Finite Group Databasebeta

← #24915522

Subgroups — #24915522

Order 10624.

430 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#14581786C211yesmaximal, characteristic
2#19127639C212yesmaximal, complemented
3#19127636C212yesmaximal
4#19127632C212yesmaximal, complemented
5#19127627C212yesmaximal
6#19127620C212yesmaximal, complemented
7#19127612C212yesmaximal
8#19127569C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C83C211yesmaximal, characteristic, complemented
10#14581786—24no—
11C2×Dic83⋊C4C2211yescharacteristic
12C2×Dic83⋊C4C2211yescharacteristic
13C2×Dic83⋊C4C2211yescharacteristic
14C2×Dic83⋊C4C2214yes—
15C2×(C2×D83)⋊C4C2211yescharacteristic
16C2×(C2×D83)⋊C4C2211yescharacteristic
17C2×(C2×D83)⋊C4C2211yescharacteristic
18C2×(C2×D83)⋊C4C2214yes—
19C83⋊#95C2212yes—
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2214yes—
23C83⋊#95C2214yes—
24C22×Q8×C83C2212yes—
25C4⋊Q8×C83—24no—
26C2×C4⋊C4×C83C2212yes—
27C2×C4⋊C4×C83C2212yes—
28C2×C42×C83C2211yescharacteristic
29C23×D83C2311yescharacteristic
30C22×Dic83C2314yes—
31C22×Dic83C2311yescharacteristic
32C22×Dic83C2311yescharacteristic
33C22×Dic83C2311yescharacteristic
34Dic83⋊C4—22no—
35Dic83⋊C4—22no—
36Dic83⋊C4—22no—
37Dic83⋊C4—44no—
38(C2×D83)⋊C4—22no—
39(C2×D83)⋊C4—22no—
40(C2×D83)⋊C4—22no—
41(C2×D83)⋊C4—44no—
42C2×Q8×C83—24no—
43C2×Q8×C83—44no—
44C22×C4×C83C2314yes—
45C22×C4×C83C2311yescharacteristic
46C22×C4×C83C2311yescharacteristic
47C22×C4×C83C2311yescharacteristic
48C4⋊C4×C83—24no—
49C4⋊C4×C83—24no—
50C42×C83D414yes—
51C23×C83C2411yescharacteristic
52C22×D83—21no—
53C22×D83—22no—
54C22×D83—22no—
55C22×D83—22no—
56Q8×C83—48no—
57C2×C4×C83—22no—
58C2×C4×C83—22no—
59C2×C4×C83—22no—
60C2×C4×C83—24no—
61C2×C4×C83—24no—
62C2×C4×C83—24no—
63C2×C4×C83C2×D412yes—
64C2×C4×C83C2×D412yes—
65C2×C4×C83C2×D412yes—
66C2×Dic83—21no—
67C2×Dic83—21no—
68C2×Dic83—21no—
69C2×Dic83—24no—
70C2×Dic83—22no—
71C2×Dic83—22no—
72C2×Dic83—22no—
73C2×Dic83—24no—
74C2×Dic83—24no—
75C22×C83C22×D411yescharacteristic
76C22×C83H(5,𝔽2)12yes—
77C22×C832−1+412yes—
78C22×C832−1+412yes—
79C2×D83—41no—
80C2×D83—41no—
81C2×D83—41no—
82C2×D83—44no—
83C4×C83—22no—
84C4×C83—22no—
85C4×C83—22no—
86C4×C83—44no—
87Dic83—41no—
88Dic83—41no—
89Dic83—41no—
90Dic83—44no—
91C2×C83#53511yescharacteristic
92C2×C83#53611yescharacteristic
93C2×C83#53611yescharacteristic
94C2×C83#56314yes—
95D83—81no—
96#2821—831nomaximal, Sylow
97C83#282111yescharacteristic, complemented, Sylow
98C2×C4⋊Q8D8311yescharacteristic, complemented
99C2×#126—831no—
100#400—832no—
101#399—832no—
102#398—832no—
103#397—832no—
104#395—832no—
105#393—832no—
106#383—831no—
107C22×Q8C2×D8312yes—
108C4⋊Q8—24no—
109#126—1664no—
110C2×C4⋊C4—834no—
111C2×C4⋊C4—831no—
112C2×C4⋊C4—831no—
113C2×C4⋊C4—831no—
114C2×C4⋊C4C2×D8312yes—
115C2×C4⋊C4C2×D8312yes—
116C2×C22⋊C4—834no—
117C2×C22⋊C4—831no—
118C2×C22⋊C4—831no—
119C2×C22⋊C4—831no—
120C2×C42C2×D8311yescharacteristic
121#95—832no—
122#95—832no—
123#95—832no—
124#95—834no—
125#95—834no—
126C24—831no—
127C2×Q8—24no—
128C2×Q8—44no—
129C22×C4—834no—
130C22×C4—831no—
131C22×C4—831no—
132C22×C4—831no—
133C22×C4C22×D8314yes—
134C22×C4C22×D8311yescharacteristic
135C22×C4C22×D8311yescharacteristic
136C22×C4C22×D8311yescharacteristic
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—3324no—
140C4⋊C4—1662no—
141C4⋊C4—1662no—
142C4⋊C4—1662no—
143C22⋊C4—3324no—
144C22⋊C4—1662no—
145C22⋊C4—1662no—
146C22⋊C4—1662no—
147C42C22⋊D8314yes—
148C23—1661no—
149C23—1662no—
150C23—1662no—
151C23—1662no—
152C23C23×D8311yescharacteristic
153Q8—48no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—1661no—
161C2×C4—1661no—
162C2×C4—1661no—
163C2×C4—1664no—
164C2×C4—1662no—
165C2×C4—1664no—
166C2×C4—1664no—
167C2×C4—1662no—
168C2×C4—1662no—
169C2×C4C2×C22⋊D8312yes—
170C2×C4C2×C22⋊D8312yes—
171C2×C4C2×C22⋊D8312yes—
172C22—3324no—
173C22—3321no—
174C22—3321no—
175C22—3321no—
176C22C22×C22⋊D8311yescharacteristic
177C22#1458183712yes—
178C22#1458184012yes—
179C22#1458184012yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—44no—
184C4—3324no—
185C4—3321no—
186C4—3321no—
187C4—3321no—
188C2—6641no—
189C2#1912826811yescharacteristic
190C2#1912827511yescharacteristic
191C2#1912827511yescharacteristic
192C2#1912845014yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C4⋊Q8×C83C2
prime radicalC23×D83C23
hypercenterC2×C4⋊Q8D83
hypocenterC83#2821
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #19127639C2semidirect
4. #19127632C2semidirect
6. #19127620C2semidirect
8. #19127569C2semidirect
9. C2×C4⋊Q8×C83C2semidirect
97. C83#2821semidirect
98. C2×C4⋊Q8D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817862
#191276392
#191276362
#191276322
#191276272
#191276202
#191276122
#191275692
C2×C4⋊Q8×C832
#282183