Subgroups — #15834427

Order 3328.

644 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123854	C₂	1	2	yes	maximal, complemented
2	C₄×#88158	C₂	1	1	yes	maximal, characteristic
3	#13113751	C₂	1	2	yes	maximal, complemented
4	#13113716	C₂	1	4	yes	maximal, complemented
5	#13113656	C₂	1	2	yes	maximal, complemented
6	#13113599	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113271	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88158	C₄	1	1	yes	characteristic
10	C₄×#6127	C₂²	1	2	yes	—
11	C₁₃×#431	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718789	—	2	4	no	—
15	#11718774	C₄	1	2	yes	—
16	#11718767	C₄	1	1	yes	characteristic
17	#11718587	C₂²	1	2	yes	—
18	#11718368	—	2	2	no	—
19	#11718366	—	2	2	no	—
20	C₄×Dic₅₂	C₄	1	1	yes	characteristic
21	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
22	C₄×D₁₀₄	C₄	1	1	yes	characteristic
23	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
24	C₄×#6095	C₄	1	2	yes	—
25	C₄×#6095	C₂²	1	2	yes	—
26	#11718349	C₂²	1	4	yes	—
27	C₈×Dic₂₆	C₂²	1	4	yes	—
28	C₁₃×#322	C₂²	1	2	yes	—
29	C₁₃×#322	C₂²	1	2	yes	—
30	#11718051	C₂²	1	4	yes	—
31	#11718050	C₂²	1	2	yes	—
32	#11718049	C₂²	1	2	yes	—
33	#11718048	C₂²	1	4	yes	—
34	C₂×#6127	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₄×C₂²⋊D₁₃	—	2	2	no	—
40	C₄×C₂²⋊D₁₃	—	2	2	no	—
41	#88180	C₂×C₄	1	2	yes	—
42	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
43	#88158	—	2	4	no	—
44	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
45	C₂×#6095	C₂×C₄	1	2	yes	—
46	C₂×#6094	—	2	4	no	—
47	C₄×D₅₂	C₂×C₄	1	2	yes	—
48	C₄×D₅₂	C₂³	1	2	yes	—
49	#88126	—	2	2	no	—
50	C₄²×D₁₃	—	2	2	no	—
51	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
52	C₄×Dic₂₆	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₈	—	2	1	no	—
54	C₁₃×C₂²⋊C₈	—	2	1	no	—
55	C₁₃×#97	D₄	1	2	yes	—
56	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	2	yes	—
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	2	yes	—
62	C₂²⋊#1055	—	2	2	no	—
63	#88061	C₂×C₄	1	2	yes	—
64	#88061	C₂×C₄	1	2	yes	—
65	#88060	—	2	4	no	—
66	#88058	C₂×C₄	1	1	yes	characteristic
67	#88058	C₂×C₄	1	1	yes	characteristic
68	#88057	C₂×C₄	1	2	yes	—
69	#88056	C₂×C₄	1	2	yes	—
70	#88056	C₂×C₄	1	2	yes	—
71	#88054	—	2	4	no	—
72	C₈×Dic₁₃	—	2	4	no	—
73	C₄⋊#1055	C₂³	1	4	yes	—
74	#88043	—	2	2	no	—
75	#18043	—	13	1	no	maximal, Sylow
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₁₃	—	2	2	no	—
80	#6127	—	2	4	no	—
81	#6127	—	4	2	no	—
82	C₂×D₅₂	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₁₃	—	2	2	no	—
84	C₂×C₄×D₁₃	—	2	2	no	—
85	C₂×C₄×D₁₃	—	2	2	no	—
86	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
87	C₁₃×M₁₆	—	4	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	4	1	no	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₃	#41	1	1	yes	characteristic
99	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₂²⋊Dic₁₃	—	2	2	no	—
101	#6103	—	2	2	no	—
102	#6103	—	2	2	no	—
103	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
104	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
105	#6101	—	2	2	no	—
