Subgroups — #16802188

Order 3328.

578 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131629	C₂	1	2	yes	maximal
2	#13131024	C₂	1	2	yes	maximal
3	#13130904	C₂	1	4	yes	maximal
4	#13130901	C₂	1	4	yes	maximal
5	#13130761	C₂	1	1	yes	maximal, characteristic
6	#13130554	C₂	1	1	yes	maximal, characteristic
7	#13125347	C₂	1	1	yes	maximal, characteristic
8	#11719433	C₂²	1	2	yes	—
9	#11719158	—	2	4	no	—
10	#11719079	C₂²	1	4	yes	—
11	#11719078	—	2	4	no	—
12	#11719072	C₂²	1	4	yes	—
13	#11719057	C₂²	1	1	yes	characteristic
14	#11719053	C₂²	1	2	yes	—
15	#11719047	C₂²	1	1	yes	characteristic
16	C₁₃×#500	C₄	1	1	yes	characteristic, complemented
17	#11718740	C₄	1	2	yes	complemented
18	#11718541	C₂²	1	2	yes	—
19	#11718541	C₂²	1	2	yes	—
20	#11718539	C₄	1	2	yes	complemented
21	#11718539	C₄	1	2	yes	complemented
22	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
23	#11718343	C₄	1	1	yes	characteristic, complemented
24	#11718268	C₂²	1	2	yes	—
25	#11718268	C₂²	1	2	yes	—
26	(C₂×#261)⋊C₈	C₂²	1	2	yes	—
27	(C₂×#261)⋊C₈	C₂²	1	2	yes	—
28	#11718247	C₂²	1	8	yes	—
29	C₂×C₄×#261	C₂³	1	2	yes	—
30	C₂×#6118	C₂³	1	4	yes	—
31	C₂×D₁₃⋊C₈	C₂³	1	2	yes	—
32	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	#88200	—	2	2	no	—
34	Q₈×Dic₁₃	—	2	2	no	—
35	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
37	#88151	—	2	2	no	—
38	#88146	C₂³	1	4	yes	—
39	#88144	C₂×C₄	1	4	yes	—
40	#88141	—	2	2	no	—
41	C₄²×D₁₃	C₂³	1	1	yes	characteristic
42	#88123	C₂×C₄	1	2	yes	—
43	(C₂×#261)⋊C₄	—	2	1	no	—
44	(C₂×#261)⋊C₄	—	2	1	no	—
45	#88112	—	2	4	no	—
46	#88112	—	2	2	no	—
47	#88112	—	2	2	no	—
48	#88110	—	2	4	no	—
49	#88110	D₄	1	2	yes	—
50	#88110	D₄	1	2	yes	—
51	C₄⋊#1057	—	2	4	no	—
52	C₄×#1057	—	2	4	no	—
53	C₁₃×#107	C₂×C₄	1	2	yes	—
54	C₁₃×#107	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
56	#88058	C₂×C₄	1	2	yes	—
57	#88058	C₂×C₄	1	2	yes	—
58	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
60	#88048	C₂×C₄	1	4	yes	—
61	#88048	C₂×C₄	1	4	yes	—
62	C₄×#1055	C₂×C₄	1	1	yes	characteristic
63	#22258	—	13	1	no	maximal, Sylow
64	C₂²×#261	—	2	2	no	—
65	C₂×Q₈×C₁₃	—	2	2	no	—
66	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
67	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
68	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
69	C₂×Dic₂₆	—	2	2	no	—
70	C₂×#1057	—	2	4	no	—
71	C₂×#1057	C₂×D₄	1	4	yes	—
72	C₄×#261	—	2	2	no	—
73	C₄×#261	—	2	2	no	—
74	#6118	—	2	4	no	—
75	#6118	—	2	4	no	—
76	D₁₃⋊C₈	—	2	2	no	—
77	D₁₃⋊C₈	—	2	2	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
82	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	#6103	—	2	4	no	—
84	C₄⋊Dic₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	—	2	4	no	—
