Subgroups — #16016286

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×#88067	C₂	1	1	yes	maximal, characteristic
2	#13123705	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116700	C₂	1	4	yes	maximal, complemented
4	C₄×#88067	C₂	1	4	yes	maximal, complemented
5	#13116618	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116555	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116552	C₂	1	1	yes	maximal, characteristic
8	#13116433	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2948	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
11	C₁₃×#518	—	2	1	no	—
12	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
13	#11719221	—	2	1	no	—
14	C₁₃×#442	—	2	2	no	—
15	C₄×C₁₃×D₈	C₂²	1	4	yes	—
16	C₁₃×#427	C₂²	1	1	yes	characteristic
17	C₁₃×#418	—	2	1	no	—
18	C₁₃×#405	C₂²	1	1	yes	characteristic
19	C₂×#88067	C₄	1	8	yes	complemented
20	C₂×#88067	C₂²	1	2	yes	—
21	C₂×#88067	C₂²	1	2	yes	—
22	#11718781	—	2	1	no	—
23	#11718767	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	C₂²×#6090	C₂²	1	1	yes	characteristic
26	#11718377	—	2	2	no	—
27	C₄×Dic₅₂	C₂²	1	4	yes	complemented
28	#11718154	C₂²	1	4	yes	—
29	#11718084	C₂²	1	4	yes	complemented
30	#11718082	C₂²	1	2	yes	complemented
31	#11718081	C₂²	1	2	yes	complemented
32	#11718053	C₂²	1	2	yes	complemented
33	C₄×#6090	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₁₃	—	2	1	no	—
35	C₂²×Dic₂₆	—	2	1	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	—	2	2	no	—
38	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
39	C₂×C₁₃×D₈	C₂³	1	2	yes	—
40	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₄×D₄×C₁₃	—	2	4	no	—
43	C₄×D₄×C₁₃	—	4	2	no	—
44	C₁₃×#117	—	2	1	no	—
45	C₁₃×#117	D₄	1	1	yes	characteristic
46	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
47	#88180	—	2	1	no	—
48	C₂×Dic₅₂	—	2	1	no	—
49	C₂×Dic₅₂	—	2	2	no	—
50	C₂×Dic₅₂	C₂×C₄	1	4	yes	complemented
51	C₂×Dic₅₂	C₂³	1	2	yes	—
52	#88141	—	4	2	no	—
53	#88131	—	4	2	no	—
54	C₄×Dic₂₆	—	2	4	no	—
55	C₁₃×#107	C₂³	1	2	yes	—
56	C₁₃×#106	—	2	1	no	—
57	C₁₃×#102	—	2	4	no	—
58	C₁₃×#97	D₄	1	2	yes	—
59	C₄×C₈×C₁₃	C₂³	1	2	yes	—
60	#88067	C₂×C₄	1	8	yes	complemented
61	#88067	C₂×C₄	1	8	yes	complemented
62	#88058	C₂³	1	2	yes	—
63	#88057	—	2	1	no	—
64	#88056	—	2	4	no	—
65	#88052	—	2	2	no	—
66	C₂×#6090	C₂×C₄	1	4	yes	complemented
67	C₂×#6090	C₂³	1	2	yes	—
68	C₂×#6090	C₂³	1	1	yes	characteristic
69	C₂×#6090	C₂³	1	1	yes	characteristic
70	#33457	—	13	1	no	maximal, Sylow
71	C₂⁴×C₁₃	—	4	1	no	—
72	C₂×D₄×C₁₃	—	2	4	no	—
73	C₂×D₄×C₁₃	—	2	2	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂×D₄×C₁₃	—	4	2	no	—
76	C₂²×C₄×C₁₃	—	4	4	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	—	4	1	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×D₈	—	2	4	no	—
84	C₁₃×D₈	—	2	2	no	—
85	C₁₃×D₈	C₂²×C₄	1	4	yes	—
86	C₁₃×M₁₆	—	2	2	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	#41	1	2	yes	—
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
94	C₁₃×C₂²⋊C₄	—	4	1	no	—
