Subgroups — #16796551

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131630	C₂	1	2	yes	maximal
2	#13130476	C₂	1	2	yes	maximal, complemented
3	#13130449	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#1057	C₂	1	2	yes	maximal, complemented
5	D₁₃⋊#335	C₂	1	1	yes	maximal, characteristic
6	#13130145	C₂	1	2	yes	maximal
7	C₈×#6118	C₂	1	1	yes	maximal, characteristic
8	#13125420	C₂	1	1	yes	maximal, characteristic
9	#11719433	C₂²	1	2	yes	—
10	#11719158	C₂²	1	4	yes	—
11	Q₈×#1057	C₂²	1	4	yes	—
12	#11719079	—	2	2	no	—
13	#11719078	—	2	4	no	—
14	#11719072	—	2	2	no	—
15	#11719051	C₂²	1	1	yes	characteristic
16	D₁₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
17	C₄×#6118	C₂²	1	1	yes	characteristic
18	C₁₃×#494	C₄	1	1	yes	characteristic
19	#11718849	C₄	1	2	yes	—
20	#11718849	C₄	1	2	yes	—
21	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
22	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
23	#11718742	C₄	1	2	yes	—
24	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
25	#11718343	C₄	1	1	yes	characteristic
26	Q₈⋊#1057	C₂²	1	4	yes	—
27	Q₈⋊#1057	C₂²	1	4	yes	—
28	#11718256	C₂²	1	2	yes	—
29	#11718254	C₂²	1	1	yes	characteristic
30	#11718254	C₂²	1	1	yes	characteristic
31	#1057⋊C₈	C₂²	1	2	yes	—
32	C₈×#1057	C₂²	1	2	yes	—
33	C₂×#6118	D₄	1	1	yes	characteristic
34	C₂×#6118	D₄	1	1	yes	characteristic
35	C₂×D₁₃⋊C₈	—	2	2	no	—
36	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
38	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	#88200	C₂×C₄	1	4	yes	—
40	Q₈×Dic₁₃	C₂³	1	4	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
43	#88151	—	2	2	no	—
44	#88146	—	2	2	no	—
45	#88144	—	2	2	no	—
46	#88141	—	2	2	no	—
47	C₄²×D₁₃	C₂³	1	1	yes	characteristic
48	#88123	C₂×C₄	1	2	yes	—
49	#88112	—	2	4	no	—
50	#88112	—	2	2	no	—
51	#88111	—	2	1	no	—
52	#88110	—	2	4	no	—
53	#88110	D₄	1	2	yes	—
54	C₄⋊#1057	—	2	4	no	—
55	C₄⋊#1057	—	2	1	no	—
56	C₄⋊#1057	C₂³	1	2	yes	—
57	C₄⋊#1057	C₂³	1	2	yes	—
58	C₄×#1057	—	2	4	no	—
59	C₄×#1057	C₂³	1	2	yes	—
60	C₄×#1057	C₂³	1	2	yes	—
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	#88075	C₂×C₄	1	4	yes	—
63	#88075	C₂×C₄	1	4	yes	—
64	#88058	C₂×C₄	1	2	yes	—
65	#88058	C₂×C₄	1	2	yes	—
66	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
67	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
68	C₄×#1055	C₂×C₄	1	1	yes	characteristic
69	#18068	—	13	1	no	maximal, Sylow
70	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
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	—	2	2	no	—
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	2	no	—
81	#6118	Q₁₆	1	4	yes	complemented
82	D₁₃⋊C₈	—	4	4	no	—
83	C₁₃×Q₁₆	—	2	4	no	—
84	C₁₃×Q₁₆	—	4	2	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	#6103	—	2	4	no	—
91	C₄⋊Dic₁₃	—	2	2	no	—
92	C₄⋊Dic₁₃	—	2	4	no	—
93	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
94	#6101	—	2	4	no	—
