Subgroups — #16018465

Order 3328.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125401	C₂	1	2	yes	maximal, complemented
2	#13117113	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117094	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117066	C₂	1	2	yes	maximal, complemented
5	#13116930	C₂	1	2	yes	maximal, complemented
6	#13116927	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116694	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116691	C₂	1	2	yes	maximal, complemented
9	D₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2215	C₂	1	1	yes	maximal, characteristic
12	#11719305	—	2	2	no	—
13	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₁₃×#466	C₂²	1	2	yes	—
16	#11718816	C₂²	1	2	yes	—
17	D₈×Dic₁₃	C₂²	1	2	yes	—
18	C₂×#88067	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	#11718702	—	2	2	no	—
23	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
24	#11718539	C₂²	1	2	yes	—
25	#11718534	C₂²	1	2	yes	—
26	#11718429	—	2	2	no	—
27	#11718419	—	2	2	no	—
28	C₁₃×#366	C₂²	1	1	yes	characteristic
29	C₁₃×#357	C₂²	1	2	yes	—
30	#11718154	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	#11718082	C₂²	1	2	yes	—
39	C₂×C₁₃×D₈	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×#121	—	2	2	no	—
42	#88190	—	2	2	no	—
43	#88189	—	4	2	no	—
44	D₄×Dic₁₃	—	2	2	no	—
45	D₄×Dic₁₃	—	4	2	no	—
46	C₂×#6105	—	2	2	no	—
47	#88178	—	2	2	no	—
48	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
49	C₂²×#1055	D₄	1	1	yes	characteristic
50	C₂×Dic₅₂	C₂³	1	2	yes	—
51	#88144	—	2	2	no	—
52	#88141	—	2	2	no	—
53	#88133	—	4	2	no	—
54	#88131	—	4	2	no	—
55	C₁₃×D₁₆	—	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×#102	—	2	2	no	—
61	#88072	—	2	2	no	—
62	#88067	—	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	#88048	—	2	2	no	—
74	Dic₁₀₄	—	2	2	no	—
75	#33735	—	13	1	no	maximal, Sylow
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	—	4	2	no	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	—	4	2	no	—
80	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
81	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
82	C₂×Dic₂₆	—	2	2	no	—
83	C₁₃×D₈	—	2	2	no	—
84	C₁₃×D₈	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	—	2	1	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₂²⋊C₄	—	4	2	no	—
91	C₂²⋊Dic₁₃	—	4	2	no	—
92	C₂²⋊Dic₁₃	—	4	2	no	—
93	C₂²⋊Dic₁₃	—	4	2	no	—
94	C₂²⋊Dic₁₃	—	4	2	no	—
95	#6105	—	4	4	no	—
96	C₄⋊Dic₁₃	—	2	2	no	—
97	C₄⋊Dic₁₃	—	4	2	no	—
98	#6101	—	4	2	no	—
99	#6101	—	4	2	no	—
100	#6101	—	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	#3168	—	13	2	no	—
116	#2253	—	13	1	no	—
117	#2249	—	13	1	no	—
118	#2242	—	13	2	no	—
119	#2217	—	13	2	no	—
120	#2216	—	13	1	no	—
121	#2215	D₁₃	1	1	yes	characteristic
122	C₄×Q₃₂	—	13	1	no	—
123	C₄×SD₃₂	—	13	2	no	—
124	C₄×D₁₆	—	13	1	no	—
125	C₂×C₄×C₁₆	—	13	1	no	—
126	C₂³×C₁₃	—	4	2	no	—
127	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
128	D₄×C₁₃	—	4	2	no	—
129	D₄×C₁₃	—	4	2	no	—
130	D₄×C₁₃	—	8	4	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₁₃	—	2	2	no	—
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	4	2	no	—
139	C₂×Dic₁₃	—	4	2	no	—
140	C₂×Dic₁₃	—	8	4	no	—
141	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
145	Dic₂₆	—	4	2	no	—
146	Dic₂₆	—	4	2	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
149	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
150	#1055	—	2	1	no	—
151	#1055	#135	1	2	yes	—
152	#529	—	26	2	no	—
153	C₂×Q₃₂	—	13	1	no	—
154	C₂×SD₃₂	—	13	2	no	—
155	C₂×D₁₆	C₂×D₁₃	1	1	yes	characteristic
156	C₂²×C₁₆	—	13	1	no	—
157	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
158	#499	—	13	2	no	—
159	#495	—	13	2	no	—
160	#488	—	26	2	no	—
161	#467	—	13	2	no	—
162	#466	C₂×D₁₃	1	2	yes	—
163	#465	—	26	2	no	—
164	C₄×Q₁₆	—	13	2	no	—
165	C₄×SD₁₆	—	26	2	no	—
166	C₄×D₈	—	13	2	no	—
167	C₂×C₄×C₈	—	13	1	no	—
168	#367	—	13	2	no	—
169	#366	—	13	1	no	—
170	#366	C₂×D₁₃	1	1	yes	characteristic
171	#358	—	13	2	no	—
172	#358	—	13	2	no	—
173	#357	—	13	2	no	—
174	#357	C₂×D₁₃	1	2	yes	—
175	C₄×C₁₆	—	13	1	no	—
176	C₄×C₁₆	—	13	1	no	—
177	C₄×C₁₆	—	13	1	no	—
178	C₄×C₁₆	—	13	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	—	8	2	no	—
