Subgroups — #16778943

Order 3328.

666 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719119	C₂	1	2	yes	maximal
2	#13129662	C₂	1	4	yes	maximal, complemented
3	C₂×M₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
6	C₂²×#6099	C₄	1	1	yes	characteristic
7	C₂×C₂²⋊#1057	—	2	2	no	—
8	#11719119	—	2	2	no	—
9	#11719119	—	2	2	no	—
10	#11719119	C₄	1	4	yes	—
11	#11719119	C₂²	1	4	yes	—
12	C₂×C₄⋊#1057	—	2	2	no	—
13	C₂×C₄×#1057	C₄	1	4	yes	—
14	M₁₆×Dic₁₃	—	2	2	no	—
15	M₁₆×Dic₁₃	—	2	2	no	—
16	C₂×#88055	C₂²	1	2	yes	—
17	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×Dic₁₃	D₄	1	1	yes	characteristic
20	C₂³×Dic₁₃	Q₈	1	1	yes	characteristic
21	C₂²×#1057	—	2	4	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	C₈	1	4	yes	complemented
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
31	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
32	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
34	C₂×#6099	—	2	2	no	—
35	C₂×#6099	—	2	2	no	—
36	C₂×#6099	C₈	1	4	yes	complemented
37	C₂²×#1055	C₂×C₄	1	2	yes	—
38	C₂²⋊#1057	—	2	4	no	—
39	C₂²⋊#1057	—	2	4	no	—
40	C₄⋊#1057	—	2	4	no	—
41	C₄⋊#1057	—	2	4	no	—
42	C₄×#1057	—	2	4	no	—
43	C₄×#1057	C₂×C₄	1	8	yes	—
44	#88055	—	2	4	no	—
45	C₈×Dic₁₃	—	2	2	no	—
46	C₈×Dic₁₃	—	2	2	no	—
47	#7860	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
55	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	—	4	2	no	—
58	C₂²×Dic₁₃	—	4	2	no	—
59	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
64	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
65	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₂×#1057	—	2	8	no	—
68	C₂×#1057	—	4	8	no	—
69	C₁₃×M₁₆	—	2	8	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	complemented
73	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
74	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
75	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
78	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
79	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
80	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
81	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
82	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
83	#6099	—	2	8	no	—
84	C₂×#1055	—	2	2	no	—
85	C₂×#1055	—	2	2	no	—
86	C₂×#1055	C₂×C₈	1	4	yes	complemented
87	C₂×#431	—	13	2	no	—
88	C₂×C₄×M₁₆	—	13	1	no	—
89	#1275	—	13	4	no	—
90	C₂³×C₁₃	—	4	1	no	—
91	C₂³×C₁₃	—	4	1	no	—
92	C₂³×C₁₃	#95	1	2	yes	—
93	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
94	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
95	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#95	1	2	yes	—
97	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
98	C₂×C₄×C₁₃	#97	1	4	yes	—
99	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
100	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
101	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
