Subgroups — #15756034

Order 3328.

494 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718349	C₂	1	1	yes	maximal, characteristic
2	#13114964	C₂	1	1	yes	maximal, characteristic, complemented
3	#13114837	C₂	1	2	yes	maximal
4	#13112720	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#6094	—	2	1	no	—
7	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88061	—	2	2	no	—
11	C₂×#88060	—	2	1	no	—
12	#11718596	C₂²	1	2	yes	—
13	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
14	#11718349	—	2	2	no	—
15	#11718349	C₄	1	4	yes	complemented
16	C₁₃×#334	—	2	2	no	—
17	#11718143	—	2	1	no	—
18	#11718143	—	2	1	no	—
19	#11718143	—	2	2	no	—
20	C₂²×D₅₂	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₁₃	—	2	1	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×#6103	—	2	1	no	—
27	C₂×C₄⋊Dic₁₃	—	2	2	no	—
28	C₂×C₄⋊Dic₁₃	—	2	2	no	—
29	C₂×C₄⋊Dic₁₃	—	2	2	no	—
30	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
31	C₂²×#1055	—	2	1	no	—
32	C₂×#6094	—	2	4	no	—
33	C₂×#6094	—	4	4	no	—
34	C₄×D₅₂	—	2	1	no	—
35	C₄×D₅₂	—	2	1	no	—
36	C₄×D₅₂	C₂×C₄	1	2	yes	—
37	C₄×D₅₂	Q₈	1	2	yes	—
38	C₁₃×C₄⋊C₈	—	2	2	no	—
39	C₁₃×C₄⋊C₈	—	2	2	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₃	D₄	1	2	yes	—
42	#88061	—	4	4	no	—
43	#88060	—	2	4	no	—
44	C₄⋊#1055	—	2	1	no	—
45	C₄⋊#1055	C₂×C₄	1	2	yes	—
46	#8656	—	13	1	no	maximal, Sylow
47	C₂³×D₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
51	C₂²×Dic₁₃	—	2	1	no	—
52	C₂²×Dic₁₃	—	4	2	no	—
53	C₂×D₅₂	—	2	2	no	—
54	C₂×D₅₂	—	2	2	no	—
55	C₂×D₅₂	—	2	1	no	—
56	C₂×D₅₂	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×D₁₃	—	2	2	no	—
58	C₂×C₄×D₁₃	—	2	2	no	—
59	C₂×C₄×D₁₃	—	4	2	no	—
60	C₂×C₄×D₁₃	—	4	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	4	2	no	—
64	C₂×C₈×C₁₃	—	4	2	no	—
65	C₂×C₈×C₁₃	—	4	2	no	—
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₁₃	#41	1	1	yes	characteristic
72	#6103	—	2	2	no	—
73	#6103	—	2	2	no	—
74	C₄⋊Dic₁₃	—	2	1	no	—
75	C₄⋊Dic₁₃	—	4	2	no	—
76	C₄⋊Dic₁₃	—	4	2	no	—
77	C₄⋊Dic₁₃	—	4	2	no	—
78	C₄⋊Dic₁₃	—	4	2	no	—
79	C₄⋊Dic₁₃	—	8	2	no	—
80	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
81	C₂×#1055	—	2	2	no	—
82	C₂×#1055	—	4	2	no	—
83	#6094	—	4	8	no	—
84	C₂×#436	—	13	1	no	—
85	#1834	—	13	1	no	—
86	#1802	—	13	2	no	—
87	C₂×#334	D₁₃	1	1	yes	characteristic, complemented
88	#1296	—	13	2	no	—
89	C₂³×C₁₃	#121	1	1	yes	characteristic
90	C₂²×D₁₃	—	2	2	no	—
91	C₂²×D₁₃	—	4	2	no	—
92	C₂²×D₁₃	—	4	1	no	—
93	C₂²×D₁₃	—	4	1	no	—
94	C₂²×D₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#135	1	2	yes	—
103	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₁₃	#137	1	2	yes	—
106	C₂×Dic₁₃	—	2	2	no	—
