Subgroups — #16018426

Order 3328.

474 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125373	C₂	1	2	yes	maximal
2	#13117123	C₂	1	2	yes	maximal, complemented
3	#13117121	C₂	1	2	yes	maximal, complemented
4	#13117113	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117111	C₂	1	1	yes	maximal, characteristic, complemented
6	#13117110	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116623	C₂	1	2	yes	maximal
8	#13116617	C₂	1	1	yes	maximal, characteristic
9	D₁₃×#366	C₂	1	1	yes	maximal, characteristic
10	#13116431	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719394	—	2	2	no	—
13	C₁₃×#499	C₂²	1	2	yes	—
14	#11718726	—	2	2	no	—
15	C₂×#88038	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
17	#11718541	C₂²	1	2	yes	—
18	D₁₃×#107	C₂²	1	2	yes	—
19	#11718539	C₂²	1	2	yes	—
20	#11718535	C₂²	1	2	yes	—
21	#11718522	—	2	2	no	—
22	#11718517	—	2	2	no	—
23	#11718350	C₂²	1	1	yes	characteristic
24	#11718343	C₂²	1	2	yes	—
25	C₁₃×#367	C₂²	1	2	yes	—
26	C₁₃×#366	C₂²	1	1	yes	characteristic
27	C₁₃×#366	C₂²	1	1	yes	characteristic
28	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
29	#11718096	C₂²	1	1	yes	characteristic
30	#11718096	C₂²	1	1	yes	characteristic
31	#11718096	C₂²	1	2	yes	—
32	#11718094	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
34	#11718082	C₂²	1	2	yes	—
35	#11718082	C₂²	1	2	yes	—
36	#11718081	C₂²	1	2	yes	—
37	#11718081	C₂²	1	2	yes	—
38	#11718053	C₂²	1	1	yes	characteristic
39	C₁₃×#125	—	2	2	no	—
40	C₂×#6094	D₄	1	1	yes	characteristic
41	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
42	#88151	—	4	2	no	—
43	#88150	—	4	2	no	—
44	#88146	—	2	2	no	—
45	C₄⋊C₄×D₁₃	—	2	2	no	—
46	#88144	—	2	2	no	—
47	#88143	—	4	2	no	—
48	#88142	—	2	2	no	—
49	#88142	—	4	2	no	—
50	#88126	D₄	1	1	yes	characteristic
51	#88123	—	2	2	no	—
52	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
54	C₁₃×#107	C₂³	1	2	yes	—
55	C₁₃×#107	C₂³	1	2	yes	—
56	C₁₃×#106	—	2	2	no	—
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	#88058	—	2	2	no	—
59	#88058	C₂³	1	2	yes	—
60	#88058	C₂³	1	2	yes	—
61	#88055	D₄	1	1	yes	characteristic
62	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
63	C₂×#6090	C₂³	1	1	yes	characteristic
64	C₂×#6090	C₂³	1	1	yes	characteristic
65	#88048	—	2	2	no	—
66	#88048	—	2	2	no	—
67	#88047	—	2	2	no	—
68	#88047	—	2	2	no	—
69	#88043	D₄	1	1	yes	characteristic
70	#88038	Q₈	1	4	yes	complemented
71	C₁₆×D₁₃	Q₈	1	4	yes	complemented
72	#33729	—	13	1	no	maximal, Sylow
73	C₂×C₄×D₁₃	—	4	2	no	—
74	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×Dic₂₆	—	4	2	no	—
76	C₂×Dic₂₆	—	4	2	no	—
77	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	4	2	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	#6103	—	4	2	no	—
85	#6103	D₈	1	1	yes	characteristic
86	#6103	D₈	1	1	yes	characteristic
87	C₄⋊Dic₁₃	—	2	2	no	—
88	C₄⋊Dic₁₃	—	2	2	no	—
89	C₄⋊Dic₁₃	—	4	2	no	—
90	C₄⋊Dic₁₃	—	4	2	no	—
91	#6101	—	4	2	no	—
92	#6101	—	4	2	no	—
93	#6101	—	4	2	no	—
94	#6101	D₈	1	1	yes	characteristic