183	C₂²×C₁₃	—	8	2	no	—
184	C₂²×C₁₃	#508	1	2	yes	—
185	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	#532	1	1	yes	characteristic
189	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
190	Dic₁₃	—	2	1	no	—
191	Dic₁₃	—	2	1	no	—
192	Dic₁₃	—	2	1	no	—
193	Dic₁₃	—	8	2	no	—
194	Dic₁₃	—	8	2	no	—
195	Dic₁₃	#508	1	2	yes	—
196	C₂×Q₁₆	—	13	2	no	—
197	C₂×SD₁₆	—	26	2	no	—
198	C₂×D₈	C₂²×D₁₃	1	2	yes	—
199	C₂²×C₈	—	13	1	no	—
200	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
201	#126	—	52	2	no	—
202	#125	—	26	2	no	—
203	#124	—	26	2	no	—
204	#123	—	52	2	no	—
205	C₂²⋊Q₈	—	26	2	no	—
206	#121	—	2	2	no	—
207	C₄×Q₈	—	26	2	no	—
208	C₄×D₄	—	26	2	no	—
209	C₄×D₄	—	52	2	no	—
210	#117	—	52	2	no	—
211	C₂×C₄²	—	13	1	no	—
212	Q₃₂	—	26	2	no	—
213	SD₃₂	—	26	4	no	—
214	D₁₆	—	2	2	no	—
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	—	13	1	no	—
217	C₂×C₁₆	—	13	1	no	—
218	C₂×C₁₆	—	26	2	no	—
219	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
220	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
221	#107	—	13	2	no	—
222	#107	C₂²×D₁₃	1	2	yes	—
223	#106	—	26	2	no	—
224	#103	—	26	2	no	—
225	#103	—	26	2	no	—
226	#102	—	2	2	no	—
227	#102	—	26	2	no	—
228	C₄×C₈	—	13	1	no	—
229	C₄×C₈	—	13	1	no	—
230	C₄×C₈	—	13	1	no	—
231	C₄×C₈	—	13	1	no	—
232	C₂×C₁₃	—	2	2	no	—
233	C₂×C₁₃	—	16	2	no	—
234	C₂×C₁₃	#3169	1	1	yes	characteristic
235	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
237	C₂×Q₈	—	26	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	52	2	no	—
243	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
244	Q₁₆	—	26	2	no	—
245	Q₁₆	—	26	2	no	—
246	SD₁₆	—	52	4	no	—
247	D₈	—	2	2	no	—
248	D₈	—	2	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	13	1	no	—
253	C₂×C₈	—	26	2	no	—
254	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	26	2	no	—
258	C₄⋊C₄	—	52	2	no	—
259	C₄⋊C₄	—	52	2	no	—
260	C₄⋊C₄	—	52	2	no	—
261	C₄⋊C₄	—	52	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	52	2	no	—
264	C₂²⋊C₄	—	52	2	no	—
265	C₂²⋊C₄	—	52	2	no	—
266	C₂²⋊C₄	—	52	2	no	—
267	C₄²	—	13	1	no	—
268	C₄²	—	13	1	no	—
269	C₄²	—	13	1	no	—
270	C₄²	—	13	1	no	—
271	C₄²	—	52	2	no	—
272	C₁₆	—	2	1	no	—
273	C₁₆	—	13	2	no	—
274	C₁₆	—	26	1	no	—
275	C₁₆	#6129	1	2	yes	—
276	C₁₃	#33735	1	1	yes	characteristic, complemented, Sylow
277	C₂³	—	4	2	no	—
278	C₂³	D₈×D₁₃	1	1	yes	characteristic
279	Q₈	—	52	2	no	—
280	Q₈	—	52	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	8	4	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	13	1	no	—
288	C₂×C₄	—	13	1	no	—
289	C₂×C₄	—	13	1	no	—
290	C₂×C₄	—	13	1	no	—
291	C₂×C₄	—	26	2	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	52	2	no	—
294	C₂×C₄	—	52	2	no	—
295	C₂×C₄	—	104	4	no	—
296	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	13	2	no	—
300	C₈	—	26	1	no	—
301	C₈	C₂×#6129	1	1	yes	characteristic
302	C₈	#41×D₁₃	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	#11718573	1	2	yes	—
309	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	2	no	—
312	C₄	—	13	2	no	—
313	C₄	—	26	1	no	—
314	C₄	—	26	1	no	—
315	C₄	—	26	1	no	—
316	C₄	—	104	2	no	—
317	C₄	—	104	2	no	—
318	C₄	#11719301	1	1	yes	characteristic
319	C₄	D₁₃×#135	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	16	2	no	—
322	C₂	#13125410	1	1	yes	characteristic
323	C₂	D₁₃×#508	1	1	yes	characteristic
324	C₂	C₂×#11718573	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₁₃×#2215	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#2215	D₁₃
hypocenter	C₁₃	#33735
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2215

Complements

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

N	H ≅ G/N	type
1. #13125401	C₂	semidirect
2. #13117113	C₂	semidirect
3. #13117094	C₂	semidirect
4. #13117066	C₂	semidirect
5. #13116930	C₂	semidirect
6. #13116927	C₂	semidirect
7. #13116694	C₂	semidirect
8. #13116691	C₂	semidirect
9. D₁₆×Dic₁₃	C₂	semidirect
10. C₂×C₁₆×Dic₁₃	C₂	semidirect
276. C₁₃	#33735	semidirect

Maximal subgroups

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

subgroup	index
#13125401	2
#13117113	2
#13117094	2
#13117066	2
#13116930	2
#13116927	2
#13116694	2
#13116691	2
D₁₆×Dic₁₃	2
C₂×C₁₆×Dic₁₃	2
C₁₃×#2215	2
#33735	13