Finite Group Databasebeta

← C83×#2123 #24903808

Subgroups — C83×#2123 #24903808

Order 10624.

234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 84 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#536×C83C211yesmaximal, characteristic
2#413×C83C211yesmaximal, characteristic
3#410×C83C211yesmaximal, characteristic
4#352×C83C214yesmaximal, complemented
5C2×#41×C83C2211yescharacteristic
6C22×Q8×C83C411yescharacteristic, complemented
7C2×M16×C83—21no—
8C4⋊Q8×C83—21no—
9#124×C83—21no—
10#123×C83C412yescomplemented
11#123×C83C2212yes—
12C22⋊Q8×C83—21no—
13C22⋊Q8×C83—21no—
14#117×C83—21no—
15#117×C83C411yescharacteristic, complemented
16#101×C83—21no—
17#100×C83C2212yes—
18#99×C83—21no—
19#99×C83C2212yes—
20#41×C83—41no—
21#41×C83—41no—
22C2×Q8×C83—21no—
23C2×Q8×C83—21no—
24C2×Q8×C83—41no—
25C2×Q8×C83—41no—
26C2×Q8×C83D411yescharacteristic
27C2×D4×C83C2×C411yescharacteristic
28C2×D4×C83D411yescharacteristic
29C2×D4×C83C2311yescharacteristic
30C22×C4×C83C2×C411yescharacteristic
31C22×C4×C83C2×C412yes—
32C22×C4×C83D411yescharacteristic
33C22×C4×C83D411yescharacteristic
34M16×C83—22no—
35M16×C83—41no—
36C2×C8×C83—41no—
37C4⋊C4×C83—22no—
38C4⋊C4×C83—21no—
39C4⋊C4×C83—22no—
40C4⋊C4×C83—41no—
41C22⋊C4×C83—22no—
42C22⋊C4×C83—22no—
43C22⋊C4×C83—22no—
44C22⋊C4×C83C2×C412yes—
45C42×C83—21no—
46C42×C83—41no—
47C23×C83C22⋊C411yescharacteristic
48C23×C83C22×C411yescharacteristic
49C23×C83C2×D411yescharacteristic
50Q8×C83—41no—
51Q8×C83—42no—
52Q8×C83—41no—
53Q8×C83—41no—
54D4×C83—41no—
55D4×C83—41no—
56D4×C83—41no—
57C2×C4×C83—21no—
58C2×C4×C83—21no—
59C2×C4×C83—21no—
60C2×C4×C83—21no—
61C2×C4×C83—21no—
62C2×C4×C83—21no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—42no—
66C2×C4×C83—42no—
67C2×C4×C83C22⋊C411yescharacteristic
68C2×C4×C83C22⋊C411yescharacteristic
69C2×C4×C83C22⋊C411yescharacteristic
70C2×C4×C83C2×D411yescharacteristic
71C8×C83—42no—
72C22×C83—21no—
73C22×C83—21no—
74C22×C83—21no—
75C22×C83—41no—
76C22×C83—41no—
77C22×C83—41no—
78C22×C83C2×C22⋊C411yescharacteristic
79C4×C83—21no—
80C4×C83—21no—
81C4×C83—21no—
82C4×C83—42no—
83C4×C83—42no—
84C4×C83—82no—
85C4×C83#9911yescharacteristic
86C4×C83#9911yescharacteristic
87C2×C83—21no—
88C2×C83—41no—
89C2×C83—41no—
90C2×C83—41no—
91C2×C83C2×#9911yescharacteristic
92#2123C8311yesmaximal, characteristic, direct product factor, Sylow
93C83#212311yescharacteristic, direct product factor, Sylow
94#536C2×C8311yescharacteristic
95#413C2×C8311yescharacteristic
96#410C2×C8311yescharacteristic
97#352C2×C8314yescomplemented
98C2×#41C22×C8311yescharacteristic
99C22×Q8C4×C8311yescharacteristic, complemented
100C2×M16—21no—
101C4⋊Q8—21no—
102#124—21no—
103#123C4×C8312yescomplemented
104#123C22×C8312yes—
105C22⋊Q8—21no—
106C22⋊Q8—21no—
107#117—21no—
108#117C4×C8311yescharacteristic, complemented
109#101—21no—
110#100C22×C8312yes—
111#99—21no—
112#99C22×C8312yes—
113#41—41no—
114#41—41no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8D4×C8311yescharacteristic
120C2×D4C2×C4×C8311yescharacteristic
121C2×D4D4×C8311yescharacteristic
122C2×D4C23×C8311yescharacteristic
123C22×C4C2×C4×C8311yescharacteristic
124C22×C4C2×C4×C8312yes—
125C22×C4D4×C8311yescharacteristic
126C22×C4D4×C8311yescharacteristic
127M16—22no—
128M16—41no—
129C2×C8—41no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—41no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—22no—
137C22⋊C4C2×C4×C8312yes—
138C42—21no—
139C42—41no—
140C23C22⋊C4×C8311yescharacteristic
141C23C22×C4×C8311yescharacteristic
142C23C2×D4×C8311yescharacteristic
143Q8—41no—
144Q8—42no—
145Q8—41no—
146Q8—41no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4C22⋊C4×C8311yescharacteristic
161C2×C4C22⋊C4×C8311yescharacteristic
162C2×C4C22⋊C4×C8311yescharacteristic
163C2×C4C2×D4×C8311yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×C22⋊C4×C8311yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—42no—
176C4—42no—
177C4—82no—
178C4#99×C8311yescharacteristic
179C4#99×C8311yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#99×C8311yescharacteristic
subgroupG/H
center, socleC2×C83C2×#99
commutatorC23C22×C4×C83
Frattini, cosocleC2×D4C23×C83
prime radicalC2×#41×C83C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C83 ◁ C2×C4×C83 ◁ C2×#41×C83 ◁ G

Complements

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

NH ≅ G/Ntype
4. #352×C83C2semidirect
6. C22×Q8×C83C4semidirect
10. #123×C83C4semidirect
15. #117×C83C4semidirect
92. #2123C83direct (N × H)
93. C83#2123direct (N × H)
97. #352C2×C83semidirect
99. C22×Q8C4×C83semidirect
103. #123C4×C83semidirect
108. #117C4×C83semidirect

Maximal subgroups

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

subgroupindex
#536×C832
#413×C832
#410×C832
#352×C832
#212383