Subgroups — #16790114

Order 3328.

542 conjugacy classes of subgroups.

The subgroup lattice has 304 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	#13129875	C₂	1	2	yes	maximal, complemented
3	#13129874	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
6	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
8	C₂²×#6092	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₄×C₈×D₁₃	—	2	4	no	—
14	#11718259	—	2	4	no	—
15	C₂²×C₄×D₁₃	C₈	1	1	yes	characteristic
16	C₂²×C₄×D₁₃	C₈	1	1	yes	characteristic
17	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂³	1	1	yes	characteristic
26	C₂×C₈×D₁₃	—	2	2	no	—
27	C₂×C₈×D₁₃	—	2	2	no	—
28	C₂×C₈×D₁₃	—	2	2	no	—
29	C₂×C₈×D₁₃	—	2	2	no	—
30	C₂×C₈×D₁₃	—	2	2	no	—
31	C₂×C₈×D₁₃	D₄	1	4	yes	—
32	C₄²×D₁₃	—	2	2	no	—
33	C₄²×D₁₃	—	2	2	no	—
34	C₂×#6092	—	2	4	no	—
35	C₂×#6092	—	2	4	no	—
36	C₂×#6092	—	2	4	no	—
37	C₄×C₈×C₁₃	—	2	2	no	—
38	C₈×Dic₁₃	—	2	2	no	—
39	C₈×Dic₁₃	—	2	2	no	—
40	C₄×#1055	—	2	2	no	—
41	#12844	—	13	1	no	maximal, Sylow
42	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×D₁₃	—	2	2	no	—
50	C₂×C₄×D₁₃	—	2	2	no	—
51	C₂×C₄×D₁₃	—	2	2	no	—
52	C₂×C₄×D₁₃	—	2	2	no	—
53	C₂×C₄×D₁₃	—	2	1	no	—
54	C₂×C₄×D₁₃	—	2	1	no	—
55	C₂×C₄×D₁₃	—	2	1	no	—
56	C₂×C₄×D₁₃	—	2	1	no	—
57	C₂×C₄×D₁₃	—	2	2	no	—
58	C₂×C₄×D₁₃	C₁₆	1	4	yes	complemented
59	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
67	C₄²×C₁₃	—	2	1	no	—
68	C₄²×C₁₃	—	2	1	no	—
69	C₄×Dic₁₃	—	2	1	no	—
70	C₄×Dic₁₃	—	2	1	no	—
71	C₄×Dic₁₃	—	2	2	no	—
72	C₄×Dic₁₃	—	2	2	no	—
73	C₂×#1055	—	2	1	no	—
74	C₂×#1055	—	2	1	no	—
75	C₂×#1055	M₁₆	1	2	yes	—
76	C₂×#1055	C₂×D₄	1	2	yes	—
77	C₂×#1055	#41	1	2	yes	—
78	C₂×#1055	#41	1	2	yes	—
79	C₈×D₁₃	—	2	8	no	—
80	C₈×D₁₃	—	2	8	no	—
81	#6092	—	4	4	no	—
82	C₂²×C₄×C₈	—	13	1	no	—
83	C₂×C₂²⋊C₁₆	—	13	2	no	—
84	#1374	—	13	2	no	—
85	#1374	—	13	2	no	—
86	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
87	C₂²×D₁₃	—	2	1	no	—
88	C₂²×D₁₃	—	2	1	no	—
89	C₂²×D₁₃	—	2	1	no	—
90	C₂²×D₁₃	—	2	1	no	—
91	C₂²×D₁₃	C₂²⋊C₈	1	2	yes	—
92	C₂²×D₁₃	C₂×C₁₆	1	2	yes	—
93	C₂²×D₁₃	M₃₂	1	2	yes	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
99	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×Dic₁₃	—	2	1	no	—
107	C₂×Dic₁₃	—	2	1	no	—
108	C₂×Dic₁₃	—	2	1	no	—
109	C₂×Dic₁₃	—	2	1	no	—
110	C₂×Dic₁₃	—	2	1	no	—
111	C₂×Dic₁₃	—	2	1	no	—
112	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
113	C₂×Dic₁₃	C₂×C₁₆	1	2	yes	—
114	C₂×Dic₁₃	M₃₂	1	2	yes	—
115	C₄×D₁₃	—	2	8	no	—
116	C₄×D₁₃	—	2	8	no	—
117	C₄×D₁₃	—	2	2	no	—
118	C₄×D₁₃	—	2	2	no	—
119	C₄×D₁₃	—	2	2	no	—
120	C₄×D₁₃	—	2	2	no	—
121	C₈×C₁₃	—	2	2	no	—
122	C₈×C₁₃	—	2	2	no	—
123	#1055	—	2	2	no	—
