Subgroups — #35320447

Order 21248.

446 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₈₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127504	C₂	1	2	yes	maximal
4	#24905554	C₂	1	2	yes	maximal
5	C₈₃×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₆₆	—	2	1	no	—
8	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
9	C₄×Dic₃₃₂	—	2	2	no	—
10	C₄×Dic₃₃₂	C₄	1	4	yes	complemented
11	C₂×#14581726	C₂²	1	1	yes	characteristic
12	C₂×#14581717	—	2	1	no	—
13	C₄⋊(C₈₃⋊C₁₆)	—	2	2	no	—
14	#19127504	—	2	4	no	—
15	C₂×#107×C₈₃	—	2	1	no	—
16	C₂×#107×C₈₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	#384×C₈₃	—	2	1	no	—
19	C₂²×Dic₁₆₆	—	2	1	no	—
20	C₂×Dic₃₃₂	—	2	1	no	—
21	C₂×Dic₃₃₂	—	2	1	no	—
22	C₂×Dic₃₃₂	—	2	1	no	—
23	C₂×Dic₃₃₂	C₂×C₄	1	2	yes	—
24	C₂×Dic₃₃₂	D₄	1	2	yes	—
25	C₂×Dic₃₃₂	Q₈	1	2	yes	—
26	C₄×Dic₁₆₆	—	2	4	no	—
27	C₄×Dic₁₆₆	—	4	2	no	—
28	C₂×C₄⋊Dic₈₃	—	2	1	no	—
29	C₂×Dic₈₃⋊C₄	—	4	1	no	—
30	C₂×C₄×Dic₈₃	—	4	1	no	—
31	C₂×C₈₃⋊C₁₆	—	2	2	no	—
32	C₂×C₈₃⋊C₁₆	—	2	2	no	—
33	C₂×C₈₃⋊C₁₆	—	2	2	no	—
34	#14581726	—	2	1	no	—
35	#14581726	C₂×C₄	1	2	yes	—
36	#14581717	—	2	4	no	—
37	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
41	C₂×C₄⋊C₄×C₈₃	—	4	1	no	—
42	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
43	#107×C₈₃	—	2	2	no	—
44	#107×C₈₃	—	2	2	no	—
45	#107×C₈₃	—	4	2	no	—
46	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₈₃	D₄	1	2	yes	—
48	C₂×Dic₁₆₆	—	2	2	no	—
49	C₂×Dic₁₆₆	—	2	2	no	—
50	C₂×Dic₁₆₆	—	2	2	no	—
51	C₂×Dic₁₆₆	—	4	1	no	—
52	C₂×Dic₁₆₆	—	4	1	no	—
53	C₂×Dic₁₆₆	—	4	1	no	—
54	C₂²×Dic₈₃	—	4	1	no	—
55	C₂²×Dic₈₃	—	4	1	no	—
56	Dic₃₃₂	—	2	2	no	—
57	Dic₃₃₂	—	2	2	no	—
58	Dic₃₃₂	—	2	2	no	—
59	Dic₃₃₂	C₄⋊C₄	1	4	yes	complemented
60	C₄⋊Dic₈₃	—	2	2	no	—
61	C₄⋊Dic₈₃	—	4	1	no	—
62	Dic₈₃⋊C₄	—	4	4	no	—
63	C₄×Dic₈₃	—	4	4	no	—
64	C₈₃⋊C₁₆	—	4	2	no	—
65	C₂²×C₄×C₈₃	—	4	2	no	—
66	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₈₃	—	2	1	no	—
70	C₂×C₈×C₈₃	—	2	1	no	—
71	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
77	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
78	C₄⋊C₄×C₈₃	—	4	2	no	—
79	C₄⋊C₄×C₈₃	—	4	2	no	—
80	C₄⋊C₄×C₈₃	—	8	2	no	—
81	C₄⋊C₄×C₈₃	—	8	2	no	—
82	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₈₃	SD₁₆	1	2	yes	—
84	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₈₃	—	2	1	no	—
86	C₂×C₄×C₈₃	—	2	1	no	—
87	C₂×C₄×C₈₃	—	8	2	no	—
88	C₂×C₄×C₈₃	—	8	2	no	—
89	C₂×C₄×C₈₃	—	8	2	no	—
90	C₂×C₄×C₈₃	#102	1	2	yes	—
91	C₂×C₄×C₈₃	#102	1	2	yes	—
92	C₂×C₄×C₈₃	SD₃₂	1	2	yes	—
93	C₂×C₄×C₈₃	Q₃₂	1	2	yes	—
94	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
100	C₂×Dic₈₃	—	4	2	no	—
101	C₂×Dic₈₃	—	4	2	no	—
102	C₂×Dic₈₃	—	4	2	no	—
103	C₂×Dic₈₃	—	4	2	no	—
104	C₂×Dic₈₃	—	8	1	no	—
105	C₂×Dic₈₃	—	8	1	no	—
106	Dic₁₆₆	—	2	4	no	—
107	Dic₁₆₆	—	4	2	no	—
108	Dic₁₆₆	—	4	2	no	—
109	Dic₁₆₆	—	4	2	no	—
110	C₈×C₈₃	—	2	2	no	—