95	#6101	D₈	1	1	yes	characteristic
96	C₄×Dic₁₃	—	4	2	no	—
97	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×#1055	C₂×D₄	1	1	yes	characteristic
99	C₂×#1055	C₂×D₄	1	1	yes	characteristic
100	#6094	Q₁₆	1	4	yes	—
101	C₈×D₁₃	C₂×Q₈	1	2	yes	—
102	C₈×D₁₃	C₂×Q₈	1	2	yes	—
103	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
104	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
105	#6090	C₂×Q₈	1	2	yes	—
106	#6090	C₂×Q₈	1	2	yes	—
107	#3164	—	13	2	no	—
108	#2255	—	13	2	no	—
109	#2255	—	13	2	no	—
110	#2253	—	13	1	no	—
111	#2253	D₁₃	1	1	yes	characteristic, complemented
112	#2252	—	13	1	no	—
113	#2252	—	13	1	no	—
114	#2159	—	13	2	no	—
115	#2158	—	13	1	no	—
116	C₂×#366	—	13	1	no	—
117	#2108	—	13	1	no	—
118	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₁₃	—	4	2	no	—
125	C₂×Dic₁₃	—	4	2	no	—
126	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
127	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
128	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
129	C₄×D₁₃	—	16	2	no	—
130	C₄×D₁₃	C₄⋊Q₈	1	2	yes	—
131	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
132	Dic₂₆	—	16	2	no	—
133	Dic₂₆	—	16	2	no	—
134	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
135	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
136	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
137	#1055	C₄⋊Q₈	1	2	yes	—
138	#1055	C₂×Q₁₆	1	2	yes	—
139	#544	—	26	2	no	—
140	C₂×M₃₂	—	13	1	no	—
141	C₂²×C₁₆	—	13	1	no	—
142	#501	—	26	2	no	—
143	#501	—	26	2	no	—
144	#500	—	13	2	no	—
145	#500	—	13	2	no	—
146	#499	—	13	2	no	—
147	#499	C₂×D₁₃	1	2	yes	—
148	#428	—	26	2	no	—
149	#427	—	13	2	no	—
150	C₂×#107	—	13	2	no	—
151	#405	—	13	1	no	—
152	#367	—	13	2	no	—
153	#367	—	13	2	no	—
154	#367	C₂×D₁₃	1	2	yes	—
155	#366	—	13	1	no	—
156	#366	—	13	1	no	—
157	#366	—	13	2	no	—
158	#366	—	13	2	no	—
159	#366	—	13	2	no	—
160	#366	C₂×D₁₃	1	1	yes	characteristic
161	#366	C₂×D₁₃	1	1	yes	characteristic
162	#346	—	13	1	no	—
163	#346	—	13	1	no	—
164	C₄×C₁₆	—	13	1	no	—
165	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
166	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₁₃	—	2	1	no	—
168	C₂×D₁₃	—	2	1	no	—
169	C₂×D₁₃	#500	1	2	yes	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	8	2	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
175	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
176	Dic₁₃	—	2	1	no	—
177	Dic₁₃	—	2	1	no	—
178	Dic₁₃	—	8	2	no	—
179	Dic₁₃	—	8	2	no	—
180	Dic₁₃	#500	1	2	yes	—
181	C₂×M₁₆	—	13	1	no	—
182	C₂²×C₈	—	13	1	no	—
183	C₄⋊Q₈	—	26	2	no	—
184	C₄⋊Q₈	—	26	2	no	—
185	C₄⋊Q₈	—	52	2	no	—
186	#125	—	2	2	no	—
187	#125	—	26	2	no	—
188	#125	—	52	2	no	—
189	C₂²⋊Q₈	—	52	2	no	—
190	C₂²⋊Q₈	—	52	2	no	—
191	#117	—	13	1	no	—
192	#117	—	26	2	no	—
193	C₂×C₄⋊C₄	—	26	2	no	—
194	M₃₂	—	13	4	no	—
195	C₂×C₁₆	—	13	1	no	—
196	C₂×C₁₆	—	13	1	no	—
197	C₂×C₁₆	—	13	4	no	—
198	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
199	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
