Subgroups — #35386774

Order 21248.

606 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917977	C₂	1	1	yes	maximal, characteristic
2	#24917503	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
4	#24906868	C₂	1	2	yes	maximal, complemented
5	#24906858	C₂	1	2	yes	maximal, complemented
6	#24906846	C₂	1	4	yes	maximal, complemented
7	#24906118	C₂	1	2	yes	maximal
8	C₈₃×#1564	C₂	1	1	yes	maximal, characteristic
9	#19128258	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
11	#19128106	C₂²	1	1	yes	characteristic
12	#19128106	C₂²	1	1	yes	characteristic
13	#19128103	—	2	1	no	—
14	#19128103	—	2	1	no	—
15	#19128099	C₂²	1	2	yes	—
16	#19128037	—	2	2	no	—
17	#19127997	C₂²	1	4	yes	—
18	#19127912	C₂²	1	4	yes	—
19	#19127859	—	2	2	no	—
20	C₈×Dic₁₆₆	—	2	1	no	—
21	C₈×Dic₁₆₆	—	2	1	no	—
22	C₈×Dic₁₆₆	C₂²	1	2	yes	—
23	C₈×Dic₁₆₆	C₂²	1	2	yes	—
24	C₄×Dic₃₃₂	—	2	4	no	—
25	#19127739	—	2	1	no	—
26	#19127739	—	2	1	no	—
27	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂×Dic₈₃⋊C₈	—	2	1	no	—
29	C₂×C₈×Dic₈₃	—	2	1	no	—
30	#19127449	C₂²	1	4	yes	—
31	#19127448	C₂²	1	4	yes	—
32	#431×C₈₃	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
34	#335×C₈₃	C₂²	1	2	yes	—
35	#334×C₈₃	C₂²	1	2	yes	—
36	C₂²×Dic₁₆₆	D₄	1	1	yes	characteristic
37	C₂×Dic₃₃₂	—	2	4	no	—
38	C₂²×C₈₃⋊C₈	—	2	1	no	—
39	#14581792	C₂³	1	2	yes	—
40	#14581792	C₂³	1	2	yes	—
41	#14581784	—	2	2	no	—
42	#14581771	—	2	2	no	—
43	#14581771	—	2	2	no	—
44	C₄×Dic₁₆₆	—	2	1	no	—
45	C₄×Dic₁₆₆	—	2	2	no	—
46	C₄×Dic₁₆₆	—	2	1	no	—
47	C₄×Dic₁₆₆	—	2	2	no	—
48	C₄×Dic₁₆₆	D₄	1	2	yes	—
49	C₄×Dic₁₆₆	C₂³	1	2	yes	—
50	#14581751	—	2	2	no	—
51	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
52	C₂×Dic₈₃⋊C₄	—	2	1	no	—
53	C₂×C₄×Dic₈₃	—	2	1	no	—
54	#14581726	—	2	4	no	—
55	#14581726	—	2	1	no	—
56	#14581726	—	2	1	no	—
57	#14581726	—	2	1	no	—
58	#14581726	—	2	1	no	—
59	#14581724	—	2	2	no	—
60	#14581724	—	2	2	no	—
61	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
62	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
63	Dic₈₃⋊C₈	—	2	2	no	—
64	Dic₈₃⋊C₈	—	2	2	no	—
65	#14581717	—	2	4	no	—
66	#14581717	—	2	4	no	—
67	C₈×Dic₈₃	—	2	2	no	—
68	C₈×Dic₈₃	—	2	2	no	—
69	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
70	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₈₃	C₂³	1	4	yes	—
73	C₂²⋊C₈×C₈₃	—	2	2	no	—
74	C₄×C₈×C₈₃	—	2	2	no	—
75	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₆₆	—	2	1	no	—
80	C₂×Dic₁₆₆	—	2	4	no	—
81	C₂×Dic₁₆₆	—	2	2	no	—
82	C₂×Dic₁₆₆	—	2	2	no	—
83	C₂×Dic₁₆₆	C₂×D₄	1	2	yes	—
84	C₂²×Dic₈₃	—	2	1	no	—
85	C₂²×Dic₈₃	—	2	1	no	—
86	Dic₃₃₂	—	4	8	no	—
87	C₂×C₈₃⋊C₈	—	2	2	no	—
88	C₂×C₈₃⋊C₈	—	4	2	no	—
89	C₄⋊Dic₈₃	—	2	1	no	—
90	C₄⋊Dic₈₃	—	2	2	no	—
91	C₄⋊Dic₈₃	—	2	2	no	—
92	C₄⋊Dic₈₃	—	2	2	no	—
93	C₄⋊Dic₈₃	—	2	2	no	—
94	C₄⋊Dic₈₃	C₂×D₄	1	2	yes	—
95	Dic₈₃⋊C₄	—	2	2	no	—
96	Dic₈₃⋊C₄	—	2	2	no	—
97	Dic₈₃⋊C₄	—	4	2	no	—
98	Dic₈₃⋊C₄	—	4	2	no	—
99	C₂²⋊Dic₈₃	—	4	2	no	—
100	C₄×Dic₈₃	—	2	2	no	—
101	C₄×Dic₈₃	—	2	2	no	—
102	C₄×Dic₈₃	—	4	2	no	—
103	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
106	C₂×C₈×C₈₃	—	2	2	no	—
107	C₂×C₈×C₈₃	—	2	4	no	—
108	C₂×C₈×C₈₃	Q₁₆	1	4	yes	complemented
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×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₃	#41	1	1	yes	characteristic
