Subgroups — #24919935

Order 10624.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128397	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128355	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128110	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128026	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128020	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128020	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128015	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128002	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127992	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127954	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127937	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127814	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127743	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#14581724	C₂	1	1	yes	maximal, characteristic, complemented
15	#435×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	—	2	1	no	—
17	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
18	#14581792	C₂²	1	1	yes	characteristic
19	#14581787	—	2	1	no	—
20	#14581784	C₂²	1	1	yes	characteristic
21	#14581776	—	2	1	no	—
22	#14581776	—	2	1	no	—
23	#14581774	—	2	1	no	—
24	#14581772	—	2	1	no	—
25	#14581766	C₂²	1	1	yes	characteristic
26	#14581766	C₂²	1	1	yes	characteristic
27	#14581766	C₂²	1	1	yes	characteristic
28	#14581766	C₂²	1	1	yes	characteristic
29	#14581764	—	2	1	no	—
30	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
31	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
32	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
33	#14581751	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
37	#14581726	C₂²	1	1	yes	characteristic
38	#14581726	C₂²	1	1	yes	characteristic
39	#14581726	C₂²	1	1	yes	characteristic
40	#14581724	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581724	C₂²	1	1	yes	characteristic
43	#14581724	C₂²	1	1	yes	characteristic
44	#14581724	C₂²	1	1	yes	characteristic
45	#14581724	C₂²	1	1	yes	characteristic
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581714	C₂²	1	1	yes	characteristic
48	#14581714	C₂²	1	1	yes	characteristic
49	#14581714	C₂²	1	1	yes	characteristic
50	#14581714	C₂²	1	1	yes	characteristic
51	#14581714	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
58	#97×C₈₃	C₂²	1	1	yes	characteristic
59	#13050589	—	4	4	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
62	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂²×Dic₈₃	—	2	1	no	—
68	C₂²×Dic₈₃	—	2	1	no	—
69	C₂²×Dic₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	—	2	1	no	—
72	D₆₆₄	—	2	2	no	—
73	C₄⋊Dic₈₃	—	2	1	no	—
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	—	2	1	no	—
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	C₂²⋊Dic₈₃	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	C₄×Dic₈₃	—	2	1	no	—
93	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
94	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
96	M₁₆×C₈₃	—	2	2	no	—
97	C₂×C₈×C₈₃	—	2	2	no	—
98	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
103	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
105	C₄²×C₈₃	C₂³	1	1	yes	characteristic
106	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₈₃	—	2	1	no	—
109	C₂²×D₈₃	—	2	1	no	—
110	D₄×C₈₃	—	2	2	no	—
111	C₂×C₄×C₈₃	—	2	1	no	—
112	C₂×C₄×C₈₃	—	2	1	no	—
113	C₂×C₄×C₈₃	—	2	1	no	—
