Subgroups — #16019705

Order 3328.

646 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	#13127979	C₂	1	2	yes	maximal, complemented
2	#13126624	C₂	1	2	yes	maximal, complemented
3	#13117097	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116967	C₂	1	2	yes	maximal, complemented
5	#13116905	C₂	1	2	yes	maximal, complemented
6	#13116815	C₂	1	2	yes	maximal, complemented
7	#13116794	C₂	1	2	yes	maximal, complemented
8	#13116674	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2178	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719476	C₂²	1	2	yes	—
11	#11719474	C₂²	1	2	yes	complemented
12	#11719456	—	2	2	no	—
13	C₁₃×#508	C₂²	1	1	yes	characteristic
14	C₁₃×#443	C₂²	1	2	yes	complemented
15	#11718856	C₂²	1	2	yes	complemented
16	#11718854	C₂²	1	2	yes	—
17	#11718802	—	2	2	no	—
18	#11718579	—	2	2	no	—
19	#11718577	—	2	2	no	—
20	D₁₃×SD₃₂	—	2	2	no	—
21	#11718572	—	2	2	no	—
22	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic, complemented
23	C₂×#88040	C₂²	1	2	yes	complemented
24	C₂×D₂₀₈	C₂²	1	1	yes	characteristic
25	#11718562	C₂²	1	2	yes	complemented
26	#11718558	C₂²	1	2	yes	—
27	#11718557	C₂²	1	2	yes	complemented
28	#11718488	—	2	2	no	—
29	#11718486	—	2	2	no	—
30	#11718361	C₂²	1	1	yes	characteristic
31	#11718357	C₂²	1	2	yes	complemented
32	C₁₃×#368	C₂²	1	1	yes	characteristic, complemented
33	C₁₃×#359	C₂²	1	2	yes	complemented
34	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
35	#11718100	C₂²	1	2	yes	complemented
36	#11718088	C₂²	1	2	yes	complemented
37	#11718087	C₂²	1	2	yes	complemented
38	#11718055	C₂²	1	1	yes	characteristic, complemented
39	#88257	—	2	2	no	—
40	#88256	—	2	2	no	—
41	C₁₃×#135	C₂³	1	2	yes	—
42	C₁₃×#131	—	2	2	no	—
43	#88205	—	2	2	no	—
44	#88203	—	2	2	no	—
45	#88173	—	2	2	no	—
46	Q₁₆×D₁₃	—	2	2	no	—
47	#88169	—	4	2	no	—
48	#88168	—	4	2	no	—
49	#88166	—	2	2	no	—
50	D₈×D₁₃	—	2	2	no	—
51	#88163	—	2	2	no	—
52	#88163	—	4	2	no	—
53	#88162	—	2	2	no	—
54	#88162	—	4	2	no	—
55	C₂×Dic₅₂	C₂³	1	2	yes	—
56	#88158	—	4	2	no	—
57	#88158	D₄	1	1	yes	characteristic
58	C₂×D₁₀₄	C₂³	1	2	yes	—
59	C₂×#6095	—	2	2	no	—
60	C₂×#6095	—	4	2	no	—
61	#88155	C₂³	1	2	yes	—
62	#88129	—	2	2	no	—
63	C₁₃×Q₃₂	D₄	1	1	yes	characteristic
64	C₁₃×SD₃₂	D₄	1	2	yes	—
65	C₁₃×D₁₆	D₄	1	1	yes	characteristic
66	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
68	C₁₃×#108	C₂³	1	2	yes	—
69	C₁₃×#104	—	2	2	no	—
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	#88069	—	2	2	no	—
72	#88068	—	2	2	no	—
73	#88067	—	2	2	no	—
74	#88066	—	2	2	no	—
75	#88064	—	2	2	no	—
76	#88063	—	2	2	no	—
77	#88059	D₄	1	1	yes	characteristic
78	#88052	C₂³	1	2	yes	—
79	#88045	D₄	1	2	yes	—
80	Dic₁₀₄	—	4	1	no	—
81	#88040	—	2	2	no	—
82	#88040	—	2	2	no	—
83	D₂₀₈	—	4	1	no	—
84	#88038	—	2	2	no	—
85	C₁₆×D₁₃	—	2	2	no	—
86	#33857	—	13	1	no	maximal, Sylow
87	C₁₃×#41	—	2	2	no	—
88	#6131	—	4	2	no	—
89	Q₈×D₁₃	—	4	2	no	—
90	#6129	—	4	2	no	—
91	#6129	—	8	2	no	—
92	D₄×D₁₃	—	4	2	no	—
93	D₄×D₁₃	—	8	2	no	—
94	#6127	—	4	2	no	—
95	#6127	—	4	2	no	—
96	#6127	C₂×D₄	1	2	yes	—
97	C₂×D₅₂	—	2	2	no	—
98	C₂×D₅₂	—	4	2	no	—
99	C₂×Dic₂₆	—	2	2	no	—
100	C₂×Dic₂₆	—	4	2	no	—
101	C₁₃×Q₁₆	C₂×D₄	1	2	yes	—
102	C₁₃×SD₁₆	—	2	2	no	—
103	C₁₃×D₈	C₂×D₄	1	2	yes	—
104	C₁₃×M₁₆	—	2	2	no	—
105	C₁₃×M₁₆	—	4	2	no	—
106	C₂×C₈×C₁₃	—	4	2	no	—
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	#6107	—	4	2	no	—
111	#6106	—	4	2	no	—
