Subgroups — #16795908

Order 3328.

560 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131744	C₂	1	2	yes	maximal
2	#13130482	C₂	1	2	yes	maximal, complemented
3	#13130475	C₂	1	2	yes	maximal, complemented
4	#13130431	C₂	1	2	yes	maximal, complemented
5	#13130424	C₂	1	2	yes	maximal, complemented
6	#13130397	C₂	1	1	yes	maximal, characteristic
7	#13130200	C₂	1	2	yes	maximal
8	#13130040	C₂	1	1	yes	maximal, characteristic
9	#13125423	C₂	1	1	yes	maximal, characteristic
10	#11719433	C₂²	1	2	yes	—
11	#11719158	—	2	4	no	—
12	#11719080	C₂²	1	4	yes	—
13	#11719077	—	2	4	no	—
14	#11719074	C₂²	1	4	yes	—
15	#11719054	C₂²	1	2	yes	—
16	#11719049	C₂²	1	1	yes	characteristic
17	#11719049	C₂²	1	1	yes	characteristic
18	C₁₃×#500	C₄	1	1	yes	characteristic
19	#11718754	C₄	1	2	yes	—
20	#11718548	C₄	1	2	yes	—
21	#11718548	C₄	1	2	yes	—
22	#11718534	C₂²	1	2	yes	—
23	#11718534	C₂²	1	2	yes	—
24	#11718360	C₄	1	1	yes	characteristic
25	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
26	#11718254	C₂²	1	2	yes	—
27	#11718253	C₂²	1	2	yes	—
28	#11718251	C₂²	1	1	yes	characteristic
29	#11718251	C₂²	1	1	yes	characteristic
30	C₈×#1057	C₂²	1	1	yes	characteristic
31	C₈×#1057	C₂²	1	1	yes	characteristic
32	#11718244	C₂²	1	4	yes	—
33	#11718244	C₂²	1	4	yes	—
34	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	#88200	—	2	2	no	—
36	Q₈×Dic₁₃	—	2	2	no	—
37	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
38	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
41	#88151	C₂×C₄	1	4	yes	—
42	#88146	—	2	2	no	—
43	#88144	—	2	2	no	—
44	#88141	C₂³	1	4	yes	—
45	C₄²×D₁₃	C₂³	1	1	yes	characteristic
46	#88123	C₂×C₄	1	2	yes	—
47	#88112	—	2	4	no	—
48	#88112	—	2	1	no	—
49	#88112	D₄	1	2	yes	—
50	#88111	—	2	2	no	—
51	#88111	—	2	1	no	—
52	#88111	D₄	1	2	yes	—
53	#88110	—	2	4	no	—
54	#88110	—	2	2	no	—
55	C₄⋊#1057	—	2	4	no	—
56	C₄⋊#1057	C₂³	1	4	yes	—
57	C₄×#1057	—	2	4	no	—
58	C₄×#1057	C₂³	1	1	yes	characteristic
59	C₄×#1057	C₂³	1	1	yes	characteristic
60	C₄×#1057	C₂³	1	1	yes	characteristic
61	C₄×#1057	C₂³	1	1	yes	characteristic
62	C₁₃×#107	C₂×C₄	1	2	yes	—
63	C₁₃×#107	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
66	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
67	#88050	C₂×C₄	1	4	yes	—
68	#88050	C₂×C₄	1	4	yes	—
69	C₄×#1055	C₂×C₄	1	1	yes	characteristic
70	#17972	—	13	1	no	maximal, Sylow
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
75	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
76	C₂×#1057	—	2	4	no	—
77	C₂×#1057	—	2	1	no	—
78	C₂×#1057	—	2	1	no	—
79	C₂×#1057	C₂×D₄	1	2	yes	—
80	C₂×#1057	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	#6103	—	2	4	no	—
87	C₄⋊Dic₁₃	—	2	2	no	—
88	C₄⋊Dic₁₃	—	2	2	no	—
89	C₄⋊Dic₁₃	—	2	4	no	—
90	#6101	—	2	4	no	—
