Finite Group Databasebeta

← #38741738

Subgroups — #38741738

Order 25216.

236 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 51 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#27161704C211yesmaximal, characteristic
2#27161415C211yesmaximal, characteristic, complemented
3#27161412C211yesmaximal, characteristic, complemented
4#27161016C211yesmaximal, characteristic
5#27160815C211yesmaximal, characteristic, complemented
6#27160802C211yesmaximal, characteristic
7#328×C197C211yesmaximal, characteristic, complemented
8#20263306—21no—
9#20263305C2211yescharacteristic
10#20263304—21no—
11#20263304—21no—
12#20263303—21no—
13#20263302—21no—
14C2×C4⋊Dic197—21no—
15C2×C4⋊Dic197C2211yescharacteristic
16C2×#15691335—21no—
17C2×#15691335C2211yescharacteristic
18C22⋊C4×D197—21no—
19#20263260—21no—
20#20263259—21no—
21#20263251—21no—
22#20263250—21no—
23C22⋊#13096802C2211yescharacteristic
24C22⋊Q8×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27C23×D197D411yescharacteristic
28C2×Dic394—21no—
29C2×Dic394D411yescharacteristic
30C22×Dic197—21no—
31C22×Dic197D411yescharacteristic
32C2×C4×D197—21no—
33C2×C4×D197—21no—
34C2×#13096802—21no—
35C4⋊Dic197—21no—
36C4⋊Dic197—21no—
37C4⋊Dic197—21no—
38C4⋊Dic197—42no—
39C4⋊Dic197D411yescharacteristic
40#15691337—21no—
41#15691337—21no—
42#15691337—21no—
43C22⋊Dic197—21no—
44#15691335—21no—
45#15691335—21no—
46#15691335—21no—
47#15691335—21no—
48#15691335—42no—
49C2×Q8×C197D411yescharacteristic
50C22×C4×C197—21no—
51C22×C4×C197C2311yescharacteristic
52C2×C8×C197—21no—
53C4⋊C4×C197—21no—
54C4⋊C4×C197D411yescharacteristic
55C22⋊C4×C197—21no—
56C22×D197—21no—
57C22×D197—21no—
58C22×D197—41no—
59C22×D197—41no—
60C22×D197Q1612yes—
61Dic394—41no—
62Dic394—81no—
63C2×Dic197—21no—
64C2×Dic197—21no—
65C2×Dic197—21no—
66C2×Dic197—41no—
67C2×Dic197—41no—
68C2×Dic197—41no—
69C2×Dic197—41no—
70C4×D197—81no—
71C4×D197—81no—
72#13096802—41no—
73C23×C197C2×D411yescharacteristic
74Q8×C197—41no—
75C2×C4×C197—22no—
76C2×C4×C197—21no—
77C2×C4×C197—21no—
78C2×C4×C197—41no—
79C2×C4×C197—41no—
80C2×C4×C197—41no—
81C2×C4×C197C2×D411yescharacteristic
82C2×C4×C197C2×D411yescharacteristic
83C8×C197—41no—
84C22×C197—21no—
85C22×C197—21no—
86C22×C197#12011yescharacteristic
87C22×C197C2×Q1611yescharacteristic
88C22×C197#13711yescharacteristic
89C2×D197—22no—
90C2×D197—41no—
91C2×D197—41no—
92C2×D197—42no—
93C2×D197—81no—
94C4×C197—21no—
95C4×C197—21no—
96C4×C197—42no—
97C4×C197—41no—
98C4×C197—41no—
99Dic197—41no—
100Dic197—41no—
101Dic197—41no—
102Dic197—81no—
103C2×C197—21no—
104C2×C197—21no—
105C2×C197#45111yescharacteristic
106C2×C197#45311yescharacteristic
107C2×C197#45811yescharacteristic
108D197—42no—
109C197#161411yescharacteristic, complemented, Sylow
110#1614—1971nomaximal, Sylow
111#543—1971no—
112#484—1971no—
113#484—1971no—
114#399—1971no—
115#328—1971no—
116#328D19711yescharacteristic, complemented
117#323—1971no—
118C22⋊Q8—3941no—
119C22⋊Q8—3941no—
120C22⋊Q8—3941no—
121C22⋊Q8—3941no—
122C22⋊Q8—3941no—
123C22⋊Q8—1971no—
124C22⋊Q8C2×D19711yescharacteristic
125C2×C4⋊C4—3941no—
126C2×C4⋊C4—1971no—
127C2×C22⋊C4—3941no—
128C2×C22⋊C4—3941no—
129C2×C22⋊C4—1971no—
130#107—3941no—
131#107—3941no—
132#103—3941no—
133#103—3941no—
134C22⋊C8—1971no—
135C22⋊C8C2×D19711yescharacteristic
136#95C2×D19711yescharacteristic
137C24—1971no—
138C2×Q8—3941no—
139C2×Q8—1971no—
140C2×Q8D78811yescharacteristic
141C22×C4—21no—
142C22×C4—3941no—
143C22×C4—3941no—
144C22×C4—3941no—
145C22×C4—1971no—
146C22×C4C22×D19711yescharacteristic
147C2×C8—21no—
148C2×C8—3941no—
149C4⋊C4—21no—
150C4⋊C4—7882no—
151C4⋊C4—3941no—
152C4⋊C4—3941no—
153C4⋊C4—3941no—
154C4⋊C4—3941no—
155C4⋊C4—3941no—
156C4⋊C4—3941no—
157C4⋊C4—1971no—
158C4⋊C4D78811yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—7882no—
161C22⋊C4—3941no—
162C22⋊C4—3941no—
163C22⋊C4—3941no—
164C22⋊C4—3941no—
165C22⋊C4—3941no—
166C23—7881no—
167C23—7881no—
168C23—3941no—
169C23—3941no—
170C23—1972no—
171C23D4×D19711yescharacteristic
172Q8—41no—
173Q8—7881no—
174Q8—15761no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—7881no—
182C2×C4—7881no—
183C2×C4—7881no—
184C2×C4—7881no—
185C2×C4—15761no—
186C2×C4—15761no—
187C2×C4—3941no—
188C2×C4—3941no—
189C2×C4—3941no—
190C2×C4D4×D19711yescharacteristic
191C2×C4C2×D78811yescharacteristic
192C8—41no—
193C8—7881no—
194C22—21no—
195C22—21no—
196C22—7882no—
197C22—7881no—
198C22—7881no—
199C22—15761no—
200C22—3942no—
201C22#2026329611yescharacteristic
202C22Q16×D19711yescharacteristic
203C22#2026335511yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—41no—
208C4—41no—
209C4—7881no—
210C4—7881no—
211C4—7881no—
212C4—15761no—
213C2—21no—
214C2—21no—
215C2—7882no—
216C2#2716123611yescharacteristic
217C2#2716124811yescharacteristic
218C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#328×C197C2
socleC22×C197#120
prime radicalC23×D197D4
hypercenter#328D197
hypocenterC197#1614
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #27161415C2semidirect
3. #27161412C2semidirect
5. #27160815C2semidirect
7. #328×C197C2semidirect
109. C197#1614semidirect
116. #328D197semidirect

Maximal subgroups

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

subgroupindex
#271617042
#271614152
#271614122
#271610162
#271608152
#271608022
#328×C1972
#1614197