112	#6105	—	4	2	no	—
113	#6104	—	4	2	no	—
114	#6103	—	8	2	no	—
115	#6099	C₂×D₄	1	2	yes	—
116	Dic₅₂	—	2	2	no	—
117	Dic₅₂	—	2	2	no	—
118	Dic₅₂	—	4	2	no	—
119	Dic₅₂	D₈	1	1	yes	characteristic
120	D₁₀₄	—	2	2	no	—
121	D₁₀₄	—	2	2	no	—
122	D₁₀₄	—	4	2	no	—
123	D₁₀₄	D₈	1	1	yes	characteristic
124	#6095	—	4	2	no	—
125	#6095	—	4	2	no	—
126	#6095	—	4	2	no	—
127	#6095	—	4	2	no	—
128	#6095	D₈	1	2	yes	—
129	#6094	—	2	2	no	—
130	C₈×D₁₃	—	2	2	no	—
131	C₁₃×C₁₆	—	2	1	no	—
132	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
133	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
134	#6090	—	2	2	no	—
135	#3416	—	13	2	no	—
136	#3290	—	13	2	no	—
137	#2249	—	13	1	no	—
138	#2223	—	13	2	no	—
139	#2212	—	13	2	no	—
140	#2195	—	13	2	no	—
141	#2192	—	13	2	no	—
142	#2178	D₁₃	1	1	yes	characteristic, complemented
143	#2170	—	13	1	no	—
144	C₂²×D₁₃	—	4	2	no	—
145	C₂²×D₁₃	—	8	2	no	—
146	Q₈×C₁₃	—	2	2	no	—
147	D₄×C₁₃	—	2	2	no	—
148	D₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	2	1	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
152	C₂²⋊D₁₃	—	2	2	no	—
153	C₂²⋊D₁₃	—	8	2	no	—
154	C₂²⋊D₁₃	—	8	2	no	—
155	C₂×Dic₁₃	—	4	2	no	—
156	C₂×Dic₁₃	—	8	2	no	—
157	D₅₂	—	4	2	no	—
158	D₅₂	—	4	2	no	—
159	D₅₂	—	8	2	no	—
160	D₅₂	C₂×D₈	1	2	yes	—
161	C₄×D₁₃	—	2	2	no	—
162	C₄×D₁₃	—	8	2	no	—
163	C₄×D₁₃	—	8	2	no	—
164	Dic₂₆	—	4	2	no	—
165	Dic₂₆	—	4	2	no	—
166	Dic₂₆	—	8	2	no	—
167	Dic₂₆	C₂×D₈	1	2	yes	—
168	C₈×C₁₃	—	4	2	no	—
169	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
170	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
171	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
172	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
173	#1055	—	2	2	no	—
174	#578	—	13	2	no	—
175	#577	—	26	2	no	—
176	#576	—	13	2	no	—
177	#510	—	13	2	no	—
178	#510	—	26	2	no	—
179	#509	—	13	2	no	—
180	#509	—	26	2	no	—
181	#508	—	26	2	no	—
182	#508	C₂×D₁₃	1	1	yes	characteristic
183	C₂×Q₃₂	—	13	1	no	—
184	C₂×SD₃₂	—	13	2	no	—
185	C₂×SD₃₂	—	26	2	no	—
186	C₂×D₁₆	—	13	1	no	—
187	#504	—	13	2	no	—
188	#495	—	13	2	no	—
189	#473	—	13	2	no	—
190	#472	—	13	2	no	—
191	#471	—	26	2	no	—
192	#455	—	26	2	no	—
193	#454	—	26	2	no	—
194	#443	—	13	1	no	—
195	#443	C₂×D₁₃	1	2	yes	complemented
196	#368	C₂×D₁₃	1	1	yes	characteristic, complemented
197	#364	—	13	1	no	—
198	#362	—	13	2	no	—
199	#361	—	13	2	no	—
200	#359	C₂×D₁₃	1	2	yes	complemented
201	#350	—	13	2	no	—
202	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
203	C₂²×C₁₃	—	4	2	no	—
204	C₂²×C₁₃	#576	1	1	yes	characteristic
205	C₂×D₁₃	—	2	2	no	—
206	C₂×D₁₃	—	8	2	no	—
207	C₂×D₁₃	—	16	2	no	—
208	C₂×D₁₃	—	32	2	no	—
209	C₄×C₁₃	—	4	1	no	—
210	C₄×C₁₃	—	4	2	no	—
211	C₄×C₁₃	D₄²	1	1	yes	characteristic
212	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
213	Dic₁₃	—	2	2	no	—
214	Dic₁₃	—	8	2	no	—
215	2₋¹⁺⁴	—	26	2	no	—
216	H(5,𝔽₂)	—	26	2	no	—
217	#137	—	26	2	no	—
218	#137	—	26	2	no	—
219	#137	—	52	2	no	—
220	#137	—	52	2	no	—
221	AGL(1,ℤ/8)	—	26	2	no	—
222	AGL(1,ℤ/8)	—	26	2	no	—
223	AGL(1,ℤ/8)	—	52	2	no	—
224	AGL(1,ℤ/8)	—	52	2	no	—
225	#135	—	13	1	no	—
226	#135	—	26	2	no	—
227	#135	—	26	2	no	—
228	#135	—	52	2	no	—
229	#135	C₂²×D₁₃	1	2	yes	—
230	C₂×Q₁₆	—	13	2	no	—
231	C₂×Q₁₆	—	26	2	no	—
232	C₂×SD₁₆	—	26	2	no	—
233	C₂×SD₁₆	—	52	2	no	—
234	C₂×D₈	—	13	2	no	—
235	C₂×D₈	—	26	2	no	—
236	#131	—	2	2	no	—
237	#131	—	13	2	no	—
238	#124	—	26	2	no	—
