Subgroups — #35422903

Order 21248.

862 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916339	C₂	1	2	yes	maximal
2	#24916063	C₂	1	2	yes	maximal, complemented
3	#24908249	C₂	1	2	yes	maximal, complemented
4	#24907691	C₂	1	4	yes	maximal, complemented
5	#24907623	C₂	1	1	yes	maximal, characteristic, complemented
6	#24907527	C₂	1	2	yes	maximal, complemented
7	#24907342	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#443×C₈₃	C₂	1	1	yes	maximal, characteristic
9	#19128184	C₂²	1	4	yes	—
10	#19127802	—	2	4	no	—
11	Q₁₆×Dic₈₃	—	2	2	no	—
12	D₈×Dic₈₃	—	2	2	no	—
13	C₂×#14581728	C₂²	1	1	yes	characteristic
14	#19127743	—	2	4	no	—
15	#19127743	C₂²	1	4	yes	—
16	#19127739	—	2	2	no	—
17	#19127739	C₂²	1	1	yes	characteristic
18	C₂×#14581726	—	2	2	no	—
19	C₂×#14581726	C₂²	1	2	yes	—
20	C₂×#14581726	C₂²	1	2	yes	—
21	#19127702	C₂²	1	4	yes	—
22	#19127656	C₂²	1	2	yes	—
23	#19127571	C₂²	1	2	yes	—
24	#19127469	C₂²	1	4	yes	—
25	#19127461	C₂²	1	2	yes	—
26	C₂×#135×C₈₃	C₂²	1	1	yes	characteristic
27	C₂×#131×C₈₃	C₂²	1	2	yes	—
28	#443×C₈₃	C₄	1	8	yes	complemented
29	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
30	C₂×#104×C₈₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
32	C₂×#13050574	C₂³	1	2	yes	—
33	Q₈×Dic₈₃	—	2	4	no	—
34	D₄×Dic₈₃	—	2	4	no	—
35	D₄×Dic₈₃	—	4	4	no	—
36	#14581751	—	2	4	no	—
37	#14581751	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₈₃	—	2	4	no	—
39	C₂×C₄⋊Dic₈₃	—	4	2	no	—
40	C₂×C₄⋊Dic₈₃	C₂³	1	4	yes	—
41	#14581728	—	2	2	no	—
42	#14581726	—	2	4	no	—
43	#14581726	—	2	2	no	—
44	#14581726	—	2	2	no	—
45	#14581726	—	2	2	no	—
46	#14581726	—	2	2	no	—
47	#14581726	D₄	1	1	yes	characteristic
48	#14581726	D₄	1	1	yes	characteristic
49	#14581724	—	2	4	no	—
50	#14581724	D₄	1	2	yes	—
51	#14581716	—	2	4	no	—
52	#14581713	—	2	4	no	—
53	#14581702	—	2	2	no	—
54	C₈×Dic₈₃	—	2	2	no	—
55	C₂×#41×C₈₃	C₂³	1	2	yes	—
56	#135×C₈₃	C₂×C₄	1	4	yes	—
57	#135×C₈₃	Q₈	1	4	yes	—
58	C₂×Q₁₆×C₈₃	D₄	1	1	yes	characteristic
59	C₂×SD₁₆×C₈₃	D₄	1	2	yes	—
60	C₂×D₈×C₈₃	D₄	1	1	yes	characteristic
61	#131×C₈₃	—	2	4	no	—
62	#131×C₈₃	C₂×C₄	1	8	yes	complemented
63	C₂×M₁₆×C₈₃	—	2	2	no	—
64	C₂×M₁₆×C₈₃	C₂³	1	2	yes	—
65	C₂²×C₈×C₈₃	—	2	2	no	—
66	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
69	#108×C₈₃	C₂×C₄	1	4	yes	complemented
70	#104×C₈₃	C₂×C₄	1	8	yes	complemented
71	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	complemented
72	C₂²×Dic₈₃	—	2	4	no	—
73	C₂²×Dic₈₃	—	4	4	no	—
74	#13050574	—	2	2	no	—
75	#13050574	—	2	2	no	—
76	C₂×C₈₃⋊C₈	—	2	2	no	—
77	C₄⋊Dic₈₃	—	2	4	no	—
78	C₄⋊Dic₈₃	—	2	4	no	—
79	C₄⋊Dic₈₃	—	4	4	no	—
80	C₄⋊Dic₈₃	—	8	4	no	—
81	C₄⋊Dic₈₃	C₂×D₄	1	2	yes	—
82	C₄⋊Dic₈₃	C₂×D₄	1	2	yes	—
83	C₂²⋊Dic₈₃	—	2	2	no	—
