Subgroups — #15753923

Order 3328.

452 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	#13123549	C₂	1	1	yes	maximal, characteristic
2	#13113496	C₂	1	1	yes	maximal, characteristic
3	#13113427	C₂	1	1	yes	maximal, characteristic
4	C₈×#6099	C₂	1	1	yes	maximal, characteristic, complemented
5	#13112727	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112726	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88155	C₄	1	2	yes	—
9	#11719182	C₂²	1	1	yes	characteristic
10	C₁₃×#431	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718771	C₄	1	2	yes	—
14	#11718586	C₂²	1	1	yes	characteristic
15	#11718585	C₂²	1	1	yes	characteristic
16	C₄×#6099	C₂²	1	1	yes	characteristic
17	#11718349	—	2	1	no	—
18	#11718349	—	2	1	no	—
19	C₈×D₅₂	—	2	1	no	—
20	C₈×D₅₂	—	2	1	no	—
21	C₁₃×#322	—	2	1	no	—
22	C₈²×C₁₃	—	2	1	no	—
23	#11718077	C₄	1	2	yes	—
24	#11718073	C₄	1	2	yes	—
25	#11718056	C₄	1	4	yes	—
26	#11718046	—	2	1	no	—
27	C₈×#1055	—	2	1	no	—
28	C₂×#6127	C₂×C₄	1	2	yes	—
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	#88184	—	2	1	no	—
34	#88184	—	2	1	no	—
35	C₂×#6099	—	2	1	no	—
36	C₂×#6099	C₂×C₄	1	2	yes	—
37	#88155	—	4	4	no	—
38	C₂×#6094	—	2	2	no	—
39	C₂×C₈×D₁₃	—	2	2	no	—
40	#88129	C₂×C₄	1	2	yes	—
41	#88128	C₂×C₄	1	1	yes	characteristic
42	C₄×D₅₂	—	2	1	no	—
43	C₄×D₅₂	—	2	1	no	—
44	#88123	C₂×C₄	1	1	yes	characteristic
45	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₁₃×C₂²⋊C₈	—	2	1	no	—
47	C₁₃×#97	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₃	—	2	1	no	—
49	C₄×C₈×C₁₃	—	2	1	no	—
50	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
54	C₂²⋊#1055	—	2	1	no	—
55	#88060	—	2	2	no	—
56	#88060	—	2	2	no	—
57	C₄⋊#1055	—	2	1	no	—
58	C₄⋊#1055	—	2	1	no	—
59	C₄⋊#1055	C₂×C₄	1	2	yes	—
60	#88043	C₂×C₄	1	2	yes	—
61	C₄×#1055	—	2	1	no	—
62	C₄×#1055	—	2	1	no	—
63	#8241	—	13	1	no	maximal, Sylow
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₃	—	2	2	no	—
68	#6127	—	4	2	no	—
69	#6127	—	4	2	no	—
70	C₂×D₅₂	—	2	1	no	—
71	C₂×D₅₂	—	2	1	no	—
72	C₂×D₅₂	C₄²	1	2	yes	—
73	C₂×C₄×D₁₃	—	2	2	no	—
74	C₂×Dic₂₆	C₄²	1	2	yes	—
75	C₁₃×M₁₆	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	4	1	no	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	#6103	—	2	2	no	—
89	#6103	—	2	2	no	—
90	C₄⋊Dic₁₃	—	2	1	no	—
91	C₄⋊Dic₁₃	—	2	1	no	—
92	#6099	—	2	4	no	—
93	#6099	—	4	2	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	—	2	2	no	—
97	#6094	—	4	4	no	—
98	C₈×D₁₃	—	4	4	no	—
99	#2931	—	13	1	no	—
100	#1543	—	13	1	no	—
101	#1525	—	13	1	no	—
102	C₈×M₁₆	—	13	1	no	—
103	C₈×M₁₆	D₁₃	1	1	yes	characteristic, complemented
104	#1299	—	13	1	no	—
105	#1299	—	13	1	no	—
106	C₂³×C₁₃	#117	1	1	yes	characteristic
107	C₂²×D₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂²⋊D₁₃	—	4	4	no	—
119	C₂×Dic₁₃	—	2	2	no	—
120	D₅₂	—	4	2	no	—
121	D₅₂	—	4	2	no	—
122	D₅₂	—	4	2	no	—
123	C₄×D₁₃	—	4	2	no	—
124	C₄×D₁₃	—	4	2	no	—
125	Dic₂₆	—	4	2	no	—
126	C₈×C₁₃	—	2	4	no	—
127	C₈×C₁₃	—	2	1	no	—
128	C₈×C₁₃	—	2	1	no	—
129	C₈×C₁₃	—	2	1	no	—
130	C₈×C₁₃	—	4	2	no	—
131	C₈×C₁₃	#100	1	2	yes	—
132	#1055	—	2	4	no	—
133	#1055	—	4	2	no	—
134	#1055	—	4	2	no	—
135	C₂×#131	—	13	2	no	—
136	#532	—	13	1	no	—
137	#436	—	26	1	no	—
138	#436	—	26	1	no	—
139	C₈×D₄	—	26	1	no	—
140	C₈×D₄	—	26	1	no	—
141	#431	—	13	1	no	—
142	#431	C₂×D₁₃	1	1	yes	characteristic
143	#423	—	13	1	no	—
144	#408	—	13	2	no	—
145	C₄×M₁₆	—	13	1	no	—
146	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
148	#333	—	13	2	no	—
149	#331	—	13	2	no	—
150	#329	—	13	4	no	—
151	#322	—	2	1	no	—
152	#322	—	26	1	no	—
153	C₈²	—	2	1	no	—
154	C₈²	—	26	1	no	—
155	C₂²×C₁₃	—	2	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×D₁₃	—	4	2	no	—
160	C₂×D₁₃	—	4	2	no	—
161	C₂×D₁₃	—	4	2	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	C₂×#100	1	1	yes	characteristic
166	C₄×C₁₃	#413	1	1	yes	characteristic
167	C₄×C₁₃	#443	1	2	yes	—
168	C₄×C₁₃	#443	1	2	yes	—
169	Dic₁₃	—	4	2	no	—
