Subgroups — #16807202

Order 3328.

674 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131598	C₂	1	2	yes	maximal, complemented
2	#13131232	C₂	1	4	yes	maximal, complemented
3	Q₃₂×#261	C₂	1	2	yes	maximal, complemented
4	D₁₆×#261	C₂	1	2	yes	maximal, complemented
5	C₂×#11718221	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131144	C₂	1	2	yes	maximal, complemented
7	#13131044	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₃×#508	C₂	1	1	yes	maximal, characteristic
9	#11719631	—	2	2	no	—
10	D₁₃×#135	C₂²	1	2	yes	—
11	#11719164	—	2	2	no	—
12	Q₁₆×#261	C₂²	1	4	yes	—
13	#11719106	—	2	4	no	—
14	D₈×#261	C₂²	1	4	yes	—
15	C₂×#88102	C₂²	1	2	yes	—
16	#11719089	C₂²	1	1	yes	characteristic
17	D₁₃⋊M₃₂	C₂²	1	1	yes	characteristic
18	C₁₃×#508	C₄	1	1	yes	characteristic, complemented
19	#11718855	C₄	1	2	yes	complemented
20	#11718580	C₄	1	1	yes	characteristic, complemented
21	D₁₃×Q₃₂	C₂²	1	1	yes	characteristic
22	#11718577	C₄	1	2	yes	complemented
23	D₁₃×SD₃₂	C₂²	1	2	yes	—
24	#11718573	C₄	1	1	yes	characteristic, complemented
25	D₁₃×D₁₆	C₂²	1	1	yes	characteristic
26	#11718565	C₄	1	1	yes	characteristic, complemented
27	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
28	#11718280	C₂²	1	4	yes	—
29	#11718276	C₂²	1	4	yes	—
30	#11718221	C₂²	1	2	yes	—
31	#11718221	C₂²	1	2	yes	—
32	#11718216	C₂²	1	2	yes	—
33	C₁₆×#261	C₂²	1	2	yes	—
34	#41×D₁₃	—	2	2	no	—
35	Q₈×#261	—	2	4	no	—
36	D₄×#261	—	2	4	no	—
37	D₄×#261	—	4	4	no	—
38	#88237	—	4	2	no	—
39	#88237	D₄	1	1	yes	characteristic
40	C₂×C₄⋊#261	—	2	2	no	—
41	C₂×C₄⋊#261	—	4	2	no	—
42	D₁₃⋊M₁₆	D₄	1	1	yes	characteristic
43	C₁₃×#135	C₂×C₄	1	2	yes	complemented
44	#88204	—	2	2	no	—
45	#88173	C₂×C₄	1	2	yes	complemented
46	Q₁₆×D₁₃	C₂³	1	2	yes	—
47	#88170	—	2	2	no	—
48	SD₁₆×D₁₃	—	2	2	no	—
49	#88166	C₂×C₄	1	2	yes	complemented
50	D₈×D₁₃	C₂³	1	2	yes	—
51	#88158	C₂×C₄	1	2	yes	complemented
52	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
53	#88117	—	2	4	no	—
54	#88115	—	2	4	no	—
55	#88106	—	2	1	no	—
56	#88102	—	2	2	no	—
57	#88102	C₂³	1	4	yes	—
58	#88100	D₄	1	2	yes	—
59	C₈×#261	C₂³	1	2	yes	—
60	#88098	—	2	1	no	—
61	D₁₃⋊C₁₆	C₂³	1	2	yes	—
62	C₁₃×Q₃₂	C₂×C₄	1	1	yes	characteristic, complemented
63	C₁₃×SD₃₂	C₂×C₄	1	2	yes	complemented
64	C₁₃×D₁₆	C₂×C₄	1	1	yes	characteristic, complemented
65	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic, complemented
66	#88069	C₂×C₄	1	2	yes	complemented
67	#88068	C₂×C₄	1	2	yes	complemented
