Subgroups — #16017661

Order 3328.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125348	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718556	C₂	1	1	yes	maximal, characteristic
3	#13117011	C₂	1	4	yes	maximal, complemented
4	#13117006	C₂	1	4	yes	maximal, complemented
5	#13116610	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718103	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116573	C₂	1	1	yes	maximal, characteristic
8	#13116571	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#88161	—	2	1	no	—
11	C₂×#88158	C₂²	1	1	yes	characteristic
12	#11719222	—	2	1	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
15	C₁₃×#427	C₂²	1	1	yes	characteristic
16	C₂×#88065	—	2	1	no	—
17	#11718767	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×#88052	C₂²	1	1	yes	characteristic
20	#11718659	—	2	1	no	—
21	#11718556	—	2	1	no	—
22	#11718556	—	2	1	no	—
23	#11718556	C₂²	1	2	yes	—
24	#11718556	C₂²	1	2	yes	—
25	#11718539	—	2	1	no	—
26	#11718535	—	2	1	no	—
27	#11718519	C₂²	1	2	yes	complemented
28	#11718516	C₂²	1	2	yes	complemented
29	C₁₃×#365	C₂²	1	4	yes	complemented
30	#11718110	C₂²	1	4	yes	—
31	#11718103	C₂²	1	2	yes	complemented
32	#11718103	C₂²	1	2	yes	complemented
33	#11718086	C₂²	1	4	yes	complemented
34	#11718080	C₂²	1	4	yes	—
35	C₂×#6127	—	2	1	no	—
36	C₂×C₁₃×M₁₆	—	2	1	no	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#117	—	2	1	no	—
39	#88180	—	2	1	no	—
40	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
41	C₂²×#1055	D₄	1	1	yes	characteristic
42	#88161	—	2	2	no	—
43	#88161	—	2	2	no	—
44	#88161	—	4	1	no	—
45	#88161	—	4	1	no	—
46	C₂×Dic₅₂	Q₈	1	1	yes	characteristic, complemented
47	#88158	—	2	1	no	—
48	#88158	—	2	1	no	—
49	#88158	Q₈	1	2	yes	complemented
50	#88158	C₂³	1	2	yes	—
51	C₂×D₁₀₄	Q₈	1	1	yes	characteristic, complemented
52	C₂×#6095	—	2	1	no	—
53	C₂×#6094	—	4	1	no	—
54	C₂×C₈×D₁₃	—	4	1	no	—
55	#88144	—	2	2	no	—
56	#88142	—	2	2	no	—
57	#88132	—	4	2	no	—
58	C₄×Dic₂₆	—	4	2	no	—
59	C₁₃×M₃₂	—	2	1	no	—
60	C₁₃×M₃₂	C₂³	1	2	yes	—
61	C₂×C₁₃×C₁₆	—	2	1	no	—
62	C₁₃×#108	—	2	1	no	—
63	C₁₃×#108	C₂³	1	2	yes	—
64	C₁₃×#107	—	2	1	no	—
65	C₁₃×#106	C₂³	1	2	yes	—
66	#88065	—	2	2	no	—
67	#88065	—	2	2	no	—
68	#88058	—	2	1	no	—
69	#88057	C₂³	1	2	yes	—
70	C₈×Dic₁₃	D₄	1	1	yes	characteristic
71	C₈×Dic₁₃	D₄	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
73	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
74	#88052	—	2	1	no	—
75	#88052	C₂³	1	2	yes	—
76	C₂×#6090	—	2	1	no	—
77	#88048	—	2	2	no	—
78	#88047	—	2	2	no	—
79	#33637	—	13	1	no	maximal, Sylow
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₁₃	—	4	1	no	—
82	C₂²×Dic₁₃	—	2	1	no	—
83	#6127	—	2	2	no	—
84	#6127	—	2	2	no	—
85	#6127	—	4	1	no	—
86	#6127	—	4	1	no	—
87	C₂×D₅₂	—	2	1	no	—
88	C₂×C₄×D₁₃	—	4	1	no	—
