Subgroups — #15754190

Order 3328.

410 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88057	C₂	1	1	yes	maximal, characteristic
2	#13114703	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
5	#13112676	C₂	1	1	yes	maximal, characteristic
6	#13112676	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic
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×#88057	C₄	1	2	yes	—
12	C₂×#88055	—	2	1	no	—
13	C₂×C₈×Dic₁₃	—	2	1	no	—
14	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
18	C₁₃×#322	C₄	1	4	yes	complemented
19	#11718143	C₄	1	2	yes	—
20	#11718142	—	2	1	no	—
21	#11718142	—	2	1	no	—
22	#11718050	C₄	1	4	yes	complemented
23	C₂²×C₈×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₁₃	—	2	1	no	—
32	C₂×C₄×Dic₁₃	—	2	1	no	—
33	C₂²×#1055	—	2	1	no	—
34	C₂²×#1055	—	2	1	no	—
35	C₄×C₈×C₁₃	—	2	1	no	—
36	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₁₃	Q₈	1	2	yes	—
41	#88071	—	2	1	no	—
42	#88071	—	2	1	no	—
43	#88057	—	2	4	no	—
44	#88055	—	4	2	no	—
45	C₈×Dic₁₃	—	4	2	no	—
46	C₄⋊#1055	—	2	1	no	—
47	C₄⋊#1055	—	2	1	no	—
48	C₄⋊#1055	C₂×C₄	1	2	yes	—
49	C₄⋊#1055	C₂×C₄	1	2	yes	—
50	#8289	—	13	1	no	maximal, Sylow
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
54	C₂²×Dic₁₃	—	2	2	no	—
55	C₂²×Dic₁₃	—	2	2	no	—
56	C₂×C₈×C₁₃	—	2	4	no	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	1	no	—
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	C₄²	1	2	yes	—
65	C₂×C₈×C₁₃	C₄²	1	2	yes	—
66	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₁₃	#41	1	1	yes	characteristic
72	C₄⋊Dic₁₃	—	2	2	no	—
73	C₄⋊Dic₁₃	—	2	2	no	—
74	C₄⋊Dic₁₃	—	2	2	no	—
75	C₄⋊Dic₁₃	M₁₆	1	4	yes	—
76	C₄×Dic₁₃	—	4	1	no	—
77	C₄×Dic₁₃	—	4	1	no	—
78	C₄×Dic₁₃	—	4	1	no	—
79	C₄×Dic₁₃	—	4	1	no	—
80	C₂×#1055	—	2	2	no	—
81	C₂×#1055	—	2	2	no	—
82	C₂×#1055	—	4	1	no	—
83	C₂×#1055	—	4	1	no	—
84	C₂×#1055	—	4	1	no	—
85	C₂×#1055	—	4	1	no	—
86	C₄×#106	—	13	1	no	—
87	#1770	—	13	1	no	—
88	C₈×C₄⋊C₄	—	13	1	no	—
89	C₂×#335	—	13	1	no	—
90	C₂×#322	D₁₃	1	1	yes	characteristic
91	#1279	—	13	1	no	—
92	#1279	—	13	1	no	—
93	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₁₃	#131	1	2	yes	—
103	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
107	C₂×Dic₁₃	—	2	4	no	—
108	C₂×Dic₁₃	—	4	2	no	—
109	C₂×Dic₁₃	—	4	2	no	—
110	C₂×Dic₁₃	—	4	2	no	—
111	C₂×Dic₁₃	—	4	2	no	—
112	C₂×Dic₁₃	—	4	2	no	—
113	C₈×C₁₃	—	2	4	no	—
114	C₈×C₁₃	—	2	2	no	—
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	—	2	2	no	—
118	#1055	—	4	2	no	—
119	#1055	—	4	2	no	—
120	C₂×#106	—	13	2	no	—
121	C₂×C₄⋊C₈	—	13	1	no	—
122	C₂×C₄⋊C₈	—	13	1	no	—
123	C₂×#97	—	26	1	no	—
124	C₂×C₄×C₈	—	26	1	no	—
125	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
128	C₄×C₄⋊C₄	—	13	1	no	—
129	C₄×C₄⋊C₄	—	13	1	no	—
130	#340	—	13	2	no	—
131	#336	—	26	1	no	—
132	#336	—	26	1	no	—
133	#335	—	13	4	no	—
134	#322	Dic₁₃	1	4	yes	complemented
135	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₁₃	#435	1	2	yes	—
137	C₂²×C₁₃	#440	1	1	yes	characteristic
138	C₂²×C₁₃	#440	1	1	yes	characteristic
139	C₂²×C₁₃	#443	1	2	yes	—
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
143	C₄×C₁₃	#405	1	1	yes	characteristic
144	C₄×C₁₃	#428	1	1	yes	characteristic
145	C₄×C₁₃	#428	1	1	yes	characteristic
146	C₄×C₁₃	#441	1	1	yes	characteristic
147	C₄×C₁₃	#442	1	1	yes	characteristic
148	C₄×C₁₃	#446	1	2	yes	—
149	Dic₁₃	—	4	4	no	—
150	Dic₁₃	—	8	2	no	—
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	—	26	1	no	—
153	C₂²×C₈	—	26	1	no	—
154	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
155	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
156	C₂²×C₈	D₅₂	1	1	yes	characteristic
157	C₂²×C₈	D₅₂	1	1	yes	characteristic
