Subgroups — #16789316

Order 3328.

482 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718270	C₂	1	2	yes	maximal
2	C₄×#88106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718231	C₂	1	1	yes	maximal, characteristic, complemented
4	#13129891	C₂	1	2	yes	maximal, complemented
5	C₄×#88161	C₂	1	1	yes	maximal, characteristic
6	C₂×#88161	C₄	1	1	yes	characteristic
7	C₂×#88161	C₂²	1	1	yes	characteristic
8	C₄×#6127	C₄	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	2	yes	—
10	C₂²×#6092	C₂²	1	2	yes	—
11	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
12	#11718783	C₄	1	1	yes	characteristic
13	#11718783	C₂²	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	#11718369	—	2	1	no	—
16	#11718369	—	2	1	no	—
17	C₄×#6094	—	2	1	no	—
18	C₄×C₈×D₁₃	—	2	1	no	—
19	#11718270	C₄	1	8	yes	complemented
20	#11718231	—	2	1	no	—
21	#11718231	—	2	1	no	—
22	#11718231	—	2	1	no	—
23	C₄×#6092	—	2	1	no	—
24	C₂×#6127	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₂²⋊D₁₃	—	2	2	no	—
29	#88180	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	C₂³	1	1	yes	characteristic
32	#88161	—	2	2	no	—
33	#88161	—	2	2	no	—
34	#88161	C₈	1	4	yes	—
35	#88161	C₂×C₄	1	4	yes	—
36	C₂×#6094	—	2	1	no	—
37	C₂×#6094	—	2	1	no	—
38	C₂×C₈×D₁₃	—	2	1	no	—
39	C₂×C₈×D₁₃	—	2	1	no	—
40	C₄×D₅₂	C₈	1	2	yes	—
41	#88126	—	2	1	no	—
42	C₄²×D₁₃	—	2	1	no	—
43	C₄×Dic₂₆	C₈	1	2	yes	—
44	#88106	—	2	2	no	—
45	#88106	C₂×C₄	1	4	yes	complemented
46	C₂×#6092	—	2	2	no	—
47	C₂×#6092	—	2	2	no	—
48	C₂×#6092	—	2	2	no	—
49	C₂×#6092	C₂×C₄	1	4	yes	complemented
50	C₁₃×#97	—	2	1	no	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	#88055	—	2	1	no	—
53	#88055	—	2	1	no	—
54	C₈×Dic₁₃	—	2	1	no	—
55	C₈×Dic₁₃	—	2	1	no	—
56	C₄×#1055	D₄	1	1	yes	characteristic
57	C₄×#1055	D₄	1	1	yes	characteristic
58	C₄×#1055	D₄	1	1	yes	characteristic
59	C₄×#1055	D₄	1	1	yes	characteristic
60	#12604	—	13	1	no	maximal, Sylow
61	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃	—	2	1	no	—
64	#6127	—	2	1	no	—
65	#6127	—	2	1	no	—
66	#6127	C₄²	1	2	yes	—
67	#6127	C₂×C₈	1	2	yes	—
68	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₁₃	—	2	1	no	—
70	C₂×C₄×D₁₃	—	2	2	no	—
71	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
72	C₁₃×M₁₆	—	2	1	no	—
73	C₁₃×M₁₆	—	2	1	no	—
74	C₁₃×M₁₆	C₄²	1	2	yes	—
75	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂×C₈	1	2	yes	—
81	C₂²⋊Dic₁₃	—	2	1	no	—
82	#6103	—	2	2	no	—
83	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
84	#6101	—	2	2	no	—
85	C₄×Dic₁₃	—	2	2	no	—
86	C₄×Dic₁₃	—	2	1	no	—
87	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×#1055	C₂×C₈	1	2	yes	—
90	C₂×#1055	M₁₆	1	2	yes	—
91	C₂×#1055	C₂²×C₄	1	2	yes	—
92	C₂×#1055	C₂×D₄	1	1	yes	characteristic
93	C₂×#1055	C₂×D₄	1	1	yes	characteristic
