Subgroups — #16016388

Order 3328.

632 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718570	C₂	1	1	yes	maximal, characteristic
2	#13123754	C₂	1	1	yes	maximal, characteristic
3	#13123705	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116729	C₂	1	8	yes	maximal, complemented
5	#13116609	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718109	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116573	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
9	#11719450	—	2	1	no	—
10	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
11	C₂×#88158	C₂²	1	1	yes	characteristic
12	#11719221	—	2	1	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₁₃×#405	C₂²	1	1	yes	characteristic
16	#11718781	—	2	1	no	—
17	C₂×#88059	C₂²	1	1	yes	characteristic
18	#11718767	C₂²	1	1	yes	characteristic
19	#11718666	—	2	1	no	—
20	#11718570	C₄	1	4	yes	complemented
21	#11718570	C₂²	1	4	yes	—
22	C₂×Dic₁₀₄	C₄	1	2	yes	complemented
23	C₂×#88040	C₄	1	2	yes	complemented
24	#11718378	C₂²	1	4	yes	—
25	#11718377	C₂²	1	4	yes	complemented
26	#11718361	—	2	2	no	—
27	C₄×Dic₅₂	—	2	2	no	—
28	C₁₃×#347	C₂²	1	4	yes	complemented
29	#11718110	C₂²	1	4	yes	—
30	#11718109	C₂²	1	4	yes	complemented
31	#11718105	C₂²	1	4	yes	complemented
32	C₂×#6127	—	2	1	no	—
33	C₂²×Dic₂₆	—	2	1	no	—
34	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×#117	D₄	1	1	yes	characteristic
37	#88180	—	2	1	no	—
38	C₂×#6099	—	2	1	no	—
39	#88161	—	4	2	no	—
40	M₁₆×D₁₃	—	4	2	no	—
41	C₂×Dic₅₂	—	2	1	no	—
42	C₂×Dic₅₂	—	2	1	no	—
43	C₂×Dic₅₂	—	2	1	no	—
44	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
45	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
46	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
47	C₂×Dic₅₂	C₂³	1	2	yes	—
48	#88158	—	2	1	no	—
49	#88158	—	2	1	no	—
50	#88158	C₂×C₄	1	2	yes	—
51	#88158	C₂³	1	2	yes	—
52	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
53	C₂×#6095	—	2	1	no	—
54	#88155	—	2	4	no	—
55	#88141	—	4	2	no	—
56	#88131	—	4	2	no	—
57	C₄×Dic₂₆	—	2	4	no	—
58	C₁₃×M₃₂	—	2	1	no	—
59	C₁₃×M₃₂	C₂×C₄	1	2	yes	—
60	C₁₃×M₃₂	C₂³	1	2	yes	—
61	C₁₃×M₃₂	C₂³	1	2	yes	—
62	C₂×C₁₃×C₁₆	—	2	1	no	—
63	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
64	C₁₃×#97	C₂³	1	2	yes	—
65	C₄×C₈×C₁₃	D₄	1	2	yes	—
66	#88059	—	2	1	no	—
67	#88059	C₂³	1	2	yes	—
68	#88058	—	2	1	no	—
69	#88057	C₂³	1	2	yes	—
70	#88056	—	2	4	no	—
71	#88045	—	2	4	no	—
72	Dic₁₀₄	C₂×C₄	1	8	yes	—
73	#88040	C₂×C₄	1	8	yes	—
74	#33470	—	13	1	no	maximal, Sylow
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₁₃	—	4	1	no	—
77	C₂²×Dic₁₃	—	4	1	no	—
78	#6127	—	2	2	no	—
79	#6127	—	2	2	no	—
80	#6127	—	4	1	no	—
81	#6127	—	4	1	no	—
82	C₂×D₅₂	—	2	1	no	—
83	C₂×C₄×D₁₃	—	4	1	no	—
