Finite Group Databasebeta

← #38743852

Subgroups — #38743852

Order 25216.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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×#20263325C211yesmaximal, characteristic
2C2×#20263251C211yesmaximal, characteristic
3C2×#20263249C211yesmaximal, characteristic
4#27160984C211yesmaximal, characteristic
5#27160841C211yesmaximal, characteristic
6#27160833C211yesmaximal, characteristic
7#340×C197C211yesmaximal, characteristic
8C22×Dic394—21no—
9C22×Dic394C2211yescharacteristic
10C22×#13096802C2211yescharacteristic
11#20263325—22no—
12#20263325—22no—
13C2×C4⋊Dic197C2211yescharacteristic
14C2×#15691337—21no—
15C2×C4×Dic197C2211yescharacteristic
16#20263251—22no—
17#20263249—22no—
18#20263233—21no—
19#20263233—21no—
20C22×C8×C197C2211yescharacteristic
21C2×C4⋊C4×C197C2211yescharacteristic
22C2×C4⋊C4×C197C2211yescharacteristic
23C2×Dic394—24no—
24C2×Dic394—21no—
25C2×Dic394—21no—
26C2×Dic394—21no—
27C2×Dic394—21no—
28C2×Dic394—21no—
29C2×Dic394—42no—
30C2×Dic394—42no—
31C2×Dic394Q812yescomplemented
32C22×Dic197—21no—
33C22×Dic197—21no—
34C22×Dic197D411yescharacteristic
35C22×Dic197D411yescharacteristic
36C2×#13096802—21no—
37C2×#13096802—21no—
38C2×#13096802—21no—
39C4⋊Dic197—21no—
40C4⋊Dic197D412yes—
41#15691337—24no—
42C4×Dic197—21no—
43C4×Dic197—21no—
44C22×C4×C197—21no—
45C22×C4×C197—21no—
46C22×C4×C197C2311yescharacteristic
47C2×C8×C197—21no—
48C2×C8×C197—21no—
49C2×C8×C197—21no—
50C4⋊C4×C197—21no—
51C4⋊C4×C197—21no—
52C4⋊C4×C197—21no—
53C4⋊C4×C197D412yes—
54Dic394—22no—
55Dic394—22no—
56Dic394—48no—
57Dic394—41no—
58Dic394—41no—
59C2×Dic197—21no—
60C2×Dic197—21no—
61C2×Dic197—21no—
62C2×Dic197—21no—
63C2×Dic197—22no—
64C2×Dic197—22no—
65C2×Dic197—41no—
66C2×Dic197—41no—
67C2×Dic197—41no—
68C2×Dic197—41no—
69C2×Dic197SD1612yes—
70C2×Dic197Q1612yes—
71#13096802—41no—
72C23×C197C2×D411yescharacteristic
73C2×C4×C197—22no—
74C2×C4×C197—41no—
75C2×C4×C197—41no—
76C2×C4×C197—41no—
77C2×C4×C197—41no—
78C2×C4×C197—41no—
79C2×C4×C197C2×D411yescharacteristic
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2×Q811yescharacteristic
82C2×C4×C197#4111yescharacteristic
83C2×C4×C197#4111yescharacteristic
84C2×C4×C197#4111yescharacteristic
85C8×C197—41no—
86C22×C197#12011yescharacteristic
87C22×C197C22⋊Q811yescharacteristic
88C22×C197#12411yescharacteristic
89C22×C197C2×SD1611yescharacteristic
90C22×C197C2×Q1611yescharacteristic
91C22×C197AGL(1,ℤ/8)11yescharacteristic
92C22×C197#13711yescharacteristic
93C4×C197—42no—
94C4×C197—81no—
95C4×C197C22⋊Q811yescharacteristic
96C4×C197C22⋊Q811yescharacteristic
97C4×C197#12411yescharacteristic
98C4×C197#12411yescharacteristic
99Dic197—22no—
100Dic197—22no—
101Dic197—42no—
102Dic197—42no—
103C2×C197#39311yescharacteristic
104C2×C197#45011yescharacteristic
105C2×C197#45111yescharacteristic
106C2×C197#47511yescharacteristic
