Subgroups — #35304439

Order 21248.

518 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14581792	C₂	1	1	yes	maximal, characteristic
2	#24907580	C₂	1	1	yes	maximal, characteristic
3	#24905303	C₂	1	4	yes	maximal
4	C₄×#97×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581792	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₆₆	—	2	1	no	—
7	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
8	#19127607	—	2	1	no	—
9	#19127599	—	2	1	no	—
10	#19127599	C₄	1	2	yes	—
11	#19127571	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₈₃	—	2	1	no	—
13	C₄×C₄⋊Dic₈₃	C₂²	1	2	yes	—
14	#19127456	—	2	1	no	—
15	#19127456	—	2	1	no	—
16	C₂×#97×C₈₃	—	2	1	no	—
17	C₂×#97×C₈₃	—	2	1	no	—
18	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
19	C₄³×C₈₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₆₆	—	2	1	no	—
21	C₂²×C₈₃⋊C₈	—	2	2	no	—
22	#14581792	—	2	4	no	—
23	C₄×Dic₁₆₆	—	4	4	no	—
24	C₂×C₄⋊Dic₈₃	—	2	1	no	—
25	C₂×C₄⋊Dic₈₃	—	2	1	no	—
26	C₂×C₄⋊Dic₈₃	—	2	1	no	—
27	C₂×C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
29	C₂×Dic₈₃⋊C₄	—	2	2	no	—
30	C₂×C₄×Dic₈₃	—	2	2	no	—
31	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
32	C₈₃⋊#95	—	2	2	no	—
33	C₈₃⋊#95	—	2	2	no	—
34	C₂²×C₈×C₈₃	—	2	2	no	—
35	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
42	#97×C₈₃	—	2	4	no	—
43	#97×C₈₃	—	2	4	no	—
44	C₄×C₈×C₈₃	—	2	2	no	—
45	C₄×C₈×C₈₃	Q₈	1	4	yes	complemented
46	C₂×Dic₁₆₆	—	2	4	no	—
47	C₂×Dic₁₆₆	—	4	2	no	—
48	C₂×Dic₁₆₆	—	4	2	no	—
49	C₂²×Dic₈₃	—	2	2	no	—
50	C₂²×Dic₈₃	—	2	2	no	—
51	C₂×C₈₃⋊C₈	—	4	2	no	—
52	C₂×C₈₃⋊C₈	—	4	2	no	—
53	C₂×C₈₃⋊C₈	—	4	2	no	—
54	C₄⋊Dic₈₃	—	2	4	no	—
55	C₄⋊Dic₈₃	—	2	2	no	—
56	C₄⋊Dic₈₃	—	2	4	no	—
57	C₄⋊Dic₈₃	—	4	2	no	—
58	C₄⋊Dic₈₃	—	4	2	no	—
59	C₄⋊Dic₈₃	M₁₆	1	4	yes	—
60	Dic₈₃⋊C₄	—	4	4	no	—
61	C₄×Dic₈₃	—	4	4	no	—
62	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₈₃	—	2	4	no	—
70	C₂×C₈×C₈₃	—	2	4	no	—
71	C₂×C₈×C₈₃	—	2	4	no	—
72	C₄²×C₈₃	—	2	2	no	—
73	C₄²×C₈₃	—	2	2	no	—
74	C₄²×C₈₃	—	2	2	no	—
75	C₄²×C₈₃	—	2	2	no	—
76	C₄²×C₈₃	—	2	2	no	—
77	C₄²×C₈₃	—	2	2	no	—
78	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
79	C₄²×C₈₃	#41	1	2	yes	—
80	C₂³×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₈₃	—	2	1	no	—
82	C₂×C₄×C₈₃	—	2	1	no	—
83	C₂×C₄×C₈₃	—	2	1	no	—
84	C₂×C₄×C₈₃	—	2	1	no	—
85	C₂×C₄×C₈₃	—	2	1	no	—
86	C₂×C₄×C₈₃	—	2	1	no	—
87	C₂×C₄×C₈₃	—	2	1	no	—
88	C₂×C₄×C₈₃	—	2	1	no	—
89	C₂×C₄×C₈₃	—	2	1	no	—
90	C₂×C₄×C₈₃	—	2	1	no	—
91	C₂×C₄×C₈₃	—	2	1	no	—
