Subgroups — #15832064

Order 3328.

730 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718585	C₂	1	2	yes	maximal
2	#13113607	C₂	1	8	yes	maximal, complemented
3	#13113592	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×#6099	—	2	2	no	—
7	C₁₃×#431	C₄	1	4	yes	—
8	C₁₃×#431	C₂²	1	4	yes	—
9	C₂×C₁₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₁₃×C₂²⋊C₈	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
12	#11718585	—	2	2	no	—
13	#11718585	—	2	2	no	—
14	#11718585	C₂²	1	4	yes	—
15	#11718585	C₂²	1	4	yes	—
16	C₂×C₄⋊#1055	—	2	2	no	—
17	C₂×C₄⋊#1055	C₂²	1	4	yes	—
18	#11718049	C₂²	1	16	yes	complemented
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₃	D₄	1	2	yes	—
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
31	C₂×#6099	—	2	4	no	—
32	C₂×#6099	—	2	4	no	—
33	C₂×#6099	—	4	8	no	—
34	C₂²×#1055	—	2	4	no	—
35	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
36	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₁₃	C₂³	1	4	yes	—
39	C₄⋊#1055	—	2	4	no	—
40	C₄⋊#1055	—	2	4	no	—
41	C₄⋊#1055	—	2	4	no	—
42	C₄⋊#1055	C₂³	1	8	yes	—
43	#17813	—	13	1	no	maximal, Sylow
44	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	—	4	2	no	—
46	C₂²×C₄×C₁₃	—	4	2	no	—
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	SD₁₆	1	8	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₁₃	—	4	4	no	—
61	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
62	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
63	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
64	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
67	C₄²×C₁₃	C₂×D₄	1	2	yes	—
68	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
71	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
72	#6099	—	4	8	no	—
73	#6099	—	4	8	no	—
74	C₂×#1055	—	2	8	no	—
75	C₂×#1055	—	4	8	no	—
76	C₂×#431	D₁₃	1	1	yes	characteristic
77	C₂×#423	—	13	2	no	—
78	#1570	—	13	8	no	—
79	#1566	—	13	4	no	—
80	C₂³×C₁₃	—	4	1	no	—
81	C₂³×C₁₃	—	4	1	no	—
82	C₂³×C₁₃	#106	1	4	yes	—
83	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	4	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	4	1	no	—
91	C₂×C₄×C₁₃	—	4	1	no	—
92	C₂×C₄×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	#106	1	4	yes	—
94	C₂×C₄×C₁₃	#106	1	8	yes	—
95	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
102	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
103	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
104	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
107	C₈×C₁₃	—	2	8	no	—
108	#1055	—	4	8	no	—
109	C₂²×M₁₆	—	26	2	no	—
110	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
111	#431	Dic₁₃	1	4	yes	—
112	#431	C₂×D₁₃	1	4	yes	—
113	#423	—	13	4	no	—
114	#423	—	13	4	no	—
115	#423	—	26	2	no	—
116	#423	—	26	2	no	—
117	C₂×C₄⋊C₈	—	13	4	no	—
118	C₂×C₄⋊C₈	—	26	2	no	—
119	C₂×C₄⋊C₈	Dic₁₃	1	2	yes	—
120	C₂×C₂²⋊C₈	Dic₁₃	1	2	yes	—
121	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
122	#334	—	13	16	no	—
123	C₂²×C₁₃	—	2	1	no	—
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	4	4	no	—
128	C₂²×C₁₃	C₂×#106	1	4	yes	—
129	C₂²×C₁₃	C₂×#106	1	2	yes	—
130	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	2	1	no	—
135	C₄×C₁₃	C₂×#106	1	4	yes	—
136	C₄×C₁₃	C₂×#106	1	2	yes	—
137	C₄×C₁₃	#430	1	2	yes	—
138	C₄×C₁₃	#568	1	2	yes	—
139	C₄×C₁₃	C₂²×SD₁₆	1	2	yes	—
140	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
141	C₂³×C₄	D₅₂	1	1	yes	characteristic
142	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
143	C₂×M₁₆	—	26	4	no	—
144	C₂×M₁₆	—	26	4	no	—
145	C₂×M₁₆	—	52	8	no	—
146	C₂²×C₈	—	26	4	no	—
