Finite Group Databasebeta

← #35291996

Subgroups — #35291996

Order 21248.

236 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 41 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#24908619C211yesmaximal, characteristic
2#24905588C211yesmaximal, characteristic
3C83×#1270C211yesmaximal, characteristic
4#19128332C411yescharacteristic, complemented
5#19127706—21no—
6C2×#14581707—21no—
7#19127470C411yescharacteristic, complemented
8#19127461—21no—
9#390×C83C2211yescharacteristic
10#336×C83—21no—
11C2×#13050574—21no—
12#14581791—41no—
13#14581783—41no—
14#14581771—41no—
15#14581751—21no—
16C2×C22⋊Dic83—41no—
17#14581713—41no—
18#14581713—41no—
19#14581707—22no—
20#14581707—41no—
21C22×D4×C83D411yescharacteristic
22C22×C8×C83—41no—
23C2×C4⋊C4×C83C2×C411yescharacteristic
24C2×C22⋊C4×C83—41no—
25C2×C42×C83D411yescharacteristic
26C2×Dic166—41no—
27C22×Dic83—81no—
28#13050574—41no—
29#13050574—41no—
30C2×C83⋊C8—41no—
31C4⋊Dic83—21no—
32C4⋊Dic83—21no—
33C4⋊Dic83—41no—
34Dic83⋊C4—42no—
35C22⋊Dic83—41no—
36C22⋊Dic83—42no—
37C22⋊Dic83—81no—
38C4×Dic83—41no—
39C24×C83—21no—
40C2×D4×C83—21no—
41C2×D4×C83—41no—
42C2×D4×C83—41no—
43C2×D4×C83SD1611yescharacteristic
44C2×D4×C83Q1611yescharacteristic
45C22×C4×C83—21no—
46C22×C4×C83—21no—
47C22×C4×C83C22⋊C411yescharacteristic
48C2×C8×C83—42no—
49C2×C8×C83—81no—
50C2×C8×C83—81no—
51C4⋊C4×C83—41no—
52C22⋊C4×C83—81no—
53C22⋊C4×C83—81no—
54C42×C83—41no—
55C23×C83—21no—
56C23×C83—21no—
57C23×C83—41no—
58C23×C83—41no—
59C23×C83—81no—
60C23×C83—81no—
61C23×C83#9911yescharacteristic
62D4×C83—41no—
63D4×C83—41no—
64D4×C83—81no—
65C2×C4×C83—21no—
66C2×C4×C83—21no—
67C2×C4×C83—21no—
68C2×C4×C83—21no—
69C2×C4×C83—41no—
70C2×C4×C83—41no—
71C2×C4×C83—41no—
72C2×C4×C83—81no—
73C2×C4×C83#10311yescharacteristic
74C2×C4×C83#10411yescharacteristic
75C2×Dic83—41no—
76C2×Dic83—41no—
77C2×Dic83—82no—
78C2×Dic83—161no—
79C2×Dic83—161no—
80Dic166—161no—
81C8×C83—82no—
82C83⋊C8—81no—
83C22×C83—21no—
84C22×C83—21no—
85C22×C83—41no—
86C22×C83—41no—
87C22×C83—81no—
88C22×C83—81no—
89C22×C83—81no—
90C22×C83—81no—
91C22×C83—161no—
92C22×C83#32811yescharacteristic
93C22×C83AΣL(1,𝔽16)11yescharacteristic
94C22×C83#35411yescharacteristic
95C4×C83—21no—
96C4×C83—21no—
97C4×C83—41no—
98C4×C83—41no—
99C4×C83—81no—
100Dic83—81no—
101Dic83—81no—
102Dic83—162no—
103#7430—831nomaximal, Sylow
104C2×C83—21no—
105C2×C83—21no—
106C2×C83—81no—
107C2×C83—81no—
108C2×C83#135111yescharacteristic
109C2×C83#140211yescharacteristic
110C2×C83#140411yescharacteristic
111#2027—831no—
112#1393—831no—
113#1270D8311yescharacteristic
114C83#743011yescharacteristic, complemented, Sylow
115#544—831no—
116#418—1661no—
117C2×#99—1661no—
118#390C2×D8311yescharacteristic
119#343—831no—
120#339—1661no—
121#336—21no—
122C22×D4D33211yescharacteristic
123C2×M16—1661no—
124C22×C8—41no—
125C4⋊Q8—3321no—
126#125—3321no—
127C22⋊Q8—3321no—
128#117—1661no—
129C2×C4⋊C4C4×D8311yescharacteristic
130C2×C22⋊C4—41no—
131C2×C22⋊C4—3321no—
132C2×C42C22⋊D8311yescharacteristic
133#102—3321no—
134#102—3321no—
135#99—3321no—
136#99—1662no—
137C24—21no—
138C2×Q8—3321no—
139C2×D4—21no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4#1305057911yescharacteristic
143C2×D4Dic33211yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—6641no—
147C22×C4(C2×D83)⋊C411yescharacteristic
148M16—3321no—
149M16—3321no—
150C2×C8—42no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8—3321no—
154C4⋊C4—41no—
155C4⋊C4—3321no—
156C4⋊C4—3322no—
157C4⋊C4—1661no—
158C4⋊C4—1661no—
159C22⋊C4—81no—
160C22⋊C4—81no—
161C22⋊C4—3321no—
162C22⋊C4—3322no—
163C22⋊C4—6641no—
164C42—41no—
165C42—3321no—
166C23—21no—
167C23—21no—
168C23—41no—
169C23—41no—
170C23—81no—
171C23—81no—
172C23#1458170611yescharacteristic
173Q8—13281no—
174D4—41no—
175D4—41no—
176D4—81no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—13281no—
186C2×C4—13281no—
187C2×C4—3321no—
188C2×C4—3321no—
189C2×C4—6642no—
190C2×C4#1458171711yescharacteristic
191C2×C4#1458171811yescharacteristic
192C8—82no—
193C8—6641no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—161no—
203C22#1912743911yescharacteristic
204C22#1912748911yescharacteristic
205C22#1912749511yescharacteristic
206C4—21no—
207C4—21no—
208C4—41no—
209C4—41no—
210C4—81no—
211C4—13282no—
212C4—6641no—
213C4—6641no—
214C2—21no—
215C2—21no—
216C2—81no—
217C2—81no—
218C2#2490548511yescharacteristic
219C2#2490560911yescharacteristic
220C2#2490561511yescharacteristic
subgroupG/H
centerC22#19127489
commutatorC2×C4⋊C4×C83C2×C4
Frattini#390C2×D83
FittingC83×#1270C2
socleC22×C83AΣL(1,𝔽16)
prime radicalC22×D4×C83D4
cosocle#390×C83C22
hypercenter#1270D83
hypocenterC83#7430
LCSG ▷ C2×C4⋊C4×C83 ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #1270

Complements

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

NH ≅ G/Ntype
4. #19128332C4semidirect
7. #19127470C4semidirect
114. C83#7430semidirect

Maximal subgroups

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

subgroupindex
#249086192
#249055882
C83×#12702
#743083