95	C₄×Dic₁₃	—	2	4	no	—
96	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄×Dic₁₃	#41	1	1	yes	characteristic
98	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
99	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
100	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
101	C₈×D₁₃	—	2	2	no	—
102	C₈×D₁₃	—	4	1	no	—
103	#3170	—	13	1	no	—
104	#2962	—	13	2	no	—
105	#1591	—	13	2	no	—
106	#1584	—	13	4	no	—
107	C₈×Q₁₆	—	13	2	no	—
108	C₂×#335	—	13	1	no	—
109	#1481	—	13	2	no	—
110	C₈×M₁₆	—	13	1	no	—
111	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
112	Q₈×C₁₃	—	2	4	no	—
113	Q₈×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×Dic₁₃	—	2	4	no	—
119	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
122	C₄×D₁₃	—	2	1	no	—
123	C₄×D₁₃	—	2	1	no	—
124	C₄×D₁₃	—	4	1	no	—
125	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
126	C₄×D₁₃	#135	1	2	yes	—
127	Dic₂₆	—	8	2	no	—
128	#1057	—	2	2	no	—
129	#1057	—	4	2	no	—
130	#1057	—	4	2	no	—
131	#1057	—	4	2	no	—
132	#1057	C₂×Q₁₆	1	4	yes	—
133	C₈×C₁₃	—	2	1	no	—
134	C₈×C₁₃	#131	1	2	yes	—
135	#1055	—	2	1	no	—
136	#1055	—	2	1	no	—
137	#533	—	13	2	no	—
138	#500	—	13	1	no	—
139	#494	#261	1	1	yes	characteristic
140	#487	—	13	2	no	—
141	#467	—	13	2	no	—
142	#467	—	13	2	no	—
143	#446	—	13	4	no	—
144	#446	—	26	4	no	—
145	C₈×Q₈	—	13	4	no	—
146	C₄×Q₁₆	—	13	2	no	—
147	C₄×Q₁₆	—	13	2	no	—
148	#431	—	13	1	no	—
149	#424	—	26	2	no	—
150	C₂×C₄⋊C₈	—	13	2	no	—
151	#405	—	26	2	no	—
152	C₄×M₁₆	—	13	1	no	—
153	C₂×C₄×C₈	—	13	1	no	—
154	#335	—	13	1	no	—
155	#335	—	13	2	no	—
156	#335	—	13	1	no	—
157	Q₈⋊C₈	—	13	4	no	—
158	Q₈⋊C₈	—	13	4	no	—
159	#322	—	13	2	no	—
160	C₈²	—	13	2	no	—
161	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂×D₁₃	—	2	1	no	—
163	C₂×D₁₃	—	2	1	no	—
164	C₂×D₁₃	C₄×Q₁₆	1	2	yes	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	4	4	no	—
167	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
168	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
169	C₄×C₁₃	#568	1	1	yes	characteristic
170	Dic₁₃	—	2	1	no	—
171	Dic₁₃	—	2	1	no	—
172	Dic₁₃	—	4	4	no	—
173	Dic₁₃	C₂²×Q₁₆	1	1	yes	characteristic
174	Dic₁₃	C₂×#135	1	1	yes	characteristic
175	C₂×Q₁₆	C₂×#261	1	2	yes	—
176	C₂×Q₁₆	C₂×#261	1	2	yes	—
177	C₂×M₁₆	—	13	1	no	—
178	C₂×M₁₆	—	13	1	no	—
179	C₂²×C₈	—	13	1	no	—
180	C₂²×C₈	—	13	1	no	—
181	C₂²×C₈	—	26	2	no	—
182	C₄⋊Q₈	—	13	2	no	—
183	C₄⋊Q₈	C₂×#261	1	2	yes	—
184	#125	—	26	2	no	—
185	C₂²⋊Q₈	—	13	4	no	—
186	C₂²⋊Q₈	—	26	2	no	—
187	C₄×Q₈	—	13	4	no	—
188	C₄×Q₈	—	26	2	no	—
189	#117	—	26	2	no	—
190	C₂×C₄²	—	13	1	no	—
191	#107	—	13	2	no	—
192	#107	—	13	2	no	—
193	C₄⋊C₈	—	13	2	no	—
194	C₄⋊C₈	—	13	2	no	—
195	C₄⋊C₈	—	26	1	no	—
196	C₄⋊C₈	—	26	2	no	—
197	C₄⋊C₈	—	26	4	no	—
198	C₄⋊C₈	—	26	4	no	—
199	#103	—	13	4	no	—
200	#103	—	13	4	no	—
201	C₂²⋊C₈	—	26	1	no	—
202	#97	—	13	2	no	—
203	#97	—	26	4	no	—
204	C₄×C₈	—	13	2	no	—