89	C₂×Dic₂₆	—	2	1	no	—
90	C₂×Dic₂₆	—	4	2	no	—
91	C₁₃×M₁₆	—	2	2	no	—
92	C₁₃×M₁₆	—	4	1	no	—
93	C₂×C₈×C₁₃	—	4	1	no	—
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×Q₈	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	2	no	—
101	C₁₃×C₂²⋊C₄	—	4	1	no	—
102	C₄²×C₁₃	—	4	1	no	—
103	C₂²⋊Dic₁₃	—	4	1	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	C₄⋊Dic₁₃	—	4	2	no	—
106	#6101	—	4	2	no	—
107	#6101	—	4	2	no	—
108	C₄×Dic₁₃	—	2	1	no	—
109	C₄×Dic₁₃	—	4	1	no	—
110	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
111	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×#1055	SD₁₆	1	1	yes	characteristic
113	C₂×#1055	SD₁₆	1	1	yes	characteristic
114	C₂×#1055	SD₁₆	1	1	yes	characteristic
115	C₂×#1055	SD₁₆	1	1	yes	characteristic
116	C₂×#1055	C₂×D₄	1	1	yes	characteristic
117	C₂×#1055	C₂×D₄	1	1	yes	characteristic
118	Dic₅₂	—	2	1	no	—
119	Dic₅₂	C₂×Q₈	1	2	yes	—
120	D₁₀₄	—	2	1	no	—
121	D₁₀₄	C₂×Q₈	1	2	yes	—
122	#6095	—	2	2	no	—
123	#6095	—	2	2	no	—
124	#6094	—	4	2	no	—
125	#6094	—	4	2	no	—
126	C₈×D₁₃	—	4	2	no	—
127	C₈×D₁₃	—	4	2	no	—
128	C₁₃×C₁₆	—	2	2	no	—
129	#6090	—	2	2	no	—
130	#3160	—	13	1	no	—
131	C₂×#443	—	13	1	no	—
132	#2231	—	13	4	no	—
133	#2230	—	13	4	no	—
134	#2155	—	13	1	no	—
135	#2155	D₁₃	1	1	yes	characteristic, complemented
136	C₂×#364	—	13	1	no	—
137	#2145	—	13	1	no	—
138	#2145	—	13	1	no	—
139	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂²×D₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₁₃	—	4	2	no	—
149	C₂²⋊D₁₃	—	4	2	no	—
150	C₂×Dic₁₃	—	2	1	no	—
151	C₂×Dic₁₃	—	2	1	no	—
152	C₂×Dic₁₃	—	4	2	no	—
153	C₂×Dic₁₃	—	4	1	no	—
154	C₂×Dic₁₃	—	4	2	no	—
155	D₅₂	—	2	2	no	—
156	D₅₂	—	4	1	no	—
157	C₄×D₁₃	—	4	2	no	—
158	C₄×D₁₃	—	4	2	no	—
159	Dic₂₆	—	2	2	no	—
160	Dic₂₆	—	4	1	no	—
161	Dic₂₆	—	8	4	no	—
162	C₈×C₁₃	—	4	2	no	—
163	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
164	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
165	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
166	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
167	#1055	C₂×SD₁₆	1	1	yes	characteristic
168	#1055	C₂×SD₁₆	1	1	yes	characteristic
169	#1055	C₂×SD₁₆	1	1	yes	characteristic
170	#1055	C₂×SD₁₆	1	1	yes	characteristic
171	C₂×#135	—	13	1	no	—
172	C₂×#131	—	26	1	no	—
173	#533	—	26	1	no	—
174	C₂×M₃₂	—	13	1	no	—
175	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
176	#500	—	26	1	no	—
177	#499	—	13	2	no	—
178	#499	—	26	1	no	—
179	#498	—	13	2	no	—
180	#443	—	13	2	no	—
181	#443	—	13	2	no	—
182	#443	—	26	1	no	—
183	#443	—	26	1	no	—
184	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
185	#427	—	13	1	no	—
186	#427	—	26	1	no	—
187	#427	C₂×D₁₃	1	1	yes	characteristic
188	C₂×#104	—	26	1	no	—
189	C₂×C₄×C₈	—	13	1	no	—
190	#365	—	13	4	no	—
191	#365	C₂×D₁₃	1	4	yes	complemented
192	#364	—	13	2	no	—
