Subgroups — #16016008

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	#13116640	C₂	1	4	yes	maximal, complemented
3	#13116632	C₂	1	4	yes	maximal, complemented
4	#13116596	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116594	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116450	C₂	1	1	yes	maximal, characteristic
7	M₃₂×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2113	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
11	#41×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×#88202	C₄	1	1	yes	characteristic
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	#11718783	C₄	1	2	yes	—
16	#11718783	C₄	1	1	yes	characteristic
17	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
18	M₁₆×Dic₁₃	C₄	1	2	yes	—
19	#11718762	C₂²	1	1	yes	characteristic
20	#11718762	C₂²	1	2	yes	—
21	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
23	C₂×#88052	C₂²	1	1	yes	characteristic
24	C₂²×#6090	C₂²	1	1	yes	characteristic
25	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
26	C₂²⋊#6090	C₂²	1	4	yes	—
27	#11718094	C₂²	1	2	yes	—
28	#11718093	C₂²	1	4	yes	—
29	#11718093	C₂²	1	4	yes	—
30	C₁₆×Dic₁₃	C₂²	1	2	yes	—
31	C₁₆×Dic₁₃	C₂²	1	2	yes	—
32	C₁₆×Dic₁₃	C₂²	1	2	yes	—
33	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
34	C₁₃×#131	—	2	2	no	—
35	C₁₃×#131	—	2	2	no	—
36	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	#88202	—	2	4	no	—
41	Q₈×Dic₁₃	C₈	1	2	yes	—
42	D₄×Dic₁₃	C₈	1	4	yes	—
43	D₄×Dic₁₃	C₈	1	2	yes	—
44	#88180	C₂×C₄	1	2	yes	—
45	#88180	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×#6099	C₂×C₄	1	1	yes	characteristic
49	C₂×#6099	C₂×C₄	1	2	yes	—
50	C₂²×#1055	C₂×C₄	1	2	yes	—
51	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
52	C₁₃×M₃₂	—	2	2	no	—
53	C₂×C₁₃×C₁₆	—	2	2	no	—
54	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
55	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
56	#88055	C₂×C₄	1	2	yes	—
57	#88055	C₂×C₄	1	2	yes	—
58	#88055	C₂×C₄	1	2	yes	—
59	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
60	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
61	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
62	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
63	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
64	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
65	C₈×Dic₁₃	C₂³	1	2	yes	—
66	#88052	—	2	2	no	—
67	C₂×#6090	—	2	2	no	—
68	C₂×#6090	C₂³	1	2	yes	—
69	C₂×#6090	C₂³	1	2	yes	—
70	#33420	—	13	1	no	maximal, Sylow
71	C₁₃×#41	—	2	2	no	—
72	C₁₃×#41	—	2	2	no	—
73	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
75	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
79	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
80	C₁₃×M₁₆	—	2	2	no	—
81	C₁₃×M₁₆	—	2	4	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	#41	1	2	yes	—
87	C₂×C₈×C₁₃	#41	1	2	yes	—
88	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
90	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
91	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
92	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
93	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
94	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
95	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
98	#6099	—	2	2	no	—
99	#6099	—	2	4	no	—
100	C₂×#1055	—	2	4	no	—
101	C₂×#1055	M₁₆	1	4	yes	—
102	C₂×#1055	C₂²×C₄	1	2	yes	—
103	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
104	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
105	C₁₃×C₁₆	—	2	2	no	—
106	C₁₃×C₁₆	—	2	2	no	—
107	#6090	—	2	2	no	—
108	#6090	—	2	2	no	—
109	C₄×#131	—	13	1	no	—
110	#2164	—	13	4	no	—
111	#2162	—	13	4	no	—
112	#2151	—	13	1	no	—
113	#2151	—	13	1	no	—
114	#2113	—	13	1	no	—
115	#2113	D₁₃	1	1	yes	characteristic
116	C₄×M₃₂	—	13	1	no	—
117	C₂×C₄×C₁₆	—	13	1	no	—
118	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
119	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
120	Q₈×C₁₃	—	2	2	no	—
121	D₄×C₁₃	—	2	4	no	—
122	D₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	#117	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×Dic₁₃	—	2	4	no	—
131	C₂×Dic₁₃	—	2	8	no	—
132	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
133	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
134	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	#117	1	2	yes	—
137	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
139	#1055	—	2	2	no	—
140	#1055	C₂×M₁₆	1	2	yes	—
141	#1055	C₂×M₁₆	1	2	yes	—
142	C₂×#131	—	13	1	no	—
143	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
144	C₄×#41	—	13	1	no	—
145	C₂×M₃₂	—	13	1	no	—
146	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
