Subgroups — #16018284

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125350	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718357	C₂	1	1	yes	maximal, characteristic
3	#13117073	C₂	1	4	yes	maximal, complemented
4	#13117068	C₂	1	4	yes	maximal, complemented
5	#13116557	C₂	1	2	yes	maximal, complemented
6	#13116553	C₂	1	2	yes	maximal
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
9	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
10	C₂²×#6095	—	2	1	no	—
11	C₂×#88129	—	2	1	no	—
12	#11719175	—	2	1	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718767	—	2	1	no	—
16	#11718767	C₂²	1	2	yes	—
17	#11718357	—	2	1	no	—
18	#11718357	—	2	1	no	—
19	#11718357	C₂²	1	2	yes	—
20	#11718357	C₂²	1	2	yes	—
21	#11718344	—	2	1	no	—
22	#11718344	C₂²	1	2	yes	complemented
23	#11718343	C₂²	1	2	yes	complemented
24	#11718341	—	2	1	no	—
25	C₁₃×#346	C₂²	1	2	yes	complemented
26	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
27	#11718101	C₂²	1	8	yes	—
28	#11718095	C₂²	1	8	yes	complemented
29	C₂²×D₅₂	—	2	1	no	—
30	C₂²×Dic₂₆	—	2	1	no	—
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×#6103	—	4	2	no	—
35	#88180	—	2	2	no	—
36	#88178	—	4	2	no	—
37	C₂×Dic₅₂	—	2	4	no	—
38	C₂×Dic₅₂	—	2	1	no	—
39	C₂×Dic₅₂	C₂³	1	2	yes	—
40	C₂×D₁₀₄	—	2	4	no	—
41	C₂×D₁₀₄	—	2	1	no	—
42	C₂×D₁₀₄	C₂³	1	2	yes	—
43	C₂×#6095	—	2	2	no	—
44	C₂×#6095	—	2	2	no	—
45	C₂×#6095	—	4	4	no	—
46	#88129	—	2	2	no	—
47	#88129	—	2	2	no	—
48	#88129	—	4	2	no	—
49	#88124	—	2	2	no	—
50	#88123	—	2	2	no	—
51	C₄×Dic₂₆	—	4	2	no	—
52	C₁₃×M₃₂	—	2	4	no	—
53	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
54	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	#88058	—	2	2	no	—
59	#88058	C₂³	1	4	yes	—
60	#88057	—	2	2	no	—
61	#88057	—	2	2	no	—
62	#33706	—	13	1	no	maximal, Sylow
63	C₂³×D₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	D₈	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₁₃	—	4	1	no	—
67	C₂×D₅₂	—	2	2	no	—
68	C₂×D₅₂	—	4	2	no	—
69	C₂×D₅₂	—	4	2	no	—
70	C₂×D₅₂	—	4	1	no	—
71	C₂×Dic₂₆	—	2	2	no	—
72	C₂×Dic₂₆	—	4	2	no	—
73	C₂×Dic₂₆	—	4	2	no	—
74	C₂×Dic₂₆	—	4	2	no	—
75	C₂×Dic₂₆	—	4	1	no	—
76	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
77	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	#41	1	4	yes	—
81	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₄²×C₁₃	D₈	1	2	yes	—
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₃	—	4	2	no	—
86	#6103	—	4	4	no	—
87	#6103	—	4	4	no	—
88	C₄⋊Dic₁₃	—	2	4	no	—
89	C₄⋊Dic₁₃	—	4	2	no	—
90	#6101	—	4	4	no	—
91	C₄×Dic₁₃	—	4	2	no	—
92	Dic₅₂	—	2	4	no	—
93	Dic₅₂	—	4	2	no	—
94	D₁₀₄	—	2	4	no	—
95	D₁₀₄	—	4	2	no	—
96	#6095	—	4	4	no	—
97	#6095	—	4	4	no	—
98	C₁₃×C₁₆	—	2	4	no	—
99	#3160	—	13	1	no	—
100	C₂×#495	—	13	1	no	—
101	#2243	—	13	4	no	—
102	#2242	—	13	4	no	—
103	#2141	—	13	2	no	—
104	#2140	—	13	2	no	—
