Subgroups — #16780922

Order 3328.

446 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#1057	C₂	1	2	yes	maximal
2	C₂×#11718256	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718255	C₂	1	1	yes	maximal, characteristic
4	#13129666	C₂	1	2	yes	maximal
5	#97×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#1057	C₂²	1	2	yes	—
7	C₂×C₄⋊#1057	C₂²	1	2	yes	—
8	C₂×C₄×#1057	—	2	2	no	—
9	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
10	C₂×#88055	C₄	1	1	yes	characteristic
11	C₂×#88055	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88043	C₄	1	1	yes	characteristic
16	#11718290	—	2	2	no	—
17	#11718256	C₄	1	4	yes	complemented
18	#11718255	C₄	1	4	yes	complemented
19	C₂²×#1057	—	2	2	no	—
20	C₂²×#1057	—	2	2	no	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
26	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
27	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
28	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₄⋊#1057	—	2	2	no	—
31	C₄⋊#1057	—	2	2	no	—
32	C₄⋊#1057	C₂×C₄	1	4	yes	—
33	C₄⋊#1057	C₂×C₄	1	4	yes	—
34	C₄×#1057	—	4	4	no	—
35	C₁₃×#97	—	2	2	no	—
36	#88055	—	2	2	no	—
37	#88055	C₈	1	4	yes	complemented
38	C₈×Dic₁₃	—	2	1	no	—
39	C₈×Dic₁₃	—	2	1	no	—
40	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
41	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
42	#88043	C₈	1	4	yes	complemented
43	#8275	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
47	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×#1057	—	2	4	no	—
50	C₂×#1057	—	2	4	no	—
51	C₂×#1057	—	4	2	no	—
52	C₂×#1057	—	4	2	no	—
53	C₂×#1057	—	4	2	no	—
54	C₂×#1057	—	4	2	no	—
55	C₂×C₈×C₁₃	—	2	1	no	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	C₄²	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
61	C₄²×C₁₃	C₂×C₈	1	2	yes	—
62	C₄²×C₁₃	M₁₆	1	2	yes	—
63	C₄×Dic₁₃	—	2	2	no	—
64	C₄×Dic₁₃	—	2	2	no	—
65	C₄×Dic₁₃	—	4	2	no	—
66	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
67	C₄×Dic₁₃	M₁₆	1	2	yes	—
68	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₄×Dic₁₃	#41	1	1	yes	characteristic
71	C₄×Dic₁₃	#41	1	1	yes	characteristic
72	C₂×#1055	—	2	1	no	—
73	C₂×#1055	—	2	1	no	—
74	C₂×#1055	C₄²	1	2	yes	—
75	C₂×#1055	C₂×C₈	1	2	yes	—
76	C₂×#1055	C₂×C₈	1	2	yes	—
77	C₂×#1055	C₂×C₈	1	2	yes	—
78	C₄×C₄⋊C₈	—	13	2	no	—
79	C₄×#97	—	13	1	no	—
80	C₂×#335	—	13	1	no	—
81	C₂×#334	—	13	1	no	—
82	#1277	—	13	2	no	—
83	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
87	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂×Dic₁₃	—	2	2	no	—
94	C₂×Dic₁₃	—	2	2	no	—
95	C₂×Dic₁₃	—	2	2	no	—
96	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
97	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
98	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
99	C₂×Dic₁₃	C₄×Q₈	1	1	yes	characteristic
100	C₂×Dic₁₃	C₄×Q₈	1	1	yes	characteristic
101	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
102	C₂×Dic₁₃	#137	1	2	yes	—
103	#1057	—	4	4	no	—
104	#1057	—	4	4	no	—
105	C₈×C₁₃	—	2	2	no	—
106	C₈×C₁₃	—	2	2	no	—
107	#1055	—	2	2	no	—
108	#1055	C₄×C₈	1	4	yes	complemented
109	C₂×C₄⋊C₈	—	13	2	no	—
110	C₂×C₄⋊C₈	—	13	2	no	—
111	C₂×#97	—	13	1	no	—
112	C₂×#97	—	13	1	no	—
113	C₂×#97	—	13	1	no	—
114	C₂×#97	#261	1	1	yes	characteristic
115	C₂×C₄×C₈	—	13	1	no	—
116	C₂×C₄×C₈	—	13	1	no	—
117	C₂×C₄×C₈	—	26	2	no	—
118	C₄³	—	13	1	no	—
119	#336	—	26	2	no	—
120	#335	—	13	4	no	—
121	#334	—	13	4	no	—
122	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
124	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₁₃	#428	1	2	yes	—
126	C₂²×C₁₃	#430	1	2	yes	—
127	C₄×C₁₃	—	2	4	no	—
128	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
129	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
