Subgroups — #16796697

Order 3328.

564 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	#13131747	C₂	1	2	yes	maximal
2	#13130482	C₂	1	2	yes	maximal, complemented
3	#13130448	C₂	1	4	yes	maximal, complemented
4	#13130431	C₂	1	2	yes	maximal, complemented
5	#13130384	C₂	1	1	yes	maximal, characteristic
6	#13130160	C₂	1	2	yes	maximal
7	#13130046	C₂	1	1	yes	maximal, characteristic
8	#13125423	C₂	1	1	yes	maximal, characteristic
9	#11719433	C₂²	1	2	yes	—
10	#11719158	—	2	2	no	—
11	Q₈×#1057	—	2	2	no	—
12	#11719080	C₂²	1	4	yes	—
13	#11719077	—	2	4	no	—
14	#11719074	C₂²	1	4	yes	—
15	#11719051	C₂²	1	2	yes	—
16	#11719051	C₂²	1	1	yes	characteristic
17	C₄×#6118	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	#11718256	C₂²	1	2	yes	—
27	#11718253	C₂²	1	1	yes	characteristic
28	#11718253	C₂²	1	1	yes	characteristic
29	#1057⋊C₈	C₂²	1	2	yes	—
30	#11718251	C₂²	1	2	yes	—
31	#11718244	C₂²	1	4	yes	—
32	#11718244	C₂²	1	4	yes	—
33	C₂×#6118	—	2	1	no	—
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	#88111	—	2	2	no	—
49	#88111	D₄	1	2	yes	—
50	#88110	—	2	4	no	—
51	#88110	—	2	1	no	—
52	C₄⋊#1057	—	2	4	no	—
53	C₄⋊#1057	D₄	1	2	yes	—
54	C₄⋊#1057	C₂³	1	2	yes	—
55	C₄⋊#1057	C₂³	1	2	yes	—
56	C₄×#1057	—	2	4	no	—
57	C₄×#1057	—	2	2	no	—
58	C₄×#1057	C₂³	1	2	yes	—
59	C₄×#1057	C₂³	1	2	yes	—
60	C₁₃×#107	C₂×C₄	1	2	yes	—
61	C₁₃×#107	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
63	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
64	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
65	#88050	C₂×C₄	1	4	yes	—
66	#88050	C₂×C₄	1	4	yes	—
67	C₄×#1055	C₂×C₄	1	1	yes	characteristic
68	#18090	—	13	1	no	maximal, Sylow
69	C₂×Q₈×C₁₃	—	2	2	no	—
70	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
73	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
74	C₂×#1057	—	2	2	no	—
75	C₂×#1057	—	2	2	no	—
76	C₂×#1057	—	2	2	no	—
77	C₂×#1057	C₂×D₄	1	4	yes	—
78	#6118	—	4	2	no	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
83	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	#6103	—	2	4	no	—
85	C₄⋊Dic₁₃	—	2	2	no	—
86	C₄⋊Dic₁₃	—	2	2	no	—
87	C₄⋊Dic₁₃	—	2	4	no	—
88	#6101	—	2	4	no	—
89	C₄×Dic₁₃	—	2	4	no	—
90	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄×Dic₁₃	#41	1	1	yes	characteristic
92	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
93	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
94	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
95	Dic₅₂	—	2	4	no	—
96	Dic₅₂	—	4	2	no	—