68	#88067	C₂×C₄	1	2	yes	complemented
69	#88066	C₂×C₄	1	2	yes	complemented
70	C₂×#6090	C₂×C₄	1	1	yes	characteristic, complemented
71	Dic₁₀₄	C₂×C₄	1	1	yes	characteristic, complemented
72	#88040	C₂×C₄	1	2	yes	complemented
73	D₂₀₈	C₂×C₄	1	1	yes	characteristic, complemented
74	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic, complemented
75	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic, complemented
76	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
77	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
78	#33459	—	13	1	no	maximal, Sylow
79	C₂²×#261	—	4	2	no	—
80	C₂²×#261	—	4	4	no	—
81	C₁₃×#41	—	2	2	no	—
82	#6131	—	2	2	no	—
83	Q₈×D₁₃	—	2	2	no	—
84	#6129	—	2	2	no	—
85	#6129	—	4	2	no	—
86	D₄×D₁₃	—	2	2	no	—
87	D₄×D₁₃	—	4	2	no	—
88	#6127	—	2	2	no	—
89	#6127	—	4	2	no	—
90	C₂×C₄×D₁₃	—	4	2	no	—
91	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂²⋊#261	—	4	4	no	—
93	C₂²⋊#261	D₈	1	2	yes	—
94	#6122	—	2	1	no	—
95	C₄⋊#261	—	2	4	no	—
96	C₄⋊#261	—	4	4	no	—
97	C₄⋊#261	—	4	2	no	—
98	C₄⋊#261	D₈	1	1	yes	characteristic
99	C₄⋊#261	D₈	1	1	yes	characteristic
100	C₄×#261	—	4	4	no	—
101	C₄×#261	C₂×D₄	1	2	yes	—
102	#6118	—	2	1	no	—
103	D₁₃⋊C₈	C₂×D₄	1	2	yes	—
104	C₁₃×Q₁₆	C₂²×C₄	1	2	yes	—
105	C₁₃×SD₁₆	—	2	2	no	—
106	C₁₃×D₈	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	#6107	—	2	2	no	—
109	#6106	—	2	2	no	—
110	#6105	—	2	2	no	—
111	#6104	—	2	2	no	—
112	C₂×#1055	#41	1	1	yes	characteristic
113	Dic₅₂	C₂²×C₄	1	2	yes	—
114	D₁₀₄	C₂²×C₄	1	2	yes	—
115	#6095	—	2	2	no	—
116	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
118	C₈×D₁₃	#41	1	1	yes	characteristic
119	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
120	#6092	—	2	2	no	—
121	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
122	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
123	#6090	C₂²×C₄	1	1	yes	characteristic
124	#6090	C₂²×C₄	1	1	yes	characteristic
125	C₂×#508	—	13	1	no	—
126	#2951	—	13	2	no	—
127	#2175	—	13	4	no	—
128	C₄×Q₃₂	—	13	2	no	—
129	C₄×D₁₆	—	13	2	no	—
130	C₂×#366	—	13	1	no	—
131	#2139	—	13	2	no	—
132	#2108	—	13	1	no	—
133	C₂²×D₁₃	—	4	2	no	—
134	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×#261	—	4	4	no	—
136	C₂×#261	—	16	4	no	—
137	C₂×#261	—	16	2	no	—
138	C₂×#261	C₂×D₈	1	2	yes	—
139	C₂×#261	C₂×D₈	1	2	yes	—
140	Q₈×C₁₃	—	2	2	no	—
141	D₄×C₁₃	—	2	2	no	—
142	D₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
145	C₂²⋊D₁₃	—	4	2	no	—
