Subgroups — #15834745

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125415	C₂	1	1	yes	maximal, characteristic
2	#13123582	C₂	1	2	yes	maximal, complemented
3	#13113751	C₂	1	2	yes	maximal, complemented
4	#13113702	C₂	1	4	yes	maximal, complemented
5	#13113661	C₂	1	2	yes	maximal, complemented
6	C₂×#11718049	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113236	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88161	—	2	2	no	—
10	#11719182	C₂²	1	2	yes	—
11	C₁₃×#431	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718790	—	2	2	no	—
15	#11718776	C₄	1	2	yes	—
16	#11718776	C₄	1	2	yes	—
17	C₂×C₄⋊#1055	C₂²	1	2	yes	—
18	#11718367	—	2	4	no	—
19	#11718356	C₄	1	1	yes	characteristic
20	#11718355	C₄	1	2	yes	—
21	C₄×D₁₀₄	C₂²	1	2	yes	—
22	C₄×D₁₀₄	C₂²	1	2	yes	—
23	#11718349	C₂²	1	4	yes	—
24	C₈×D₅₂	C₂²	1	4	yes	—
25	#11718343	C₄	1	1	yes	characteristic
26	C₁₃×#322	C₂²	1	2	yes	—
27	C₁₃×#322	C₂²	1	2	yes	—
28	#11718051	C₂²	1	4	yes	—
29	#11718051	C₂²	1	4	yes	—
30	#11718049	C₂²	1	1	yes	characteristic
31	#11718049	C₂²	1	2	yes	—
32	#11718049	C₂²	1	1	yes	characteristic
33	C₂×#6127	—	2	2	no	—
34	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	#88184	—	2	2	no	—
39	#88184	C₂×C₄	1	4	yes	—
40	C₂²×#1055	—	2	2	no	—
41	#88161	—	4	8	no	—
42	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
43	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
44	C₂×#6094	—	2	4	no	—
45	C₂×C₈×D₁₃	—	2	4	no	—
46	#88129	—	2	2	no	—
47	#88128	C₂×C₄	1	2	yes	—
48	C₄×D₅₂	—	2	2	no	—
49	C₄×D₅₂	C₂³	1	4	yes	—
50	#88123	C₂×C₄	1	2	yes	—
51	C₁₃×C₄⋊C₈	—	2	1	no	—
52	C₁₃×C₂²⋊C₈	—	2	1	no	—
53	C₁₃×#97	D₄	1	2	yes	—
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	2	yes	—
60	#88061	C₂×C₄	1	4	yes	—
61	#88061	C₂×C₄	1	4	yes	—
62	#88060	—	2	4	no	—
63	#88060	—	2	4	no	—
64	#88058	C₂×C₄	1	2	yes	—
65	#88058	C₂×C₄	1	2	yes	—
66	C₄⋊#1055	—	2	2	no	—
67	C₄⋊#1055	C₂³	1	2	yes	—
68	C₄⋊#1055	C₂³	1	2	yes	—
69	#18073	—	13	1	no	maximal, Sylow
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
73	C₂×C₂²⋊D₁₃	—	2	4	no	—
74	#6127	—	4	4	no	—
75	#6127	—	4	4	no	—
76	C₂×D₅₂	—	2	2	no	—
77	C₂×D₅₂	—	2	2	no	—
78	C₂×D₅₂	C₂²×C₄	1	4	yes	—
79	C₂×C₄×D₁₃	—	2	4	no	—
80	C₂×Dic₂₆	—	2	2	no	—
81	C₁₃×M₁₆	—	2	4	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	4	1	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
90	C₄²×C₁₃	C₂²×C₄	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	#6103	—	2	4	no	—
95	#6103	—	2	4	no	—
96	C₄⋊Dic₁₃	—	2	2	no	—
97	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	4	4	no	—
101	D₁₀₄	—	2	4	no	—
102	D₁₀₄	—	4	2	no	—
103	#6094	—	4	8	no	—
104	C₈×D₁₃	—	4	8	no	—
105	#3169	—	13	1	no	—
106	#2935	—	13	2	no	—
107	#1594	—	13	2	no	—
108	#1586	—	13	4	no	—
109	#1580	—	13	2	no	—
110	C₂×#334	—	13	1	no	—
111	#1480	—	13	2	no	—
112	#1451	D₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂²×D₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂²⋊D₁₃	—	4	8	no	—
126	C₂×Dic₁₃	—	2	4	no	—
127	D₅₂	—	2	4	no	—
128	D₅₂	—	2	4	no	—
129	D₅₂	—	4	4	no	—
130	D₅₂	—	8	2	no	—
131	C₄×D₁₃	—	4	4	no	—
132	C₄×D₁₃	—	4	4	no	—
133	Dic₂₆	—	4	4	no	—
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	—	2	1	no	—
137	C₈×C₁₃	—	2	4	no	—
138	C₈×C₁₃	—	4	2	no	—
139	C₈×C₁₃	#131	1	2	yes	complemented
140	#1055	—	4	2	no	—
141	#1055	—	4	2	no	—
142	C₂×#131	—	26	2	no	—
143	#532	—	13	2	no	—
144	#500	—	13	1	no	—
145	#493	—	13	1	no	—
146	#486	—	13	2	no	—
147	#466	—	13	2	no	—
148	#466	—	13	2	no	—
149	C₄×D₈	—	13	2	no	—
150	C₄×D₈	—	13	2	no	—
151	#436	—	13	4	no	—
152	#435	—	26	4	no	—
153	C₈×D₄	—	13	4	no	—
154	#431	C₂×D₁₃	1	1	yes	characteristic
155	C₂×C₄⋊C₈	—	13	2	no	—
156	#408	—	26	2	no	—
157	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
159	#334	—	13	1	no	—
160	#334	—	13	2	no	—
161	#334	—	13	1	no	—
162	D₄⋊C₈	—	13	4	no	—
163	D₄⋊C₈	—	13	4	no	—
164	#322	C₂×D₁₃	1	2	yes	—
165	#322	C₂×D₁₃	1	2	yes	—
166	C₂²×C₁₃	—	2	1	no	—
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	#442	1	2	yes	—
169	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×D₁₃	—	4	4	no	—
171	C₂×D₁₃	—	4	4	no	—
172	C₂×D₁₃	—	4	4	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	#442	1	2	yes	—
177	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