111	C₈×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈×C₈₃	#117	1	1	yes	characteristic
113	C₈×C₈₃	#121	1	1	yes	characteristic
114	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
116	C₂²×C₈₃	#459	1	1	yes	characteristic
117	C₂²×C₈₃	#474	1	1	yes	characteristic
118	C₂²×C₈₃	C₂×SD₃₂	1	1	yes	characteristic
119	C₂²×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
120	C₂²×C₈₃	#509	1	1	yes	characteristic
121	C₂²×C₈₃	#510	1	1	yes	characteristic
122	C₄×C₈₃	—	2	2	no	—
123	C₄×C₈₃	—	16	2	no	—
124	C₄×C₈₃	#358	1	4	yes	complemented
125	C₄×C₈₃	#380	1	1	yes	characteristic
126	C₄×C₈₃	#417	1	1	yes	characteristic
127	C₄×C₈₃	#461	1	1	yes	characteristic
128	C₄×C₈₃	#476	1	1	yes	characteristic
129	Dic₈₃	—	4	4	no	—
130	Dic₈₃	—	8	2	no	—
131	#12812	—	83	1	no	maximal, Sylow
132	C₂×C₈₃	#1802	1	1	yes	characteristic
133	C₂×C₈₃	C₂×#358	1	1	yes	characteristic
134	C₂×C₈₃	#2139	1	1	yes	characteristic
135	C₂×C₈₃	#2207	1	1	yes	characteristic
136	C₂×C₈₃	#2209	1	1	yes	characteristic
137	C₂×C₈₃	#2225	1	1	yes	characteristic
138	C₂×C₈₃	#2227	1	1	yes	characteristic
139	C₂×C₄×Q₁₆	—	83	1	no	—
140	C₂×C₄⋊C₁₆	—	83	1	no	—
141	C₂×#358	—	83	2	no	—
142	#1845	D₈₃	1	1	yes	characteristic
143	#1380	—	83	2	no	—
144	C₈₃	#12812	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₁₆	—	83	1	no	—
146	C₂×C₄×Q₈	—	166	1	no	—
147	C₂²×C₁₆	—	83	2	no	—
148	C₄×Q₁₆	—	83	4	no	—
149	C₄×Q₁₆	—	166	2	no	—
150	C₂×#107	—	2	1	no	—
151	C₂×#107	—	83	1	no	—
152	C₂×#107	C₂×D₈₃	1	2	yes	—
153	C₂×#103	—	166	1	no	—
154	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
155	#384	—	2	1	no	—
156	C₄⋊C₁₆	—	166	2	no	—
157	#358	—	166	4	no	—
158	C₂²×Q₈	—	166	1	no	—
159	C₂×Q₁₆	—	83	2	no	—
160	C₂×Q₁₆	—	83	2	no	—
161	C₂×Q₁₆	—	83	2	no	—
162	C₂×Q₁₆	—	166	1	no	—
163	C₂×Q₁₆	—	166	1	no	—
164	C₂×Q₁₆	—	166	1	no	—
165	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
167	C₄×Q₈	—	332	2	no	—
168	C₄×Q₈	—	166	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	4	1	no	—
171	C₂×C₄⋊C₄	—	4	1	no	—
172	C₂×C₄⋊C₄	—	332	1	no	—
173	C₂×C₄⋊C₄	—	166	1	no	—
174	C₂×C₄²	—	332	1	no	—
175	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
176	C₂×C₁₆	—	166	2	no	—
177	C₂×C₁₆	—	166	2	no	—
178	C₂×C₁₆	—	166	2	no	—
179	#107	—	2	2	no	—
180	#107	—	2	2	no	—
181	#107	—	4	2	no	—
182	#107	—	83	2	no	—
183	#107	—	166	1	no	—
184	#103	—	166	4	no	—
185	C₄×C₈	C₄×D₈₃	1	2	yes	—
186	C₄×C₈	D₃₃₂	1	2	yes	—
187	C₂×Q₈	—	332	1	no	—
188	C₂×Q₈	—	332	1	no	—
189	C₂×Q₈	—	332	1	no	—
190	C₂×Q₈	—	166	2	no	—
191	C₂×Q₈	—	166	2	no	—
192	C₂×Q₈	—	166	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	332	1	no	—
195	C₂²×C₄	—	332	1	no	—
196	C₂²×C₄	#13050575	1	1	yes	characteristic
197	C₂²×C₄	#13050575	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
199	Q₁₆	—	83	4	no	—
200	Q₁₆	—	166	2	no	—
201	Q₁₆	—	166	2	no	—
202	Q₁₆	—	166	2	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