193	#364	—	13	2	no	—
194	#360	—	13	4	no	—
195	#360	—	13	4	no	—
196	C₂²×C₁₃	—	4	1	no	—
197	C₂²×C₁₃	#476	1	1	yes	characteristic
198	C₂²×C₁₃	#476	1	1	yes	characteristic
199	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂×D₁₃	—	4	2	no	—
201	C₂×D₁₃	—	8	2	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	#476	1	1	yes	characteristic
204	C₄×C₁₃	#476	1	1	yes	characteristic
205	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
206	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
207	Dic₁₃	—	2	2	no	—
208	Dic₁₃	—	4	1	no	—
209	Dic₁₃	—	4	2	no	—
210	Dic₁₃	—	8	2	no	—
211	C₂×#41	—	26	1	no	—
212	#135	—	13	2	no	—
213	#135	—	13	2	no	—
214	#135	—	26	1	no	—
215	#135	—	26	1	no	—
216	C₂×Q₁₆	—	13	1	no	—
217	C₂×SD₁₆	—	26	1	no	—
218	C₂×D₈	—	13	1	no	—
219	#131	—	26	2	no	—
220	#131	—	26	2	no	—
221	#131	—	52	1	no	—
222	#131	—	52	1	no	—
223	C₂×M₁₆	—	2	1	no	—
224	C₂×M₁₆	—	52	1	no	—
225	C₂²×C₈	—	13	1	no	—
226	C₂²×C₈	—	52	1	no	—
227	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
228	C₄⋊Q₈	—	26	2	no	—
229	#125	—	26	2	no	—
230	C₂²⋊Q₈	—	52	2	no	—
231	C₄×Q₈	—	52	2	no	—
232	#117	—	2	1	no	—
233	#117	—	26	1	no	—
234	C₂×C₄²	—	13	1	no	—
235	M₃₂	—	2	1	no	—
236	M₃₂	—	13	2	no	—
237	M₃₂	—	26	1	no	—
238	M₃₂	C₂²×D₁₃	1	2	yes	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	26	1	no	—
241	#108	—	2	1	no	—
242	#108	C₂²×D₁₃	1	2	yes	—
243	#107	—	2	1	no	—
244	#107	—	26	2	no	—
245	#107	—	26	1	no	—
246	#106	—	13	2	no	—
247	#106	—	26	2	no	—
248	#106	C₂²×D₁₃	1	2	yes	—
249	#104	—	26	2	no	—
250	#104	—	26	2	no	—
251	C₄×C₈	—	13	1	no	—
252	C₄×C₈	—	13	1	no	—
253	C₄×C₈	—	13	1	no	—
254	C₄×C₈	—	13	1	no	—
255	C₂×C₁₃	—	2	1	no	—
256	C₂×C₁₃	—	2	1	no	—
257	C₂×C₁₃	—	2	1	no	—
258	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
259	D₁₃	—	8	2	no	—
260	#41	—	26	2	no	—
261	#41	—	26	2	no	—
262	#41	—	52	1	no	—
263	#41	—	52	1	no	—
264	C₂×Q₈	—	26	1	no	—
265	C₂×Q₈	—	52	2	no	—
266	C₂×D₄	—	26	1	no	—
267	C₂×D₄	—	52	1	no	—
268	C₂²×C₄	—	26	1	no	—
269	C₂²×C₄	—	52	1	no	—
270	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
271	Q₁₆	—	13	2	no	—
272	Q₁₆	—	26	1	no	—
273	SD₁₆	—	26	2	no	—
274	SD₁₆	—	26	2	no	—
275	D₈	—	13	2	no	—
276	D₈	—	26	1	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	4	1	no	—
279	M₁₆	—	52	2	no	—
280	M₁₆	—	52	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	13	1	no	—
283	C₂×C₈	—	13	1	no	—
284	C₂×C₈	—	13	1	no	—
285	C₂×C₈	—	13	1	no	—
286	C₂×C₈	—	13	1	no	—
287	C₂×C₈	—	13	1	no	—
288	C₂×C₈	—	52	2	no	—
289	C₂×C₈	—	52	2	no	—
290	C₂×C₈	#6127	1	1	yes	characteristic
291	C₂×C₈	#6127	1	1	yes	characteristic
292	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
293	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
294	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