146	C₂²⋊D₁₃	—	4	2	no	—
147	C₂²⋊D₁₃	—	4	2	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	#135	1	1	yes	characteristic
150	D₅₂	—	2	2	no	—
151	D₅₂	—	4	2	no	—
152	C₄×D₁₃	—	4	2	no	—
153	C₄×D₁₃	—	4	2	no	—
154	C₄×D₁₃	—	4	2	no	—
155	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
156	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
157	C₄×D₁₃	#135	1	1	yes	characteristic
158	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
159	Dic₂₆	—	2	2	no	—
160	Dic₂₆	—	4	2	no	—
161	#1057	—	2	2	no	—
162	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
163	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
164	#1055	C₄×D₄	1	1	yes	characteristic
165	#1055	C₂×#41	1	1	yes	characteristic
166	C₂×#135	—	13	2	no	—
167	#520	—	26	2	no	—
168	#508	—	13	2	no	—
169	#508	—	13	1	no	—
170	#508	—	13	2	no	—
171	#508	—	13	1	no	—
172	#508	—	13	1	no	—
173	#508	#261	1	1	yes	characteristic, complemented
174	C₂×Q₃₂	—	13	1	no	—
175	C₂×SD₃₂	—	13	2	no	—
176	C₂×D₁₆	—	13	1	no	—
177	C₂×M₃₂	—	13	1	no	—
178	C₂²×C₁₆	—	13	1	no	—
179	#440	—	26	4	no	—
180	C₄×Q₁₆	—	13	4	no	—
181	C₄×D₈	—	13	4	no	—
182	C₂×#107	—	13	2	no	—
183	#417	—	26	2	no	—
184	#405	—	13	1	no	—
185	#366	—	13	2	no	—
186	#366	—	13	2	no	—
187	#358	—	13	4	no	—
188	#357	—	13	4	no	—
189	#346	—	13	2	no	—
190	C₄×C₁₆	—	13	2	no	—
191	C₂²×C₁₃	—	4	2	no	—
192	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
193	C₂×D₁₃	—	2	1	no	—
194	C₂×D₁₃	—	2	1	no	—
195	C₂×D₁₃	—	4	2	no	—
196	C₂×D₁₃	—	8	2	no	—
197	C₂×D₁₃	—	8	2	no	—
198	C₂×D₁₃	C₄×D₈	1	1	yes	characteristic
199	C₂×D₁₃	C₂²×D₈	1	1	yes	characteristic
200	#261	—	2	2	no	—
201	#261	—	2	2	no	—
202	#261	—	8	4	no	—
203	C₄×C₁₃	—	4	2	no	—
204	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
205	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
206	Dic₁₃	—	4	2	no	—
207	Dic₁₃	C₄×D₈	1	1	yes	characteristic
208	Dic₁₃	C₂×#135	1	1	yes	characteristic
209	C₂×#41	—	26	2	no	—
210	#135	—	13	2	no	—
211	#135	—	13	2	no	—
212	#135	—	13	2	no	—
213	#135	—	26	2	no	—
214	#135	—	26	2	no	—
215	#135	C₂×#261	1	2	yes	complemented
216	C₂×Q₁₆	—	13	2	no	—
217	C₂×SD₁₆	—	26	2	no	—
218	C₂×D₈	—	13	2	no	—
219	C₂×M₁₆	—	13	1	no	—
220	C₂²×C₈	—	13	1	no	—
221	C₄×Q₈	—	26	4	no	—
222	C₄×D₄	—	26	4	no	—
223	C₄×D₄	—	52	4	no	—
224	#117	—	13	1	no	—
225	#117	—	52	2	no	—
226	C₂×C₄⋊C₄	—	26	2	no	—
227	C₂×C₄⋊C₄	—	52	2	no	—
228	Q₃₂	—	13	2	no	—
229	Q₃₂	—	13	1	no	—
230	Q₃₂	C₂×#261	1	1	yes	characteristic, complemented
