Subgroups — #16018222

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125264	C₂	1	1	yes	maximal, characteristic
2	#13117070	C₂	1	4	yes	maximal, complemented
3	#13117065	C₂	1	2	yes	maximal, complemented
4	#13117061	C₂	1	2	yes	maximal, complemented
5	#13116541	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#533	—	2	1	no	—
9	C₂×#88158	—	2	1	no	—
10	C₂×#88158	C₂²	1	2	yes	—
11	#11719182	—	2	1	no	—
12	C₁₃×#500	C₂²	1	2	yes	—
13	C₁₃×#499	—	2	1	no	—
14	C₁₃×#499	C₂²	1	2	yes	—
15	C₁₃×#498	—	2	1	no	—
16	C₁₃×#427	—	2	1	no	—
17	C₁₃×#427	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂²×#6090	C₂²	1	2	yes	—
20	#11718360	C₂²	1	1	yes	characteristic
21	#11718357	—	2	1	no	—
22	#11718357	C₂²	1	2	yes	—
23	#11718356	C₂²	1	1	yes	characteristic
24	#11718354	—	2	1	no	—
25	#11718084	C₂²	1	8	yes	—
26	#11718083	C₂²	1	8	yes	—
27	C₄×#6090	C₂²	1	2	yes	—
28	C₄×#6090	C₂²	1	2	yes	—
29	C₂×#6127	—	2	2	no	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×C₄⋊Q₈	—	2	2	no	—
33	C₁₃×#125	—	2	2	no	—
34	C₁₃×C₂²⋊Q₈	—	4	2	no	—
35	C₄×Q₈×C₁₃	—	4	2	no	—
36	C₁₃×#117	—	2	2	no	—
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	#88184	—	4	2	no	—
39	C₂×Dic₅₂	—	2	1	no	—
40	C₂×Dic₅₂	C₂³	1	2	yes	—
41	#88158	—	2	8	no	—
42	#88158	—	4	4	no	—
43	C₂×D₁₀₄	—	2	1	no	—
44	C₂×D₁₀₄	C₂³	1	2	yes	—
45	C₂×#6095	—	2	2	no	—
46	C₂×#6095	—	2	2	no	—
47	#88129	—	2	2	no	—
48	#88128	—	2	1	no	—
49	C₄×D₅₂	—	4	2	no	—
50	#88123	—	2	1	no	—
51	C₁₃×#107	—	2	2	no	—
52	C₁₃×#107	C₂³	1	4	yes	—
53	C₁₃×#106	—	2	2	no	—
54	C₁₃×#106	—	2	2	no	—
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₂×#6090	—	2	4	no	—
60	C₂×#6090	C₂³	1	4	yes	—
61	#33697	—	13	1	no	maximal, Sylow
62	C₂×Q₈×C₁₃	—	4	2	no	—
63	C₂²×C₄×C₁₃	D₈	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₃	—	4	2	no	—
66	#6127	—	4	4	no	—
67	#6127	—	4	4	no	—
68	C₂×D₅₂	—	2	2	no	—
69	C₂×D₅₂	—	4	2	no	—
70	C₂×C₄×D₁₃	—	4	2	no	—
71	C₂×Dic₂₆	—	2	2	no	—
72	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	#41	1	4	yes	—
77	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	—	2	4	no	—
79	C₁₃×C₄⋊C₄	—	4	4	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₂²⋊C₄	—	4	2	no	—
82	C₄²×C₁₃	—	4	2	no	—
83	C₄²×C₁₃	D₈	1	2	yes	—
84	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
86	#6103	—	4	4	no	—
87	C₄⋊Dic₁₃	—	4	2	no	—
88	Dic₅₂	—	2	4	no	—
89	Dic₅₂	—	4	2	no	—
90	D₁₀₄	—	2	4	no	—
91	D₁₀₄	—	4	2	no	—
92	#6095	—	4	4	no	—
93	#6095	—	4	4	no	—
94	#6090	—	2	4	no	—
95	#3160	D₁₃	1	1	yes	characteristic, complemented
96	#3147	—	13	1	no	—
97	#2242	—	13	4	no	—
98	#2241	—	13	2	no	—
99	#2240	—	13	2	no	—
100	#2138	—	13	4	no	—
101	C₂×C₄×C₁₆	—	13	1	no	—
102	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
103	C₂²×D₁₃	—	4	2	no	—
104	Q₈×C₁₃	—	8	4	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	#124	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
112	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂²⋊D₁₃	—	8	4	no	—
114	C₂×Dic₁₃	—	4	2	no	—
115	D₅₂	—	4	2	no	—
116	D₅₂	—	4	2	no	—
117	D₅₂	—	8	4	no	—
118	C₄×D₁₃	—	8	4	no	—
119	Dic₂₆	—	4	2	no	—
120	Dic₂₆	—	4	2	no	—
121	C₈×C₁₃	#124	1	2	yes	—
122	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
123	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
124	C₈×C₁₃	C₂×#41	1	2	yes	—
125	C₂×#135	—	13	2	no	—
126	C₂×#135	—	26	1	no	—
127	#533	—	2	1	no	—
128	#532	—	26	1	no	—
129	C₂²×C₁₆	—	13	2	no	—
130	#500	C₂×D₁₃	1	2	yes	—
131	#499	—	2	1	no	—
132	#499	C₂×D₁₃	1	2	yes	—
133	#498	—	2	1	no	—
134	#495	—	13	2	no	—
135	#495	—	26	1	no	—
136	#494	—	13	1	no	—
137	#493	—	13	1	no	—
138	#492	—	26	1	no	—
139	#427	—	2	1	no	—
140	#427	C₂×D₁₃	1	2	yes	—
141	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
142	#358	—	13	8	no	—
143	#357	—	13	8	no	—
144	C₄×C₁₆	—	13	2	no	—