107	C₂×Dic₁₃	—	4	2	no	—
108	C₂×Dic₁₃	—	8	2	no	—
109	C₂×Dic₁₃	—	8	2	no	—
110	C₂×Dic₁₃	—	8	2	no	—
111	D₅₂	—	2	4	no	—
112	D₅₂	—	4	2	no	—
113	C₄×D₁₃	—	4	2	no	—
114	C₄×D₁₃	—	4	2	no	—
115	C₄×D₁₃	—	4	4	no	—
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	—	8	2	no	—
119	#1055	—	4	2	no	—
120	C₂²×M₁₆	—	26	1	no	—
121	C₂×C₄×D₄	—	13	1	no	—
122	#436	—	13	4	no	—
123	#436	—	26	2	no	—
124	C₂×C₄⋊C₈	—	13	1	no	—
125	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
126	C₂×#102	—	26	2	no	—
127	C₂×C₂²⋊C₈	—	26	1	no	—
128	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
129	#384	—	13	2	no	—
130	#340	—	26	2	no	—
131	#340	—	26	1	no	—
132	#340	—	26	1	no	—
133	#334	—	2	2	no	—
134	C₂²×C₁₃	#380	1	1	yes	characteristic
135	C₂²×C₁₃	#463	1	2	yes	—
136	C₂²×C₁₃	#469	1	1	yes	characteristic
137	C₂²×C₁₃	#470	1	1	yes	characteristic
138	C₂²×C₁₃	#471	1	2	yes	—
139	C₂×D₁₃	—	4	2	no	—
140	C₂×D₁₃	—	4	2	no	—
141	C₂×D₁₃	—	4	2	no	—
142	C₂×D₁₃	—	8	2	no	—
143	C₂×D₁₃	—	8	2	no	—
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	#416	1	1	yes	characteristic
147	C₄×C₁₃	#418	1	1	yes	characteristic
148	C₄×C₁₃	#428	1	2	yes	—
149	C₄×C₁₃	#468	1	2	yes	—
150	C₄×C₁₃	#478	1	2	yes	—
151	Dic₁₃	—	4	2	no	—
152	Dic₁₃	—	16	2	no	—
153	C₂²×D₄	—	13	1	no	—
154	C₂³×C₄	—	26	1	no	—
155	C₂×M₁₆	—	26	4	no	—
156	C₂×M₁₆	—	52	4	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	26	1	no	—
159	C₂²×C₈	D₅₂	1	1	yes	characteristic
160	C₂²×C₈	D₅₂	1	1	yes	characteristic
161	C₄×D₄	—	13	2	no	—
162	C₄×D₄	—	13	2	no	—
163	C₄×D₄	—	26	1	no	—
164	C₄×D₄	—	26	1	no	—
165	C₂×C₄⋊C₄	—	13	1	no	—
166	C₂×C₄⋊C₄	—	26	2	no	—
167	C₂×C₄⋊C₄	—	26	2	no	—
168	C₂×C₄⋊C₄	—	26	2	no	—
169	C₂×C₂²⋊C₄	—	26	1	no	—
170	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	13	2	no	—
174	C₄⋊C₈	—	26	1	no	—
175	#102	—	52	4	no	—
176	C₂²⋊C₈	—	26	4	no	—
177	C₄×C₈	C₄×D₁₃	1	2	yes	—
178	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
179	C₂×C₁₃	#1701	1	2	yes	—
180	C₂×C₁₃	#1703	1	2	yes	—
181	C₂×C₁₃	#1804	1	1	yes	characteristic
182	C₂×C₁₃	#1809	1	1	yes	characteristic
183	C₂×C₁₃	#1815	1	1	yes	characteristic
184	D₁₃	—	8	2	no	—
185	C₂⁴	—	26	1	no	—
186	C₂×D₄	—	13	2	no	—
187	C₂×D₄	—	26	2	no	—
188	C₂×D₄	—	26	2	no	—
189	C₂×D₄	—	26	1	no	—
190	C₂²×C₄	—	26	1	no	—
191	C₂²×C₄	—	26	2	no	—
192	C₂²×C₄	—	26	2	no	—
193	C₂²×C₄	—	52	2	no	—
194	C₂²×C₄	—	52	2	no	—
195	C₂²×C₄	—	52	2	no	—
196	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
197	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
198	C₂²×C₄	#6131	1	1	yes	characteristic
199	M₁₆	—	52	8	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	26	2	no	—
206	C₂×C₈	—	52	2	no	—