124	#1055	#123	1	4	yes	—
125	C₂³×C₈	—	13	1	no	—
126	C₂³×C₈	—	13	1	no	—
127	C₂²×C₄²	—	13	1	no	—
128	C₂²×C₁₆	—	13	4	no	—
129	C₂×C₄×C₈	—	13	1	no	—
130	C₂×C₄×C₈	—	13	1	no	—
131	C₂×C₄×C₈	—	13	1	no	—
132	C₂×C₄×C₈	—	26	4	no	—
133	C₂×C₄×C₈	#261	1	1	yes	characteristic
134	C₂²⋊C₁₆	—	26	4	no	—
135	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₂²×C₁₃	#431	1	1	yes	characteristic
137	C₂²×C₁₃	#431	1	1	yes	characteristic
138	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
139	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
140	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
141	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
142	C₂×D₁₃	—	2	2	no	—
143	C₂×D₁₃	—	2	2	no	—
144	C₂×D₁₃	—	2	2	no	—
145	C₂×D₁₃	—	2	2	no	—
146	C₂×D₁₃	—	2	2	no	—
147	C₂×D₁₃	—	2	2	no	—
148	C₂×D₁₃	C₂²⋊C₁₆	1	4	yes	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	#381	1	1	yes	characteristic
152	C₄×C₁₃	#407	1	1	yes	characteristic
153	C₄×C₁₃	#432	1	1	yes	characteristic
154	C₄×C₁₃	#432	1	1	yes	characteristic
155	Dic₁₃	—	2	2	no	—
156	Dic₁₃	—	2	2	no	—
157	Dic₁₃	—	2	2	no	—
158	C₂³×C₄	—	13	1	no	—
159	C₂³×C₄	—	13	1	no	—
160	C₂³×C₄	—	13	1	no	—
161	C₂²×C₈	—	13	1	no	—
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	4	no	—
164	C₂²×C₈	—	26	2	no	—
165	C₂²×C₈	—	26	2	no	—
166	C₂²×C₈	—	26	2	no	—
167	C₂²×C₈	—	26	2	no	—
168	C₂²×C₈	—	26	2	no	—
169	C₂²×C₈	C₂×#261	1	1	yes	characteristic
170	C₂²×C₈	C₂×#261	1	1	yes	characteristic
171	C₂×C₄²	—	13	1	no	—
172	C₂×C₄²	—	13	1	no	—
173	C₂×C₄²	—	13	1	no	—
174	C₂×C₄²	—	26	2	no	—
175	C₂×C₄²	—	26	2	no	—
176	C₂×C₄²	C₂×#261	1	1	yes	characteristic
177	C₂×C₁₆	—	26	4	no	—
178	C₂×C₁₆	—	26	4	no	—
179	C₂×C₁₆	—	26	4	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	26	2	no	—
182	C₄×C₈	—	26	2	no	—
183	C₄×C₈	—	26	2	no	—
184	C₂×C₁₃	#1817	1	1	yes	characteristic
185	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
186	C₂×C₁₃	#2112	1	1	yes	characteristic
187	C₂×C₁₃	#2162	1	1	yes	characteristic
188	C₂×C₁₃	#2162	1	1	yes	characteristic
189	C₂×C₁₃	#2163	1	1	yes	characteristic
190	C₂×C₁₃	#2163	1	1	yes	characteristic
191	D₁₃	—	2	4	no	—
192	C₂⁴	—	13	1	no	—
193	C₂²×C₄	—	13	1	no	—
194	C₂²×C₄	—	13	1	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	4	no	—
197	C₂²×C₄	—	13	2	no	—
198	C₂²×C₄	—	13	2	no	—
199	C₂²×C₄	—	26	2	no	—
200	C₂²×C₄	—	26	2	no	—
201	C₂²×C₄	—	26	2	no	—
202	C₂²×C₄	—	26	2	no	—
203	C₂²×C₄	—	26	2	no	—
204	C₂²×C₄	—	26	1	no	—
205	C₂²×C₄	—	26	1	no	—
206	C₂²×C₄	—	26	1	no	—
207	C₂²×C₄	—	26	1	no	—
208	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
209	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
