Finite Group Databasebeta

← #24914877

Subgroups — #24914877

Order 10624.

424 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#19127643C212yesmaximal, complemented
2#19127621C212yesmaximal, complemented
3#19127613C211yesmaximal, characteristic
4#19127613C211yesmaximal, characteristic
5#19127605C212yesmaximal, complemented
6#19127594C212yesmaximal
7#19127574C212yesmaximal
8#19127566C212yesmaximal, complemented
9#385×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C22×Dic166C2211yescharacteristic
12C2×C4⋊Dic83C2212yes—
13C2×Dic83⋊C4C2212yes—
14C2×Dic83⋊C4C2212yes—
15C2×C22⋊Dic83C2212yes—
16C2×C22⋊Dic83C2212yes—
17C2×C4×Dic83C2212yes—
18C83⋊#95C2211yescharacteristic
19C83⋊#95C2212yes—
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C23×C4×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2212yes—
28C2×C22⋊C4×C83C2211yescharacteristic
29C2×C22⋊C4×C83C2211yescharacteristic
30#95×C83C2212yes—
31C2×Dic166—24no—
32C2×Dic166—41no—
33C2×Dic166—42no—
34C2×Dic166—41no—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C4⋊Dic83—42no—
40Dic83⋊C4—42no—
41Dic83⋊C4—42no—
42C22⋊Dic83—24no—
43C22⋊Dic83—42no—
44C4×Dic83—24no—
45C24×C83C2311yescharacteristic
46C22×C4×C83—42no—
47C22×C4×C83—41no—
48C22×C4×C83C2311yescharacteristic
49C22×C4×C83C2311yescharacteristic
50C22×C4×C83C2312yes—
51C22×C4×C83C2312yes—
52C4⋊C4×C83—42no—
53C22⋊C4×C83—41no—
54C22⋊C4×C83Q814yescomplemented
55C23×C83—22no—
56C23×C83—21no—
57C23×C83—22no—
58C23×C83—21no—
59C23×C83C2×Q812yes—
60C23×C83C2411yescharacteristic
61C2×C4×C83—22no—
62C2×C4×C83—22no—
63C2×C4×C83—22no—
64C2×C4×C83—21no—
65C2×C4×C83—21no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—44no—
71C2×C4×C83C2×Q814yes—
72C2×Dic83—22no—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83#4114yes—
84Dic166—44no—
85Dic166—44no—
86C22×C83—22no—
87C22×C83—24no—
88C22×C83—42no—
89C22×C83—41no—
90C22×C83—41no—
91C22×C83C22×Q811yescharacteristic
92C22×C83C2×#4112yes—
93C22×C83H(5,𝔽2)12yes—
94C22×C83H(5,𝔽2)11yescharacteristic
95C22×C832−1+411yescharacteristic
96C4×C83—24no—
97C4×C83—41no—
98C4×C83—41no—
99C4×C83—42no—
100Dic83—24no—
101Dic83—42no—
102Dic83—42no—
103Dic83—42no—
104C2×C83—42no—
105C2×C83#54212yes—
106C2×C83#54411yescharacteristic
107C2×C83#55111yescharacteristic
108C2×C83#55412yes—
109C2×C83H(3,𝔽4)11yescharacteristic
110#2763—831nomaximal, Sylow
111C83#276311yescharacteristic, complemented, Sylow
112#400—832no—
113#395—832no—
114#393—831no—
115#393—831no—
116#391—832no—
117#388—832no—
118#385—832no—
119#385D8311yescharacteristic
120#382—832no—
121C22×Q8—831no—
122C22×Q8—831no—
123C23×C4C2×D8311yescharacteristic
124C2×C4⋊C4—832no—
125C2×C4⋊C4—832no—
126C2×C4⋊C4—832no—
127C2×C4⋊C4C2×D8312yes—
128C2×C22⋊C4—832no—
129C2×C22⋊C4—832no—
130C2×C22⋊C4C2×D8311yescharacteristic
131C2×C22⋊C4C2×D8311yescharacteristic
132C2×C42—832no—
133#95—831no—
134#95—832no—
135#95—832no—
136#95—832no—
137#95—832no—
138#95—832no—
139#95—832no—
140#95—831no—
141#95C2×D8312yes—
142C24C22×D8311yescharacteristic
143C2×Q8—3321no—
144C2×Q8—3321no—
145C2×Q8—3322no—
146C2×Q8—1664no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—832no—
150C22×C4—832no—
151C22×C4—832no—
152C22×C4—832no—
153C22×C4C22×D8311yescharacteristic
154C22×C4C22×D8311yescharacteristic
155C22×C4C22×D8312yes—
156C22×C4C22×D8312yes—
157C4⋊C4—42no—
158C4⋊C4—3322no—
159C4⋊C4—3322no—
160C4⋊C4—3322no—
161C22⋊C4—41no—
162C22⋊C4—3322no—
163C22⋊C4—1664no—
164C22⋊C4Dic16614yescomplemented
165C42—1664no—
166C23—22no—
167C23—21no—
168C23—22no—
169C23—21no—
170C23C2×Dic16612yes—
171C23C23×D8311yescharacteristic
172Q8—3324no—
173Q8—3324no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—834no—
185C2×C4—1662no—
186C2×C4—1662no—
187C2×C4—1662no—
188C2×C4—1662no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4—1662no—
192C2×C4—1662no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4—1662no—
196C2×C4C2×Dic16614yes—
197C22—22no—
198C22—24no—
199C22—42no—
200C22—41no—
201C22—41no—
202C22C22×Dic16611yescharacteristic
203C22#41×D8312yes—
204C22#1458183712yes—
205C22#1458183711yescharacteristic
206C22#1458184011yescharacteristic
207C4—24no—
208C4—41no—
209C4—42no—
210C4—41no—
211C4—3322no—
212C4—3322no—
213C4—3322no—
214C4—1664no—
215C2—42no—
216C2#1912832112yes—
217C2#1912833211yescharacteristic
218C2#1912837911yescharacteristic
219C2#1912841012yes—
220C2#1912844411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#385×C83C2
prime radicalC24×C83C23
hypercenter#385D83
hypocenterC83#2763
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #19127643C2semidirect
2. #19127621C2semidirect
5. #19127605C2semidirect
8. #19127566C2semidirect
54. C22⋊C4×C83Q8semidirect
111. C83#2763semidirect
164. C22⋊C4Dic166semidirect

Maximal subgroups

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

subgroupindex
#191276432
#191276212
#191276132
#191276052
#191275942
#191275742
#191275662
#385×C832
#276383