Subgroups — #16019338

Order 3328.

488 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125486	C₂	1	1	yes	maximal, characteristic
2	#13117000	C₂	1	2	yes	maximal, complemented
3	#13116990	C₂	1	2	yes	maximal, complemented
4	#13116951	C₂	1	2	yes	maximal, complemented
5	#13116934	C₂	1	2	yes	maximal, complemented
6	#13116878	C₂	1	2	yes	maximal, complemented
7	#13116876	C₂	1	2	yes	maximal, complemented
8	#13116595	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3177	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#541	—	2	1	no	—
11	#11719240	—	2	1	no	—
12	C₁₃×#499	C₂²	1	1	yes	characteristic
13	C₁₃×#494	C₂²	1	1	yes	characteristic
14	C₁₃×#491	—	2	1	no	—
15	C₁₃×#470	—	2	1	no	—
16	C₁₃×#467	C₂²	1	2	yes	—
17	C₁₃×#441	—	2	1	no	—
18	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
19	C₁₃×#405	C₂²	1	1	yes	characteristic
20	C₂×#88068	C₂²	1	2	yes	—
21	C₂×#88068	C₂²	1	2	yes	—
22	#11718796	—	2	1	no	—
23	#11718776	C₂²	1	2	yes	—
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	C₂²×#6090	C₂²	1	1	yes	characteristic
26	#11718374	—	2	1	no	—
27	#11718372	—	2	1	no	—
28	#11718356	C₂²	1	1	yes	characteristic
29	C₄×D₁₀₄	C₂²	1	2	yes	—
30	#11718344	C₂²	1	1	yes	characteristic
31	#11718156	C₂²	1	2	yes	—
32	#11718156	C₂²	1	2	yes	—
33	#11718083	C₂²	1	2	yes	—
34	#11718083	C₂²	1	2	yes	—
35	#11718082	C₂²	1	2	yes	—
36	#11718081	C₂²	1	2	yes	—
37	C₄⋊#6090	C₂²	1	2	yes	—
38	C₄⋊#6090	C₂²	1	2	yes	—
39	C₂×C₁₃×Q₁₆	—	2	1	no	—
40	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
41	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×#126	—	4	2	no	—
45	C₁₃×#125	—	2	1	no	—
46	C₁₃×#125	—	4	2	no	—
47	C₁₃×C₂²⋊Q₈	—	2	2	no	—
48	C₄×Q₈×C₁₃	—	2	2	no	—
49	C₁₃×#117	D₄	1	1	yes	characteristic
50	#88184	—	2	2	no	—
51	C₂×D₁₀₄	—	2	1	no	—
52	C₂×D₁₀₄	C₂³	1	2	yes	—
53	#88148	—	4	2	no	—
54	#88138	—	4	2	no	—
55	#88128	—	2	1	no	—
56	C₄×D₅₂	—	2	2	no	—
57	#88124	—	2	1	no	—
58	C₁₃×#107	C₂³	1	2	yes	—
59	C₁₃×#106	—	2	1	no	—
60	C₁₃×#103	—	2	2	no	—
61	C₁₃×#103	—	2	2	no	—
62	C₁₃×#97	D₄	1	1	yes	characteristic
63	C₁₃×#97	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	#88068	—	4	2	no	—
67	#88068	—	4	2	no	—
68	#88061	—	2	2	no	—
69	#88061	—	2	2	no	—
70	#88058	C₂³	1	2	yes	—
71	#88057	—	2	1	no	—
72	#88052	—	4	1	no	—
73	C₂×#6090	—	4	1	no	—
74	C₂×#6090	C₂³	1	2	yes	—
75	C₂×#6090	C₂³	1	2	yes	—
76	#33826	—	13	1	no	maximal, Sylow
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₁₃	—	4	2	no	—
80	C₂×D₅₂	—	2	2	no	—
81	C₂×D₅₂	—	4	2	no	—
82	C₂×C₄×D₁₃	—	4	2	no	—
83	C₁₃×Q₁₆	—	2	2	no	—
84	C₁₃×Q₁₆	—	2	2	no	—
85	C₁₃×Q₁₆	—	4	2	no	—
86	C₁₃×M₁₆	—	4	1	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	—	4	2	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
99	C₁₃×C₂²⋊C₄	—	4	2	no	—
100	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
101	C₄²×C₁₃	—	4	2	no	—
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	#6103	—	4	2	no	—
105	#6103	—	4	2	no	—
106	C₄⋊Dic₁₃	—	2	2	no	—
107	#6101	—	4	2	no	—
108	D₁₀₄	—	2	2	no	—
109	D₁₀₄	—	2	2	no	—
110	D₁₀₄	—	4	2	no	—
111	#6090	—	2	2	no	—
112	#6090	—	2	2	no	—
113	#3177	D₁₃	1	1	yes	characteristic, complemented
114	#3176	—	13	1	no	—
115	#2229	—	13	2	no	—
116	#2228	—	13	2	no	—
117	#2220	—	13	2	no	—
118	#2217	—	13	2	no	—
119	#2208	—	13	2	no	—
120	#2207	—	13	2	no	—
121	#2151	—	13	1	no	—
122	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₁₃	—	4	2	no	—
124	Q₈×C₁₃	—	4	2	no	—
125	Q₈×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₁₃	—	16	2	no	—
135	C₂×Dic₁₃	—	4	2	no	—
136	D₅₂	—	4	2	no	—
137	D₅₂	—	4	2	no	—
138	D₅₂	—	16	2	no	—
139	C₄×D₁₃	—	16	2	no	—
140	C₈×C₁₃	—	2	1	no	—
141	C₈×C₁₃	—	2	1	no	—
142	C₈×C₁₃	—	2	1	no	—
143	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
145	#541	—	2	1	no	—
146	#540	—	26	1	no	—
147	C₂×SD₃₂	—	13	2	no	—
148	C₂×SD₃₂	—	13	2	no	—
149	C₂×M₃₂	—	13	1	no	—
150	C₂²×C₁₆	—	13	1	no	—
151	#499	—	13	1	no	—
152	#499	C₂×D₁₃	1	1	yes	characteristic
153	#494	C₂×D₁₃	1	1	yes	characteristic
154	#493	—	13	1	no	—
155	#491	—	2	1	no	—
156	#490	—	26	1	no	—
157	#470	—	2	1	no	—
158	#469	—	26	1	no	—
159	#467	C₂×D₁₃	1	2	yes	—
160	#466	—	13	2	no	—
161	#442	—	26	1	no	—
162	#441	—	2	1	no	—
163	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