105	C₄×M₃₂	D₁₃	1	1	yes	characteristic, complemented
106	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₁₃	—	4	2	no	—
108	C₂²×D₁₃	—	8	4	no	—
109	C₂²×D₁₃	—	8	2	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	#124	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₁₃	—	4	4	no	—
118	C₂×Dic₁₃	—	4	2	no	—
119	C₂×Dic₁₃	—	8	2	no	—
120	D₅₂	—	4	2	no	—
121	D₅₂	—	4	2	no	—
122	D₅₂	—	8	2	no	—
123	Dic₂₆	—	4	2	no	—
124	Dic₂₆	—	4	2	no	—
125	Dic₂₆	—	8	4	no	—
126	Dic₂₆	—	8	2	no	—
127	C₈×C₁₃	#124	1	2	yes	—
128	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
129	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
130	C₈×C₁₃	C₂×#41	1	2	yes	—
131	C₂²×Q₁₆	—	13	1	no	—
132	C₂²×SD₁₆	—	26	1	no	—
133	C₂²×D₈	—	13	1	no	—
134	#533	—	26	1	no	—
135	C₂×#124	—	26	1	no	—
136	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
137	#500	—	13	2	no	—
138	#499	—	13	2	no	—
139	#499	—	26	1	no	—
140	#498	—	26	1	no	—
141	#495	—	13	2	no	—
142	#495	—	13	2	no	—
143	#495	—	26	1	no	—
144	#495	—	26	1	no	—
145	#427	—	13	2	no	—
146	#427	—	26	1	no	—
147	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
148	#358	—	13	8	no	—
149	#357	—	13	8	no	—
150	#346	C₂×D₁₃	1	2	yes	complemented
151	C₄×C₁₆	C₂×D₁₃	1	2	yes	complemented
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	#486	1	2	yes	—
154	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₁₃	—	8	2	no	—
156	C₂×D₁₃	—	8	4	no	—
157	C₂×D₁₃	—	16	4	no	—
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	#486	1	2	yes	—
161	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
162	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
163	Dic₁₃	—	8	2	no	—
164	Dic₁₃	—	8	4	no	—
165	C₂²×Q₈	—	26	1	no	—
166	C₂²×D₄	—	26	1	no	—
167	C₂×Q₁₆	—	13	2	no	—
168	C₂×Q₁₆	—	26	1	no	—
169	C₂×Q₁₆	—	26	4	no	—
170	C₂×SD₁₆	—	26	2	no	—
171	C₂×SD₁₆	—	26	2	no	—
172	C₂×SD₁₆	—	52	4	no	—
173	C₂×D₈	—	13	2	no	—
174	C₂×D₈	—	26	1	no	—
175	C₂×D₈	—	26	4	no	—
176	C₂²×C₈	D₅₂	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₄⋊Q₈	—	26	2	no	—
179	#125	—	26	2	no	—
180	#124	—	26	2	no	—
181	#124	—	26	2	no	—
182	#124	—	52	2	no	—
183	C₂²⋊Q₈	—	52	2	no	—
184	C₄×Q₈	—	52	2	no	—
185	#117	—	26	2	no	—
186	C₂×C₂²⋊C₄	—	52	2	no	—
187	C₂×C₄²	D₅₂	1	1	yes	characteristic
188	M₃₂	—	2	4	no	—
189	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
190	#107	—	13	4	no	—
191	#107	—	26	2	no	—
192	#106	—	26	2	no	—
193	#106	—	26	2	no	—
194	C₄×C₈	D₅₂	1	1	yes	characteristic
195	C₄×C₈	D₅₂	1	1	yes	characteristic
196	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
197	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
198	C₂×C₁₃	—	2	2	no	—
199	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
200	C₂×C₁₃	#3414	1	2	yes	—
201	D₁₃	—	16	2	no	—
202	C₂⁴	—	52	1	no	—
203	C₂×Q₈	—	26	2	no	—
204	C₂×Q₈	—	52	2	no	—
205	C₂×Q₈	—	52	1	no	—
206	C₂×Q₈	—	52	2	no	—
