Subgroups — #16780285

Order 3328.

478 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88110	C₂	1	1	yes	maximal, characteristic
2	C₄²×#1057	C₂	1	1	yes	maximal, characteristic
3	#13129658	C₂	1	2	yes	maximal
4	#13129658	C₂	1	2	yes	maximal
5	#13114580	C₂	1	1	yes	maximal, characteristic
6	C₂×#88110	C₄	1	2	yes	—
7	C₂×#88110	C₄	1	2	yes	—
8	C₂×C₄×#1057	C₄	1	2	yes	—
9	C₂×C₄×#1057	C₄	1	2	yes	—
10	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
11	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1057	C₂²	1	2	yes	—
13	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
14	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#336	C₄	1	1	yes	characteristic
16	#11718142	C₄	1	2	yes	—
17	#11718116	C₄	1	1	yes	characteristic
18	C₂²×#1057	C₂×C₄	1	4	yes	—
19	C₂²×#1057	C₂×C₄	1	2	yes	—
20	C₂²×#1057	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	2	yes	—
30	#88110	—	2	4	no	—
31	#88110	C₈	1	8	yes	complemented
32	C₄×#1057	—	2	4	no	—
33	C₄×#1057	—	2	4	no	—
34	C₄×#1057	—	2	2	no	—
35	C₄×#1057	—	2	2	no	—
36	C₄×#1057	—	2	2	no	—
37	C₄×#1057	—	2	2	no	—
38	C₈×Dic₁₃	—	2	2	no	—
39	C₈×Dic₁₃	—	2	2	no	—
40	#8132	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
45	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×#1057	—	2	4	no	—
49	C₂×#1057	—	2	2	no	—
50	C₂×#1057	—	2	2	no	—
51	C₂×#1057	—	2	2	no	—
52	C₂×#1057	—	2	2	no	—
53	C₂×#1057	—	2	4	no	—
54	C₂×#1057	C₂×C₈	1	8	yes	—
55	C₂×#1057	M₁₆	1	4	yes	—
56	C₂×#1057	M₁₆	1	4	yes	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₄²×C₁₃	—	2	2	no	—
61	C₄×Dic₁₃	—	2	1	no	—
62	C₄×Dic₁₃	—	2	1	no	—
63	C₄×Dic₁₃	—	2	2	no	—
64	C₄×Dic₁₃	—	2	2	no	—
65	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
66	C₄×Dic₁₃	M₁₆	1	4	yes	—
67	C₄×Dic₁₃	#41	1	2	yes	—
68	C₄×Dic₁₃	#41	1	2	yes	—
69	C₂×#1055	—	2	2	no	—
70	C₂×#1055	—	2	2	no	—
71	C₂×#1055	—	2	2	no	—
72	#1744	—	13	1	no	—
73	C₄×#97	—	13	1	no	—
74	C₄²×C₈	—	13	1	no	—
75	#1273	—	13	2	no	—
76	#1273	—	13	2	no	—
77	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
83	C₂×C₄×C₁₃	#97	1	2	yes	—
84	C₂×C₄×C₁₃	#117	1	2	yes	—
85	C₂×C₄×C₁₃	#131	1	2	yes	—
86	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
88	C₂×Dic₁₃	—	2	1	no	—
89	C₂×Dic₁₃	—	2	1	no	—
90	C₂×Dic₁₃	—	2	1	no	—
91	C₂×Dic₁₃	—	2	1	no	—
92	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
93	C₂×Dic₁₃	#97	1	2	yes	—
94	C₂×Dic₁₃	#117	1	2	yes	—
95	C₂×Dic₁₃	#117	1	1	yes	characteristic
96	C₂×Dic₁₃	#117	1	1	yes	characteristic
97	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
98	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
99	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
100	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
101	#1057	—	2	8	no	—
102	#1057	—	2	4	no	—
103	#1057	—	2	4	no	—
104	C₈×C₁₃	—	4	2	no	—
105	#1055	—	4	2	no	—
106	C₂×#97	—	13	2	no	—
107	C₂×#97	—	13	2	no	—
108	C₂×C₄×C₈	—	13	2	no	—
109	C₂×C₄×C₈	—	13	2	no	—
110	C₂×C₄×C₈	—	13	1	no	—
111	C₂×C₄×C₈	—	13	1	no	—
112	C₂×C₄×C₈	—	13	2	no	—
113	C₂×C₄×C₈	—	13	2	no	—
114	C₄³	—	13	1	no	—
115	#336	—	13	2	no	—
116	#336	—	13	1	no	—
117	#336	#261	1	1	yes	characteristic
118	C₂²×C₁₃	#376	1	1	yes	characteristic
119	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
120	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
121	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
122	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
123	C₂²×C₁₃	#405	1	1	yes	characteristic
124	C₂²×C₁₃	#405	1	1	yes	characteristic
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	#433	1	2	yes	—
128	C₄×C₁₃	#433	1	2	yes	—
