Finite Group Databasebeta

← #35316355

Subgroups — #35316355

Order 21248.

220 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 71 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#24909327C211yesmaximal, characteristic
2#24906373C211yesmaximal, characteristic, complemented
3#24905528C211yesmaximal, characteristic, complemented
4#24905432C211yesmaximal, characteristic, complemented
5#24905430C211yesmaximal, characteristic
6#24905384C211yesmaximal, characteristic, complemented
7C83×#1580C211yesmaximal, characteristic
8C2×#14581736C411yescharacteristic
9#19127831—21no—
10C4×Dic332C2211yescharacteristic
11#19127526C411yescharacteristic
12#19127504C411yescharacteristic
13#19127502C411yescharacteristic
14#19127477C2211yescharacteristic
15#19127477C2211yescharacteristic
16#19127452C2211yescharacteristic
17#19127433—21no—
18C4×D8×C83C2211yescharacteristic
19C8×D4×C83—21no—
20#334×C83C2211yescharacteristic
21D4⋊C8×C83—21no—
22#322×C83C2211yescharacteristic
23C2×Dic332C2×C411yescharacteristic
24C4×Dic166—21no—
25#14581736—22no—
26C2×C83⋊C16—21no—
27C2×C83⋊C16C2×C411yescharacteristic
28C2×C83⋊C16C2×C411yescharacteristic
29#14581726C2×C411yescharacteristic
30C4⋊(C83⋊C8)—21no—
31Dic83⋊C8—41no—
32#14581717—21no—
33#14581702—41no—
34C2×D8×C83C2×C411yescharacteristic
35C22×C8×C83—41no—
36C4×D4×C83—21no—
37#107×C83C2×C411yescharacteristic
38C4⋊C8×C83—21no—
39#102×C83—21no—
40C22⋊C8×C83—41no—
41C4×C8×C83D411yescharacteristic
42C4×C8×C83D411yescharacteristic
43C4×C8×C83C2311yescharacteristic
44C2×Dic166—21no—
45Dic332—21no—
46Dic332—21no—
47C2×C83⋊C8—41no—
48C4⋊Dic83—21no—
49Dic83⋊C4—41no—
50C4×Dic83—41no—
51C83⋊C16—22no—
52C83⋊C16—21no—
53C83⋊C16—21no—
54C2×D4×C83—21no—
55C22×C4×C83—41no—
56D8×C83—21no—
57D8×C83M1612yes—
58C2×C8×C83—21no—
59C2×C8×C83—21no—
60C2×C8×C83—41no—
61C2×C8×C83—82no—
62C2×C8×C83C22×C411yescharacteristic
63C2×C8×C83#4111yescharacteristic
64C4⋊C4×C83—21no—
65C22⋊C4×C83—41no—
66C42×C83C2×D411yescharacteristic
67C23×C83—41no—
68D4×C83—22no—
69D4×C83—41no—
70C2×C4×C83—41no—
71C2×C4×C83—81no—
72C2×C4×C83—81no—
73C2×C4×C83C4×D411yescharacteristic
74C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
75C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
76C2×Dic83—41no—
77C2×Dic83—41no—
78Dic166—41no—
79Dic166—41no—
80C8×C83—21no—
81C8×C83—41no—
82C8×C83—41no—
83C8×C83—81no—
84C8×C83C2×M1611yescharacteristic
85C8×C83#13111yescharacteristic
86C83⋊C8—81no—
87C22×C83—42no—
88C22×C83—81no—
89C22×C83—81no—
90C22×C83#44211yescharacteristic
91C4×C83—21no—
92C4×C83—21no—
93C4×C83—81no—
94C4×C83#43511yescharacteristic
95C4×C83#44311yescharacteristic
96Dic83—81no—
97Dic83—81no—
98#12077—831nomaximal, Sylow
99C2×C83—82no—
100C2×C83#158611yescharacteristic
101C2×C83#218011yescharacteristic
102C2×C83AGL(1,ℤ/16)11yescharacteristic
103#2175—831no—
104#1593—831no—
105#1580D8311yescharacteristic
106#1368—831no—
107#1334—831no—
108#1333—831no—
109#1313—831no—
110C83#1207711yescharacteristic, complemented, Sylow
111C2×SD32—831no—
112#446—1661no—
113C4×Q16—831no—
114C4×D8C2×D8311yescharacteristic
115C8×D4—21no—
116#366—831no—
117#358—831no—
118#357—831no—
119#346—831no—
120#346—831no—
121#334—831no—
122#334C2×D8311yescharacteristic
123Q8⋊C8—1661no—
124D4⋊C8—21no—
125#322C2×D8311yescharacteristic
126C2×Q16—831no—
127C2×D8C4×D8311yescharacteristic
128C22×C8—41no—
129C4×Q8—1661no—
130C4×D4—21no—
131SD32—1662no—
132C2×C16—831no—
133C2×C16—831no—
134C2×C16—1661no—
135#107—831no—
136#107C4×D8311yescharacteristic
137C4⋊C8—21no—
138C4⋊C8—3321no—
139C4⋊C8—1661no—
140#103—1661no—
141#102—21no—
142C22⋊C8—41no—
143#97—3321no—
144C4×C8C22⋊D8311yescharacteristic
145C4×C8C22⋊D8311yescharacteristic
146C4×C8C22×D8311yescharacteristic
147C2×Q8—1661no—
148C2×D4—21no—
149C22×C4—41no—
150Q16—1661no—
151Q16—1661no—
152D8—21no—
153D8#1305057312yes—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—41no—
157C2×C8—82no—
158C2×C8—3321no—
159C2×C8C2×C4×D8311yescharacteristic
160C2×C8#1305059011yescharacteristic
161C4⋊C4—21no—
162C4⋊C4—3321no—
163C4⋊C4—1661no—
164C22⋊C4—41no—
165C42—3321no—
166C42C2×C22⋊D8311yescharacteristic
167C16—1662no—
168C16—1661no—
169C16—1661no—
170C23—41no—
171Q8—3321no—
172Q8—3321no—
173D4—22no—
174D4—41no—
175C2×C4—41no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—3321no—
179C2×C4—3321no—
180C2×C4C4×C22⋊D8311yescharacteristic
181C2×C4#1458181711yescharacteristic
182C2×C4#1458181811yescharacteristic
183C8—21no—
184C8—41no—
185C8—41no—
186C8—81no—
187C8—6641no—
188C8C2×#1305057311yescharacteristic
189C8#1458179811yescharacteristic
190C22—42no—
191C22—81no—
192C22—81no—
193C22#1912781111yescharacteristic
194C4—21no—
195C4—21no—
196C4—81no—
197C4—6641no—
198C4—6641no—
199C4#1912777211yescharacteristic
200C4#1912781611yescharacteristic
201C2—82no—
202C2#2490632711yescharacteristic
203C2#2490935411yescharacteristic
204C2#2490935911yescharacteristic
subgroupG/H
centerC22#19127811
commutatorC2×C8×C83C22×C4
FrattiniC4×C8C22×D83
FittingC83×#1580C2
socleC22×C83#442
prime radicalC2×D8×C83C2×C4
cosocleC4×C8×C83C23
hypercenter#1580D83
hypocenterC83#12077
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #322 ◁ #1580

Complements

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

NH ≅ G/Ntype
2. #24906373C2semidirect
3. #24905528C2semidirect
4. #24905432C2semidirect
6. #24905384C2semidirect
110. C83#12077semidirect

Maximal subgroups

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

subgroupindex
#249093272
#249063732
#249055282
#249054322
#249054302
#249053842
C83×#15802
#1207783