Finite Group Databasebeta

← #24918412

Subgroups — #24918412

Order 10624.

384 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128256C211yesmaximal, characteristic
2C4⋊Q8×D83C211yesmaximal, characteristic
3#19128081C211yesmaximal, characteristic, complemented
4#19128080C211yesmaximal, characteristic, complemented
5#19128053C211yesmaximal, characteristic, complemented
6#19127910C214yesmaximal, complemented
7#19127858C214yesmaximal
8#19127752C211yesmaximal, characteristic
9#487×C83C211yesmaximal, characteristic, complemented
10C2×Q8×D83C2212yes—
11C2×Dic332C2214yes—
12C2×#13050580C2214yes—
13#14581792C2211yescharacteristic
14#14581792C2211yescharacteristic
15#14581791—21no—
16#14581791—21no—
17#14581790C2212yes—
18#14581783—21no—
19#14581772C2212yes—
20#14581768—21no—
21Q8×Dic83—21no—
22#14581757C2212yes—
23#14581750—21no—
24C4⋊C4×D83—21no—
25C4⋊C4×D83—21no—
26C42×D83C2211yescharacteristic
27Dic83⋊C8C2212yes—
28#14581715C2214yes—
29(C2×D83)⋊C8C2212yes—
30C4×C83⋊C8C2211yescharacteristic
31C4⋊Q8×C83C2211yescharacteristic
32C4⋊Q8×C83C2211yescharacteristic
33#103×C83C2214yes—
34C4×C8×C83C2211yescharacteristic
35C2×Dic166—22no—
36C2×Dic166C2312yes—
37C2×Dic166C2312yes—
38Q8×D83—44no—
39C2×C4×D83—22no—
40C2×C4×D83D411yescharacteristic
41C2×C4×D83D411yescharacteristic
42C2×C4×D83C2311yescharacteristic
43Dic332—44no—
44#13050580—44no—
45C2×C83⋊C8C2312yes—
46C4⋊Dic83—21no—
47C4⋊Dic83—21no—
48C4⋊Dic83—21no—
49C4⋊Dic83—21no—
50C4⋊Dic83—22no—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54(C2×D83)⋊C4—22no—
55C4×Dic83—22no—
56C4×Dic83D411yescharacteristic
57C4×Dic83D411yescharacteristic
58C4×Dic83C2311yescharacteristic
59C2×Q8×C83—21no—
60C2×Q8×C83C2312yes—
61C2×C8×C83C2312yes—
62C4⋊C4×C83—21no—
63C4⋊C4×C83—21no—
64C4⋊C4×C83—21no—
65C4⋊C4×C83C2312yes—
66C42×C83C2311yescharacteristic
67C22×D83C2×D411yescharacteristic
68Q8×C83—24no—
69Q8×C83—42no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2411yescharacteristic
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83C2×D411yescharacteristic
78C2×Dic83C2×D411yescharacteristic
79C2×Dic83C2×D411yescharacteristic
80C4×D83—22no—
81C4×D83—22no—
82C4×D83—44no—
83C4×D83Q1614yescomplemented
84Dic166—24no—
85Dic166—24no—
86Dic166—44no—
87C8×C83—22no—
88C83⋊C8—22no—
89C22×C83C22×D411yescharacteristic
90C2×D83—21no—
91C2×D83—21no—
92C2×D83C2×Q1612yes—
93C4×C83—42no—
94C4×C83—42no—
95C4×C83C2×Q1612yes—
96C4×C83#13712yes—
97C4×C83H(5,𝔽2)12yes—
98Dic83—21no—
99Dic83—21no—
100Dic83—42no—
101Dic83—42no—
102Dic83C2×Q1612yes—
103C2×C83#53511yescharacteristic
104C2×C83C22×Q1611yescharacteristic
105C2×C83C2×#13711yescharacteristic
106D83—22no—
107#3209—831nomaximal, Sylow
108C83#320911yescharacteristic, complemented, Sylow
109#533—831no—
110C2×C4⋊Q8—831no—
111#494—831no—
112#494—831no—
113#487—831no—
114#487D8311yescharacteristic, complemented
115#462—834no—
116#451—834no—
117#431—831no—
118C22×Q8—832no—
119C2×Q16—834no—
120C2×Q16—834no—
121C4⋊Q8—832no—
122C4⋊Q8—831no—
123C4⋊Q8—831no—
124C4⋊Q8—1661no—
125C4⋊Q8—1661no—
126C4⋊Q8C2×D8311yescharacteristic
127C4⋊Q8C2×D8311yescharacteristic
128#125—1661no—
129C22⋊Q8—832no—
130C22⋊Q8—1661no—
131C4×Q8—832no—
132C4×Q8—1661no—
133#117—1661no—
134C2×C4⋊C4—1661no—
135C2×C4⋊C4—1661no—
136C2×C42—831no—
137C4⋊C8—832no—
138#103—834no—
139#103C2×D8314yes—
140C22⋊C8—832no—
141C4×C8—831no—
142C4×C8C2×D8311yescharacteristic
143C2×Q8—21no—
144C2×Q8—3324no—
145C2×Q8—832no—
146C2×Q8—832no—
147C2×Q8—1662no—
148C2×Q8C22×D8312yes—
149C22×C4—831no—
150C22×C4—831no—
151C22×C4—831no—
152C22×C4—1662no—
153Q16—3324no—
154Q16—3324no—
155C2×C8—832no—
156C2×C8C22×D8312yes—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—1662no—
161C4⋊C4—1662no—
162C4⋊C4—1662no—
163C4⋊C4—1662no—
164C4⋊C4—1661no—
165C4⋊C4—1661no—
166C4⋊C4—1661no—
167C4⋊C4—1661no—
168C4⋊C4C22×D8312yes—
169C22⋊C4—1662no—
170C42—831no—
171C42—831no—
172C42—831no—
173C42—1662no—
174C42C22×D8311yescharacteristic
175C23—831no—
176Q8—24no—
177Q8—42no—
178Q8—3324no—
179Q8—1664no—
180Q8—1664no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—3324no—
184C2×C4—831no—
185C2×C4—834no—
186C2×C4—831no—
187C2×C4—831no—
188C2×C4—1662no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4—1662no—
192C2×C4D4×D8311yescharacteristic
193C2×C4D4×D8311yescharacteristic
194C2×C4C23×D8311yescharacteristic
195C8—22no—
196C8—1662no—
197C22—832no—
198C22—1661no—
199C22—1661no—
200C22C2×D4×D8311yescharacteristic
201C4—42no—
202C4—42no—
203C4—3322no—
204C4—3322no—
205C4—832no—
206C4—1661no—
207C4—1661no—
208C4Q16×D8312yes—
209C4#1458182112yes—
210C4#1458183812yes—
211C2—1662no—
212C2#1912826911yescharacteristic
213C2C2×Q16×D8311yescharacteristic
214C2C2×#1458182111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#487×C83C2
socleC22×C83C22×D4
prime radicalC22×D83C2×D4
hypercenter#487D83
hypocenterC83#3209
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #19128081C2semidirect
4. #19128080C2semidirect
5. #19128053C2semidirect
6. #19127910C2semidirect
9. #487×C83C2semidirect
83. C4×D83Q16semidirect
108. C83#3209semidirect
114. #487D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
C4⋊Q8×D832
#191280812
#191280802
#191280532
#191279102
#191278582
#191277522
#487×C832
#320983