231	SD₃₂	—	13	2	no	—
232	SD₃₂	—	13	2	no	—
233	SD₃₂	—	13	2	no	—
234	SD₃₂	C₂×#261	1	2	yes	complemented
235	D₁₆	—	13	2	no	—
236	D₁₆	—	13	1	no	—
237	D₁₆	C₂×#261	1	1	yes	characteristic, complemented
238	M₃₂	—	26	1	no	—
239	M₃₂	—	26	1	no	—
240	C₂×C₁₆	—	13	1	no	—
241	C₂×C₁₆	—	13	1	no	—
242	C₂×C₁₆	—	13	1	no	—
243	C₂×C₁₆	—	13	1	no	—
244	C₂×C₁₆	—	13	1	no	—
245	C₂×C₁₆	—	13	2	no	—
246	C₂×C₁₆	C₂×#261	1	1	yes	characteristic, complemented
247	#107	—	13	4	no	—
248	#107	—	26	2	no	—
249	#103	—	26	4	no	—
250	#102	—	26	4	no	—
251	#97	—	13	2	no	—
252	C₄×C₈	—	13	2	no	—
253	C₂×C₁₃	—	2	1	no	—
254	C₂×C₁₃	—	8	2	no	—
255	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
256	D₁₃	—	2	1	no	—
257	D₁₃	—	8	2	no	—
258	D₁₃	#3415	1	1	yes	characteristic
259	D₁₃	#3417	1	1	yes	characteristic
260	#41	—	2	2	no	—
261	#41	—	26	2	no	—
262	#41	—	26	2	no	—
263	#41	—	26	2	no	—
264	#41	—	52	2	no	—
265	#41	—	52	2	no	—
266	C₂×Q₈	—	26	2	no	—
267	C₂×D₄	—	26	2	no	—
268	C₂×D₄	—	52	2	no	—
269	C₂²×C₄	—	13	1	no	—
270	C₂²×C₄	—	52	2	no	—
271	C₂²×C₄	—	52	4	no	—
272	C₂²×C₄	—	52	2	no	—
273	Q₁₆	—	13	2	no	—
274	Q₁₆	—	26	2	no	—
275	Q₁₆	C₂²×#261	1	2	yes	—
276	SD₁₆	—	2	2	no	—
277	SD₁₆	—	26	2	no	—
278	SD₁₆	—	26	2	no	—
279	SD₁₆	—	26	2	no	—
280	D₈	—	13	2	no	—
281	D₈	—	26	2	no	—
282	D₈	C₂²×#261	1	2	yes	—
283	M₁₆	—	26	1	no	—
284	M₁₆	—	26	1	no	—
285	C₂×C₈	—	13	1	no	—
286	C₂×C₈	—	13	1	no	—
287	C₂×C₈	—	13	1	no	—
288	C₂×C₈	—	13	1	no	—
289	C₂×C₈	—	13	1	no	—
290	C₂×C₈	—	13	2	no	—
291	C₂×C₈	C₂²×#261	1	1	yes	characteristic
292	C₄⋊C₄	—	13	1	no	—
293	C₄⋊C₄	—	13	1	no	—
294	C₄⋊C₄	—	26	4	no	—
295	C₄⋊C₄	—	52	2	no	—
296	C₄⋊C₄	—	52	4	no	—
297	C₂²⋊C₄	—	13	2	no	—
298	C₂²⋊C₄	—	52	4	no	—
299	C₄²	—	13	2	no	—
300	C₄²	—	52	4	no	—
301	C₁₆	—	13	1	no	—
302	C₁₆	—	13	1	no	—
303	C₁₆	—	26	2	no	—
304	C₁₆	C₂²×#261	1	1	yes	characteristic
305	C₁₆	C₂²×#261	1	1	yes	characteristic
306	C₁₃	#33459	1	1	yes	characteristic, complemented, Sylow
307	C₂³	—	13	1	no	—
308	C₂³	—	52	2	no	—
309	Q₈	—	2	2	no	—
310	Q₈	—	26	2	no	—
311	Q₈	—	52	2	no	—
312	D₄	—	2	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	26	2	no	—
315	D₄	—	52	2	no	—
316	D₄	—	52	2	no	—
317	D₄	—	52	2	no	—
318	D₄	—	52	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	13	1	no	—
321	C₂×C₄	—	13	1	no	—
322	C₂×C₄	—	13	1	no	—
323	C₂×C₄	—	13	1	no	—
324	C₂×C₄	—	13	1	no	—
325	C₂×C₄	—	13	2	no	—
326	C₂×C₄	—	13	2	no	—
327	C₂×C₄	—	52	2	no	—
328	C₂×C₄	—	52	2	no	—
329	C₂×C₄	—	52	2	no	—
330	C₂×C₄	—	52	2	no	—
