Subgroups — #16018474

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	#13125409	C₂	1	2	yes	maximal, complemented
2	#13117113	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117098	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117069	C₂	1	2	yes	maximal, complemented
5	#13116935	C₂	1	2	yes	maximal, complemented
6	#13116924	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116923	C₂	1	1	yes	maximal, characteristic
8	C₄×#88068	C₂	1	2	yes	maximal, complemented
9	C₄×D₂₀₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719285	—	2	2	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
15	#11718852	C₂²	1	2	yes	—
16	#11718849	C₂²	1	2	yes	—
17	C₂×#88068	C₂²	1	2	yes	—
18	C₄×#6106	—	2	2	no	—
19	C₂×D₂₀₈	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
22	#11718544	C₂²	1	2	yes	—
23	#11718539	C₂²	1	2	yes	—
24	#11718468	—	2	2	no	—
25	#11718461	—	2	2	no	—
26	C₄×D₁₀₄	C₂²	1	2	yes	—
27	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#366	C₂²	1	1	yes	characteristic
29	C₁₃×#358	C₂²	1	2	yes	—
30	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
31	#11718156	C₂²	1	2	yes	—
32	#11718101	C₂²	1	2	yes	—
33	#11718096	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
36	#11718083	C₂²	1	2	yes	—
37	#11718082	C₂²	1	2	yes	—
38	C₄×#6090	C₂²	1	1	yes	characteristic
39	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
40	C₄×Q₈×C₁₃	—	2	2	no	—
41	#88201	—	2	2	no	—
42	#88200	—	2	2	no	—
43	C₂×#6106	—	2	2	no	—
44	C₂×D₁₀₄	C₂³	1	2	yes	—
45	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
46	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
47	#88152	—	4	2	no	—
48	#88149	—	2	2	no	—
49	#88148	—	4	2	no	—
50	#88144	—	2	2	no	—
51	C₄×D₅₂	—	2	2	no	—
52	C₄×D₅₂	—	4	2	no	—
53	#88126	—	4	2	no	—
54	C₄²×D₁₃	D₄	1	1	yes	characteristic
55	C₁₃×Q₃₂	—	2	2	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₁₃×#107	C₂³	1	2	yes	—
59	C₁₃×#103	—	2	2	no	—
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	#88075	—	2	2	no	—
62	#88068	—	2	4	no	—
63	#88061	—	2	2	no	—
64	#88058	C₂³	1	2	yes	—
65	C₈×Dic₁₃	D₄	1	1	yes	characteristic
66	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
67	C₂×#6090	C₂³	1	1	yes	characteristic
68	C₂×#6090	C₂³	1	1	yes	characteristic
69	#88049	—	2	2	no	—
70	#88048	—	2	2	no	—
71	C₄×#1055	D₄	1	1	yes	characteristic
72	D₂₀₈	—	2	2	no	—
73	C₁₆×D₁₃	—	2	2	no	—
74	C₁₆×D₁₃	—	2	2	no	—
75	#33735	—	13	1	no	maximal, Sylow
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×D₅₂	—	2	2	no	—
78	C₂×D₅₂	—	4	2	no	—
79	C₂×C₄×D₁₃	—	4	2	no	—
80	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
81	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
82	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
83	C₁₃×Q₁₆	—	2	2	no	—
84	C₁₃×Q₁₆	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	4	2	no	—
89	C₄²×C₁₃	—	4	2	no	—
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	#6106	—	4	4	no	—
92	#6103	—	4	2	no	—
93	#6103	—	4	2	no	—
94	#6103	—	4	2	no	—
95	#6103	—	4	2	no	—
96	#6103	—	4	2	no	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	4	2	no	—
99	#6101	—	4	2	no	—
100	#6101	—	4	2	no	—
101	C₄×Dic₁₃	D₈	1	1	yes	characteristic
102	C₄×Dic₁₃	D₈	1	1	yes	characteristic
103	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×#1055	C₂×D₄	1	1	yes	characteristic
105	C₂×#1055	C₂×D₄	1	1	yes	characteristic
106	D₁₀₄	—	2	2	no	—
107	D₁₀₄	—	2	2	no	—
108	C₈×D₁₃	—	2	1	no	—
109	C₈×D₁₃	—	2	1	no	—
110	C₈×D₁₃	—	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	—	13	1	no	—
122	C₄×Q₃₂	D₁₃	1	1	yes	characteristic, complemented
123	C₄×SD₃₂	—	13	2	no	—
124	C₄×D₁₆	—	13	1	no	—
125	C₂×C₄×C₁₆	—	13	1	no	—
126	C₂²×D₁₃	—	4	2	no	—
127	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₁₃	—	4	2	no	—
136	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
139	D₅₂	—	4	2	no	—
140	D₅₂	—	4	2	no	—
141	D₅₂	—	8	4	no	—
142	C₄×D₁₃	—	2	2	no	—
143	C₄×D₁₃	—	2	1	no	—
144	C₄×D₁₃	—	2	1	no	—
145	C₄×D₁₃	—	2	2	no	—
146	C₄×D₁₃	—	8	4	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	C₂×D₁₃	1	1	yes	characteristic
154	C₂×SD₃₂	—	13	2	no	—
155	C₂×D₁₆	—	13	1	no	—
156	C₂²×C₁₆	—	13	1	no	—
157	C₂²×C₁₆	—	13	1	no	—
158	#499	—	13	2	no	—
159	#495	—	13	2	no	—
160	#488	—	26	2	no	—
161	#467	—	13	2	no	—
162	#466	—	13	2	no	—
163	#465	—	26	2	no	—