207	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
208	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
209	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
210	C₂×C₈	#13050589	1	1	yes	characteristic
211	C₂×C₈	#13050591	1	1	yes	characteristic
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	8	2	no	—
215	C₄⋊C₄	—	8	2	no	—
216	C₄⋊C₄	—	332	1	no	—
217	C₄⋊C₄	—	332	4	no	—
218	C₄⋊C₄	—	166	2	no	—
219	C₄²	—	332	4	no	—
220	C₄²	(C₂×D₈₃)⋊C₄	1	2	yes	—
221	C₄²	#13050577	1	2	yes	—
222	C₁₆	—	332	2	no	—
223	C₂³	C₂×#13050575	1	1	yes	characteristic
224	Q₈	—	332	2	no	—
225	Q₈	—	332	2	no	—
226	Q₈	—	332	2	no	—
227	Q₈	—	166	4	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	332	2	no	—
234	C₂×C₄	—	332	2	no	—
235	C₂×C₄	—	332	2	no	—
236	C₂×C₄	—	332	2	no	—
237	C₂×C₄	—	664	1	no	—
238	C₂×C₄	—	664	1	no	—
239	C₂×C₄	#14581712	1	2	yes	—
240	C₂×C₄	#14581712	1	2	yes	—
241	C₂×C₄	#14581736	1	2	yes	—
242	C₂×C₄	#14581738	1	2	yes	—
243	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
244	C₂×C₄	#14581763	1	1	yes	characteristic
245	C₂×C₄	#14581768	1	1	yes	characteristic
246	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
247	C₂×C₄	#14581818	1	1	yes	characteristic
248	C₂×C₄	#14581823	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	C₄⋊C₄×D₈₃	1	1	yes	characteristic
251	C₈	#14581750	1	1	yes	characteristic
252	C₈	#14581767	1	1	yes	characteristic
253	C₈	#14581772	1	1	yes	characteristic
254	C₂²	C₂×#14581712	1	1	yes	characteristic
255	C₂²	#19127893	1	1	yes	characteristic
256	C₂²	#19127978	1	1	yes	characteristic
257	C₂²	C₂×#14581736	1	1	yes	characteristic
258	C₂²	C₂×#14581738	1	1	yes	characteristic
259	C₂²	#19128137	1	1	yes	characteristic
260	C₂²	#19128142	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	16	2	no	—
263	C₄	—	332	4	no	—
264	C₄	—	664	2	no	—
265	C₄	#19127504	1	4	yes	complemented
266	C₄	#19127558	1	1	yes	characteristic
267	C₄	#19127700	1	1	yes	characteristic
268	C₄	#19127903	1	1	yes	characteristic
269	C₄	#19127988	1	1	yes	characteristic
270	C₂	#24907460	1	1	yes	characteristic
271	C₂	C₂×#19127504	1	1	yes	characteristic
272	C₂	#24909173	1	1	yes	characteristic
273	C₂	#24909500	1	1	yes	characteristic
274	C₂	#24909510	1	1	yes	characteristic
275	C₂	#24909602	1	1	yes	characteristic
276	C₂	#24909612	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13050575
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1845	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1845	D₈₃
hypocenter	C₈₃	#12812
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₃₃₂	C₄	semidirect
59. Dic₃₃₂	C₄⋊C₄	semidirect
124. C₄×C₈₃	#358	semidirect
144. C₈₃	#12812	semidirect
265. C₄	#19127504	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₃₂	2
C₂×C₄⋊(C₈₃⋊C₁₆)	2
C₂×#19127504	2
#24905554	2
C₈₃×#1845	2
#12812	83