84	C₂²⋊Dic₈₃	—	4	4	no	—
85	C₄×Dic₈₃	—	2	2	no	—
86	C₄×Dic₈₃	—	4	4	no	—
87	#41×C₈₃	—	2	2	no	—
88	#41×C₈₃	—	2	2	no	—
89	#41×C₈₃	C₂²×C₄	1	4	yes	—
90	#41×C₈₃	C₂×Q₈	1	4	yes	—
91	C₂×Q₈×C₈₃	C₂×D₄	1	2	yes	—
92	C₂×D₄×C₈₃	—	2	2	no	—
93	C₂×D₄×C₈₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₈₃	—	2	1	no	—
95	C₂²×C₄×C₈₃	—	2	2	no	—
96	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
97	Q₁₆×C₈₃	C₄⋊C₄	1	4	yes	complemented
98	SD₁₆×C₈₃	C₄⋊C₄	1	8	yes	complemented
99	D₈×C₈₃	C₄⋊C₄	1	4	yes	complemented
100	M₁₆×C₈₃	—	2	2	no	—
101	M₁₆×C₈₃	—	2	8	no	—
102	M₁₆×C₈₃	C₂²×C₄	1	4	yes	—
103	C₂×C₈×C₈₃	—	2	2	no	—
104	C₂×C₈×C₈₃	—	2	8	no	—
105	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
112	C₂×C₈×C₈₃	#41	1	2	yes	—
113	C₄²×C₈₃	—	2	1	no	—
114	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
115	C₂³×C₈₃	—	2	2	no	—
116	C₂³×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
117	Q₈×C₈₃	—	2	2	no	—
118	Q₈×C₈₃	C₂×C₄⋊C₄	1	4	yes	—
119	D₄×C₈₃	—	2	4	no	—
120	D₄×C₈₃	—	2	4	no	—
121	D₄×C₈₃	—	2	2	no	—
122	D₄×C₈₃	C₂×C₄⋊C₄	1	4	yes	—
123	C₂×C₄×C₈₃	—	2	2	no	—
124	C₂×C₄×C₈₃	—	2	2	no	—
125	C₂×C₄×C₈₃	—	2	2	no	—
126	C₂×C₄×C₈₃	—	2	2	no	—
127	C₂×C₄×C₈₃	—	2	4	no	—
128	C₂×C₄×C₈₃	—	2	4	no	—
129	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×Dic₈₃	—	2	2	no	—
136	C₂×Dic₈₃	—	2	2	no	—
137	C₂×Dic₈₃	—	4	4	no	—
138	C₂×Dic₈₃	—	8	4	no	—
139	C₂×Dic₈₃	—	16	4	no	—
140	C₈×C₈₃	—	2	4	no	—
141	C₈×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₈×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
143	C₈×C₈₃	C₄×D₄	1	2	yes	—
144	C₈×C₈₃	C₄×D₄	1	2	yes	—
145	C₈₃⋊C₈	—	4	2	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	—	2	1	no	—
148	C₂²×C₈₃	—	2	4	no	—
149	C₂²×C₈₃	—	4	2	no	—
150	C₂²×C₈₃	—	4	2	no	—
151	C₂²×C₈₃	#520	1	1	yes	characteristic
152	C₂²×C₈₃	D₄²	1	1	yes	characteristic
153	C₂²×C₈₃	#554	1	1	yes	characteristic
154	C₄×C₈₃	—	2	4	no	—
155	C₄×C₈₃	—	2	4	no	—
156	C₄×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₈₃	#547	1	1	yes	characteristic
159	C₄×C₈₃	D₄×Q₈	1	1	yes	characteristic
160	Dic₈₃	—	4	2	no	—
161	Dic₈₃	—	4	2	no	—
162	Dic₈₃	—	8	4	no	—
163	#22692	—	83	1	no	maximal, Sylow
164	C₂×C₈₃	—	2	1	no	—
165	C₂×C₈₃	—	2	1	no	—
166	C₂×C₈₃	—	4	4	no	—
167	C₂×C₈₃	D₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₈₃	#3292	1	1	yes	characteristic
169	C₂×C₈₃	#3293	1	1	yes	characteristic
170	C₂×#443	D₈₃	1	1	yes	characteristic
171	#2951	—	83	2	no	—
172	#2913	—	83	2	no	—
173	#1952	—	83	2	no	—
174	#1859	—	83	4	no	—
175	#1845	—	83	1	no	—
176	#1816	—	83	2	no	—
177	#1772	—	83	1	no	—
178	C₈₃	#22692	1	1	yes	characteristic, complemented, Sylow
179	C₂×#135	C₂×D₈₃	1	1	yes	characteristic
180	C₂×#131	C₂×D₈₃	1	2	yes	—