178	C₄×C₁₃	#568	1	1	yes	characteristic
179	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
180	Dic₁₃	—	4	4	no	—
181	C₂×#41	—	26	2	no	—
182	C₂×D₈	—	13	2	no	—
183	C₂×D₈	—	13	2	no	—
184	#131	—	52	8	no	—
185	C₂×M₁₆	—	26	4	no	—
186	C₂×M₁₆	D₅₂	1	2	yes	—
187	C₂²×C₈	—	26	2	no	—
188	C₂²×C₈	—	26	4	no	—
189	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
190	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
191	C₄⋊Q₈	—	13	2	no	—
192	#127	—	13	2	no	—
193	#124	—	26	2	no	—
194	#121	—	13	4	no	—
195	#121	—	26	2	no	—
196	C₄×D₄	—	13	4	no	—
197	C₄×D₄	—	26	2	no	—
198	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
199	#107	—	13	2	no	—
200	#107	—	13	2	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	13	2	no	—
203	C₄⋊C₈	—	13	2	no	—
204	C₄⋊C₈	—	26	2	no	—
205	#102	—	13	4	no	—
206	#102	—	13	4	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	C₂²⋊C₈	—	26	4	no	—
209	C₂²⋊C₈	—	26	4	no	—
210	#97	D₅₂	1	2	yes	—
211	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
212	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₁₃	1	2	yes	—
214	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₁₃	1	2	yes	—
217	C₂×C₁₃	—	2	2	no	—
218	C₂×C₁₃	#2935	1	1	yes	characteristic
219	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
220	C₂×C₁₃	#2957	1	1	yes	characteristic
221	D₁₃	—	8	4	no	—
222	#41	—	52	4	no	—
223	#41	—	52	4	no	—
224	C₂×Q₈	—	26	2	no	—
225	C₂×D₄	—	13	4	no	—
226	C₂×D₄	—	26	4	no	—
227	C₂×D₄	—	26	2	no	—
228	C₂×D₄	—	26	2	no	—
229	C₂²×C₄	—	26	4	no	—
230	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
232	C₂²×C₄	#6127	1	1	yes	characteristic
233	D₈	—	26	4	no	—
234	D₈	—	52	2	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	52	8	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	26	2	no	—
241	C₂×C₈	—	26	2	no	—
242	C₂×C₈	—	52	8	no	—
243	C₂×C₈	—	52	4	no	—
244	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×D₅₂	1	4	yes	—
249	C₄⋊C₄	—	13	4	no	—
250	C₄⋊C₄	—	26	2	no	—
251	C₂²⋊C₄	—	26	4	no	—
252	C₂²⋊C₄	—	26	4	no	—
253	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
254	C₄²	C₂×D₅₂	1	1	yes	characteristic
255	C₄²	#6127	1	1	yes	characteristic
256	C₄²	C₂³×D₁₃	1	1	yes	characteristic
257	C₁₃	#18073	1	1	yes	characteristic, complemented, Sylow
258	C₂³	—	26	4	no	—
259	C₂³	C₄×D₅₂	1	1	yes	characteristic
260	Q₈	—	52	4	no	—
261	D₄	—	26	4	no	—
262	D₄	—	26	4	no	—
263	D₄	—	52	8	no	—
264	D₄	—	52	4	no	—
265	D₄	—	104	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	26	4	no	—
270	C₂×C₄	—	52	4	no	—
271	C₂×C₄	—	52	4	no	—
272	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
273	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
274	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
275	C₂×C₄	#88162	1	4	yes	—
276	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
278	C₂×C₄	C₂×#6127	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	52	2	no	—
285	C₈	—	52	2	no	—
286	C₈	#88161	1	2	yes	complemented
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	52	4	no	—
290	C₂²	—	52	4	no	—
291	C₂²	—	52	4	no	—
292	C₂²	#11718372	1	2	yes	—
293	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	52	4	no	—
298	C₄	#11718372	1	2	yes	—
299	C₄	C₂×#88161	1	1	yes	characteristic
300	C₄	#11719450	1	1	yes	characteristic
301	C₄	C₂×#88162	1	2	yes	—
302	C₂	—	2	2	no	—
303	C₂	—	104	4	no	—
304	C₂	#13123582	1	1	yes	characteristic
305	C₂	C₂×#11718372	1	1	yes	characteristic
306	C₂	#13123754	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1451	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#13125415	C₂
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1451	D₁₃
hypocenter	C₁₃	#18073
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
2. #13123582	C₂	semidirect
3. #13113751	C₂	semidirect
4. #13113702	C₂	semidirect
5. #13113661	C₂	semidirect
6. C₂×#11718049	C₂	semidirect
7. #13113236	C₂	semidirect
8. C₁₃×#1451	C₂	semidirect
112. #1451	D₁₃	semidirect
139. C₈×C₁₃	#131	semidirect
257. C₁₃	#18073	semidirect
286. C₈	#88161	semidirect

Maximal subgroups

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

subgroup	index
#13125415	2
#13123582	2
#13113751	2
#13113702	2
#13113661	2
C₂×#11718049	2
#13113236	2
C₁₃×#1451	2
#18073	13