106	#6101	—	2	2	no	—
107	C₄×Dic₁₃	—	2	2	no	—
108	C₄×Dic₁₃	—	2	2	no	—
109	C₄×Dic₁₃	—	2	2	no	—
110	C₂×#1055	—	2	4	no	—
111	Dic₅₂	M₁₆	1	4	yes	complemented
112	D₁₀₄	—	2	2	no	—
113	#6095	—	2	4	no	—
114	#6094	—	4	8	no	—
115	#2971	—	13	2	no	—
116	C₄×#135	—	13	1	no	—
117	#1594	—	13	2	no	—
118	#1588	—	13	4	no	—
119	#1579	—	13	2	no	—
120	#1568	—	13	1	no	—
121	#1488	—	13	2	no	—
122	#1451	D₁₃	1	1	yes	characteristic, complemented
123	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂²×D₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	4	1	no	—
129	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂²⋊D₁₃	—	4	4	no	—
136	C₂×Dic₁₃	—	2	2	no	—
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	2	2	no	—
139	D₅₂	—	2	2	no	—
140	D₅₂	—	2	2	no	—
141	C₄×D₁₃	—	4	4	no	—
142	C₄×D₁₃	—	4	4	no	—
143	C₄×D₁₃	—	8	2	no	—
144	Dic₂₆	—	2	2	no	—
145	Dic₂₆	C₂×M₁₆	1	4	yes	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	4	no	—
150	C₈×C₁₃	—	4	2	no	—
151	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
152	#1055	—	4	4	no	—
153	C₂×#135	—	13	1	no	—
154	C₄×#41	—	13	2	no	—
155	C₈×Q₈	—	13	4	no	—
156	C₄×Q₁₆	—	13	1	no	—
157	C₄×Q₁₆	—	13	1	no	—
158	C₄×SD₁₆	—	13	2	no	—
159	C₄×SD₁₆	—	13	2	no	—
160	C₄×D₈	—	13	1	no	—
161	C₄×D₈	—	13	1	no	—
162	#436	—	13	4	no	—
163	#436	—	26	4	no	—
164	#432	—	13	2	no	—
165	#432	—	26	2	no	—
166	#431	C₂×D₁₃	1	1	yes	characteristic
167	#427	—	13	1	no	—
168	#416	—	13	2	no	—
169	C₄×M₁₆	—	26	2	no	—
170	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
172	#335	—	13	2	no	—
173	#334	—	13	2	no	—
174	Q₈⋊C₈	—	13	4	no	—
175	D₄⋊C₈	—	13	4	no	—
176	#322	C₂×D₁₃	1	2	yes	—
177	#322	C₂×D₁₃	1	2	yes	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
182	C₂×D₁₃	—	4	2	no	—
183	C₂×D₁₃	—	4	2	no	—
184	C₂×D₁₃	—	4	2	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
190	C₄×C₁₃	#568	1	1	yes	characteristic
191	C₄×C₁₃	#575	1	2	yes	—
192	Dic₁₃	—	2	4	no	—
193	Dic₁₃	—	4	2	no	—
194	Dic₁₃	—	4	2	no	—
195	C₂×#41	—	13	2	no	—
196	#135	—	26	4	no	—
197	C₂×Q₁₆	—	13	1	no	—
198	C₂×SD₁₆	—	13	2	no	—
199	C₂×D₈	—	13	1	no	—
200	C₂×M₁₆	—	26	4	no	—
201	C₂×M₁₆	D₅₂	1	2	yes	—
202	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
203	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
204	C₄×Q₈	—	13	2	no	—
205	C₄×Q₈	—	13	2	no	—
206	C₄×D₄	—	13	2	no	—
207	C₄×D₄	—	13	2	no	—
208	C₄×D₄	—	26	2	no	—
209	C₄×D₄	—	26	2	no	—
210	#117	—	13	2	no	—
211	#117	—	26	2	no	—
212	C₂×C₄²	—	26	2	no	—
213	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
214	#107	—	13	1	no	—
215	#107	—	13	1	no	—
216	#106	—	13	2	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	13	4	no	—
219	C₄⋊C₈	—	26	4	no	—
220	#103	—	13	2	no	—
221	#103	—	13	2	no	—
222	#102	—	13	2	no	—
223	#102	—	13	2	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	26	2	no	—
226	C₂²⋊C₈	—	26	4	no	—
227	#97	—	26	2	no	—
228	#97	D₅₂	1	2	yes	—
229	C₄×C₈	—	26	4	no	—
230	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
231	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₁₃	1	2	yes	—
233	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₁₃	1	2	yes	—
