Subgroups — #15845998

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125959	C₂	1	2	yes	maximal
2	C₂×#11718360	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123549	C₂	1	1	yes	maximal, characteristic
4	#13114326	C₂	1	4	yes	maximal, complemented
5	#13114202	C₂	1	4	yes	maximal, complemented
6	#13113507	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₅₂	—	2	1	no	—
9	C₂²×Dic₅₂	—	2	1	no	—
10	C₂×#88155	—	2	1	no	—
11	#11719182	C₂²	1	1	yes	characteristic
12	C₂×#88123	C₂²	1	2	yes	—
13	C₁₃×#431	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718771	—	2	1	no	—
16	#11718585	C₂²	1	1	yes	characteristic
17	#11718507	C₂²	1	8	yes	—
18	#11718397	—	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	#11718355	—	2	2	no	—
24	#11718349	—	2	1	no	—
25	#11718349	C₂²	1	2	yes	—
26	C₈×D₅₂	—	2	1	no	—
27	C₈×D₅₂	C₂²	1	2	yes	—
28	#11718343	—	2	2	no	—
29	C₁₃×#335	C₂²	1	2	yes	—
30	C₁₃×#334	C₂²	1	2	yes	—
31	#11718078	C₂²	1	8	yes	—
32	C₂×#6127	—	2	1	no	—
33	C₂²×Dic₂₆	—	2	2	no	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	#88184	—	2	1	no	—
38	#88184	D₄	1	2	yes	—
39	C₂×C₄⋊Dic₁₃	—	2	2	no	—
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
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	#88155	—	4	4	no	—
49	C₂×#6094	—	2	2	no	—
50	C₂×C₈×D₁₃	—	2	2	no	—
51	#88129	—	2	1	no	—
52	#88128	D₄	1	1	yes	characteristic
53	C₄×D₅₂	—	2	1	no	—
54	C₄×D₅₂	C₂³	1	2	yes	—
55	#88123	—	2	2	no	—
56	#88123	—	2	2	no	—
57	#88123	—	2	2	no	—
58	#88123	D₄	1	1	yes	characteristic
59	#88123	C₂³	1	4	yes	—
60	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
61	C₁₃×C₂²⋊C₈	—	2	2	no	—
62	C₄×C₈×C₁₃	—	2	2	no	—
63	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	#88061	—	2	8	no	—
68	#88060	—	2	2	no	—
69	#88060	—	2	2	no	—
70	#88058	—	2	8	no	—
71	C₄⋊#1055	—	2	1	no	—
72	C₄⋊#1055	C₂³	1	2	yes	—
73	#19188	—	13	1	no	maximal, Sylow
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
77	C₂×C₂²⋊D₁₃	—	2	2	no	—
78	C₂²×Dic₁₃	—	4	2	no	—
79	#6127	—	4	2	no	—
80	#6127	—	4	2	no	—
81	C₂×D₅₂	—	2	1	no	—
82	C₂×D₅₂	—	2	1	no	—
83	C₂×D₅₂	C₂×D₄	1	2	yes	—
84	C₂×C₄×D₁₃	—	2	2	no	—
85	C₂×Dic₂₆	—	2	1	no	—
86	C₂×Dic₂₆	—	2	4	no	—
87	C₂×Dic₂₆	—	4	4	no	—
88	C₂×Dic₂₆	—	4	4	no	—
89	C₂×Dic₂₆	—	4	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	4	no	—
92	C₂×C₈×C₁₃	D₈	1	4	yes	—
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
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	#6103	—	2	2	no	—
102	#6103	—	2	2	no	—
103	C₄⋊Dic₁₃	—	2	1	no	—
104	C₄⋊Dic₁₃	—	2	4	no	—
105	C₄⋊Dic₁₃	—	2	4	no	—
106	C₄⋊Dic₁₃	—	4	2	no	—
107	C₄⋊Dic₁₃	—	4	2	no	—
108	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
109	#6099	—	4	2	no	—
110	C₂×#1055	—	2	2	no	—
111	Dic₅₂	—	4	4	no	—
112	Dic₅₂	—	4	4	no	—
113	Dic₅₂	—	4	4	no	—
114	Dic₅₂	—	4	4	no	—
115	#6094	—	4	4	no	—
116	C₈×D₁₃	—	4	4	no	—
117	#3226	—	13	2	no	—
118	C₂×#494	—	13	1	no	—
119	#2931	—	13	1	no	—
120	#1689	—	13	4	no	—
121	#1669	—	13	4	no	—
122	#1564	D₁₃	1	1	yes	characteristic, complemented
123	#1546	—	13	2	no	—