129	Dic₁₃	—	2	2	no	—
130	Dic₁₃	—	2	2	no	—
131	Dic₁₃	#431	1	4	yes	—
132	Dic₁₃	#432	1	2	yes	—
133	Dic₁₃	#432	1	2	yes	—
134	C₂²×C₈	—	13	2	no	—
135	C₂²×C₈	—	13	2	no	—
136	C₂²×C₈	—	13	4	no	—
137	C₂²×C₈	—	13	2	no	—
138	C₂²×C₈	C₂×#261	1	2	yes	—
139	C₂×C₄²	—	13	1	no	—
140	C₂×C₄²	—	13	1	no	—
141	C₂×C₄²	—	13	1	no	—
142	C₂×C₄²	—	13	1	no	—
143	C₂×C₄²	—	13	1	no	—
144	C₂×C₄²	—	13	1	no	—
145	C₂×C₄²	C₂×#261	1	1	yes	characteristic
146	#97	—	13	8	no	—
147	#97	—	26	4	no	—
148	C₄×C₈	—	26	2	no	—
149	C₄×C₈	—	26	2	no	—
150	C₄×C₈	—	26	2	no	—
151	C₄×C₈	—	26	2	no	—
152	C₄×C₈	—	26	2	no	—
153	C₄×C₈	—	26	2	no	—
154	C₄×C₈	—	26	4	no	—
155	C₄×C₈	—	26	4	no	—
156	C₂×C₁₃	C₈×M₁₆	1	1	yes	characteristic
157	C₂×C₁₃	C₈×M₁₆	1	1	yes	characteristic
158	C₂×C₁₃	#1451	1	1	yes	characteristic
159	C₂×C₁₃	#1451	1	1	yes	characteristic
160	C₂×C₁₃	#1744	1	1	yes	characteristic
161	C₂×C₁₃	#1745	1	1	yes	characteristic
162	C₂×C₁₃	#1746	1	1	yes	characteristic
163	C₂²×C₄	—	13	1	no	—
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	—	13	1	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	C₄×#261	1	1	yes	characteristic
168	C₂²×C₄	C₄×#261	1	1	yes	characteristic
169	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	13	4	no	—
174	C₂×C₈	—	13	4	no	—
175	C₂×C₈	—	13	8	no	—
176	C₂×C₈	—	26	2	no	—
177	C₂×C₈	—	26	2	no	—
178	C₂×C₈	—	26	4	no	—
179	C₂×C₈	—	26	2	no	—
180	C₂×C₈	—	26	2	no	—
181	C₂×C₈	—	26	4	no	—
182	C₂×C₈	—	26	2	no	—
183	C₂×C₈	—	26	2	no	—
184	C₂×C₈	—	26	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	13	4	no	—
187	C₄²	—	13	4	no	—
188	C₄²	—	13	2	no	—
189	C₄²	—	13	2	no	—
190	C₄²	—	26	2	no	—
191	C₄²	—	26	2	no	—
192	C₄²	—	26	1	no	—
193	C₄²	—	26	1	no	—
194	C₁₃	#8132	1	1	yes	characteristic, complemented, Sylow
195	C₂³	C₂×C₄×#261	1	1	yes	characteristic
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	13	2	no	—
201	C₂×C₄	—	13	1	no	—
202	C₂×C₄	—	13	1	no	—
203	C₂×C₄	—	13	2	no	—
204	C₂×C₄	—	13	2	no	—
205	C₂×C₄	—	13	2	no	—
206	C₂×C₄	—	13	2	no	—
207	C₂×C₄	—	13	2	no	—
208	C₂×C₄	—	13	2	no	—
209	C₂×C₄	—	26	1	no	—
210	C₂×C₄	—	26	1	no	—
211	C₂×C₄	—	26	1	no	—
212	C₂×C₄	—	26	1	no	—
213	C₂×C₄	C₈×#261	1	2	yes	—
214	C₂×C₄	#88100	1	2	yes	—
215	C₂×C₄	#88237	1	2	yes	—
216	C₂×C₄	#88238	1	1	yes	characteristic
217	C₂×C₄	#88238	1	2	yes	—
218	C₂×C₄	#88240	1	1	yes	characteristic
219	C₈	—	4	2	no	—
220	C₈	—	26	8	no	—
221	C₈	—	26	4	no	—
222	C₈	—	26	4	no	—
223	C₈	—	52	2	no	—
224	C₂²	#11719057	1	1	yes	characteristic
225	C₂²	#11719062	1	1	yes	characteristic
226	C₂²	#11719072	1	1	yes	characteristic
227	C₂²	C₂×C₈×#261	1	1	yes	characteristic
228	C₂²	C₂×#88100	1	1	yes	characteristic
229	C₂²	M₁₆×#261	1	1	yes	characteristic
230	C₂²	M₁₆×#261	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	13	4	no	—
234	C₄	—	13	2	no	—
235	C₄	—	13	2	no	—
236	C₄	—	26	2	no	—
237	C₄	—	26	2	no	—
238	C₄	#11719068	1	2	yes	—
239	C₄	#11719077	1	2	yes	—
240	C₂	#13130037	1	1	yes	characteristic
241	C₂	#13130038	1	1	yes	characteristic
242	C₂	#13130047	1	1	yes	characteristic
243	C₂	#13130048	1	1	yes	characteristic
244	C₂	#13130549	1	1	yes	characteristic
245	C₂	#13130552	1	1	yes	characteristic
246	C₂	#13130555	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₁₃×#336	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₄²
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#336	#261
hypocenter	C₁₃	#8132
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
31. #88110	C₈	semidirect
194. C₁₃	#8132	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88110	2
C₄²×#1057	2
#13129658	2
#13114580	2
#8132	13