Subgroups — #15762731

Order 3328.

694 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718356	C₂	1	1	yes	maximal, characteristic
2	#13115642	C₂	1	2	yes	maximal, complemented
3	#13115642	C₂	1	2	yes	maximal, complemented
4	#13114964	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₄	—	2	2	no	—
7	C₂×#88128	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88061	—	2	4	no	—
12	C₂×#88061	—	2	4	no	—
13	#11718596	C₂²	1	2	yes	—
14	#11718356	—	2	2	no	—
15	#11718356	C₄	1	4	yes	complemented
16	C₁₃×#322	—	2	2	no	—
17	#11718143	—	2	2	no	—
18	C₂²×D₅₂	—	2	2	no	—
19	C₂²×D₅₂	—	2	2	no	—
20	C₂²×D₅₂	—	2	2	no	—
21	C₂²×C₈×C₁₃	D₄	1	2	yes	—
22	C₂²×C₈×C₁₃	D₄	1	2	yes	—
23	C₂²×C₈×C₁₃	D₄	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₃	—	2	2	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂×C₄⋊Dic₁₃	—	2	2	no	—
28	C₂×D₁₀₄	—	2	8	no	—
29	C₂×D₁₀₄	—	4	8	no	—
30	#88128	—	2	2	no	—
31	#88128	—	2	2	no	—
32	#88128	—	2	2	no	—
33	#88128	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₁₃	—	2	1	no	—
35	C₄×C₈×C₁₃	—	2	1	no	—
36	C₄×C₈×C₁₃	—	2	1	no	—
37	C₄×C₈×C₁₃	—	2	1	no	—
38	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₃	D₄	1	2	yes	—
40	#88061	—	4	8	no	—
41	#88061	—	4	8	no	—
42	#9832	—	13	1	no	maximal, Sylow
43	C₂³×D₁₃	—	4	2	no	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	—	4	2	no	—
48	C₂×D₅₂	—	2	4	no	—
49	C₂×D₅₂	—	2	4	no	—
50	C₂×D₅₂	—	2	4	no	—
51	C₂×D₅₂	—	4	4	no	—
52	C₂×D₅₂	—	4	4	no	—
53	C₂×D₅₂	—	4	4	no	—
54	C₂×D₅₂	—	4	4	no	—
55	C₂×D₅₂	—	4	4	no	—
56	C₂×D₅₂	—	4	4	no	—
57	C₂×D₅₂	—	4	2	no	—
58	C₂×D₅₂	—	4	2	no	—
59	C₂×D₅₂	—	4	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	4	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	2	4	no	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈×C₁₃	D₈	1	4	yes	complemented
67	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₃	#41	1	1	yes	characteristic
70	C₄²×C₁₃	#41	1	1	yes	characteristic
71	C₄⋊Dic₁₃	—	4	4	no	—
72	C₄⋊Dic₁₃	—	4	4	no	—
73	C₄⋊Dic₁₃	—	4	4	no	—
74	D₁₀₄	—	4	16	no	—
75	C₂×#493	—	13	1	no	—
76	#1957	—	13	2	no	—
77	#1957	—	13	2	no	—
78	#1834	—	13	1	no	—
79	C₂×#322	D₁₃	1	1	yes	characteristic, complemented
80	C₂³×C₁₃	#127	1	1	yes	characteristic
81	C₂²×D₁₃	—	4	4	no	—
82	C₂²×D₁₃	—	8	2	no	—
83	C₂²×D₁₃	—	8	2	no	—
84	C₂²×D₁₃	—	8	2	no	—
85	C₂²×D₁₃	—	8	2	no	—
86	C₂²×D₁₃	—	8	2	no	—
87	C₂²×D₁₃	—	8	2	no	—
88	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
92	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₁₃	#135	1	2	yes	—
