Finite Group Databasebeta

← C83×#10574 #35238398

Subgroups — C83×#10574 #35238398

Order 21248.

494 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C83×#2551C211yesmaximal, characteristic
2C83×#1948C212yesmaximal
3C83×#1844C211yesmaximal, characteristic
4C83×#1296C211yesmaximal, characteristic
5C83×#1279C212yesmaximal
6C2×C4⋊Q8×C83—22no—
7C2×#106×C83—22no—
8C2×#106×C83—22no—
9C2×C4⋊C8×C83C2212yes—
10C2×C4×C8×C83C2211yescharacteristic
11#395×C83—22no—
12#389×C83—22no—
13#384×C83C2211yescharacteristic
14#384×C83C2211yescharacteristic
15C4×C4⋊C4×C83C2212yes—
16#340×C83—22no—
17C22×Q8×C83—22no—
18C22×C8×C83—21no—
19C22×C8×C83—22no—
20C4⋊Q8×C83—44no—
21C4⋊Q8×C83—44no—
22C2×C4⋊C4×C83—24no—
23C2×C4⋊C4×C83—22no—
24C2×C4⋊C4×C83—22no—
25C2×C4⋊C4×C83—21no—
26C2×C4⋊C4×C83D412yes—
27C2×C4⋊C4×C83D412yes—
28C2×C4⋊C4×C83D412yes—
29C2×C42×C83—22no—
30C2×C42×C83C2311yescharacteristic
31#106×C83—44no—
32#106×C83—44no—
33C4⋊C8×C83—22no—
34C4⋊C8×C83Q814yescomplemented
35C4×C8×C83—21no—
36C4×C8×C83Q812yescomplemented
37#95×C83—22no—
38#95×C83—22no—
39C2×Q8×C83—48no—
40C2×Q8×C83—82no—
41C22×C4×C83—22no—
42C22×C4×C83—22no—
43C22×C4×C83—22no—
44C22×C4×C83—41no—
45C22×C4×C83C2×D411yescharacteristic
46C22×C4×C83C2×D412yes—
47C2×C8×C83—22no—
48C2×C8×C83—22no—
49C2×C8×C83—22no—
50C2×C8×C83—24no—
51C2×C8×C83—42no—
52C2×C8×C83—42no—
53C4⋊C4×C83—22no—
54C4⋊C4×C83—22no—
55C4⋊C4×C83—42no—
56C4⋊C4×C83—48no—
57C4⋊C4×C83—42no—
58C4⋊C4×C83—42no—
59C4⋊C4×C83—42no—
60C4⋊C4×C83—42no—
61C4⋊C4×C83—42no—
62C4⋊C4×C83SD1614yes—
63C4⋊C4×C83SD1614yes—
64C42×C83—42no—
65C42×C83—42no—
66C42×C83C2×Q811yescharacteristic
67C42×C83C2×Q812yes—
68C42×C83#4111yescharacteristic
69C23×C83#12011yescharacteristic
70Q8×C83—88no—
71C2×C4×C83—21no—
72C2×C4×C83—21no—
73C2×C4×C83—24no—
74C2×C4×C83—24no—
75C2×C4×C83—42no—
76C2×C4×C83—42no—
77C2×C4×C83—42no—
78C2×C4×C83—42no—
79C2×C4×C83—44no—
80C2×C4×C83—42no—
81C2×C4×C83—82no—
82C2×C4×C83—81no—
83C2×C4×C83C22⋊Q811yescharacteristic
84C2×C4×C83C22⋊Q812yes—
85C2×C4×C83C4⋊Q811yescharacteristic
86C2×C4×C83C4⋊Q812yes—
87C2×C4×C83C2×SD1612yes—
88C2×C4×C83C2×SD1612yes—
89C2×C4×C83AGL(1,ℤ/8)12yes—
90C2×C4×C83#13712yes—
91C8×C83—24no—
92C8×C83—44no—
93C22×C83#39511yescharacteristic
94C22×C83#44812yes—
95C22×C83#45012yes—
96C22×C83#45311yescharacteristic
97C22×C83#45611yescharacteristic
98C4×C83—21no—
99C4×C83—21no—
100C4×C83—44no—
101C4×C83—44no—
102C4×C83—82no—
103C4×C83—161no—
104C4×C83#47512yes—
105C4×C83#47612yes—
106C4×C83#49812yes—
107C4×C83#50112yes—
108#10574C8311yesmaximal, characteristic, direct product factor, Sylow
109C2×C83#162311yescharacteristic
110C2×C83#162911yescharacteristic