94	C₂×#1055	#41	1	2	yes	—
95	#6094	—	2	4	no	—
96	#6094	—	2	4	no	—
97	C₈×D₁₃	—	2	4	no	—
98	C₈×D₁₃	—	2	4	no	—
99	#6092	—	2	4	no	—
100	#6092	—	2	4	no	—
101	C₄×#131	—	13	1	no	—
102	C₂×#350	—	13	2	no	—
103	C₄×M₃₂	—	13	1	no	—
104	C₂×#346	—	13	1	no	—
105	#1379	—	13	2	no	—
106	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂²×D₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
117	C₂²⋊D₁₃	—	2	2	no	—
118	C₂²⋊D₁₃	—	2	2	no	—
119	C₂×Dic₁₃	—	2	1	no	—
120	C₂×Dic₁₃	—	2	2	no	—
121	D₅₂	—	2	1	no	—
122	D₅₂	C₄×C₈	1	2	yes	—
123	C₄×D₁₃	—	2	2	no	—
124	C₄×D₁₃	—	2	2	no	—
125	C₄×D₁₃	—	4	4	no	—
126	Dic₂₆	—	2	1	no	—
127	Dic₂₆	C₄×C₈	1	2	yes	—
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	—	2	2	no	—
130	#1055	C₂²⋊C₈	1	2	yes	—
131	#1055	C₂²⋊C₈	1	2	yes	—
132	#1055	C₄×D₄	1	2	yes	—
133	#1055	C₄×D₄	1	2	yes	—
134	C₂×#131	—	13	1	no	—
135	C₂×#131	—	13	1	no	—
136	C₄×#41	—	13	1	no	—
137	C₂×M₃₂	—	13	2	no	—
138	C₂²×C₁₆	—	13	2	no	—
139	#405	—	13	1	no	—
140	#405	—	13	1	no	—
141	#405	—	26	1	no	—
142	#405	—	26	1	no	—
143	C₄×M₁₆	—	26	1	no	—
144	C₄×M₁₆	#261	1	1	yes	characteristic
145	C₂×C₄×C₈	—	13	1	no	—
146	C₂×C₄×C₈	—	26	1	no	—
147	#350	—	13	8	no	—
148	#346	—	26	1	no	—
149	#346	—	26	1	no	—
150	#346	—	26	1	no	—
151	C₄×C₁₆	—	26	1	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
154	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
155	C₂²×C₁₃	#431	1	1	yes	characteristic
156	C₂×D₁₃	—	2	2	no	—
157	C₂×D₁₃	—	4	2	no	—
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
161	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
162	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₄×C₁₃	#431	1	1	yes	characteristic
164	Dic₁₃	—	2	2	no	—
165	Dic₁₃	—	4	2	no	—
166	C₂×#41	—	13	1	no	—
167	#131	—	13	4	no	—
168	#131	—	13	4	no	—
169	#131	—	26	2	no	—
170	#131	—	26	2	no	—
171	C₂×M₁₆	—	26	1	no	—
172	C₂×M₁₆	—	26	1	no	—
173	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
174	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
175	C₂²×C₈	—	13	1	no	—
176	C₂²×C₈	—	13	1	no	—
177	C₂²×C₈	—	26	1	no	—
178	C₂²×C₈	—	26	1	no	—
179	C₄×Q₈	—	13	2	no	—
180	C₄×D₄	—	13	2	no	—
181	C₄×D₄	—	26	2	no	—
182	#117	—	13	1	no	—
183	#117	—	26	1	no	—
184	C₂×C₄²	—	26	1	no	—
185	C₂×C₄²	C₂×#261	1	1	yes	characteristic
186	M₃₂	—	13	4	no	—
187	M₃₂	—	26	2	no	—
188	C₂×C₁₆	—	13	4	no	—
189	C₂×C₁₆	—	26	2	no	—
190	C₂×C₁₆	—	26	2	no	—
191	C₂×C₁₆	—	26	2	no	—
192	#97	—	2	1	no	—
193	#97	—	26	1	no	—
194	#97	—	26	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	13	1	no	—
197	C₄×C₈	—	13	1	no	—
198	C₄×C₈	—	13	1	no	—
199	C₄×C₈	—	13	1	no	—
200	C₄×C₈	—	26	1	no	—
201	C₄×C₈	—	26	1	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
205	C₂×C₁₃	#2117	1	2	yes	—
206	D₁₃	—	4	2	no	—
207	#41	—	13	2	no	—
208	#41	—	13	2	no	—