331	C₂×C₄	—	52	4	no	—
332	C₂×C₄	—	208	4	no	—
333	C₂×C₄	—	208	2	no	—
334	C₂×C₄	D₄×#261	1	1	yes	characteristic
335	C₈	—	13	1	no	—
336	C₈	—	13	1	no	—
337	C₈	—	26	2	no	—
338	C₈	D₄×#261	1	1	yes	characteristic
339	C₈	C₂³×#261	1	1	yes	characteristic
340	C₂²	—	4	2	no	—
341	C₂²	—	13	1	no	—
342	C₂²	—	13	1	no	—
343	C₂²	—	26	1	no	—
344	C₂²	—	26	1	no	—
345	C₂²	—	52	2	no	—
346	C₂²	—	104	2	no	—
347	C₂²	—	104	2	no	—
348	C₂²	D₈×#261	1	1	yes	characteristic
349	C₄	—	4	2	no	—
350	C₄	—	13	1	no	—
351	C₄	—	13	1	no	—
352	C₄	—	26	2	no	—
353	C₄	—	26	2	no	—
354	C₄	—	52	2	no	—
355	C₄	—	104	4	no	—
356	C₄	D₈×#261	1	1	yes	characteristic
357	C₄	C₂×D₄×#261	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	8	2	no	—
360	C₂	—	13	1	no	—
361	C₂	—	13	1	no	—
362	C₂	—	26	1	no	—
363	C₂	—	104	2	no	—
364	C₂	C₂×D₈×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#261
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₈	C₂³×#261
Fitting	C₁₃×#508	C₄
socle	C₂×C₁₃	C₂×C₄×D₈
prime radical	D₁₃×#508	C₂
cosocle	C₈×D₁₃	C₂⁴
hypercenter	#508	#261
hypocenter	C₁₃	#33459
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #13131598	C₂	semidirect
2. #13131232	C₂	semidirect
3. Q₃₂×#261	C₂	semidirect
4. D₁₆×#261	C₂	semidirect
5. C₂×#11718221	C₂	semidirect
6. #13131144	C₂	semidirect
7. #13131044	C₂	semidirect
18. C₁₃×#508	C₄	semidirect
19. #11718855	C₄	semidirect
20. #11718580	C₄	semidirect
22. #11718577	C₄	semidirect
24. #11718573	C₄	semidirect
26. #11718565	C₄	semidirect
43. C₁₃×#135	C₂×C₄	semidirect
45. #88173	C₂×C₄	semidirect
49. #88166	C₂×C₄	semidirect
51. #88158	C₂×C₄	semidirect
62. C₁₃×Q₃₂	C₂×C₄	semidirect
63. C₁₃×SD₃₂	C₂×C₄	semidirect
64. C₁₃×D₁₆	C₂×C₄	semidirect
65. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
66. #88069	C₂×C₄	semidirect
67. #88068	C₂×C₄	semidirect
68. #88067	C₂×C₄	semidirect
69. #88066	C₂×C₄	semidirect
70. C₂×#6090	C₂×C₄	semidirect
71. Dic₁₀₄	C₂×C₄	semidirect
72. #88040	C₂×C₄	semidirect
73. D₂₀₈	C₂×C₄	semidirect
74. C₁₆×D₁₃	C₂×C₄	semidirect
75. C₁₆×D₁₃	C₂×C₄	semidirect
173. #508	#261	semidirect
215. #135	C₂×#261	semidirect
230. Q₃₂	C₂×#261	semidirect
234. SD₃₂	C₂×#261	semidirect
237. D₁₆	C₂×#261	semidirect
246. C₂×C₁₆	C₂×#261	semidirect
306. C₁₃	#33459	semidirect

Maximal subgroups

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

subgroup	index
#13131598	2
#13131232	2
Q₃₂×#261	2
D₁₆×#261	2
C₂×#11718221	2
#13131144	2
#13131044	2
D₁₃×#508	2
#33459	13