200	#107	—	13	2	no	—
201	#107	—	13	2	no	—
202	#107	—	26	2	no	—
203	#107	—	26	2	no	—
204	#107	—	26	2	no	—
205	#107	C₂²×D₁₃	1	2	yes	—
206	#107	C₂²×D₁₃	1	2	yes	—
207	#106	—	2	2	no	—
208	#106	—	26	2	no	—
209	#106	—	26	2	no	—
210	#97	—	13	1	no	—
211	#97	—	13	1	no	—
212	C₄×C₈	—	13	1	no	—
213	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
215	C₂×C₁₃	#3415	1	1	yes	characteristic
216	C₂×C₁₃	#3417	1	1	yes	characteristic
217	D₁₃	—	2	2	no	—
218	C₂×Q₈	—	52	2	no	—
219	C₂×Q₈	—	52	2	no	—
220	C₂²×C₄	—	13	1	no	—
221	C₂²×C₄	—	52	2	no	—
222	M₁₆	—	13	4	no	—
223	C₂×C₈	—	13	1	no	—
224	C₂×C₈	—	13	1	no	—
225	C₂×C₈	—	13	2	no	—
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
228	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	13	1	no	—
234	C₄⋊C₄	—	13	1	no	—
235	C₄⋊C₄	—	26	2	no	—
236	C₄⋊C₄	—	26	2	no	—
237	C₄⋊C₄	—	52	2	no	—
238	C₄⋊C₄	—	52	2	no	—
239	C₄⋊C₄	—	52	2	no	—
240	C₄⋊C₄	—	52	2	no	—
241	C₄⋊C₄	—	52	2	no	—
242	C₂²⋊C₄	—	13	1	no	—
243	C₂²⋊C₄	—	13	1	no	—
244	C₂²⋊C₄	—	52	2	no	—
245	C₄²	—	13	1	no	—
246	C₄²	—	52	2	no	—
247	C₄²	D₄×D₁₃	1	1	yes	characteristic
248	C₁₆	—	13	2	no	—
249	C₁₆	—	13	2	no	—
250	C₁₆	Q₈×D₁₃	1	2	yes	—
251	C₁₆	Q₈×D₁₃	1	2	yes	—
252	C₁₃	#33729	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	13	1	no	—
254	Q₈	—	208	2	no	—
255	Q₈	—	208	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	13	1	no	—
259	C₂×C₄	—	13	2	no	—
260	C₂×C₄	—	13	2	no	—
261	C₂×C₄	—	13	1	no	—
262	C₂×C₄	—	13	1	no	—
263	C₂×C₄	—	52	2	no	—
264	C₂×C₄	—	52	2	no	—
265	C₂×C₄	—	208	2	no	—
266	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
267	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
269	C₈	—	13	2	no	—
270	C₈	—	13	2	no	—
271	C₈	Q₁₆×D₁₃	1	2	yes	—
272	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
273	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
274	C₂²	—	13	2	no	—
275	C₂²	—	26	1	no	—
276	C₂²	—	26	1	no	—
277	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	13	2	no	—
283	C₄	—	26	1	no	—
284	C₄	—	26	1	no	—
285	C₄	—	104	2	no	—
286	C₄	—	104	2	no	—
287	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
288	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
289	C₂	—	26	2	no	—
290	C₂	D₁₃×#500	1	1	yes	characteristic
291	C₂	#13127974	1	1	yes	characteristic
292	C₂	#13127986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2253	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂²×D₁₃	C₂×D₈
hypercenter	#2253	D₁₃
hypocenter	C₁₃	#33729
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. #13117123	C₂	semidirect
3. #13117121	C₂	semidirect
4. #13117113	C₂	semidirect
5. #13117111	C₂	semidirect
6. #13117110	C₂	semidirect
11. C₁₃×#2253	C₂	semidirect
70. #88038	Q₈	semidirect
71. C₁₆×D₁₃	Q₈	semidirect
111. #2253	D₁₃	semidirect
252. C₁₃	#33729	semidirect

Maximal subgroups

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

subgroup	index
#13125373	2
#13117123	2
#13117121	2
#13117113	2
#13117111	2
#13117110	2
#13116623	2
#13116617	2
D₁₃×#366	2
#13116431	2
C₁₃×#2253	2
#33729	13