Finite Group Databasebeta

← #35318670

Subgroups — #35318670

Order 21248.

232 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 65 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#24909718C211yesmaximal, characteristic
2#24909365C211yesmaximal, characteristic, complemented
3#24906834C211yesmaximal, characteristic, complemented
4#24906820C211yesmaximal, characteristic
5#24905536C211yesmaximal, characteristic, complemented
6#24905513C211yesmaximal, characteristic, complemented
7C83×#1361C211yesmaximal, characteristic
8C2×#14581736—21no—
9C2×#14581736—21no—
10#19128081C2211yescharacteristic
11#19127987—21no—
12#19127898—21no—
13C8×Dic166C2211yescharacteristic
14Dic83⋊C16—21no—
15C4×C83⋊C16C2211yescharacteristic
16C16×Dic83—21no—
17#19127454C2211yescharacteristic
18#19127446—21no—
19#19127445—21no—
20#493×C83C2211yescharacteristic
21C4⋊C16×C83C2211yescharacteristic
22#335×C83C2211yescharacteristic
23C2×Dic332—21no—
24C2×Dic332—21no—
25C2×#13050578—41no—
26#14581792—21no—
27C4×Dic166D411yescharacteristic
28#14581736—42no—
29#14581736—42no—
30C2×C83⋊C16D411yescharacteristic
31C2×C83⋊C16D411yescharacteristic
32#14581724—41no—
33C4⋊(C83⋊C8)—21no—
34C4⋊(C83⋊C8)D411yescharacteristic
35Dic83⋊C8—21no—
36#14581717—41no—
37#14581713—41no—
38C8×Dic83—21no—
39C2×D8×C83—21no—
40C2×D8×C83—21no—
41#127×C83—21no—
42C2×C16×C83—21no—
43C4⋊C8×C83—21no—
44C4×C8×C83C2311yescharacteristic
45C2×Dic166—41no—
46C2×Dic166D811yescharacteristic
47Dic332—41no—
48Dic332—41no—
49Dic332—81no—
50#13050578—82no—
51C2×C83⋊C8—21no—
52C2×C83⋊C8—41no—
53C4⋊Dic83—41no—
54C4⋊Dic83—41no—
55C4⋊Dic83D811yescharacteristic
56Dic83⋊C4—21no—
57C4×Dic83—21no—
58C83⋊C16—21no—
59C83⋊C16SD1612yescomplemented
60C2×D4×C83—41no—
61C2×D4×C83—41no—
62C2×D4×C83—41no—
63D8×C83—41no—
64D8×C83—41no—
65D8×C83—81no—
66C2×C8×C83—41no—
67C2×C8×C83C2×D411yescharacteristic
68C2×C8×C83#4111yescharacteristic
69C42×C83C2×D411yescharacteristic
70C16×C83—41no—
71C23×C83—81no—
72D4×C83—81no—
73D4×C83—81no—
74D4×C83—81no—
75D4×C83—81no—
76D4×C83—161no—
77C2×C4×C83#12111yescharacteristic
78C2×C4×C83C2×D811yescharacteristic
79C2×C4×C83#13511yescharacteristic
80C2×Dic83—21no—
81C2×Dic83—21no—
82C2×Dic83—81no—
83Dic166—21no—
84Dic166—81no—
85Dic166—81no—
86Dic166SD3212yescomplemented
87C8×C83—21no—
88C8×C83—81no—
89C8×C83C2×SD1611yescharacteristic
90C8×C83AGL(1,ℤ/8)11yescharacteristic
91C83⋊C8—41no—
92C83⋊C8—81no—
93C22×C83—161no—
94C22×C83—161no—
95C22×C83—161no—
96C22×C83#46611yescharacteristic
97C4×C83—21no—
98C4×C83#46011yescharacteristic
99C4×C83#47211yescharacteristic
100C4×C83C2×SD3211yescharacteristic
101C4×C83#50811yescharacteristic
102Dic83—22no—
103Dic83—41no—
104Dic83—161no—
105#12441—831nomaximal, Sylow
106C2×C83—321no—
107C2×C83#166811yescharacteristic
108C2×C83#221711yescharacteristic
109C2×C83#222311yescharacteristic
110#2248—831no—
111Q8×C16—831no—
112#1670—831no—
113#1668—831no—
114#1372—831no—
115#1362—831no—
116#1361D8311yescharacteristic
117C83#1244111yescharacteristic, complemented, Sylow
118C2×SD32—1661no—
119C2×SD32—1661no—
120#494—831no—
121#493C2×D8311yescharacteristic
122#475—1661no—
123#460—1661no—
124C8×Q8—831no—
125C4⋊C16—1661no—
126C4⋊C16C2×D8311yescharacteristic
127C4×C16—831no—
128C4×C16—1661no—
129#335—831no—
130#335C2×D8311yescharacteristic
131#332—1661no—
132#331—1661no—
133C2×Q16—1661no—
134C2×Q16—1661no—
135C2×SD16—3321no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—1661no—
139#127—21no—
140C4×Q8—831no—
141SD32—3322no—
142SD32—3322no—
143C2×C16—21no—
144C2×C16—831no—
145C2×C16—831no—
146#106—3321no—
147C4⋊C8—21no—
148C4⋊C8—831no—
149C4⋊C8—1661no—
150C4⋊C8—1661no—
151#103—3321no—
152#102—3321no—
153C4×C8—1661no—
154C4×C8C22×D8311yescharacteristic
155C2×Q8—3321no—
156C2×Q8—831no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—3321no—
161Q16—3321no—
162Q16—6641no—
163SD16—6642no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—41no—
168C2×C8—3321no—
169C2×C8—1661no—
170C2×C8D4×D8311yescharacteristic
171C2×C8#1305059011yescharacteristic
172C4⋊C4—3321no—
173C4⋊C4—3321no—
174C4⋊C4—831no—
175C4⋊C4—1661no—
176C42—1661no—
177C42D4×D8311yescharacteristic
178C16—41no—
179C16—832no—
180C16—1661no—
181C23—81no—
182Q8—832no—
183Q8—6641no—
184Q8—6641no—
185Q8—1661no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—6641no—
192C2×C4—1661no—
193C2×C4—1661no—
194C2×C4#1458176511yescharacteristic
195C2×C4D8×D8311yescharacteristic
196C2×C4#1458181311yescharacteristic
197C8—21no—
198C8—81no—
199C8—3321no—
200C8—6641no—
201C8SD16×D8311yescharacteristic
202C8#1458181611yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1912793611yescharacteristic
207C4—21no—
208C4—13281no—
209C4—3321no—
210C4—1662no—
211C4#1912790011yescharacteristic
212C4#1912797011yescharacteristic
213C4SD32×D8311yescharacteristic
214C4#1912813011yescharacteristic
215C2—321no—
216C2#2490682211yescharacteristic
217C2#2490955811yescharacteristic
218C2#2490959211yescharacteristic
subgroupG/H
centerC22#19127936
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#1361C2
socleC22×C83#466
prime radical#493×C83C22
hypercenter#1361D83
hypocenterC83#12441
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1361

Complements

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

NH ≅ G/Ntype
2. #24909365C2semidirect
3. #24906834C2semidirect
5. #24905536C2semidirect
6. #24905513C2semidirect
59. C83⋊C16SD16semidirect
86. Dic166SD32semidirect
117. C83#12441semidirect

Maximal subgroups

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

subgroupindex
#249097182
#249093652
#249068342
#249068202
#249055362
#249055132
C83×#13612
#1244183