Subgroups — #508×#6923 #30624284

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×#6923	C₂	1	2	yes	maximal, complemented
2	Q₃₂×#6923	C₂	1	2	yes	maximal, complemented
3	SD₃₂×#6923	C₂	1	4	yes	maximal, complemented
4	D₁₆×#6923	C₂	1	2	yes	maximal, complemented
5	#21462170	C₂	1	1	yes	maximal, characteristic, complemented
6	#21462126	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₆×#6923	C₂	1	1	yes	maximal, characteristic, complemented
8	#508×D₆₁	C₂	1	1	yes	maximal, characteristic
9	#41×#6923	—	2	2	no	—
10	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	Q₁₆×#6923	C₂²	1	4	yes	—
12	SD₁₆×#6923	—	2	4	no	—
13	D₈×#6923	C₂²	1	4	yes	—
14	#16840772	C₂²	1	2	yes	—
15	#16840752	—	2	2	no	—
16	C₂×C₈×#6923	C₂²	1	1	yes	characteristic
17	#16840689	C₂²	1	2	yes	—
18	#16840688	C₂²	1	2	yes	—
19	#16840675	C₂²	1	4	yes	—
20	#16840674	C₂²	1	4	yes	—
21	C₁₆×#6923	C₂²	1	2	yes	—
22	C₁₆×#6923	C₂²	1	2	yes	—
23	#135×D₆₁	C₂²	1	2	yes	—
24	#16840195	C₄	1	1	yes	characteristic, complemented
25	#16840194	C₄	1	1	yes	characteristic, complemented
26	#16840193	C₄	1	2	yes	complemented
27	#16840192	C₄	1	2	yes	complemented
28	#16840191	C₄	1	1	yes	characteristic, complemented
29	Q₃₂×D₆₁	C₂²	1	1	yes	characteristic
30	SD₃₂×D₆₁	C₂²	1	2	yes	—
31	D₁₆×D₆₁	C₂²	1	1	yes	characteristic
32	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
33	C₆₁×#508	C₄	1	1	yes	characteristic, complemented
34	C₂×D₆₁⋊C₈	D₄	1	1	yes	characteristic
35	Q₈×#6923	—	2	4	no	—
36	D₄×#6923	—	2	4	no	—
37	D₄×#6923	—	4	4	no	—
38	#14519856	—	2	2	no	—
39	#14519856	—	4	2	no	—
40	C₂×C₄×#6923	—	4	2	no	—
41	C₂×C₄×#6923	D₄	1	1	yes	characteristic
42	D₆₁⋊C₁₆	—	2	1	no	—
43	D₆₁⋊C₁₆	C₂³	1	2	yes	—
44	C₂×#11758187	—	2	1	no	—
45	#14519846	C₂³	1	4	yes	—
46	#14519845	—	2	2	no	—
47	#14519840	—	2	4	no	—
48	#14519839	—	2	4	no	—
49	C₈×#6923	D₄	1	2	yes	—
50	C₈×#6923	C₂³	1	2	yes	—
51	#41×D₆₁	—	2	2	no	—
52	#14519800	C₂×C₄	1	2	yes	complemented
53	#14519799	C₂×C₄	1	2	yes	complemented
54	#14519798	—	2	2	no	—
55	#14519797	—	2	2	no	—
56	#14519796	C₂×C₄	1	2	yes	complemented
57	Q₁₆×D₆₁	C₂³	1	2	yes	—
58	SD₁₆×D₆₁	—	2	2	no	—
59	D₈×D₆₁	C₂³	1	2	yes	—
60	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
61	Dic₄₈₈	C₂×C₄	1	1	yes	characteristic, complemented
62	#14519723	C₂×C₄	1	2	yes	complemented
63	#14519722	C₂×C₄	1	2	yes	complemented
64	#14519721	C₂×C₄	1	2	yes	complemented
65	#14519720	C₂×C₄	1	2	yes	complemented
66	D₉₇₆	C₂×C₄	1	1	yes	characteristic, complemented
67	#14519718	C₂×C₄	1	2	yes	complemented
