Subgroups — #15767167

Order 3328.

422 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115785	C₂	1	2	yes	maximal
2	#13115554	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718138	C₂	1	2	yes	maximal
4	#13112671	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₁₃×#500	—	2	4	no	—
8	C₂×C₁₃×#107	—	2	1	no	—
9	C₂×C₁₃×#107	—	2	1	no	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88054	—	2	1	no	—
12	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
15	#11718142	—	2	1	no	—
16	#11718138	—	2	4	no	—
17	#11718121	—	2	2	no	—
18	#11718121	—	2	2	no	—
19	C₂²×Q₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₁₃×C₄⋊Q₈	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	2	no	—
24	C₁₃×C₄⋊Q₈	—	2	2	no	—
25	C₁₃×C₄⋊Q₈	D₄	1	4	yes	—
26	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	—	2	1	no	—
32	C₂²×#1055	—	2	1	no	—
33	C₂²×#1055	—	2	2	no	—
34	C₁₃×#107	—	2	4	no	—
35	C₁₃×#107	—	2	4	no	—
36	C₄×C₈×C₁₃	—	2	2	no	—
37	#88071	—	2	1	no	—
38	#88054	—	4	2	no	—
39	C₄⋊#1055	—	2	2	no	—
40	C₄⋊#1055	—	2	2	no	—
41	C₄⋊#1055	—	2	2	no	—
42	#10543	—	13	1	no	maximal, Sylow
43	C₂×Q₈×C₁₃	—	2	4	no	—
44	C₂×Q₈×C₁₃	—	4	2	no	—
45	C₂×Q₈×C₁₃	—	8	2	no	—
46	C₂×Q₈×C₁₃	—	8	2	no	—
47	C₂²×C₄×C₁₃	—	4	2	no	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
51	C₂²×Dic₁₃	—	2	1	no	—
52	C₂²×Dic₁₃	—	2	1	no	—
53	C₂×C₈×C₁₃	—	2	1	no	—
54	C₂×C₈×C₁₃	—	2	1	no	—
55	C₂×C₈×C₁₃	—	2	1	no	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
58	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
59	C₁₃×C₄⋊C₄	—	2	4	no	—
60	C₁₃×C₄⋊C₄	—	2	4	no	—
61	C₁₃×C₄⋊C₄	—	4	2	no	—
62	C₁₃×C₄⋊C₄	—	4	2	no	—
63	C₄²×C₁₃	C₂×D₄	1	2	yes	—
64	C₄²×C₁₃	#41	1	2	yes	—
65	C₄⋊Dic₁₃	—	2	1	no	—
66	C₄⋊Dic₁₃	—	2	2	no	—
67	C₄⋊Dic₁₃	Q₁₆	1	2	yes	—
68	C₄×Dic₁₃	—	4	1	no	—
69	C₄×Dic₁₃	—	4	1	no	—
70	C₂×#1055	—	4	1	no	—
71	C₂×#1055	—	4	2	no	—
72	C₂×#1055	—	4	1	no	—
73	C₂×#1055	—	4	2	no	—
74	C₂×#1055	—	4	1	no	—
75	C₂×#1055	—	4	2	no	—
76	C₂×#500	D₁₃	1	1	yes	characteristic
77	#1987	—	13	2	no	—
78	#1939	—	13	1	no	—
79	C₂×#333	—	13	2	no	—
80	#1277	—	13	1	no	—
81	C₂³×C₁₃	#121	1	1	yes	characteristic
82	Q₈×C₁₃	—	8	4	no	—
83	Q₈×C₁₃	—	8	4	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	4	4	no	—
87	C₂×C₄×C₁₃	—	8	2	no	—
88	C₂×C₄×C₁₃	—	8	2	no	—
89	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#121	1	2	yes	—
91	C₂×C₄×C₁₃	#123	1	2	yes	—
92	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#137	1	2	yes	—
100	C₂×Dic₁₃	—	2	2	no	—
101	C₂×Dic₁₃	—	4	1	no	—
102	C₂×Dic₁₃	—	4	1	no	—
103	C₂×Dic₁₃	—	4	1	no	—
104	C₂×Dic₁₃	—	4	1	no	—
105	C₂×Dic₁₃	—	4	1	no	—
106	C₈×C₁₃	—	2	2	no	—
107	C₈×C₁₃	—	2	2	no	—
108	#1055	—	8	1	no	—
109	#1055	—	8	2	no	—
110	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
111	#500	—	2	4	no	—
112	C₂×#107	—	2	1	no	—
113	C₂×#107	—	2	1	no	—
114	C₂×C₄⋊C₈	—	13	1	no	—
115	C₂×C₄⋊C₈	—	13	2	no	—
116	C₂×C₄⋊C₈	—	26	1	no	—
117	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
118	C₄×C₄⋊C₄	—	13	1	no	—
119	#340	—	26	2	no	—
120	#340	—	26	2	no	—
121	#336	—	26	1	no	—
122	#333	—	26	4	no	—
123	C₂²×C₁₃	#394	1	1	yes	characteristic
124	C₂²×C₁₃	#461	1	1	yes	characteristic
125	C₂²×C₁₃	#462	1	1	yes	characteristic
126	C₂²×C₁₃	#467	1	1	yes	characteristic
127	C₂²×C₁₃	#470	1	1	yes	characteristic
128	C₂²×C₁₃	#472	1	1	yes	characteristic
129	C₂²×C₁₃	#472	1	1	yes	characteristic
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	—	8	4	no	—
132	C₄×C₁₃	#448	1	1	yes	characteristic
133	C₄×C₁₃	#451	1	1	yes	characteristic
134	C₄×C₁₃	#462	1	2	yes	—
135	C₄×C₁₃	#464	1	2	yes	—
136	C₄×C₁₃	#487	1	1	yes	characteristic
137	C₄×C₁₃	#491	1	1	yes	characteristic
138	Dic₁₃	—	4	2	no	—
139	Dic₁₃	—	8	1	no	—
140	C₂²×Q₈	—	2	2	no	—
141	C₂²×C₈	—	26	2	no	—
142	C₂²×C₈	—	26	1	no	—
