Subgroups — #16784431

Order 3328.

404 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	#13130768	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1057	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718254	C₂	1	2	yes	maximal
4	#13129666	C₂	1	2	yes	maximal
5	C₄⋊C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#1057	—	2	1	no	—
7	C₂×C₄⋊#1057	—	2	1	no	—
8	C₂×C₄⋊#1057	C₂²	1	2	yes	—
9	C₂×C₄⋊#1057	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#1057	C₂²	1	1	yes	characteristic
11	C₂×C₄×#1057	—	2	1	no	—
12	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
14	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
16	#11718290	—	2	1	no	—
17	#11718254	—	2	4	no	—
18	#11718142	C₄	1	2	yes	—
19	C₂²×#1057	—	2	1	no	—
20	C₂²×#1057	—	2	1	no	—
21	C₂²×#1057	—	2	2	no	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
30	C₂²×#1055	C₂×C₄	1	2	yes	—
31	C₄⋊#1057	—	2	2	no	—
32	C₄⋊#1057	—	2	1	no	—
33	C₄⋊#1057	—	2	1	no	—
34	C₄⋊#1057	—	2	2	no	—
35	C₄⋊#1057	—	4	2	no	—
36	C₄⋊#1057	—	4	2	no	—
37	C₄⋊#1057	D₄	1	2	yes	—
38	C₄⋊#1057	Q₈	1	2	yes	—
39	C₄×#1057	—	4	2	no	—
40	C₁₃×C₄⋊C₈	C₈	1	4	yes	complemented
41	C₈×Dic₁₃	—	2	2	no	—
42	C₈×Dic₁₃	—	2	2	no	—
43	C₄⋊#1055	—	2	2	no	—
44	#9601	—	13	1	no	maximal, Sylow
45	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×#1057	—	2	2	no	—
53	C₂×#1057	—	2	2	no	—
54	C₂×#1057	—	4	1	no	—
55	C₂×#1057	—	4	1	no	—
56	C₂×#1057	—	4	1	no	—
57	C₂×#1057	—	4	1	no	—
58	C₂×#1057	—	4	2	no	—
59	C₂×#1057	—	4	2	no	—
60	C₂×#1057	—	4	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
64	C₄²×C₁₃	C₂×C₈	1	2	yes	—
65	C₄²×C₁₃	M₁₆	1	2	yes	—
66	C₄×Dic₁₃	—	2	2	no	—
67	C₄×Dic₁₃	—	2	1	no	—
68	C₄×Dic₁₃	—	2	1	no	—
69	C₄×Dic₁₃	—	2	1	no	—
70	C₄×Dic₁₃	—	2	1	no	—
71	C₄×Dic₁₃	—	4	1	no	—
72	C₄×Dic₁₃	D₈	1	2	yes	—
73	C₄×Dic₁₃	Q₁₆	1	2	yes	—
74	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
75	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
76	C₄×Dic₁₃	#41	1	1	yes	characteristic
77	C₄×Dic₁₃	#41	1	1	yes	characteristic
78	C₂×#1055	—	2	2	no	—
79	C₂×#1055	—	2	2	no	—
80	C₂×#1055	—	2	2	no	—
81	#1842	—	13	1	no	—
82	C₄×C₄⋊C₈	—	13	1	no	—
83	C₄×C₄⋊C₈	—	13	1	no	—
84	C₂×#335	—	13	2	no	—
85	#1277	—	13	2	no	—
86	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
90	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₁₃	#107	1	2	yes	—
92	C₂×C₄×C₁₃	#108	1	2	yes	—
93	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
99	C₂×Dic₁₃	—	2	1	no	—
100	C₂×Dic₁₃	—	2	1	no	—
101	C₂×Dic₁₃	—	2	1	no	—
102	C₂×Dic₁₃	—	2	1	no	—
103	C₂×Dic₁₃	—	2	1	no	—
104	C₂×Dic₁₃	—	2	1	no	—
105	C₂×Dic₁₃	—	2	1	no	—
106	C₂×Dic₁₃	#107	1	2	yes	—
107	C₂×Dic₁₃	#121	1	1	yes	characteristic
108	C₂×Dic₁₃	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×Dic₁₃	#125	1	1	yes	characteristic
