Subgroups — #16021010

Order 3328.

504 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127825	C₂	1	2	yes	maximal, complemented
2	#13117107	C₂	1	2	yes	maximal, complemented
3	#13117097	C₂	1	1	yes	maximal, characteristic
4	#13116899	C₂	1	2	yes	maximal, complemented
5	#13116892	C₂	1	2	yes	maximal
6	#13116887	C₂	1	2	yes	maximal, complemented
7	#13116702	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719289	—	2	2	no	—
11	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₀₄	C₂²	1	2	yes	—
15	C₂×#88040	C₂²	1	2	yes	—
16	C₂×#88040	C₂²	1	2	yes	—
17	C₂×D₂₀₈	C₂²	1	1	yes	characteristic
18	#11718506	—	2	2	no	—
19	#11718506	—	2	2	no	—
20	#11718502	—	2	2	no	—
21	#11718360	C₂²	1	2	yes	—
22	C₄×Dic₅₂	C₂²	1	2	yes	—
23	C₄×Dic₅₂	C₂²	1	2	yes	—
24	#11718357	C₂²	1	2	yes	—
25	#11718357	C₂²	1	2	yes	—
26	C₄×D₁₀₄	C₂²	1	2	yes	—
27	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
28	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₁₃×#346	C₂²	1	2	yes	—
30	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
31	#11718101	C₂²	1	2	yes	—
32	#11718096	C₂²	1	1	yes	characteristic
33	#11718095	C₂²	1	2	yes	—
34	#11718095	C₂²	1	2	yes	—
35	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
36	#88200	—	4	2	no	—
37	C₂×Dic₅₂	—	2	2	no	—
38	C₂×Dic₅₂	C₂³	1	2	yes	—
39	C₂×Dic₅₂	C₂³	1	2	yes	—
40	C₂×D₁₀₄	C₂³	1	2	yes	—
41	C₂×#6095	—	2	2	no	—
42	C₂×#6095	—	2	2	no	—
43	#88152	—	4	2	no	—
44	#88152	—	4	2	no	—
45	#88146	—	4	2	no	—
46	#88129	—	2	2	no	—
47	#88129	—	2	2	no	—
48	C₄×D₅₂	—	2	2	no	—
49	#88123	—	2	2	no	—
50	C₄×Dic₂₆	—	2	2	no	—
51	C₄×Dic₂₆	—	2	2	no	—
52	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
54	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
55	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
56	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	2	yes	—
59	#88061	—	2	2	no	—
60	#88058	C₂³	1	2	yes	—
61	#88056	—	2	2	no	—
62	#88056	—	2	2	no	—
63	Dic₁₀₄	—	2	2	no	—
64	Dic₁₀₄	—	4	1	no	—
65	Dic₁₀₄	—	4	2	no	—
66	#88040	—	2	4	no	—
67	#88040	—	4	2	no	—
68	D₂₀₈	—	4	1	no	—
69	#33969	—	13	1	no	maximal, Sylow
70	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
71	C₂×D₅₂	—	2	2	no	—
72	C₂×C₄×D₁₃	—	4	2	no	—
73	C₂×Dic₂₆	—	2	2	no	—
74	C₂×Dic₂₆	—	2	2	no	—
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
79	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
80	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₃	C₂×D₄	1	2	yes	—
82	#6103	—	4	2	no	—
83	#6103	—	4	2	no	—
84	#6103	—	4	2	no	—
85	#6103	—	4	2	no	—
86	#6103	—	4	2	no	—
87	C₄⋊Dic₁₃	—	2	2	no	—
88	C₄⋊Dic₁₃	—	4	2	no	—
89	C₄⋊Dic₁₃	—	4	2	no	—
90	#6101	—	4	2	no	—
91	#6101	—	4	2	no	—
92	C₄×Dic₁₃	—	4	2	no	—
93	C₄×Dic₁₃	—	4	2	no	—
94	Dic₅₂	—	2	2	no	—
95	Dic₅₂	—	2	2	no	—
96	Dic₅₂	—	2	2	no	—
97	Dic₅₂	—	2	2	no	—
98	Dic₅₂	—	8	2	no	—
99	D₁₀₄	—	2	2	no	—
