Finite Group Databasebeta

← #24907997

Subgroups — #24907997

Order 10624.

314 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 79 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#19128272C211yesmaximal, characteristic
2#19128178C211yesmaximal, characteristic, complemented
3#19127718C211yesmaximal, characteristic
4C2×#14581710C211yesmaximal, characteristic, complemented
5#19127458C211yesmaximal, characteristic, complemented
6#19127457C211yesmaximal, characteristic, complemented
7#421×C83C211yesmaximal, characteristic
8C22×Dic166C2211yescharacteristic
9C2×#13050574C2211yescharacteristic
10#14581790C412yescomplemented
11#14581779C412yescomplemented
12#14581773—22no—
13#14581771—22no—
14C4×Dic166—22no—
15#14581757—22no—
16#14581751C2211yescharacteristic
17#14581751C2211yescharacteristic
18#14581748—22no—
19#14581719—22no—
20#14581710—22no—
21C2×#41×C83C2211yescharacteristic
22C2×M16×C83C2211yescharacteristic
23#117×C83C2211yescharacteristic
24#104×C83—22no—
25C2×Dic166—21no—
26C2×Dic166—21no—
27C2×Dic166—21no—
28C2×Dic166—21no—
29C2×Dic166C2×C412yes—
30C2×Dic166D412yes—
31C22×Dic83D412yes—
32#13050574—21no—
33#13050574—21no—
34C2×C83⋊C8—21no—
35C4⋊Dic83—21no—
36C4⋊Dic83—21no—
37Dic83⋊C4—24no—
38Dic83⋊C4—24no—
39C22⋊Dic83—21no—
40C22⋊Dic83—21no—
41C22⋊Dic83—24no—
42C4×Dic83—24no—
43C4×Dic83C2×C412yes—
44C4×Dic83D412yes—
45#41×C83—21no—
46#41×C83—21no—
47#41×C83—21no—
48#41×C83—21no—
49C2×Q8×C83C2×C411yescharacteristic
50C2×D4×C83—21no—
51C2×D4×C83C2×C411yescharacteristic
52C22×C4×C83—21no—
53C22×C4×C83C2311yescharacteristic
54M16×C83—21no—
55M16×C83—21no—
56C2×C8×C83—21no—
57C4⋊C4×C83—21no—
58C22⋊C4×C83—21no—
59C42×C83D412yes—
60C23×C83—21no—
61C23×C83C2×D411yescharacteristic
62Q8×C83—21no—
63Q8×C83—21no—
64D4×C83—21no—
65D4×C83—21no—
66D4×C83—41no—
67D4×C83—41no—
68C2×C4×C83—21no—
69C2×C4×C83—22no—
70C2×C4×C83—41no—
71C2×C4×C83—41no—
72C2×C4×C83C22×C411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83#4111yescharacteristic
77C2×C4×C83#4111yescharacteristic
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83C22⋊C414yes—
83Dic166—22no—
84Dic166—22no—
85Dic166—22no—
86Dic166—22no—
87C8×C83—41no—
88C83⋊C8—41no—
89C22×C83—41no—
90C22×C83—42no—
91C22×C83—41no—
92C22×C83C2×C22⋊C411yescharacteristic
93C22×C83#12011yescharacteristic
94C22×C83#12311yescharacteristic
95C4×C83—42no—
96C4×C83—41no—
97C4×C83C4×D411yescharacteristic
98C4×C83C4×D411yescharacteristic
99C4×C83#12111yescharacteristic
100C4×C83#12111yescharacteristic
101Dic83—24no—
102Dic83—42no—
103Dic83—42no—
104C2×C83—21no—
105C2×C83—21no—
106C2×C83—21no—
107C2×C83—81no—
108C2×C83#38611yescharacteristic
109#1912—831nomaximal, Sylow
110C83#191211yescharacteristic, complemented, Sylow
111#536—831no—
112#519—831no—
113#421—831no—
114#421D8311yescharacteristic
115C2×#101—831no—
116H(3,ℤ/4)—831no—
117H(3,ℤ/4)—831no—
118C2×#41C2×D8311yescharacteristic
119C22×Q8—831no—
120C2×M16—831no—
121C2×M16C2×D8311yescharacteristic
122C4⋊Q8—832no—
123#124—832no—
124#123—1662no—
125C22⋊Q8—1662no—
126C4×Q8—1662no—
127C4×Q8—1662no—
128#117—831no—
129#117—831no—
130#117—1662no—
131#117C2×D8311yescharacteristic
132#104—22no—
133#104—1662no—
134#101—1662no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—832no—
140C2×Q8—832no—
141C2×Q8—1661no—
142C2×Q8—1661no—
143C2×Q8—1661no—
144C2×Q8—1661no—
145C2×Q8C4×D8311yescharacteristic
146C2×D4—21no—
147C2×D4C4×D8311yescharacteristic
148C22×C4—21no—
149C22×C4—832no—
150C22×C4C22×D8311yescharacteristic
151M16—21no—
152M16—21no—
153M16—1661no—
154M16—1661no—
155C2×C8—21no—
156C2×C8—1661no—
157C4⋊C4—21no—
158C4⋊C4—1664no—
159C4⋊C4—1664no—
160C4⋊C4—1661no—
161C4⋊C4—1661no—
162C22⋊C4—21no—
163C22⋊C4—1664no—
164C22⋊C4—1661no—
165C22⋊C4—1661no—
166C42—832no—
167C42—832no—
168C42—1664no—
169C42C22⋊D8312yes—
170C23—21no—
171C23D4×D8311yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—1662no—
175Q8—1662no—
176Q8—1662no—
177Q8—1662no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—834no—
187C2×C4—1662no—
188C2×C4—1662no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4C2×C4×D8311yescharacteristic
192C2×C4D4×D8311yescharacteristic
193C2×C4D4×D8311yescharacteristic
194C2×C4C2×C22⋊D8311yescharacteristic
195C2×C4#1305058911yescharacteristic
196C2×C4#1305059011yescharacteristic
197C8—41no—
198C8—3321no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22⋊C4×D8311yescharacteristic
203C22#1458176011yescharacteristic
204C22#1458177411yescharacteristic
205C4—42no—
206C4—41no—
207C4—3322no—
208C4—3322no—
209C4—1664no—
210C4(C22⋊D83)⋊C411yescharacteristic
211C4C4×C22⋊D8311yescharacteristic
212C4#1458176311yescharacteristic
213C4#1458176511yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#1912757711yescharacteristic
subgroupG/H
centerC2#19127577
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#421×C83C2
socleC2×C83#386
prime radicalC2×#41×C83C22
cosocleC22×C4×C83C23
hypercenter#421D83
hypocenterC83#1912
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #19128178C2semidirect
4. C2×#14581710C2semidirect
5. #19127458C2semidirect
6. #19127457C2semidirect
10. #14581790C4semidirect
11. #14581779C4semidirect
110. C83#1912semidirect

Maximal subgroups

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

subgroupindex
#191282722
#191281782
#191277182
C2×#145817102
#191274582
#191274572
#421×C832
#191283