Subgroups — #16019099

Order 3328.

458 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718344	C₂	1	1	yes	maximal, characteristic
2	#13117119	C₂	1	2	yes	maximal, complemented
3	#13117081	C₂	1	2	yes	maximal, complemented
4	#13117036	C₂	1	2	yes	maximal
5	#13117031	C₂	1	2	yes	maximal
6	#13117000	C₂	1	4	yes	maximal, complemented
7	#13116639	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#533	—	2	1	no	—
10	C₂×#88124	—	2	1	no	—
11	C₁₃×#500	C₂²	1	1	yes	characteristic
12	C₁₃×#494	C₂²	1	1	yes	characteristic
13	C₁₃×#487	—	2	1	no	—
14	C₁₃×#467	—	2	1	no	—
15	C₁₃×#467	C₂²	1	2	yes	—
16	C₄×C₁₃×Q₁₆	—	2	1	no	—
17	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×#88058	C₂²	1	1	yes	characteristic
20	C₂×#88058	C₂²	1	1	yes	characteristic
21	C₂×#88057	—	2	1	no	—
22	#11718344	—	2	1	no	—
23	#11718344	—	2	1	no	—
24	#11718344	C₂²	1	2	yes	—
25	#11718344	C₂²	1	2	yes	—
26	#11718156	C₂²	1	4	yes	—
27	#11718156	C₂²	1	4	yes	—
28	C₂²⋊#6090	C₂²	1	2	yes	—
29	#11718082	C₂²	1	4	yes	—
30	#11718081	C₂²	1	4	yes	—
31	C₄⋊#6090	C₂²	1	2	yes	—
32	#11718053	C₂²	1	2	yes	—
33	C₂×C₁₃×Q₁₆	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×C₄⋊Q₈	—	2	1	no	—
38	C₁₃×C₄⋊Q₈	—	2	1	no	—
39	C₁₃×#125	—	4	1	no	—
40	C₁₃×C₂²⋊Q₈	—	2	2	no	—
41	C₁₃×C₂²⋊Q₈	—	4	1	no	—
42	C₄×Q₈×C₁₃	—	2	2	no	—
43	C₄×Q₈×C₁₃	—	4	1	no	—
44	C₁₃×#117	—	4	1	no	—
45	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
46	C₂×C₄⋊Dic₁₃	—	2	1	no	—
47	C₂×C₄⋊Dic₁₃	—	2	1	no	—
48	C₂×#6101	—	4	1	no	—
49	C₂×#6101	—	4	1	no	—
50	#88124	—	2	2	no	—
51	#88124	—	2	2	no	—
52	#88124	—	4	1	no	—
53	#88124	—	4	1	no	—
54	C₁₃×#107	—	2	1	no	—
55	C₁₃×#107	C₂³	1	2	yes	—
56	C₁₃×#103	—	2	2	no	—
57	C₁₃×#103	—	2	2	no	—
58	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	#88058	—	2	1	no	—
63	#88058	—	2	1	no	—
64	#88058	C₂³	1	2	yes	—
65	#88058	C₂³	1	2	yes	—
66	#88057	—	2	2	no	—
67	#88057	—	2	2	no	—
68	C₂×#6090	C₂³	1	4	yes	—
69	#33802	—	13	1	no	maximal, Sylow
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×Q₈×C₁₃	—	4	1	no	—
72	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×Dic₁₃	—	4	1	no	—
76	C₂²×Dic₁₃	—	4	1	no	—
77	C₁₃×Q₁₆	—	2	4	no	—
78	C₁₃×Q₁₆	—	4	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₄⋊C₄	—	4	1	no	—
87	C₁₃×C₄⋊C₄	—	4	1	no	—
88	C₁₃×C₄⋊C₄	—	4	2	no	—
89	C₁₃×C₄⋊C₄	—	4	2	no	—
90	C₁₃×C₄⋊C₄	—	4	1	no	—
91	C₁₃×C₂²⋊C₄	—	4	2	no	—
92	C₄²×C₁₃	—	4	2	no	—
93	C₄²×C₁₃	D₈	1	1	yes	characteristic
94	C₄²×C₁₃	D₈	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	—	4	1	no	—
100	C₄⋊Dic₁₃	—	4	1	no	—
101	#6101	—	4	2	no	—
102	#6101	—	4	2	no	—
103	#6101	—	4	2	no	—
104	#6101	—	4	2	no	—
105	#6090	—	2	4	no	—
106	#3170	D₁₃	1	1	yes	characteristic
107	C₂×#499	—	13	1	no	—
108	#2255	—	13	2	no	—
109	#2245	—	13	2	no	—
110	#2236	—	13	2	no	—
111	#2235	—	13	2	no	—
112	#2229	—	13	4	no	—
113	#2163	—	13	1	no	—
114	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₁₃	—	4	2	no	—
116	Q₈×C₁₃	—	4	2	no	—
117	Q₈×C₁₃	—	8	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₁₃	—	4	2	no	—
131	C₂×Dic₁₃	—	4	2	no	—
132	C₂×Dic₁₃	—	8	2	no	—
133	C₂×Dic₁₃	—	8	2	no	—
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	#137	1	2	yes	—
137	C₈×C₁₃	2₋¹⁺⁴	1	2	yes	—
138	#533	—	2	1	no	—
139	C₂×#125	—	26	1	no	—
140	#500	C₂×D₁₃	1	1	yes	characteristic
141	#499	—	13	2	no	—
142	#499	—	13	2	no	—
143	#499	—	26	1	no	—
144	#499	—	26	1	no	—
145	#494	C₂×D₁₃	1	1	yes	characteristic
146	#487	—	2	1	no	—
147	#467	—	2	1	no	—
148	#467	C₂×D₁₃	1	2	yes	—
149	C₄×Q₁₆	—	2	1	no	—
150	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
151	C₂×#107	—	13	1	no	—
152	C₂×#107	—	13	1	no	—
153	C₂×#106	—	26	1	no	—
154	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
155	#367	—	13	4	no	—
