277	C₈	C₂×Dic₅₂	1	2	yes	—
278	C₈	C₂×Dic₅₂	1	2	yes	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	#11718510	1	2	yes	—
282	C₂²	C₂×#88149	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	52	2	no	—
286	C₄	—	52	2	no	—
287	C₄	—	104	4	no	—
288	C₄	#11718510	1	2	yes	—
289	C₄	#11718557	1	2	yes	—
290	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
291	C₄	C₂×#88163	1	1	yes	characteristic
292	C₄	#11719474	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	C₂×#11718510	1	1	yes	characteristic
295	C₂	#13124485	1	1	yes	characteristic
296	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	#19208
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. #13114355	C₂	semidirect
5. #13114231	C₂	semidirect
82. C₂×C₈×C₁₃	Q₁₆	semidirect
232. C₂×C₈	Dic₅₂	semidirect
253. C₁₃	#19208	semidirect

Maximal subgroups

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

subgroup	index
#13125786	2
C₂×#11718343	2
#13123770	2
#13114355	2
#13114231	2
#13113520	2
C₁₃×#1564	2
#19208	13