209	#41	—	26	1	no	—
210	#41	—	26	1	no	—
211	C₂×Q₈	—	13	1	no	—
212	C₂×D₄	—	13	1	no	—
213	C₂×D₄	—	26	1	no	—
214	C₂²×C₄	—	26	1	no	—
215	C₂²×C₄	—	26	2	no	—
216	C₂²×C₄	#6118	1	2	yes	—
217	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
218	M₁₆	—	2	1	no	—
219	M₁₆	—	2	1	no	—
220	M₁₆	—	26	4	no	—
221	M₁₆	—	26	4	no	—
222	M₁₆	D₁₃⋊C₈	1	2	yes	—
223	M₁₆	C₄×#261	1	2	yes	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	—	13	2	no	—
228	C₂×C₈	—	13	2	no	—
229	C₂×C₈	—	13	2	no	—
230	C₂×C₈	—	13	1	no	—
231	C₂×C₈	—	13	1	no	—
232	C₂×C₈	—	13	1	no	—
233	C₂×C₈	—	13	1	no	—
234	C₂×C₈	—	26	4	no	—
235	C₂×C₈	—	26	4	no	—
236	C₄⋊C₄	—	13	2	no	—
237	C₄⋊C₄	—	26	2	no	—
238	C₂²⋊C₄	—	26	1	no	—
239	C₂²⋊C₄	—	26	2	no	—
240	C₄²	—	26	2	no	—
241	C₄²	—	26	1	no	—
242	C₄²	D₁₃⋊C₈	1	2	yes	—
243	C₄²	C₂²⋊#261	1	1	yes	characteristic
244	C₄²	C₂²⋊#261	1	1	yes	characteristic
245	C₁₆	—	26	4	no	—
246	C₁₆	—	26	4	no	—
247	C₁₃	#12604	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	26	1	no	—
249	C₂³	C₂×#6118	1	1	yes	characteristic
250	Q₈	—	13	2	no	—
251	Q₈	—	26	1	no	—
252	D₄	—	13	2	no	—
253	D₄	—	26	2	no	—
254	D₄	—	26	2	no	—
255	D₄	—	26	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	26	1	no	—
258	C₂×C₄	—	26	2	no	—
259	C₂×C₄	—	26	2	no	—
260	C₂×C₄	—	26	2	no	—
261	C₂×C₄	—	52	4	no	—
262	C₂×C₄	#88111	1	2	yes	—
263	C₂×C₄	#88111	1	2	yes	—
264	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
265	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
266	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
268	C₂×C₄	#88237	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	—	13	2	no	—
273	C₈	—	13	2	no	—
274	C₈	—	13	2	no	—
275	C₈	—	13	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	26	2	no	—
278	C₂²	—	52	2	no	—
279	C₂²	C₄×#6118	1	1	yes	characteristic
280	C₂²	#11719051	1	1	yes	characteristic
281	C₂²	C₂×#88111	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	26	2	no	—
285	C₄	—	52	2	no	—
286	C₄	C₄×D₁₃⋊C₈	1	1	yes	characteristic
287	C₄	#11719049	1	1	yes	characteristic
288	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
289	C₄	C₄×C₂²⋊#261	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	52	2	no	—
293	C₂	C₄×#88111	1	1	yes	characteristic
294	C₂	#13131080	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₄×C₁₃×M₁₆	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂²×#1055	C₂³
hypercenter	C₄×M₁₆	#261
hypocenter	C₁₃	#12604
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#88106	C₂	semidirect
3. C₂×#11718231	C₂	semidirect
4. #13129891	C₂	semidirect
19. #11718270	C₄	semidirect
45. #88106	C₂×C₄	semidirect
49. C₂×#6092	C₂×C₄	semidirect
247. C₁₃	#12604	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718270	2
C₄×#88106	2
C₂×#11718231	2
#13129891	2
C₄×#88161	2
#12604	13