Subgroups — #16016290

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88040	C₂	1	1	yes	maximal, characteristic
2	#13123705	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123689	C₂	1	1	yes	maximal, characteristic
4	#13116702	C₂	1	4	yes	maximal, complemented
5	C₄×#88040	C₂	1	4	yes	maximal, complemented
6	#13116620	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116557	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116553	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
11	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
12	#11719226	—	2	1	no	—
13	#11719221	—	2	1	no	—
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#405	C₂²	1	1	yes	characteristic
17	#11718792	—	2	1	no	—
18	#11718781	—	2	1	no	—
19	#11718767	C₂²	1	1	yes	characteristic
20	#11718767	C₂²	1	1	yes	characteristic
21	C₂×#88040	C₄	1	8	yes	complemented
22	C₂×#88040	C₂²	1	2	yes	—
23	C₂×#88040	C₂²	1	2	yes	—
24	#11718377	—	2	2	no	—
25	#11718372	—	2	2	no	—
26	C₄×Dic₅₂	C₂²	1	4	yes	complemented
27	C₄×D₁₀₄	C₂²	1	4	yes	—
28	C₁₃×#346	C₂²	1	2	yes	complemented
29	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
30	#11718101	C₂²	1	4	yes	—
31	#11718097	C₂²	1	2	yes	complemented
32	#11718096	C₂²	1	2	yes	complemented
33	#11718095	C₂²	1	4	yes	complemented
34	C₂²×D₅₂	—	2	1	no	—
35	C₂²×Dic₂₆	—	2	1	no	—
36	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#117	D₄	1	1	yes	characteristic
39	#88180	—	2	1	no	—
40	#88180	—	2	1	no	—
41	C₂×Dic₅₂	—	2	1	no	—
42	C₂×Dic₅₂	—	2	2	no	—
43	C₂×Dic₅₂	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₅₂	C₂³	1	2	yes	—
45	C₂×D₁₀₄	—	2	1	no	—
46	C₂×D₁₀₄	—	2	2	no	—
47	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
48	C₂×D₁₀₄	C₂³	1	2	yes	—
49	#88147	—	4	2	no	—
50	#88141	—	4	2	no	—
51	C₂²⋊C₄×D₁₃	—	4	2	no	—
52	#88131	—	4	2	no	—
53	C₄×D₅₂	—	2	4	no	—
54	C₄×Dic₂₆	—	2	4	no	—
55	C₁₃×M₃₂	—	2	2	no	—
56	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
57	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₁₃×#97	D₄	1	2	yes	—
61	C₄×C₈×C₁₃	C₂³	1	2	yes	—
62	#88061	—	2	4	no	—
63	#88058	C₂³	1	2	yes	—
64	#88058	C₂³	1	2	yes	—
65	#88057	—	2	1	no	—
66	#88057	—	2	1	no	—
67	#88056	—	2	4	no	—
68	#88040	C₂×C₄	1	8	yes	complemented
69	#88040	C₂×C₄	1	8	yes	complemented
70	#33457	—	13	1	no	maximal, Sylow
71	C₂³×D₁₃	—	4	1	no	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₁₃	—	4	1	no	—
74	C₂×D₅₂	—	2	4	no	—
75	C₂×D₅₂	—	2	2	no	—
76	C₂×D₅₂	—	4	1	no	—
77	C₂×D₅₂	—	4	2	no	—
78	C₂×C₄×D₁₃	—	4	4	no	—
79	C₂×Dic₂₆	—	2	4	no	—
80	C₂×Dic₂₆	—	2	2	no	—
81	C₂×Dic₂₆	—	4	1	no	—
82	C₂×Dic₂₆	—	4	2	no	—
83	C₁₃×M₁₆	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	#41	1	2	yes	—
88	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
90	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
91	C₄²×C₁₃	C₂×D₄	1	2	yes	—
92	C₂²⋊Dic₁₃	—	4	1	no	—
93	C₂²⋊Dic₁₃	—	4	1	no	—
94	#6103	—	4	4	no	—
95	C₄⋊Dic₁₃	—	2	2	no	—
96	C₄⋊Dic₁₃	—	2	2	no	—
97	#6101	—	4	4	no	—
98	C₄×Dic₁₃	—	4	4	no	—
99	C₄×Dic₁₃	—	4	1	no	—
100	C₄×Dic₁₃	—	4	1	no	—
101	Dic₅₂	—	2	4	no	—
102	Dic₅₂	—	2	2	no	—
103	Dic₅₂	C₂²×C₄	1	4	yes	—
104	D₁₀₄	—	2	4	no	—
105	D₁₀₄	—	2	2	no	—
106	D₁₀₄	C₂²×C₄	1	4	yes	—
107	C₁₃×C₁₆	—	2	2	no	—
108	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
