Subgroups — #16789295

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718259	C₂	1	2	yes	maximal
2	C₂×#11718258	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718257	C₂	1	1	yes	maximal, characteristic, complemented
4	#13129874	C₂	1	2	yes	maximal, complemented
5	C₂×#11718369	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6094	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88126	C₄	1	1	yes	characteristic
9	C₂²×#6092	C₂²	1	2	yes	—
10	C₂²×#6092	C₂²	1	2	yes	—
11	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
12	C₂×#88055	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	#11718369	—	2	2	no	—
16	#11718369	—	2	2	no	—
17	#11718259	C₄	1	8	yes	complemented
18	#11718258	—	2	2	no	—
19	#11718257	—	2	2	no	—
20	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
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×#6103	C₈	1	2	yes	—
25	C₂×#6101	C₈	1	2	yes	—
26	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂³	1	1	yes	characteristic
29	C₂×#6094	—	2	2	no	—
30	C₂×#6094	—	2	2	no	—
31	C₂×#6094	C₈	1	4	yes	—
32	C₂×C₈×D₁₃	—	2	1	no	—
33	C₂×C₈×D₁₃	—	2	1	no	—
34	C₂×C₈×D₁₃	—	2	2	no	—
35	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
36	#88126	—	2	2	no	—
37	#88126	—	2	2	no	—
38	C₂×#6092	—	2	2	no	—
39	C₂×#6092	—	2	2	no	—
40	C₂×#6092	—	2	2	no	—
41	C₂×#6092	—	2	2	no	—
42	C₂×#6092	C₂×C₄	1	4	yes	—
43	C₂×#6092	C₂×C₄	1	4	yes	—
44	C₁₃×#97	—	2	2	no	—
45	#88055	—	2	2	no	—
46	C₈×Dic₁₃	D₄	1	2	yes	—
47	C₈×Dic₁₃	D₄	1	2	yes	—
48	C₄×#1055	—	2	1	no	—
49	C₄×#1055	—	2	1	no	—
50	#12600	—	13	1	no	maximal, Sylow
51	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₁₃	—	2	1	no	—
57	C₂×C₄×D₁₃	—	2	1	no	—
58	C₂×C₄×D₁₃	—	2	1	no	—
59	C₂×C₄×D₁₃	—	2	1	no	—
60	C₂×C₄×D₁₃	C₄²	1	2	yes	—
61	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	C₄²	1	2	yes	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
68	C₄²×C₁₃	—	2	1	no	—
69	C₄²×C₁₃	—	2	1	no	—
70	#6103	—	2	4	no	—
71	#6101	—	2	4	no	—
72	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
73	C₄×Dic₁₃	M₁₆	1	2	yes	—
74	C₂×#1055	—	2	1	no	—
75	C₂×#1055	—	2	1	no	—
76	C₂×#1055	C₂×C₈	1	2	yes	—
77	C₂×#1055	C₂²×C₄	1	2	yes	—
78	C₂×#1055	C₂×D₄	1	1	yes	characteristic
79	C₂×#1055	C₂×D₄	1	1	yes	characteristic
80	C₂×#1055	#41	1	2	yes	—
81	#6094	—	2	8	no	—
82	C₈×D₁₃	—	2	4	no	—
83	C₈×D₁₃	—	2	4	no	—
84	#6092	—	2	4	no	—
85	#6092	—	2	4	no	—
86	C₂×#405	—	13	1	no	—
87	C₂×C₂²⋊C₁₆	—	13	2	no	—
88	C₂×#346	—	13	1	no	—
89	C₂×C₄×C₁₆	—	13	1	no	—
90	#1374	—	13	2	no	—
91	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