95	C₁₃×C₂²⋊C₄	—	4	4	no	—
96	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
97	C₄²×C₁₃	—	4	1	no	—
98	C₄²×C₁₃	C₂×D₄	1	2	yes	—
99	C₂²⋊Dic₁₃	—	4	1	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	#6101	—	4	4	no	—
102	C₄×Dic₁₃	—	4	4	no	—
103	C₄×Dic₁₃	—	4	1	no	—
104	Dic₅₂	—	2	4	no	—
105	Dic₅₂	—	2	2	no	—
106	Dic₅₂	C₂²×C₄	1	4	yes	—
107	#6090	—	2	2	no	—
108	#6090	C₂²×C₄	1	2	yes	—
109	#6090	C₂²×C₄	1	2	yes	—
110	C₂²×SD₃₂	—	13	1	no	—
111	#2950	—	13	1	no	—
112	#2948	D₁₃	1	1	yes	characteristic
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	—	13	1	no	—
119	C₂³×C₁₃	—	4	2	no	—
120	C₂³×C₁₃	—	4	4	no	—
121	C₂³×C₁₃	—	16	2	no	—
122	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
123	D₄×C₁₃	—	2	4	no	—
124	D₄×C₁₃	—	4	2	no	—
125	D₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	16	4	no	—
128	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#135	1	2	yes	—
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₁₃	—	4	2	no	—
134	C₂×Dic₁₃	—	4	4	no	—
135	C₂×Dic₁₃	—	4	2	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	C₂×D₁₃	1	1	yes	characteristic
145	#519	—	26	1	no	—
146	#518	—	2	1	no	—
147	C₂×SD₃₂	—	13	8	no	—
148	C₂×SD₃₂	—	13	2	no	—
149	C₂×SD₃₂	—	13	2	no	—
150	C₂×M₃₂	—	13	1	no	—
151	C₂²×C₁₆	—	13	1	no	—
152	#442	—	2	2	no	—
153	#441	—	26	2	no	—
154	C₄×Q₁₆	—	13	4	no	—
155	C₄×D₈	C₂×D₁₃	1	4	yes	—
156	#427	—	13	1	no	—
157	#427	C₂×D₁₃	1	1	yes	characteristic
158	#419	—	26	1	no	—
159	#418	—	2	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	—	13	2	no	—
166	C₄×C₁₆	—	13	2	no	—
167	C₂²×C₁₃	—	2	2	no	—
168	C₂²×C₁₃	—	4	4	no	—
169	C₂²×C₁₃	—	8	4	no	—
170	C₂²×C₁₃	—	16	4	no	—
171	C₂²×C₁₃	C₄×D₈	1	2	yes	—
172	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
173	C₄×C₁₃	—	2	4	no	—
174	C₄×C₁₃	—	8	2	no	—
175	C₄×C₁₃	C₄×D₈	1	2	yes	—
176	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
177	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
178	Dic₁₃	—	4	4	no	—
179	Dic₁₃	—	8	2	no	—
180	C₂²×Q₈	—	26	1	no	—
181	C₂²×D₄	—	2	1	no	—
182	C₂×Q₁₆	—	13	4	no	—
183	C₂×Q₁₆	—	13	2	no	—
184	C₂×Q₁₆	—	26	2	no	—
185	C₂×Q₁₆	—	26	1	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	C₄×D₁₃	1	4	yes	—
189	C₂×D₈	C₂²×D₁₃	1	2	yes	—
190	C₂×M₁₆	C₂²⋊D₁₃	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	—	2	4	no	—
195	C₄×D₄	—	4	2	no	—
196	#117	—	2	1	no	—
197	#117	—	26	1	no	—
198	#117	—	52	2	no	—
199	#117	C₂²⋊D₁₃	1	1	yes	characteristic
200	C₂×C₂²⋊C₄	—	4	2	no	—
201	SD₃₂	—	13	8	no	—
202	SD₃₂	—	13	8	no	—
203	M₃₂	—	26	2	no	—
204	C₂×C₁₆	—	13	4	no	—
205	C₂×C₁₆	—	13	1	no	—
206	C₂×C₁₆	—	13	1	no	—
207	C₂×C₁₆	—	13	2	no	—
208	#107	—	13	2	no	—
209	#107	C₂²×D₁₃	1	2	yes	—
210	#106	—	2	1	no	—
211	#106	—	26	1	no	—
212	#103	—	26	4	no	—
213	#102	—	2	4	no	—
214	#97	C₂²⋊D₁₃	1	2	yes	—
215	C₄×C₈	C₂²×D₁₃	1	2	yes	—
216	C₂×C₁₃	—	2	2	no	—
217	C₂×C₁₃	—	8	4	no	—
218	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
219	C₂×C₁₃	#3416	1	2	yes	—