97	C₈×D₁₃	—	2	2	no	—
98	C₈×D₁₃	—	4	1	no	—
99	#3170	—	13	1	no	—
100	#2986	—	13	2	no	—
101	#1593	—	13	2	no	—
102	#1584	—	13	4	no	—
103	#1579	—	13	2	no	—
104	#1564	—	13	1	no	—
105	#1485	—	13	2	no	—
106	#1451	—	13	1	no	—
107	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
108	Q₈×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×Dic₁₃	—	2	4	no	—
114	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
117	C₄×D₁₃	—	2	1	no	—
118	C₄×D₁₃	—	2	1	no	—
119	C₄×D₁₃	—	2	2	no	—
120	C₄×D₁₃	—	2	2	no	—
121	Dic₂₆	—	2	4	no	—
122	Dic₂₆	—	2	4	no	—
123	#1057	—	2	4	no	—
124	#1057	—	4	2	no	—
125	#1057	—	4	2	no	—
126	#1057	—	4	2	no	—
127	C₈×C₁₃	—	2	1	no	—
128	C₈×C₁₃	#131	1	2	yes	—
129	#1055	—	2	1	no	—
130	#1055	—	2	1	no	—
131	#533	—	13	2	no	—
132	#500	#261	1	1	yes	characteristic
133	#494	—	13	1	no	—
134	#487	—	13	2	no	—
135	#467	—	13	2	no	—
136	#467	—	13	2	no	—
137	#446	—	13	4	no	—
138	#446	—	26	2	no	—
139	C₈×Q₈	—	13	4	no	—
140	C₈×Q₈	—	26	2	no	—
141	C₄×Q₁₆	—	13	2	no	—
142	C₄×Q₁₆	—	13	2	no	—
143	#433	—	26	4	no	—
144	#431	—	13	1	no	—
145	#431	—	13	2	no	—
146	C₄×M₁₆	—	13	1	no	—
147	C₂×C₄×C₈	—	13	1	no	—
148	#335	—	13	2	no	—
149	#334	—	13	1	no	—
150	#334	—	13	1	no	—
151	Q₈⋊C₈	—	13	4	no	—
152	Q₈⋊C₈	—	13	4	no	—
153	#322	—	13	2	no	—
154	#322	—	13	2	no	—
155	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂×D₁₃	—	2	1	no	—
157	C₂×D₁₃	—	2	1	no	—
158	C₂×D₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	1	no	—
160	C₄×C₁₃	—	4	4	no	—
161	C₄×C₁₃	#444	1	2	yes	—
162	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
163	C₄×C₁₃	#568	1	1	yes	characteristic
164	Dic₁₃	—	2	1	no	—
165	Dic₁₃	—	2	1	no	—
166	Dic₁₃	—	4	4	no	—
167	Dic₁₃	#578	1	2	yes	—
168	C₂×Q₁₆	—	13	2	no	—
169	C₂×Q₁₆	—	13	2	no	—
170	C₂×M₁₆	—	26	1	no	—
171	C₂²×C₈	—	13	1	no	—
172	C₂²×C₈	—	13	1	no	—
173	C₄⋊Q₈	—	13	2	no	—
174	C₄⋊Q₈	C₂×#261	1	2	yes	—
175	#125	—	26	2	no	—
176	C₂²⋊Q₈	—	13	4	no	—
177	C₂²⋊Q₈	—	26	2	no	—
178	C₄×Q₈	—	13	4	no	—
179	C₄×Q₈	—	26	2	no	—
180	#117	—	26	2	no	—
181	C₂×C₄²	—	13	1	no	—
182	#107	C₂×#261	1	2	yes	—
183	#107	C₂×#261	1	2	yes	—
184	C₄⋊C₈	—	13	2	no	—
185	C₄⋊C₈	—	13	2	no	—
186	C₄⋊C₈	—	13	2	no	—
187	C₄⋊C₈	—	26	4	no	—
188	C₄⋊C₈	—	26	4	no	—
189	#103	—	13	4	no	—
190	#103	—	13	4	no	—
191	C₂²⋊C₈	—	13	2	no	—
192	C₂²⋊C₈	—	26	2	no	—
193	#97	—	26	1	no	—
194	#97	—	26	4	no	—
195	C₄×C₈	—	13	2	no	—
196	C₄×C₈	—	13	1	no	—
197	C₄×C₈	—	13	1	no	—
198	C₄×C₈	—	13	2	no	—
199	C₄×C₈	—	13	1	no	—
200	C₄×C₈	—	26	2	no	—
201	C₄×C₈	—	26	4	no	—