116	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
117	C₂³×C₈₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₈₃	—	2	2	no	—
119	C₂×C₄×C₈₃	—	4	1	no	—
120	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₈₃	#137	1	2	yes	—
125	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₈₃	—	2	4	no	—
129	C₂×Dic₈₃	—	2	2	no	—
130	C₂×Dic₈₃	—	2	2	no	—
131	C₂×Dic₈₃	—	4	2	no	—
132	C₂×Dic₈₃	—	4	2	no	—
133	C₂×Dic₈₃	—	4	2	no	—
134	Dic₁₆₆	—	2	4	no	—
135	Dic₁₆₆	—	2	2	no	—
136	Dic₁₆₆	—	4	4	no	—
137	Dic₁₆₆	—	8	2	no	—
138	Dic₁₆₆	#135	1	4	yes	—
139	C₈×C₈₃	—	2	1	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₈×C₈₃	—	4	4	no	—
142	C₈×C₈₃	C₂×Q₁₆	1	2	yes	—
143	C₈×C₈₃	C₂×Q₁₆	1	2	yes	—
144	C₈₃⋊C₈	—	4	2	no	—
145	C₂²×C₈₃	—	2	1	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	#462	1	2	yes	—
148	C₂²×C₈₃	C₂×#121	1	1	yes	characteristic
149	C₄×C₈₃	—	2	1	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	2	1	no	—
152	C₄×C₈₃	#462	1	2	yes	—
153	C₄×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
154	C₄×C₈₃	C₂×#135	1	2	yes	—
155	C₄×C₈₃	C₂×#137	1	1	yes	characteristic
156	Dic₈₃	—	2	4	no	—
157	Dic₈₃	—	4	2	no	—
158	Dic₈₃	—	8	2	no	—
159	Dic₈₃	—	8	2	no	—
160	#19118	—	83	1	no	maximal, Sylow
161	C₂×C₈₃	—	2	2	no	—
162	C₂×C₈₃	C₂×#462	1	1	yes	characteristic
163	C₂×C₈₃	#3050	1	1	yes	characteristic
164	C₂×C₈₃	#3067	1	1	yes	characteristic
165	#3160	—	83	1	no	—
166	#3104	—	83	2	no	—
167	C₂×C₈×Q₈	—	83	1	no	—
168	#1676	—	83	2	no	—
169	#1674	—	83	2	no	—
170	#1672	—	83	4	no	—
171	#1564	D₈₃	1	1	yes	characteristic
172	#1542	—	83	2	no	—
173	C₈₃	#19118	1	1	yes	characteristic, complemented, Sylow
174	#533	—	83	2	no	—
175	C₂×C₄×Q₈	—	83	1	no	—
176	#500	—	83	1	no	—
177	#500	—	83	1	no	—
178	#499	—	166	1	no	—
179	#499	—	166	1	no	—
180	#498	—	83	2	no	—
181	#484	—	166	2	no	—
182	#477	—	83	4	no	—
183	#462	—	83	4	no	—
184	#451	—	166	2	no	—
185	C₈×Q₈	—	83	2	no	—
186	C₈×Q₈	—	83	2	no	—
187	C₈×Q₈	—	166	1	no	—
188	C₈×Q₈	—	166	1	no	—
189	C₄×Q₁₆	—	166	4	no	—
190	#431	C₂×D₈₃	1	2	yes	—
191	#427	—	166	1	no	—
192	#427	—	166	1	no	—
193	C₂×C₄⋊C₈	—	83	1	no	—
194	C₂×C₄⋊C₈	—	166	1	no	—
195	C₂×C₄×C₈	—	166	1	no	—
196	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
197	#335	C₂×D₈₃	1	2	yes	—
198	#334	C₂×D₈₃	1	2	yes	—
199	#333	—	83	4	no	—
200	#332	—	83	4	no	—
201	C₂²×Q₈	—	83	1	no	—
202	C₂×Q₁₆	—	166	4	no	—
203	C₂²×C₈	—	166	1	no	—
204	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
205	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
206	C₄⋊Q₈	—	83	2	no	—
207	C₄⋊Q₈	—	83	2	no	—
208	#125	—	166	2	no	—
209	C₂²⋊Q₈	—	166	2	no	—
210	C₂²⋊Q₈	—	166	2	no	—
211	C₄×Q₈	—	83	2	no	—
212	C₄×Q₈	—	83	2	no	—
213	C₄×Q₈	—	166	2	no	—
214	C₄×Q₈	—	166	2	no	—
215	C₄×Q₈	—	166	1	no	—
216	C₄×Q₈	—	166	1	no	—
217	#117	—	166	2	no	—
218	C₂×C₄⋊C₄	—	83	1	no	—
219	C₂×C₄⋊C₄	—	166	1	no	—
220	C₂×C₄²	—	166	1	no	—
221	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
222	#107	—	166	4	no	—
223	#107	—	166	1	no	—
224	#107	—	166	1	no	—
225	#107	—	166	1	no	—
226	#107	—	166	1	no	—
227	#106	—	166	2	no	—
228	#106	—	166	2	no	—
229	C₄⋊C₈	—	83	2	no	—
230	C₄⋊C₈	—	166	2	no	—
231	C₄⋊C₈	—	166	2	no	—
232	C₄⋊C₈	—	166	1	no	—
233	C₄⋊C₈	C₂²×D₈₃	1	4	yes	—
234	#103	—	166	4	no	—