220	C₂⁴	—	4	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	—	2	4	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	4	1	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	52	1	no	—
231	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
232	Q₁₆	—	13	4	no	—
233	Q₁₆	—	26	4	no	—
234	Q₁₆	—	26	2	no	—
235	D₈	—	2	4	no	—
236	D₈	—	2	2	no	—
237	D₈	C₂×C₄×D₁₃	1	4	yes	—
238	M₁₆	—	2	2	no	—
239	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
240	C₂×C₈	#6127	1	2	yes	—
241	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	26	2	no	—
246	C₄⋊C₄	—	52	4	no	—
247	C₄⋊C₄	#6104	1	2	yes	—
248	C₂²⋊C₄	—	4	1	no	—
249	C₂²⋊C₄	—	4	4	no	—
250	C₂²⋊C₄	—	52	1	no	—
251	C₂²⋊C₄	#6104	1	2	yes	—
252	C₄²	—	4	1	no	—
253	C₄²	—	52	1	no	—
254	C₄²	—	52	4	no	—
255	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
256	C₁₆	—	13	2	no	—
257	C₁₆	—	13	2	no	—
258	C₁₆	—	26	2	no	—
259	C₁₃	#33457	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	4	2	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	16	2	no	—
263	C₂³	C₂×#6104	1	1	yes	characteristic
264	Q₈	—	26	4	no	—
265	Q₈	—	52	4	no	—
266	Q₈	—	52	2	no	—
267	D₄	—	2	4	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	16	4	no	—
272	C₂×C₄	—	52	4	no	—
273	C₂×C₄	—	52	2	no	—
274	C₂×C₄	—	52	2	no	—
275	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
276	C₂×C₄	C₂×#6104	1	4	yes	—
277	C₂×C₄	C₂×#6104	1	1	yes	characteristic
278	C₂×C₄	#88204	1	2	yes	—
279	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	C₄×C₂²⋊D₁₃	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	—	4	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	16	4	no	—
288	C₂²	C₄×#6104	1	2	yes	—
289	C₂²	C₂²×#6104	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	8	2	no	—
292	C₄	—	52	4	no	—
293	C₄	—	104	2	no	—
294	C₄	C₄×#6104	1	2	yes	—
295	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
296	C₄	C₂×#88204	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	8	4	no	—
299	C₂	C₂×C₄×#6104	1	1	yes	characteristic
300	C₂	#13127983	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2948	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×C₁₃×D₈	C₂²
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2948	D₁₃
hypocenter	C₁₃	#33457
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2948

Complements

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

N	H ≅ G/N	type
2. #13123705	C₂	semidirect
3. #13116700	C₂	semidirect
4. C₄×#88067	C₂	semidirect
5. #13116618	C₂	semidirect
6. #13116555	C₂	semidirect
8. #13116433	C₂	semidirect
19. C₂×#88067	C₄	semidirect
27. C₄×Dic₅₂	C₂²	semidirect
29. #11718084	C₂²	semidirect
30. #11718082	C₂²	semidirect
31. #11718081	C₂²	semidirect
32. #11718053	C₂²	semidirect
33. C₄×#6090	C₂²	semidirect
50. C₂×Dic₅₂	C₂×C₄	semidirect
60. #88067	C₂×C₄	semidirect
61. #88067	C₂×C₄	semidirect
66. C₂×#6090	C₂×C₄	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×#88067	2
#13123705	2
#13116700	2
C₄×#88067	2
#13116618	2
#13116555	2
#13116552	2
#13116433	2
C₁₃×#2948	2
#33457	13