92	C₂²×D₁₃	—	2	1	no	—
93	C₂²×D₁₃	—	2	1	no	—
94	C₂²×D₁₃	—	2	4	no	—
95	C₂²×D₁₃	C₄×C₈	1	2	yes	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	4	no	—
98	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×Dic₁₃	—	2	1	no	—
105	C₂×Dic₁₃	—	2	1	no	—
106	C₂×Dic₁₃	—	2	4	no	—
107	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
108	C₂×Dic₁₃	C₂²⋊C₈	1	4	yes	—
109	C₄×D₁₃	—	2	2	no	—
110	C₄×D₁₃	—	2	2	no	—
111	C₄×D₁₃	—	2	2	no	—
112	C₄×D₁₃	—	2	2	no	—
113	C₈×C₁₃	—	2	2	no	—
114	C₈×C₁₃	C₂²⋊C₈	1	4	yes	—
115	#1055	—	2	2	no	—
116	#1055	C₄×D₄	1	2	yes	—
117	#1055	C₄×D₄	1	2	yes	—
118	C₂²×M₁₆	—	13	1	no	—
119	C₂³×C₈	—	13	1	no	—
120	C₂×#117	—	13	1	no	—
121	C₂²×C₁₆	—	13	2	no	—
122	C₂²×C₁₆	—	13	2	no	—
123	#405	—	26	2	no	—
124	#405	—	26	2	no	—
125	C₂×#97	—	13	1	no	—
126	C₂×#97	#261	1	1	yes	characteristic
127	C₂×C₄×C₈	—	13	1	no	—
128	C₂×C₄×C₈	—	13	1	no	—
129	C₂²⋊C₁₆	—	13	8	no	—
130	#346	—	26	2	no	—
131	C₄×C₁₆	—	26	2	no	—
132	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
133	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₁₃	#431	1	1	yes	characteristic
135	C₂²×C₁₃	#504	1	4	yes	—
136	C₂×D₁₃	—	2	4	no	—
137	C₂×D₁₃	—	2	2	no	—
138	C₂×D₁₃	—	4	2	no	—
139	C₂×D₁₃	—	4	2	no	—
140	C₄×C₁₃	—	4	2	no	—
141	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
143	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₄×C₁₃	#431	1	1	yes	characteristic
145	Dic₁₃	—	2	4	no	—
146	Dic₁₃	—	2	2	no	—
147	C₂³×C₄	—	13	1	no	—
148	C₂×M₁₆	—	13	4	no	—
149	C₂×M₁₆	—	26	2	no	—
150	C₂×M₁₆	—	26	2	no	—
151	C₂²×C₈	—	13	1	no	—
152	C₂²×C₈	—	13	1	no	—
153	C₂²×C₈	—	13	4	no	—
154	C₂²×C₈	—	26	2	no	—
155	C₂²×C₈	—	26	1	no	—
156	C₂²×C₈	—	26	1	no	—
157	C₂²×C₈	C₂×#261	1	1	yes	characteristic
158	C₂²×C₈	C₂×#261	1	1	yes	characteristic
159	#117	—	26	2	no	—
160	#117	—	26	2	no	—
161	C₂×C₄⋊C₄	—	13	2	no	—
162	C₂×C₂²⋊C₄	—	13	2	no	—
163	C₂×C₄²	—	13	1	no	—
164	C₂×C₄²	C₂×#261	1	1	yes	characteristic
165	C₂×C₁₆	—	13	4	no	—
166	C₂×C₁₆	—	13	4	no	—
167	C₂×C₁₆	—	26	2	no	—
168	C₂×C₁₆	—	26	2	no	—
169	C₂×C₁₆	—	26	2	no	—
170	C₂×C₁₆	—	26	2	no	—
171	#97	—	2	2	no	—
172	#97	—	26	2	no	—
173	C₄×C₈	—	13	2	no	—
174	C₄×C₈	—	13	2	no	—
175	C₄×C₈	—	26	1	no	—
176	C₄×C₈	—	26	1	no	—
177	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
178	C₂×C₁₃	#2108	1	2	yes	—
179	C₂×C₁₃	#2113	1	2	yes	—
180	C₂×C₁₃	#2164	1	2	yes	—
181	D₁₃	—	4	2	no	—
182	C₂⁴	—	13	1	no	—
183	C₂²×C₄	—	13	2	no	—
184	C₂²×C₄	—	13	1	no	—
185	C₂²×C₄	—	13	2	no	—
186	C₂²×C₄	—	13	2	no	—
187	C₂²×C₄	—	26	1	no	—
188	C₂²×C₄	—	26	1	no	—