181	#520	—	83	4	no	—
182	#443	Dic₈₃	1	8	yes	complemented
183	#440	—	166	4	no	—
184	C₄×Q₁₆	—	166	2	no	—
185	C₄×D₈	—	166	2	no	—
186	C₂×#108	—	83	1	no	—
187	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
188	#428	—	83	4	no	—
189	#428	—	166	4	no	—
190	#427	—	83	1	no	—
191	#427	—	166	2	no	—
192	C₂×#107	—	83	2	no	—
193	C₂×#107	—	83	2	no	—
194	C₂×#107	—	166	2	no	—
195	C₂×#104	C₂×D₈₃	1	2	yes	—
196	#417	—	83	4	no	—
197	#405	—	83	2	no	—
198	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
199	#384	—	83	2	no	—
200	#343	—	83	4	no	—
201	#339	—	83	2	no	—
202	C₂×#41	C₂²×D₈₃	1	2	yes	—
203	#135	Dic₁₆₆	1	4	yes	—
204	#135	C₂×Dic₈₃	1	4	yes	—
205	C₂×Q₁₆	D₃₃₂	1	1	yes	characteristic
206	C₂×SD₁₆	D₃₃₂	1	2	yes	—
207	C₂×D₈	D₃₃₂	1	1	yes	characteristic
208	#131	—	2	4	no	—
209	#131	C₂×Dic₈₃	1	8	yes	complemented
210	C₂×M₁₆	—	2	2	no	—
211	C₂×M₁₆	—	83	2	no	—
212	C₂×M₁₆	C₂²×D₈₃	1	2	yes	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
216	C₄×Q₈	—	166	4	no	—
217	C₄×D₄	—	332	4	no	—
218	C₄×D₄	—	166	4	no	—
219	#117	—	83	2	no	—
220	#117	—	166	4	no	—
221	C₂×C₄⋊C₄	—	332	2	no	—
222	C₂×C₄⋊C₄	—	83	4	no	—
223	C₂×C₄⋊C₄	—	166	4	no	—
224	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
225	#108	—	166	2	no	—
226	#108	C₂×Dic₈₃	1	4	yes	complemented
227	#107	—	83	1	no	—
228	#107	—	83	1	no	—
229	#107	—	166	4	no	—
230	#107	—	166	2	no	—
231	#107	—	166	2	no	—
232	#107	—	166	2	no	—
233	#107	—	166	2	no	—
234	#106	—	83	2	no	—
235	#106	—	166	4	no	—
236	#104	C₂×Dic₈₃	1	8	yes	complemented
237	#103	—	166	4	no	—
238	#102	—	166	4	no	—
239	#97	—	166	2	no	—
240	C₄×C₈	—	166	2	no	—
241	C₄×C₈	C₂×Dic₈₃	1	4	yes	complemented
242	#41	—	2	2	no	—
243	#41	—	2	2	no	—
244	#41	C₂²×Dic₈₃	1	4	yes	—
245	#41	C₂×Dic₁₆₆	1	4	yes	—
246	C₂×Q₈	C₂×D₃₃₂	1	2	yes	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	C₂×D₃₃₂	1	2	yes	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	332	4	no	—
252	C₂²×C₄	—	166	4	no	—
253	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
254	Q₁₆	C₄⋊Dic₈₃	1	4	yes	complemented
255	SD₁₆	C₄⋊Dic₈₃	1	8	yes	complemented
256	D₈	C₄⋊Dic₈₃	1	4	yes	complemented
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	8	no	—
259	M₁₆	—	166	2	no	—
260	M₁₆	—	166	2	no	—
261	M₁₆	C₂²×Dic₈₃	1	4	yes	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	8	no	—
264	C₂×C₈	—	166	2	no	—
265	C₂×C₈	C₂²×Dic₈₃	1	2	yes	—
266	C₂×C₈	C₂²×Dic₈₃	1	2	yes	—
267	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
268	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
269	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
270	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
271	C₂×C₈	C₂×Dic₁₆₆	1	2	yes	—
272	C₂×C₈	#13050589	1	2	yes	—