91	C₄×Dic₁₃	—	2	4	no	—
92	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
93	C₄×Dic₁₃	#41	1	1	yes	characteristic
94	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
95	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
96	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
97	Dic₅₂	—	2	4	no	—
98	Dic₅₂	—	2	4	no	—
99	C₈×D₁₃	—	2	2	no	—
100	C₈×D₁₃	—	2	2	no	—
101	#3170	—	13	1	no	—
102	#2985	—	13	2	no	—
103	#1593	—	13	2	no	—
104	#1591	—	13	2	no	—
105	#1579	—	13	2	no	—
106	C₈×Q₁₆	—	13	2	no	—
107	#1568	—	13	1	no	—
108	#1497	—	13	2	no	—
109	#1450	—	13	1	no	—
110	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
111	Q₈×C₁₃	—	8	2	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×Dic₁₃	—	2	4	no	—
117	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
120	C₄×D₁₃	—	2	1	no	—
121	C₄×D₁₃	—	2	1	no	—
122	C₄×D₁₃	—	2	2	no	—
123	C₄×D₁₃	—	4	1	no	—
124	Dic₂₆	—	2	4	no	—
125	Dic₂₆	—	2	4	no	—
126	#1057	—	2	2	no	—
127	#1057	—	2	2	no	—
128	#1057	—	4	1	no	—
129	#1057	—	4	1	no	—
130	#1057	—	4	4	no	—
131	C₈×C₁₃	#131	1	2	yes	—
132	C₈×C₁₃	#131	1	2	yes	—
133	#1055	—	2	1	no	—
134	#1055	—	2	1	no	—
135	#533	—	13	2	no	—
136	#500	#261	1	1	yes	characteristic
137	#494	—	13	1	no	—
138	#487	—	13	2	no	—
139	#467	—	13	2	no	—
140	#467	—	13	2	no	—
141	#446	—	13	4	no	—
142	#446	—	26	4	no	—
143	C₈×Q₈	—	13	4	no	—
144	C₄×Q₁₆	—	13	2	no	—
145	C₄×Q₁₆	—	13	2	no	—
146	#433	—	26	4	no	—
147	#432	—	13	2	no	—
148	#431	—	13	1	no	—
149	#431	—	13	1	no	—
150	C₂×C₄×C₈	—	13	1	no	—
151	#335	—	13	2	no	—
152	#334	—	13	2	no	—
153	Q₈⋊C₈	—	13	4	no	—
154	Q₈⋊C₈	—	13	4	no	—
155	#322	—	13	1	no	—
156	#322	—	13	1	no	—
157	C₈²	—	13	1	no	—
158	C₈²	—	13	1	no	—
159	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂×D₁₃	—	2	1	no	—
161	C₂×D₁₃	—	2	1	no	—
162	C₂×D₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	4	4	no	—
166	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
167	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
168	Dic₁₃	—	2	1	no	—
169	Dic₁₃	—	2	1	no	—
170	Dic₁₃	—	4	4	no	—
171	Dic₁₃	#578	1	2	yes	—
172	C₂×Q₁₆	—	13	2	no	—
173	C₂×Q₁₆	—	13	2	no	—
174	C₂²×C₈	—	13	1	no	—
175	C₂²×C₈	—	13	1	no	—
176	C₄⋊Q₈	—	13	2	no	—
177	C₄⋊Q₈	C₂×#261	1	2	yes	—
178	#125	—	26	2	no	—
179	C₂²⋊Q₈	—	13	4	no	—
180	C₂²⋊Q₈	—	26	2	no	—
181	C₄×Q₈	—	13	4	no	—
182	C₄×Q₈	—	26	2	no	—
183	#117	—	26	2	no	—
184	C₂×C₄²	—	13	1	no	—
185	#107	C₂×#261	1	2	yes	—
186	#107	C₂×#261	1	2	yes	—
187	C₄⋊C₈	—	13	2	no	—
188	C₄⋊C₈	—	13	4	no	—
189	C₄⋊C₈	—	26	1	no	—
190	C₄⋊C₈	—	26	4	no	—
191	C₄⋊C₈	—	26	4	no	—
192	#103	—	13	4	no	—
193	#103	—	13	4	no	—
194	C₂²⋊C₈	—	13	2	no	—
195	C₂²⋊C₈	—	26	1	no	—
196	C₂²⋊C₈	—	26	2	no	—
197	#97	—	26	2	no	—
198	#97	—	26	4	no	—
199	C₄×C₈	—	13	1	no	—
200	C₄×C₈	—	13	1	no	—
