Finite Group Databasebeta

← #24915653

Subgroups — #24915653

Order 10624.

380 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 117 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#19127642C212yesmaximal, complemented
2#19127641C212yesmaximal, complemented
3#19127640C212yesmaximal, complemented
4#19127629C212yesmaximal
5#19127622C212yesmaximal, complemented
6#19127613C211yesmaximal, characteristic
7#19127576C212yesmaximal
8#19127573C211yesmaximal, characteristic
9#397×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C23×Dic83C2211yescharacteristic
12C2×C4⋊Dic83C2212yes—
13C2×C4⋊Dic83C2212yes—
14C2×Dic83⋊C4C2212yes—
15C2×Dic83⋊C4C2212yes—
16C2×C22⋊Dic83C2211yescharacteristic
17C2×C22⋊Dic83C2212yes—
18C83⋊#95C2212yes—
19C83⋊#95C2212yes—
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2211yescharacteristic
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C22⋊C4×C83C2212yes—
28C2×C22⋊C4×C83C2211yescharacteristic
29#95×C83C2212yes—
30#95×C83C2211yescharacteristic
31C2×Dic166—42no—
32C2×Dic166—41no—
33C22×Dic83—42no—
34C22×Dic83—41no—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C4⋊Dic83—42no—
40C4⋊Dic83—42no—
41Dic83⋊C4—42no—
42Dic83⋊C4—42no—
43C22⋊Dic83—22no—
44C22⋊Dic83—42no—
45C24×C83C2311yescharacteristic
46C22×C4×C83C2311yescharacteristic
47C22×C4×C83C2312yes—
48C22×C4×C83C2311yescharacteristic
49C22×C4×C83C2312yes—
50C4⋊C4×C83—41no—
51C22⋊C4×C83—24no—
52C22⋊C4×C83—41no—
53C23×C83—21no—
54C23×C83—22no—
55C23×C83—22no—
56C23×C83—21no—
57C23×C83#4112yes—
58C23×C83C2411yescharacteristic
59C2×C4×C83—22no—
60C2×C4×C83—22no—
61C2×C4×C83—21no—
62C2×C4×C83—21no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×Dic83—22no—
70C2×Dic83—22no—
71C2×Dic83—22no—
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—44no—
82Dic166—44no—
83C22×C83—24no—
84C22×C83—22no—
85C22×C83—41no—
86C22×C83—42no—
87C22×C83—41no—
88C22×C83C2×#4111yescharacteristic
89C22×C83H(5,𝔽2)11yescharacteristic
90C22×C83H(5,𝔽2)12yes—
91C22×C832−1+411yescharacteristic
92C22×C832−1+412yes—
93C4×C83—42no—
94C4×C83—42no—
95C4×C83—41no—
96C4×C83—41no—
97Dic83—42no—
98Dic83—42no—
99Dic83—42no—
100Dic83—42no—
101C2×C83—42no—
102C2×C83#53912yes—
103C2×C83#53911yescharacteristic
104C2×C83H(3,𝔽4)11yescharacteristic
105C2×C83#56312yes—
106C2×C83#56311yescharacteristic
107#2835—831nomaximal, Sylow
108C83#283511yescharacteristic, complemented, Sylow
109#400—832no—
110#400—832no—
111#400—832no—
112#397—832no—
113#397D8311yescharacteristic
114#395—832no—
115#393—831no—
116#385—831no—
117#385—832no—
118C22×Q8—831no—
119C23×C4—831no—
120C2×C4⋊C4—832no—
121C2×C4⋊C4—832no—
122C2×C4⋊C4—832no—
123C2×C4⋊C4—832no—
124C2×C4⋊C4C2×D8311yescharacteristic
125C2×C22⋊C4—832no—
126C2×C22⋊C4—831no—
127C2×C22⋊C4C2×D8312yes—
128C2×C22⋊C4C2×D8311yescharacteristic
129#95—832no—
130#95—832no—
131#95—832no—
132#95—831no—
133#95—832no—
134#95—832no—
135#95—832no—
136#95—832no—
137#95C2×D8312yes—
138#95C2×D8311yescharacteristic
139C24C22×D8311yescharacteristic
140C2×Q8—3321no—
141C2×Q8—3322no—
142C22×C4—3321no—
143C22×C4—3322no—
144C22×C4—832no—
145C22×C4—832no—
146C22×C4—832no—
147C22×C4—832no—
148C22×C4C22×D8311yescharacteristic
149C22×C4C22×D8312yes—
150C22×C4C22×D8312yes—
151C22×C4C22×D8311yescharacteristic
152C4⋊C4—41no—
153C4⋊C4—3322no—
154C4⋊C4—3322no—
155C4⋊C4—3322no—
156C4⋊C4—3322no—
157C22⋊C4—24no—
158C22⋊C4—41no—
159C22⋊C4—3322no—
160C22⋊C4—1662no—
161C23—21no—
162C23—22no—
163C23—22no—
164C23—21no—
165C23#1305058912yes—
166C23C23×D8311yescharacteristic
167Q8—3324no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—3324no—
179C2×C4—1662no—
180C2×C4—1662no—
181C2×C4—1662no—
182C2×C4—1662no—
183C2×C4—1662no—
184C2×C4—1662no—
185C2×C4—1662no—
186C2×C4—1662no—
187C2×C4—1662no—
188C2×C4—1662no—
189C2×C4—1662no—
190C2×C4—1662no—
191C22—24no—
192C22—22no—
193C22—41no—
194C22—42no—
195C22—41no—
196C22C2×#1305058911yescharacteristic
197C22#1458183711yescharacteristic
198C22#1458183812yes—
199C22#1458183911yescharacteristic
200C22#1458183912yes—
201C4—41no—
202C4—41no—
203C4—42no—
204C4—42no—
205C4—3322no—
206C4—3322no—
207C4—3322no—
208C4—3322no—
209C2—42no—
210C2#1912829512yes—
211C2#1912829611yescharacteristic
212C2#1912844311yescharacteristic
213C2#1912845312yes—
214C2#1912845411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC24×C83C23
hypercenter#397D83
hypocenterC83#2835
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. #19127642C2semidirect
2. #19127641C2semidirect
3. #19127640C2semidirect
5. #19127622C2semidirect
108. C83#2835semidirect

Maximal subgroups

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

subgroupindex
#191276422
#191276412
#191276402
#191276292
#191276222
#191276132
#191275762
#191275732
#397×C832
#283583