Subgroups — #16018458

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125417	C₂	1	2	yes	maximal
2	#13117110	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117087	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117062	C₂	1	2	yes	maximal, complemented
5	#13116927	C₂	1	1	yes	maximal, characteristic
6	#13116925	C₂	1	2	yes	maximal
7	Q₃₂×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116694	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116692	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2216	C₂	1	1	yes	maximal, characteristic
12	#11719327	—	2	2	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₁₃×#467	C₂²	1	2	yes	—
16	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
17	#11718842	C₂²	1	2	yes	—
18	C₂×#88069	C₂²	1	2	yes	—
19	#11718777	C₂²	1	2	yes	—
20	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂²×#6090	C₂²	1	1	yes	characteristic
22	#11718712	—	2	2	no	—
23	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
24	#11718535	C₂²	1	2	yes	—
25	#11718534	C₂²	1	2	yes	—
26	#11718453	—	2	2	no	—
27	#11718451	—	2	2	no	—
28	C₁₃×#366	C₂²	1	1	yes	characteristic
29	C₁₃×#358	C₂²	1	2	yes	—
30	#11718156	C₂²	1	2	yes	—
31	#11718096	C₂²	1	1	yes	characteristic
32	#11718095	C₂²	1	2	yes	—
33	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
37	#11718084	C₂²	1	2	yes	—
38	#11718081	C₂²	1	2	yes	—
39	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₂²⋊Q₈	—	2	2	no	—
42	Q₈×Dic₁₃	—	2	2	no	—
43	#88198	—	2	2	no	—
44	C₂×#6107	—	2	2	no	—
45	#88178	—	2	2	no	—
46	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
47	C₂²×#1055	D₄	1	1	yes	characteristic
48	C₂×Dic₅₂	C₂³	1	2	yes	—
49	#88143	—	4	2	no	—
50	#88142	—	2	2	no	—
51	#88141	—	2	2	no	—
52	#88141	—	4	2	no	—
53	#88132	—	4	2	no	—
54	#88131	—	4	2	no	—
55	C₁₃×Q₃₂	—	2	2	no	—
56	C₂×C₁₃×C₁₆	—	2	2	no	—
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₁₃×#107	C₂³	1	2	yes	—
60	C₁₃×#103	—	2	2	no	—
61	#88075	—	2	2	no	—
62	#88069	—	2	4	no	—
63	#88058	C₂³	1	2	yes	—
64	#88056	—	2	2	no	—
65	C₈×Dic₁₃	D₄	1	1	yes	characteristic
66	C₈×Dic₁₃	D₄	1	1	yes	characteristic
67	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
68	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
69	C₂×#6090	—	2	2	no	—
70	C₂×#6090	C₂³	1	1	yes	characteristic
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	#88050	—	2	2	no	—
73	#88047	—	2	2	no	—
74	Dic₁₀₄	—	2	2	no	—
75	#33734	—	13	1	no	maximal, Sylow
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
79	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
80	C₂×Dic₂₆	—	2	2	no	—
81	C₂×Dic₂₆	—	4	2	no	—
82	C₁₃×Q₁₆	—	2	2	no	—
83	C₁₃×Q₁₆	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	4	2	no	—
90	C₁₃×C₂²⋊C₄	—	4	2	no	—
91	C₂²⋊Dic₁₃	—	4	2	no	—
92	#6107	—	4	4	no	—
93	C₄⋊Dic₁₃	—	2	2	no	—
94	C₄⋊Dic₁₃	—	4	2	no	—
95	#6101	—	4	2	no	—
96	#6101	—	4	2	no	—
97	#6101	—	4	2	no	—
98	#6101	—	4	2	no	—
99	#6101	—	4	2	no	—
100	C₄×Dic₁₃	—	4	2	no	—
101	C₄×Dic₁₃	—	4	2	no	—
102	C₄×Dic₁₃	D₈	1	1	yes	characteristic