68	C₂×#11758185	C₂×C₄	1	1	yes	characteristic, complemented
69	C₁₆×D₆₁	C₂×C₄	1	1	yes	characteristic, complemented
70	C₁₆×D₆₁	C₂×C₄	1	1	yes	characteristic, complemented
71	C₁₆×D₆₁	C₂³	1	1	yes	characteristic
72	C₁₆×D₆₁	C₂³	1	1	yes	characteristic
73	#135×C₆₁	C₂×C₄	1	2	yes	complemented
74	Q₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
75	SD₃₂×C₆₁	C₂×C₄	1	2	yes	complemented
76	D₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
77	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
78	C₂²×#6923	—	4	4	no	—
79	C₂²×#6923	D₈	1	2	yes	—
80	#41×C₆₁	—	2	2	no	—
81	#11758226	—	2	2	no	—
82	Q₈×D₆₁	—	2	2	no	—
83	#11758224	—	2	2	no	—
84	#11758224	—	4	2	no	—
85	D₄×D₆₁	—	2	2	no	—
86	D₄×D₆₁	—	4	2	no	—
87	#11758222	—	2	2	no	—
88	#11758222	—	4	2	no	—
89	C₂×C₄×D₆₁	—	4	2	no	—
90	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂²⋊#6923	—	4	2	no	—
92	C₂²⋊#6923	—	4	4	no	—
93	C₂×#92420	—	2	1	no	—
94	C₄⋊#6923	—	2	4	no	—
95	C₄⋊#6923	—	4	4	no	—
96	C₄×#6923	—	4	4	no	—
97	C₄×#6923	—	4	2	no	—
98	C₄×#6923	D₈	1	1	yes	characteristic
99	C₄×#6923	D₈	1	1	yes	characteristic
100	C₄×#6923	C₂×D₄	1	2	yes	—
101	D₆₁⋊C₈	—	2	1	no	—
102	D₆₁⋊C₈	C₂×D₄	1	2	yes	—
103	Q₁₆×C₆₁	C₂²×C₄	1	2	yes	—
104	SD₁₆×C₆₁	—	2	2	no	—
105	D₈×C₆₁	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
107	#11758202	—	2	2	no	—
108	#11758201	—	2	2	no	—
109	#11758200	—	2	2	no	—
110	#11758199	—	2	2	no	—
111	C₂×#92418	#41	1	1	yes	characteristic
112	Dic₂₄₄	C₂²×C₄	1	2	yes	—
113	D₄₈₈	C₂²×C₄	1	2	yes	—
114	#11758190	—	2	2	no	—
115	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
116	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
117	C₈×D₆₁	#41	1	1	yes	characteristic
118	C₈×D₆₁	C₂⁴	1	1	yes	characteristic
119	#11758187	—	2	2	no	—
120	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
121	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
122	#11758185	C₂²×C₄	1	1	yes	characteristic
123	#11758185	C₂²×C₄	1	1	yes	characteristic
124	C₂²×D₆₁	—	4	2	no	—
125	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
126	C₂×#6923	—	2	4	no	—
127	C₂×#6923	—	4	4	no	—
128	C₂×#6923	—	8	8	no	—
129	C₂×#6923	C₂×D₈	1	2	yes	—
130	C₂×#6923	C₂×D₈	1	2	yes	—
131	Q₈×C₆₁	—	2	2	no	—
132	D₄×C₆₁	—	2	2	no	—
133	D₄×C₆₁	—	4	2	no	—
134	C₂×C₄×C₆₁	—	4	2	no	—
135	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
136	C₂²⋊D₆₁	—	4	2	no	—
137	C₂²⋊D₆₁	—	4	2	no	—
138	C₂²⋊D₆₁	—	4	2	no	—
139	C₂×Dic₆₁	—	4	2	no	—
140	C₂×Dic₆₁	#135	1	1	yes	characteristic
141	D₂₄₄	—	2	2	no	—
