Subgroups — #15781473

Order 3328.

332 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125350	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718055	C₂	1	1	yes	maximal, characteristic
3	#13112950	C₂	1	2	yes	maximal
4	#13112937	C₂	1	2	yes	maximal
5	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic
6	#11719175	—	2	1	no	—
7	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
8	C₁₃×#427	C₂²	1	1	yes	characteristic
9	C₁₃×#405	—	2	1	no	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	#11718767	—	2	1	no	—
12	#11718767	C₂²	1	2	yes	—
13	C₂×#88052	C₂²	1	2	yes	—
14	#11718344	—	2	1	no	—
15	#11718344	C₄	1	2	yes	complemented
16	#11718343	—	2	1	no	—
17	#11718341	C₄	1	2	yes	complemented
18	C₁₃×#339	—	2	1	no	—
19	#11718055	—	2	1	no	—
20	#11718055	—	2	1	no	—
21	C₂×C₁₃×M₁₆	—	2	1	no	—
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	C₁₃×#117	—	2	1	no	—
25	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
26	#88180	—	2	2	no	—
27	#88178	—	4	2	no	—
28	#88124	—	2	2	no	—
29	#88123	—	2	2	no	—
30	C₄×Dic₂₆	—	4	2	no	—
31	C₁₃×#108	—	2	1	no	—
32	C₁₃×#108	Q₈	1	2	yes	complemented
33	C₁₃×#107	—	2	1	no	—
34	C₁₃×#106	Q₈	1	2	yes	complemented
35	C₁₃×#97	—	4	1	no	—
36	C₄×C₈×C₁₃	—	4	1	no	—
37	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
41	#88058	—	2	2	no	—
42	#88058	—	2	2	no	—
43	#88057	—	2	2	no	—
44	#88057	C₂×C₄	1	4	yes	—
45	#88052	—	2	2	no	—
46	#88052	—	2	2	no	—
47	C₂×#6090	—	2	2	no	—
48	#13271	—	13	1	no	maximal, Sylow
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₂₆	—	4	2	no	—
52	C₁₃×M₁₆	—	2	2	no	—
53	C₁₃×M₁₆	—	4	1	no	—
54	C₂×C₈×C₁₃	—	4	1	no	—
55	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
58	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
59	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
60	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
66	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	—	4	1	no	—
69	C₄²×C₁₃	—	2	1	no	—
70	C₄²×C₁₃	—	4	1	no	—
71	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²⋊Dic₁₃	—	4	2	no	—
74	C₄⋊Dic₁₃	—	2	4	no	—
75	C₄⋊Dic₁₃	—	4	2	no	—
76	#6101	—	4	4	no	—
77	C₄×Dic₁₃	—	4	2	no	—
78	#6090	—	4	2	no	—
79	#3160	—	13	1	no	—
80	C₂×#364	—	13	1	no	—
81	#1772	D₁₃	1	1	yes	characteristic
82	#1389	—	13	2	no	—
83	#1384	—	13	2	no	—
84	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×Dic₁₃	—	4	4	no	—
96	Dic₂₆	—	8	4	no	—
97	C₈×C₁₃	—	4	2	no	—
98	C₈×C₁₃	#103	1	1	yes	characteristic
99	C₈×C₁₃	#103	1	1	yes	characteristic
100	C₈×C₁₃	#103	1	1	yes	characteristic
101	C₈×C₁₃	#103	1	1	yes	characteristic
102	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
103	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
104	C₈×C₁₃	#124	1	1	yes	characteristic
105	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
106	#533	—	26	1	no	—
107	C₂×M₃₂	—	13	2	no	—
108	#500	—	26	1	no	—
109	#499	—	13	2	no	—
110	#499	—	26	1	no	—
111	#498	—	13	2	no	—
112	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
113	#427	—	13	2	no	—
114	#427	—	26	1	no	—
115	#427	C₂×D₁₃	1	1	yes	characteristic
116	#405	—	2	1	no	—
117	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
118	#364	—	26	1	no	—
119	#364	—	26	1	no	—
120	#339	—	2	1	no	—
121	C₂²×C₁₃	—	4	1	no	—
122	C₂²×C₁₃	#418	1	1	yes	characteristic
123	C₂²×C₁₃	#476	1	1	yes	characteristic
124	C₂²×C₁₃	#477	1	1	yes	characteristic
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	4	1	no	—
127	C₄×C₁₃	—	8	2	no	—
128	C₄×C₁₃	#391	1	1	yes	characteristic
129	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
130	C₄×C₁₃	#476	1	1	yes	characteristic
131	C₄×C₁₃	#477	1	1	yes	characteristic
132	Dic₁₃	—	8	4	no	—
133	C₂×M₁₆	—	2	1	no	—