170	C₂×#41	—	13	2	no	—
171	#131	—	52	4	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	13	2	no	—
174	C₂×M₁₆	—	26	1	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂²×C₈	—	26	2	no	—
177	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
179	C₄⋊Q₈	—	13	1	no	—
180	#127	—	13	1	no	—
181	#124	—	13	2	no	—
182	#121	—	26	1	no	—
183	#121	—	26	1	no	—
184	C₄×D₄	—	26	1	no	—
185	C₄×D₄	—	26	1	no	—
186	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
187	C₄⋊C₈	—	13	2	no	—
188	C₄⋊C₈	—	26	1	no	—
189	C₄⋊C₈	—	26	1	no	—
190	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
191	C₂²⋊C₈	—	2	1	no	—
192	C₂²⋊C₈	—	26	1	no	—
193	C₂²⋊C₈	—	26	2	no	—
194	C₂²⋊C₈	—	26	2	no	—
195	#97	—	13	2	no	—
196	#97	C₄×D₁₃	1	2	yes	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	26	1	no	—
200	C₄×C₈	—	26	1	no	—
201	C₄×C₈	D₅₂	1	1	yes	characteristic
202	C₄×C₈	D₅₂	1	1	yes	characteristic
203	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
204	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
205	C₂×C₁₃	—	4	1	no	—
206	C₂×C₁₃	C₄×#100	1	1	yes	characteristic
207	C₂×C₁₃	#1764	1	1	yes	characteristic
208	C₂×C₁₃	#1764	1	1	yes	characteristic
209	D₁₃	—	8	2	no	—
210	#41	—	52	2	no	—
211	#41	—	52	2	no	—
212	C₂×Q₈	—	13	2	no	—
213	C₂×D₄	—	13	2	no	—
214	C₂×D₄	—	26	2	no	—
215	C₂×D₄	—	26	1	no	—
216	C₂×D₄	—	26	1	no	—
217	C₂²×C₄	—	26	2	no	—
218	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
219	C₂²×C₄	#6127	1	1	yes	characteristic
220	C₂²×C₄	#6127	1	1	yes	characteristic
221	M₁₆	—	2	4	no	—
222	M₁₆	—	26	4	no	—
223	M₁₆	—	52	4	no	—
224	M₁₆	—	52	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	26	2	no	—
230	C₂×C₈	—	26	2	no	—
231	C₂×C₈	—	26	2	no	—
232	C₂×C₈	—	52	4	no	—
233	C₂×C₈	#6103	1	1	yes	characteristic
234	C₂×C₈	#6103	1	1	yes	characteristic
235	C₂×C₈	#6103	1	1	yes	characteristic
236	C₂×C₈	#6103	1	1	yes	characteristic
237	C₄⋊C₄	—	26	1	no	—
238	C₄⋊C₄	—	26	1	no	—
239	C₂²⋊C₄	—	26	2	no	—
240	C₂²⋊C₄	—	26	2	no	—
241	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
242	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
243	C₄²	C₂×D₅₂	1	1	yes	characteristic
244	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
245	C₁₃	#8241	1	1	yes	characteristic, complemented, Sylow
246	C₂³	—	26	2	no	—
247	C₂³	#88126	1	1	yes	characteristic
248	Q₈	—	52	2	no	—
249	D₄	—	52	2	no	—
250	D₄	—	52	4	no	—
251	D₄	—	52	2	no	—
252	D₄	—	52	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	—	52	2	no	—
259	C₂×C₄	—	52	2	no	—
260	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
261	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
262	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
263	C₂×C₄	C₂×#6103	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	4	2	no	—
271	C₈	—	26	4	no	—
272	C₈	—	52	2	no	—
273	C₈	—	52	2	no	—
274	C₈	#88063	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	52	2	no	—
279	C₂²	—	52	2	no	—
280	C₂²	—	52	2	no	—
281	C₂²	C₄×#6103	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	52	2	no	—
286	C₄	#11718361	1	2	yes	—
287	C₄	C₂×#88063	1	1	yes	characteristic
288	C₄	#11718793	1	1	yes	characteristic
289	C₄	#11718805	1	2	yes	—
290	C₂	—	4	1	no	—
291	C₂	—	104	2	no	—
292	C₂	C₄×#88063	1	1	yes	characteristic
293	C₂	#13114673	1	1	yes	characteristic
294	C₂	#13114675	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#6103
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₄×C₂²⋊C₄
prime radical	#11719182	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#8241
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₈×#6099	C₂	semidirect
5. #13112727	C₂	semidirect
6. #13112726	C₂	semidirect
7. C₈×C₁₃×M₁₆	C₂	semidirect
103. C₈×M₁₆	D₁₃	semidirect
245. C₁₃	#8241	semidirect

Maximal subgroups

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

subgroup	index
#13123549	2
#13113496	2
#13113427	2
C₈×#6099	2
#13112727	2
#13112726	2
C₈×C₁₃×M₁₆	2
#8241	13