207	C₂×C₈	#6103	1	2	yes	—
208	C₂×C₈	#6103	1	2	yes	—
209	C₄⋊C₄	—	13	2	no	—
210	C₄⋊C₄	—	26	1	no	—
211	C₄⋊C₄	—	52	2	no	—
212	C₄⋊C₄	—	52	2	no	—
213	C₄⋊C₄	—	52	2	no	—
214	C₄⋊C₄	—	52	2	no	—
215	C₄⋊C₄	—	104	2	no	—
216	C₂²⋊C₄	—	26	2	no	—
217	C₂²⋊C₄	—	26	2	no	—
218	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
219	C₄²	#6129	1	1	yes	characteristic
220	C₄²	Q₈×D₁₃	1	1	yes	characteristic
221	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
222	C₁₃	#8656	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	26	2	no	—
224	C₂³	—	52	2	no	—
225	C₂³	—	52	1	no	—
226	C₂³	—	52	1	no	—
227	C₂³	—	52	2	no	—
228	C₂³	#88149	1	1	yes	characteristic
229	D₄	—	26	4	no	—
230	D₄	—	52	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	26	2	no	—
233	C₂×C₄	—	52	2	no	—
234	C₂×C₄	—	52	4	no	—
235	C₂×C₄	—	52	2	no	—
236	C₂×C₄	—	52	2	no	—
237	C₂×C₄	—	104	2	no	—
238	C₂×C₄	—	104	2	no	—
239	C₂×C₄	—	104	2	no	—
240	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
241	C₂×C₄	#88146	1	1	yes	characteristic
242	C₂×C₄	#88151	1	1	yes	characteristic
243	C₂×C₄	#88158	1	2	yes	—
244	C₂×C₄	#88162	1	2	yes	—
245	C₂×C₄	#88168	1	2	yes	—
246	C₂×C₄	#88169	1	2	yes	—
247	C₂×C₄	C₂×#6103	1	1	yes	characteristic
248	C₂×C₄	#88192	1	1	yes	characteristic
249	C₂×C₄	#88200	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	8	2	no	—
253	C₈	—	52	2	no	—
254	C₂²	—	52	2	no	—
255	C₂²	—	52	2	no	—
256	C₂²	—	52	2	no	—
257	C₂²	—	104	2	no	—
258	C₂²	—	104	2	no	—
259	C₂²	#11718505	1	2	yes	—
260	C₂²	#11718526	1	1	yes	characteristic
261	C₂²	#11718529	1	1	yes	characteristic
262	C₂²	#11718559	1	2	yes	—
263	C₂²	#11718648	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	52	2	no	—
267	C₄	—	208	2	no	—
268	C₄	#11718508	1	2	yes	—
269	C₄	#11718522	1	2	yes	—
270	C₄	#11718774	1	1	yes	characteristic
271	C₄	#11718792	1	1	yes	characteristic
272	C₄	#11718835	1	2	yes	—
273	C₂	—	104	2	no	—
274	C₂	#13114398	1	2	yes	—
275	C₂	#13114412	1	2	yes	—
276	C₂	#13114847	1	1	yes	characteristic
277	C₂	#13114866	1	1	yes	characteristic
278	C₂	#13114892	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88149
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#334	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂²×D₅₂	D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#334	D₁₃
hypocenter	C₁₃	#8656
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
2. #13114964	C₂	semidirect
4. #13112720	C₂	semidirect
5. C₂×C₁₃×#334	C₂	semidirect
15. #11718349	C₄	semidirect
87. C₂×#334	D₁₃	semidirect
222. C₁₃	#8656	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718349	2
#13114964	2
#13114837	2
#13112720	2
C₂×C₁₃×#334	2
#8656	13