92	C₂×C₄×C₈₃	—	2	1	no	—
93	C₂×C₄×C₈₃	#104	1	2	yes	—
94	C₂×C₄×C₈₃	#104	1	2	yes	—
95	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₈₃	#131	1	2	yes	—
101	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
102	C₂×C₄×C₈₃	#137	1	2	yes	—
103	C₂×Dic₈₃	—	2	4	no	—
104	C₂×Dic₈₃	—	4	2	no	—
105	C₂×Dic₈₃	—	4	2	no	—
106	C₂×Dic₈₃	—	4	2	no	—
107	C₂×Dic₈₃	—	4	2	no	—
108	C₂×Dic₈₃	—	4	2	no	—
109	Dic₁₆₆	—	4	8	no	—
110	C₈×C₈₃	—	2	8	no	—
111	C₈₃⋊C₈	—	8	2	no	—
112	C₂²×C₈₃	#391	1	1	yes	characteristic
113	C₂²×C₈₃	#407	1	1	yes	characteristic
114	C₂²×C₈₃	#408	1	1	yes	characteristic
115	C₂²×C₈₃	#418	1	1	yes	characteristic
116	C₂²×C₈₃	#419	1	1	yes	characteristic
117	C₂²×C₈₃	C₂×#104	1	1	yes	characteristic
118	C₂²×C₈₃	C₂×#104	1	1	yes	characteristic
119	C₄×C₈₃	—	2	2	no	—
120	C₄×C₈₃	—	2	2	no	—
121	C₄×C₈₃	—	2	2	no	—
122	C₄×C₈₃	—	4	1	no	—
123	C₄×C₈₃	—	4	1	no	—
124	C₄×C₈₃	#446	1	4	yes	—
125	C₄×C₈₃	#489	1	2	yes	—
126	C₄×C₈₃	#501	1	2	yes	—
127	Dic₈₃	—	4	4	no	—
128	Dic₈₃	—	8	2	no	—
129	#9877	—	83	1	no	maximal, Sylow
130	C₂×C₈₃	#1484	1	1	yes	characteristic
131	C₂×C₈₃	#1485	1	1	yes	characteristic
132	C₂×C₈₃	#1490	1	1	yes	characteristic
133	C₂×C₈₃	#1491	1	1	yes	characteristic
134	C₂×C₈₃	#1985	1	1	yes	characteristic
135	C₂×C₈₃	#1993	1	1	yes	characteristic
136	C₂×C₈₃	#1994	1	1	yes	characteristic
137	C₄×C₄⋊Q₈	—	83	1	no	—
138	#1831	—	83	1	no	—
139	C₄×#97	D₈₃	1	1	yes	characteristic
140	#1279	—	83	4	no	—
141	C₈₃	#9877	1	1	yes	characteristic, complemented, Sylow
142	C₂×C₄⋊Q₈	—	83	1	no	—
143	C₂×C₄×Q₈	—	166	1	no	—
144	C₂×C₄⋊C₈	—	83	2	no	—
145	C₂×#97	—	2	1	no	—
146	C₂×#97	—	2	1	no	—
147	C₂×C₄×C₈	C₂×D₈₃	1	2	yes	—
148	#391	—	166	1	no	—
149	#389	—	83	2	no	—
150	#389	—	166	1	no	—
151	#384	—	83	1	no	—
152	C₄×C₄⋊C₄	—	83	2	no	—
153	C₄×C₄⋊C₄	—	166	1	no	—
154	C₄³	C₂×D₈₃	1	1	yes	characteristic
155	#336	—	166	1	no	—
156	#336	—	166	1	no	—
157	C₂²×Q₈	—	166	1	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	166	2	no	—
160	C₄⋊Q₈	—	166	4	no	—
161	C₄×Q₈	—	332	4	no	—
162	C₂×C₄⋊C₄	—	83	2	no	—
163	C₂×C₄⋊C₄	—	83	2	no	—
164	C₂×C₄⋊C₄	—	166	1	no	—
165	C₂×C₄⋊C₄	—	166	2	no	—
166	C₂×C₄⋊C₄	—	166	1	no	—
167	C₂×C₄⋊C₄	—	166	1	no	—
168	C₂×C₄²	—	166	2	no	—
169	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
171	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
172	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
176	C₄⋊C₈	—	166	4	no	—
177	#97	—	2	4	no	—
