Finite Group Databasebeta

← C83×#8327 #35236151

Subgroups — C83×#8327 #35236151

Order 21248.

684 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 158 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×#541×C83C211yesmaximal, characteristic
2C83×#1479C214yesmaximal, complemented
3C83×#1458C212yesmaximal
4#541×C83—28no—
5C2×C4⋊Q8×C83C411yescharacteristic
6C2×#126×C83—22no—
7C2×#125×C83—22no—
8C2×#125×C83C411yescharacteristic
9C2×C22⋊Q8×C83C412yes—
10C2×C4×Q8×C83C2212yes—
11C2×#117×C83C2211yescharacteristic
12#431×C83C2214yes—
13Q8⋊C8×C83—28no—
14#324×C83—22no—
15#323×C83—22no—
16C22×Q8×C83C2×C412yes—
17C23×C4×C83C2×C411yescharacteristic
18C4⋊Q8×C83—24no—
19#126×C83—216no—
20#125×C83—216no—
21#125×C83—24no—
22C22⋊Q8×C83—24no—
23C22⋊Q8×C83—24no—
24C4×Q8×C83—24no—
25C4×Q8×C83D418yes—
26#117×C83—24no—
27C2×C4⋊C4×C83—22no—
28C2×C4⋊C4×C83—22no—
29C2×C4⋊C4×C83—22no—
30C2×C4⋊C4×C83—22no—
31C2×C4⋊C4×C83C2×C412yes—
32C2×C4⋊C4×C83D412yes—
33C2×C22⋊C4×C83—22no—
34C2×C22⋊C4×C83D412yes—
35C2×C42×C83—22no—
36C2×C42×C83C2×C411yescharacteristic
37C2×C42×C83C2311yescharacteristic
38C4⋊C8×C83—24no—
39C22⋊C8×C83—24no—
40C4×C8×C83—24no—
41C24×C83C22⋊C411yescharacteristic
42C2×Q8×C83—22no—
43C2×Q8×C83—28no—
44C2×Q8×C83C22⋊C414yes—
45C22×C4×C83—22no—
46C22×C4×C83—22no—
47C22×C4×C83—22no—
48C22×C4×C83—24no—
49C22×C4×C83C22⋊C411yescharacteristic
50C22×C4×C83C22⋊C412yes—
51C22×C4×C83C22×C411yescharacteristic
52C22×C4×C83C2×D412yes—
53C2×C8×C83—44no—
54C4⋊C4×C83—28no—
55C4⋊C4×C83—28no—
56C4⋊C4×C83—28no—
57C4⋊C4×C83—22no—
58C4⋊C4×C83—28no—
59C4⋊C4×C83—24no—
60C4⋊C4×C83C22⋊C414yes—
61C22⋊C4×C83—28no—
62C22⋊C4×C83—24no—
63C42×C83—22no—
64C42×C83—28no—
65C42×C83C2×D414yes—
66C23×C83—21no—
67C23×C83—22no—
68C23×C83—42no—
69C23×C83C2×C22⋊C411yescharacteristic
70Q8×C83—28no—
71Q8×C83—44no—
72C2×C4×C83—22no—
73C2×C4×C83—24no—
74C2×C4×C83—24no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—42no—
78C2×C4×C83—44no—
79C2×C4×C83—44no—
80C2×C4×C83—44no—
81C2×C4×C83—42no—
82C2×C4×C83C2×C22⋊C412yes—
83C2×C4×C83#12014yes—
84C2×C4×C83#13114yes—
85C8×C83—84no—
86C22×C83—22no—
87C22×C83—42no—
88C22×C83—41no—
89C22×C83—82no—
90C22×C83#37911yescharacteristic
91C22×C83#40812yes—
92C4×C83—24no—
93C4×C83—42no—
94C4×C83—44no—
95C4×C83—41no—
96C4×C83—44no—
97C4×C83#40814yes—
98#8327C8311yesmaximal, characteristic, direct product factor, Sylow
99C2×C83—22no—
100C2×C83—81no—
101C2×C83#178011yescharacteristic
102C2×C83#179612yes—
103C2×#541C2×C8311yescharacteristic
104#1479C2×C8314yescomplemented
105#1458C2×C8312yes—
106C83#832711yescharacteristic, direct product factor, Sylow
107#541—28no—
108C2×C4⋊Q8C4×C8311yescharacteristic
109C2×#126—22no—
110C2×#125—22no—
111C2×#125C4×C8311yescharacteristic
112C2×C22⋊Q8C4×C8312yes—
113C2×C4×Q8C22×C8312yes—
114C2×#117C22×C8311yescharacteristic
115#431C22×C8314yes—
116Q8⋊C8—28no—
117#324—22no—
118#323—22no—
119C22×Q8C2×C4×C8312yes—
120C23×C4C2×C4×C8311yescharacteristic
121C4⋊Q8—24no—
122#126—216no—
123#125—216no—
124#125—24no—
125C22⋊Q8—24no—
126C22⋊Q8—24no—
127C4×Q8—24no—
128C4×Q8D4×C8318yes—
129#117—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4C2×C4×C8312yes—
135C2×C4⋊C4D4×C8312yes—
136C2×C22⋊C4—22no—
137C2×C22⋊C4D4×C8312yes—
138C2×C42—22no—
139C2×C42C2×C4×C8311yescharacteristic
140C2×C42C23×C8311yescharacteristic
141C4⋊C8—24no—
142C22⋊C8—24no—
143C4×C8—24no—
144C24C22⋊C4×C8311yescharacteristic
145C2×Q8—22no—
146C2×Q8—28no—
147C2×Q8C22⋊C4×C8314yes—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4C22⋊C4×C8311yescharacteristic
153C22×C4C22⋊C4×C8312yes—
154C22×C4C22×C4×C8311yescharacteristic
155C22×C4C2×D4×C8312yes—
156C2×C8—44no—
157C4⋊C4—28no—
158C4⋊C4—28no—
159C4⋊C4—28no—
160C4⋊C4—22no—
161C4⋊C4—28no—
162C4⋊C4—24no—
163C4⋊C4C22⋊C4×C8314yes—
164C22⋊C4—28no—
165C22⋊C4—24no—
166C42—22no—
167C42—28no—
168C42C2×D4×C8314yes—
169C23—21no—
170C23—22no—
171C23—42no—
172C23C2×C22⋊C4×C8311yescharacteristic
173Q8—28no—
174Q8—44no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4C2×C22⋊C4×C8312yes—
186C2×C4#120×C8314yes—
187C2×C4#131×C8314yes—
188C8—84no—
189C22—22no—
190C22—41no—
191C22—42no—
192C22—82no—
193C22#379×C8311yescharacteristic
194C22#408×C8312yes—
195C4—24no—
196C4—41no—
197C4—44no—
198C4—44no—
199C4—42no—
200C4#408×C8314yes—
201C2—22no—
202C2—81no—
203C2C83×#178011yescharacteristic
204C2C83×#179612yes—
subgroupG/H
center, socleC22×C83#379
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC24×C83C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#1479C2semidirect
98. #8327C83direct (N × H)
104. #1479C2×C83semidirect
106. C83#8327direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#541×C832
C83×#14792
C83×#14582
#832783