142	D₂₄₄	—	4	2	no	—
143	C₄×D₆₁	—	4	2	no	—
144	C₄×D₆₁	—	4	2	no	—
145	C₄×D₆₁	—	4	2	no	—
146	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
147	C₄×D₆₁	C₂×D₈	1	1	yes	characteristic
148	C₄×D₆₁	#135	1	1	yes	characteristic
149	C₄×D₆₁	C₂²×D₄	1	1	yes	characteristic
150	Dic₁₂₂	—	2	2	no	—
151	Dic₁₂₂	—	4	2	no	—
152	#92420	—	2	2	no	—
153	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
154	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
155	#92418	C₄×D₄	1	1	yes	characteristic
156	#92418	C₂×#41	1	1	yes	characteristic
157	C₄×#508	—	61	1	no	maximal, Sylow
158	C₂²×C₆₁	—	4	2	no	—
159	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
160	C₂×D₆₁	—	2	2	no	—
161	C₂×D₆₁	—	4	2	no	—
162	C₂×D₆₁	—	8	4	no	—
163	C₂×D₆₁	C₄×D₈	1	1	yes	characteristic
164	C₂×D₆₁	C₂²×D₈	1	1	yes	characteristic
165	#6923	—	2	2	no	—
166	#6923	—	8	4	no	—
167	#6923	#508	1	4	yes	direct product factor
168	C₄×C₆₁	—	4	2	no	—
169	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
170	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
171	Dic₆₁	—	4	2	no	—
172	Dic₆₁	C₄×D₈	1	1	yes	characteristic
173	Dic₆₁	C₂×#135	1	1	yes	characteristic
174	C₂×#508	—	61	1	no	—
175	C₄×#135	—	61	2	no	—
176	C₄×Q₃₂	—	61	2	no	—
177	C₄×SD₃₂	—	61	4	no	—
178	C₄×D₁₆	—	61	2	no	—
179	#2158	—	61	1	no	—
180	#2138	—	61	2	no	—
181	C₂×C₄×C₁₆	—	61	1	no	—
182	C₂×C₆₁	—	2	1	no	—
183	C₂×C₆₁	—	8	2	no	—
184	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
185	D₆₁	—	2	1	no	—
186	D₆₁	—	8	2	no	—
187	D₆₁	C₂×#508	1	2	yes	—
188	C₂×#135	—	61	2	no	—
189	C₄×#41	—	122	2	no	—
190	#508	—	61	2	no	—
191	#508	—	61	1	no	—
192	#508	—	61	2	no	—
193	#508	—	61	1	no	—
194	#508	—	61	1	no	—
195	#508	#6923	1	1	yes	characteristic, direct product factor
196	C₂×Q₃₂	—	61	1	no	—
197	C₂×SD₃₂	—	61	2	no	—
198	C₂×D₁₆	—	61	1	no	—
199	C₂²×C₁₆	—	61	1	no	—
200	C₂²×C₁₆	—	61	1	no	—
201	C₄×Q₁₆	—	61	4	no	—
202	C₄×SD₁₆	—	122	4	no	—
203	C₄×D₈	—	61	4	no	—
204	#427	—	61	2	no	—
205	#416	—	122	2	no	—
206	C₂×C₄×C₈	—	61	1	no	—
207	#367	—	61	2	no	—
208	#366	—	61	2	no	—
209	#358	—	61	4	no	—
210	#357	—	61	4	no	—
211	C₄×C₁₆	—	61	2	no	—
212	C₄×C₁₆	—	61	2	no	—
213	C₆₁	C₄×#508	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	122	2	no	—
215	#135	—	61	2	no	—
216	#135	—	61	2	no	—
217	#135	—	61	2	no	—
218	#135	—	122	2	no	—
219	#135	—	122	2	no	—
220	#135	C₂×#6923	1	2	yes	complemented
221	C₂×Q₁₆	—	61	2	no	—
222	C₂×SD₁₆	—	122	2	no	—
223	C₂×D₈	—	61	2	no	—
224	C₂²×C₈	—	61	1	no	—
225	C₂²×C₈	—	61	1	no	—
226	C₄×Q₈	—	122	4	no	—
