Subgroups — #15833780

Order 3328.

544 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123931	C₂	1	2	yes	maximal
2	#13113708	C₂	1	4	yes	maximal, complemented
3	#13113687	C₂	1	4	yes	maximal, complemented
4	#13113599	C₂	1	1	yes	maximal, characteristic
5	#13113309	C₂	1	2	yes	maximal
6	#13113128	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	C₂²	1	2	yes	—
9	C₁₃×#500	C₄	1	1	yes	characteristic
10	C₁₃×#494	C₄	1	1	yes	characteristic
11	C₁₃×#487	C₄	1	2	yes	—
12	C₁₃×#467	C₄	1	2	yes	—
13	C₁₃×#467	C₄	1	2	yes	—
14	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
15	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#11718685	—	2	4	no	—
18	#11718685	C₂²	1	4	yes	—
19	Q₈×#1055	C₂²	1	4	yes	—
20	#11718661	—	2	4	no	—
21	#11718587	C₂²	1	2	yes	—
22	#11718586	C₂²	1	1	yes	characteristic
23	#11718586	C₂²	1	1	yes	characteristic
24	Q₈⋊#1055	C₂²	1	8	yes	—
25	#11718050	C₂²	1	2	yes	—
26	#11718049	C₂²	1	2	yes	—
27	#11718046	C₂²	1	2	yes	—
28	#11718046	C₂²	1	2	yes	—
29	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₁₃×#125	—	2	2	no	—
36	C₁₃×C₂²⋊Q₈	—	2	2	no	—
37	C₁₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
38	C₄×Q₈×C₁₃	—	2	2	no	—
39	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
40	C₁₃×#117	—	2	2	no	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#107	C₂×C₄	1	2	yes	—
43	C₁₃×#107	C₂×C₄	1	2	yes	—
44	C₁₃×#103	C₂×C₄	1	4	yes	—
45	C₁₃×#103	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂²⋊#1055	—	2	2	no	—
51	C₂²⋊#1055	—	2	1	no	—
52	C₂²⋊#1055	D₄	1	2	yes	—
53	C₄⋊#1055	—	2	1	no	—
54	C₄⋊#1055	—	2	4	no	—
55	C₄⋊#1055	—	2	4	no	—
56	C₄⋊#1055	D₄	1	2	yes	—
57	C₄⋊#1055	C₂³	1	4	yes	—
58	#88043	—	2	2	no	—
59	#88043	—	2	4	no	—
60	C₄×#1055	—	2	4	no	—
61	C₄×#1055	C₂³	1	2	yes	—
62	C₄×#1055	C₂³	1	2	yes	—
63	#17991	—	13	1	no	maximal, Sylow
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
69	C₁₃×Q₁₆	—	4	2	no	—
70	C₁₃×Q₁₆	—	4	2	no	—
71	C₂×C₈×C₁₃	—	4	1	no	—
72	C₂×C₈×C₁₃	—	4	1	no	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	2	4	no	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
83	C₁₃×C₂²⋊C₄	—	2	4	no	—
84	C₄²×C₁₃	—	2	4	no	—
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃	#41	1	1	yes	characteristic
88	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	2	2	no	—
91	C₂×#1055	C₂×D₄	1	4	yes	—
92	#3170	D₁₃	1	1	yes	characteristic
93	#2985	—	13	2	no	—
94	#1587	—	13	4	no	—
95	#1584	—	13	4	no	—
96	#1568	—	13	1	no	—
97	#1497	—	13	2	no	—
98	#1452	—	13	1	no	—
99	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	4	no	—
101	Q₈×C₁₃	—	2	4	no	—
102	Q₈×C₁₃	—	8	2	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	—	2	1	no	—
118	C₈×C₁₃	—	2	1	no	—
119	#1055	—	2	4	no	—
120	#1055	—	4	2	no	—
121	#1055	—	4	4	no	—
122	#533	C₂×D₁₃	1	2	yes	—
123	#500	Dic₁₃	1	1	yes	characteristic
124	#494	Dic₁₃	1	1	yes	characteristic
125	#487	Dic₁₃	1	2	yes	—
126	#467	Dic₁₃	1	2	yes	—
127	#467	Dic₁₃	1	2	yes	—
128	#446	—	13	4	no	—
129	#446	—	26	4	no	—
130	C₈×Q₈	—	13	4	no	—
131	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
132	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
133	#433	—	26	4	no	—
134	#432	—	13	2	no	—
135	#431	—	13	1	no	—
136	#431	—	13	1	no	—
137	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
138	#335	—	13	2	no	—
139	#334	—	13	2	no	—
140	Q₈⋊C₈	—	13	8	no	—
141	#322	—	13	2	no	—
