Subgroups — #16016058

Order 3328.

524 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116652	C₂	1	4	yes	maximal, complemented
2	#13116648	C₂	1	4	yes	maximal, complemented
3	#13116602	C₂	1	2	yes	maximal, complemented
4	C₂×#11718145	C₂	1	2	yes	maximal
5	#13116441	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2873	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
10	C₁₃×#405	C₄	1	1	yes	characteristic
11	C₁₃×#405	C₄	1	2	yes	—
12	C₁₃×#405	C₄	1	1	yes	characteristic
13	C₁₃×#405	C₂²	1	1	yes	characteristic
14	C₁₃×#405	C₂²	1	2	yes	—
15	C₁₃×#405	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×M₁₆	C₄	1	2	yes	—
17	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
18	C₂×#88052	C₂²	1	4	yes	—
19	#11718145	C₂²	1	4	yes	—
20	#11718145	C₂²	1	4	yes	—
21	#11718058	C₂²	1	8	yes	complemented
22	#11718055	C₂²	1	8	yes	complemented
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	—	2	4	no	—
25	C₂×C₁₃×M₁₆	—	2	4	no	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
32	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×#117	—	2	1	no	—
37	C₁₃×#117	—	2	1	no	—
38	C₁₃×#117	C₂×C₄	1	2	yes	—
39	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
40	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
43	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
44	C₁₃×#97	C₂×C₄	1	2	yes	—
45	C₁₃×#97	C₂×C₄	1	2	yes	—
46	C₁₃×#97	C₂×C₄	1	2	yes	—
47	C₁₃×#97	C₂³	1	2	yes	—
48	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	2	yes	—
52	#88052	—	2	4	no	—
53	#88052	C₂³	1	8	yes	—
54	C₂×#6090	—	2	4	no	—
55	#33427	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₁₃×M₁₆	—	2	2	no	—
67	C₁₃×M₁₆	—	2	2	no	—
68	C₁₃×M₁₆	—	2	2	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	2	2	no	—
71	C₁₃×M₁₆	—	2	2	no	—
72	C₁₃×M₁₆	M₁₆	1	4	yes	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃	#41	1	4	yes	—
81	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	—	2	1	no	—
84	C₁₃×C₄⋊C₄	—	2	1	no	—
85	C₁₃×C₂²⋊C₄	—	2	2	no	—
86	C₁₃×C₂²⋊C₄	—	2	1	no	—
87	C₁₃×C₂²⋊C₄	—	2	1	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
91	#6090	—	2	8	no	—
92	#2873	D₁₃	1	1	yes	characteristic
93	#2166	—	13	4	no	—
94	#2165	—	13	4	no	—
95	#2153	—	13	2	no	—
96	C₂×#349	—	13	2	no	—
97	#2110	—	13	2	no	—
98	C₂³×C₁₃	—	2	2	no	—
99	C₂³×C₁₃	—	8	1	no	—
100	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	8	2	no	—
113	C₂×C₄×C₁₃	#117	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₁₃	—	2	1	no	—
120	C₈×C₁₃	—	2	1	no	—
121	C₈×C₁₃	—	2	2	no	—
122	C₈×C₁₃	#117	1	2	yes	—
123	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
124	C₈×C₁₃	C₂×#41	1	2	yes	—
125	C₂²×M₁₆	Dic₁₃	1	1	yes	characteristic
126	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
127	C₂×#117	Dic₁₃	1	1	yes	characteristic
128	C₂×M₃₂	—	13	4	no	—
129	#405	Dic₁₃	1	1	yes	characteristic
130	#405	Dic₁₃	1	2	yes	—
131	#405	Dic₁₃	1	1	yes	characteristic
132	#405	C₂×D₁₃	1	2	yes	—
133	#405	C₂×D₁₃	1	1	yes	characteristic
134	#405	C₂×D₁₃	1	1	yes	characteristic
135	C₄×M₁₆	Dic₁₃	1	2	yes	—
136	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
137	#364	—	13	8	no	—
138	#349	—	13	4	no	—
139	#349	—	13	4	no	—
140	#347	—	13	8	no	—
141	C₂²×C₁₃	—	2	1	no	—
142	C₂²×C₁₃	—	2	1	no	—
143	C₂²×C₁₃	—	4	1	no	—
144	C₂²×C₁₃	—	4	1	no	—
145	C₂²×C₁₃	—	4	1	no	—
146	C₂²×C₁₃	—	8	2	no	—
147	C₂²×C₁₃	#432	1	2	yes	—
148	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	4	2	no	—
152	C₄×C₁₃	—	4	2	no	—
153	C₄×C₁₃	#432	1	2	yes	—
154	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
155	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
156	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
