Subgroups — #15763073

Order 3328.

506 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123234	C₂	1	1	yes	maximal, characteristic
2	#13114689	C₂	1	1	yes	maximal, characteristic
3	#13112720	C₂	1	4	yes	maximal
4	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	—	2	1	no	—
6	C₂×#88123	—	2	1	no	—
7	C₂×#88123	C₄	1	2	yes	—
8	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	#11718596	—	2	1	no	—
11	#11718596	—	2	1	no	—
12	#11718596	C₄	1	2	yes	—
13	#11718596	C₂²	1	2	yes	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₂×#88043	—	2	1	no	—
16	C₁₃×#336	—	2	1	no	—
17	C₁₃×#336	—	2	1	no	—
18	#11718116	—	2	1	no	—
19	C₂²×Dic₂₆	—	2	2	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₃	—	2	2	no	—
29	C₂×C₄⋊Dic₁₃	—	2	2	no	—
30	C₂×C₄⋊Dic₁₃	—	2	2	no	—
31	C₂×C₄⋊Dic₁₃	—	2	2	no	—
32	C₂×C₄⋊Dic₁₃	—	2	2	no	—
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
34	C₂²×#1055	—	2	2	no	—
35	#88123	—	4	4	no	—
36	#88123	—	4	2	no	—
37	#88123	—	4	2	no	—
38	#88123	—	4	4	no	—
39	C₁₃×C₄⋊C₈	—	2	4	no	—
40	C₄⋊#1055	—	2	2	no	—
41	C₄⋊#1055	Q₈	1	4	yes	complemented
42	#88043	—	4	2	no	—
43	#9905	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₁₃	—	2	4	no	—
52	C₂×Dic₂₆	—	4	4	no	—
53	C₂×Dic₂₆	—	4	4	no	—
54	C₂×Dic₂₆	—	4	4	no	—
55	C₂×C₈×C₁₃	—	4	2	no	—
56	C₂×C₈×C₁₃	—	4	2	no	—
57	C₂×C₈×C₁₃	—	4	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	2	2	no	—
60	C₄²×C₁₃	—	2	2	no	—
61	C₄²×C₁₃	—	2	2	no	—
62	C₄²×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	D₈	1	2	yes	—
64	C₄²×C₁₃	SD₁₆	1	2	yes	—
65	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
66	C₄²×C₁₃	#41	1	2	yes	—
67	C₄⋊Dic₁₃	—	4	4	no	—
68	C₄⋊Dic₁₃	—	4	4	no	—
69	C₄⋊Dic₁₃	—	4	4	no	—
70	C₄⋊Dic₁₃	—	4	2	no	—
71	C₄⋊Dic₁₃	—	4	2	no	—
72	C₄⋊Dic₁₃	—	4	4	no	—
73	C₄⋊Dic₁₃	—	4	4	no	—
74	C₂×#1055	—	2	4	no	—
75	C₂×#1055	—	4	2	no	—
76	C₂×#1055	—	4	2	no	—
77	#2868	—	13	1	no	—
78	#1831	D₁₃	1	1	yes	characteristic
79	#1767	—	13	1	no	—
80	#1296	—	13	4	no	—
81	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	#100	1	2	yes	—
93	C₂×C₄×C₁₃	#101	1	2	yes	—
94	C₂×C₄×C₁₃	#102	1	2	yes	—
95	C₂×C₄×C₁₃	#102	1	2	yes	—
96	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
98	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₁₃	#137	1	2	yes	—
104	C₂×C₄×C₁₃	#137	1	2	yes	—
105	C₂×Dic₁₃	—	4	4	no	—
106	C₂×Dic₁₃	—	4	4	no	—
107	C₂×Dic₁₃	—	4	4	no	—
108	Dic₂₆	—	8	8	no	—
109	C₈×C₁₃	—	8	2	no	—
110	#1055	—	4	4	no	—
111	C₂×C₄⋊Q₈	—	13	2	no	—
112	C₂×C₄⋊Q₈	—	26	1	no	—
113	C₂×C₄⋊Q₈	—	26	1	no	—
114	C₂×C₄⋊C₈	—	13	2	no	—
115	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
116	C₂×#97	—	26	1	no	—
117	#384	—	13	2	no	—
118	#384	—	13	2	no	—
119	#384	—	26	1	no	—
120	#384	—	26	1	no	—
121	C₄³	C₂×D₁₃	1	1	yes	characteristic
122	#336	—	2	1	no	—
123	#336	—	2	1	no	—
124	#336	—	26	1	no	—
125	C₂²×C₁₃	#391	1	1	yes	characteristic
126	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
129	C₂²×C₁₃	#418	1	1	yes	characteristic
130	C₂²×C₁₃	#419	1	1	yes	characteristic
131	C₂²×C₁₃	#419	1	1	yes	characteristic
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	#474	1	2	yes	—
138	C₄×C₁₃	#475	1	2	yes	—
139	C₄×C₁₃	#489	1	2	yes	—
140	C₄×C₁₃	#501	1	2	yes	—
141	Dic₁₃	—	8	4	no	—
142	C₂²×Q₈	—	26	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	26	2	no	—