109	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
110	C₂²×SD₃₂	—	13	1	no	—
111	#2950	—	13	1	no	—
112	#2948	—	13	1	no	—
113	#2175	—	13	4	no	—
114	C₄×SD₃₂	—	13	4	no	—
115	#2158	—	13	1	no	—
116	#2141	—	13	1	no	—
117	#2140	—	13	1	no	—
118	#2108	D₁₃	1	1	yes	characteristic, complemented
119	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
120	C₂²×D₁₃	—	4	2	no	—
121	C₂²×D₁₃	—	4	4	no	—
122	C₂²×D₁₃	—	16	2	no	—
123	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
125	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#135	1	2	yes	—
127	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₁₃	—	4	2	no	—
129	C₂×Dic₁₃	—	4	4	no	—
130	C₂×Dic₁₃	—	4	2	no	—
131	C₂×Dic₁₃	—	4	2	no	—
132	D₅₂	—	2	4	no	—
133	D₅₂	—	4	2	no	—
134	D₅₂	—	4	4	no	—
135	C₄×D₁₃	—	16	4	no	—
136	Dic₂₆	—	2	4	no	—
137	Dic₂₆	—	4	2	no	—
138	Dic₂₆	—	4	4	no	—
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	C₄×D₄	1	2	yes	—
141	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
143	C₂²×Q₁₆	—	13	1	no	—
144	C₂²×D₈	—	13	1	no	—
145	#519	—	26	1	no	—
146	#518	—	26	1	no	—
147	C₂×SD₃₂	—	13	8	no	—
148	C₂×SD₃₂	—	13	2	no	—
149	C₂×SD₃₂	—	13	2	no	—
150	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
151	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
152	#442	—	26	2	no	—
153	#441	—	26	2	no	—
154	C₄×Q₁₆	—	13	4	no	—
155	C₄×D₈	—	13	4	no	—
156	#427	—	13	1	no	—
157	#427	—	13	1	no	—
158	#419	—	26	1	no	—
159	#418	—	26	1	no	—
160	#405	C₂×D₁₃	1	1	yes	characteristic
161	#367	—	13	2	no	—
162	#366	—	13	2	no	—
163	#358	—	13	4	no	—
164	#357	—	13	4	no	—
165	#346	C₂×D₁₃	1	2	yes	complemented
166	C₄×C₁₆	C₂×D₁₃	1	2	yes	complemented
167	C₂²×C₁₃	—	2	2	no	—
168	C₂²×C₁₃	C₄×D₈	1	2	yes	—
169	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
170	C₂×D₁₃	—	4	4	no	—
171	C₂×D₁₃	—	8	4	no	—
172	C₂×D₁₃	—	16	4	no	—
173	C₄×C₁₃	—	2	4	no	—
174	C₄×C₁₃	C₄×D₈	1	2	yes	—
175	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
177	Dic₁₃	—	4	4	no	—
178	Dic₁₃	—	8	2	no	—
179	Dic₁₃	—	8	2	no	—
180	C₂²×Q₈	—	26	1	no	—
181	C₂²×D₄	—	26	1	no	—
182	C₂×Q₁₆	—	13	4	no	—
183	C₂×Q₁₆	—	13	2	no	—
184	C₂×Q₁₆	—	26	1	no	—
185	C₂×Q₁₆	—	26	2	no	—
186	C₂×D₈	—	13	4	no	—
187	C₂×D₈	—	13	2	no	—
188	C₂×D₈	—	26	1	no	—
189	C₂×D₈	—	26	2	no	—
190	C₂×M₁₆	D₅₂	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
192	C₄×Q₈	—	26	4	no	—
193	C₄×Q₈	—	52	2	no	—
194	C₄×D₄	—	26	4	no	—
195	C₄×D₄	—	52	2	no	—
196	#117	—	26	1	no	—
197	#117	—	26	1	no	—
198	#117	—	52	2	no	—
199	#117	D₅₂	1	1	yes	characteristic
200	C₂×C₂²⋊C₄	—	52	2	no	—
201	SD₃₂	—	13	8	no	—
202	SD₃₂	—	13	8	no	—
203	M₃₂	—	2	2	no	—
204	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
205	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
206	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
207	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
208	#107	—	13	2	no	—
209	#107	—	13	2	no	—
210	#106	—	26	1	no	—
211	#106	—	26	1	no	—
212	#103	—	26	4	no	—
213	#102	—	26	4	no	—
214	#97	D₅₂	1	2	yes	—
215	C₄×C₈	C₂²×D₁₃	1	2	yes	—
216	C₂×C₁₃	—	2	2	no	—
217	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
218	C₂×C₁₃	#3416	1	2	yes	—
219	D₁₃	—	8	4	no	—
220	C₂⁴	—	52	1	no	—
221	C₂×Q₈	—	26	4	no	—
222	C₂×Q₈	—	26	2	no	—
223	C₂×Q₈	—	52	1	no	—
224	C₂×Q₈	—	52	2	no	—
