Subgroups — #15756053

Order 3328.

466 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114974	C₂	1	1	yes	maximal, characteristic, complemented
2	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113261	C₂	1	2	yes	maximal
4	#13112733	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	—	2	1	no	—
7	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₁₃	—	2	1	no	—
9	C₈×D₄×C₁₃	—	2	1	no	—
10	C₈×D₄×C₁₃	C₄	1	2	yes	—
11	C₈×D₄×C₁₃	C₄	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718586	C₂²	1	2	yes	—
16	#11718585	C₂²	1	2	yes	—
17	#11718144	—	2	1	no	—
18	#11718144	—	2	1	no	—
19	D₄⋊#1055	—	2	4	no	—
20	#11718050	—	2	1	no	—
21	#11718049	—	2	1	no	—
22	C₂²×D₄×C₁₃	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	—	2	1	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	4	4	no	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₄×D₄×C₁₃	—	2	1	no	—
31	C₄×D₄×C₁₃	—	2	1	no	—
32	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
33	C₄×D₄×C₁₃	D₄	1	2	yes	—
34	C₂×C₁₃×C₄⋊C₄	Q₈	1	1	yes	characteristic
35	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×#6099	—	2	2	no	—
38	C₁₃×C₄⋊C₈	—	2	1	no	—
39	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₁₃×C₂²⋊C₈	—	2	2	no	—
41	C₁₃×C₂²⋊C₈	—	2	2	no	—
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₂²⋊#1055	—	2	2	no	—
47	C₄⋊#1055	—	2	2	no	—
48	C₄⋊#1055	—	2	2	no	—
49	C₄⋊#1055	—	2	2	no	—
50	C₄×#1055	—	2	2	no	—
51	#8660	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	—	2	1	no	—
53	C₂×D₄×C₁₃	—	2	4	no	—
54	C₂×D₄×C₁₃	—	2	1	no	—
55	C₂×D₄×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	4	4	no	—
68	C₂×C₈×C₁₃	—	4	4	no	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	—	2	1	no	—
75	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
76	C₁₃×C₂²⋊C₄	—	2	2	no	—
77	C₁₃×C₂²⋊C₄	—	2	2	no	—
78	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₃	#41	1	1	yes	characteristic
82	#6099	—	4	2	no	—
83	#6099	—	4	2	no	—
84	C₂×#1055	—	4	2	no	—
85	C₂×#1055	—	4	2	no	—
86	C₂×C₈×D₄	D₁₃	1	1	yes	characteristic
87	#1837	—	13	1	no	—
88	#1564	—	13	1	no	—
89	#1486	—	13	2	no	—
90	#1301	—	13	2	no	—
91	C₂³×C₁₃	—	2	4	no	—
92	C₂³×C₁₃	—	2	2	no	—
93	C₂³×C₁₃	—	4	2	no	—
94	C₂³×C₁₃	—	4	2	no	—
95	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
96	D₄×C₁₃	—	2	4	no	—
97	D₄×C₁₃	—	2	2	no	—
98	D₄×C₁₃	#108	1	4	yes	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	4	1	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	4	no	—
106	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#135	1	2	yes	—
113	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
114	C₈×C₁₃	—	2	1	no	—
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	—	2	1	no	—
118	C₈×C₁₃	—	4	2	no	—
119	#1055	—	8	2	no	—
120	#1055	—	8	2	no	—
121	C₂³×C₈	—	2	1	no	—
122	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
123	C₈×D₄	—	2	1	no	—
124	C₈×D₄	—	2	1	no	—
125	C₈×D₄	Dic₁₃	1	2	yes	—
126	C₈×D₄	Dic₁₃	1	2	yes	—
127	#431	—	13	2	no	—
128	#423	—	13	2	no	—
129	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
130	C₂×C₂²⋊C₈	—	2	1	no	—
131	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
132	#341	—	26	1	no	—
133	#341	—	26	1	no	—
134	#335	—	26	1	no	—
135	#334	—	26	1	no	—
136	D₄⋊C₈	—	26	4	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	—	2	4	no	—
140	C₂²×C₁₃	—	4	8	no	—
141	C₂²×C₁₃	—	4	2	no	—
142	C₂²×C₁₃	—	4	2	no	—
143	C₂²×C₁₃	#380	1	1	yes	characteristic
144	C₂²×C₁₃	#478	1	2	yes	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	4	2	no	—
149	C₄×C₁₃	#416	1	1	yes	characteristic
150	C₄×C₁₃	#418	1	1	yes	characteristic
151	C₄×C₁₃	C₂×#108	1	2	yes	—
152	C₄×C₁₃	#463	1	2	yes	—
153	C₂²×D₄	Dic₂₆	1	1	yes	characteristic
154	C₂³×C₄	—	2	1	no	—
155	C₂×M₁₆	—	26	2	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	4	4	no	—
160	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	D₅₂	1	2	yes	—
165	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
166	C₂×C₄⋊C₄	Dic₂₆	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	—	2	1	no	—
168	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	26	2	no	—
171	C₄⋊C₈	—	26	2	no	—
172	C₄⋊C₈	—	26	2	no	—
173	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	26	2	no	—
177	C₄×C₈	—	26	2	no	—
178	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
179	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
182	C₂×C₁₃	—	2	2	no	—
183	C₂×C₁₃	—	4	4	no	—
184	C₂×C₁₃	#1804	1	1	yes	characteristic
185	C₂×C₁₃	#1810	1	1	yes	characteristic
186	C₂×C₁₃	#1814	1	1	yes	characteristic
187	C₂⁴	—	2	1	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	C₄⋊Dic₁₃	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
197	C₂²×C₄	#6127	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
199	M₁₆	—	52	2	no	—
200	M₁₆	—	52	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	4	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	52	2	no	—
208	C₂×C₈	—	52	2	no	—
209	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₄²	C₂×D₅₂	1	1	yes	characteristic
218	C₄²	#6127	1	1	yes	characteristic
219	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
221	C₁₃	#8660	1	1	yes	characteristic, complemented, Sylow
222	C₂³	—	2	4	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	#88178	1	1	yes	characteristic
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	#88059	1	4	yes	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	#88178	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
239	C₂×C₄	#88180	1	1	yes	characteristic
240	C₂×C₄	#88184	1	1	yes	characteristic
241	C₂×C₄	#88184	1	1	yes	characteristic
242	C₂×C₄	#88186	1	2	yes	—
243	C₂×C₄	#88204	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	2	no	—
250	C₈	—	104	2	no	—
251	C₈	—	104	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	8	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	#11718670	1	2	yes	—
259	C₂²	#11718873	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	#11718677	1	2	yes	—
265	C₄	C₂×#88059	1	2	yes	—
266	C₄	#11718876	1	1	yes	characteristic
267	C₄	#11718894	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	4	4	no	—
270	C₂	#13114846	1	1	yes	characteristic
271	C₂	#13114874	1	1	yes	characteristic
272	C₂	#13114890	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₈×D₄×C₁₃	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂²×D₄×C₁₃	Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₈×D₄	D₁₃
hypocenter	C₁₃	#8660
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #13114974	C₂	semidirect
2. #13113584	C₂	semidirect
4. #13112733	C₂	semidirect
221. C₁₃	#8660	semidirect

Maximal subgroups

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

subgroup	index
#13114974	2
#13113584	2
#13113261	2
#13112733	2
C₂×C₈×D₄×C₁₃	2
#8660	13