124	C₂³×C₁₃	#121	1	1	yes	characteristic
125	C₂²×D₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
136	C₂²⋊D₁₃	—	4	4	no	—
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	4	4	no	—
139	C₂×Dic₁₃	—	8	4	no	—
140	D₅₂	—	2	4	no	—
141	D₅₂	—	4	2	no	—
142	D₅₂	—	4	2	no	—
143	C₄×D₁₃	—	4	2	no	—
144	C₄×D₁₃	—	4	2	no	—
145	Dic₂₆	—	4	2	no	—
146	Dic₂₆	—	4	4	no	—
147	Dic₂₆	—	4	4	no	—
148	Dic₂₆	—	8	4	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	—	4	4	no	—
152	C₈×C₁₃	C₂×D₈	1	2	yes	—
153	C₈×C₁₃	C₂×D₈	1	2	yes	—
154	#1055	—	4	2	no	—
155	C₂²×Q₁₆	—	26	1	no	—
156	C₂²×Q₁₆	—	26	1	no	—
157	C₂×#131	—	26	1	no	—
158	#532	—	13	1	no	—
159	C₂×C₄⋊Q₈	—	13	2	no	—
160	#500	—	26	2	no	—
161	#494	—	13	2	no	—
162	#494	—	13	2	no	—
163	#494	—	26	1	no	—
164	#494	—	26	1	no	—
165	#486	—	26	2	no	—
166	#480	—	26	4	no	—
167	#474	—	13	8	no	—
168	#436	—	13	2	no	—
169	#436	—	26	1	no	—
170	C₈×D₄	—	13	2	no	—
171	C₈×D₄	—	26	1	no	—
172	#431	C₂×D₁₃	1	2	yes	—
173	#423	—	13	1	no	—
174	#408	—	26	1	no	—
175	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
176	#335	C₂×D₁₃	1	2	yes	—
177	#334	C₂×D₁₃	1	2	yes	—
178	#332	—	13	8	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	#459	1	2	yes	—
182	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
183	C₂×D₁₃	—	4	2	no	—
184	C₂×D₁₃	—	4	2	no	—
185	C₂×D₁₃	—	4	2	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	#459	1	2	yes	—
189	C₄×C₁₃	#473	1	2	yes	—
190	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
191	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₄×C₁₃	#578	1	2	yes	—
193	Dic₁₃	—	4	2	no	—
194	Dic₁₃	—	8	4	no	—
195	C₂×#41	—	26	1	no	—
196	C₂²×Q₈	—	26	2	no	—
197	C₂×Q₁₆	—	26	2	no	—
198	C₂×Q₁₆	—	26	2	no	—
199	C₂×Q₁₆	—	26	2	no	—
200	C₂×Q₁₆	—	26	2	no	—
201	C₂×Q₁₆	—	52	4	no	—
202	C₂×Q₁₆	—	52	4	no	—
203	#131	—	52	4	no	—
204	C₂×M₁₆	—	26	1	no	—
205	C₂×M₁₆	—	26	2	no	—
206	C₂²×C₈	—	26	2	no	—
207	C₂²×C₈	D₅₂	1	1	yes	characteristic
208	C₂²×C₈	D₅₂	1	1	yes	characteristic
209	C₄⋊Q₈	—	13	1	no	—
210	C₄⋊Q₈	—	13	4	no	—
211	C₄⋊Q₈	—	26	2	no	—
212	C₄⋊Q₈	—	26	2	no	—
213	C₄⋊Q₈	—	26	2	no	—
214	#127	—	13	1	no	—
215	#124	—	26	1	no	—
216	#121	—	13	2	no	—
217	#121	—	26	1	no	—
218	C₄×D₄	—	13	2	no	—
219	C₄×D₄	—	26	1	no	—
220	C₂×C₄⋊C₄	—	26	2	no	—
221	C₂×C₄⋊C₄	—	26	2	no	—
222	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
223	#107	—	26	8	no	—
224	C₄⋊C₈	—	13	2	no	—
225	C₄⋊C₈	—	26	1	no	—
226	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
227	#102	—	26	8	no	—
228	C₂²⋊C₈	—	2	2	no	—
229	C₂²⋊C₈	—	26	2	no	—
230	C₂²⋊C₈	—	26	2	no	—
231	C₄×C₈	—	2	2	no	—
232	C₄×C₈	D₅₂	1	1	yes	characteristic
233	C₄×C₈	D₅₂	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
236	C₂×C₁₃	—	2	2	no	—
237	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
238	C₂×C₁₃	#3059	1	1	yes	characteristic
239	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
240	D₁₃	—	8	2	no	—
241	#41	—	52	2	no	—
242	#41	—	52	2	no	—
243	C₂×Q₈	—	26	1	no	—
244	C₂×Q₈	—	26	4	no	—
245	C₂×Q₈	—	52	2	no	—
246	C₂×Q₈	—	52	4	no	—