142	#322	—	13	2	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	4	4	no	—
152	C₄×C₁₃	—	4	4	no	—
153	C₄×C₁₃	#568	1	1	yes	characteristic
154	C₄×C₁₃	#568	1	1	yes	characteristic
155	C₄×C₁₃	#578	1	2	yes	—
156	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
157	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
158	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
159	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
160	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
161	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
162	#125	—	2	2	no	—
163	C₂²⋊Q₈	—	2	2	no	—
164	C₂²⋊Q₈	C₂×Dic₁₃	1	4	yes	—
165	C₄×Q₈	—	2	2	no	—
166	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
167	#117	—	2	2	no	—
168	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
169	#107	C₂×Dic₁₃	1	2	yes	—
170	#107	C₂×Dic₁₃	1	2	yes	—
171	C₄⋊C₈	—	13	2	no	—
172	C₄⋊C₈	—	13	4	no	—
173	C₄⋊C₈	—	26	4	no	—
174	C₄⋊C₈	—	26	4	no	—
175	C₄⋊C₈	—	26	1	no	—
176	#103	C₂×Dic₁₃	1	4	yes	—
177	#103	C₂×Dic₁₃	1	4	yes	—
178	C₂²⋊C₈	—	13	2	no	—
179	C₂²⋊C₈	—	26	2	no	—
180	C₂²⋊C₈	—	26	1	no	—
181	#97	—	26	4	no	—
182	#97	—	26	2	no	—
183	C₄×C₈	—	13	2	no	—
184	C₄×C₈	—	13	2	no	—
185	C₄×C₈	—	26	4	no	—
186	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
187	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
190	C₂×C₁₃	—	4	1	no	—
191	C₂×C₁₃	#2932	1	1	yes	characteristic
192	C₂×C₁₃	#2950	1	1	yes	characteristic
193	C₂×C₁₃	#2957	1	1	yes	characteristic
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
196	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
197	C₂²×C₄	#6129	1	1	yes	characteristic
198	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
199	Q₁₆	—	4	2	no	—
200	Q₁₆	—	4	2	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	13	4	no	—
204	C₂×C₈	—	26	4	no	—
205	C₂×C₈	—	26	2	no	—
206	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
207	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
208	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
209	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₄²	—	2	4	no	—
218	C₄²	D₄×D₁₃	1	1	yes	characteristic
219	C₄²	#6129	1	1	yes	characteristic
220	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
221	C₄²	C₂³×D₁₃	1	1	yes	characteristic
222	C₁₃	#17991	1	1	yes	characteristic, complemented, Sylow
223	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	2	4	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
234	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
235	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂×#6129	1	1	yes	characteristic
238	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	26	4	no	—
244	C₈	—	52	2	no	—
245	C₈	—	52	4	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	4	no	—
255	C₄	—	4	4	no	—
256	C₄	#11719471	1	2	yes	—
257	C₄	#11719524	1	1	yes	characteristic
258	C₄	#11719524	1	1	yes	characteristic
259	C₂	—	4	1	no	—
260	C₂	#13123554	1	1	yes	characteristic
261	C₂	#13123704	1	1	yes	characteristic
262	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	#17991
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. #13113708	C₂	semidirect
3. #13113687	C₂	semidirect
222. C₁₃	#17991	semidirect

Maximal subgroups

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

subgroup	index
#13123931	2
#13113708	2
#13113687	2
#13113599	2
#13113309	2
#13113128	2
C₁₃×#3170	2
#17991	13