143	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
144	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
145	C₄⋊Q₈	—	2	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	C₂²⋊D₁₃	1	4	yes	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	13	1	no	—
152	C₂×C₄⋊C₄	—	13	1	no	—
153	C₂×C₄²	—	26	1	no	—
154	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
155	#107	—	2	4	no	—
156	#107	—	2	4	no	—
157	C₄⋊C₈	—	26	2	no	—
158	C₄⋊C₈	—	26	2	no	—
159	C₄⋊C₈	—	26	2	no	—
160	C₄⋊C₈	—	52	2	no	—
161	C₄×C₈	—	2	2	no	—
162	#95	—	26	1	no	—
163	C₂×C₁₃	#1675	1	1	yes	characteristic
164	C₂×C₁₃	#1676	1	1	yes	characteristic
165	C₂×C₁₃	#1695	1	1	yes	characteristic
166	C₂×C₁₃	#1696	1	1	yes	characteristic
167	C₂×C₁₃	#2043	1	1	yes	characteristic
168	C₂×C₁₃	#2045	1	1	yes	characteristic
169	C₂×C₁₃	#2056	1	1	yes	characteristic
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	8	2	no	—
173	C₂×Q₈	—	8	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	26	1	no	—
176	C₂²×C₄	—	26	1	no	—
177	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
178	C₂²×C₄	#6129	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	52	2	no	—
185	C₂×C₈	—	52	2	no	—
186	C₂×C₈	—	52	2	no	—
187	C₂×C₈	—	52	1	no	—
188	C₂×C₈	—	52	1	no	—
189	C₂×C₈	—	52	1	no	—
190	C₂×C₈	#6106	1	2	yes	—
191	C₂×C₈	#6107	1	2	yes	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	13	2	no	—
197	C₄⋊C₄	—	26	1	no	—
198	C₄⋊C₄	—	26	2	no	—
199	C₄²	—	52	1	no	—
200	C₄²	—	52	1	no	—
201	C₄²	#6129	1	2	yes	—
202	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
203	C₁₃	#10543	1	1	yes	characteristic, complemented, Sylow
204	C₂³	#88192	1	1	yes	characteristic
205	Q₈	—	8	4	no	—
206	Q₈	—	8	4	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	2	no	—
212	C₂×C₄	—	26	2	no	—
213	C₂×C₄	—	52	1	no	—
214	C₂×C₄	—	52	1	no	—
215	C₂×C₄	—	52	1	no	—
216	C₂×C₄	—	52	1	no	—
217	C₂×C₄	—	52	1	no	—
218	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
219	C₂×C₄	#88172	1	2	yes	—
220	C₂×C₄	#88173	1	1	yes	characteristic
221	C₂×C₄	#88189	1	2	yes	—
222	C₂×C₄	#88190	1	1	yes	characteristic
223	C₂×C₄	#88193	1	2	yes	—
224	C₂×C₄	C₂×#6106	1	1	yes	characteristic
225	C₂×C₄	#88196	1	1	yes	characteristic
226	C₂×C₄	#88196	1	1	yes	characteristic
227	C₂×C₄	C₂×#6107	1	1	yes	characteristic
228	C₂×C₄	#88207	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	104	2	no	—
232	C₈	—	104	1	no	—
233	C₂²	#11718744	1	1	yes	characteristic
234	C₂²	#11718751	1	1	yes	characteristic
235	C₂²	#11718821	1	1	yes	characteristic
236	C₂²	#11718821	1	1	yes	characteristic
237	C₂²	#11718849	1	1	yes	characteristic
238	C₂²	#11718850	1	1	yes	characteristic
239	C₂²	#11718883	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	8	4	no	—
242	C₄	—	52	2	no	—
243	C₄	—	104	1	no	—
244	C₄	#11718742	1	1	yes	characteristic
245	C₄	#11718743	1	1	yes	characteristic
246	C₄	#11718843	1	2	yes	—
247	C₄	#11718846	1	2	yes	—
248	C₄	#11718888	1	1	yes	characteristic
249	C₄	#11718889	1	1	yes	characteristic
250	C₂	#13114232	1	1	yes	characteristic
251	C₂	#13114237	1	1	yes	characteristic
252	C₂	#13114356	1	1	yes	characteristic
253	C₂	#13114361	1	1	yes	characteristic
254	C₂	#13116079	1	1	yes	characteristic
255	C₂	#13116089	1	1	yes	characteristic
256	C₂	#13116156	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88192
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle, prime radical	C₂³×C₁₃	#121
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	#10543
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
203. C₁₃	#10543	semidirect

Maximal subgroups

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

subgroup	index
#13115785	2
#13115554	2
C₂×#11718138	2
#13112671	2
C₂×C₁₃×#500	2
#10543	13