Finite Group Databasebeta

← #35318565

Subgroups — #35318565

Order 21248.

234 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 63 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#24909597C211yesmaximal, characteristic
2#24909509C211yesmaximal, characteristic
3#24907061C211yesmaximal, characteristic, complemented
4#24906257C211yesmaximal, characteristic
5#24905526C211yesmaximal, characteristic, complemented
6#24905511C211yesmaximal, characteristic, complemented
7C83×#1364C211yesmaximal, characteristic, complemented
8C2×#14581737—21no—
9#19128097—21no—
10#19128081C2211yescharacteristic
11#19128051—21no—
12C4×D664C2211yescharacteristic
13#19127766—21no—
14#19127526—21no—
15C4⋊(C83⋊C16)C2211yescharacteristic
16#19127506—21no—
17#19127501—21no—
18#19127428—21no—
19C8×C83⋊C8C2211yescharacteristic
20#500×C83C2211yescharacteristic
21C4⋊C16×C83C2211yescharacteristic
22#335×C83C2211yescharacteristic
23C2×Dic332—21no—
24C2×Dic332—21no—
25C2×D664D411yescharacteristic
26C2×C8×D83—41no—
27#14581792—21no—
28C4×D332—21no—
29#14581737—42no—
30C2×C83⋊C16—21no—
31#14581726D411yescharacteristic
32#14581723—42no—
33#14581716—41no—
34#14581716—41no—
35#14581714—21no—
36(C2×D83)⋊C8—41no—
37C4×C83⋊C8D411yescharacteristic
38C4×C83⋊C8D411yescharacteristic
39C4⋊Q8×C83—21no—
40C2×C16×C83—21no—
41#107×C83—21no—
42#107×C83—21no—
43C4⋊C8×C83—21no—
44C4×C8×C83C2311yescharacteristic
45C2×Dic166—41no—
46C2×D332—21no—
47C2×C4×D83—41no—
48Dic332—41no—
49Dic332—41no—
50Dic332—81no—
51D664—21no—
52D664SD1612yes—
53C2×C83⋊C8—21no—
54C2×C83⋊C8D811yescharacteristic
55C2×C83⋊C8D811yescharacteristic
56C8×D83—82no—
57C4⋊Dic83—21no—
58C4⋊Dic83—41no—
59C4⋊Dic83—41no—
60(C2×D83)⋊C4—41no—
61C83⋊C16—41no—
62C2×Q8×C83—41no—
63C2×C8×C83—41no—
64C2×C8×C83C2×D411yescharacteristic
65C2×C8×C83#4111yescharacteristic
66C4⋊C4×C83—41no—
67C4⋊C4×C83—41no—
68C42×C83C2×D411yescharacteristic
69C16×C83—41no—
70C22×D83—41no—
71Q8×C83—81no—
72Q8×C83—81no—
73C2×C4×C83—81no—
74C2×C4×C83#12111yescharacteristic
75C2×C4×C83C2×D811yescharacteristic
76C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
77C2×Dic83—41no—
78C2×Dic83—81no—
79D332—22no—
80D332—41no—
81C4×D83—81no—
82C4×D83—81no—
83Dic166—81no—
84Dic166—81no—
85C8×C83—21no—
86C8×C83—81no—
87C8×C83C2×SD1611yescharacteristic
88C8×C83#13511yescharacteristic
89C83⋊C8—22no—
90C83⋊C8—21no—
91C83⋊C8—21no—
92C22×C83#45911yescharacteristic
93C2×D83—42no—
94C2×D83—81no—
95C2×D83—81no—
96C4×C83—21no—
97C4×C83—161no—
98C4×C83#46511yescharacteristic
99C4×C83#47111yescharacteristic
100C4×C83#50811yescharacteristic
101C4×C83#50911yescharacteristic
102Dic83—81no—
103Dic83—161no—
104#12422—831nomaximal, Sylow
105C2×C83#166511yescharacteristic
106C2×C83#221011yescharacteristic
107C2×C83#221411yescharacteristic
108D83—82no—
109#2224—831no—
110#2208—831no—
111#1707—831no—
112C8×D8—831no—
113#1367—831no—
114#1364D8311yescharacteristic, complemented
115#1362—831no—
116C83#1242211yescharacteristic, complemented, Sylow
117C2×SD32—1661no—
118#500C2×D8311yescharacteristic
119#498—1661no—
120#494—831no—
121#487—1661no—
122C4×D8—831no—
123C8×D4—1661no—
124#366—1661no—
125C4⋊C16—831no—
126C4⋊C16C2×D8311yescharacteristic
127#358—1661no—
128#357—1661no—
129#335C2×D8311yescharacteristic
130D4⋊C8—1661no—
131C82—831no—
132C2×Q16—1661no—
133C2×Q16—1661no—
134C2×D8—831no—
135C22×C8—3321no—
136C4⋊Q8—21no—
137C4⋊Q8—1661no—
138C4×D4—1661no—
139SD32—3322no—
140C2×C16—21no—
141C2×C16—1661no—
142#107—21no—
143#107—21no—
144#107—831no—
145#106—3322no—
146C4⋊C8—21no—
147#103—3321no—
148#103—3321no—
149#102—1661no—
150C22⋊C8—3321no—
151C4×C8—831no—
152C4×C8—831no—
153C4×C8C22×D8311yescharacteristic
154C2×Q8—41no—
155C2×Q8—3321no—
156C2×D4—1661no—
157C22×C4—3321no—
158Q16—3321no—
159Q16—3321no—
160Q16—6641no—
161D8—832no—
162D8—1661no—
163C2×C8—41no—
164C2×C8—831no—
165C2×C8—831no—
166C2×C8—6642no—
167C2×C8—1661no—
168C2×C8D4×D8311yescharacteristic
169C2×C8#1305059011yescharacteristic
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—3321no—
173C4⋊C4—3321no—
174C4⋊C4—1661no—
175C22⋊C4—3321no—
176C42D4×D8311yescharacteristic
177C16—41no—
178C16—3321no—
179C23—3321no—
180Q8—81no—
181Q8—81no—
182Q8—6641no—
183Q8—6641no—
184D4—3321no—
185D4—1662no—
186C2×C4—81no—
187C2×C4—3321no—
188C2×C4—6641no—
189C2×C4—6641no—
190C2×C4—6641no—
191C2×C4#1458176511yescharacteristic
192C2×C4D8×D8311yescharacteristic
193C2×C4#1458181911yescharacteristic
194C8—21no—
195C8—81no—
196C8—1662no—
197C8—1661no—
198C8—1661no—
199C8SD16×D8311yescharacteristic
200C8#1458181511yescharacteristic
201C22—3322no—
202C22—6641no—
203C22—6641no—
204C22#1912789511yescharacteristic
205C4—21no—
206C4—161no—
207C4—13281no—
208C4—6641no—
209C4#1912793311yescharacteristic
210C4#1912796711yescharacteristic
211C4#1912813211yescharacteristic
212C4#1912813511yescharacteristic
213C2—6642no—
214C2#2490680711yescharacteristic
215C2#2490951711yescharacteristic
216C2#2490954311yescharacteristic
subgroupG/H
centerC22#19127895
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#1364C2
socleC22×C83#459
prime radicalC2×D664D4
hypercenter#1364D83
hypocenterC83#12422
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
3. #24907061C2semidirect
5. #24905526C2semidirect
6. #24905511C2semidirect
7. C83×#1364C2semidirect
114. #1364D83semidirect
116. C83#12422semidirect

Maximal subgroups

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

subgroupindex
#249095972
#249095092
#249070612
#249062572
#249055262
#249055112
C83×#13642
#1242283