225	C₂×D₄	—	26	4	no	—
226	C₂×D₄	—	26	2	no	—
227	C₂×D₄	—	52	1	no	—
228	C₂×D₄	—	52	2	no	—
229	C₂²×C₄	—	52	1	no	—
230	C₂²×C₄	—	52	4	no	—
231	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
232	Q₁₆	—	13	4	no	—
233	Q₁₆	—	26	4	no	—
234	Q₁₆	—	26	2	no	—
235	D₈	—	13	4	no	—
236	D₈	—	26	4	no	—
237	D₈	—	26	2	no	—
238	M₁₆	—	2	2	no	—
239	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
240	C₂×C₈	C₂×D₅₂	1	2	yes	—
241	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
242	C₂×C₈	#6127	1	2	yes	—
243	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
244	C₄⋊C₄	—	26	2	no	—
245	C₄⋊C₄	—	26	2	no	—
246	C₄⋊C₄	—	52	4	no	—
247	C₄⋊C₄	D₁₀₄	1	2	yes	—
248	C₂²⋊C₄	—	52	1	no	—
249	C₂²⋊C₄	—	52	1	no	—
250	C₂²⋊C₄	—	52	4	no	—
251	C₂²⋊C₄	D₁₀₄	1	2	yes	—
252	C₄²	—	52	1	no	—
253	C₄²	—	52	1	no	—
254	C₄²	—	52	4	no	—
255	C₄²	C₂×D₅₂	1	2	yes	—
256	C₁₆	—	2	2	no	—
257	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
258	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
259	C₁₃	#33457	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	52	2	no	—
261	C₂³	—	52	4	no	—
262	C₂³	—	208	2	no	—
263	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
264	Q₈	—	26	4	no	—
265	Q₈	—	52	2	no	—
266	Q₈	—	52	4	no	—
267	D₄	—	26	4	no	—
268	D₄	—	52	2	no	—
269	D₄	—	52	4	no	—
270	C₂×C₄	—	52	2	no	—
271	C₂×C₄	—	52	4	no	—
272	C₂×C₄	—	52	2	no	—
273	C₂×C₄	—	52	2	no	—
274	C₂×C₄	—	208	4	no	—
275	C₂×C₄	C₄×D₅₂	1	2	yes	—
276	C₂×C₄	C₂×D₁₀₄	1	4	yes	—
277	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
278	C₂×C₄	#88158	1	2	yes	—
279	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	C₄×D₅₂	1	2	yes	—
282	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
283	C₈	C₂×#6127	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	52	4	no	—
286	C₂²	—	104	4	no	—
287	C₂²	—	208	4	no	—
288	C₂²	C₄×D₁₀₄	1	2	yes	—
289	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	52	4	no	—
292	C₄	—	104	2	no	—
293	C₄	—	104	2	no	—
294	C₄	C₄×D₁₀₄	1	2	yes	—
295	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
296	C₄	C₂×#88158	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	104	4	no	—
299	C₂	C₂×C₄×D₁₀₄	1	1	yes	characteristic
300	C₂	#13127984	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2108	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₀₄	C₂²
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2108	D₁₃
hypocenter	C₁₃	#33457
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #13123705	C₂	semidirect
4. #13116702	C₂	semidirect
5. C₄×#88040	C₂	semidirect
6. #13116620	C₂	semidirect
7. #13116557	C₂	semidirect
9. C₁₃×#2108	C₂	semidirect
21. C₂×#88040	C₄	semidirect
26. C₄×Dic₅₂	C₂²	semidirect
28. C₁₃×#346	C₂²	semidirect
29. C₄×C₁₃×C₁₆	C₂²	semidirect
31. #11718097	C₂²	semidirect
32. #11718096	C₂²	semidirect
33. #11718095	C₂²	semidirect
43. C₂×Dic₅₂	C₂×C₄	semidirect
56. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
68. #88040	C₂×C₄	semidirect
69. #88040	C₂×C₄	semidirect
118. #2108	D₁₃	semidirect
165. #346	C₂×D₁₃	semidirect
166. C₄×C₁₆	C₂×D₁₃	semidirect
204. C₂×C₁₆	C₄×D₁₃	semidirect
259. C₁₃	#33457	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88040	2
#13123705	2
#13123689	2
#13116702	2
C₄×#88040	2
#13116620	2
#13116557	2
#13116553	2
C₁₃×#2108	2
#33457	13