295	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	26	2	no	—
298	C₄⋊C₄	—	52	2	no	—
299	C₄⋊C₄	—	52	2	no	—
300	C₄⋊C₄	—	52	2	no	—
301	C₂²⋊C₄	—	4	1	no	—
302	C₂²⋊C₄	—	52	1	no	—
303	C₄²	—	4	1	no	—
304	C₄²	—	13	1	no	—
305	C₄²	—	13	1	no	—
306	C₄²	—	26	1	no	—
307	C₄²	—	52	1	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	26	2	no	—
310	C₁₃	#33637	1	1	yes	characteristic, complemented, Sylow
311	C₂³	—	52	1	no	—
312	C₂³	#88168	1	1	yes	characteristic
313	Q₈	—	26	2	no	—
314	Q₈	—	52	1	no	—
315	Q₈	—	104	4	no	—
316	D₄	—	26	2	no	—
317	D₄	—	52	2	no	—
318	D₄	—	52	2	no	—
319	D₄	—	52	1	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	26	1	no	—
322	C₂×C₄	—	26	1	no	—
323	C₂×C₄	—	52	2	no	—
324	C₂×C₄	—	52	1	no	—
325	C₂×C₄	—	52	2	no	—
326	C₂×C₄	—	52	2	no	—
327	C₂×C₄	—	52	2	no	—
328	C₂×C₄	#88150	1	1	yes	characteristic
329	C₂×C₄	#88150	1	1	yes	characteristic
330	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
331	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
332	C₂×C₄	#88168	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
334	C₈	—	4	2	no	—
335	C₈	—	13	1	no	—
336	C₈	—	13	1	no	—
337	C₈	—	13	1	no	—
338	C₈	—	13	1	no	—
339	C₈	#88150	1	1	yes	characteristic
340	C₈	#88150	1	1	yes	characteristic
341	C₈	C₂×#6127	1	1	yes	characteristic
342	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	52	2	no	—
345	C₂²	—	104	2	no	—
346	C₂²	#11718531	1	1	yes	characteristic
347	C₂²	#11718531	1	1	yes	characteristic
348	C₂²	C₂×#88168	1	1	yes	characteristic
349	C₄	—	8	2	no	—
350	C₄	—	26	2	no	—
351	C₄	—	52	1	no	—
352	C₄	—	52	2	no	—
353	C₄	—	104	2	no	—
354	C₄	#11718531	1	1	yes	characteristic
355	C₄	#11718531	1	1	yes	characteristic
356	C₄	C₂×#88150	1	1	yes	characteristic
357	C₄	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	104	2	no	—
362	C₂	C₂×#11718531	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11718531
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2155	C₂
socle	C₂×C₁₃	C₂×#476
prime radical	C₂×#88158	C₂²
hypercenter	#2155	D₁₃
hypocenter	C₁₃	#33637
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
1. #13125348	C₂	semidirect
3. #13117011	C₂	semidirect
4. #13117006	C₂	semidirect
5. #13116610	C₂	semidirect
6. C₂×#11718103	C₂	semidirect
9. C₁₃×#2155	C₂	semidirect
27. #11718519	C₂²	semidirect
28. #11718516	C₂²	semidirect
29. C₁₃×#365	C₂²	semidirect
31. #11718103	C₂²	semidirect
32. #11718103	C₂²	semidirect
33. #11718086	C₂²	semidirect
46. C₂×Dic₅₂	Q₈	semidirect
49. #88158	Q₈	semidirect
51. C₂×D₁₀₄	Q₈	semidirect
135. #2155	D₁₃	semidirect
191. #365	C₂×D₁₃	semidirect
310. C₁₃	#33637	semidirect

Maximal subgroups

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

subgroup	index
#13125348	2
C₂×#11718556	2
#13117011	2
#13117006	2
#13116610	2
C₂×#11718103	2
#13116573	2
#13116571	2
C₁₃×#2155	2
#33637	13