Finite Group Databasebeta

← #24916989

Subgroups — #24916989

Order 10624.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#14581822C211yesmaximal, characteristic
2C22×#13050580C211yesmaximal, characteristic, complemented
3#19128178C211yesmaximal, characteristic, complemented
4#19127920C214yesmaximal, complemented
5#19127908C214yesmaximal, complemented
6#19127728C211yesmaximal, characteristic, complemented
7#19127710C211yesmaximal, characteristic, complemented
8#19127697C211yesmaximal, characteristic
9#536×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11#14581822—24no—
12C2×#13050580—24no—
13C2×#13050580C2212yes—
14C2×#13050580C2214yes—
15C2×#13050578C2212yes—
16C2×#13050574C2211yescharacteristic
17C22×C83⋊C8C2211yescharacteristic
18C4×Dic166C2214yes—
19#14581757—22no—
20#14581751C2211yescharacteristic
21#14581748—22no—
22C4⋊(C83⋊C8)C2214yes—
23#14581716C2212yes—
24#14581716C2214yes—
25#14581713C2212yes—
26C2×#41×C83C2211yescharacteristic
27C22×Q8×C83C2211yescharacteristic
28C4⋊Q8×C83C2212yes—
29#124×C83C2212yes—
30#123×C83—22no—
31C22⋊Q8×C83—22no—
32#117×C83C2211yescharacteristic
33C2×Dic166—21no—
34C2×Dic166—22no—
35C2×Dic166D414yes—
36C2×Dic166C2312yes—
37C22×Dic83—21no—
38#13050580—28no—
39#13050580—24no—
40#13050578—24no—
41#13050574—22no—
42C2×C83⋊C8—22no—
43C2×C83⋊C8C2312yes—
44C2×C83⋊C8C2312yes—
45C4⋊Dic83C2312yes—
46Dic83⋊C4—24no—
47C22⋊Dic83—21no—
48C4×Dic83—24no—
49C4×Dic83—21no—
50#41×C83—22no—
51#41×C83—22no—
52C2×Q8×C83—21no—
53C2×Q8×C83—22no—
54C2×Q8×C83—22no—
55C2×Q8×C83C2312yes—
56C2×Q8×C83C2311yescharacteristic
57C2×D4×C83—21no—
58C2×D4×C83C2311yescharacteristic
59C22×C4×C83—21no—
60C22×C4×C83—21no—
61C22×C4×C83C2311yescharacteristic
62C4⋊C4×C83—24no—
63C4⋊C4×C83D412yes—
64C22⋊C4×C83—24no—
65C22⋊C4×C83D412yes—
66C42×C83C2312yes—
67C23×C83—21no—
68C23×C83C2×D411yescharacteristic
69Q8×C83—22no—
70Q8×C83—21no—
71Q8×C83—22no—
72Q8×C83—21no—
73Q8×C83—42no—
74D4×C83—21no—
75D4×C83—21no—
76D4×C83—42no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—42no—
80C2×C4×C83—42no—
81C2×C4×C83C2×D412yes—
82C2×C4×C83C2×D414yes—
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83#4112yes—
85C2×C4×C83C2411yescharacteristic
86C2×Dic83—24no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89Dic166—24no—
90Dic166—22no—
91Dic166C2×D414yes—
92C83⋊C8—22no—
93C83⋊C8—22no—
94C22×C83—22no—
95C22×C83—41no—
96C22×C83—42no—
97C22×C83#12112yes—
98C22×C83C22×D411yescharacteristic
99C4×C83—24no—
100C4×C83—41no—
101C4×C83—42no—
102C4×C83#12112yes—
103C4×C83C22×D411yescharacteristic
104C4×C83C2×#4111yescharacteristic
105Dic83—24no—
106Dic83—42no—
107C2×C83—22no—
108C2×C83—81no—
109C2×C83C2×#12111yescharacteristic
110C2×C83#57812yes—
111#3044—831nomaximal, Sylow
112C83#304411yescharacteristic, complemented, Sylow
113C2×#137—831no—
114C22×Q16—831no—
115#536D8311yescharacteristic
116#519—831no—
117#464—834no—
118#462—834no—
119#424—831no—
120#419—831no—
121#416—831no—
122C2×#41C2×D8311yescharacteristic
123C22×Q8—831no—
124C22×Q8C2×D8311yescharacteristic
125#137—1664no—
126C2×Q16—834no—
127C2×Q16—832no—
128C2×Q16—1664no—
129C2×SD16—832no—
130C2×M16—831no—
131C22×C8—831no—
132C4⋊Q8C2×D8312yes—
133#124C2×D8312yes—
134#123—22no—
135C22⋊Q8—22no—
136C4×Q8—834no—
137C4×Q8—1662no—
138#117—831no—
139#117—1662no—
140#117C2×D8311yescharacteristic
141C4⋊C8—834no—
142#103—832no—
143#103—834no—
144#102—832no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—834no—
151C2×Q8—832no—
152C2×Q8—1661no—
153C2×Q8—1662no—
154C2×Q8C22×D8312yes—
155C2×Q8C22×D8311yescharacteristic
156C2×D4—21no—
157C2×D4C22×D8311yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—1661no—
161C22×C4C22×D8311yescharacteristic
162Q16—1668no—
163Q16—1664no—
164SD16—1664no—
165M16—1662no—
166C2×C8—832no—
167C2×C8—832no—
168C2×C8—1662no—
169C4⋊C4—24no—
170C4⋊C4—832no—
171C4⋊C4—1664no—
172C4⋊C4C22⋊D8312yes—
173C22⋊C4—24no—
174C22⋊C4—1661no—
175C22⋊C4C22⋊D8312yes—
176C42—1661no—
177C42—1664no—
178C42C22×D8312yes—
179C23—21no—
180C23C2×C22⋊D8311yescharacteristic
181Q8—22no—
182Q8—21no—
183Q8—22no—
184Q8—21no—
185Q8—42no—
186Q8—834no—
187Q8—1664no—
188Q8—1662no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—1662no—
197C2×C4—1664no—
198C2×C4—1662no—
199C2×C4D4×D8312yes—
200C2×C4C2×C22⋊D8314yes—
201C2×C4C2×C22⋊D8311yescharacteristic
202C2×C4#1305058912yes—
203C2×C4C23×D8311yescharacteristic
204C8—1662no—
205C8—1662no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22#1458176312yes—
210C22C22×C22⋊D8311yescharacteristic
211C4—24no—
212C4—41no—
213C4—42no—
214C4—3322no—
215C4—1664no—
216C4#1458176312yes—
217C4C2×D4×D8311yescharacteristic
218C4C2×#1305058911yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1458176311yescharacteristic
222C2#1912851612yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#536×C83C2
socleC22×C83C22×D4
prime radicalC2×#41×C83C22
hypercenter#536D83
hypocenterC83#3044
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C22×#13050580C2semidirect
3. #19128178C2semidirect
4. #19127920C2semidirect
5. #19127908C2semidirect
6. #19127728C2semidirect
7. #19127710C2semidirect
112. C83#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#145818222
C22×#130505802
#191281782
#191279202
#191279082
#191277282
#191277102
#191276972
#536×C832
#304483