160	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
161	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
162	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
163	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
164	C₂×M₁₆	C₂²×D₁₃	1	1	yes	characteristic
165	C₂×M₁₆	C₂²×D₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
167	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
169	#117	—	2	1	no	—
170	#117	—	2	1	no	—
171	#117	C₂×Dic₁₃	1	2	yes	—
172	#117	C₂×Dic₁₃	1	1	yes	characteristic
173	#117	C₂×Dic₁₃	1	1	yes	characteristic
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₂²⋊C₄	—	2	1	no	—
176	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
177	M₃₂	—	13	8	no	—
178	M₃₂	—	26	4	no	—
179	C₂×C₁₆	—	26	4	no	—
180	#97	C₂×Dic₁₃	1	2	yes	—
181	#97	C₂×Dic₁₃	1	2	yes	—
182	#97	C₂×Dic₁₃	1	2	yes	—
183	#97	C₂²×D₁₃	1	2	yes	—
184	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
185	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
186	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
187	C₄×C₈	C₂²×D₁₃	1	2	yes	—
188	C₂×C₁₃	—	2	1	no	—
189	C₂×C₁₃	—	2	2	no	—
190	C₂×C₁₃	—	4	1	no	—
191	C₂×C₁₃	—	4	1	no	—
192	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
193	C₂⁴	#6099	1	1	yes	characteristic
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	#6099	1	1	yes	characteristic
198	C₂²×C₄	#6099	1	1	yes	characteristic
199	C₂²×C₄	#6099	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
203	M₁₆	—	2	2	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	#6099	1	4	yes	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	#6127	1	4	yes	—
213	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
214	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
216	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
217	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
218	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₄²	—	2	2	no	—
226	C₄²	C₂²×Dic₁₃	1	2	yes	—
227	C₄²	C₂²×Dic₁₃	1	2	yes	—
228	C₁₆	—	26	8	no	—
229	C₁₃	#33427	1	1	yes	characteristic, complemented, Sylow
230	C₂³	—	2	2	no	—
231	C₂³	—	8	1	no	—
232	C₂³	C₂×#6099	1	1	yes	characteristic
233	C₂³	C₂×#6099	1	1	yes	characteristic
234	C₂³	C₂×#6099	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	C₂×#6099	1	1	yes	characteristic
246	C₂×C₄	C₂×#6099	1	1	yes	characteristic
247	C₂×C₄	C₂×#6099	1	2	yes	—
248	C₂×C₄	C₂×#6099	1	1	yes	characteristic
249	C₂×C₄	#88180	1	2	yes	—
250	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	C₂×#6099	1	4	yes	—
255	C₈	#88180	1	2	yes	—
256	C₈	C₂×#6127	1	2	yes	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	#11718587	1	2	yes	—
264	C₂²	C₂²×#6099	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	#11718587	1	2	yes	—
270	C₄	C₂²×#6099	1	1	yes	characteristic
271	C₄	C₂×#88180	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	2	no	—
274	C₂	—	4	1	no	—
275	C₂	—	4	1	no	—
276	C₂	C₂×#11718587	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88180
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2873	C₂
socle	C₂×C₁₃	C₂×#432
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2873	D₁₃
hypocenter	C₁₃	#33427
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ #405 ◁ #2873

Complements

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

N	H ≅ G/N	type
1. #13116652	C₂	semidirect
2. #13116648	C₂	semidirect
3. #13116602	C₂	semidirect
5. #13116441	C₂	semidirect
21. #11718058	C₂²	semidirect
22. #11718055	C₂²	semidirect
229. C₁₃	#33427	semidirect

Maximal subgroups

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

subgroup	index
#13116652	2
#13116648	2
#13116602	2
C₂×#11718145	2
#13116441	2
C₁₃×#2873	2
#33427	13