84	C₂×Dic₂₆	—	2	1	no	—
85	C₂×Dic₂₆	—	2	2	no	—
86	C₂×Dic₂₆	—	2	2	no	—
87	C₂×Dic₂₆	—	2	2	no	—
88	C₂×Dic₂₆	—	4	1	no	—
89	C₂×Dic₂₆	—	4	1	no	—
90	C₂×Dic₂₆	—	4	1	no	—
91	C₁₃×M₁₆	—	2	1	no	—
92	C₁₃×M₁₆	—	2	1	no	—
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	2	yes	—
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₁₃×C₄⋊C₄	—	2	1	no	—
101	C₁₃×C₂²⋊C₄	—	2	1	no	—
102	C₄²×C₁₃	C₂×D₄	1	2	yes	—
103	C₂²⋊Dic₁₃	—	4	1	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	#6101	—	4	4	no	—
106	C₄×Dic₁₃	—	4	4	no	—
107	C₄×Dic₁₃	—	4	1	no	—
108	#6099	—	2	2	no	—
109	#6099	—	4	1	no	—
110	C₂×#1055	—	4	1	no	—
111	Dic₅₂	—	2	1	no	—
112	Dic₅₂	—	2	2	no	—
113	Dic₅₂	—	2	2	no	—
114	Dic₅₂	—	2	2	no	—
115	Dic₅₂	C₂²×C₄	1	2	yes	—
116	Dic₅₂	C₂²×C₄	1	4	yes	—
117	D₁₀₄	—	2	1	no	—
118	D₁₀₄	C₂²×C₄	1	2	yes	—
119	#6095	—	2	2	no	—
120	#6095	—	2	2	no	—
121	#6094	—	4	4	no	—
122	C₈×D₁₃	—	4	4	no	—
123	C₁₃×C₁₆	—	2	2	no	—
124	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
125	C₂×#510	—	13	1	no	—
126	#2957	—	13	1	no	—
127	#2950	—	13	1	no	—
128	#2180	—	13	8	no	—
129	#2155	—	13	1	no	—
130	C₂×#362	—	13	1	no	—
131	#2145	—	13	1	no	—
132	#2110	D₁₃	1	1	yes	characteristic, complemented
133	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂²×D₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₁₃	—	4	2	no	—
143	C₂²⋊D₁₃	—	4	2	no	—
144	C₂×Dic₁₃	—	4	2	no	—
145	C₂×Dic₁₃	—	4	2	no	—
146	C₂×Dic₁₃	—	4	2	no	—
147	C₂×Dic₁₃	—	4	1	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	D₅₂	—	2	2	no	—
150	D₅₂	—	4	1	no	—
151	C₄×D₁₃	—	4	2	no	—
152	C₄×D₁₃	—	4	2	no	—
153	Dic₂₆	—	2	2	no	—
154	Dic₂₆	—	2	4	no	—
155	Dic₂₆	—	4	2	no	—
156	Dic₂₆	—	4	2	no	—
157	Dic₂₆	—	4	2	no	—
158	Dic₂₆	—	4	1	no	—
159	C₈×C₁₃	—	2	2	no	—
160	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
161	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
162	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
163	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
164	#1055	—	4	2	no	—
165	C₂×#135	—	13	1	no	—
166	C₂²×Q₁₆	—	13	1	no	—
167	#568	—	26	1	no	—
168	#519	—	26	1	no	—
169	#510	—	13	4	no	—
170	#510	—	13	4	no	—
171	C₂×Q₃₂	—	13	2	no	—
172	C₂×SD₃₂	—	13	2	no	—
173	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
174	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
175	#444	—	13	4	no	—
176	#443	—	26	2	no	—
177	#441	—	13	4	no	—
178	C₄×Q₁₆	—	26	2	no	—
179	C₂×#108	—	13	1	no	—
180	#427	—	13	1	no	—
181	#421	—	26	1	no	—
182	#419	—	26	1	no	—
183	#405	C₂×D₁₃	1	1	yes	characteristic
184	#365	—	13	4	no	—
185	#362	—	13	4	no	—
186	#360	—	13	4	no	—
187	#347	C₂×D₁₃	1	4	yes	complemented
188	C₂²×C₁₃	—	4	1	no	—
189	C₂²×C₁₃	#442	1	1	yes	characteristic