110	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
113	C₂×Dic₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×Dic₁₃	#137	1	1	yes	characteristic
115	#1057	—	4	2	no	—
116	#1057	—	4	2	no	—
117	#1057	—	8	2	no	—
118	C₈×C₁₃	—	2	4	no	—
119	#1055	—	4	2	no	—
120	C₂×C₄⋊C₈	—	13	1	no	—
121	C₂×C₄⋊C₈	—	13	1	no	—
122	C₂×C₄⋊C₈	—	13	2	no	—
123	C₂×C₄⋊C₈	—	13	1	no	—
124	C₂×C₄⋊C₈	—	26	1	no	—
125	C₂×C₄⋊C₈	—	26	1	no	—
126	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
127	C₂×C₄×C₈	—	13	2	no	—
128	C₂×C₄×C₈	—	26	1	no	—
129	C₄³	—	13	1	no	—
130	#336	—	13	2	no	—
131	#336	—	26	1	no	—
132	#335	—	26	4	no	—
133	C₂²×C₁₃	#389	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
135	C₂²×C₁₃	#424	1	1	yes	characteristic
136	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
137	C₂²×C₁₃	#428	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
139	C₂²×C₁₃	#430	1	1	yes	characteristic
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	#335	1	4	yes	complemented
142	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
143	C₄×C₁₃	#436	1	1	yes	characteristic
144	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
145	C₄×C₁₃	#446	1	1	yes	characteristic
146	Dic₁₃	—	2	2	no	—
147	Dic₁₃	—	2	1	no	—
148	Dic₁₃	—	2	1	no	—
149	Dic₁₃	—	4	1	no	—
150	Dic₁₃	#459	1	1	yes	characteristic
151	Dic₁₃	#462	1	1	yes	characteristic
152	Dic₁₃	#474	1	1	yes	characteristic
153	Dic₁₃	#477	1	1	yes	characteristic
154	C₂²×C₈	—	13	2	no	—
155	C₂²×C₈	—	26	1	no	—
156	C₂²×C₈	—	26	1	no	—
157	C₂²×C₈	—	26	2	no	—
158	C₂²×C₈	C₂×#261	1	2	yes	—
159	C₂×C₄²	—	13	1	no	—
160	C₂×C₄²	—	13	1	no	—
161	C₂×C₄²	—	13	1	no	—
162	C₂×C₄²	—	13	1	no	—
163	C₂×C₄²	—	13	1	no	—
164	C₂×C₄²	—	13	1	no	—
165	C₂×C₄²	C₂×#261	1	1	yes	characteristic
166	C₄⋊C₈	—	13	2	no	—
167	C₄⋊C₈	—	13	2	no	—
168	C₄⋊C₈	—	26	1	no	—
169	C₄⋊C₈	—	26	1	no	—
170	C₄⋊C₈	—	26	2	no	—
171	C₄⋊C₈	—	26	2	no	—
172	C₄⋊C₈	—	26	2	no	—
173	C₄⋊C₈	—	52	2	no	—
174	C₄⋊C₈	—	52	2	no	—
175	C₄⋊C₈	#1057	1	4	yes	complemented
176	C₄×C₈	—	26	2	no	—
177	C₄×C₈	—	26	2	no	—
178	C₄×C₈	—	52	2	no	—
179	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
180	C₂×C₁₃	#1565	1	1	yes	characteristic
181	C₂×C₁₃	#1571	1	1	yes	characteristic
182	C₂×C₁₃	#1574	1	1	yes	characteristic
183	C₂×C₁₃	#1939	1	1	yes	characteristic
184	C₂×C₁₃	#1947	1	1	yes	characteristic
185	C₂×C₁₃	#1950	1	1	yes	characteristic
186	C₂²×C₄	—	13	1	no	—
187	C₂²×C₄	—	13	1	no	—
188	C₂²×C₄	—	13	1	no	—
189	C₂²×C₄	—	13	1	no	—
190	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
191	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
192	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	26	2	no	—
196	C₂×C₈	—	26	2	no	—
197	C₂×C₈	—	26	2	no	—
198	C₂×C₈	—	26	2	no	—
199	C₂×C₈	—	26	2	no	—
200	C₂×C₈	—	52	1	no	—
201	C₂×C₈	—	52	1	no	—
202	C₂×C₈	—	52	1	no	—
203	C₂×C₈	—	52	1	no	—