235	#103	—	166	4	no	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	—	166	2	no	—
239	C₄×C₈	—	166	2	no	—
240	C₄×C₈	D₃₃₂	1	1	yes	characteristic
241	C₄×C₈	D₃₃₂	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
244	C₂×Q₈	—	83	2	no	—
245	C₂×Q₈	—	166	4	no	—
246	C₂×Q₈	—	166	2	no	—
247	C₂×Q₈	—	166	2	no	—
248	C₂×Q₈	—	166	1	no	—
249	C₂²×C₄	—	166	1	no	—
250	C₂²×C₄	—	166	1	no	—
251	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
253	C₂²×C₄	#13050591	1	1	yes	characteristic
254	Q₁₆	—	332	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	332	2	no	—
258	C₂×C₈	—	166	2	no	—
259	C₂×C₈	Dic₃₃₂	1	4	yes	complemented
260	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
261	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
262	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
263	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
264	C₄⋊C₄	—	332	2	no	—
265	C₄⋊C₄	—	332	2	no	—
266	C₄⋊C₄	—	83	2	no	—
267	C₄⋊C₄	—	166	2	no	—
268	C₄⋊C₄	—	166	2	no	—
269	C₄⋊C₄	—	166	2	no	—
270	C₄⋊C₄	—	166	2	no	—
271	C₄⋊C₄	—	166	1	no	—
272	C₄⋊C₄	—	166	2	no	—
273	C₄⋊C₄	—	166	2	no	—
274	C₂²⋊C₄	—	332	2	no	—
275	C₄²	—	332	2	no	—
276	C₄²	—	166	2	no	—
277	C₄²	—	166	2	no	—
278	C₄²	D₄×D₈₃	1	1	yes	characteristic
279	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
280	C₄²	#13050591	1	1	yes	characteristic
281	C₄²	C₂³×D₈₃	1	1	yes	characteristic
282	C₂³	#14581767	1	1	yes	characteristic
283	Q₈	—	332	4	no	—
284	Q₈	—	83	4	no	—
285	Q₈	—	664	2	no	—
286	Q₈	—	166	4	no	—
287	Q₈	—	166	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	332	2	no	—
291	C₂×C₄	—	332	2	no	—
292	C₂×C₄	—	332	2	no	—
293	C₂×C₄	—	166	2	no	—
294	C₂×C₄	—	166	4	no	—
295	C₂×C₄	—	166	2	no	—
296	C₂×C₄	#14581767	1	1	yes	characteristic
297	C₂×C₄	#14581767	1	1	yes	characteristic
298	C₂×C₄	#14581767	1	1	yes	characteristic
299	C₂×C₄	C₂×Dic₃₃₂	1	2	yes	—
300	C₂×C₄	#14581825	1	2	yes	—
301	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
303	C₂×C₄	C₂×#13050591	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	4	4	no	—
307	C₈	—	332	2	no	—
308	C₈	C₂×Dic₃₃₂	1	2	yes	—
309	C₈	C₂×Dic₃₃₂	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#19127912	1	2	yes	—
313	C₂²	C₂×#14581767	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	332	2	no	—
318	C₄	—	664	2	no	—
319	C₄	—	664	2	no	—
320	C₄	—	166	4	no	—
321	C₄	#19127912	1	2	yes	—
322	C₄	C₂²×Dic₃₃₂	1	1	yes	characteristic
323	C₄	#135×D₈₃	1	2	yes	—
324	C₄	C₂×#14581825	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	C₂×#19127912	1	1	yes	characteristic
327	C₂	#24917045	1	1	yes	characteristic
328	C₂	#24917184	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581767
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#1564	C₂
socle	C₂²×C₈₃	C₂×#121
prime radical	C₂³×C₈₃	#121
hypercenter	#1564	D₈₃
hypocenter	C₈₃	#19118
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
4. #24906868	C₂	semidirect
5. #24906858	C₂	semidirect
6. #24906846	C₂	semidirect
108. C₂×C₈×C₈₃	Q₁₆	semidirect
173. C₈₃	#19118	semidirect
259. C₂×C₈	Dic₃₃₂	semidirect

Maximal subgroups

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

subgroup	index
#24917977	2
#24917503	2
C₂×C₈×Dic₁₆₆	2
#24906868	2
#24906858	2
#24906846	2
#24906118	2
C₈₃×#1564	2
#19118	83