158	C₂×C₄⋊C₄	—	13	2	no	—
159	C₂×C₄⋊C₄	—	13	2	no	—
160	C₂×C₄²	—	26	1	no	—
161	C₂×C₄²	—	26	1	no	—
162	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
163	#106	—	26	4	no	—
164	C₄⋊C₈	—	13	2	no	—
165	C₄⋊C₈	—	13	2	no	—
166	C₄⋊C₈	—	26	1	no	—
167	C₄⋊C₈	—	26	1	no	—
168	#97	—	52	2	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	—	52	2	no	—
171	C₄×C₈	Dic₂₆	1	2	yes	—
172	C₄×C₈	C₄×D₁₃	1	2	yes	—
173	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
174	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
175	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
176	#95	—	26	1	no	—
177	#95	—	26	1	no	—
178	C₂×C₁₃	#1585	1	2	yes	—
179	C₂×C₁₃	#1588	1	2	yes	—
180	C₂×C₁₃	#1771	1	1	yes	characteristic
181	C₂×C₁₃	#1772	1	1	yes	characteristic
182	C₂×C₁₃	#1776	1	1	yes	characteristic
183	C₂²×C₄	—	26	2	no	—
184	C₂²×C₄	—	26	2	no	—
185	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
187	C₂²×C₄	#6127	1	1	yes	characteristic
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	26	2	no	—
197	C₂×C₈	—	26	2	no	—
198	C₂×C₈	—	52	1	no	—
199	C₂×C₈	—	52	1	no	—
200	C₂×C₈	—	52	1	no	—
201	C₂×C₈	—	52	1	no	—
202	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
203	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
204	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
205	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
206	C₄⋊C₄	—	13	4	no	—
207	C₄⋊C₄	—	26	2	no	—
208	C₄⋊C₄	—	26	2	no	—
209	C₄⋊C₄	—	26	2	no	—
210	C₄²	—	52	1	no	—
211	C₄²	—	52	1	no	—
212	C₄²	—	52	1	no	—
213	C₄²	—	52	1	no	—
214	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
215	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₄²	#6127	1	1	yes	characteristic
217	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
218	C₁₃	#8289	1	1	yes	characteristic, complemented, Sylow
219	C₂³	C₄×D₅₂	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	26	4	no	—
222	C₂×C₄	—	52	2	no	—
223	C₂×C₄	—	52	2	no	—
224	C₂×C₄	—	52	2	no	—
225	C₂×C₄	—	52	2	no	—
226	C₂×C₄	—	52	2	no	—
227	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
228	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
229	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
230	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
231	C₂×C₄	#88161	1	2	yes	—
232	C₂×C₄	#88162	1	1	yes	characteristic
233	C₂×C₄	#88162	1	1	yes	characteristic
234	C₂×C₄	#88163	1	1	yes	characteristic
235	C₂×C₄	#88163	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
238	C₂×C₄	#88180	1	1	yes	characteristic
239	C₈	—	2	4	no	—
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	52	2	no	—
245	C₈	—	52	2	no	—
246	C₂²	#11718361	1	2	yes	—
247	C₂²	#11718367	1	2	yes	—
248	C₂²	#11718371	1	1	yes	characteristic
249	C₂²	#11718371	1	1	yes	characteristic
250	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	52	4	no	—
254	C₄	—	104	2	no	—
255	C₄	#11718362	1	2	yes	—
256	C₄	#11718372	1	1	yes	characteristic
257	C₄	#11718377	1	1	yes	characteristic
258	C₄	M₁₆×Dic₁₃	1	1	yes	characteristic
259	C₄	#11718782	1	1	yes	characteristic
260	C₄	#11718782	1	1	yes	characteristic
261	C₄	#11718783	1	1	yes	characteristic
262	C₂	#13113695	1	2	yes	—
263	C₂	#13113716	1	2	yes	—
264	C₂	#13114708	1	1	yes	characteristic
265	C₂	#13114715	1	1	yes	characteristic
266	C₂	#13114733	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#8289
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
18. C₁₃×#322	C₄	semidirect
22. #11718050	C₄	semidirect
134. #322	Dic₁₃	semidirect
218. C₁₃	#8289	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88057	2
#13114703	2
C₈×C₄⋊Dic₁₃	2
C₂×#11718050	2
#13112676	2
C₂×C₁₃×#322	2
#8289	13