130	C₄×C₁₃	#431	1	1	yes	characteristic
131	C₄×C₁₃	#431	1	1	yes	characteristic
132	Dic₁₃	—	2	4	no	—
133	Dic₁₃	—	2	1	no	—
134	Dic₁₃	—	2	1	no	—
135	Dic₁₃	#440	1	2	yes	—
136	Dic₁₃	#441	1	1	yes	characteristic
137	Dic₁₃	#442	1	1	yes	characteristic
138	C₂²×C₈	—	13	1	no	—
139	C₂²×C₈	—	13	1	no	—
140	C₂²×C₈	—	26	2	no	—
141	C₂²×C₈	—	26	2	no	—
142	C₂²×C₈	C₂×#261	1	1	yes	characteristic
143	C₂²×C₈	C₂×#261	1	1	yes	characteristic
144	C₂×C₄²	—	13	1	no	—
145	C₂×C₄²	—	13	1	no	—
146	C₂×C₄²	—	13	2	no	—
147	C₂×C₄²	—	13	2	no	—
148	C₂×C₄²	C₂×#261	1	1	yes	characteristic
149	C₄⋊C₈	—	13	4	no	—
150	C₄⋊C₈	—	13	4	no	—
151	C₄⋊C₈	—	26	2	no	—
152	C₄⋊C₈	—	26	2	no	—
153	#97	—	2	2	no	—
154	#97	—	13	4	no	—
155	#97	—	13	4	no	—
156	#97	—	26	2	no	—
157	C₄×C₈	—	13	2	no	—
158	C₄×C₈	—	13	2	no	—
159	C₄×C₈	—	26	1	no	—
160	C₄×C₈	—	26	1	no	—
161	C₄×C₈	—	52	4	no	—
162	C₂×C₁₃	#1565	1	2	yes	—
163	C₂×C₁₃	#1565	1	2	yes	—
164	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
165	C₂×C₁₃	#1776	1	1	yes	characteristic
166	C₂×C₁₃	#1778	1	1	yes	characteristic
167	C₂²×C₄	—	13	1	no	—
168	C₂²×C₄	—	13	2	no	—
169	C₂²×C₄	—	13	1	no	—
170	C₂²×C₄	C₄⋊#261	1	2	yes	—
171	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	13	2	no	—
177	C₂×C₈	—	13	2	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	13	2	no	—
180	C₂×C₈	—	26	1	no	—
181	C₂×C₈	—	26	1	no	—
182	C₂×C₈	—	26	4	no	—
183	C₂×C₈	—	26	4	no	—
184	C₂×C₈	—	52	2	no	—
185	C₂×C₈	—	52	2	no	—
186	C₂×C₈	—	52	2	no	—
187	C₂×C₈	—	52	2	no	—
188	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
189	C₂×C₈	C₄×#261	1	2	yes	—
190	C₄²	—	13	2	no	—
191	C₄²	—	13	2	no	—
192	C₄²	—	13	1	no	—
193	C₄²	—	13	1	no	—
194	C₄²	—	13	1	no	—
195	C₄²	—	13	1	no	—
196	C₄²	—	26	2	no	—
197	C₄²	—	26	2	no	—
198	C₄²	—	52	2	no	—
199	C₄²	D₁₃⋊C₈	1	2	yes	—
200	C₄²	#6122	1	2	yes	—
201	C₁₃	#8275	1	1	yes	characteristic, complemented, Sylow
202	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	13	1	no	—
206	C₂×C₄	—	13	1	no	—
207	C₂×C₄	—	13	2	no	—
208	C₂×C₄	—	13	2	no	—
209	C₂×C₄	—	13	4	no	—
210	C₂×C₄	—	13	1	no	—
211	C₂×C₄	—	13	1	no	—
212	C₂×C₄	—	26	2	no	—
213	C₂×C₄	—	26	2	no	—
214	C₂×C₄	—	26	2	no	—
215	C₂×C₄	#88112	1	4	yes	—
216	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
217	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
218	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
220	C₂×C₄	#88237	1	1	yes	characteristic
221	C₂×C₄	C₂×#6122	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	2	2	no	—
224	C₈	—	13	4	no	—
225	C₈	—	26	2	no	—
226	C₈	—	52	4	no	—
227	C₈	—	52	4	no	—
228	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
229	C₂²	C₄×C₄⋊#261	1	1	yes	characteristic
230	C₂²	#11719098	1	2	yes	—
231	C₂²	#11719100	1	2	yes	—
232	C₂²	C₂×#88112	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	13	2	no	—
235	C₄	—	13	1	no	—
236	C₄	—	13	1	no	—
237	C₄	—	26	4	no	—
238	C₄	—	26	1	no	—
239	C₄	—	26	1	no	—
240	C₄	C₄×D₁₃⋊C₈	1	1	yes	characteristic
241	C₄	#11719049	1	1	yes	characteristic
242	C₄	C₄×#6122	1	1	yes	characteristic
243	C₄	#11719125	1	1	yes	characteristic
244	C₂	#13130388	1	2	yes	—
245	C₂	#13130389	1	2	yes	—
246	C₂	C₄×#88112	1	1	yes	characteristic
247	C₂	#13130650	1	1	yes	characteristic
248	C₂	#13130656	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#97	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×#97	#261
hypocenter	C₁₃	#8275
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
17. #11718256	C₄	semidirect
18. #11718255	C₄	semidirect
37. #88055	C₈	semidirect
42. #88043	C₈	semidirect
108. #1055	C₄×C₈	semidirect
201. C₁₃	#8275	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#1057	2
C₂×#11718256	2
C₂×#11718255	2
#13129666	2
#97×Dic₁₃	2
#8275	13