239	Q₃₂	—	26	2	no	—
240	Q₃₂	—	52	1	no	—
241	Q₃₂	D₅₂	1	1	yes	characteristic
242	SD₃₂	—	26	2	no	—
243	SD₃₂	—	26	2	no	—
244	SD₃₂	—	26	2	no	—
245	SD₃₂	—	26	2	no	—
246	SD₃₂	D₅₂	1	2	yes	—
247	D₁₆	—	26	2	no	—
248	D₁₆	—	52	1	no	—
249	D₁₆	D₅₂	1	1	yes	characteristic
250	M₃₂	—	13	2	no	—
251	M₃₂	—	26	2	no	—
252	C₂×C₁₆	—	26	2	no	—
253	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
254	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
255	#108	—	13	1	no	—
256	#108	C₂²×D₁₃	1	2	yes	—
257	#104	—	2	2	no	—
258	#104	—	13	2	no	—
259	#101	—	26	2	no	—
260	#100	—	26	2	no	—
261	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
262	C₂×C₁₃	—	2	1	no	—
263	C₂×C₁₃	—	8	2	no	—
264	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
265	D₁₃	—	4	2	no	—
266	D₁₃	—	16	2	no	—
267	#41	—	2	2	no	—
268	#41	—	13	2	no	—
269	#41	—	52	2	no	—
270	#41	—	52	2	no	—
271	#41	—	52	2	no	—
272	#41	—	52	2	no	—
273	#41	—	104	2	no	—
274	C₂×Q₈	—	26	2	no	—
275	C₂×Q₈	—	52	2	no	—
276	C₂×Q₈	—	52	2	no	—
277	C₂×D₄	—	26	2	no	—
278	C₂×D₄	—	52	2	no	—
279	C₂×D₄	—	52	2	no	—
280	C₂×D₄	—	104	2	no	—
281	Q₁₆	—	13	1	no	—
282	Q₁₆	—	26	2	no	—
283	Q₁₆	—	26	2	no	—
284	Q₁₆	—	52	2	no	—
285	Q₁₆	—	52	2	no	—
286	Q₁₆	C₂×D₅₂	1	2	yes	—
287	SD₁₆	—	2	2	no	—
288	SD₁₆	—	13	2	no	—
289	SD₁₆	—	52	2	no	—
290	SD₁₆	—	52	2	no	—
291	SD₁₆	—	52	2	no	—
292	SD₁₆	—	52	2	no	—
293	SD₁₆	—	52	2	no	—
294	SD₁₆	—	52	2	no	—
295	D₈	—	13	1	no	—
296	D₈	—	26	2	no	—
297	D₈	—	26	2	no	—
298	D₈	—	52	2	no	—
299	D₈	—	52	2	no	—
300	D₈	C₂×D₅₂	1	2	yes	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	4	2	no	—
303	M₁₆	—	13	2	no	—
304	M₁₆	—	26	2	no	—
305	C₂×C₈	—	4	2	no	—
306	C₂×C₈	—	26	2	no	—
307	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
309	C₂²⋊C₄	—	104	2	no	—
310	C₄²	D₄×D₁₃	1	1	yes	characteristic
311	C₁₆	—	2	1	no	—
312	C₁₆	—	26	2	no	—
313	C₁₆	C₂×D₅₂	1	1	yes	characteristic
314	C₁₆	C₂×D₅₂	1	1	yes	characteristic
315	C₁₃	#33857	1	1	yes	characteristic, complemented, Sylow
316	C₂³	—	52	2	no	—
317	C₂³	—	104	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	13	2	no	—
320	Q₈	—	52	2	no	—
321	Q₈	—	52	2	no	—
322	Q₈	—	104	2	no	—
323	D₄	—	2	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	13	2	no	—
326	D₄	—	26	2	no	—
327	D₄	—	52	2	no	—
328	D₄	—	52	2	no	—
329	D₄	—	104	2	no	—
330	D₄	—	104	2	no	—
331	D₄	—	104	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	26	2	no	—
335	C₂×C₄	—	52	2	no	—
336	C₂×C₄	—	104	2	no	—
337	C₂×C₄	—	104	2	no	—
338	C₂×C₄	—	104	2	no	—
339	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
340	C₈	—	4	2	no	—
341	C₈	—	26	2	no	—
342	C₈	D₈×D₁₃	1	1	yes	characteristic
343	C₈	D₈×D₁₃	1	1	yes	characteristic
344	C₈	C₂²×D₅₂	1	1	yes	characteristic
345	C₈	#88256	1	1	yes	characteristic
346	C₂²	—	4	2	no	—
347	C₂²	—	26	2	no	—
348	C₂²	—	104	2	no	—
349	C₂²	—	416	2	no	—
350	C₂²	—	208	2	no	—
351	C₂²	#11719474	1	1	yes	characteristic
352	C₄	—	4	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	26	2	no	—
355	C₄	—	104	2	no	—
356	C₄	D₄×D₅₂	1	1	yes	characteristic
357	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	8	2	no	—
360	C₂	—	52	2	no	—
361	C₂	—	208	2	no	—
362	C₂	D₈×D₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₅₂
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2178	C₂
socle	C₂×C₁₃	D₄×D₈
hypercenter	#2178	D₁₃
hypocenter	C₁₃	#33857
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2178