190	C₂²×C₁₃	#442	1	1	yes	characteristic
191	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₂×D₁₃	—	4	2	no	—
193	C₂×D₁₃	—	16	1	no	—
194	C₄×C₁₃	—	4	2	no	—
195	C₄×C₁₃	#442	1	1	yes	characteristic
196	C₄×C₁₃	#442	1	1	yes	characteristic
197	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
198	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
199	Dic₁₃	—	4	4	no	—
200	Dic₁₃	—	4	2	no	—
201	Dic₁₃	—	8	2	no	—
202	C₂×#41	—	26	1	no	—
203	C₂²×Q₈	—	26	1	no	—
204	#135	—	13	2	no	—
205	#135	—	13	2	no	—
206	#135	—	26	1	no	—
207	#135	—	26	1	no	—
208	C₂×Q₁₆	—	13	2	no	—
209	C₂×Q₁₆	—	13	2	no	—
210	C₂×Q₁₆	—	13	1	no	—
211	C₂×Q₁₆	—	13	2	no	—
212	C₂×Q₁₆	—	26	1	no	—
213	C₂×Q₁₆	—	26	1	no	—
214	C₂×Q₁₆	—	26	1	no	—
215	C₂×SD₁₆	—	26	1	no	—
216	C₂×D₈	—	13	1	no	—
217	#131	—	26	4	no	—
218	#131	—	52	2	no	—
219	C₂×M₁₆	—	26	1	no	—
220	C₂×M₁₆	—	52	2	no	—
221	C₂×M₁₆	D₅₂	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
223	C₄×Q₈	—	26	4	no	—
224	C₄×Q₈	—	52	2	no	—
225	#117	—	26	1	no	—
226	#117	—	52	2	no	—
227	#117	D₅₂	1	1	yes	characteristic
228	Q₃₂	—	13	8	no	—
229	SD₃₂	—	13	8	no	—
230	M₃₂	—	2	1	no	—
231	M₃₂	C₄×D₁₃	1	2	yes	—
232	M₃₂	C₂²×D₁₃	1	2	yes	—
233	M₃₂	C₂²×D₁₃	1	2	yes	—
234	C₂×C₁₆	—	2	1	no	—
235	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
236	#108	—	13	2	no	—
237	#108	—	26	1	no	—
238	#107	—	26	1	no	—
239	#106	—	13	2	no	—
240	#104	—	26	4	no	—
241	#103	—	26	4	no	—
242	#97	C₂²×D₁₃	1	2	yes	—
243	C₄×C₈	D₅₂	1	2	yes	—
244	C₂×C₁₃	—	2	1	no	—
245	C₂×C₁₃	—	2	1	no	—
246	C₂×C₁₃	—	2	1	no	—
247	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
248	D₁₃	—	8	2	no	—
249	#41	—	26	2	no	—
250	#41	—	26	2	no	—
251	#41	—	52	1	no	—
252	#41	—	52	1	no	—
253	C₂×Q₈	—	26	1	no	—
254	C₂×Q₈	—	26	2	no	—
255	C₂×Q₈	—	26	2	no	—
256	C₂×Q₈	—	26	2	no	—
257	C₂×Q₈	—	52	1	no	—
258	C₂×Q₈	—	52	1	no	—
259	C₂×Q₈	—	52	1	no	—
260	C₂×D₄	—	26	1	no	—
261	C₂×D₄	—	52	1	no	—
262	C₂²×C₄	—	52	1	no	—
263	C₂²×C₄	—	52	1	no	—
264	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
265	Q₁₆	—	13	4	no	—
266	Q₁₆	—	13	2	no	—
267	Q₁₆	—	26	2	no	—
268	Q₁₆	—	26	2	no	—
269	Q₁₆	—	26	1	no	—
270	Q₁₆	—	26	2	no	—
271	SD₁₆	—	26	2	no	—
272	SD₁₆	—	26	2	no	—
273	D₈	—	13	2	no	—
274	D₈	—	26	1	no	—
275	M₁₆	—	2	1	no	—
276	M₁₆	—	2	1	no	—
277	M₁₆	—	26	2	no	—
278	M₁₆	—	52	4	no	—
279	M₁₆	—	52	1	no	—
280	C₂×C₈	—	52	4	no	—
281	C₂×C₈	—	52	1	no	—
282	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
283	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
284	C₂×C₈	C₂×D₅₂	1	2	yes	—
285	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
286	C₂×C₈	#6127	1	1	yes	characteristic
287	C₂×C₈	#6127	1	1	yes	characteristic
288	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	26	2	no	—