103	C₄×Dic₁₃	D₈	1	1	yes	characteristic
104	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
105	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×#1055	—	2	2	no	—
107	C₂×#1055	C₂×D₄	1	1	yes	characteristic
108	C₂×#1055	C₂×D₄	1	1	yes	characteristic
109	Dic₅₂	—	2	2	no	—
110	Dic₅₂	—	2	2	no	—
111	C₁₃×C₁₆	—	2	1	no	—
112	C₁₃×C₁₆	#41	1	2	yes	—
113	#6090	—	2	1	no	—
114	#6090	#41	1	2	yes	—
115	#3170	—	13	2	no	—
116	#2252	—	13	1	no	—
117	#2247	—	13	1	no	—
118	#2241	—	13	2	no	—
119	#2216	—	13	1	no	—
120	#2216	—	13	2	no	—
121	#2216	D₁₃	1	1	yes	characteristic
122	C₄×Q₃₂	—	13	1	no	—
123	C₄×Q₃₂	—	13	1	no	—
124	C₄×Q₃₂	—	13	2	no	—
125	C₂×C₄×C₁₆	—	13	1	no	—
126	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₁₃	—	2	2	no	—
136	C₂×Dic₁₃	—	4	2	no	—
137	C₂×Dic₁₃	—	4	2	no	—
138	C₂×Dic₁₃	Q₃₂	1	4	yes	complemented
139	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
140	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
141	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
143	Dic₂₆	—	4	2	no	—
144	Dic₂₆	—	4	2	no	—
145	Dic₂₆	—	8	4	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
148	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
149	#1055	—	2	1	no	—
150	#1055	#135	1	2	yes	—
151	#533	—	26	2	no	—
152	C₂×Q₃₂	—	13	2	no	—
153	C₂×Q₃₂	—	13	1	no	—
154	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic
155	C₂²×C₁₆	—	13	1	no	—
156	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
157	#500	—	13	2	no	—
158	#494	—	13	2	no	—
159	#487	—	26	2	no	—
160	#467	—	13	2	no	—
161	#467	—	26	2	no	—
162	#467	C₂×D₁₃	1	2	yes	—
163	C₄×Q₁₆	—	13	2	no	—
164	C₄×Q₁₆	—	13	2	no	—
165	C₄×Q₁₆	—	26	2	no	—
166	C₂×C₄×C₈	—	13	1	no	—
167	#366	—	13	1	no	—
168	#366	—	13	2	no	—
169	#366	C₂×D₁₃	1	1	yes	characteristic
170	#358	—	13	2	no	—
171	#358	—	13	2	no	—
172	#358	—	13	2	no	—
173	#358	C₂×D₁₃	1	2	yes	—
174	C₄×C₁₆	—	13	1	no	—
175	C₄×C₁₆	—	13	1	no	—
176	C₄×C₁₆	—	13	1	no	—
177	C₄×C₁₆	—	13	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
181	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	8	2	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	#532	1	1	yes	characteristic
186	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
187	Dic₁₃	—	2	1	no	—
188	Dic₁₃	—	8	2	no	—
189	Dic₁₃	—	8	2	no	—
190	Dic₁₃	C₂×Q₃₂	1	2	yes	—
191	Dic₁₃	C₂×Q₃₂	1	2	yes	—
192	Dic₁₃	#508	1	2	yes	—
193	C₂×Q₁₆	—	13	2	no	—
194	C₂×Q₁₆	—	26	2	no	—
195	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
196	C₂²×C₈	—	13	1	no	—
197	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
198	C₄⋊Q₈	—	26	2	no	—
199	C₄⋊Q₈	—	26	2	no	—
200	#125	—	52	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	C₂²⋊Q₈	—	26	2	no	—
203	C₂²⋊Q₈	—	52	2	no	—
204	C₄×Q₈	—	26	2	no	—
205	C₄×Q₈	—	26	2	no	—
206	C₄×Q₈	—	52	2	no	—
207	#117	—	52	2	no	—
208	C₂×C₄²	—	13	1	no	—
209	Q₃₂	—	2	2	no	—
210	Q₃₂	—	26	4	no	—
211	Q₃₂	—	26	2	no	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	—	13	1	no	—
214	C₂×C₁₆	—	13	1	no	—
215	C₂×C₁₆	—	26	2	no	—