100	D₁₀₄	—	2	2	no	—
101	#6095	—	8	2	no	—
102	#6095	—	8	2	no	—
103	C₁₃×C₁₆	—	2	1	no	—
104	C₁₃×C₁₆	—	2	2	no	—
105	C₁₃×C₁₆	#41	1	2	yes	—
106	#3393	—	13	2	no	—
107	#2251	—	13	2	no	—
108	#2249	—	13	1	no	—
109	#2211	—	13	2	no	—
110	#2210	—	13	2	no	—
111	#2209	—	13	2	no	—
112	#2183	D₁₃	1	1	yes	characteristic, complemented
113	#2175	—	13	2	no	—
114	C₄×Q₃₂	—	13	1	no	—
115	C₂²×D₁₃	—	4	2	no	—
116	Q₈×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃	—	4	2	no	—
123	C₂×Dic₁₃	—	4	2	no	—
124	C₂×Dic₁₃	—	4	2	no	—
125	D₅₂	—	4	2	no	—
126	D₅₂	—	4	2	no	—
127	C₄×D₁₃	—	8	4	no	—
128	Dic₂₆	—	4	2	no	—
129	Dic₂₆	—	4	2	no	—
130	Dic₂₆	—	4	2	no	—
131	Dic₂₆	—	4	2	no	—
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	—	2	2	no	—
134	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
136	#556	—	26	2	no	—
137	C₂×Q₃₂	—	13	1	no	—
138	C₂×Q₃₂	—	13	1	no	—
139	C₂×Q₃₂	—	13	2	no	—
140	C₂×SD₃₂	—	13	2	no	—
141	C₂×SD₃₂	—	13	2	no	—
142	C₂×D₁₆	—	13	1	no	—
143	#495	—	13	2	no	—
144	#495	—	13	2	no	—
145	#494	—	13	2	no	—
146	#464	—	26	2	no	—
147	#464	—	26	2	no	—
148	#461	—	26	2	no	—
149	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
150	C₄×Q₁₆	—	13	2	no	—
151	C₄×Q₁₆	—	13	2	no	—
152	C₄×D₈	—	13	2	no	—
153	#366	—	13	1	no	—
154	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
155	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
156	#358	—	13	2	no	—
157	#358	—	13	2	no	—
158	#357	—	13	2	no	—
159	#346	C₂×D₁₃	1	2	yes	—
160	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
161	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₁₃	—	8	2	no	—
163	C₂×D₁₃	—	8	2	no	—
164	C₂×D₁₃	—	8	2	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	#550	1	1	yes	characteristic
170	C₄×C₁₃	#576	1	1	yes	characteristic
171	Dic₁₃	—	8	2	no	—
172	Dic₁₃	—	8	2	no	—
173	Dic₁₃	—	8	2	no	—
174	C₂×Q₁₆	—	13	2	no	—
175	C₂×Q₁₆	—	13	2	no	—
176	C₂×Q₁₆	—	26	2	no	—
177	C₂×SD₁₆	—	26	2	no	—
178	C₂×SD₁₆	—	26	2	no	—
179	C₂×D₈	—	13	2	no	—
180	C₄⋊Q₈	—	26	2	no	—
181	#126	—	52	2	no	—
182	#126	—	52	2	no	—
183	#124	—	26	2	no	—
184	#124	—	26	2	no	—
185	C₂²⋊Q₈	—	52	2	no	—
186	C₄×Q₈	—	26	2	no	—
187	C₄×Q₈	—	26	2	no	—
188	C₄×Q₈	D₅₂	1	1	yes	characteristic
189	C₄×D₄	—	26	2	no	—
190	#117	—	52	2	no	—
191	Q₃₂	—	26	2	no	—
192	Q₃₂	—	52	1	no	—
193	Q₃₂	—	52	2	no	—
194	SD₃₂	—	26	4	no	—
195	SD₃₂	—	52	2	no	—
196	D₁₆	—	52	1	no	—
197	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
198	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
199	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
200	#107	—	13	2	no	—
201	C₄⋊C₈	D₅₂	1	2	yes	—
202	C₄⋊C₈	D₅₂	1	1	yes	characteristic
203	#103	—	26	2	no	—
204	#103	—	26	2	no	—
205	#102	—	26	2	no	—
206	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₁₃	1	2	yes	—
208	C₂×C₁₃	#3391	1	1	yes	characteristic
209	C₂×C₁₃	#3414	1	1	yes	characteristic
