Subgroups — #15779872

Order 3328.

530 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	C₂×#11718777	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718084	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#6090	C₂	1	1	yes	maximal, characteristic
4	#13112927	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	—	2	1	no	—
8	C₂×#88178	—	2	1	no	—
9	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
10	C₁₃×#427	—	2	4	no	—
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	C₂×C₁₃×#106	—	2	1	no	—
13	#11718777	—	2	2	no	—
14	#11718777	C₄	1	4	yes	complemented
15	C₂×#88058	C₂²	1	1	yes	characteristic
16	C₂×#88056	—	2	1	no	—
17	C₂²×#6090	C₂²	1	2	yes	—
18	C₂²⋊#6090	—	2	2	no	—
19	#11718084	—	2	4	no	—
20	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×Dic₂₆	—	2	1	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	D₄	1	2	yes	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×#117	—	2	4	no	—
29	C₁₃×#117	—	4	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₁₃	—	4	1	no	—
33	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
34	C₂×C₄⋊Dic₁₃	—	2	1	no	—
35	#88178	—	2	4	no	—
36	#88178	—	4	2	no	—
37	C₂×#6101	—	4	1	no	—
38	C₂×Dic₅₂	—	2	2	no	—
39	C₂×Dic₅₂	—	2	2	no	—
40	C₂×Dic₅₂	—	2	1	no	—
41	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
42	C₁₃×#107	—	2	2	no	—
43	C₁₃×#107	D₄	1	4	yes	—
44	C₁₃×#106	—	2	4	no	—
45	#88058	—	2	1	no	—
46	#88058	C₂×C₄	1	2	yes	—
47	#88056	—	2	4	no	—
48	C₂×#6090	—	2	2	no	—
49	C₂×#6090	—	2	2	no	—
50	C₂×#6090	—	2	2	no	—
51	#12994	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₁₃	—	4	1	no	—
63	C₂²×Dic₁₃	—	4	1	no	—
64	C₂×Dic₂₆	—	2	2	no	—
65	C₂×Dic₂₆	—	4	1	no	—
66	C₂×Dic₂₆	—	4	1	no	—
67	C₂×Dic₂₆	—	4	1	no	—
68	C₂×Dic₂₆	—	4	1	no	—
69	C₂×Dic₂₆	—	4	1	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	#41	1	2	yes	—
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₂²⋊C₄	—	4	4	no	—
82	C₄²×C₁₃	—	4	4	no	—
83	C₂²⋊Dic₁₃	—	4	4	no	—
84	C₄⋊Dic₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	—	4	1	no	—
86	#6101	—	4	4	no	—
87	Dic₅₂	—	2	4	no	—
88	Dic₅₂	—	4	2	no	—
89	#6090	—	4	2	no	—
90	C₂×#467	—	13	1	no	—
91	C₂×#427	D₁₃	1	1	yes	characteristic
92	C₂×#358	—	13	2	no	—
93	C₂×C₂²⋊C₁₆	—	13	1	no	—
94	#1380	—	13	2	no	—
95	C₂³×C₁₃	—	2	1	no	—
96	C₂³×C₁₃	—	2	1	no	—
97	C₂³×C₁₃	—	2	2	no	—
98	C₂³×C₁₃	—	2	2	no	—
99	C₂³×C₁₃	#102	1	2	yes	—
100	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	8	2	no	—
109	C₂×C₄×C₁₃	—	8	2	no	—
110	C₂×C₄×C₁₃	#102	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
117	C₂×Dic₁₃	—	4	2	no	—
118	C₂×Dic₁₃	—	4	2	no	—
119	C₂×Dic₁₃	—	8	1	no	—
120	C₂×Dic₁₃	—	8	1	no	—
121	C₂×Dic₁₃	—	8	2	no	—
122	Dic₂₆	—	4	2	no	—
123	Dic₂₆	—	4	2	no	—
124	Dic₂₆	—	8	1	no	—
125	Dic₂₆	—	8	1	no	—
126	C₈×C₁₃	—	2	2	no	—
127	C₈×C₁₃	C₄×D₄	1	2	yes	—
128	C₈×C₁₃	#121	1	2	yes	—
129	C₂²×Q₁₆	—	13	1	no	—
130	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
131	C₂×C₂²⋊Q₈	—	26	1	no	—
132	C₂×#117	—	2	1	no	—
133	C₂²×C₁₆	—	13	2	no	—
134	#467	—	13	4	no	—
135	#467	—	26	2	no	—
136	#427	—	2	4	no	—
137	C₂×#107	—	13	1	no	—
138	C₂×#107	C₂×D₁₃	1	2	yes	—
139	C₂×#106	—	2	1	no	—
140	C₂×#103	—	26	1	no	—
141	#358	—	26	4	no	—
142	C₂²⋊C₁₆	—	26	2	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	4	2	no	—
147	C₂²×C₁₃	—	4	2	no	—
148	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
149	C₂²×C₁₃	#447	1	1	yes	characteristic
150	C₂²×C₁₃	#480	1	1	yes	characteristic
151	C₂²×C₁₃	#508	1	2	yes	—
152	C₂²×C₁₃	#510	1	2	yes	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	8	4	no	—
155	C₄×C₁₃	#386	1	1	yes	characteristic
156	C₄×C₁₃	#417	1	1	yes	characteristic
157	C₄×C₁₃	#448	1	1	yes	characteristic
158	C₄×C₁₃	#481	1	1	yes	characteristic
