Subgroups — #15750978

Order 3328.

478 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88123	C₂	1	1	yes	maximal, characteristic
2	#13114953	C₂	1	1	yes	maximal, characteristic
3	#13112658	C₂	1	4	yes	maximal
4	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₂₆	—	2	1	no	—
7	C₂×C₄×C₈×C₁₃	—	2	1	no	—
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	#11718596	C₄	1	1	yes	characteristic
10	C₄×C₄⋊Dic₁₃	—	2	1	no	—
11	C₄×C₄⋊Dic₁₃	C₄	1	2	yes	—
12	#11718590	—	2	1	no	—
13	#11718589	—	2	1	no	—
14	#11718589	C₂²	1	2	yes	—
15	C₂×C₄⋊#1055	—	2	1	no	—
16	C₁₃×#336	—	2	1	no	—
17	C₁₃×#336	C₂²	1	2	yes	—
18	#11718116	—	2	1	no	—
19	#11718116	C₂²	1	2	yes	—
20	C₂²×Dic₂₆	—	2	1	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	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	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	—	2	1	no	—
30	C₂×C₄⋊Dic₁₃	—	2	1	no	—
31	C₂×C₄⋊Dic₁₃	—	2	1	no	—
32	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
34	C₂×#6101	—	2	2	no	—
35	C₂×C₄×Dic₁₃	—	2	2	no	—
36	C₂²×#1055	—	2	2	no	—
37	#88123	—	2	2	no	—
38	#88123	C₈	1	4	yes	complemented
39	C₄×Dic₂₆	—	4	2	no	—
40	C₄×Dic₂₆	—	4	2	no	—
41	C₄×C₈×C₁₃	—	2	4	no	—
42	#88071	—	2	2	no	—
43	#88071	—	2	2	no	—
44	C₄⋊#1055	—	4	2	no	—
45	#7588	—	13	1	no	maximal, Sylow
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	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₁₃	—	2	2	no	—
54	C₂²×Dic₁₃	—	2	2	no	—
55	C₂×Dic₂₆	—	2	4	no	—
56	C₂×Dic₂₆	—	4	2	no	—
57	C₂×Dic₂₆	—	4	2	no	—
58	C₂×C₈×C₁₃	—	2	4	no	—
59	C₂×C₈×C₁₃	—	4	2	no	—
60	C₂×C₈×C₁₃	—	4	2	no	—
61	C₄²×C₁₃	—	2	2	no	—
62	C₄²×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	—	2	2	no	—
64	C₄²×C₁₃	—	4	1	no	—
65	C₄²×C₁₃	—	4	1	no	—
66	C₄²×C₁₃	C₂×C₈	1	2	yes	—
67	C₄²×C₁₃	M₁₆	1	2	yes	—
68	C₄²×C₁₃	D₈	1	2	yes	—
69	C₄²×C₁₃	SD₁₆	1	2	yes	—
70	C₄⋊Dic₁₃	—	2	4	no	—
71	C₄⋊Dic₁₃	—	2	2	no	—
72	C₄⋊Dic₁₃	—	4	2	no	—
73	C₄⋊Dic₁₃	—	4	2	no	—
74	C₄⋊Dic₁₃	—	4	2	no	—
75	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
76	#6101	—	4	2	no	—
77	#6101	—	4	2	no	—
78	C₄×Dic₁₃	—	4	4	no	—
79	C₂×#1055	—	4	2	no	—
80	C₂×#1055	—	4	2	no	—
81	C₂×#1055	—	4	2	no	—
82	C₄×C₄⋊Q₈	—	13	1	no	—
83	#1831	—	13	1	no	—
84	#1744	D₁₃	1	1	yes	characteristic
85	#1271	—	13	4	no	—
86	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
100	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
101	C₂×C₄×C₁₃	#101	1	2	yes	—
102	C₂×C₄×C₁₃	#102	1	2	yes	—
103	C₂×C₄×C₁₃	#102	1	2	yes	—
104	C₂×C₄×C₁₃	#104	1	2	yes	—
105	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#137	1	2	yes	—
110	C₂×Dic₁₃	—	2	4	no	—
111	C₂×Dic₁₃	—	4	2	no	—
112	C₂×Dic₁₃	—	4	2	no	—
113	C₂×Dic₁₃	—	4	2	no	—
114	C₂×Dic₁₃	—	4	2	no	—
115	C₂×Dic₁₃	—	4	2	no	—
116	Dic₂₆	—	4	8	no	—
117	C₈×C₁₃	—	4	4	no	—
118	#1055	—	8	2	no	—
119	C₂×C₄⋊Q₈	—	13	1	no	—
120	C₂×C₄×Q₈	—	26	1	no	—
121	C₂×C₄⋊C₈	—	26	1	no	—
122	C₂×C₄×C₈	—	2	1	no	—
123	#391	—	26	1	no	—
124	#389	—	13	2	no	—
125	#389	—	26	1	no	—
126	#384	—	13	1	no	—
127	C₄×C₄⋊C₄	—	13	2	no	—
128	C₄×C₄⋊C₄	—	26	1	no	—
129	C₄³	C₂×D₁₃	1	1	yes	characteristic
130	#336	—	2	1	no	—
131	#336	—	13	2	no	—
132	#336	—	26	1	no	—
133	#336	C₂×D₁₃	1	2	yes	—
134	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₂²×C₁₃	#410	1	1	yes	characteristic
136	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
137	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
138	C₂²×C₁₃	#419	1	1	yes	characteristic
139	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
140	C₂²×C₁₃	#421	1	1	yes	characteristic