134	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	26	2	no	—
137	#125	—	26	2	no	—
138	C₂²⋊Q₈	—	52	2	no	—
139	C₄×Q₈	—	52	2	no	—
140	#117	—	2	1	no	—
141	#117	—	26	2	no	—
142	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
143	M₃₂	—	26	2	no	—
144	M₃₂	—	26	2	no	—
145	C₂×C₁₆	—	26	2	no	—
146	#108	—	2	1	no	—
147	#108	Dic₂₆	1	2	yes	complemented
148	#107	—	2	1	no	—
149	#107	—	26	2	no	—
150	#107	—	26	2	no	—
151	#106	—	13	4	no	—
152	#106	—	26	2	no	—
153	#106	Dic₂₆	1	2	yes	complemented
154	#97	—	4	1	no	—
155	C₄×C₈	—	4	1	no	—
156	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
157	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
158	C₄×C₈	D₅₂	1	1	yes	characteristic
159	C₄×C₈	D₅₂	1	1	yes	characteristic
160	C₂×C₁₃	—	2	1	no	—
161	C₂×C₁₃	—	2	1	no	—
162	C₂×C₁₃	—	2	1	no	—
163	C₂×C₁₃	#1987	1	1	yes	characteristic
164	C₂×Q₈	—	52	2	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
167	M₁₆	—	2	2	no	—
168	M₁₆	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	#6103	1	1	yes	characteristic
171	C₂×C₈	#6103	1	1	yes	characteristic
172	C₂×C₈	#6105	1	1	yes	characteristic
173	C₂×C₈	#6105	1	1	yes	characteristic
174	C₂×C₈	#6107	1	1	yes	characteristic
175	C₂×C₈	#6107	1	1	yes	characteristic
176	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
177	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
178	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
179	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
180	C₂×C₈	#6127	1	1	yes	characteristic
181	C₂×C₈	#6127	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	26	4	no	—
184	C₄⋊C₄	—	52	4	no	—
185	C₄⋊C₄	—	52	2	no	—
186	C₂²⋊C₄	—	4	1	no	—
187	C₂²⋊C₄	—	52	2	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	4	1	no	—
190	C₄²	—	52	2	no	—
191	C₄²	#6103	1	1	yes	characteristic
192	C₄²	#6103	1	1	yes	characteristic
193	C₁₆	—	52	2	no	—
194	C₁₃	#13271	1	1	yes	characteristic, complemented, Sylow
195	C₂³	#88203	1	1	yes	characteristic
196	Q₈	—	104	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	52	4	no	—
202	C₂×C₄	#88178	1	1	yes	characteristic
203	C₂×C₄	#88178	1	1	yes	characteristic
204	C₂×C₄	C₂×#6103	1	1	yes	characteristic
205	C₂×C₄	C₂×#6105	1	1	yes	characteristic
206	C₂×C₄	C₂×#6107	1	1	yes	characteristic
207	C₂×C₄	#88203	1	1	yes	characteristic
208	C₈	—	4	2	no	—
209	C₈	#88050	1	1	yes	characteristic
210	C₈	#88050	1	1	yes	characteristic
211	C₈	#88050	1	1	yes	characteristic
212	C₈	#88050	1	1	yes	characteristic
213	C₈	C₄×Dic₂₆	1	1	yes	characteristic
214	C₈	C₄×Dic₂₆	1	1	yes	characteristic
215	C₈	#88123	1	1	yes	characteristic
216	C₈	#88129	1	1	yes	characteristic
217	C₂²	—	4	1	no	—
218	C₂²	#11718662	1	1	yes	characteristic
219	C₂²	#11718669	1	1	yes	characteristic
220	C₂²	#11718682	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	8	2	no	—
224	C₄	—	104	4	no	—
225	C₄	#11718590	1	1	yes	characteristic
226	C₄	C₂×#88050	1	1	yes	characteristic
227	C₄	#11718669	1	1	yes	characteristic
228	C₄	#11718682	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#13115784	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11718669
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1772	C₂
socle	C₂×C₁₃	#1987
prime radical	C₂³×C₁₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1772	D₁₃
hypocenter	C₁₃	#13271
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
15. #11718344	C₄	semidirect
17. #11718341	C₄	semidirect
32. C₁₃×#108	Q₈	semidirect
34. C₁₃×#106	Q₈	semidirect
147. #108	Dic₂₆	semidirect
153. #106	Dic₂₆	semidirect
194. C₁₃	#13271	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
C₂×#11718055	2
#13112950	2
#13112937	2
C₁₃×#1772	2
#13271	13