107C2×C197#47711yescharacteristic
108C2×C197#48711yescharacteristic
109C2×C197#48911yescharacteristic
110C197#202411yescharacteristic, complemented, Sylow
111#2024—1971nomaximal, Sylow
112C2×C4⋊Q8—1971no—
113C2×#103—1971no—
114C2×#103—1971no—
115#391—1971no—
116#340—1971no—
117#340D19711yescharacteristic
118#336—1971no—
119C22×Q8—3941no—
120C22×Q8—1971no—
121C22×C8—1971no—
122C22×C8C2×D19711yescharacteristic
123C4⋊Q8—3942no—
124C4⋊Q8—3942no—
125C2×C4⋊C4—3941no—
126C2×C4⋊C4—1971no—
127C2×C4⋊C4C2×D19711yescharacteristic
128C2×C4⋊C4C2×D19711yescharacteristic
129C2×C42—1971no—
130#103—3942no—
131#103—3942no—
132#95—3941no—
133#95—3941no—
134C2×Q8—7882no—
135C2×Q8—7882no—
136C2×Q8—3944no—
137C2×Q8—3941no—
138C2×Q8—3941no—
139C2×Q8—3941no—
140C2×Q8—3941no—
141C2×Q8—3941no—
142C2×Q8—1972no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—3941no—
146C22×C4—3941no—
147C22×C4—1971no—
148C22×C4—1971no—
149C22×C4C22×D19711yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—3941no—
154C2×C8—3941no—
155C2×C8—3941no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—3944no—
160C4⋊C4—3941no—
161C4⋊C4—1972no—
162C4⋊C4D78812yes—
163C42—3941no—
164C42—3941no—
165C23D4×D19711yescharacteristic
166Q8—7888no—
167Q8—7881no—
168Q8—7881no—
169Q8—3942no—
170Q8—3942no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—7881no—
178C2×C4—7881no—
179C2×C4—7881no—
180C2×C4—7881no—
181C2×C4—3942no—
182C2×C4—3942no—
183C2×C4—3941no—
184C2×C4—3941no—
185C2×C4—3941no—
186C2×C4—3941no—
187C2×C4—1972no—
188C2×C4—1972no—
189C2×C4D4×D19711yescharacteristic
190C2×C4C2×D78811yescharacteristic
191C2×C4Q8×D19711yescharacteristic
192C2×C4#1569135711yescharacteristic
193C2×C4#1569135811yescharacteristic
194C2×C4#1569135811yescharacteristic
195C8—41no—
196C8—7881no—
197C22#2026329611yescharacteristic
198C22#2026330411yescharacteristic
199C22#2026331211yescharacteristic
200C22SD16×D19711yescharacteristic
201C22Q16×D19711yescharacteristic
202C22#2026335011yescharacteristic
203C22#2026335511yescharacteristic
204C4—42no—
205C4—81no—
206C4—7882no—
207C4—7882no—
208C4—3942no—
209C4—3942no—
210C4#2026330411yescharacteristic
211C4#2026330611yescharacteristic
212C4#2026331211yescharacteristic
213C4#2026331411yescharacteristic
214C2#2716099211yescharacteristic
215C2#2716123111yescharacteristic
216C2#2716123611yescharacteristic
217C2#2716136311yescharacteristic
218C2#2716137311yescharacteristic
219C2#2716142811yescharacteristic
220C2#2716143711yescharacteristic
subgroupG/H
centerC23D4×D197
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#340×C197C2
socle, prime radicalC23×C197C2×D4
hypercenter#340D197
hypocenterC197#2024
LCSG ▷ C22×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
31. C2×Dic394Q8semidirect
110. C197#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#202633252
C2×#202632512
C2×#202632492
#271609842
#271608412
#271608332
#340×C1972
#2024197