236	C₂×C₁₃	—	4	1	no	—
237	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
238	C₂×C₁₃	#2943	1	1	yes	characteristic
239	C₂×C₁₃	#2957	1	1	yes	characteristic
240	D₁₃	—	8	2	no	—
241	#41	—	26	4	no	—
242	#41	—	52	2	no	—
243	C₂×Q₈	—	13	2	no	—
244	C₂×D₄	—	13	2	no	—
245	C₂×D₄	—	26	2	no	—
246	C₂²×C₄	—	26	2	no	—
247	C₂²×C₄	—	26	2	no	—
248	C₂²×C₄	—	26	2	no	—
249	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
251	C₂²×C₄	#6127	1	1	yes	characteristic
252	Q₁₆	—	13	4	no	—
253	SD₁₆	—	26	4	no	—
254	D₈	—	26	2	no	—
255	M₁₆	—	4	2	no	—
256	M₁₆	—	52	8	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	26	4	no	—
261	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
263	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
265	C₂×C₈	C₂×D₅₂	1	4	yes	—
266	C₄⋊C₄	—	13	2	no	—
267	C₄⋊C₄	—	13	2	no	—
268	C₄⋊C₄	—	26	2	no	—
269	C₄⋊C₄	—	26	2	no	—
270	C₂²⋊C₄	—	26	2	no	—
271	C₂²⋊C₄	—	26	2	no	—
272	C₂²⋊C₄	—	26	2	no	—
273	C₄²	—	26	2	no	—
274	C₄²	—	26	2	no	—
275	C₄²	—	26	2	no	—
276	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
277	C₄²	C₂×D₅₂	1	1	yes	characteristic
278	C₄²	#6127	1	1	yes	characteristic
279	C₄²	C₂³×D₁₃	1	1	yes	characteristic
280	C₁₃	#18043	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	26	2	no	—
282	C₂³	C₄×D₅₂	1	1	yes	characteristic
283	Q₈	—	13	4	no	—
284	Q₈	—	26	2	no	—
285	D₄	—	26	2	no	—
286	D₄	—	26	2	no	—
287	D₄	—	52	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	26	2	no	—
295	C₂×C₄	—	52	4	no	—
296	C₂×C₄	—	52	4	no	—
297	C₂×C₄	—	104	2	no	—
298	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
299	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
300	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
303	C₂×C₄	C₂×#6127	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	4	no	—
308	C₈	—	4	2	no	—
309	C₈	—	52	4	no	—
310	C₈	M₁₆×D₁₃	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	52	2	no	—
315	C₂²	—	52	2	no	—
316	C₂²	—	52	2	no	—
317	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	26	4	no	—
323	C₄	—	52	2	no	—
324	C₄	—	52	2	no	—
325	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
326	C₄	#11719450	1	1	yes	characteristic
327	C₄	#11719453	1	2	yes	—
328	C₂	—	4	1	no	—
329	C₂	—	104	2	no	—
330	C₂	M₁₆×D₅₂	1	1	yes	characteristic
331	C₂	#13123638	1	1	yes	characteristic
332	C₂	#13123754	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1451	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88158	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1451	D₁₃
hypocenter	C₁₃	#18043
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
1. #13123854	C₂	semidirect
3. #13113751	C₂	semidirect
4. #13113716	C₂	semidirect
5. #13113656	C₂	semidirect
6. #13113599	C₂	semidirect
7. #13113271	C₂	semidirect
8. C₁₃×#1451	C₂	semidirect
111. Dic₅₂	M₁₆	semidirect
122. #1451	D₁₃	semidirect
280. C₁₃	#18043	semidirect

Maximal subgroups

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

subgroup	index
#13123854	2
C₄×#88158	2
#13113751	2
#13113716	2
#13113656	2
#13113599	2
#13113271	2
C₁₃×#1451	2
#18043	13