210	C₂×C₁₃	#3417	1	1	yes	characteristic
211	D₁₃	—	16	2	no	—
212	C₂×Q₈	—	26	2	no	—
213	C₂×Q₈	—	26	2	no	—
214	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
215	C₂×D₄	—	26	2	no	—
216	C₂²×C₄	—	52	2	no	—
217	Q₁₆	—	26	2	no	—
218	Q₁₆	—	26	2	no	—
219	Q₁₆	—	26	2	no	—
220	Q₁₆	—	26	2	no	—
221	Q₁₆	—	104	2	no	—
222	SD₁₆	—	104	2	no	—
223	SD₁₆	—	104	2	no	—
224	D₈	—	26	2	no	—
225	D₈	—	26	2	no	—
226	C₂×C₈	C₂×D₅₂	1	2	yes	—
227	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
228	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
229	C₄⋊C₄	—	26	2	no	—
230	C₄⋊C₄	—	52	2	no	—
231	C₄⋊C₄	—	52	2	no	—
232	C₄⋊C₄	—	52	2	no	—
233	C₄⋊C₄	—	52	2	no	—
234	C₄⋊C₄	D₁₀₄	1	2	yes	—
235	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
236	C₂²⋊C₄	—	52	2	no	—
237	C₂²⋊C₄	—	52	2	no	—
238	C₂²⋊C₄	—	52	2	no	—
239	C₂²⋊C₄	—	52	2	no	—
240	C₂²⋊C₄	—	52	2	no	—
241	C₄²	—	52	2	no	—
242	C₄²	—	52	2	no	—
243	C₄²	C₂×D₅₂	1	2	yes	—
244	C₄²	C₂×D₅₂	1	1	yes	characteristic
245	C₁₆	—	2	1	no	—
246	C₁₆	—	2	2	no	—
247	C₁₆	#6131	1	2	yes	—
248	C₁₃	#33969	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	52	2	no	—
250	Q₈	—	4	1	no	—
251	Q₈	—	52	2	no	—
252	Q₈	—	52	2	no	—
253	Q₈	—	52	2	no	—
254	Q₈	—	52	2	no	—
255	D₄	—	52	2	no	—
256	D₄	—	52	2	no	—
257	C₂×C₄	—	52	2	no	—
258	C₂×C₄	—	52	2	no	—
259	C₂×C₄	—	52	2	no	—
260	C₂×C₄	—	104	4	no	—
261	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
262	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
263	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
264	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
265	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	2	no	—
268	C₈	C₂×#6131	1	1	yes	characteristic
269	C₈	#88256	1	1	yes	characteristic
270	C₂²	—	104	2	no	—
271	C₂²	—	104	2	no	—
272	C₂²	—	104	2	no	—
273	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	104	2	no	—
279	C₄	—	104	2	no	—
280	C₄	—	104	2	no	—
281	C₄	#11719282	1	1	yes	characteristic
282	C₄	#11719457	1	1	yes	characteristic
283	C₂	—	208	2	no	—
284	C₂	#13127811	1	1	yes	characteristic
285	C₂	#13127972	1	1	yes	characteristic
286	C₂	#13127989	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2183	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×D₂₀₈	C₂²
hypercenter	#2183	D₁₃
hypocenter	C₁₃	#33969
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
1. #13127825	C₂	semidirect
2. #13117107	C₂	semidirect
4. #13116899	C₂	semidirect
6. #13116887	C₂	semidirect
7. #13116702	C₂	semidirect
8. C₄×Dic₁₀₄	C₂	semidirect
9. C₁₃×#2183	C₂	semidirect
112. #2183	D₁₃	semidirect
248. C₁₃	#33969	semidirect

Maximal subgroups

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

subgroup	index
#13127825	2
#13117107	2
#13117097	2
#13116899	2
#13116892	2
#13116887	2
#13116702	2
C₄×Dic₁₀₄	2
C₁₃×#2183	2
#33969	13