205	C₄×C₈	—	13	1	no	—
206	C₄×C₈	—	13	1	no	—
207	C₄×C₈	—	13	2	no	—
208	C₄×C₈	—	13	1	no	—
209	C₄×C₈	—	26	4	no	—
210	C₄×C₈	C₂×#261	1	1	yes	characteristic
211	C₂×C₁₃	#2935	1	1	yes	characteristic
212	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
213	C₂×C₁₃	#2957	1	1	yes	characteristic
214	D₁₃	—	2	2	no	—
215	C₂×Q₈	—	26	2	no	—
216	C₂×Q₈	C₂²×#261	1	4	yes	—
217	C₂²×C₄	—	13	1	no	—
218	C₂²×C₄	—	13	1	no	—
219	C₂²×C₄	—	13	1	no	—
220	Q₁₆	—	2	4	no	—
221	Q₁₆	—	4	2	no	—
222	M₁₆	—	13	4	no	—
223	M₁₆	—	26	2	no	—
224	C₂×C₈	—	13	2	no	—
225	C₂×C₈	—	13	2	no	—
226	C₂×C₈	—	13	1	no	—
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	—	26	2	no	—
230	C₂×C₈	—	26	2	no	—
231	C₂×C₈	—	26	2	no	—
232	C₂×C₈	—	52	1	no	—
233	C₂×C₈	—	52	4	no	—
234	C₂×C₈	C₂²×#261	1	1	yes	characteristic
235	C₂×C₈	C₂²×#261	1	1	yes	characteristic
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	13	4	no	—
239	C₄⋊C₄	—	26	4	no	—
240	C₄⋊C₄	—	26	4	no	—
241	C₄⋊C₄	—	26	2	no	—
242	C₂²⋊C₄	—	26	4	no	—
243	C₄²	—	13	1	no	—
244	C₄²	—	13	1	no	—
245	C₄²	—	13	1	no	—
246	C₄²	—	26	4	no	—
247	C₄²	C₂²×#261	1	1	yes	characteristic
248	C₁₃	#18068	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	13	1	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	2	4	no	—
252	Q₈	—	104	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	13	1	no	—
255	C₂×C₄	—	13	2	no	—
256	C₂×C₄	—	13	2	no	—
257	C₂×C₄	—	13	1	no	—
258	C₂×C₄	—	13	1	no	—
259	C₂×C₄	—	26	1	no	—
260	C₂×C₄	—	26	1	no	—
261	C₂×C₄	—	26	4	no	—
262	C₂×C₄	—	52	1	no	—
263	C₂×C₄	D₄×#261	1	1	yes	characteristic
264	C₂×C₄	D₄×#261	1	1	yes	characteristic
265	C₂×C₄	C₂³×#261	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	13	4	no	—
268	C₈	—	26	2	no	—
269	C₈	—	26	1	no	—
270	C₈	—	26	1	no	—
271	C₈	—	52	2	no	—
272	C₈	—	52	2	no	—
273	C₈	—	52	2	no	—
274	C₈	#88238	1	2	yes	—
275	C₂²	—	13	2	no	—
276	C₂²	—	26	1	no	—
277	C₂²	—	26	1	no	—
278	C₂²	C₂×D₄×#261	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	4	4	no	—
281	C₄	—	13	1	no	—
282	C₄	—	13	1	no	—
283	C₄	—	26	1	no	—
284	C₄	—	26	1	no	—
285	C₄	—	52	4	no	—
286	C₄	Q₁₆×#261	1	2	yes	—
287	C₄	C₂×#88238	1	1	yes	characteristic
288	C₄	#11719621	1	1	yes	characteristic
289	C₂	—	26	2	no	—
290	C₂	#13131579	1	1	yes	characteristic
291	C₂	C₂×Q₁₆×#261	1	1	yes	characteristic
292	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×#494	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₃	C₄×D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#494	#261
hypocenter	C₁₃	#18068
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #13130476	C₂	semidirect
3. #13130449	C₂	semidirect
4. Q₁₆×#1057	C₂	semidirect
81. #6118	Q₁₆	semidirect
248. C₁₃	#18068	semidirect

Maximal subgroups

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

subgroup	index
#13131630	2
#13130476	2
#13130449	2
Q₁₆×#1057	2
D₁₃⋊#335	2
#13130145	2
C₈×#6118	2
#13125420	2
#18068	13