Subgroups — #15834735

Order 3328.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123785	C₂	1	2	yes	maximal
2	#13113745	C₂	1	2	yes	maximal, complemented
3	#13113708	C₂	1	4	yes	maximal, complemented
4	#13113655	C₂	1	2	yes	maximal, complemented
5	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
6	#13113239	C₂	1	2	yes	maximal
7	#13113125	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#533	C₂²	1	2	yes	—
10	C₁₃×#500	C₄	1	1	yes	characteristic
11	C₁₃×#494	C₄	1	1	yes	characteristic
12	C₁₃×#487	C₄	1	2	yes	—
13	C₁₃×#467	C₄	1	2	yes	—
14	C₁₃×#467	C₄	1	2	yes	—
15	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	#11718685	—	2	4	no	—
19	#11718685	C₂²	1	4	yes	—
20	Q₈×#1055	C₂²	1	4	yes	—
21	#11718660	—	2	2	no	—
22	#11718658	—	2	2	no	—
23	#11718586	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#1055	C₂²	1	2	yes	—
25	C₄×#6099	C₂²	1	1	yes	characteristic
26	Q₈⋊#1055	C₂²	1	4	yes	—
27	Q₈⋊#1055	C₂²	1	4	yes	—
28	#11718049	C₂²	1	1	yes	characteristic
29	#11718049	C₂²	1	2	yes	—
30	#11718049	C₂²	1	1	yes	characteristic
31	#11718047	C₂²	1	2	yes	—
32	#11718046	C₂²	1	2	yes	—
33	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₁₃×#125	—	2	2	no	—
40	C₁₃×C₂²⋊Q₈	—	2	2	no	—
41	C₁₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
42	C₄×Q₈×C₁₃	—	2	2	no	—
43	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
44	C₁₃×#117	—	2	2	no	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×#6099	D₄	1	2	yes	—
47	C₂²×#1055	—	2	2	no	—
48	C₁₃×#107	C₂×C₄	1	2	yes	—
49	C₁₃×#107	C₂×C₄	1	2	yes	—
50	C₁₃×#103	C₂×C₄	1	4	yes	—
51	C₁₃×#103	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₂²⋊#1055	—	2	1	no	—
57	C₄⋊#1055	—	2	2	no	—
58	C₄⋊#1055	—	2	1	no	—
59	C₄⋊#1055	—	2	4	no	—
60	C₄⋊#1055	—	2	4	no	—
61	C₄⋊#1055	C₂³	1	2	yes	—
62	C₄⋊#1055	C₂³	1	2	yes	—
63	#88043	—	2	4	no	—
64	#88043	D₄	1	2	yes	—
65	C₄×#1055	—	2	4	no	—
66	C₄×#1055	C₂³	1	2	yes	—
67	C₄×#1055	C₂³	1	2	yes	—
68	#18072	—	13	1	no	maximal, Sylow
69	C₂×Q₈×C₁₃	—	2	2	no	—
70	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
74	C₁₃×Q₁₆	—	2	4	no	—
75	C₁₃×Q₁₆	—	4	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	4	1	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	—	2	4	no	—
87	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
88	C₁₃×C₂²⋊C₄	—	2	4	no	—
89	C₄²×C₁₃	—	2	4	no	—
90	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₃	#41	1	1	yes	characteristic
93	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
94	#6099	—	2	4	no	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	—	2	2	no	—
97	C₂×#1055	—	2	2	no	—
98	C₂×#1055	—	4	4	no	—
99	C₂×#1055	C₂×D₄	1	4	yes	—
100	#3170	D₁₃	1	1	yes	characteristic
101	#2962	—	13	2	no	—
102	#1593	—	13	2	no	—
103	#1587	—	13	4	no	—
104	#1579	—	13	2	no	—
105	C₂×#334	—	13	1	no	—
106	#1481	—	13	2	no	—
107	#1451	—	13	1	no	—
108	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
109	Q₈×C₁₃	—	2	4	no	—
110	Q₈×C₁₃	—	2	4	no	—
111	Q₈×C₁₃	—	8	2	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#137	1	4	yes	—
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₈×C₁₃	—	2	1	no	—
125	C₈×C₁₃	—	2	1	no	—
126	C₈×C₁₃	—	2	1	no	—
127	C₈×C₁₃	#131	1	2	yes	—
128	#1055	—	2	4	no	—
129	#1055	—	4	2	no	—
130	#1055	—	4	2	no	—
131	#1055	—	4	2	no	—
132	#533	C₂×D₁₃	1	2	yes	—
133	#500	Dic₁₃	1	1	yes	characteristic
134	#494	Dic₁₃	1	1	yes	characteristic
135	#487	Dic₁₃	1	2	yes	—
136	#467	Dic₁₃	1	2	yes	—
137	#467	Dic₁₃	1	2	yes	—
138	#446	—	13	4	no	—
139	#446	—	26	4	no	—
140	C₈×Q₈	—	13	4	no	—
141	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
142	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
143	#431	—	13	1	no	—
144	#424	—	26	2	no	—
145	C₂×C₄⋊C₈	—	13	2	no	—
146	#405	—	26	2	no	—
147	C₄×M₁₆	—	13	1	no	—
148	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