291	C₄⋊C₄	—	52	4	no	—
292	C₂²⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	52	1	no	—
294	C₄²	—	52	4	no	—
295	C₄²	—	52	1	no	—
296	C₄²	C₂×D₅₂	1	2	yes	—
297	C₁₆	—	2	2	no	—
298	C₁₆	C₂×C₄×D₁₃	1	4	yes	—
299	C₁₃	#33470	1	1	yes	characteristic, complemented, Sylow
300	C₂³	—	52	1	no	—
301	C₂³	#88162	1	1	yes	characteristic
302	Q₈	—	26	2	no	—
303	Q₈	—	26	4	no	—
304	Q₈	—	52	2	no	—
305	Q₈	—	52	2	no	—
306	Q₈	—	52	2	no	—
307	Q₈	—	52	1	no	—
308	D₄	—	26	2	no	—
309	D₄	—	52	2	no	—
310	D₄	—	52	2	no	—
311	D₄	—	52	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	52	2	no	—
314	C₂×C₄	—	52	1	no	—
315	C₂×C₄	—	52	2	no	—
316	C₂×C₄	—	52	2	no	—
317	C₂×C₄	—	52	2	no	—
318	C₂×C₄	—	52	2	no	—
319	C₂×C₄	—	52	2	no	—
320	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
321	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
322	C₂×C₄	#88162	1	1	yes	characteristic
323	C₂×C₄	#88162	1	1	yes	characteristic
324	C₂×C₄	#88162	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	52	2	no	—
328	C₈	C₄×D₅₂	1	1	yes	characteristic
329	C₈	C₄×D₅₂	1	1	yes	characteristic
330	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
331	C₈	C₂×#6127	1	1	yes	characteristic
332	C₂²	—	4	1	no	—
333	C₂²	—	52	2	no	—
334	C₂²	—	208	1	no	—
335	C₂²	#11718372	1	1	yes	characteristic
336	C₂²	#11718372	1	1	yes	characteristic
337	C₂²	C₂×#88162	1	1	yes	characteristic
338	C₄	—	4	2	no	—
339	C₄	—	52	2	no	—
340	C₄	—	52	4	no	—
341	C₄	—	104	2	no	—
342	C₄	#11718372	1	1	yes	characteristic
343	C₄	#11718372	1	1	yes	characteristic
344	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
345	C₄	C₂×#88162	1	1	yes	characteristic
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	104	2	no	—
350	C₂	C₂×#11718372	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11718372
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2110	C₂
socle	C₂×C₁₃	C₂×#442
prime radical	C₂×#88158	C₂²
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2110	D₁₃
hypocenter	C₁₃	#33470
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2110

Complements

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

N	H ≅ G/N	type
3. #13123705	C₂	semidirect
4. #13116729	C₂	semidirect
5. #13116609	C₂	semidirect
6. C₂×#11718109	C₂	semidirect
8. C₁₃×#2110	C₂	semidirect
20. #11718570	C₄	semidirect
22. C₂×Dic₁₀₄	C₄	semidirect
23. C₂×#88040	C₄	semidirect
25. #11718377	C₂²	semidirect
28. C₁₃×#347	C₂²	semidirect
30. #11718109	C₂²	semidirect
31. #11718105	C₂²	semidirect
132. #2110	D₁₃	semidirect
187. #347	C₂×D₁₃	semidirect
299. C₁₃	#33470	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718570	2
#13123754	2
#13123705	2
#13116729	2
#13116609	2
C₂×#11718109	2
#13116573	2
C₁₃×#2110	2
#33470	13