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	D₄⋊C₈	1	4	yes	—
150	Dic₁₃	—	4	4	no	—
151	Dic₁₃	—	8	2	no	—
152	C₂²×Q₈	—	26	1	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	26	2	no	—
155	C₄⋊Q₈	—	13	4	no	—
156	C₄⋊Q₈	—	26	2	no	—
157	C₄×Q₈	—	52	2	no	—
158	C₄×Q₈	—	52	2	no	—
159	C₂×C₄⋊C₄	—	13	2	no	—
160	C₂×C₄⋊C₄	—	13	2	no	—
161	C₂×C₄⋊C₄	—	26	1	no	—
162	C₂×C₄⋊C₄	—	26	1	no	—
163	C₂×C₄⋊C₄	—	26	1	no	—
164	C₂×C₄⋊C₄	—	26	2	no	—
165	C₂×C₄²	—	26	2	no	—
166	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
167	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
168	C₂×C₄²	D₅₂	1	1	yes	characteristic
169	C₂×C₄²	D₅₂	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
173	C₄⋊C₈	—	52	2	no	—
174	C₄×C₈	—	2	4	no	—
175	#95	—	26	2	no	—
176	#95	—	26	2	no	—
177	C₂×C₁₃	#1461	1	1	yes	characteristic
178	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
179	C₂×C₁₃	#1479	1	1	yes	characteristic
180	C₂×C₁₃	#1506	1	1	yes	characteristic
181	C₂×C₁₃	#1517	1	1	yes	characteristic
182	C₂×C₁₃	#1531	1	1	yes	characteristic
183	C₂×C₁₃	#1546	1	1	yes	characteristic
184	C₂×Q₈	—	26	4	no	—
185	C₂×Q₈	—	52	2	no	—
186	C₂×Q₈	—	52	2	no	—
187	C₂²×C₄	—	26	2	no	—
188	C₂²×C₄	—	26	2	no	—
189	C₂²×C₄	#6103	1	1	yes	characteristic
190	C₂²×C₄	#6103	1	1	yes	characteristic
191	C₂²×C₄	#6103	1	1	yes	characteristic
192	C₂²×C₄	#6103	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	52	2	no	—
200	C₂×C₈	—	52	2	no	—
201	C₂×C₈	—	52	2	no	—
202	C₄⋊C₄	—	13	4	no	—
203	C₄⋊C₄	—	26	2	no	—
204	C₄⋊C₄	—	26	4	no	—
205	C₄⋊C₄	—	52	2	no	—
206	C₄⋊C₄	—	52	2	no	—
207	C₄⋊C₄	—	52	2	no	—
208	C₄⋊C₄	—	52	2	no	—
209	C₄⋊C₄	—	52	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	—	52	4	no	—
216	C₄²	C₈×D₁₃	1	2	yes	—
217	C₄²	#6094	1	2	yes	—
218	C₄²	#6095	1	2	yes	—
219	C₄²	D₁₀₄	1	2	yes	—
220	C₁₃	#7588	1	1	yes	characteristic, complemented, Sylow
221	C₂³	C₂×#6103	1	1	yes	characteristic
222	Q₈	—	52	8	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	26	4	no	—
236	C₂×C₄	—	52	2	no	—
237	C₂×C₄	—	52	2	no	—
238	C₂×C₄	—	52	2	no	—
239	C₂×C₄	—	52	2	no	—
240	C₂×C₄	—	52	2	no	—
241	C₂×C₄	#88045	1	2	yes	—
242	C₂×C₄	#88060	1	2	yes	—
243	C₂×C₄	#88060	1	2	yes	—
244	C₂×C₄	#88061	1	2	yes	—
245	C₂×C₄	#88061	1	2	yes	—
246	C₂×C₄	#88064	1	2	yes	—
247	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂×#6094	1	1	yes	characteristic
249	C₂×C₄	C₂×#6095	1	1	yes	characteristic
250	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
251	C₂×C₄	#88205	1	2	yes	—
252	C₈	—	4	4	no	—
253	C₈	—	104	2	no	—
254	C₂²	C₂×#88045	1	1	yes	characteristic
255	C₂²	#11718665	1	1	yes	characteristic
256	C₂²	#11718667	1	1	yes	characteristic
257	C₂²	C₂×#88060	1	1	yes	characteristic
258	C₂²	C₂×#88061	1	1	yes	characteristic
259	C₂²	C₂×#88064	1	1	yes	characteristic
260	C₂²	#11718800	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	52	4	no	—
270	C₄	—	104	2	no	—
271	C₄	#11718051	1	4	yes	—
272	C₂	#13113159	1	1	yes	characteristic
273	C₂	C₂×#11718051	1	1	yes	characteristic
274	C₂	#13113230	1	1	yes	characteristic
275	C₂	#13113347	1	1	yes	characteristic
276	C₂	#13113397	1	1	yes	characteristic
277	C₂	#13113450	1	1	yes	characteristic
278	C₂	#13113507	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1744	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1744	D₁₃
hypocenter	C₁₃	#7588
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
38. #88123	C₈	semidirect
220. C₁₃	#7588	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88123	2
#13114953	2
#13112658	2
C₁₃×#1744	2
#7588	13