210	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	13	2	no	—
216	C₂×C₈	—	13	2	no	—
217	C₂×C₈	—	13	2	no	—
218	C₂×C₈	—	13	2	no	—
219	C₂×C₈	—	26	8	no	—
220	C₂×C₈	—	26	8	no	—
221	C₂×C₈	—	26	1	no	—
222	C₂×C₈	—	26	1	no	—
223	C₂×C₈	#6118	1	2	yes	—
224	C₂×C₈	C₂²⋊#261	1	2	yes	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	26	2	no	—
228	C₄²	—	26	2	no	—
229	C₄²	—	26	1	no	—
230	C₄²	—	26	1	no	—
231	C₁₆	—	52	4	no	—
232	C₁₃	#12844	1	1	yes	characteristic, complemented, Sylow
233	C₂³	—	13	2	no	—
234	C₂³	—	13	2	no	—
235	C₂³	—	13	2	no	—
236	C₂³	—	26	1	no	—
237	C₂³	—	26	1	no	—
238	C₂³	—	26	1	no	—
239	C₂³	—	26	1	no	—
240	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	13	2	no	—
246	C₂×C₄	—	13	2	no	—
247	C₂×C₄	—	13	2	no	—
248	C₂×C₄	—	26	8	no	—
249	C₂×C₄	—	26	8	no	—
250	C₂×C₄	—	26	1	no	—
251	C₂×C₄	—	26	1	no	—
252	C₂×C₄	—	26	1	no	—
253	C₂×C₄	—	26	1	no	—
254	C₂×C₄	—	26	1	no	—
255	C₂×C₄	—	26	1	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	—	26	2	no	—
259	C₂×C₄	—	26	2	no	—
260	C₂×C₄	D₁₃⋊C₁₆	1	2	yes	—
261	C₂×C₄	#88098	1	2	yes	—
262	C₂×C₄	C₂×#6118	1	1	yes	characteristic
263	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
264	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
265	C₂×C₄	#88237	1	1	yes	characteristic
266	C₂×C₄	#88237	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	13	4	no	—
271	C₈	—	26	2	no	—
272	C₂²	—	13	4	no	—
273	C₂²	—	26	2	no	—
274	C₂²	—	26	2	no	—
275	C₂²	—	26	2	no	—
276	C₂²	—	26	2	no	—
277	C₂²	—	26	2	no	—
278	C₂²	—	26	2	no	—
279	C₂²	#11719049	1	1	yes	characteristic
280	C₂²	#11719051	1	1	yes	characteristic
281	C₂²	C₂×D₁₃⋊C₁₆	1	1	yes	characteristic
282	C₂²	C₂×#88098	1	1	yes	characteristic
283	C₂²	D₁₃⋊M₃₂	1	1	yes	characteristic
284	C₂²	D₁₃⋊M₃₂	1	1	yes	characteristic
285	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	26	2	no	—
289	C₄	—	26	2	no	—
290	C₄	—	26	2	no	—
291	C₄	#11719054	1	1	yes	characteristic
292	C₄	#11719055	1	1	yes	characteristic
293	C₄	#11719127	1	1	yes	characteristic
294	C₄	#11719135	1	1	yes	characteristic
295	C₂	—	26	4	no	—
296	C₂	#13130728	1	1	yes	characteristic
297	C₂	D₁₃⋊(C₂²⋊C₁₆)	1	1	yes	characteristic
298	C₂	#13131064	1	1	yes	characteristic
299	C₂	#13131195	1	1	yes	characteristic
300	C₂	#13131196	1	1	yes	characteristic
301	C₂	#13131199	1	1	yes	characteristic
302	C₂	#13131200	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#12844
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13129875	C₂	semidirect
3. #13129874	C₂	semidirect
58. C₂×C₄×D₁₃	C₁₆	semidirect
232. C₁₃	#12844	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
#13129875	2
#13129874	2
C₂×C₄×C₈×D₁₃	2
#12844	13