102	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×Dic₁₃	—	2	4	no	—
110	C₂×Dic₁₃	—	2	2	no	—
111	C₂×Dic₁₃	—	2	2	no	—
112	C₂×Dic₁₃	—	2	4	no	—
113	C₂×Dic₁₃	—	4	2	no	—
114	C₂×Dic₁₃	—	4	2	no	—
115	C₂×Dic₁₃	—	4	1	no	—
116	C₂×Dic₁₃	—	4	1	no	—
117	C₂×Dic₁₃	#95	1	1	yes	characteristic
118	C₂×Dic₁₃	#95	1	1	yes	characteristic
119	C₂×Dic₁₃	#95	1	1	yes	characteristic
120	C₂×Dic₁₃	#95	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂×C₂²⋊C₄	1	2	yes	—
122	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
123	#1057	—	4	8	no	—
124	C₈×C₁₃	—	2	4	no	—
125	#1055	—	2	4	no	—
126	C₂²×M₁₆	—	13	1	no	—
127	C₂²×M₁₆	#261	1	1	yes	characteristic
128	C₂²×C₄²	—	13	1	no	—
129	#431	—	13	4	no	—
130	#431	—	13	4	no	—
131	#431	—	26	2	no	—
132	#431	—	26	2	no	—
133	C₂×C₄⋊C₈	—	26	2	no	—
134	C₂×C₂²⋊C₈	—	26	2	no	—
135	C₄×M₁₆	—	26	2	no	—
136	C₄×M₁₆	—	26	2	no	—
137	C₂×#97	—	13	2	no	—
138	C₂×C₄×C₈	—	13	4	no	—
139	C₂×C₄×C₈	—	13	2	no	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	2	1	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	—	4	2	no	—
145	C₂²×C₁₃	—	4	2	no	—
146	C₂²×C₁₃	#336	1	4	yes	—
147	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
150	C₄×C₁₃	#336	1	4	yes	—
151	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
152	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
153	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
155	Dic₁₃	—	2	2	no	—
156	Dic₁₃	—	2	1	no	—
157	Dic₁₃	—	2	1	no	—
158	Dic₁₃	—	2	2	no	—
159	Dic₁₃	H(3,ℤ/4)	1	2	yes	—
160	Dic₁₃	#338	1	2	yes	—
161	C₂³×C₄	—	13	1	no	—
162	C₂³×C₄	—	13	1	no	—
163	C₂³×C₄	C₂×#261	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	13	4	no	—
167	C₂×M₁₆	—	26	2	no	—
168	C₂×M₁₆	—	26	2	no	—
169	C₂×M₁₆	#1057	1	4	yes	complemented
170	C₂²×C₈	—	13	2	no	—
171	C₂²×C₈	—	26	4	no	—
172	C₂²×C₈	C₂×#261	1	2	yes	—
173	C₂×C₄²	—	13	2	no	—
174	C₂×C₄²	—	13	2	no	—
175	C₂×C₄²	—	13	1	no	—
176	C₂×C₄²	—	13	1	no	—
177	C₂×C₄²	—	13	2	no	—
178	C₂×C₄²	—	13	2	no	—
179	C₂×C₄²	—	13	1	no	—
180	C₂×C₄²	—	13	1	no	—
181	C₄⋊C₈	—	26	4	no	—
182	C₄⋊C₈	—	26	4	no	—
183	C₂²⋊C₈	—	26	4	no	—
184	C₂²⋊C₈	—	26	4	no	—
185	#97	—	26	4	no	—
186	C₄×C₈	—	13	8	no	—
187	C₄×C₈	—	26	2	no	—
188	C₄×C₈	—	26	2	no	—
189	C₄×C₈	—	26	4	no	—
190	C₂×C₁₃	—	2	4	no	—
191	C₂×C₁₃	—	2	1	no	—
192	C₂×C₁₃	—	2	1	no	—
193	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
194	C₂×C₁₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
195	C₂×C₁₃	C₂×#338	1	1	yes	characteristic
196	C₂⁴	C₄⋊#261	1	1	yes	characteristic
197	C₂²×C₄	—	13	2	no	—
198	C₂²×C₄	—	13	2	no	—
199	C₂²×C₄	—	13	1	no	—
200	C₂²×C₄	—	13	1	no	—
201	C₂²×C₄	—	13	2	no	—
202	C₂²×C₄	—	13	2	no	—
203	C₂²×C₄	—	13	1	no	—
204	C₂²×C₄	—	13	1	no	—
205	C₂²×C₄	—	52	2	no	—
206	C₂²×C₄	—	52	2	no	—
207	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
208	C₂²×C₄	#6118	1	2	yes	—
209	C₂²×C₄	C₄×#261	1	2	yes	—
210	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
211	C₂²×C₄	C₂×#1057	1	2	yes	—
212	C₂²×C₄	#6122	1	2	yes	—
