Subgroups — #16796674

Order 3328.

572 conjugacy classes of subgroups.

The subgroup lattice has 290 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	#13130475	C₂	1	2	yes	maximal, complemented
3	#13130460	C₂	1	4	yes	maximal, complemented
4	#13130424	C₂	1	2	yes	maximal, complemented
5	#13130386	C₂	1	1	yes	maximal, characteristic
6	#13130160	C₂	1	2	yes	maximal
7	C₈×#6118	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	#11718255	C₂²	1	2	yes	—
27	#11718254	C₂²	1	1	yes	characteristic
28	#11718254	C₂²	1	1	yes	characteristic
29	#1057⋊C₈	C₂²	1	2	yes	—
30	C₈×#1057	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	1	yes	characteristic
54	C₄⋊#1057	D₄	1	1	yes	characteristic
55	C₄⋊#1057	C₂³	1	2	yes	—
56	C₄⋊#1057	C₂³	1	2	yes	—
57	C₄×#1057	—	2	4	no	—
58	C₄×#1057	—	2	2	no	—
59	C₄×#1057	C₂³	1	2	yes	—
60	C₄×#1057	C₂³	1	2	yes	—
61	C₁₃×#107	C₂×C₄	1	2	yes	—
62	C₁₃×#107	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
64	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
65	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
66	#88050	C₂×C₄	1	4	yes	—
67	#88050	C₂×C₄	1	4	yes	—
68	C₄×#1055	C₂×C₄	1	1	yes	characteristic
69	#18086	—	13	1	no	maximal, Sylow
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
74	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
75	C₂×#1057	—	2	2	no	—
76	C₂×#1057	—	2	2	no	—
77	C₂×#1057	—	2	2	no	—
78	C₂×#1057	C₂×D₄	1	2	yes	—
79	C₂×#1057	C₂×D₄	1	2	yes	—
80	#6118	—	2	4	no	—
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	—	4	2	no	—
99	C₈×D₁₃	—	2	2	no	—
100	C₈×D₁₃	—	4	1	no	—
101	#3170	—	13	1	no	—
102	#2986	—	13	2	no	—
103	#1591	—	13	2	no	—
104	#1587	—	13	4	no	—
105	C₈×Q₁₆	—	13	2	no	—
106	#1564	—	13	1	no	—
107	#1485	—	13	2	no	—
108	C₈×M₁₆	—	13	1	no	—
109	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
110	Q₈×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×Dic₁₃	—	2	4	no	—
116	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
119	C₄×D₁₃	—	2	1	no	—
120	C₄×D₁₃	—	2	1	no	—
121	C₄×D₁₃	—	2	2	no	—
122	C₄×D₁₃	—	2	1	no	—
123	C₄×D₁₃	—	2	1	no	—
124	Dic₂₆	—	2	4	no	—
125	Dic₂₆	—	2	4	no	—
126	#1057	—	2	4	no	—
127	#1057	—	2	2	no	—
128	#1057	—	2	2	no	—
129	#1057	—	4	2	no	—
130	#1057	—	4	2	no	—
131	C₈×C₁₃	—	2	1	no	—
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	2	no	—
143	C₈×Q₈	—	13	4	no	—
144	C₈×Q₈	—	26	2	no	—
145	C₄×Q₁₆	—	13	2	no	—
146	C₄×Q₁₆	—	13	2	no	—
147	#433	—	26	4	no	—
148	#431	—	13	1	no	—
149	#431	—	13	2	no	—
150	C₄×M₁₆	—	13	1	no	—
151	C₂×C₄×C₈	—	13	1	no	—
152	#335	—	13	1	no	—
153	#335	—	13	1	no	—
154	#334	—	13	2	no	—
155	Q₈⋊C₈	—	13	4	no	—
156	Q₈⋊C₈	—	13	4	no	—
157	#322	—	13	2	no	—
158	C₈²	—	13	2	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	—	4	4	no	—
165	C₄×C₁₃	#443	1	2	yes	—
166	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
167	C₄×C₁₃	#568	1	1	yes	characteristic
168	Dic₁₃	—	2	1	no	—
169	Dic₁₃	—	2	1	no	—
170	Dic₁₃	—	4	4	no	—
171	Dic₁₃	#578	1	1	yes	characteristic
172	Dic₁₃	#578	1	1	yes	characteristic
173	C₂×Q₁₆	—	13	2	no	—
174	C₂×Q₁₆	—	13	2	no	—
175	C₂×M₁₆	—	26	1	no	—
176	C₂²×C₈	—	13	1	no	—
177	C₂²×C₈	—	13	1	no	—
178	C₄⋊Q₈	—	13	2	no	—
179	C₄⋊Q₈	C₂×#261	1	2	yes	—
180	#125	—	26	2	no	—
