Subgroups — #15756065

Order 3328.

374 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718340	C₂	1	1	yes	maximal, characteristic
2	#13113584	C₂	1	1	yes	maximal, characteristic
3	#13113264	C₂	1	2	yes	maximal
4	#13112731	C₂	1	2	yes	maximal
5	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#108	—	2	1	no	—
8	C₂×C₁₃×#108	—	2	1	no	—
9	C₁₃×#423	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88055	—	2	1	no	—
12	C₂×#88054	—	2	1	no	—
13	#11718586	C₂²	1	2	yes	—
14	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
15	#11718340	—	2	1	no	—
16	#11718340	—	2	1	no	—
17	#11718340	C₄	1	2	yes	—
18	#11718340	C₄	1	2	yes	—
19	#11718076	—	2	4	no	—
20	#11718050	—	2	1	no	—
21	#11718049	—	2	1	no	—
22	C₂²×Dic₂₆	Q₈	1	1	yes	characteristic
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
28	C₂×#6101	—	2	1	no	—
29	C₂×C₄×Dic₁₃	—	2	1	no	—
30	C₂²×#1055	—	2	1	no	—
31	C₄×Dic₂₆	—	2	1	no	—
32	C₄×Dic₂₆	—	2	1	no	—
33	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
34	C₄×Dic₂₆	D₄	1	2	yes	—
35	C₁₃×#108	—	4	2	no	—
36	C₁₃×#108	—	4	2	no	—
37	C₁₃×C₄⋊C₈	—	2	2	no	—
38	C₁₃×C₄⋊C₈	—	2	2	no	—
39	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂²⋊#1055	—	2	2	no	—
44	#88055	—	2	2	no	—
45	#88055	—	2	2	no	—
46	#88054	—	2	2	no	—
47	#88054	—	2	2	no	—
48	C₄⋊#1055	—	2	1	no	—
49	C₄⋊#1055	—	2	2	no	—
50	C₄⋊#1055	C₂×C₄	1	2	yes	—
51	C₄×#1055	—	2	2	no	—
52	#8663	—	13	1	no	maximal, Sylow
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	2	1	no	—
57	C₂²×Dic₁₃	—	2	1	no	—
58	C₂×Dic₂₆	—	2	1	no	—
59	C₂×Dic₂₆	—	4	2	no	—
60	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
61	C₁₃×M₁₆	—	4	2	no	—
62	C₁₃×M₁₆	—	4	2	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	—	4	2	no	—
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
71	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
72	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₃	#41	1	1	yes	characteristic
76	C₄⋊Dic₁₃	—	2	1	no	—
77	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
78	#6101	—	2	2	no	—
79	#6101	—	2	2	no	—
80	C₄×Dic₁₃	—	2	2	no	—
81	C₄×Dic₁₃	—	2	2	no	—
82	C₂×#1055	—	2	2	no	—
83	C₂×#1055	—	4	1	no	—
84	C₂×#1055	—	4	1	no	—
85	C₂×#1055	—	4	2	no	—
86	C₂×#446	—	13	1	no	—
87	#1848	D₁₃	1	1	yes	characteristic
88	#1564	—	13	1	no	—
89	#1487	—	13	2	no	—
90	#1301	—	13	2	no	—
91	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	4	1	no	—
94	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
104	C₂×Dic₁₃	—	2	2	no	—
105	C₂×Dic₁₃	—	2	2	no	—
106	C₂×Dic₁₃	—	4	2	no	—
107	C₂×Dic₁₃	—	4	2	no	—
108	Dic₂₆	—	2	4	no	—
109	Dic₂₆	—	4	2	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	—	2	1	no	—
112	C₈×C₁₃	—	8	2	no	—
113	C₈×C₁₃	#103	1	2	yes	—
114	C₈×C₁₃	#103	1	2	yes	—
115	#1055	—	4	2	no	—
116	#1055	—	8	2	no	—
117	C₂×C₄×Q₈	—	13	1	no	—
118	#446	—	13	2	no	—
119	#446	—	13	2	no	—
120	#446	—	26	1	no	—
121	#446	—	26	1	no	—
122	#431	—	13	2	no	—
123	C₂×#108	—	2	1	no	—
124	C₂×#108	—	2	1	no	—
125	#423	C₂×D₁₃	1	2	yes	—
126	C₂×C₄⋊C₈	—	13	1	no	—
127	C₂×C₄⋊C₈	—	26	1	no	—
128	C₂×#97	—	26	1	no	—
129	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
130	#335	—	26	1	no	—
131	#334	—	26	1	no	—
132	Q₈⋊C₈	—	26	4	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	#380	1	1	yes	characteristic
136	C₂²×C₁₃	#475	1	1	yes	characteristic
137	C₂²×C₁₃	#477	1	1	yes	characteristic
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
142	C₄×C₁₃	#417	1	1	yes	characteristic
143	C₄×C₁₃	#430	1	2	yes	—
144	C₄×C₁₃	#460	1	1	yes	characteristic
145	C₄×C₁₃	#462	1	1	yes	characteristic
146	Dic₁₃	—	4	2	no	—
147	Dic₁₃	—	4	2	no	—
148	C₂²×Q₈	—	13	1	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂²×C₈	—	26	1	no	—