86	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
87	#6101	—	2	4	no	—
88	C₄×Dic₁₃	—	2	4	no	—
89	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
93	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
94	C₈×D₁₃	—	2	2	no	—
95	C₈×D₁₃	—	2	2	no	—
96	#3160	—	13	1	no	—
97	#2961	—	13	2	no	—
98	#1995	—	13	2	no	—
99	#1922	—	13	4	no	—
100	#1922	—	13	4	no	—
101	#1837	—	13	1	no	—
102	#1746	—	13	1	no	—
103	C₂²×D₁₃	C₂×#41	1	1	yes	characteristic
104	C₂×#261	—	4	2	no	—
105	C₂×#261	—	4	2	no	—
106	C₂×#261	—	4	2	no	—
107	Q₈×C₁₃	—	8	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×Dic₁₃	—	2	4	no	—
113	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₄×D₁₃	—	4	1	no	—
117	C₄×D₁₃	—	4	1	no	—
118	C₄×D₁₃	2₋¹⁺⁴	1	2	yes	—
119	C₄×D₁₃	2₋¹⁺⁴	1	2	yes	—
120	Dic₂₆	—	8	2	no	—
121	#1057	—	2	4	no	—
122	#1057	—	4	4	no	—
123	C₈×C₁₃	—	2	1	no	—
124	C₈×C₁₃	—	2	1	no	—
125	#1055	—	2	1	no	—
126	#1055	—	2	1	no	—
127	#533	—	13	2	no	—
128	#500	—	13	1	no	—
129	#500	#261	1	1	yes	characteristic, complemented
130	#499	—	13	2	no	—
131	#499	—	13	2	no	—
132	#498	—	13	2	no	—
133	#446	—	26	4	no	—
134	#446	—	26	4	no	—
135	#427	—	13	2	no	—
136	#427	—	13	2	no	—
137	#424	—	13	4	no	—
138	#423	—	13	2	no	—
139	#405	—	13	4	no	—
140	C₂×#97	—	13	1	no	—
141	C₂×C₄×C₈	—	13	1	no	—
142	#376	—	13	1	no	—
143	#341	—	13	2	no	—
144	#341	—	13	2	no	—
145	#339	—	13	8	no	—
146	#336	—	13	2	no	—
147	#336	—	13	2	no	—
148	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×D₁₃	—	2	1	no	—
150	C₂×D₁₃	—	2	1	no	—
151	C₂×D₁₃	#541	1	2	yes	—
152	#261	—	8	2	no	—
153	C₄×C₁₃	—	2	1	no	—
154	C₄×C₁₃	—	2	1	no	—
155	C₄×C₁₃	—	4	4	no	—
156	C₄×C₁₃	#519	1	1	yes	characteristic
157	C₄×C₁₃	#519	1	1	yes	characteristic
158	Dic₁₃	—	2	1	no	—
159	Dic₁₃	—	2	1	no	—
160	Dic₁₃	—	4	4	no	—
161	Dic₁₃	#536	1	2	yes	—
162	C₂×M₁₆	—	13	4	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	—	13	1	no	—
165	C₂²×C₈	—	13	1	no	—
166	C₄⋊Q₈	—	13	2	no	—
167	C₄⋊Q₈	C₂×#261	1	2	yes	—
168	#125	—	13	4	no	—
169	C₂²⋊Q₈	—	26	2	no	—
170	C₂²⋊Q₈	—	26	2	no	—
171	C₄×Q₈	—	26	2	no	—
172	C₄×Q₈	—	26	2	no	—
173	#117	—	13	4	no	—
174	C₂×C₄²	—	13	1	no	—
175	C₂×C₄²	—	13	2	no	—
176	#107	—	13	2	no	—
177	#107	—	13	2	no	—
178	#107	C₂×#261	1	2	yes	—
179	#107	C₂×#261	1	2	yes	—
180	#106	—	13	4	no	—
181	#106	—	13	4	no	—
182	C₄⋊C₈	—	26	2	no	—
183	C₄⋊C₈	—	26	2	no	—
184	C₄⋊C₈	—	26	4	no	—
185	C₄⋊C₈	—	26	4	no	—
186	#97	—	13	2	no	—
187	#97	—	13	2	no	—
188	#97	—	26	4	no	—
189	C₄×C₈	—	13	1	no	—
190	C₄×C₈	—	13	1	no	—
191	C₄×C₈	—	13	1	no	—
192	C₄×C₈	—	26	4	no	—
193	C₄×C₈	C₂×#261	1	1	yes	characteristic
194	#95	—	26	1	no	—
195	#95	—	26	1	no	—
196	C₂×C₁₃	#2327	1	1	yes	characteristic