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₈₃	—	4	2	no	—
121	C₂²⋊D₈₃	—	4	2	no	—
122	C₂²⋊D₈₃	—	4	2	no	—
123	C₂²⋊D₈₃	—	8	1	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	4	2	no	—
131	C₂×Dic₈₃	—	4	2	no	—
132	C₂×Dic₈₃	—	8	1	no	—
133	D₃₃₂	—	2	1	no	—
134	D₃₃₂	—	2	1	no	—
135	D₃₃₂	—	2	1	no	—
136	D₃₃₂	—	2	1	no	—
137	C₄×D₈₃	—	4	2	no	—
138	C₄×D₈₃	—	8	1	no	—
139	Dic₁₆₆	—	4	2	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈×C₈₃	—	2	1	no	—
143	C₈×C₈₃	#41	1	2	yes	complemented
144	C₂²×C₈₃	—	2	1	no	—
145	C₂²×C₈₃	—	2	1	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	—	2	1	no	—
148	C₂²×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
150	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₈₃	—	4	1	no	—
152	C₂×D₈₃	—	4	1	no	—
153	C₂×D₈₃	—	4	1	no	—
154	C₂×D₈₃	—	4	1	no	—
155	C₂×D₈₃	—	4	1	no	—
156	C₂×D₈₃	—	4	1	no	—
157	C₄×C₈₃	—	2	1	no	—
158	C₄×C₈₃	—	2	1	no	—
159	C₄×C₈₃	—	2	1	no	—
160	C₄×C₈₃	—	2	1	no	—
161	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
162	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
163	Dic₈₃	—	4	1	no	—
164	Dic₈₃	—	4	1	no	—
165	Dic₈₃	—	4	1	no	—
166	Dic₈₃	—	4	1	no	—
167	Dic₈₃	—	4	1	no	—
168	Dic₈₃	—	4	1	no	—
169	C₂×C₈₃	—	2	2	no	—
170	C₂×C₈₃	—	4	1	no	—
171	C₂×C₈₃	#547	1	1	yes	characteristic
172	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₂×C₈₃	#577	1	1	yes	characteristic
174	D₈₃	—	8	1	no	—
175	D₈₃	—	8	1	no	—
176	#3345	—	83	1	no	maximal, Sylow
177	C₈₃	#3345	1	1	yes	characteristic, complemented, Sylow
178	#553	—	83	1	no	—
179	#547	—	83	1	no	—
180	#501	—	83	1	no	—
181	#482	—	83	1	no	—
182	#481	—	83	1	no	—
183	#481	—	83	1	no	—
184	#480	—	83	1	no	—
185	#478	—	83	1	no	—
186	#476	—	83	1	no	—
187	#469	—	83	1	no	—
188	#466	—	83	1	no	—
189	#442	—	83	1	no	—
190	#435	D₈₃	1	1	yes	characteristic, complemented
191	#428	—	83	1	no	—
192	C₂×#106	—	83	1	no	—
193	C₂×#41	—	166	1	no	—
194	C₂×D₈	—	83	1	no	—
195	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
196	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	83	1	no	—
198	#126	—	166	1	no	—
199	#125	—	83	1	no	—
200	#123	—	166	1	no	—
201	#123	—	166	1	no	—
202	#123	—	166	1	no	—
203	C₂²⋊Q₈	—	166	1	no	—
204	#121	—	83	1	no	—
205	#121	—	83	1	no	—
206	#121	—	83	1	no	—
207	#121	—	83	1	no	—
208	#121	—	166	1	no	—
209	C₄×D₄	—	83	1	no	—
210	C₄×D₄	—	83	1	no	—
211	C₄×D₄	—	166	1	no	—
212	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
213	#117	—	83	1	no	—
214	C₂×C₄⋊C₄	—	83	1	no	—
215	C₂×C₄⋊C₄	—	83	1	no	—
216	C₂×C₄⋊C₄	—	83	1	no	—
217	#107	—	83	1	no	—
218	#107	—	83	1	no	—
219	#107	—	83	1	no	—
220	#106	—	83	1	no	—
221	#106	—	83	1	no	—
222	#106	—	83	1	no	—
223	#106	—	83	1	no	—
224	#106	—	83	1	no	—
225	#106	—	83	1	no	—
226	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
227	#102	—	83	1	no	—
228	#102	—	83	1	no	—
229	#102	—	83	1	no	—
230	#102	—	83	1	no	—
231	#102	—	83	1	no	—
232	#102	—	83	1	no	—
233	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
234	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
235	#97	C₂×D₈₃	1	1	yes	characteristic
236	#41	—	332	4	no	—
237	C₂×Q₈	—	166	1	no	—
238	C₂×D₄	—	83	1	no	—
239	C₂×D₄	—	83	1	no	—
240	C₂×D₄	—	166	1	no	—
241	C₂×D₄	—	166	1	no	—
242	C₂×D₄	—	166	1	no	—
243	C₂×D₄	—	166	1	no	—
244	C₂×D₄	D₃₃₂	1	1	yes	characteristic
245	C₂²×C₄	—	166	1	no	—
246	C₂²×C₄	—	166	1	no	—
247	C₂²×C₄	—	166	1	no	—
248	C₂²×C₄	—	166	1	no	—
249	C₂²×C₄	—	166	1	no	—