216	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
217	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
218	#107	—	13	2	no	—
219	#107	—	26	2	no	—
220	#107	C₂²×D₁₃	1	2	yes	—
221	#103	—	2	2	no	—
222	#103	—	26	2	no	—
223	#103	—	26	2	no	—
224	#103	—	26	2	no	—
225	C₄×C₈	—	13	1	no	—
226	C₄×C₈	—	13	1	no	—
227	C₄×C₈	—	13	1	no	—
228	C₄×C₈	—	13	1	no	—
229	C₂×C₁₃	—	2	2	no	—
230	C₂×C₁₃	#3169	1	1	yes	characteristic
231	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
232	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	26	2	no	—
235	C₂×Q₈	—	52	2	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	26	2	no	—
242	Q₁₆	—	26	2	no	—
243	Q₁₆	—	52	4	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	13	1	no	—
247	C₂×C₈	—	13	1	no	—
248	C₂×C₈	—	26	2	no	—
249	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	26	2	no	—
254	C₄⋊C₄	—	52	2	no	—
255	C₄⋊C₄	—	52	2	no	—
256	C₄⋊C₄	—	52	2	no	—
257	C₄⋊C₄	—	52	2	no	—
258	C₄⋊C₄	—	52	2	no	—
259	C₄⋊C₄	—	52	2	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	52	2	no	—
262	C₄²	—	13	1	no	—
263	C₄²	—	13	1	no	—
264	C₄²	—	13	1	no	—
265	C₄²	—	13	1	no	—
266	C₄²	—	52	2	no	—
267	C₄²	—	52	2	no	—
268	C₁₆	—	2	1	no	—
269	C₁₆	—	13	2	no	—
270	C₁₆	—	26	1	no	—
271	C₁₆	#6129	1	2	yes	—
272	C₁₃	#33734	1	1	yes	characteristic, complemented, Sylow
273	C₂³	D₈×D₁₃	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	52	2	no	—
277	Q₈	—	52	2	no	—
278	Q₈	—	104	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	13	1	no	—
284	C₂×C₄	—	13	1	no	—
285	C₂×C₄	—	13	1	no	—
286	C₂×C₄	—	13	4	no	—
287	C₂×C₄	—	13	1	no	—
288	C₂×C₄	—	26	2	no	—
289	C₂×C₄	—	52	2	no	—
290	C₂×C₄	—	52	2	no	—
291	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	13	2	no	—
295	C₈	—	26	1	no	—
296	C₈	C₂×#6129	1	1	yes	characteristic
297	C₈	#41×D₁₃	1	1	yes	characteristic
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	D₁₃×Q₃₂	1	2	yes	—
301	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	13	2	no	—
306	C₄	—	13	2	no	—
307	C₄	—	13	2	no	—
308	C₄	—	26	1	no	—
309	C₄	—	104	2	no	—
310	C₄	—	104	2	no	—
311	C₄	#11719301	1	1	yes	characteristic
312	C₄	D₁₃×#135	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	#13125410	1	1	yes	characteristic
315	C₂	C₂×D₁₃×Q₃₂	1	1	yes	characteristic
316	C₂	D₁₃×#508	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×#2216	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#2216	D₁₃
hypocenter	C₁₃	#33734
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

Complements

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

N	H ≅ G/N	type
2. #13117110	C₂	semidirect
3. #13117087	C₂	semidirect
4. #13117062	C₂	semidirect
7. Q₃₂×Dic₁₃	C₂	semidirect
8. #13116694	C₂	semidirect
9. #13116692	C₂	semidirect
138. C₂×Dic₁₃	Q₃₂	semidirect
272. C₁₃	#33734	semidirect

Maximal subgroups

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

subgroup	index
#13125417	2
#13117110	2
#13117087	2
#13117062	2
#13116927	2
#13116925	2
Q₃₂×Dic₁₃	2
#13116694	2
#13116692	2
C₂×C₁₆×Dic₁₃	2
C₁₃×#2216	2
#33734	13