Subgroups — #15749869

Order 3328.

386 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#6090	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718142	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
5	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₄	1	1	yes	characteristic
7	C₂²×C₁₃×C₁₆	—	2	1	no	—
8	C₂²×#6090	—	2	1	no	—
9	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
10	C₁₃×C₂²⋊C₁₆	C₄	1	2	yes	complemented
11	#11718142	—	2	1	no	—
12	#11718142	—	2	1	no	—
13	C₂²⋊#6090	—	2	1	no	—
14	C₂²⋊#6090	C₄	1	2	yes	complemented
15	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×Dic₁₃	C₈	1	2	yes	—
17	C₂²×C₈×C₁₃	—	4	1	no	—
18	C₂²×C₈×C₁₃	—	4	1	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	—	2	1	no	—
26	C₂×C₄×Dic₁₃	—	2	1	no	—
27	C₂×C₄×Dic₁₃	—	4	1	no	—
28	C₂×C₄×Dic₁₃	—	4	1	no	—
29	C₂²×#1055	—	2	1	no	—
30	C₂²×#1055	—	2	1	no	—
31	C₂²×#1055	—	4	1	no	—
32	C₂²×#1055	—	4	1	no	—
33	C₂²×#1055	C₈	1	2	yes	—
34	C₂×C₁₃×C₁₆	—	2	2	no	—
35	C₂×C₁₃×C₁₆	—	4	1	no	—
36	C₂×C₁₃×C₁₆	—	4	1	no	—
37	C₂×#6090	—	2	2	no	—
38	C₂×#6090	—	4	1	no	—
39	C₂×#6090	—	4	1	no	—
40	#7183	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	1	no	—
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
52	C₂²×Dic₁₃	—	2	2	no	—
53	C₂²×Dic₁₃	—	2	2	no	—
54	C₂²×Dic₁₃	—	4	2	no	—
55	C₂²×Dic₁₃	—	4	2	no	—
56	C₂²×Dic₁₃	C₁₆	1	4	yes	complemented
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	—	2	1	no	—
61	C₂×C₈×C₁₃	—	4	1	no	—
62	C₂×C₈×C₁₃	—	4	1	no	—
63	C₂×C₈×C₁₃	—	4	1	no	—
64	C₂×C₈×C₁₃	—	4	1	no	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
69	C₄×Dic₁₃	—	4	2	no	—
70	C₄×Dic₁₃	—	4	1	no	—
71	C₄×Dic₁₃	—	4	1	no	—
72	C₂×#1055	—	2	2	no	—
73	C₂×#1055	—	4	2	no	—
74	C₂×#1055	—	4	1	no	—
75	C₂×#1055	—	4	1	no	—
76	C₂×#1055	—	4	2	no	—
77	C₁₃×C₁₆	—	4	2	no	—
78	#6090	—	4	2	no	—
79	C₂×C₂²⋊C₁₆	—	13	1	no	—
80	C₂×C₂²⋊C₁₆	D₁₃	1	1	yes	characteristic
81	C₂×#336	—	13	1	no	—
82	C₂³×C₁₃	—	2	1	no	—
83	C₂³×C₁₃	—	2	1	no	—
84	C₂³×C₁₃	—	2	2	no	—
85	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
86	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
87	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
88	C₂³×C₁₃	C₂×C₁₆	1	2	yes	—
89	C₂³×C₁₃	M₃₂	1	2	yes	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#101	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
108	C₂×Dic₁₃	—	2	4	no	—
109	C₂×Dic₁₃	—	4	4	no	—
110	C₂×Dic₁₃	—	4	2	no	—
111	C₂×Dic₁₃	—	4	2	no	—
112	C₂×Dic₁₃	—	4	4	no	—
113	C₈×C₁₃	—	2	1	no	—
114	C₈×C₁₃	—	2	1	no	—
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	—	2	1	no	—
117	#1055	—	4	2	no	—
118	C₂³×C₈	—	13	1	no	—
119	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
120	C₂²×C₄²	—	13	1	no	—
121	C₂²×C₁₆	—	2	1	no	—
122	C₂²×C₁₆	—	26	1	no	—
123	C₂²⋊C₁₆	—	2	1	no	—
124	C₂²⋊C₁₆	—	13	2	no	—
125	C₂²⋊C₁₆	—	26	1	no	—
126	C₂²⋊C₁₆	Dic₁₃	1	2	yes	complemented
127	#336	—	26	1	no	—
128	#336	—	26	1	no	—
129	C₂²×C₁₃	—	2	1	no	—
130	C₂²×C₁₃	—	2	1	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	#323	1	1	yes	characteristic
138	C₂²×C₁₃	#324	1	1	yes	characteristic
139	C₂²×C₁₃	#336	1	1	yes	characteristic
140	C₂²×C₁₃	C₄×C₁₆	1	1	yes	characteristic
141	C₂²×C₁₃	#346	1	1	yes	characteristic
142	C₂²×C₁₃	C₂²⋊C₁₆	1	2	yes	—
143	C₂²×C₁₃	#349	1	2	yes	—
144	C₂²×C₁₃	C₄⋊C₁₆	1	2	yes	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	#323	1	1	yes	characteristic
148	C₄×C₁₃	#324	1	1	yes	characteristic
149	C₄×C₁₃	#342	1	1	yes	characteristic
150	C₄×C₁₃	#343	1	1	yes	characteristic
151	Dic₁₃	—	4	4	no	—
152	C₂³×C₄	—	13	2	no	—
153	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
154	C₂²×C₈	—	4	1	no	—
155	C₂²×C₈	—	4	1	no	—
156	C₂²×C₈	—	13	2	no	—
157	C₂²×C₈	—	26	1	no	—
158	C₂²×C₈	—	26	1	no	—
159	C₂²×C₈	—	52	1	no	—
160	C₂²×C₈	—	52	1	no	—