96	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
97	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
98	C₂×Dic₁₃	—	8	2	no	—
99	C₂×Dic₁₃	—	8	2	no	—
100	C₂×Dic₁₃	—	8	2	no	—
101	D₅₂	—	4	8	no	—
102	D₅₂	—	4	8	no	—
103	D₅₂	—	4	8	no	—
104	D₅₂	—	8	4	no	—
105	D₅₂	—	8	4	no	—
106	D₅₂	—	8	4	no	—
107	C₈×C₁₃	—	2	4	no	—
108	C₈×C₁₃	—	2	4	no	—
109	C₈×C₁₃	—	4	2	no	—
110	C₂²×D₈	—	26	2	no	—
111	C₂×#127	—	13	2	no	—
112	#493	—	13	4	no	—
113	#493	—	26	2	no	—
114	C₂×#102	—	26	4	no	—
115	C₂×#102	—	26	4	no	—
116	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
119	#384	—	13	2	no	—
120	#340	—	26	2	no	—
121	#322	—	2	2	no	—
122	C₂²×C₁₃	#390	1	1	yes	characteristic
123	C₂²×C₁₃	#493	1	1	yes	characteristic
124	C₂²×C₁₃	#495	1	1	yes	characteristic
125	C₂²×C₁₃	#496	1	2	yes	—
126	C₂²×C₁₃	#496	1	2	yes	—
127	C₂×D₁₃	—	8	4	no	—
128	C₂×D₁₃	—	8	4	no	—
129	C₂×D₁₃	—	8	4	no	—
130	C₂×D₁₃	—	16	2	no	—
131	C₂×D₁₃	—	16	2	no	—
132	C₂×D₁₃	—	16	2	no	—
133	C₂×D₁₃	—	16	2	no	—
134	C₄×C₁₃	#418	1	2	yes	—
135	C₄×C₁₃	C₄×D₈	1	2	yes	—
136	C₄×C₁₃	#442	1	2	yes	—
137	C₄×C₁₃	#466	1	2	yes	—
138	C₄×C₁₃	#469	1	2	yes	—
139	C₄×C₁₃	#490	1	2	yes	—
140	Dic₁₃	—	16	2	no	—
141	C₂²×D₄	—	26	2	no	—
142	C₂²×D₄	—	26	2	no	—
143	C₂²×D₄	—	26	2	no	—
144	C₂×D₈	—	26	8	no	—
145	C₂×D₈	—	52	8	no	—
146	C₂²×C₈	D₅₂	1	2	yes	—
147	C₂²×C₈	D₅₂	1	2	yes	—
148	C₂²×C₈	D₅₂	1	2	yes	—
149	#127	—	13	4	no	—
150	#127	—	26	2	no	—
151	#127	—	26	2	no	—
152	#127	—	26	2	no	—
153	C₂×C₄⋊C₄	—	26	2	no	—
154	C₂×C₄⋊C₄	—	26	2	no	—
155	C₂×C₄⋊C₄	—	26	2	no	—
156	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
157	#102	—	52	8	no	—
158	#102	—	52	8	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	2	1	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	C₄×D₁₃	1	2	yes	—
164	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
165	C₂×C₁₃	#1717	1	2	yes	—
166	C₂×C₁₃	#1721	1	2	yes	—
167	C₂×C₁₃	#1961	1	1	yes	characteristic
168	C₂×C₁₃	#1970	1	1	yes	characteristic
169	C₂×C₁₃	#1972	1	1	yes	characteristic
170	D₁₃	—	16	4	no	—
171	C₂⁴	—	52	2	no	—
172	C₂×D₄	—	26	4	no	—
173	C₂×D₄	—	26	4	no	—
174	C₂×D₄	—	26	4	no	—
175	C₂×D₄	—	52	2	no	—
176	C₂×D₄	—	52	4	no	—
177	C₂×D₄	—	52	4	no	—
178	C₂×D₄	—	52	4	no	—
179	C₂×D₄	—	52	4	no	—
180	C₂×D₄	—	52	4	no	—
181	C₂×D₄	—	52	4	no	—
182	C₂×D₄	—	52	2	no	—
183	C₂×D₄	—	52	2	no	—
184	C₂²×C₄	—	52	2	no	—
185	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
188	D₈	—	52	16	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	D₁₀₄	1	4	yes	complemented