151	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
153	C₄×Q₈	—	13	2	no	—
154	C₄×Q₈	—	13	2	no	—
155	C₄×Q₈	—	26	1	no	—
156	C₄×Q₈	—	26	1	no	—
157	C₂×C₄⋊C₄	—	13	1	no	—
158	C₂×C₄⋊C₄	—	26	1	no	—
159	C₂×C₄²	—	26	1	no	—
160	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
161	#108	—	4	2	no	—
162	#108	—	4	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	13	2	no	—
166	C₄⋊C₈	—	26	2	no	—
167	C₄⋊C₈	—	26	2	no	—
168	C₄⋊C₈	—	26	1	no	—
169	C₄⋊C₈	—	26	2	no	—
170	C₂²⋊C₈	—	26	2	no	—
171	#97	—	26	2	no	—
172	#97	—	26	2	no	—
173	C₄×C₈	—	26	2	no	—
174	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
175	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
176	C₄×C₈	D₅₂	1	1	yes	characteristic
177	C₄×C₈	D₅₂	1	1	yes	characteristic
178	C₂×C₁₃	—	2	1	no	—
179	C₂×C₁₃	—	2	1	no	—
180	C₂×C₁₃	#1803	1	1	yes	characteristic
181	C₂×C₁₃	#1811	1	1	yes	characteristic
182	C₂×C₁₃	#1814	1	1	yes	characteristic
183	C₂×Q₈	—	13	2	no	—
184	C₂×Q₈	—	26	1	no	—
185	C₂×Q₈	—	52	2	no	—
186	C₂²×C₄	—	26	1	no	—
187	C₂²×C₄	—	26	1	no	—
188	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
189	C₂²×C₄	#6131	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
191	M₁₆	—	4	2	no	—
192	M₁₆	—	4	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	26	2	no	—
197	C₂×C₈	—	52	1	no	—
198	C₂×C₈	—	52	1	no	—
199	C₂×C₈	—	52	2	no	—
200	C₂×C₈	#6103	1	1	yes	characteristic
201	C₂×C₈	#6103	1	1	yes	characteristic
202	C₂×C₈	#6103	1	1	yes	characteristic
203	C₂×C₈	#6103	1	1	yes	characteristic
204	C₂×C₈	#6105	1	2	yes	—
205	C₂×C₈	#6107	1	2	yes	—
206	C₄⋊C₄	—	13	2	no	—
207	C₄⋊C₄	—	26	2	no	—
208	C₄⋊C₄	—	26	2	no	—
209	C₄⋊C₄	—	26	1	no	—
210	C₄²	—	26	2	no	—
211	C₄²	—	26	2	no	—
212	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
213	C₄²	C₂×D₅₂	1	1	yes	characteristic
214	C₄²	D₄×D₁₃	1	1	yes	characteristic
215	C₄²	#6129	1	1	yes	characteristic
216	C₁₃	#8663	1	1	yes	characteristic, complemented, Sylow
217	C₂³	#88200	1	1	yes	characteristic
218	Q₈	—	26	4	no	—
219	Q₈	—	52	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	26	2	no	—
224	C₂×C₄	—	52	2	no	—
225	C₂×C₄	—	52	2	no	—
226	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
227	C₂×C₄	#88146	1	1	yes	characteristic
228	C₂×C₄	#88149	1	1	yes	characteristic
229	C₂×C₄	#88151	1	1	yes	characteristic
230	C₂×C₄	C₂×#6103	1	1	yes	characteristic
231	C₂×C₄	#88186	1	1	yes	characteristic
232	C₂×C₄	C₂×#6105	1	1	yes	characteristic
233	C₂×C₄	#88192	1	1	yes	characteristic
234	C₂×C₄	#88196	1	1	yes	characteristic
235	C₂×C₄	C₂×#6107	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	—	52	2	no	—
240	C₈	—	104	2	no	—
241	C₈	#88050	1	2	yes	—
242	C₈	#88050	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	#11718648	1	1	yes	characteristic
246	C₂²	#11718752	1	1	yes	characteristic
247	C₂²	#11718755	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	52	2	no	—
252	C₄	—	52	2	no	—
253	C₄	#11718554	1	2	yes	—
254	C₄	#11718627	1	1	yes	characteristic
255	C₄	C₂×#88050	1	1	yes	characteristic
256	C₄	#11718736	1	1	yes	characteristic
257	C₄	#11718751	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#13114838	1	1	yes	characteristic
261	C₂	#13114875	1	1	yes	characteristic
262	C₂	#13114891	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88192
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1848	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂³×C₁₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1848	D₁₃
hypocenter	C₁₃	#8663
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
216. C₁₃	#8663	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718340	2
#13113584	2
#13113264	2
#13112731	2
C₁₃×#1848	2
#8663	13