178	#97	—	2	4	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	Dic₁₆₆	1	4	yes	complemented
181	#95	—	166	2	no	—
182	#95	—	166	2	no	—
183	C₂×Q₈	—	332	2	no	—
184	C₂×Q₈	—	332	2	no	—
185	C₂×Q₈	—	166	4	no	—
186	C₂²×C₄	—	166	2	no	—
187	C₂²×C₄	—	166	2	no	—
188	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	332	2	no	—
199	C₂×C₈	—	332	2	no	—
200	C₂×C₈	—	332	2	no	—
201	C₄⋊C₄	—	332	4	no	—
202	C₄⋊C₄	—	332	2	no	—
203	C₄⋊C₄	—	332	2	no	—
204	C₄⋊C₄	—	83	4	no	—
205	C₄⋊C₄	—	166	4	no	—
206	C₄⋊C₄	—	166	4	no	—
207	C₄⋊C₄	—	166	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	332	4	no	—
215	C₄²	C₂×Dic₁₆₆	1	2	yes	—
216	C₄²	#13050590	1	2	yes	—
217	C₂³	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
218	Q₈	—	332	8	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	332	2	no	—
232	C₂×C₄	—	332	2	no	—
233	C₂×C₄	—	332	2	no	—
234	C₂×C₄	—	332	2	no	—
235	C₂×C₄	—	332	2	no	—
236	C₂×C₄	—	166	4	no	—
237	C₂×C₄	#14581720	1	2	yes	—
238	C₂×C₄	#14581720	1	2	yes	—
239	C₂×C₄	C₄×Dic₁₆₆	1	2	yes	—
240	C₂×C₄	#14581771	1	2	yes	—
241	C₂×C₄	#14581780	1	1	yes	characteristic
242	C₂×C₄	#14581792	1	1	yes	characteristic
243	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
244	C₂×C₄	#14581800	1	2	yes	—
245	C₂×C₄	#14581820	1	2	yes	—
246	C₂×C₄	#14581825	1	2	yes	—
247	C₈	—	2	8	no	—
248	C₈	—	664	2	no	—
249	C₂²	#19127607	1	1	yes	characteristic
250	C₂²	#19127664	1	1	yes	characteristic
251	C₂²	#19127669	1	1	yes	characteristic
252	C₂²	#19127707	1	1	yes	characteristic
253	C₂²	#19127711	1	1	yes	characteristic
254	C₂²	C₂×#14581720	1	1	yes	characteristic
255	C₂²	C₂×#14581720	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	1	no	—
261	C₄	—	332	4	no	—
262	C₄	—	664	2	no	—
263	C₄	#19127829	1	4	yes	—
264	C₄	#19128061	1	2	yes	—
265	C₄	#19128110	1	2	yes	—
266	C₂	#24905881	1	1	yes	characteristic
267	C₂	#24905885	1	1	yes	characteristic
268	C₂	#24905907	1	1	yes	characteristic
269	C₂	#24905911	1	1	yes	characteristic
270	C₂	#24908403	1	1	yes	characteristic
271	C₂	#24908436	1	1	yes	characteristic
272	C₂	#24908440	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₃₂
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₄×#97×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₄×#97	D₈₃
hypocenter	C₈₃	#9877
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
45. C₄×C₈×C₈₃	Q₈	semidirect
141. C₈₃	#9877	semidirect
180. C₄×C₈	Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14581792	2
#24907580	2
#24905303	2
C₄×#97×C₈₃	2
#9877	83