Finite Group Databasebeta

← #24920039

Subgroups — #24920039

Order 10624.

336 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 113 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#19128392C212yesmaximal
2#19128103C211yesmaximal, characteristic, complemented
3#19128041C212yesmaximal, complemented
4#19128035C212yesmaximal, complemented
5#19128008C212yesmaximal, complemented
6#19127956C212yesmaximal, complemented
7#19127780C211yesmaximal, characteristic
8#19127742C212yesmaximal, complemented
9C4×Q16×C83C211yesmaximal, characteristic, complemented
10C2×#13050573C2212yes—
11#14581786—22no—
12#14581784C2212yes—
13#14581782—22no—
14#14581777—22no—
15#14581768C2212yes—
16#14581768C2212yes—
17C4×D332C2211yescharacteristic
18#14581750C2212yes—
19#14581749—22no—
20C4⋊C4×D83C2212yes—
21#14581726C2211yescharacteristic
22#14581726C2211yescharacteristic
23#14581725C2212yes—
24#14581724C2212yes—
25#14581723C2212yes—
26C4⋊(C83⋊C8)C2211yescharacteristic
27#14581716C2212yes—
28#14581716C2212yes—
29(C2×D83)⋊C8C2212yes—
30C2×Q16×C83C2211yescharacteristic
31C4×Q8×C83C2212yes—
32#107×C83C2211yescharacteristic
33#103×C83C2212yes—
34C4×C8×C83C2211yescharacteristic
35C2×D332D411yescharacteristic
36C2×C4×D83—22no—
37C2×C4×D83C2312yes—
38#13050573—24no—
39C2×C83⋊C8C2312yes—
40C4⋊Dic83D411yescharacteristic
41C4⋊Dic83C2312yes—
42C4⋊Dic83C2312yes—
43Dic83⋊C4—22no—
44Dic83⋊C4—22no—
45Dic83⋊C4—22no—
46Dic83⋊C4—22no—
47Dic83⋊C4—22no—
48(C2×D83)⋊C4—22no—
49(C2×D83)⋊C4—22no—
50(C2×D83)⋊C4—22no—
51(C2×D83)⋊C4D412yes—
52C4×Dic83—22no—
53C2×Q8×C83C2312yes—
54Q16×C83—22no—
55C2×C8×C83C2311yescharacteristic
56C2×C8×C83C2311yescharacteristic
57C4⋊C4×C83—22no—
58C4⋊C4×C83C2312yes—
59C42×C83—22no—
60C42×C83C2311yescharacteristic
61C22×D83C2×D412yes—
62Q8×C83—22no—
63Q8×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83C2×D411yescharacteristic
67C2×C4×C83C2×D411yescharacteristic
68C2×C4×C83C2411yescharacteristic
69C2×Dic83—22no—
70C2×Dic83—22no—
71C2×Dic83C2×D412yes—
72D332—41no—
73C4×D83—22no—
74C4×D83—22no—
75C4×D83—44no—
76C8×C83—21no—
77C8×C83#4112yes—
78C83⋊C8—22no—
79C22×C83C22×D411yescharacteristic
80C2×D83—22no—
81C2×D83—22no—
82C2×D83—22no—
83C4×C83—21no—
84C4×C83—21no—
85C4×C83—42no—
86C4×C83—42no—
87C4×C83C2×#4111yescharacteristic
88C4×C832−1+411yescharacteristic
89Dic83—22no—
90Dic83—42no—
91Dic83—42no—
92C2×C83#54711yescharacteristic
93C2×C83#57511yescharacteristic
94C2×C83#57811yescharacteristic
95D83—42no—
96#3352—831nomaximal, Sylow
97C83#335211yescharacteristic, complemented, Sylow
98#552—832no—
99#499—831no—
100#485—832no—
101#484—832no—
102#479—832no—
103#470—832no—
104C4×Q16D8311yescharacteristic, complemented
105#436—831no—
106#428—832no—
107C2×Q16C2×D8311yescharacteristic
108C2×M16—832no—
109#126—1662no—
110#125—832no—
111#125—1662no—
112#123—1662no—
113C22⋊Q8—832no—
114C22⋊Q8—832no—
115C4×Q8C2×D8312yes—
116C4×D4—831no—
117#117—832no—
118#117—1662no—
119C2×C4⋊C4—832no—
120#107—831no—
121#107—831no—
122#107—832no—
123#107C2×D8311yescharacteristic
124#106—832no—
125#106—832no—
126C4⋊C8—831no—
127#103—832no—
128#103—832no—
129#103C2×D8312yes—
130C22⋊C8—832no—
131C4×C8C2×D8311yescharacteristic
132C2×Q8C22×D8312yes—
133C2×D4—831no—
134C22×C4—832no—
135C22×C4—1662no—
136Q16—22no—
137M16—1664no—
138C2×C8—832no—
139C2×C8C22×D8311yescharacteristic
140C2×C8C22×D8311yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—831no—
143C4⋊C4—832no—
144C4⋊C4—832no—
145C4⋊C4—1662no—
146C4⋊C4—1662no—
147C4⋊C4—1662no—
148C4⋊C4—1662no—
149C4⋊C4—1662no—
150C4⋊C4C22×D8312yes—
151C22⋊C4—832no—
152C22⋊C4—1662no—
153C22⋊C4—1662no—
154C22⋊C4—1662no—
155C42—22no—
156C42—1662no—
157C42C22×D8311yescharacteristic
158C23—832no—
159Q8—22no—
160Q8—22no—
161D4—3321no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—3324no—
165C2×C4—832no—
166C2×C4—1662no—
167C2×C4—1662no—
168C2×C4—1662no—
169C2×C4—1662no—
170C2×C4D4×D8311yescharacteristic
171C2×C4D4×D8311yescharacteristic
172C2×C4C23×D8311yescharacteristic
173C8—21no—
174C8—1662no—
175C8#1305059012yes—
176C22—1662no—
177C22—1662no—
178C22—1662no—
179C22C2×D4×D8311yescharacteristic
180C4—21no—
181C4—21no—
182C4—42no—
183C4—42no—
184C4—3322no—
185C4—3322no—
186C4—1662no—
187C4C2×#1305059011yescharacteristic
188C4#1458183911yescharacteristic
189C2—3322no—
190C2#1912835611yescharacteristic
191C2#1912849811yescharacteristic
192C2#1912851411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×Q16×C83C2
socleC22×C83C22×D4
prime radicalC2×D332D4
hypercenterC4×Q16D83
hypocenterC83#3352
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. #19128103C2semidirect
3. #19128041C2semidirect
4. #19128035C2semidirect
5. #19128008C2semidirect
6. #19127956C2semidirect
8. #19127742C2semidirect
9. C4×Q16×C83C2semidirect
97. C83#3352semidirect
104. C4×Q16D83semidirect

Maximal subgroups

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

subgroupindex
#191283922
#191281032
#191280412
#191280352
#191280082
#191279562
#191277802
#191277422
C4×Q16×C832
#335283