149	#334	—	13	1	no	—
150	#334	—	13	2	no	—
151	#334	—	13	1	no	—
152	Q₈⋊C₈	—	13	4	no	—
153	Q₈⋊C₈	—	13	4	no	—
154	#322	—	13	2	no	—
155	#322	—	13	2	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	#441	1	2	yes	—
159	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	4	4	no	—
164	C₄×C₁₃	—	4	4	no	—
165	C₄×C₁₃	#441	1	2	yes	—
166	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
167	C₄×C₁₃	#568	1	1	yes	characteristic
168	C₄×C₁₃	C₂×#137	1	2	yes	—
169	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
170	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
171	C₂×M₁₆	—	13	2	no	—
172	C₂²×C₈	—	26	2	no	—
173	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
174	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
175	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
176	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
177	#125	—	2	2	no	—
178	C₂²⋊Q₈	—	2	2	no	—
179	C₂²⋊Q₈	C₂×Dic₁₃	1	4	yes	—
180	C₄×Q₈	—	2	2	no	—
181	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
182	#117	—	2	2	no	—
183	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
184	#107	C₂×Dic₁₃	1	2	yes	—
185	#107	C₂×Dic₁₃	1	2	yes	—
186	C₄⋊C₈	—	13	2	no	—
187	C₄⋊C₈	—	13	2	no	—
188	C₄⋊C₈	—	26	4	no	—
189	C₄⋊C₈	—	26	4	no	—
190	C₄⋊C₈	—	26	2	no	—
191	C₄⋊C₈	—	26	1	no	—
192	#103	C₂×Dic₁₃	1	4	yes	—
193	#103	C₂×Dic₁₃	1	4	yes	—
194	C₂²⋊C₈	—	26	1	no	—
195	#97	—	13	2	no	—
196	#97	—	26	4	no	—
197	C₄×C₈	—	13	2	no	—
198	C₄×C₈	—	13	2	no	—
199	C₄×C₈	—	26	4	no	—
200	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
201	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	#2935	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
207	C₂×C₁₃	#2957	1	1	yes	characteristic
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
210	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
211	C₂²×C₄	#6129	1	1	yes	characteristic
212	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	4	2	no	—
215	M₁₆	—	26	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	13	4	no	—
219	C₂×C₈	—	26	2	no	—
220	C₂×C₈	—	26	2	no	—
221	C₂×C₈	—	26	2	no	—
222	C₂×C₈	—	52	4	no	—
223	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	D₄×D₁₃	1	1	yes	characteristic
236	C₄²	#6129	1	1	yes	characteristic
237	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
238	C₄²	C₂³×D₁₃	1	1	yes	characteristic
239	C₁₃	#18072	1	1	yes	characteristic, complemented, Sylow
240	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	8	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	#88172	1	4	yes	—
250	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
251	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
252	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#6129	1	1	yes	characteristic
255	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	26	4	no	—
260	C₈	—	52	2	no	—
261	C₈	—	52	2	no	—
262	C₈	—	52	2	no	—
263	C₈	#88202	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#11718845	1	2	yes	—
267	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	4	no	—
272	C₄	—	4	4	no	—
273	C₄	#11718845	1	2	yes	—
274	C₄	C₂×#88172	1	2	yes	—
275	C₄	C₂×#88202	1	1	yes	characteristic
276	C₄	#11719524	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	#13123580	1	1	yes	characteristic
279	C₂	C₂×#11718845	1	1	yes	characteristic
280	C₂	#13123753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#18072
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13113745	C₂	semidirect
3. #13113708	C₂	semidirect
4. #13113655	C₂	semidirect
239. C₁₃	#18072	semidirect

Maximal subgroups

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

subgroup	index
#13123785	2
#13113745	2
#13113708	2
#13113655	2
C₂×#11718049	2
#13113239	2
#13113125	2
C₁₃×#3170	2
#18072	13