145	C₄×C₁₆	—	13	2	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	#486	1	2	yes	—
148	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₁₃	—	8	2	no	—
150	C₂×D₁₃	—	8	4	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	8	4	no	—
153	C₄×C₁₃	#486	1	2	yes	—
154	C₄×C₁₃	#508	1	4	yes	—
155	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
156	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
157	Dic₁₃	—	8	2	no	—
158	C₂×#41	—	26	2	no	—
159	#135	—	26	8	no	—
160	#135	—	52	4	no	—
161	C₂×Q₁₆	—	13	2	no	—
162	C₂×Q₁₆	—	26	1	no	—
163	C₂×SD₁₆	—	26	2	no	—
164	C₂×SD₁₆	—	26	2	no	—
165	C₂×D₈	—	13	2	no	—
166	C₂×D₈	—	26	1	no	—
167	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	—	26	1	no	—
171	#127	—	26	1	no	—
172	#125	—	2	2	no	—
173	#124	—	26	2	no	—
174	C₂²⋊Q₈	—	4	2	no	—
175	#121	—	52	2	no	—
176	C₄×Q₈	—	4	2	no	—
177	C₄×D₄	—	52	2	no	—
178	#117	—	2	2	no	—
179	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
180	C₂×C₁₆	—	13	4	no	—
181	C₂×C₁₆	—	26	4	no	—
182	#107	—	2	2	no	—
183	#107	C₂²×D₁₃	1	4	yes	—
184	#106	—	2	2	no	—
185	#106	—	2	2	no	—
186	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
190	C₂×C₁₃	—	2	2	no	—
191	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
192	C₂×C₁₃	C₂×#508	1	2	yes	—
193	D₁₃	—	16	2	no	—
194	#41	—	52	4	no	—
195	#41	—	52	4	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	—	26	2	no	—
198	C₂×D₄	—	26	2	no	—
199	C₂×D₄	—	52	2	no	—
200	C₂×D₄	—	52	2	no	—
201	C₂²×C₄	—	52	2	no	—
202	C₂²×C₄	#6104	1	2	yes	—
203	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
204	Q₁₆	—	26	4	no	—
205	Q₁₆	—	52	2	no	—
206	SD₁₆	—	52	4	no	—
207	SD₁₆	—	52	4	no	—
208	D₈	—	26	4	no	—
209	D₈	—	52	2	no	—
210	C₂×C₈	#6106	1	4	yes	—
211	C₂×C₈	#6131	1	4	yes	—
212	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
214	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	52	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	52	4	no	—
222	C₄²	—	4	2	no	—
223	C₄²	#6104	1	2	yes	—
224	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
226	C₁₆	—	26	4	no	—
227	C₁₃	#33697	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	52	2	no	—
229	C₂³	C₂×#6104	1	1	yes	characteristic
230	Q₈	—	8	4	no	—
231	Q₈	—	52	2	no	—
232	Q₈	—	52	2	no	—
233	D₄	—	52	2	no	—
234	D₄	—	52	2	no	—
235	D₄	—	104	4	no	—
236	D₄	—	104	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	52	2	no	—
240	C₂×C₄	—	104	4	no	—
241	C₂×C₄	C₂×#6104	1	2	yes	—
242	C₂×C₄	C₂×#6104	1	2	yes	—
243	C₂×C₄	C₂×#6104	1	1	yes	characteristic
244	C₂×C₄	C₂×#6106	1	2	yes	—
245	C₂×C₄	#88201	1	2	yes	—
246	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
247	C₈	C₂×#6106	1	2	yes	—
248	C₈	C₂×#6106	1	2	yes	—
249	C₈	#88201	1	2	yes	—
250	C₈	C₂×#6131	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	104	4	no	—
253	C₂²	—	104	2	no	—
254	C₂²	#11718749	1	2	yes	—
255	C₂²	C₂²×#6104	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	8	4	no	—
258	C₄	—	104	2	no	—
259	C₄	#11718749	1	2	yes	—
260	C₄	#11718855	1	4	yes	—
261	C₄	C₂²×#6106	1	1	yes	characteristic
262	C₄	C₂×#88201	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	208	2	no	—
265	C₂	C₂×#11718749	1	1	yes	characteristic
266	C₂	C₂×#11718855	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3160	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13125264	C₂
hypercenter	#3160	D₁₃
hypocenter	C₁₃	#33697
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #13117070	C₂	semidirect
3. #13117065	C₂	semidirect
4. #13117061	C₂	semidirect
5. #13116541	C₂	semidirect
6. C₂×C₄×#6090	C₂	semidirect
7. C₁₃×#3160	C₂	semidirect
95. #3160	D₁₃	semidirect
227. C₁₃	#33697	semidirect

Maximal subgroups

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

subgroup	index
#13125264	2
#13117070	2
#13117065	2
#13117061	2
#13116541	2
C₂×C₄×#6090	2
C₁₃×#3160	2
#33697	13