Finite Group Databasebeta

← #38755319

Subgroups — #38755319

Order 25216.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#27161779C212yesmaximal, complemented
2#27161481C211yesmaximal, characteristic, complemented
3#27161401C212yesmaximal, complemented
4#27161391C212yesmaximal, complemented
5#27161379C212yesmaximal, complemented
6#27161329C212yesmaximal, complemented
7#27161158C211yesmaximal, characteristic, complemented
8#27161120C212yesmaximal, complemented
9C4×D8×C197C211yesmaximal, characteristic, complemented
10C2×#15691341C2212yes—
11#20263319—22no—
12#20263318C2212yes—
13#20263309—22no—
14#20263299—22no—
15#20263297C2212yes—
16#20263297C2212yes—
17C4×D788C2211yescharacteristic
18C4×C22⋊D197—22no—
19#20263284C2212yes—
20C4⋊C4×D197C2212yes—
21#20263260C2211yescharacteristic
22#20263260C2211yescharacteristic
23#20263259C2212yes—
24#20263258C2212yes—
25#20263257C2212yes—
26C4⋊#13096802C2211yescharacteristic
27#20263247C2212yes—
28#20263247C2212yes—
29#20263238C2212yes—
30C2×D8×C197C2211yescharacteristic
31C4×D4×C197C2212yes—
32#107×C197C2211yescharacteristic
33#102×C197C2212yes—
34C4×C8×C197C2211yescharacteristic
35C2×D788D411yescharacteristic
36C2×C22⋊D197—22no—
37C2×C22⋊D197—22no—
38C2×C4×D197—22no—
39C2×C4×D197C2312yes—
40#15691341—24no—
41C2×#13096802C2312yes—
42C4⋊Dic197D411yescharacteristic
43C4⋊Dic197C2312yes—
44C4⋊Dic197C2312yes—
45#15691337—22no—
46#15691337—22no—
47#15691337—22no—
48C22⋊Dic197—22no—
49C22⋊Dic197—22no—
50#15691335—22no—
51#15691335D412yes—
52C4×Dic197—22no—
53C2×D4×C197C2312yes—
54C22×C4×C197—22no—
55D8×C197—22no—
56C2×C8×C197C2311yescharacteristic
57C2×C8×C197C2311yescharacteristic
58C4⋊C4×C197C2312yes—
59C22⋊C4×C197—22no—
60C42×C197C2311yescharacteristic
61C22×D197C2×D412yes—
62D788—41no—
63C22⋊D197—44no—
64C22⋊D197—82no—
65C2×Dic197—22no—
66C2×Dic197—22no—
67C2×Dic197C2×D412yes—
68C4×D197—22no—
69C4×D197—22no—
70C4×D197—82no—
71#13096802—22no—
72C23×C197—22no—
73D4×C197—22no—
74D4×C197—22no—
75C2×C4×C197—22no—
76C2×C4×C197—82no—
77C2×C4×C197C2×D411yescharacteristic
78C2×C4×C197C2×D411yescharacteristic
79C2×C4×C197C2411yescharacteristic
80C8×C197—21no—
81C8×C197#4112yes—
82C22×C197—42no—
83C22×C197—42no—
84C22×C197—42no—
85C22×C197C22×D411yescharacteristic
86C2×D197—22no—
87C2×D197—22no—
88C2×D197—22no—
89C4×C197—21no—
90C4×C197—21no—
91C4×C197—42no—
92C4×C197C2×#4111yescharacteristic
93C4×C1972−1+411yescharacteristic
94Dic197—22no—
95Dic197—42no—
96Dic197—42no—
97C2×C197—82no—
98C2×C197#54711yescharacteristic
99C2×C197#57511yescharacteristic
100C2×C197#57611yescharacteristic
101D197—42no—
102C197#335511yescharacteristic, complemented, Sylow
103#3355—1971nomaximal, Sylow
104#553—1972no—
105#499—1971no—
106#482—1972no—
107#480—1972no—
108#478—1972no—
109#469—1972no—
110C4×D8D19711yescharacteristic, complemented
111#436—1971no—
112#428—1972no—
113C2×D8C2×D19711yescharacteristic
114C2×M16—1972no—
115#126—3942no—
116#125—1972no—
117#123—3942no—
118#121—3942no—
119#121—1972no—
120#121—1972no—
121C4×D4—3942no—
122C4×D4—1971no—
123C4×D4C2×D19712yes—
124#117—1972no—
125C2×C4⋊C4—1972no—
126#107—1971no—
127#107—1971no—
128#107—1972no—
129#107C2×D19711yescharacteristic
130#106—1972no—
131#106—1972no—
132C4⋊C8—1971no—
133#102—1972no—
134#102—1972no—
135#102C2×D19712yes—
136C22⋊C8—1972no—
137C4×C8C2×D19711yescharacteristic
138C2×D4—3942no—
139C2×D4—3942no—
140C2×D4—1971no—
141C2×D4C22×D19712yes—
142C22×C4—22no—
143C22×C4—3942no—
144C22×C4—1972no—
145D8—22no—
146M16—3944no—
147C2×C8—1972no—
148C2×C8C22×D19711yescharacteristic
149C2×C8C22×D19711yescharacteristic
150C4⋊C4—3942no—
151C4⋊C4—3942no—
152C4⋊C4—3942no—
153C4⋊C4—1972no—
154C4⋊C4—1972no—
155C4⋊C4—1971no—
156C4⋊C4C22×D19712yes—
157C22⋊C4—22no—
158C22⋊C4—3942no—
159C22⋊C4—3942no—
160C22⋊C4—3942no—
161C22⋊C4—1972no—
162C42—3942no—
163C42C22×D19711yescharacteristic
164C23—22no—
165C23—1972no—
166D4—22no—
167D4—22no—
168D4—7881no—
169D4—7884no—
170D4—15762no—
171C2×C4—22no—
172C2×C4—82no—
173C2×C4—15762no—
174C2×C4—3942no—
175C2×C4—3942no—
176C2×C4—3942no—
177C2×C4—3942no—
178C2×C4—1972no—
179C2×C4D4×D19711yescharacteristic
180C2×C4D4×D19711yescharacteristic
181C2×C4C23×D19711yescharacteristic
182C8—21no—
183C8—3942no—
184C8#1569135812yes—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—3942no—
189C22—3942no—
190C22—3942no—
191C22C2×D4×D19711yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—7882no—
196C4—7882no—
197C4—3942no—
198C4C2×#1569135811yescharacteristic
199C4#2026337311yescharacteristic
200C2—82no—
201C2—7882no—
202C2#2716173411yescharacteristic
203C2#2716187611yescharacteristic
204C2#2716188011yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC4×D8×C197C2
socleC22×C197C22×D4
hypercenterC4×D8D197
hypocenterC197#3355
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #27161779C2semidirect
2. #27161481C2semidirect
3. #27161401C2semidirect
4. #27161391C2semidirect
5. #27161379C2semidirect
6. #27161329C2semidirect
7. #27161158C2semidirect
8. #27161120C2semidirect
9. C4×D8×C197C2semidirect
102. C197#3355semidirect
110. C4×D8D197semidirect

Maximal subgroups

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

subgroupindex
#271617792
#271614812
#271614012
#271613912
#271613792
#271613292
#271611582
#271611202
C4×D8×C1972
#3355197