147	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
148	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
149	C₂×C₄²	Dic₂₆	1	2	yes	—
150	C₂×C₄²	D₅₂	1	2	yes	—
151	C₂×C₄²	D₅₂	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
154	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
155	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
156	C₄⋊C₈	—	13	8	no	—
157	C₄⋊C₈	—	26	4	no	—
158	C₄⋊C₈	—	26	4	no	—
159	C₄⋊C₈	—	26	4	no	—
160	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
161	C₂²⋊C₈	C₂×Dic₁₃	1	8	yes	—
162	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
163	C₄×C₈	C₂²×D₁₃	1	4	yes	—
164	C₂×C₁₃	—	2	2	no	—
165	C₂×C₁₃	—	2	4	no	—
166	C₂×C₁₃	#2905	1	1	yes	characteristic
167	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
168	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
169	C₂⁴	C₄⋊Dic₁₃	1	1	yes	characteristic
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	#6095	1	8	yes	—
173	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
174	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
175	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
176	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
179	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
180	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
181	C₂²×C₄	C₂×D₅₂	1	2	yes	—
182	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
183	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
185	M₁₆	—	52	8	no	—
186	M₁₆	—	52	8	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	26	8	no	—
189	C₂×C₈	—	52	8	no	—
190	C₂×C₈	C₂²×Dic₁₃	1	8	yes	—
191	C₄²	C₄⋊Dic₁₃	1	2	yes	—
192	C₄²	C₄⋊Dic₁₃	1	2	yes	—
193	C₄²	C₄⋊Dic₁₃	1	2	yes	—
194	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
195	C₄²	C₂×Dic₂₆	1	2	yes	—
196	C₄²	C₂×D₅₂	1	2	yes	—
197	C₄²	C₂×D₅₂	1	1	yes	characteristic
198	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
199	C₄²	C₂²×Dic₁₃	1	2	yes	—
200	C₄²	C₂³×D₁₃	1	1	yes	characteristic
201	C₁₃	#17813	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	4	1	no	—
203	C₂³	—	4	1	no	—
204	C₂³	#88057	1	4	yes	—
205	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
206	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	#88057	1	4	yes	—
216	C₂×C₄	#88057	1	8	yes	—
217	C₂×C₄	C₂×#6095	1	4	yes	—
218	C₂×C₄	C₂×#6095	1	4	yes	—
219	C₂×C₄	C₂×#6095	1	4	yes	—
220	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
221	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
223	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
225	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
227	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
228	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
229	C₈	—	2	8	no	—
230	C₈	—	52	8	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	4	4	no	—
236	C₂²	C₂×#88057	1	4	yes	—
237	C₂²	C₂×#88057	1	2	yes	—
238	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	1	no	—
243	C₄	C₂×#88057	1	4	yes	—
244	C₄	C₂×#88057	1	2	yes	—
245	C₄	#11718786	1	2	yes	—
246	C₄	C₂²×#6095	1	2	yes	—
247	C₄	#11719524	1	2	yes	—
248	C₂	—	2	2	no	—
249	C₂	—	2	4	no	—
250	C₂	#13123400	1	1	yes	characteristic
251	C₂	C₂²×#88057	1	1	yes	characteristic
252	C₂	C₂×#11718786	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#431	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#431	D₁₃
hypocenter	C₁₃	#17813
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. #13113607	C₂	semidirect
3. #13113592	C₂	semidirect
18. #11718049	C₂²	semidirect
201. C₁₃	#17813	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718585	2
#13113607	2
#13113592	2
C₂×C₁₃×#431	2
#17813	13