247	C₂×Q₈	—	52	4	no	—
248	C₂×D₄	—	13	2	no	—
249	C₂×D₄	—	26	2	no	—
250	C₂×D₄	—	26	1	no	—
251	C₂×D₄	—	26	1	no	—
252	C₂²×C₄	—	26	2	no	—
253	C₂²×C₄	—	52	2	no	—
254	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
256	C₂²×C₄	#6131	1	1	yes	characteristic
257	Q₁₆	—	52	4	no	—
258	Q₁₆	—	52	4	no	—
259	Q₁₆	—	52	4	no	—
260	Q₁₆	—	52	4	no	—
261	M₁₆	—	52	2	no	—
262	M₁₆	—	52	4	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	26	2	no	—
266	C₂×C₈	—	52	4	no	—
267	C₂×C₈	D₁₀₄	1	4	yes	—
268	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
269	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
270	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
271	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
272	C₄⋊C₄	—	13	2	no	—
273	C₄⋊C₄	—	26	1	no	—
274	C₄⋊C₄	—	26	4	no	—
275	C₄⋊C₄	—	26	4	no	—
276	C₄⋊C₄	—	52	2	no	—
277	C₄⋊C₄	—	52	2	no	—
278	C₂²⋊C₄	—	26	2	no	—
279	C₂²⋊C₄	—	26	2	no	—
280	C₄²	C₂×D₅₂	1	1	yes	characteristic
281	C₄²	D₄×D₁₃	1	1	yes	characteristic
282	C₄²	#6131	1	1	yes	characteristic
283	C₄²	C₂³×D₁₃	1	1	yes	characteristic
284	C₁₃	#19188	1	1	yes	characteristic, complemented, Sylow
285	C₂³	—	26	2	no	—
286	C₂³	#88149	1	1	yes	characteristic
287	Q₈	—	52	2	no	—
288	Q₈	—	52	4	no	—
289	Q₈	—	52	4	no	—
290	Q₈	—	104	4	no	—
291	D₄	—	26	4	no	—
292	D₄	—	52	4	no	—
293	D₄	—	52	2	no	—
294	D₄	—	52	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	26	2	no	—
298	C₂×C₄	—	52	2	no	—
299	C₂×C₄	—	52	2	no	—
300	C₂×C₄	—	52	4	no	—
301	C₂×C₄	—	104	4	no	—
302	C₂×C₄	#88149	1	1	yes	characteristic
303	C₂×C₄	#88149	1	1	yes	characteristic
304	C₂×C₄	#88149	1	1	yes	characteristic
305	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
306	C₂×C₄	#88162	1	2	yes	—
307	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂×#6131	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	—	52	2	no	—
314	C₈	C₂×D₁₀₄	1	2	yes	—
315	C₈	C₂×D₁₀₄	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	52	2	no	—
319	C₂²	—	52	2	no	—
320	C₂²	—	52	2	no	—
321	C₂²	#11718504	1	2	yes	—
322	C₂²	C₂×#88149	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	52	2	no	—
326	C₄	—	104	4	no	—
327	C₄	#11718504	1	2	yes	—
328	C₄	#11718557	1	2	yes	—
329	C₄	C₂²×D₁₀₄	1	1	yes	characteristic
330	C₄	C₂×#88162	1	1	yes	characteristic
331	C₄	#11719476	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	104	2	no	—
334	C₂	C₂×#11718504	1	1	yes	characteristic
335	C₂	#13124492	1	1	yes	characteristic
336	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	#11719182	C₂²
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#19188
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
2. C₂×#11718360	C₂	semidirect
4. #13114326	C₂	semidirect
5. #13114202	C₂	semidirect
6. #13113507	C₂	semidirect
7. C₁₃×#1564	C₂	semidirect
122. #1564	D₁₃	semidirect
284. C₁₃	#19188	semidirect

Maximal subgroups

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

subgroup	index
#13125959	2
C₂×#11718360	2
#13123549	2
#13114326	2
#13114202	2
#13113507	2
C₁₃×#1564	2
#19188	13