156	#366	—	13	4	no	—
157	C₄⋊C₁₆	—	13	2	no	—
158	#358	—	13	4	no	—
159	#358	—	13	4	no	—
160	C₂²⋊C₁₆	—	13	2	no	—
161	#346	—	13	2	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	—	2	1	no	—
164	C₂²×C₁₃	#510	1	2	yes	—
165	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	2	1	no	—
169	C₄×C₁₃	—	2	1	no	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	8	2	no	—
172	C₄×C₁₃	—	8	2	no	—
173	C₄×C₁₃	#536	1	1	yes	characteristic
174	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
175	Dic₁₃	—	8	2	no	—
176	Dic₁₃	—	8	2	no	—
177	C₂×Q₁₆	—	2	1	no	—
178	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
179	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
181	C₄⋊Q₈	—	2	1	no	—
182	C₄⋊Q₈	—	2	1	no	—
183	#125	—	4	1	no	—
184	#125	—	26	2	no	—
185	#125	—	26	2	no	—
186	#125	—	52	1	no	—
187	#125	—	52	1	no	—
188	C₂²⋊Q₈	—	2	2	no	—
189	C₂²⋊Q₈	—	4	1	no	—
190	C₄×Q₈	—	2	2	no	—
191	C₄×Q₈	—	4	1	no	—
192	#117	—	4	1	no	—
193	C₂×C₄⋊C₄	—	26	1	no	—
194	C₂×C₄⋊C₄	—	26	1	no	—
195	C₂×C₄⋊C₄	—	52	1	no	—
196	C₂×C₄⋊C₄	—	52	1	no	—
197	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
198	C₂×C₁₆	—	13	4	no	—
199	#107	—	2	1	no	—
200	#107	—	13	2	no	—
201	#107	—	13	2	no	—
202	#107	—	26	1	no	—
203	#107	—	26	1	no	—
204	#107	C₂²×D₁₃	1	2	yes	—
205	#106	—	26	2	no	—
206	#106	—	26	2	no	—
207	#103	—	2	2	no	—
208	#103	—	2	2	no	—
209	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
210	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
213	C₂×C₁₃	—	2	2	no	—
214	C₂×C₁₃	#3226	1	1	yes	characteristic
215	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
216	C₂×C₁₃	#3414	1	1	yes	characteristic
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	4	1	no	—
219	C₂²×C₄	—	52	1	no	—
220	C₂²×C₄	—	52	1	no	—
221	C₂²×C₄	#6104	1	1	yes	characteristic
222	C₂²×C₄	#6104	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
224	Q₁₆	—	2	4	no	—
225	Q₁₆	—	4	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	26	2	no	—
239	C₄⋊C₄	—	26	2	no	—
240	C₄⋊C₄	—	52	1	no	—
241	C₄⋊C₄	—	52	1	no	—
242	C₄⋊C₄	—	52	2	no	—
243	C₄⋊C₄	—	52	2	no	—
244	C₄⋊C₄	—	52	2	no	—
245	C₄⋊C₄	—	52	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	#6104	1	1	yes	characteristic
249	C₄²	#6104	1	1	yes	characteristic
250	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
252	C₁₆	—	26	4	no	—
253	C₁₃	#33802	1	1	yes	characteristic, complemented, Sylow
254	C₂³	C₂×#6104	1	1	yes	characteristic
255	Q₈	—	4	2	no	—
256	Q₈	—	4	2	no	—
257	Q₈	—	8	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	52	2	no	—
265	C₂×C₄	—	52	2	no	—
266	C₂×C₄	—	104	2	no	—
267	C₂×C₄	—	104	2	no	—
268	C₂×C₄	C₂×#6104	1	1	yes	characteristic
269	C₂×C₄	C₂×#6104	1	1	yes	characteristic
270	C₂×C₄	C₂×#6104	1	1	yes	characteristic
271	C₂×C₄	C₂×#6104	1	1	yes	characteristic
272	C₂×C₄	C₂×#6104	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	#88205	1	2	yes	—
277	C₈	#88257	1	2	yes	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	#11718840	1	2	yes	—
281	C₂²	C₂²×#6104	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	8	2	no	—
288	C₄	—	8	2	no	—
289	C₄	—	104	2	no	—
290	C₄	—	104	2	no	—
291	C₄	C₂×#88205	1	1	yes	characteristic
292	C₄	#11719530	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	#13125954	1	1	yes	characteristic
295	C₂	C₂×#11718840	1	1	yes	characteristic
296	C₂	#13127970	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#33802
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13117119	C₂	semidirect
3. #13117081	C₂	semidirect
6. #13117000	C₂	semidirect
253. C₁₃	#33802	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718344	2
#13117119	2
#13117081	2
#13117036	2
#13117031	2
#13117000	2
#13116639	2
C₁₃×#3170	2
#33802	13