Subgroups — #15834676

Order 3328.

722 conjugacy classes of subgroups.

The subgroup lattice has 320 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	#13123579	C₂	1	2	yes	maximal, complemented
3	#13113720	C₂	1	2	yes	maximal, complemented
4	#13113664	C₂	1	4	yes	maximal, complemented
5	#13113630	C₂	1	2	yes	maximal, complemented
6	#13113234	C₂	1	2	yes	maximal, complemented
7	C₈×#6099	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88155	—	2	2	no	—
10	#11719182	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718776	C₄	1	2	yes	—
14	#11718776	C₄	1	2	yes	—
15	#11718771	—	2	2	no	—
16	#11718586	C₂²	1	1	yes	characteristic
17	C₄×#6099	C₂²	1	1	yes	characteristic
18	#11718499	C₂²	1	4	yes	—
19	#11718497	—	2	4	no	—
20	#11718495	C₂²	1	4	yes	—
21	#11718356	C₄	1	1	yes	characteristic
22	#11718355	C₄	1	2	yes	—
23	C₄×D₁₀₄	C₂²	1	2	yes	—
24	C₄×D₁₀₄	C₂²	1	2	yes	—
25	#11718343	C₄	1	1	yes	characteristic
26	C₁₃×#335	C₂²	1	1	yes	characteristic
27	C₁₃×#335	C₂²	1	2	yes	—
28	C₁₃×#335	C₂²	1	1	yes	characteristic
29	#11718075	C₂²	1	4	yes	—
30	#11718075	C₂²	1	4	yes	—
31	#11718046	C₂²	1	2	yes	—
32	C₈×#1055	C₂²	1	2	yes	—
33	C₂×#6127	—	2	2	no	—
34	C₂²×C₈×C₁₃	—	2	2	no	—
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	C₂×#6099	D₄	1	1	yes	characteristic
41	C₂×#6099	D₄	1	1	yes	characteristic
42	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
43	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
44	#88155	—	4	8	no	—
45	C₂×#6094	—	2	4	no	—
46	C₂×C₈×D₁₃	—	2	4	no	—
47	#88129	—	2	2	no	—
48	#88128	C₂×C₄	1	2	yes	—
49	C₄×D₅₂	—	2	2	no	—
50	C₄×D₅₂	C₂³	1	4	yes	—
51	#88123	C₂×C₄	1	2	yes	—
52	C₁₃×C₄⋊C₈	—	2	2	no	—
53	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
54	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
55	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₂²⋊#1055	—	2	1	no	—
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	1	no	—
67	#88043	D₄	1	2	yes	—
68	C₄×#1055	C₂³	1	2	yes	—
69	C₄×#1055	C₂³	1	2	yes	—
70	#18066	—	13	1	no	maximal, Sylow
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
74	C₂×C₂²⋊D₁₃	—	2	4	no	—
75	#6127	—	4	4	no	—
76	#6127	—	4	4	no	—
77	C₂×D₅₂	—	2	2	no	—
78	C₂×D₅₂	—	2	2	no	—
79	C₂×D₅₂	C₂²×C₄	1	4	yes	—
80	C₂×C₄×D₁₃	—	2	4	no	—
81	C₂×Dic₂₆	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	4	1	no	—
86	C₂×C₈×C₁₃	—	4	4	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃	#41	1	1	yes	characteristic
94	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
95	#6103	—	2	4	no	—
96	#6103	—	2	4	no	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
99	#6099	—	2	2	no	—
100	#6099	D₈	1	4	yes	complemented
101	C₂×#1055	—	2	2	no	—
102	C₂×#1055	C₂×D₄	1	2	yes	—
103	C₂×#1055	C₂×D₄	1	2	yes	—
104	D₁₀₄	—	2	4	no	—
105	D₁₀₄	—	4	2	no	—
106	#6094	—	4	8	no	—
107	C₈×D₁₃	—	4	8	no	—
108	#3169	—	13	1	no	—
109	#2935	—	13	2	no	—
110	#1589	—	13	2	no	—
111	#1581	—	13	4	no	—
112	C₈×D₈	—	13	2	no	—
113	C₂×#335	D₁₃	1	1	yes	characteristic, complemented
114	#1480	—	13	2	no	—
115	C₈×M₁₆	—	13	1	no	—
116	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₁₃	#135	1	2	yes	—