227	C₄×D₄	—	122	4	no	—
228	C₄×D₄	—	244	4	no	—
229	#117	—	122	2	no	—
230	#117	—	244	2	no	—
231	C₂×C₄²	—	61	1	no	—
232	C₂×C₄²	—	244	2	no	—
233	Q₃₂	—	61	2	no	—
234	Q₃₂	—	61	1	no	—
235	Q₃₂	C₂×#6923	1	1	yes	characteristic, complemented
236	SD₃₂	—	61	2	no	—
237	SD₃₂	—	61	2	no	—
238	SD₃₂	—	61	2	no	—
239	SD₃₂	C₂×#6923	1	2	yes	complemented
240	D₁₆	—	61	2	no	—
241	D₁₆	—	61	1	no	—
242	D₁₆	C₂×#6923	1	1	yes	characteristic, complemented
243	C₂×C₁₆	—	61	1	no	—
244	C₂×C₁₆	—	61	1	no	—
245	C₂×C₁₆	—	61	1	no	—
246	C₂×C₁₆	—	61	1	no	—
247	C₂×C₁₆	—	61	1	no	—
248	C₂×C₁₆	—	61	2	no	—
249	C₂×C₁₆	—	122	1	no	—
250	C₂×C₁₆	—	122	1	no	—
251	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic, complemented
252	#107	—	61	4	no	—
253	#106	—	122	2	no	—
254	#103	—	122	4	no	—
255	#102	—	122	4	no	—
256	C₄×C₈	—	61	2	no	—
257	C₄×C₈	—	61	2	no	—
258	#41	—	2	2	no	—
259	#41	—	122	2	no	—
260	#41	—	122	2	no	—
261	#41	—	122	2	no	—
262	#41	—	244	2	no	—
263	#41	—	244	2	no	—
264	C₂×Q₈	—	122	2	no	—
265	C₂×D₄	—	122	2	no	—
266	C₂×D₄	—	244	2	no	—
267	C₂²×C₄	—	61	1	no	—
268	C₂²×C₄	—	61	2	no	—
269	C₂²×C₄	—	244	2	no	—
270	C₂²×C₄	—	244	4	no	—
271	Q₁₆	—	61	2	no	—
272	Q₁₆	—	122	2	no	—
273	Q₁₆	C₂²×#6923	1	2	yes	—
274	SD₁₆	—	2	2	no	—
275	SD₁₆	—	122	2	no	—
276	SD₁₆	—	122	2	no	—
277	SD₁₆	—	122	2	no	—
278	D₈	—	61	2	no	—
279	D₈	—	122	2	no	—
280	D₈	C₂²×#6923	1	2	yes	—
281	C₂×C₈	—	61	1	no	—
282	C₂×C₈	—	61	1	no	—
283	C₂×C₈	—	61	1	no	—
284	C₂×C₈	—	61	1	no	—
285	C₂×C₈	—	61	1	no	—
286	C₂×C₈	—	61	2	no	—
287	C₂×C₈	—	122	1	no	—
288	C₂×C₈	—	122	1	no	—
289	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
290	C₄⋊C₄	—	122	4	no	—
291	C₄⋊C₄	—	244	4	no	—
292	C₂²⋊C₄	—	244	4	no	—
293	C₂²⋊C₄	—	244	2	no	—
294	C₄²	—	61	2	no	—
295	C₄²	—	61	1	no	—
296	C₄²	—	61	1	no	—
297	C₄²	—	244	2	no	—
298	C₄²	—	244	4	no	—
299	C₁₆	—	61	1	no	—
300	C₁₆	—	61	1	no	—
301	C₁₆	—	122	2	no	—
302	C₁₆	C₂²×#6923	1	1	yes	characteristic
303	C₁₆	C₂²×#6923	1	1	yes	characteristic
304	C₂³	—	61	1	no	—
305	C₂³	—	244	2	no	—
306	Q₈	—	2	2	no	—
307	Q₈	—	122	2	no	—
308	Q₈	—	244	2	no	—
309	D₄	—	2	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	122	2	no	—
312	D₄	—	244	2	no	—
313	D₄	—	244	2	no	—
314	D₄	—	244	2	no	—
315	D₄	—	244	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	61	2	no	—
318	C₂×C₄	—	61	1	no	—
319	C₂×C₄	—	61	1	no	—
320	C₂×C₄	—	61	1	no	—
321	C₂×C₄	—	61	1	no	—
322	C₂×C₄	—	61	1	no	—
323	C₂×C₄	—	61	2	no	—
324	C₂×C₄	—	122	4	no	—
325	C₂×C₄	—	488	8	no	—
326	C₂×C₄	—	244	2	no	—