181	C₂²⋊Q₈	—	13	4	no	—
182	C₂²⋊Q₈	—	26	2	no	—
183	C₄×Q₈	—	13	4	no	—
184	C₄×Q₈	—	26	2	no	—
185	#117	—	26	2	no	—
186	C₂×C₄²	—	13	1	no	—
187	#107	C₂×#261	1	2	yes	—
188	#107	C₂×#261	1	2	yes	—
189	C₄⋊C₈	—	13	1	no	—
190	C₄⋊C₈	—	13	1	no	—
191	C₄⋊C₈	—	13	2	no	—
192	C₄⋊C₈	—	13	2	no	—
193	C₄⋊C₈	—	26	4	no	—
194	C₄⋊C₈	—	26	4	no	—
195	#103	—	13	4	no	—
196	#103	—	13	4	no	—
197	C₂²⋊C₈	—	13	2	no	—
198	C₂²⋊C₈	—	26	2	no	—
199	#97	—	26	1	no	—
200	#97	—	26	4	no	—
201	C₄×C₈	—	13	2	no	—
202	C₄×C₈	—	13	1	no	—
203	C₄×C₈	—	13	1	no	—
204	C₄×C₈	—	13	2	no	—
205	C₄×C₈	—	13	1	no	—
206	C₄×C₈	—	26	2	no	—
207	C₄×C₈	—	26	4	no	—
208	C₄×C₈	C₂×#261	1	1	yes	characteristic
209	C₂×C₁₃	#2935	1	1	yes	characteristic
210	C₂×C₁₃	#2950	1	1	yes	characteristic
211	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
212	D₁₃	—	4	1	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	13	4	no	—
215	C₂²×C₄	—	13	1	no	—
216	C₂²×C₄	—	13	1	no	—
217	C₂²×C₄	—	13	1	no	—
218	Q₁₆	—	26	4	no	—
219	Q₁₆	—	52	2	no	—
220	M₁₆	—	26	4	no	—
221	C₂×C₈	—	13	2	no	—
222	C₂×C₈	—	13	2	no	—
223	C₂×C₈	—	13	1	no	—
224	C₂×C₈	—	13	1	no	—
225	C₂×C₈	—	26	2	no	—
226	C₂×C₈	—	26	2	no	—
227	C₂×C₈	—	26	2	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	—	52	1	no	—
230	C₂×C₈	C₂²×#261	1	1	yes	characteristic
231	C₂×C₈	C₂²×#261	1	1	yes	characteristic
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	26	4	no	—
234	C₄⋊C₄	—	26	4	no	—
235	C₄⋊C₄	—	26	2	no	—
236	C₄⋊C₄	—	26	2	no	—
237	C₄⋊C₄	C₂²×#261	1	4	yes	—
238	C₂²⋊C₄	—	26	4	no	—
239	C₄²	—	13	1	no	—
240	C₄²	—	13	1	no	—
241	C₄²	—	13	1	no	—
242	C₄²	—	26	4	no	—
243	C₄²	C₂²×#261	1	1	yes	characteristic
244	C₁₃	#18086	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	13	1	no	—
246	Q₈	—	4	4	no	—
247	Q₈	—	26	4	no	—
248	Q₈	—	26	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	13	1	no	—
251	C₂×C₄	—	13	1	no	—
252	C₂×C₄	—	13	1	no	—
253	C₂×C₄	—	26	1	no	—
254	C₂×C₄	—	26	1	no	—
255	C₂×C₄	—	26	2	no	—
256	C₂×C₄	—	26	1	no	—
257	C₂×C₄	—	26	1	no	—
258	C₂×C₄	—	26	4	no	—
259	C₂×C₄	D₄×#261	1	1	yes	characteristic
260	C₂×C₄	D₄×#261	1	1	yes	characteristic
261	C₂×C₄	C₂³×#261	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	26	2	no	—
264	C₈	—	26	2	no	—
265	C₈	—	26	4	no	—
266	C₈	—	26	1	no	—
267	C₈	—	26	1	no	—
268	C₈	—	52	2	no	—
269	C₈	—	52	2	no	—
270	C₈	#88240	1	2	yes	—
271	C₂²	—	26	1	no	—
272	C₂²	—	26	1	no	—
273	C₂²	—	26	1	no	—
274	C₂²	C₂×D₄×#261	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	—	13	1	no	—
278	C₄	—	13	1	no	—
279	C₄	—	26	1	no	—
280	C₄	—	26	1	no	—
281	C₄	—	52	4	no	—
282	C₄	#11719103	1	2	yes	—
283	C₄	C₂×#88240	1	1	yes	characteristic
284	C₄	#11719625	1	1	yes	characteristic
285	C₂	—	52	1	no	—
286	C₂	#13131582	1	1	yes	characteristic
287	C₂	#13131597	1	1	yes	characteristic
288	C₂	C₂×#11719103	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	#18086
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. #13130475	C₂	semidirect
3. #13130460	C₂	semidirect
4. #13130424	C₂	semidirect
244. C₁₃	#18086	semidirect

Maximal subgroups

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

subgroup	index
#13131747	2
#13130475	2
#13130460	2
#13130424	2
#13130386	2
#13130160	2
C₈×#6118	2
#13125423	2
#18086	13