126	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
129	C₂²⋊D₁₃	—	4	8	no	—
130	C₂×Dic₁₃	—	2	4	no	—
131	D₅₂	—	2	4	no	—
132	D₅₂	—	2	4	no	—
133	D₅₂	—	4	4	no	—
134	D₅₂	—	8	2	no	—
135	C₄×D₁₃	—	4	4	no	—
136	C₄×D₁₃	—	4	4	no	—
137	Dic₂₆	—	4	4	no	—
138	C₈×C₁₃	—	2	1	no	—
139	C₈×C₁₃	—	2	1	no	—
140	C₈×C₁₃	—	2	1	no	—
141	C₈×C₁₃	—	4	2	no	—
142	C₈×C₁₃	—	4	2	no	—
143	C₈×C₁₃	#131	1	2	yes	complemented
144	#1055	—	2	2	no	—
145	#1055	—	4	2	no	—
146	#1055	C₂×D₈	1	4	yes	—
147	C₂×#131	—	26	2	no	—
148	#532	—	13	2	no	—
149	#500	—	13	1	no	—
150	#493	—	13	1	no	—
151	#486	—	13	2	no	—
152	#466	—	13	2	no	—
153	#466	—	13	2	no	—
154	C₄×D₈	—	13	2	no	—
155	C₄×D₈	—	13	2	no	—
156	#436	—	13	4	no	—
157	#435	—	26	4	no	—
158	C₈×D₄	—	13	4	no	—
159	#431	—	13	1	no	—
160	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
161	#408	—	26	2	no	—
162	C₄×M₁₆	—	13	1	no	—
163	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
164	#335	C₂×D₁₃	1	1	yes	characteristic
165	#335	C₂×D₁₃	1	2	yes	—
166	#335	C₂×D₁₃	1	1	yes	characteristic
167	D₄⋊C₈	—	13	4	no	—
168	D₄⋊C₈	—	13	4	no	—
169	#322	—	13	2	no	—
170	C₈²	—	13	2	no	—
171	C₂²×C₁₃	—	2	1	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	C₄×D₈	1	2	yes	—
174	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×D₁₃	—	4	4	no	—
176	C₂×D₁₃	—	4	4	no	—
177	C₂×D₁₃	—	4	4	no	—
178	C₄×C₁₃	—	2	1	no	—
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	C₄×D₈	1	2	yes	—
182	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
183	C₄×C₁₃	#568	1	1	yes	characteristic
184	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
185	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
186	Dic₁₃	—	4	4	no	—
187	C₂×#41	—	26	2	no	—
188	C₂×D₈	—	13	2	no	—
189	C₂×D₈	—	13	2	no	—
190	#131	—	52	8	no	—
191	C₂×M₁₆	—	13	1	no	—
192	C₂×M₁₆	—	13	1	no	—
193	C₂×M₁₆	—	26	4	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	26	4	no	—
196	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
197	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
198	C₄⋊Q₈	—	13	2	no	—
199	#127	—	13	2	no	—
200	#124	—	26	2	no	—
201	#121	—	13	4	no	—
202	#121	—	26	2	no	—
203	C₄×D₄	—	13	4	no	—
204	C₄×D₄	—	26	2	no	—
205	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
206	#107	—	13	2	no	—
207	#107	—	13	2	no	—
208	C₄⋊C₈	—	2	2	no	—
209	C₄⋊C₈	—	26	1	no	—
210	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
211	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
212	#102	—	13	4	no	—
213	#102	—	13	4	no	—
214	C₂²⋊C₈	—	26	4	no	—
215	C₂²⋊C₈	—	26	4	no	—
216	C₂²⋊C₈	—	26	1	no	—
217	#97	—	13	2	no	—
218	C₄×C₈	—	13	2	no	—
219	C₄×C₈	—	13	2	no	—
220	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
221	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×C₁₃	—	2	2	no	—
225	C₂×C₁₃	#2935	1	1	yes	characteristic
226	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
227	C₂×C₁₃	#2957	1	1	yes	characteristic
228	D₁₃	—	8	4	no	—
229	#41	—	52	4	no	—
230	#41	—	52	4	no	—
231	C₂×Q₈	—	26	2	no	—
232	C₂×D₄	—	13	4	no	—
233	C₂×D₄	—	26	4	no	—
234	C₂×D₄	—	26	2	no	—