161	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
162	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
163	C₂²×C₈	D₅₂	1	1	yes	characteristic
164	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
167	C₂×C₄²	—	26	1	no	—
168	C₂×C₄²	—	26	1	no	—
169	C₂×C₄²	—	52	1	no	—
170	C₂×C₄²	—	52	1	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	4	1	no	—
173	C₂×C₁₆	—	4	1	no	—
174	C₂×C₁₆	—	26	2	no	—
175	C₂×C₁₆	—	52	1	no	—
176	C₂×C₁₆	—	52	1	no	—
177	C₂×C₁₃	—	2	1	no	—
178	C₂×C₁₃	—	2	1	no	—
179	C₂×C₁₃	—	2	2	no	—
180	C₂×C₁₃	#1280	1	1	yes	characteristic
181	C₂×C₁₃	#1314	1	2	yes	—
182	C₂×C₁₃	#1322	1	2	yes	—
183	C₂×C₁₃	#1374	1	1	yes	characteristic
184	C₂×C₁₃	#1377	1	1	yes	characteristic
185	C₂⁴	C₈×D₁₃	1	1	yes	characteristic
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	13	4	no	—
191	C₂²×C₄	—	26	2	no	—
192	C₂²×C₄	—	26	2	no	—
193	C₂²×C₄	—	52	2	no	—
194	C₂²×C₄	—	52	2	no	—
195	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
196	C₂²×C₄	#6094	1	1	yes	characteristic
197	C₂²×C₄	#6094	1	1	yes	characteristic
198	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
199	C₂²×C₄	#6101	1	1	yes	characteristic
200	C₂²×C₄	#6103	1	1	yes	characteristic
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	26	2	no	—
210	C₂×C₈	—	52	2	no	—
211	C₂×C₈	—	52	1	no	—
212	C₂×C₈	—	52	1	no	—
213	C₂×C₈	—	52	2	no	—
214	C₂×C₈	#6101	1	1	yes	characteristic
215	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
216	C₂×C₈	#6103	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
218	C₄²	—	52	2	no	—
219	C₄²	—	52	1	no	—
220	C₄²	—	52	1	no	—
221	C₁₆	—	4	2	no	—
222	C₁₆	—	52	2	no	—
223	C₁₃	#7183	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	2	no	—
227	C₂³	C₁₆×D₁₃	1	2	yes	—
228	C₂³	#88038	1	2	yes	—
229	C₂³	C₈×Dic₁₃	1	1	yes	characteristic
230	C₂³	#88054	1	1	yes	characteristic
231	C₂³	#88060	1	1	yes	characteristic
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	26	4	no	—
243	C₂×C₄	—	52	4	no	—
244	C₂×C₄	—	52	2	no	—
245	C₂×C₄	—	52	2	no	—
246	C₂×C₄	—	52	4	no	—
247	C₂×C₄	#88046	1	1	yes	characteristic
248	C₂×C₄	#88046	1	1	yes	characteristic
249	C₂×C₄	#88054	1	1	yes	characteristic
250	C₂×C₄	#88055	1	1	yes	characteristic
251	C₂×C₄	#88060	1	1	yes	characteristic
252	C₂×C₄	#88063	1	1	yes	characteristic
253	C₂×C₄	#88064	1	1	yes	characteristic
254	C₂×C₄	#88071	1	1	yes	characteristic
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	52	2	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	#11718060	1	1	yes	characteristic
269	C₂²	#11718061	1	1	yes	characteristic
270	C₂²	C₁₆×Dic₁₃	1	1	yes	characteristic
271	C₂²	#11718093	1	2	yes	—
272	C₂²	#11718094	1	1	yes	characteristic
273	C₂²	#11718099	1	2	yes	—
274	C₂²	#11718106	1	2	yes	—
275	C₂²	#11718142	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	52	4	no	—
279	C₄	#11718060	1	1	yes	characteristic
280	C₄	#11718061	1	1	yes	characteristic
281	C₄	#11718119	1	1	yes	characteristic
282	C₄	#11718146	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	2	no	—
285	C₂	—	2	1	no	—
286	C₂	#13112678	1	1	yes	characteristic
287	C₂	#13112759	1	2	yes	—
288	C₂	#13112775	1	2	yes	—
289	C₂	#13112913	1	1	yes	characteristic
290	C₂	#13112919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#6103
commutator	C₂²×C₁₃	C₄×C₁₆
Frattini	C₂³×C₈	C₂×D₁₃
Fitting	C₂×C₁₃×C₂²⋊C₁₆	C₂
socle	C₂³×C₁₃	C₂²⋊C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂³×C₈×C₁₃	C₂²
hypercenter	C₂×C₂²⋊C₁₆	D₁₃
hypocenter	C₁₃	#7183
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
10. C₁₃×C₂²⋊C₁₆	C₄	semidirect
14. C₂²⋊#6090	C₄	semidirect
56. C₂²×Dic₁₃	C₁₆	semidirect
126. C₂²⋊C₁₆	Dic₁₃	semidirect
223. C₁₃	#7183	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#6090	2
C₂×#11718142	2
C₂×C₁₃×C₂²⋊C₁₆	2
#7183	13