327	C₂×C₄	—	244	2	no	—
328	C₂×C₄	—	244	2	no	—
329	C₂×C₄	—	244	4	no	—
330	C₂×C₄	—	244	2	no	—
331	C₂×C₄	D₄×#6923	1	1	yes	characteristic
332	C₈	—	61	1	no	—
333	C₈	—	61	1	no	—
334	C₈	—	122	2	no	—
335	C₈	D₄×#6923	1	1	yes	characteristic
336	C₈	C₂³×#6923	1	1	yes	characteristic
337	C₂²	—	4	2	no	—
338	C₂²	—	61	1	no	—
339	C₂²	—	61	1	no	—
340	C₂²	—	122	2	no	—
341	C₂²	—	488	4	no	—
342	C₂²	—	244	2	no	—
343	C₂²	D₈×#6923	1	1	yes	characteristic
344	C₄	—	4	2	no	—
345	C₄	—	61	4	no	—
346	C₄	—	61	1	no	—
347	C₄	—	61	1	no	—
348	C₄	—	122	2	no	—
349	C₄	—	488	4	no	—
350	C₄	—	244	2	no	—
351	C₄	D₈×#6923	1	1	yes	characteristic
352	C₄	C₂×D₄×#6923	1	1	yes	characteristic
353	C₂	—	2	1	no	—
354	C₂	—	8	2	no	—
355	C₂	—	61	2	no	—
356	C₂	—	122	1	no	—
357	C₂	—	488	2	no	—
358	C₂	C₂×D₈×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×#6923
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₈	C₂³×#6923
Fitting	C₆₁×#508	C₄
socle	C₂×C₆₁	C₂×C₄×D₈
prime radical	#508×D₆₁	C₂
cosocle	C₈×D₆₁	C₂⁴
hypercenter	#508	#6923
hypocenter	C₆₁	C₄×#508
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #135×#6923	C₂	semidirect
2. Q₃₂×#6923	C₂	semidirect
3. SD₃₂×#6923	C₂	semidirect
4. D₁₆×#6923	C₂	semidirect
5. #21462170	C₂	semidirect
6. #21462126	C₂	semidirect
7. C₂×C₁₆×#6923	C₂	semidirect
24. #16840195	C₄	semidirect
25. #16840194	C₄	semidirect
26. #16840193	C₄	semidirect
27. #16840192	C₄	semidirect
28. #16840191	C₄	semidirect
33. C₆₁×#508	C₄	semidirect
52. #14519800	C₂×C₄	semidirect
53. #14519799	C₂×C₄	semidirect
56. #14519796	C₂×C₄	semidirect
61. Dic₄₈₈	C₂×C₄	semidirect
62. #14519723	C₂×C₄	semidirect
63. #14519722	C₂×C₄	semidirect
64. #14519721	C₂×C₄	semidirect
65. #14519720	C₂×C₄	semidirect
66. D₉₇₆	C₂×C₄	semidirect
67. #14519718	C₂×C₄	semidirect
68. C₂×#11758185	C₂×C₄	semidirect
69. C₁₆×D₆₁	C₂×C₄	semidirect
70. C₁₆×D₆₁	C₂×C₄	semidirect
73. #135×C₆₁	C₂×C₄	semidirect
74. Q₃₂×C₆₁	C₂×C₄	semidirect
75. SD₃₂×C₆₁	C₂×C₄	semidirect
76. D₁₆×C₆₁	C₂×C₄	semidirect
77. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
167. #6923	#508	direct (N × H)
195. #508	#6923	direct (N × H)
213. C₆₁	C₄×#508	semidirect
220. #135	C₂×#6923	semidirect
235. Q₃₂	C₂×#6923	semidirect
239. SD₃₂	C₂×#6923	semidirect
242. D₁₆	C₂×#6923	semidirect
251. C₂×C₁₆	C₂×#6923	semidirect

Maximal subgroups

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

subgroup	index
#135×#6923	2
Q₃₂×#6923	2
SD₃₂×#6923	2
D₁₆×#6923	2
#21462170	2
#21462126	2
C₂×C₁₆×#6923	2
#508×D₆₁	2
C₄×#508	61