164	C₄×D₈	—	13	2	no	—
165	#405	C₂×D₁₃	1	1	yes	characteristic
166	#367	—	13	2	no	—
167	#366	—	13	2	no	—
168	C₄⋊C₁₆	—	13	2	no	—
169	C₄⋊C₁₆	—	13	2	no	—
170	#358	—	13	2	no	—
171	#358	—	13	2	no	—
172	#357	—	13	2	no	—
173	#357	—	13	2	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
177	C₂×D₁₃	—	8	2	no	—
178	C₂×D₁₃	—	8	2	no	—
179	C₂×D₁₃	—	8	2	no	—
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	2	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	8	2	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	#537	1	1	yes	characteristic
186	C₄×C₁₃	#575	1	1	yes	characteristic
187	Dic₁₃	—	8	2	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
190	C₂×D₈	—	13	2	no	—
191	C₂×D₈	—	26	1	no	—
192	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	1	no	—
195	#127	—	26	1	no	—
196	#126	—	4	2	no	—
197	#125	—	2	1	no	—
198	#125	—	4	2	no	—
199	#125	—	26	1	no	—
200	#123	—	52	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	#121	—	26	2	no	—
203	#121	—	52	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×D₄	—	26	2	no	—
206	#117	C₂²⋊D₁₃	1	1	yes	characteristic
207	SD₃₂	—	52	2	no	—
208	SD₃₂	—	52	2	no	—
209	M₃₂	—	52	1	no	—
210	C₂×C₁₆	—	13	2	no	—
211	C₂×C₁₆	—	13	2	no	—
212	C₂×C₁₆	—	52	1	no	—
213	#107	—	13	2	no	—
214	#107	C₂²×D₁₃	1	2	yes	—
215	#106	—	2	1	no	—
216	#106	—	26	1	no	—
217	#103	—	2	2	no	—
218	#103	—	2	2	no	—
219	#102	—	26	2	no	—
220	#102	—	26	2	no	—
221	#97	C₂²⋊D₁₃	1	1	yes	characteristic
222	#97	C₂²⋊D₁₃	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
224	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
225	C₂×C₁₃	—	4	1	no	—
226	C₂×C₁₃	#3245	1	1	yes	characteristic
227	C₂×C₁₃	#3416	1	2	yes	—
228	D₁₃	—	16	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×D₄	—	26	2	no	—
231	C₂×D₄	—	52	2	no	—
232	C₂×D₄	—	52	2	no	—
233	C₂²×C₄	—	52	2	no	—
234	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	Q₁₆	—	4	2	no	—
238	D₈	—	26	2	no	—
239	D₈	—	26	2	no	—
240	D₈	—	52	2	no	—
241	M₁₆	—	4	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
246	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	26	2	no	—
254	C₄⋊C₄	—	52	2	no	—
255	C₄⋊C₄	#6104	1	2	yes	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	52	2	no	—
258	C₂²⋊C₄	—	52	2	no	—
259	C₂²⋊C₄	#6104	1	2	yes	—
260	C₄²	—	4	2	no	—
261	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
262	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
263	C₁₆	—	26	2	no	—
264	C₁₆	—	26	2	no	—
265	C₁₃	#33826	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	52	2	no	—
267	C₂³	C₂×#6104	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	D₄	—	52	2	no	—
271	D₄	—	52	2	no	—
272	D₄	—	208	2	no	—
273	D₄	—	208	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	52	2	no	—
279	C₂×C₄	—	208	2	no	—
280	C₂×C₄	C₂×#6104	1	2	yes	—
281	C₂×C₄	C₂×#6104	1	2	yes	—
282	C₂×C₄	C₂×#6104	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	#88256	1	1	yes	characteristic
288	C₈	#88257	1	1	yes	characteristic
289	C₂²	—	2	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	104	2	no	—
292	C₂²	—	104	2	no	—
293	C₂²	—	104	2	no	—
294	C₂²	C₂²×#6104	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	104	2	no	—
301	C₄	#11719527	1	1	yes	characteristic
302	C₄	#11719538	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	208	2	no	—
305	C₂	#13126143	1	1	yes	characteristic
306	C₂	#13127980	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3177	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13125486	C₂
hypercenter	#3177	D₁₃
hypocenter	C₁₃	#33826
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3177

Complements

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

N	H ≅ G/N	type
2. #13117000	C₂	semidirect
3. #13116990	C₂	semidirect
4. #13116951	C₂	semidirect
5. #13116934	C₂	semidirect
6. #13116878	C₂	semidirect
7. #13116876	C₂	semidirect
8. #13116595	C₂	semidirect
9. C₁₃×#3177	C₂	semidirect
113. #3177	D₁₃	semidirect
265. C₁₃	#33826	semidirect

Maximal subgroups

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

subgroup	index
#13125486	2
#13117000	2
#13116990	2
#13116951	2
#13116934	2
#13116878	2
#13116876	2
#13116595	2
C₁₃×#3177	2
#33826	13