273	C₄⋊C₄	—	332	4	no	—
274	C₄⋊C₄	—	83	2	no	—
275	C₄⋊C₄	—	83	2	no	—
276	C₄⋊C₄	—	664	4	no	—
277	C₄⋊C₄	—	166	4	no	—
278	C₄⋊C₄	—	166	4	no	—
279	C₂²⋊C₄	—	332	4	no	—
280	C₂²⋊C₄	—	166	2	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	332	4	no	—
283	C₄²	—	166	2	no	—
284	C₄²	C₂²×Dic₈₃	1	2	yes	—
285	C₂³	—	2	2	no	—
286	C₂³	#14581838	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	C₂×C₄⋊Dic₈₃	1	4	yes	—
289	D₄	—	2	4	no	—
290	D₄	—	2	4	no	—
291	D₄	—	2	2	no	—
292	D₄	C₂×C₄⋊Dic₈₃	1	4	yes	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	1328	4	no	—
300	C₂×C₄	—	332	4	no	—
301	C₂×C₄	—	664	4	no	—
302	C₂×C₄	—	166	2	no	—
303	C₂×C₄	—	166	2	no	—
304	C₂×C₄	C₂³×Dic₈₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
307	C₂×C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
308	C₂×C₄	C₂×#13050589	1	1	yes	characteristic
309	C₂×C₄	#14581839	1	1	yes	characteristic
310	C₈	—	2	4	no	—
311	C₈	—	332	2	no	—
312	C₈	C₂×C₄⋊Dic₈₃	1	2	yes	—
313	C₈	C₂×C₄⋊Dic₈₃	1	2	yes	—
314	C₈	D₄×Dic₈₃	1	2	yes	—
315	C₈	D₄×Dic₈₃	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	#19128184	1	1	yes	characteristic
322	C₂²	D₄×D₃₃₂	1	1	yes	characteristic
323	C₂²	#19128408	1	1	yes	characteristic
324	C₄	—	2	4	no	—
325	C₄	—	2	4	no	—
326	C₄	—	332	2	no	—
327	C₄	—	332	2	no	—
328	C₄	—	664	4	no	—
329	C₄	C₂²×C₄⋊Dic₈₃	1	1	yes	characteristic
330	C₄	C₂×D₄×Dic₈₃	1	1	yes	characteristic
331	C₄	#19128355	1	1	yes	characteristic
332	C₄	D₄×Dic₁₆₆	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	4	4	no	—
336	C₂	D₄×C₄⋊Dic₈₃	1	1	yes	characteristic
337	C₂	#24919263	1	1	yes	characteristic
338	C₂	#24919268	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₂×#443×C₈₃	C₂
socle	C₂²×C₈₃	D₄²
prime radical	C₂×#135×C₈₃	C₂²
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	C₂×#443	D₈₃
hypocenter	C₈₃	#22692
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
2. #24916063	C₂	semidirect
3. #24908249	C₂	semidirect
4. #24907691	C₂	semidirect
5. #24907623	C₂	semidirect
6. #24907527	C₂	semidirect
7. #24907342	C₂	semidirect
28. #443×C₈₃	C₄	semidirect
62. #131×C₈₃	C₂×C₄	semidirect
69. #108×C₈₃	C₂×C₄	semidirect
70. #104×C₈₃	C₂×C₄	semidirect
71. C₄×C₈×C₈₃	C₂×C₄	semidirect
97. Q₁₆×C₈₃	C₄⋊C₄	semidirect
98. SD₁₆×C₈₃	C₄⋊C₄	semidirect
99. D₈×C₈₃	C₄⋊C₄	semidirect
178. C₈₃	#22692	semidirect
182. #443	Dic₈₃	semidirect
209. #131	C₂×Dic₈₃	semidirect
226. #108	C₂×Dic₈₃	semidirect
236. #104	C₂×Dic₈₃	semidirect
241. C₄×C₈	C₂×Dic₈₃	semidirect
254. Q₁₆	C₄⋊Dic₈₃	semidirect
255. SD₁₆	C₄⋊Dic₈₃	semidirect
256. D₈	C₄⋊Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24916339	2
#24916063	2
#24908249	2
#24907691	2
#24907623	2
#24907527	2
#24907342	2
C₂×#443×C₈₃	2
#22692	83