250	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
252	D₈	—	166	2	no	—
253	M₁₆	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
259	C₄⋊C₄	—	83	1	no	—
260	C₄⋊C₄	—	83	1	no	—
261	C₄⋊C₄	—	83	1	no	—
262	C₄⋊C₄	—	83	1	no	—
263	C₄⋊C₄	—	83	1	no	—
264	C₄⋊C₄	—	83	1	no	—
265	C₄⋊C₄	—	166	1	no	—
266	C₄⋊C₄	—	166	1	no	—
267	C₄⋊C₄	—	166	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	—	166	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
272	C₂²⋊C₄	—	166	1	no	—
273	C₂²⋊C₄	—	166	1	no	—
274	C₂²⋊C₄	—	166	1	no	—
275	C₂²⋊C₄	—	166	1	no	—
276	C₂²⋊C₄	—	166	1	no	—
277	C₂²⋊C₄	—	166	1	no	—
278	C₂²⋊C₄	—	166	1	no	—
279	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
280	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
281	C₄²	—	166	1	no	—
282	C₄²	C₂²×D₈₃	1	1	yes	characteristic
283	C₂³	—	166	1	no	—
284	C₂³	—	166	1	no	—
285	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
286	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
287	Q₈	—	332	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	332	2	no	—
290	D₄	—	332	2	no	—
291	D₄	—	332	2	no	—
292	D₄	—	664	1	no	—
293	D₄	—	166	1	no	—
294	D₄	—	166	1	no	—
295	D₄	—	166	1	no	—
296	D₄	—	166	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	332	2	no	—
302	C₂×C₄	—	332	2	no	—
303	C₂×C₄	—	332	2	no	—
304	C₂×C₄	—	664	1	no	—
305	C₂×C₄	—	664	1	no	—
306	C₂×C₄	—	166	1	no	—
307	C₂×C₄	—	166	1	no	—
308	C₂×C₄	—	166	1	no	—
309	C₂×C₄	—	166	1	no	—
310	C₂×C₄	—	166	1	no	—
311	C₂×C₄	—	166	1	no	—
312	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
313	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
314	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
315	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	#13050589	1	2	yes	complemented
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	332	1	no	—
327	C₂²	—	332	1	no	—
328	C₂²	—	332	1	no	—
329	C₂²	—	332	1	no	—
330	C₂²	—	332	1	no	—
331	C₂²	—	332	1	no	—
332	C₂²	#14581820	1	2	yes	—
333	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	332	1	no	—
339	C₄	—	332	1	no	—
340	C₄	—	332	1	no	—
341	C₄	—	332	1	no	—
342	C₄	—	332	1	no	—
343	C₄	—	332	1	no	—
344	C₄	C₂×#13050589	1	1	yes	characteristic
345	C₄	#14581839	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	4	1	no	—
348	C₂	—	664	1	no	—
349	C₂	—	664	1	no	—
350	C₂	#19128355	1	1	yes	characteristic
351	C₂	C₂×#14581820	1	1	yes	characteristic
352	C₂	#19128513	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#435×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#435	D₈₃
hypocenter	C₈₃	#3345
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
1. #19128397	C₂	semidirect
2. #19128355	C₂	semidirect
3. #19128110	C₂	semidirect
4. #19128026	C₂	semidirect
5. #19128020	C₂	semidirect
6. #19128020	C₂	semidirect
7. #19128015	C₂	semidirect
8. #19128002	C₂	semidirect
9. #19127992	C₂	semidirect
10. #19127954	C₂	semidirect
11. #19127937	C₂	semidirect
12. #19127814	C₂	semidirect
13. #19127743	C₂	semidirect
14. C₂×#14581724	C₂	semidirect
15. #435×C₈₃	C₂	semidirect
143. C₈×C₈₃	#41	semidirect
177. C₈₃	#3345	semidirect
190. #435	D₈₃	semidirect
320. C₈	#13050589	semidirect

Maximal subgroups

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

subgroup	index
#19128397	2
#19128355	2
#19128110	2
#19128026	2
#19128020	2
#19128015	2
#19128002	2
#19127992	2
#19127954	2
#19127937	2
#19127814	2
#19127743	2
C₂×#14581724	2
#435×C₈₃	2
#3345	83