111C2×C83#165112yes—
112C2×C83#205812yes—
113C2×C83#206011yescharacteristic
114#2551C2×C8311yescharacteristic
115#1948C2×C8312yes—
116#1844C2×C8311yescharacteristic
117#1296C2×C8311yescharacteristic
118#1279C2×C8312yes—
119C83#1057411yescharacteristic, direct product factor, Sylow
120C2×C4⋊Q8—22no—
121C2×#106—22no—
122C2×#106—22no—
123C2×C4⋊C8C22×C8312yes—
124C2×C4×C8C22×C8311yescharacteristic
125#395—22no—
126#389—22no—
127#384C22×C8311yescharacteristic
128#384C22×C8311yescharacteristic
129C4×C4⋊C4C22×C8312yes—
130#340—22no—
131C22×Q8—22no—
132C22×C8—21no—
133C22×C8—22no—
134C4⋊Q8—44no—
135C4⋊Q8—44no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4D4×C8312yes—
141C2×C4⋊C4D4×C8312yes—
142C2×C4⋊C4D4×C8312yes—
143C2×C42—22no—
144C2×C42C23×C8311yescharacteristic
145#106—44no—
146#106—44no—
147C4⋊C8—22no—
148C4⋊C8Q8×C8314yescomplemented
149C4×C8—21no—
150C4×C8Q8×C8312yescomplemented
151#95—22no—
152#95—22no—
153C2×Q8—48no—
154C2×Q8—82no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—41no—
159C22×C4C2×D4×C8311yescharacteristic
160C22×C4C2×D4×C8312yes—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—24no—
165C2×C8—42no—
166C2×C8—42no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—42no—
170C4⋊C4—48no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4SD16×C8314yes—
177C4⋊C4SD16×C8314yes—
178C42—42no—
179C42—42no—
180C42C2×Q8×C8311yescharacteristic
181C42C2×Q8×C8312yes—
182C42#41×C8311yescharacteristic
183C23#120×C8311yescharacteristic
184Q8—88no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4C22⋊Q8×C8311yescharacteristic
198C2×C4C22⋊Q8×C8312yes—
199C2×C4C4⋊Q8×C8311yescharacteristic
200C2×C4C4⋊Q8×C8312yes—
201C2×C4C2×SD16×C8312yes—
202C2×C4C2×SD16×C8312yes—
203C2×C4AGL(1,ℤ/8)×C8312yes—
204C2×C4#137×C8312yes—
205C8—24no—
206C8—44no—
207C22#395×C8311yescharacteristic
208C22#448×C8312yes—
209C22#450×C8312yes—
210C22#453×C8311yescharacteristic
211C22#456×C8311yescharacteristic
212C4—21no—
213C4—21no—
214C4—44no—
215C4—44no—
216C4—82no—
217C4—161no—
218C4#475×C8312yes—
219C4#476×C8312yes—
220C4#498×C8312yes—
221C4#501×C8312yes—
222C2C83×#162311yescharacteristic
223C2C83×#162911yescharacteristic
224C2C83×#165112yes—
225C2C83×#205812yes—
226C2C83×#206011yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C83#120
commutator, Frattini, cosocleC2×C42C23×C83
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
34. C4⋊C8×C83Q8semidirect
36. C4×C8×C83Q8semidirect
108. #10574C83direct (N × H)
119. C83#10574direct (N × H)
148. C4⋊C8Q8×C83semidirect
150. C4×C8Q8×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#25512
C83×#19482
C83×#18442
C83×#12962
C83×#12792
#1057483