Finite Group Databasebeta

← #24911106

Subgroups — #24911106

Order 10624.

800 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 193 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×D83C211yesmaximal, characteristic
2C2×#14581792C211yesmaximal, characteristic, complemented
3C2×#14581790C211yesmaximal, characteristic, complemented
4C2×#14581772C211yesmaximal, characteristic
5#19127622C214yesmaximal, complemented
6#19127606C211yesmaximal, characteristic, complemented
7#19127575C214yesmaximal
8C4×(C2×D83)⋊C4C211yesmaximal, characteristic
9#391×C83C211yesmaximal, characteristic, complemented
10C22×Dic166C2212yes—
11C22×Dic166C2211yescharacteristic
12C22×Dic166C2211yescharacteristic
13C2×Q8×D83—24no—
14C2×Q8×D83—24no—
15C2×Q8×D83—24no—
16C22×C4×D83C2212yes—
17C22×C4×D83C2211yescharacteristic
18#14581792—24no—
19#14581790—24no—
20#14581772—24no—
21C2×C4⋊Dic83C2214yes—
22C2×C4⋊Dic83C2212yes—
23C2×Dic83⋊C4C2214yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×C4×Dic83C2211yescharacteristic
27C83⋊#95C2214yes—
28C83⋊#95C2212yes—
29C22×Q8×C83C2211yescharacteristic
30C2×C4⋊C4×C83C2211yescharacteristic
31C2×C42×C83C2211yescharacteristic
32#95×C83C2214yes—
33C23×D83C2311yescharacteristic
34C2×Dic166—22no—
35C2×Dic166—24no—
36C2×Dic166—22no—
37C2×Dic166—24no—
38C2×Dic166—24no—
39C2×Dic166—22no—
40C2×Dic166—22no—
41C2×Dic166—22no—
42C2×Dic166D414yes—
43Q8×D83—232no—
44C22×Dic83C2312yes—
45C22×Dic83C2312yes—
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2312yes—
48C2×C4×D83—24no—
49C2×C4×D83—22no—
50C2×C4×D83—24no—
51C2×C4×D83—22no—
52C2×C4×D83—24no—
53C2×C4×D83D414yes—
54C4⋊Dic83—24no—
55C4⋊Dic83—28no—
56Dic83⋊C4—28no—
57(C2×D83)⋊C4—24no—
58(C2×D83)⋊C4Q818yescomplemented
59C4×Dic83—22no—
60C2×Q8×C83—22no—
61C2×Q8×C83—22no—
62C2×Q8×C83D414yes—
63C22×C4×C83C2312yes—
64C22×C4×C83C2312yes—
65C22×C4×C83C2312yes—
66C22×C4×C83C2311yescharacteristic
67C4⋊C4×C83—22no—
68C42×C83—22no—
69C23×C83C2411yescharacteristic
70C22×D83—21no—
71C22×D83—21no—
72C22×D83—22no—
73C22×D83C2×D412yes—
74C22×D83C2×Q814yes—
75Q8×C83—28no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—24no—
82C2×C4×C83C2×D412yes—
83C2×C4×C83C2×D412yes—
84C2×C4×C83C2×D414yes—
85C2×C4×C83C2×D412yes—
86C2×C4×C83C2×Q814yescomplemented
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—21no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83C2×D414yes—
97C2×Dic83C2×D412yes—
98C2×Dic83C2×Q814yescomplemented
99C4×D83—216no—
100C4×D83—28no—
101Dic166—216no—
102Dic166—28no—
103Dic166—44no—
104C22×C83C22×D411yescharacteristic
105C22×C83C22×D411yescharacteristic
106C22×C83C22×D411yescharacteristic
107C22×C83C22×Q812yes—
108C22×C832−1+412yes—
109C2×D83—22no—
110C2×D83—22no—
111C2×D83—24no—
112C2×D83—24no—
113C2×D83C4⋊Q814yescomplemented
114C4×C83—24no—
115C4×C83—24no—
116C4×C83—42no—
117C4×C83#12014yes—
118Dic83—22no—
119Dic83—24no—
120Dic83—24no—
121Dic83—42no—
122C2×C83C2×#12011yescharacteristic
123C2×C83C2×C4⋊Q811yescharacteristic
124C2×C83#53611yescharacteristic
125C2×C83D4×Q812yes—
126C2×C83D4×Q812yes—
127D83—24no—
128#2434—831nomaximal, Sylow