207	C₂×Q₈	—	52	2	no	—
208	C₂×D₄	—	26	2	no	—
209	C₂×D₄	—	52	2	no	—
210	C₂×D₄	—	52	2	no	—
211	C₂×D₄	—	52	1	no	—
212	C₂²×C₄	—	52	1	no	—
213	C₂²×C₄	D₁₀₄	1	2	yes	—
214	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
215	Q₁₆	—	26	4	no	—
216	Q₁₆	—	52	2	no	—
217	SD₁₆	—	52	4	no	—
218	SD₁₆	—	52	4	no	—
219	D₈	—	26	4	no	—
220	D₈	—	52	2	no	—
221	C₂×C₈	#6095	1	4	yes	—
222	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
223	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
224	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
225	C₂×C₈	#6127	1	4	yes	—
226	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
227	C₄⋊C₄	—	26	4	no	—
228	C₄⋊C₄	—	52	4	no	—
229	C₄⋊C₄	—	52	2	no	—
230	C₂²⋊C₄	—	52	2	no	—
231	C₂²⋊C₄	—	52	4	no	—
232	C₂²⋊C₄	—	52	4	no	—
233	C₄²	—	52	2	no	—
234	C₄²	D₁₀₄	1	2	yes	—
235	C₄²	C₂×D₅₂	1	1	yes	characteristic
236	C₄²	C₂×D₅₂	1	1	yes	characteristic
237	C₁₆	—	2	4	no	—
238	C₁₃	#33706	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	52	2	no	—
240	C₂³	—	104	4	no	—
241	C₂³	—	104	2	no	—
242	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
243	Q₈	—	52	2	no	—
244	Q₈	—	52	2	no	—
245	Q₈	—	104	4	no	—
246	Q₈	—	104	2	no	—
247	D₄	—	52	2	no	—
248	D₄	—	52	2	no	—
249	D₄	—	104	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	52	4	no	—
252	C₂×C₄	—	52	2	no	—
253	C₂×C₄	—	104	2	no	—
254	C₂×C₄	#88129	1	2	yes	—
255	C₂×C₄	C₂×#6095	1	2	yes	—
256	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
257	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
258	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
260	C₈	#88129	1	2	yes	—
261	C₈	C₂×#6095	1	2	yes	—
262	C₈	C₂×#6095	1	2	yes	—
263	C₈	C₂×#6127	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	104	2	no	—
266	C₂²	—	104	4	no	—
267	C₂²	—	208	4	no	—
268	C₂²	#11718355	1	2	yes	—
269	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	104	2	no	—
273	C₄	—	104	4	no	—
274	C₄	#11718355	1	2	yes	—
275	C₄	C₂×#88129	1	1	yes	characteristic
276	C₄	C₂²×#6095	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	208	2	no	—
279	C₂	C₂×#11718355	1	1	yes	characteristic
280	C₂	#13127972	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₀₄	C₂²
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#33706
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
1. #13125350	C₂	semidirect
3. #13117073	C₂	semidirect
4. #13117068	C₂	semidirect
5. #13116557	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
22. #11718344	C₂²	semidirect
23. #11718343	C₂²	semidirect
25. C₁₃×#346	C₂²	semidirect
26. C₄×C₁₃×C₁₆	C₂²	semidirect
28. #11718095	C₂²	semidirect
105. C₄×M₃₂	D₁₃	semidirect
150. #346	C₂×D₁₃	semidirect
151. C₄×C₁₆	C₂×D₁₃	semidirect
238. C₁₃	#33706	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
C₂×#11718357	2
#13117073	2
#13117068	2
#13116557	2
#13116553	2
C₄×C₁₃×M₃₂	2
#33706	13