201	C₄×C₈	—	13	1	no	—
202	C₄×C₈	—	13	1	no	—
203	C₄×C₈	—	13	1	no	—
204	C₄×C₈	—	13	1	no	—
205	C₄×C₈	—	13	1	no	—
206	C₄×C₈	—	26	4	no	—
207	C₄×C₈	C₂×#261	1	1	yes	characteristic
208	C₂×C₁₃	#2931	1	1	yes	characteristic
209	C₂×C₁₃	#2950	1	1	yes	characteristic
210	C₂×C₁₃	#2957	1	1	yes	characteristic
211	D₁₃	—	4	1	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	13	4	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	—	13	1	no	—
216	C₂²×C₄	—	13	1	no	—
217	Q₁₆	—	26	4	no	—
218	Q₁₆	—	26	4	no	—
219	C₂×C₈	—	13	2	no	—
220	C₂×C₈	—	13	2	no	—
221	C₂×C₈	—	13	1	no	—
222	C₂×C₈	—	13	1	no	—
223	C₂×C₈	—	26	2	no	—
224	C₂×C₈	—	26	2	no	—
225	C₂×C₈	—	26	1	no	—
226	C₂×C₈	—	26	1	no	—
227	C₂×C₈	—	26	4	no	—
228	C₂×C₈	C₂²×#261	1	1	yes	characteristic
229	C₂×C₈	C₂²×#261	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	26	4	no	—
232	C₄⋊C₄	—	26	4	no	—
233	C₄⋊C₄	—	26	2	no	—
234	C₄⋊C₄	—	26	2	no	—
235	C₄⋊C₄	C₂²×#261	1	4	yes	—
236	C₂²⋊C₄	—	26	4	no	—
237	C₄²	—	13	1	no	—
238	C₄²	—	13	1	no	—
239	C₄²	—	13	1	no	—
240	C₄²	—	26	4	no	—
241	C₄²	C₂²×#261	1	1	yes	characteristic
242	C₁₃	#17972	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	13	1	no	—
244	Q₈	—	8	2	no	—
245	Q₈	—	26	4	no	—
246	Q₈	—	26	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	13	1	no	—
249	C₂×C₄	—	13	1	no	—
250	C₂×C₄	—	13	1	no	—
251	C₂×C₄	—	26	2	no	—
252	C₂×C₄	—	26	1	no	—
253	C₂×C₄	—	26	1	no	—
254	C₂×C₄	—	26	4	no	—
255	C₂×C₄	—	52	1	no	—
256	C₂×C₄	D₄×#261	1	1	yes	characteristic
257	C₂×C₄	D₄×#261	1	1	yes	characteristic
258	C₂×C₄	C₂³×#261	1	1	yes	characteristic
259	C₈	—	26	2	no	—
260	C₈	—	26	2	no	—
261	C₈	—	26	1	no	—
262	C₈	—	26	1	no	—
263	C₈	—	52	4	no	—
264	C₈	—	52	1	no	—
265	C₈	—	52	1	no	—
266	C₈	#88240	1	2	yes	—
267	C₈	#88240	1	2	yes	—
268	C₂²	—	26	1	no	—
269	C₂²	—	26	1	no	—
270	C₂²	—	26	1	no	—
271	C₂²	C₂×D₄×#261	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	4	no	—
275	C₄	—	13	2	no	—
276	C₄	—	26	1	no	—
277	C₄	—	26	1	no	—
278	C₄	—	52	4	no	—
279	C₄	C₂×#88240	1	1	yes	characteristic
280	C₄	C₂×#88240	1	1	yes	characteristic
281	C₂	—	52	1	no	—
282	C₂	#13131561	1	1	yes	characteristic
283	C₂	#13131597	1	1	yes	characteristic
284	C₂	#13131612	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	C₄×D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#500	#261
hypocenter	C₁₃	#17972
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
2. #13130482	C₂	semidirect
3. #13130475	C₂	semidirect
4. #13130431	C₂	semidirect
5. #13130424	C₂	semidirect
242. C₁₃	#17972	semidirect

Maximal subgroups

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

subgroup	index
#13131744	2
#13130482	2
#13130475	2
#13130431	2
#13130424	2
#13130397	2
#13130200	2
#13130040	2
#13125423	2
#17972	13