213	C₂²×C₄	C₂²⋊#261	1	2	yes	—
214	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
215	M₁₆	—	2	8	no	—
216	M₁₆	—	26	8	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	13	4	no	—
220	C₂×C₈	—	26	2	no	—
221	C₂×C₈	—	26	2	no	—
222	C₂×C₈	—	26	8	no	—
223	C₂×C₈	—	52	8	no	—
224	C₂×C₈	C₂×#1057	1	4	yes	complemented
225	C₄²	—	13	2	no	—
226	C₄²	—	13	2	no	—
227	C₄²	—	13	2	no	—
228	C₄²	—	13	2	no	—
229	C₄²	—	13	2	no	—
230	C₄²	—	13	1	no	—
231	C₄²	—	13	1	no	—
232	C₄²	—	13	1	no	—
233	C₄²	—	13	1	no	—
234	C₄²	—	13	2	no	—
235	C₁₃	#7860	1	1	yes	characteristic, complemented, Sylow
236	C₂³	—	4	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	C₄⋊#1057	1	2	yes	—
239	C₂³	#88112	1	2	yes	—
240	C₂³	(C₂×#261)⋊C₄	1	2	yes	—
241	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
242	C₂×C₄	—	13	1	no	—
243	C₂×C₄	—	13	1	no	—
244	C₂×C₄	—	13	2	no	—
245	C₂×C₄	—	13	1	no	—
246	C₂×C₄	—	13	1	no	—
247	C₂×C₄	—	13	2	no	—
248	C₂×C₄	—	26	4	no	—
249	C₂×C₄	—	26	2	no	—
250	C₂×C₄	—	26	4	no	—
251	C₂×C₄	—	26	2	no	—
252	C₂×C₄	—	52	1	no	—
253	C₂×C₄	—	52	2	no	—
254	C₂×C₄	—	52	2	no	—
255	C₂×C₄	—	52	1	no	—
256	C₂×C₄	C₄×#1057	1	4	yes	—
257	C₂×C₄	C₄⋊#1057	1	2	yes	—
258	C₂×C₄	#88110	1	4	yes	—
259	C₂×C₄	#88111	1	4	yes	—
260	C₂×C₄	#88112	1	2	yes	—
261	C₂×C₄	(C₂×#261)⋊C₄	1	2	yes	—
262	C₂×C₄	C₂²⋊#1057	1	4	yes	—
263	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
264	C₂×C₄	C₂×#6118	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
266	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
267	C₂×C₄	C₂×#6122	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
269	C₈	—	2	4	no	—
270	C₈	—	26	4	no	—
271	C₈	—	52	8	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	#11718290	1	4	yes	—
279	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
280	C₂²	C₂×#88112	1	1	yes	characteristic
281	C₂²	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
282	C₄	—	13	2	no	—
283	C₄	—	13	2	no	—
284	C₄	—	26	2	no	—
285	C₄	—	26	1	no	—
286	C₄	—	26	1	no	—
287	C₄	—	26	2	no	—
288	C₄	#11718290	1	4	yes	—
289	C₄	C₂×C₄×#1057	1	1	yes	characteristic
290	C₄	C₂×#88110	1	1	yes	characteristic
291	C₄	C₂×#88111	1	1	yes	characteristic
292	C₄	C₂×C₂²⋊#1057	1	1	yes	characteristic
293	C₂	—	2	4	no	—
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	C₂×#11718290	1	1	yes	characteristic
297	C₂	#13130508	1	1	yes	characteristic
298	C₂	D₁₃⋊#338	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂²×C₁₃×M₁₆	C₄
socle	C₂²×C₁₃	C₂×#95
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂²×M₁₆	#261
hypocenter	C₁₃	#7860
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13129662	C₂	semidirect
24. C₂×C₁₃×M₁₆	C₈	semidirect
36. C₂×#6099	C₈	semidirect
72. C₂×C₈×C₁₃	C₂×C₈	semidirect
86. C₂×#1055	C₂×C₈	semidirect
169. C₂×M₁₆	#1057	semidirect
224. C₂×C₈	C₂×#1057	semidirect
235. C₁₃	#7860	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719119	2
#13129662	2
C₂×M₁₆×Dic₁₃	2
#7860	13