202	C₄×C₈	C₂×#261	1	1	yes	characteristic
203	C₂×C₁₃	#2935	1	1	yes	characteristic
204	C₂×C₁₃	#2950	1	1	yes	characteristic
205	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
206	D₁₃	—	4	1	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	13	4	no	—
209	C₂²×C₄	—	13	1	no	—
210	C₂²×C₄	—	13	1	no	—
211	C₂²×C₄	—	13	1	no	—
212	Q₁₆	—	26	4	no	—
213	Q₁₆	—	52	2	no	—
214	M₁₆	—	52	2	no	—
215	C₂×C₈	—	13	4	no	—
216	C₂×C₈	—	13	1	no	—
217	C₂×C₈	—	13	1	no	—
218	C₂×C₈	—	26	2	no	—
219	C₂×C₈	—	26	2	no	—
220	C₂×C₈	—	26	2	no	—
221	C₂×C₈	—	26	2	no	—
222	C₂×C₈	—	52	1	no	—
223	C₂×C₈	C₂²×#261	1	1	yes	characteristic
224	C₂×C₈	C₂²×#261	1	1	yes	characteristic
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	26	4	no	—
227	C₄⋊C₄	—	26	4	no	—
228	C₄⋊C₄	—	26	2	no	—
229	C₄⋊C₄	—	26	2	no	—
230	C₄⋊C₄	C₂²×#261	1	4	yes	—
231	C₂²⋊C₄	—	26	4	no	—
232	C₄²	—	13	1	no	—
233	C₄²	—	13	1	no	—
234	C₄²	—	13	1	no	—
235	C₄²	—	26	4	no	—
236	C₄²	C₂²×#261	1	1	yes	characteristic
237	C₁₃	#18090	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	13	1	no	—
239	Q₈	—	4	4	no	—
240	Q₈	—	26	4	no	—
241	Q₈	—	26	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	13	1	no	—
244	C₂×C₄	—	13	1	no	—
245	C₂×C₄	—	13	1	no	—
246	C₂×C₄	—	26	2	no	—
247	C₂×C₄	—	26	2	no	—
248	C₂×C₄	—	26	1	no	—
249	C₂×C₄	—	26	1	no	—
250	C₂×C₄	—	26	4	no	—
251	C₂×C₄	D₄×#261	1	1	yes	characteristic
252	C₂×C₄	D₄×#261	1	1	yes	characteristic
253	C₂×C₄	C₂³×#261	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	26	4	no	—
256	C₈	—	26	1	no	—
257	C₈	—	26	1	no	—
258	C₈	—	52	2	no	—
259	C₈	—	52	2	no	—
260	C₈	—	52	2	no	—
261	C₈	#88240	1	2	yes	—
262	C₂²	—	26	1	no	—
263	C₂²	—	26	1	no	—
264	C₂²	—	26	1	no	—
265	C₂²	C₂×D₄×#261	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	4	4	no	—
268	C₄	—	13	2	no	—
269	C₄	—	26	1	no	—
270	C₄	—	26	1	no	—
271	C₄	—	52	4	no	—
272	C₄	#11719104	1	2	yes	—
273	C₄	C₂×#88240	1	1	yes	characteristic
274	C₄	#11719625	1	1	yes	characteristic
275	C₂	—	52	1	no	—
276	C₂	#13131582	1	1	yes	characteristic
277	C₂	#13131597	1	1	yes	characteristic
278	C₂	C₂×#11719104	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	#18090
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. #13130448	C₂	semidirect
4. #13130431	C₂	semidirect
237. C₁₃	#18090	semidirect

Maximal subgroups

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

subgroup	index
#13131747	2
#13130482	2
#13130448	2
#13130431	2
#13130384	2
#13130160	2
#13130046	2
#13125423	2
#18090	13