159	Dic₁₃	—	8	2	no	—
160	Dic₁₃	—	8	2	no	—
161	C₂²×Q₈	—	26	1	no	—
162	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
163	C₂×Q₁₆	—	13	2	no	—
164	C₂×Q₁₆	—	26	1	no	—
165	C₂×Q₁₆	—	26	2	no	—
166	C₂×Q₁₆	—	26	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	C₄×D₁₃	1	2	yes	—
170	C₂²×C₈	D₅₂	1	2	yes	—
171	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
173	C₂²⋊Q₈	—	26	4	no	—
174	C₂²⋊Q₈	—	52	2	no	—
175	#117	—	2	4	no	—
176	#117	—	4	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	26	1	no	—
179	C₂×C₄⋊C₄	—	52	1	no	—
180	C₂×C₂²⋊C₄	—	4	1	no	—
181	C₂×C₂²⋊C₄	—	52	1	no	—
182	C₂×C₄²	—	4	1	no	—
183	C₂×C₁₆	—	26	2	no	—
184	C₂×C₁₆	—	26	2	no	—
185	C₂×C₁₆	—	26	2	no	—
186	#107	—	2	2	no	—
187	#107	—	13	2	no	—
188	#107	—	26	1	no	—
189	#107	C₂²⋊D₁₃	1	4	yes	—
190	#106	—	2	4	no	—
191	#103	—	26	4	no	—
192	C₂×C₁₃	—	4	2	no	—
193	C₂×C₁₃	#1874	1	1	yes	characteristic
194	C₂×C₁₃	#2138	1	1	yes	characteristic
195	C₂×C₁₃	#2141	1	1	yes	characteristic
196	C₂×C₁₃	#2189	1	2	yes	—
197	C₂×C₁₃	#2237	1	2	yes	—
198	C₂⁴	#6104	1	1	yes	characteristic
199	C₂×Q₈	—	26	2	no	—
200	C₂×Q₈	—	52	1	no	—
201	C₂×Q₈	—	52	1	no	—
202	C₂×Q₈	—	52	1	no	—
203	C₂×Q₈	—	52	1	no	—
204	C₂×Q₈	—	52	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	52	1	no	—
211	C₂²×C₄	—	52	1	no	—
212	C₂²×C₄	#6103	1	2	yes	—
213	C₂²×C₄	#6104	1	1	yes	characteristic
214	C₂²×C₄	#6106	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
216	Q₁₆	—	26	4	no	—
217	Q₁₆	—	52	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	#6103	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
225	C₂×C₈	#6127	1	2	yes	—
226	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	26	2	no	—
230	C₄⋊C₄	—	52	1	no	—
231	C₄⋊C₄	—	52	4	no	—
232	C₂²⋊C₄	—	4	4	no	—
233	C₂²⋊C₄	—	52	4	no	—
234	C₄²	—	4	4	no	—
235	C₁₆	—	52	2	no	—
236	C₁₃	#12994	1	1	yes	characteristic, complemented, Sylow
237	C₂³	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	#88049	1	2	yes	—
242	C₂³	C₂×#6104	1	1	yes	characteristic
243	Q₈	—	52	2	no	—
244	Q₈	—	52	2	no	—
245	Q₈	—	104	1	no	—
246	Q₈	—	104	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	52	2	no	—
257	C₂×C₄	—	52	2	no	—
258	C₂×C₄	—	104	2	no	—
259	C₂×C₄	—	104	1	no	—
260	C₂×C₄	—	104	1	no	—
261	C₂×C₄	#88049	1	2	yes	—
262	C₂×C₄	C₂×#6103	1	1	yes	characteristic
263	C₂×C₄	#88183	1	1	yes	characteristic
264	C₂×C₄	#88186	1	1	yes	characteristic
265	C₂×C₄	C₂×#6106	1	1	yes	characteristic
266	C₂×C₄	#88196	1	1	yes	characteristic
267	C₂×C₄	#88207	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
270	C₈	#88184	1	2	yes	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₂×#88049	1	1	yes	characteristic
277	C₂²	#11718628	1	1	yes	characteristic
278	C₂²	#11718855	1	2	yes	—
279	C₂²	#11718856	1	2	yes	—
280	C₂²	#11718878	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	8	4	no	—
283	C₄	—	104	2	no	—
284	C₄	—	104	2	no	—
285	C₄	#11718627	1	1	yes	characteristic
286	C₄	#11718629	1	1	yes	characteristic
287	C₄	#11718874	1	1	yes	characteristic
288	C₄	#11718888	1	1	yes	characteristic
289	C₂	—	4	2	no	—
290	C₂	#13115153	1	1	yes	characteristic
291	C₂	#13116541	1	1	yes	characteristic
292	C₂	#13116555	1	1	yes	characteristic
293	C₂	#13116776	1	2	yes	—
294	C₂	#13117044	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#427	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂⁴×C₁₃	D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#427	D₁₃
hypocenter	C₁₃	#12994
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. C₂×#11718084	C₂	semidirect
4. #13112927	C₂	semidirect
14. #11718777	C₄	semidirect
236. C₁₃	#12994	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718777	2
C₂×#11718084	2
C₂×C₂²⋊#6090	2
#13112927	2
C₂×C₁₃×#427	2
#12994	13