Complements

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

N	H ≅ G/N	type
1. #13127979	C₂	semidirect
2. #13126624	C₂	semidirect
3. #13117097	C₂	semidirect
4. #13116967	C₂	semidirect
5. #13116905	C₂	semidirect
6. #13116815	C₂	semidirect
7. #13116794	C₂	semidirect
8. #13116674	C₂	semidirect
9. C₁₃×#2178	C₂	semidirect
11. #11719474	C₂²	semidirect
14. C₁₃×#443	C₂²	semidirect
15. #11718856	C₂²	semidirect
22. C₂×Dic₁₀₄	C₂²	semidirect
23. C₂×#88040	C₂²	semidirect
25. #11718562	C₂²	semidirect
27. #11718557	C₂²	semidirect
31. #11718357	C₂²	semidirect
32. C₁₃×#368	C₂²	semidirect
33. C₁₃×#359	C₂²	semidirect
34. C₄×C₁₃×C₁₆	C₂²	semidirect
35. #11718100	C₂²	semidirect
36. #11718088	C₂²	semidirect
37. #11718087	C₂²	semidirect
38. #11718055	C₂²	semidirect
142. #2178	D₁₃	semidirect
195. #443	C₂×D₁₃	semidirect
196. #368	C₂×D₁₃	semidirect
200. #359	C₂×D₁₃	semidirect
202. C₄×C₁₆	C₂×D₁₃	semidirect
315. C₁₃	#33857	semidirect

Maximal subgroups

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

subgroup	index
#13127979	2
#13126624	2
#13117097	2
#13116967	2
#13116905	2
#13116815	2
#13116794	2
#13116674	2
C₁₃×#2178	2
#33857	13