189	C₂²×C₄	—	26	1	no	—
190	C₂²×C₄	—	26	1	no	—
191	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
192	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
193	M₁₆	—	26	8	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	13	2	no	—
197	C₂×C₈	—	13	2	no	—
198	C₂×C₈	—	13	1	no	—
199	C₂×C₈	—	13	1	no	—
200	C₂×C₈	—	13	2	no	—
201	C₂×C₈	—	26	4	no	—
202	C₂×C₈	—	26	4	no	—
203	C₂×C₈	—	26	1	no	—
204	C₂×C₈	—	26	1	no	—
205	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
206	C₂×C₈	#6118	1	2	yes	—
207	C₂×C₈	C₄×#261	1	2	yes	—
208	C₂×C₈	C₂²⋊#261	1	2	yes	—
209	C₄⋊C₄	—	26	4	no	—
210	C₂²⋊C₄	—	26	4	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	13	2	no	—
214	C₄²	—	13	2	no	—
215	C₁₆	—	26	4	no	—
216	C₁₆	—	26	4	no	—
217	C₁₃	#12600	1	1	yes	characteristic, complemented, Sylow
218	C₂³	—	13	2	no	—
219	C₂³	—	26	1	no	—
220	C₂³	—	26	4	no	—
221	C₂³	—	26	1	no	—
222	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	13	2	no	—
226	C₂×C₄	—	13	4	no	—
227	C₂×C₄	—	26	1	no	—
228	C₂×C₄	—	26	4	no	—
229	C₂×C₄	—	26	1	no	—
230	C₂×C₄	—	26	2	no	—
231	C₂×C₄	—	26	2	no	—
232	C₂×C₄	—	26	2	no	—
233	C₂×C₄	—	26	2	no	—
234	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
235	C₂×C₄	C₂×#6118	1	1	yes	characteristic
236	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
238	C₂×C₄	#88237	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	13	2	no	—
242	C₈	—	13	2	no	—
243	C₈	—	26	2	no	—
244	C₈	#88111	1	4	yes	—
245	C₂²	—	26	4	no	—
246	C₂²	—	26	2	no	—
247	C₂²	—	52	2	no	—
248	C₂²	—	52	2	no	—
249	C₂²	C₄×D₁₃⋊C₈	1	1	yes	characteristic
250	C₂²	#11719051	1	1	yes	characteristic
251	C₂²	#11719096	1	4	yes	—
252	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
253	C₄	—	4	2	no	—
254	C₄	—	26	4	no	—
255	C₄	—	26	2	no	—
256	C₄	C₄×#6118	1	1	yes	characteristic
257	C₄	#11719049	1	1	yes	characteristic
258	C₄	C₂×#88111	1	1	yes	characteristic
259	C₄	C₄×C₂²⋊#261	1	1	yes	characteristic
260	C₂	—	52	2	no	—
261	C₂	C₄×#88111	1	1	yes	characteristic
262	C₂	#13131046	1	2	yes	—
263	C₂	#13131067	1	2	yes	—
264	C₂	#13131204	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₁₃×#97	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#97	#261
hypocenter	C₁₃	#12600
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#11718258	C₂	semidirect
3. C₂×#11718257	C₂	semidirect
4. #13129874	C₂	semidirect
17. #11718259	C₄	semidirect
217. C₁₃	#12600	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718259	2
C₂×#11718258	2
C₂×#11718257	2
#13129874	2
C₂×#11718369	2
#12600	13