129C83#243411yescharacteristic, complemented, Sylow
130C23×Q8—831no—
131C2×C4⋊Q8—831no—
132C2×C4⋊Q8—831no—
133C2×C22⋊Q8—831no—
134#395—834no—
135#391—831no—
136#391D8311yescharacteristic, complemented
137#385—834no—
138C4×C22⋊C4—831no—
139C22×Q8—831no—
140C22×Q8—831no—
141C22×Q8—832no—
142C22×Q8—1664no—
143C22×Q8—1664no—
144C22×Q8—1664no—
145C22×Q8C2×D8311yescharacteristic
146C23×C4—831no—
147C23×C4—832no—
148C4⋊Q8—1664no—
149C4⋊Q8—1664no—
150C22⋊Q8—1664no—
151C2×C4⋊C4—832no—
152C2×C4⋊C4—834no—
153C2×C4⋊C4—834no—
154C2×C4⋊C4C2×D8311yescharacteristic
155C2×C22⋊C4—832no—
156C2×C22⋊C4—832no—
157C2×C42—831no—
158C2×C42C2×D8311yescharacteristic
159#95—832no—
160#95—834no—
161#95C2×D8314yes—
162C24—831no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—834no—
166C2×Q8—1664no—
167C2×Q8—1664no—
168C2×Q8—16632no—
169C2×Q8—1662no—
170C2×Q8—1662no—
171C2×Q8—1662no—
172C2×Q8—1662no—
173C2×Q8—1664no—
174C2×Q8—1662no—
175C2×Q8D33214yes—
176C22×C4—832no—
177C22×C4—831no—
178C22×C4—832no—
179C22×C4—834no—
180C22×C4—832no—
181C22×C4—1664no—
182C22×C4—1664no—
183C22×C4—1662no—
184C22×C4—1664no—
185C22×C4—1662no—
186C22×C4C22×D8312yes—
187C22×C4C22×D8312yes—
188C22×C4C22×D8312yes—
189C22×C4C22×D8311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—1664no—
192C4⋊C4—1668no—
193C4⋊C4—1668no—
194C22⋊C4—838no—
195C22⋊C4—1664no—
196C42—22no—
197C42—1662no—
198C23—832no—
199C23—834no—
200C23—1661no—
201C23—1661no—
202C23—1662no—
203C23C23×D8311yescharacteristic
204Q8—28no—
205Q8—3324no—
206Q8—16616no—
207Q8—1668no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—832no—
215C2×C4—834no—
216C2×C4—834no—
217C2×C4—16616no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4—1661no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4—1661no—
225C2×C4—1662no—
226C2×C4—1662no—
227C2×C4—1668no—
228C2×C4D4×D8312yes—
229C2×C4D4×D8312yes—
230C2×C4C2×D33214yes—
231C2×C4C2×D33212yes—
232C2×C4Q8×D8314yescomplemented
233C22—834no—
234C22—1664no—
235C22—1664no—
236C22—1662no—
237C22—1662no—
238C22C2×D4×D8311yescharacteristic
239C22C2×D4×D8311yescharacteristic
240C22C22×D33211yescharacteristic
241C22C2×Q8×D8312yes—
242C22#1458183912yes—
243C4—24no—
244C4—24no—
245C4—42no—
246C4—3322no—
247C4—1664no—
248C4—1662no—
249C4—1664no—
250C4#1458176214yes—
251C2—1664no—
252C2C2×#1458176211yescharacteristic
253C2C4⋊Q8×D8311yescharacteristic
254C2#1912827111yescharacteristic
255C2#1912836412yes—
256C2Q8×D33212yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#391×C83C2
prime radicalC23×D83C23
hypercenter#391D83
hypocenterC83#2434
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#14581792C2semidirect
3. C2×#14581790C2semidirect
5. #19127622C2semidirect
6. #19127606C2semidirect
9. #391×C83C2semidirect
58. (C2×D83)⋊C4Q8semidirect
86. C2×C4×C83C2×Q8semidirect
98. C2×Dic83C2×Q8semidirect
113. C2×D83C4⋊Q8semidirect
129. C83#2434semidirect
136. #391D83semidirect
232. C2×C4Q8×D83semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D832
C2×#145817922
C2×#145817902
C2×#145817722
#191276222
#191276062
#191275752
C4×(C2×D83)⋊C42
#391×C832
#243483