204	C₂×C₈	—	52	2	no	—
205	C₂×C₈	—	52	2	no	—
206	C₂×C₈	—	52	2	no	—
207	C₂×C₈	C₂×#1057	1	4	yes	—
208	C₄²	—	13	2	no	—
209	C₄²	—	13	2	no	—
210	C₄²	—	13	1	no	—
211	C₄²	—	13	1	no	—
212	C₄²	—	13	1	no	—
213	C₄²	—	13	1	no	—
214	C₄²	—	26	1	no	—
215	C₄²	—	26	1	no	—
216	C₄²	—	26	2	no	—
217	C₄²	—	26	1	no	—
218	C₄²	—	26	1	no	—
219	C₄²	—	52	1	no	—
220	C₄²	#6118	1	2	yes	—
221	C₄²	C₂×#1057	1	2	yes	—
222	C₁₃	#9601	1	1	yes	characteristic, complemented, Sylow
223	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	13	1	no	—
227	C₂×C₄	—	13	1	no	—
228	C₂×C₄	—	13	1	no	—
229	C₂×C₄	—	13	1	no	—
230	C₂×C₄	—	13	1	no	—
231	C₂×C₄	—	13	1	no	—
232	C₂×C₄	—	13	1	no	—
233	C₂×C₄	—	13	1	no	—
234	C₂×C₄	—	13	2	no	—
235	C₂×C₄	—	26	1	no	—
236	C₂×C₄	—	26	1	no	—
237	C₂×C₄	—	26	1	no	—
238	C₂×C₄	—	26	1	no	—
239	C₂×C₄	—	26	1	no	—
240	C₂×C₄	—	26	1	no	—
241	C₂×C₄	—	26	1	no	—
242	C₂×C₄	#88102	1	2	yes	—
243	C₂×C₄	#88104	1	2	yes	—
244	C₂×C₄	C₄⋊#1057	1	2	yes	—
245	C₂×C₄	C₄⋊#1057	1	2	yes	—
246	C₂×C₄	C₂×#6118	1	1	yes	characteristic
247	C₂×C₄	#88238	1	1	yes	characteristic
248	C₂×C₄	D₄×#261	1	1	yes	characteristic
249	C₂×C₄	#88240	1	1	yes	characteristic
250	C₂×C₄	Q₈×#261	1	1	yes	characteristic
251	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
252	C₈	—	2	4	no	—
253	C₈	—	52	2	no	—
254	C₈	—	52	2	no	—
255	C₈	—	52	2	no	—
256	C₈	—	104	2	no	—
257	C₂²	#11719079	1	1	yes	characteristic
258	C₂²	#11719083	1	1	yes	characteristic
259	C₂²	C₂×#88102	1	1	yes	characteristic
260	C₂²	C₂×#88104	1	1	yes	characteristic
261	C₂²	#11719098	1	1	yes	characteristic
262	C₂²	#11719100	1	1	yes	characteristic
263	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	13	1	no	—
266	C₄	—	13	1	no	—
267	C₄	—	13	1	no	—
268	C₄	—	13	1	no	—
269	C₄	—	26	2	no	—
270	C₄	—	26	1	no	—
271	C₄	—	26	1	no	—
272	C₄	—	52	1	no	—
273	C₄	#11718254	1	4	yes	complemented
274	C₄	#11719076	1	1	yes	characteristic
275	C₄	#11719080	1	1	yes	characteristic
276	C₄	D₄×#1057	1	1	yes	characteristic
277	C₄	Q₈×#1057	1	1	yes	characteristic
278	C₂	C₂×#11718254	1	1	yes	characteristic
279	C₂	#13130387	1	1	yes	characteristic
280	C₂	#13130406	1	1	yes	characteristic
281	C₂	#13130414	1	1	yes	characteristic
282	C₂	#13130933	1	1	yes	characteristic
283	C₂	#13130947	1	1	yes	characteristic
284	C₂	#13130950	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×C₄⋊C₈	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×C₄⋊C₈	#261
hypocenter	C₁₃	#9601
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
40. C₁₃×C₄⋊C₈	C₈	semidirect
141. C₄×C₁₃	#335	semidirect
175. C₄⋊C₈	#1057	semidirect
222. C₁₃	#9601	semidirect
273. C₄	#11718254	semidirect

Maximal subgroups

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

subgroup	index
#13130768	2
C₄×C₄⋊#1057	2
C₂×#11718254	2
#13129666	2
C₄⋊C₈×Dic₁₃	2
#9601	13