195	C₂×C₈	#6103	1	4	yes	—
196	C₄⋊C₄	—	52	4	no	—
197	C₄⋊C₄	—	52	4	no	—
198	C₄⋊C₄	—	52	4	no	—
199	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
200	C₄²	#6127	1	1	yes	characteristic
201	C₄²	#6127	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
203	C₁₃	#9832	1	1	yes	characteristic, complemented, Sylow
204	C₂³	—	52	4	no	—
205	C₂³	—	104	2	no	—
206	C₂³	—	104	2	no	—
207	C₂³	—	104	2	no	—
208	C₂³	—	104	2	no	—
209	C₂³	—	104	2	no	—
210	C₂³	—	104	2	no	—
211	C₂³	#88128	1	1	yes	characteristic
212	D₄	—	52	8	no	—
213	D₄	—	52	8	no	—
214	D₄	—	52	8	no	—
215	D₄	—	104	4	no	—
216	D₄	—	104	4	no	—
217	D₄	—	104	4	no	—
218	C₂×C₄	—	104	2	no	—
219	C₂×C₄	—	104	2	no	—
220	C₂×C₄	—	104	2	no	—
221	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
222	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
223	C₂×C₄	#88129	1	1	yes	characteristic
224	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
225	C₂×C₄	#88158	1	2	yes	—
226	C₂×C₄	#88162	1	4	yes	—
227	C₂×C₄	#88162	1	4	yes	—
228	C₂×C₄	C₂×#6103	1	1	yes	characteristic
229	C₂×C₄	#88184	1	1	yes	characteristic
230	C₂×C₄	#88184	1	1	yes	characteristic
231	C₈	—	2	4	no	—
232	C₈	—	2	4	no	—
233	C₈	—	4	2	no	—
234	C₂²	—	104	4	no	—
235	C₂²	—	104	4	no	—
236	C₂²	—	104	4	no	—
237	C₂²	—	208	2	no	—
238	C₂²	—	208	2	no	—
239	C₂²	—	208	2	no	—
240	C₂²	—	208	2	no	—
241	C₂²	#11718356	1	1	yes	characteristic
242	C₂²	#11718357	1	1	yes	characteristic
243	C₂²	#11718373	1	2	yes	—
244	C₂²	#11718373	1	2	yes	—
245	C₂²	#11718600	1	1	yes	characteristic
246	C₄	—	208	2	no	—
247	C₄	C₄×D₁₀₄	1	2	yes	—
248	C₄	#11718372	1	2	yes	—
249	C₄	#11718374	1	2	yes	—
250	C₄	#11718776	1	2	yes	—
251	C₄	#11718792	1	2	yes	—
252	C₄	#11718796	1	2	yes	—
253	C₂	—	208	4	no	—
254	C₂	#13114475	1	2	yes	—
255	C₂	#13114502	1	2	yes	—
256	C₂	#13115660	1	1	yes	characteristic
257	C₂	#13115702	1	1	yes	characteristic
258	C₂	#13115712	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#11718356	C₂
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#9832
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #13115642	C₂	semidirect
3. #13115642	C₂	semidirect
4. #13114964	C₂	semidirect
5. C₂×C₁₃×#322	C₂	semidirect
15. #11718356	C₄	semidirect
66. C₂×C₈×C₁₃	D₈	semidirect
79. C₂×#322	D₁₃	semidirect
194. C₂×C₈	D₁₀₄	semidirect
203. C₁₃	#9832	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718356	2
#13115642	2
#13114964	2
C₂×C₁₃×#322	2
#9832	13