235	C₂×D₄	—	26	2	no	—
236	C₂²×C₄	—	26	4	no	—
237	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
239	C₂²×C₄	#6131	1	1	yes	characteristic
240	D₈	—	26	4	no	—
241	D₈	—	52	2	no	—
242	M₁₆	—	13	4	no	—
243	M₁₆	—	26	2	no	—
244	M₁₆	—	52	8	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	4	no	—
250	C₂×C₈	—	13	2	no	—
251	C₂×C₈	—	13	2	no	—
252	C₂×C₈	—	26	2	no	—
253	C₂×C₈	—	52	8	no	—
254	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
255	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
257	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
258	C₄⋊C₄	—	13	4	no	—
259	C₄⋊C₄	—	26	2	no	—
260	C₂²⋊C₄	—	26	4	no	—
261	C₂²⋊C₄	—	26	4	no	—
262	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
263	C₄²	D₄×D₁₃	1	1	yes	characteristic
264	C₄²	#6131	1	1	yes	characteristic
265	C₄²	C₂³×D₁₃	1	1	yes	characteristic
266	C₁₃	#18066	1	1	yes	characteristic, complemented, Sylow
267	C₂³	—	26	4	no	—
268	C₂³	#88147	1	1	yes	characteristic
269	Q₈	—	52	4	no	—
270	D₄	—	26	4	no	—
271	D₄	—	26	4	no	—
272	D₄	—	52	8	no	—
273	D₄	—	52	4	no	—
274	D₄	—	104	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	26	4	no	—
279	C₂×C₄	—	52	4	no	—
280	C₂×C₄	—	52	4	no	—
281	C₂×C₄	#88147	1	1	yes	characteristic
282	C₂×C₄	#88147	1	1	yes	characteristic
283	C₂×C₄	#88147	1	1	yes	characteristic
284	C₂×C₄	D₈×D₁₃	1	2	yes	—
285	C₂×C₄	#88173	1	2	yes	—
286	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
287	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#6131	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	4	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	13	4	no	—
295	C₈	—	26	2	no	—
296	C₈	—	52	2	no	—
297	C₈	#88155	1	2	yes	complemented
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	52	4	no	—
301	C₂²	—	52	4	no	—
302	C₂²	—	52	4	no	—
303	C₂²	#11718533	1	2	yes	—
304	C₂²	C₂×#88147	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	52	4	no	—
309	C₄	#11718533	1	2	yes	—
310	C₄	C₂×#88155	1	1	yes	characteristic
311	C₄	#11719450	1	1	yes	characteristic
312	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
313	C₄	C₂×#88173	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	104	4	no	—
316	C₂	#13123579	1	1	yes	characteristic
317	C₂	C₂×#11718533	1	1	yes	characteristic
318	C₂	#13123752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#335	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#13125415	C₂
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#335	D₁₃
hypocenter	C₁₃	#18066
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. #13123579	C₂	semidirect
3. #13113720	C₂	semidirect
4. #13113664	C₂	semidirect
5. #13113630	C₂	semidirect
6. #13113234	C₂	semidirect
7. C₈×#6099	C₂	semidirect
8. C₂×C₁₃×#335	C₂	semidirect
100. #6099	D₈	semidirect
113. C₂×#335	D₁₃	semidirect
143. C₈×C₁₃	#131	semidirect
266. C₁₃	#18066	semidirect
297. C₈	#88155	semidirect

Maximal subgroups

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

subgroup	index
#13125415	2
#13123579	2
#13113720	2
#13113664	2
#13113630	2
#13113234	2
C₈×#6099	2
C₂×C₁₃×#335	2
#18066	13