164	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
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	C₂×D₁₃	1	2	yes	—
173	#357	—	13	2	no	—
174	#357	—	13	2	no	—
175	C₄×C₁₆	—	13	1	no	—
176	C₄×C₁₆	—	13	1	no	—
177	C₄×C₁₆	—	13	1	no	—
178	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
179	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
180	C₂×D₁₃	—	2	1	no	—
181	C₂×D₁₃	—	2	1	no	—
182	C₂×D₁₃	—	8	2	no	—
183	C₂×D₁₃	—	8	2	no	—
184	C₂×D₁₃	—	8	2	no	—
185	C₂×D₁₃	#508	1	2	yes	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	—	8	2	no	—
189	C₄×C₁₃	#508	1	2	yes	—
190	C₄×C₁₃	#532	1	1	yes	characteristic
191	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
192	Dic₁₃	—	2	1	no	—
193	Dic₁₃	—	2	1	no	—
194	Dic₁₃	—	2	1	no	—
195	Dic₁₃	—	8	2	no	—
196	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
197	C₂×SD₁₆	—	26	2	no	—
198	C₂×D₈	—	13	2	no	—
199	C₂²×C₈	—	13	1	no	—
200	C₂²×C₈	—	13	1	no	—
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	—	26	2	no	—
207	C₄×Q₈	—	2	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	—	2	2	no	—
213	SD₃₂	—	26	4	no	—
214	D₁₆	—	26	2	no	—
215	C₂×C₁₆	—	13	1	no	—
216	C₂×C₁₆	—	13	1	no	—
217	C₂×C₁₆	—	26	2	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	—	2	2	no	—
225	#103	—	26	2	no	—
226	#102	—	26	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	C₂²×D₁₃	1	1	yes	characteristic
232	C₂×C₁₃	#3169	1	1	yes	characteristic
233	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
234	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
235	D₁₃	—	2	2	no	—
236	D₁₃	—	16	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×D₄	—	26	2	no	—
239	C₂×D₄	—	52	2	no	—
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	13	1	no	—
243	C₂²×C₄	—	52	2	no	—
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	2	2	no	—
246	SD₁₆	—	52	4	no	—
247	D₈	—	26	2	no	—
248	D₈	—	26	2	no	—
249	C₂×C₈	—	13	1	no	—
250	C₂×C₈	—	13	1	no	—
251	C₂×C₈	—	26	2	no	—
252	C₂×C₈	—	26	1	no	—
253	C₂×C₈	—	26	1	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	—	4	2	no	—
258	C₄⋊C₄	—	26	2	no	—
259	C₄⋊C₄	—	52	2	no	—
260	C₄⋊C₄	—	52	2	no	—
261	C₄⋊C₄	—	52	2	no	—
262	C₂²⋊C₄	—	52	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	—	4	2	no	—
268	C₄²	—	13	1	no	—
269	C₄²	—	13	1	no	—
270	C₄²	—	13	1	no	—
271	C₄²	D₄×D₁₃	1	1	yes	characteristic
272	C₁₆	—	2	1	no	—
273	C₁₆	—	13	2	no	—
274	C₁₆	—	26	1	no	—
275	C₁₆	#6127	1	2	yes	—
276	C₁₃	#33735	1	1	yes	characteristic, complemented, Sylow
277	C₂³	—	13	1	no	—
278	C₂³	—	52	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	52	2	no	—
282	D₄	—	52	2	no	—
283	D₄	—	104	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	13	1	no	—
287	C₂×C₄	—	13	1	no	—
288	C₂×C₄	—	13	1	no	—
289	C₂×C₄	—	26	2	no	—
290	C₂×C₄	—	26	1	no	—
291	C₂×C₄	—	26	1	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	52	2	no	—
294	C₂×C₄	—	104	4	no	—
295	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
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×#6127	1	1	yes	characteristic
302	C₈	#41×D₁₃	1	1	yes	characteristic
303	C₂²	—	13	2	no	—
304	C₂²	—	26	1	no	—
305	C₂²	—	26	1	no	—
306	C₂²	—	104	2	no	—
307	C₂²	—	104	2	no	—
308	C₂²	—	104	2	no	—
309	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	26	1	no	—
314	C₄	—	26	1	no	—
315	C₄	—	26	1	no	—
316	C₄	—	104	2	no	—
317	C₄	#11718580	1	2	yes	—
318	C₄	#11719253	1	1	yes	characteristic
319	C₄	D₁₃×#135	1	1	yes	characteristic
320	C₂	—	26	2	no	—
321	C₂	—	208	2	no	—
322	C₂	#13125412	1	1	yes	characteristic
323	C₂	D₁₃×#508	1	1	yes	characteristic
324	C₂	C₂×#11718580	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×Q₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13116923	C₂
hypercenter	C₄×Q₃₂	D₁₃
hypocenter	C₁₃	#33735
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
1. #13125409	C₂	semidirect
2. #13117113	C₂	semidirect
3. #13117098	C₂	semidirect
4. #13117069	C₂	semidirect
5. #13116935	C₂	semidirect
6. #13116924	C₂	semidirect
8. C₄×#88068	C₂	semidirect
9. C₄×D₂₀₈	C₂	semidirect
10. C₄×C₁₆×D₁₃	C₂	semidirect
11. C₄×C₁₃×Q₃₂	C₂	semidirect
122. C₄×Q₃₂	D₁₃	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
#13125409	2
#13117113	2
#13117098	2
#13117069	2
#13116935	2
#13116924	2
#13116923	2
C₄×#88068	2
C₄×D₂₀₈	2
C₄×C₁₆×D₁₃	2
C₄×C₁₃×Q₃₂	2
#33735	13