145	C₄⋊Q₈	—	52	4	no	—
146	C₄⋊Q₈	—	52	4	no	—
147	C₄⋊Q₈	—	52	2	no	—
148	C₄⋊Q₈	—	52	2	no	—
149	C₂×C₄⋊C₄	—	13	4	no	—
150	C₂×C₄⋊C₄	—	26	2	no	—
151	C₂×C₄⋊C₄	—	26	2	no	—
152	C₂×C₄⋊C₄	—	26	2	no	—
153	C₂×C₄⋊C₄	—	26	2	no	—
154	C₂×C₄⋊C₄	—	26	2	no	—
155	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
156	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
157	C₂×C₄²	D₅₂	1	1	yes	characteristic
158	C₂×C₄²	D₅₂	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
160	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
162	C₄⋊C₈	—	2	4	no	—
163	C₄⋊C₈	—	13	4	no	—
164	C₄⋊C₈	—	26	2	no	—
165	#97	—	52	2	no	—
166	C₂×C₁₃	#1545	1	1	yes	characteristic
167	C₂×C₁₃	#1546	1	1	yes	characteristic
168	C₂×C₁₃	#1556	1	1	yes	characteristic
169	C₂×C₁₃	#1558	1	1	yes	characteristic
170	C₂×C₁₃	#1988	1	1	yes	characteristic
171	C₂×C₁₃	#1993	1	1	yes	characteristic
172	C₂×C₁₃	#1997	1	1	yes	characteristic
173	C₂×Q₈	—	52	4	no	—
174	C₂×Q₈	—	52	4	no	—
175	C₂×Q₈	—	52	4	no	—
176	C₂²×C₄	—	26	4	no	—
177	C₂²×C₄	#6103	1	1	yes	characteristic
178	C₂²×C₄	#6103	1	1	yes	characteristic
179	C₂²×C₄	#6103	1	1	yes	characteristic
180	C₂²×C₄	#6103	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	26	4	no	—
188	C₂×C₈	—	52	2	no	—
189	C₂×C₈	—	52	2	no	—
190	C₄⋊C₄	—	52	4	no	—
191	C₄⋊C₄	—	52	4	no	—
192	C₄⋊C₄	—	52	2	no	—
193	C₄⋊C₄	—	52	2	no	—
194	C₄⋊C₄	—	52	4	no	—
195	C₄⋊C₄	—	52	4	no	—
196	C₄⋊C₄	—	52	4	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	#6095	1	2	yes	—
203	C₄²	D₁₀₄	1	2	yes	—
204	C₄²	#6129	1	2	yes	—
205	C₄²	Q₈×D₁₃	1	2	yes	—
206	C₁₃	#9905	1	1	yes	characteristic, complemented, Sylow
207	C₂³	C₂×#6103	1	1	yes	characteristic
208	Q₈	—	104	8	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	52	4	no	—
220	C₂×C₄	—	52	4	no	—
221	C₂×C₄	—	52	4	no	—
222	C₂×C₄	#88061	1	2	yes	—
223	C₂×C₄	#88061	1	2	yes	—
224	C₂×C₄	#88063	1	2	yes	—
225	C₂×C₄	#88064	1	2	yes	—
226	C₂×C₄	#88141	1	2	yes	—
227	C₂×C₄	#88151	1	2	yes	—
228	C₂×C₄	C₂×#6095	1	1	yes	characteristic
229	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
230	C₂×C₄	#88163	1	2	yes	—
231	C₂×C₄	#88190	1	1	yes	characteristic
232	C₂×C₄	#88198	1	1	yes	characteristic
233	C₂×C₄	#88203	1	2	yes	—
234	C₂×C₄	#88205	1	2	yes	—
235	C₈	—	8	2	no	—
236	C₈	—	52	4	no	—
237	C₂²	#11718635	1	1	yes	characteristic
238	C₂²	#11718662	1	1	yes	characteristic
239	C₂²	#11718665	1	1	yes	characteristic
240	C₂²	C₂×#88061	1	1	yes	characteristic
241	C₂²	#11718781	1	1	yes	characteristic
242	C₂²	C₂×#88063	1	1	yes	characteristic
243	C₂²	C₂×#88064	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	104	4	no	—
250	C₄	#11718507	1	2	yes	—
251	C₄	#11718512	1	2	yes	—
252	C₄	#11718716	1	2	yes	—
253	C₄	#11718726	1	2	yes	—
254	C₂	#13113503	1	1	yes	characteristic
255	C₂	#13113507	1	1	yes	characteristic
256	C₂	#13113554	1	1	yes	characteristic
257	C₂	#13113561	1	1	yes	characteristic
258	C₂	#13115787	1	1	yes	characteristic
259	C₂	#13115808	1	1	yes	characteristic
260	C₂	#13115824	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#9905
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
41. C₄⋊#1055	Q₈	semidirect
206. C₁₃	#9905	semidirect

Maximal subgroups

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

subgroup	index
#13123234	2
#13114689	2
#13112720	2
C₁₃×#1831	2
#9905	13