197	C₂×C₁₃	#2957	1	1	yes	characteristic
198	C₂×C₁₃	#2957	1	1	yes	characteristic
199	D₁₃	—	2	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	26	2	no	—
202	C₂²×C₄	—	13	1	no	—
203	C₂²×C₄	—	13	1	no	—
204	C₂²×C₄	—	13	1	no	—
205	C₂²×C₄	—	26	2	no	—
206	M₁₆	—	26	4	no	—
207	M₁₆	—	26	4	no	—
208	C₂×C₈	—	13	4	no	—
209	C₂×C₈	—	13	1	no	—
210	C₂×C₈	—	13	1	no	—
211	C₂×C₈	—	26	2	no	—
212	C₂×C₈	—	26	2	no	—
213	C₂×C₈	—	26	2	no	—
214	C₂×C₈	—	26	2	no	—
215	C₂×C₈	—	26	4	no	—
216	C₂×C₈	C₂²×#261	1	1	yes	characteristic
217	C₂×C₈	C₂²×#261	1	1	yes	characteristic
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	13	4	no	—
220	C₄⋊C₄	—	26	4	no	—
221	C₄⋊C₄	—	26	4	no	—
222	C₄⋊C₄	—	26	2	no	—
223	C₄⋊C₄	C₂²×#261	1	4	yes	—
224	C₂²⋊C₄	—	26	4	no	—
225	C₄²	—	13	1	no	—
226	C₄²	—	13	1	no	—
227	C₄²	—	13	1	no	—
228	C₄²	—	26	4	no	—
229	C₄²	—	26	2	no	—
230	C₄²	—	26	2	no	—
231	C₄²	C₂²×#261	1	1	yes	characteristic
232	C₁₃	#22258	1	1	yes	characteristic, complemented, Sylow
233	C₂³	—	13	1	no	—
234	Q₈	—	8	2	no	—
235	Q₈	—	104	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	13	1	no	—
238	C₂×C₄	—	13	2	no	—
239	C₂×C₄	—	13	2	no	—
240	C₂×C₄	—	13	1	no	—
241	C₂×C₄	—	13	1	no	—
242	C₂×C₄	—	26	4	no	—
243	C₂×C₄	—	52	1	no	—
244	C₂×C₄	—	52	1	no	—
245	C₂×C₄	—	52	2	no	—
246	C₂×C₄	—	52	2	no	—
247	C₂×C₄	—	52	2	no	—
248	C₂×C₄	D₄×#261	1	1	yes	characteristic
249	C₂×C₄	D₄×#261	1	1	yes	characteristic
250	C₂×C₄	C₂³×#261	1	1	yes	characteristic
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	26	4	no	—
254	C₈	—	26	1	no	—
255	C₈	—	26	1	no	—
256	C₈	—	52	4	no	—
257	C₂²	—	13	2	no	—
258	C₂²	—	26	1	no	—
259	C₂²	—	26	1	no	—
260	C₂²	C₂×D₄×#261	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	4	no	—
264	C₄	—	13	2	no	—
265	C₄	—	26	1	no	—
266	C₄	—	26	1	no	—
267	C₄	—	52	4	no	—
268	C₄	—	104	2	no	—
269	C₄	#11719627	1	1	yes	characteristic
270	C₄	#11719627	1	1	yes	characteristic
271	C₂	—	26	2	no	—
272	C₂	#13131327	1	1	yes	characteristic
273	C₂	#13131612	1	1	yes	characteristic
274	C₂	#13131616	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#500	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₃	C₂×#41
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	#500	#261
hypocenter	C₁₃	#22258
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
16. C₁₃×#500	C₄	semidirect
17. #11718740	C₄	semidirect
20. #11718539	C₄	semidirect
21. #11718539	C₄	semidirect
23. #11718343	C₄	semidirect
129. #500	#261	semidirect
232. C₁₃	#22258	semidirect

Maximal subgroups

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

subgroup	index
#13131629	2
#13131024	2
#13130904	2
#13130901	2
#13130761	2
#13130554	2
#13125347	2
#22258	13