147	C₂²×C₁₆	—	13	1	no	—
148	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
149	#405	—	13	1	no	—
150	#405	—	13	2	no	—
151	#405	—	13	1	no	—
152	#405	—	13	2	no	—
153	C₄×M₁₆	—	13	2	no	—
154	C₄×M₁₆	—	13	1	no	—
155	C₂×C₄×C₈	—	13	1	no	—
156	C₂×C₄×C₈	—	13	2	no	—
157	C₄⋊C₁₆	—	13	4	no	—
158	C₄⋊C₁₆	—	13	4	no	—
159	C₂²⋊C₁₆	—	13	4	no	—
160	C₂²⋊C₁₆	C₂×D₁₃	1	4	yes	—
161	#346	—	13	2	no	—
162	C₄×C₁₆	—	13	2	no	—
163	C₄×C₁₆	—	13	2	no	—
164	C₄×C₁₆	—	13	2	no	—
165	C₂²×C₁₃	—	2	2	no	—
166	C₂²×C₁₃	—	2	4	no	—
167	C₂²×C₁₃	—	2	2	no	—
168	C₂²×C₁₃	#431	1	2	yes	—
169	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	#431	1	2	yes	—
172	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
173	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
174	Dic₁₃	—	2	2	no	—
175	Dic₁₃	—	2	4	no	—
176	Dic₁₃	#504	1	4	yes	—
177	C₂×#41	C₄×D₁₃	1	1	yes	characteristic
178	#131	—	2	2	no	—
179	#131	—	2	2	no	—
180	#131	—	26	4	no	—
181	C₂×M₁₆	—	13	2	no	—
182	C₂×M₁₆	—	13	1	no	—
183	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
184	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
185	C₂²×C₈	—	13	1	no	—
186	C₂²×C₈	—	13	2	no	—
187	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
188	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₄×Q₈	—	13	2	no	—
190	C₄×D₄	—	13	4	no	—
191	C₄×D₄	—	13	2	no	—
192	#117	—	13	2	no	—
193	#117	—	13	1	no	—
194	C₂×C₄²	—	13	2	no	—
195	C₂×C₄²	—	13	1	no	—
196	M₃₂	—	2	2	no	—
197	M₃₂	—	26	2	no	—
198	C₂×C₁₆	—	2	2	no	—
199	C₂×C₁₆	—	13	2	no	—
200	C₂×C₁₆	—	13	2	no	—
201	C₂×C₁₆	—	26	2	no	—
202	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
203	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
204	#97	—	13	2	no	—
205	#97	—	13	2	no	—
206	#97	—	13	2	no	—
207	C₄×C₈	—	13	1	no	—
208	C₄×C₈	—	13	2	no	—
209	C₄×C₈	—	13	2	no	—
210	C₄×C₈	—	13	1	no	—
211	C₄×C₈	—	13	1	no	—
212	C₄×C₈	—	13	1	no	—
213	C₄×C₈	—	13	2	no	—
214	C₂×C₁₃	—	2	2	no	—
215	C₂×C₁₃	—	4	2	no	—
216	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
217	C₂×C₁₃	C₂×#504	1	2	yes	—
218	#41	—	2	2	no	—
219	#41	—	2	2	no	—
220	C₂×Q₈	C₈×D₁₃	1	1	yes	characteristic
221	C₂×D₄	C₈×D₁₃	1	2	yes	—
222	C₂×D₄	C₈×D₁₃	1	1	yes	characteristic
223	C₂²×C₄	—	13	4	no	—
224	C₂²×C₄	—	13	2	no	—
225	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
226	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	26	4	no	—
230	M₁₆	—	26	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	13	4	no	—
234	C₂×C₈	—	13	1	no	—
235	C₂×C₈	—	13	1	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	26	4	no	—
238	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
239	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
240	C₂×C₈	#6129	1	2	yes	—
241	C₂×C₈	#6129	1	2	yes	—
242	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
243	C₄⋊C₄	—	13	4	no	—
244	C₄⋊C₄	—	13	2	no	—
245	C₂²⋊C₄	—	13	4	no	—
246	C₂²⋊C₄	—	13	2	no	—
247	C₄²	—	13	4	no	—
248	C₄²	—	13	2	no	—
249	C₄²	—	13	1	no	—
250	C₄²	—	13	1	no	—
251	C₄²	—	13	2	no	—
252	C₁₆	—	2	2	no	—
253	C₁₆	—	2	2	no	—
254	C₁₆	—	26	2	no	—
255	C₁₆	—	26	2	no	—
256	C₁₃	#33420	1	1	yes	characteristic, complemented, Sylow
257	C₂³	C₂×C₈×D₁₃	1	2	yes	—
258	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
259	Q₈	—	2	2	no	—
260	D₄	—	2	4	no	—
261	D₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	13	2	no	—
265	C₂×C₄	—	13	4	no	—
266	C₂×C₄	—	13	2	no	—
267	C₂×C₄	—	26	4	no	—
268	C₂×C₄	—	26	8	no	—
269	C₂×C₄	#88131	1	2	yes	—
270	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
271	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
272	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	13	2	no	—
276	C₈	—	13	2	no	—
277	C₈	—	26	2	no	—
278	C₈	#88131	1	2	yes	—
279	C₈	C₂×#6129	1	1	yes	characteristic
280	C₈	C₂×#6129	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	#11718379	1	2	yes	—
285	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	13	4	no	—
288	C₄	—	26	2	no	—
289	C₄	—	26	4	no	—
290	C₄	#11718379	1	2	yes	—
291	C₄	C₂×#88131	1	1	yes	characteristic
292	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	C₂×#11718379	1	1	yes	characteristic
296	C₂	D₁₃×#504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2113	C₂
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2113	D₁₃
hypocenter	C₁₃	#33420
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
2. #13116640	C₂	semidirect
3. #13116632	C₂	semidirect
4. #13116596	C₂	semidirect
5. #13116594	C₂	semidirect
7. M₃₂×Dic₁₃	C₂	semidirect
8. C₂×C₁₆×Dic₁₃	C₂	semidirect
256. C₁₃	#33420	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₁₃	2
#13116640	2
#13116632	2
#13116596	2
#13116594	2
#13116450	2
M₃₂×Dic₁₃	2
C₂×C₁₆×Dic₁₃	2
C₁₃×#2113	2
#33420	13