Finite Group Databasebeta

← #28372614

Subgroups — #28372614

Order 13952.

520 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16812721C211yesmaximal, characteristic, complemented
2C22×C8×D109C211yesmaximal, characteristic, complemented
3#20295538C212yesmaximal, complemented
4#20295268C212yesmaximal, complemented
5#20295263C212yesmaximal, complemented
6#20295256C214yesmaximal, complemented
7#20295020C212yesmaximal, complemented
8#427×C109C211yesmaximal, characteristic, complemented
9C2×#13182029C2212yes—
10C22×C4×D109C2211yescharacteristic
11#16812721—24no—
12C2×Dic436C2211yescharacteristic, complemented
13C2×#13182018C2212yescomplemented
14C2×D872C2211yescharacteristic, complemented
15C22×#11724784C2211yescharacteristic
16C2×C8×D109—24no—
17C2×C8×D109C2212yes—
18C2×C8×D109C2212yes—
19#16812683—22no—
20#16812679C2214yes—
21#16812674C2214yes—
22#16812669—22no—
23C4×D436—24no—
24#16812622C2214yescomplemented
25#16812619C2214yescomplemented
26C22×C8×C109C2211yescharacteristic
27#117×C109C2212yes—
28#107×C109C2212yescomplemented
29#106×C109C2212yescomplemented
30C23×D109D411yescharacteristic
31#13182029—24no—
32#13182029—24no—
33C2×Dic218C2312yes—
34C2×D436—24no—
35C2×D436C2312yes—
36C2×C22⋊D109—22no—
37C22×Dic109D411yescharacteristic
38C2×C4×D109—22no—
39C2×C4×D109—22no—
40C2×C4×D109—22no—
41C2×C4×D109D412yes—
42C2×C4×D109C2312yes—
43Dic436—22no—
44#13182018—24no—
45D872—22no—
46C2×#11724784—22no—
47C2×#11724784C2312yes—
48C8×D109—24no—
49C8×D109—24no—
50C4⋊Dic109—24no—
51#13182006—24no—
52#13182006—24no—
53C22×C4×C109C2311yescharacteristic
54C2×C8×C109D414yescomplemented
55C2×C8×C109C2311yescharacteristic
56C2×C8×C109C2311yescharacteristic
57C4⋊C4×C109C2314yes—
58C22⋊C4×C109—22no—
59C42×C109—22no—
60C23×C109C2×D411yescharacteristic
61C22×D109—22no—
62C22×D109—24no—
63C22×D109—22no—
64C22×D109C2×D412yes—
65C2×C4×C109—24no—
66C2×C4×C109C2×D411yescharacteristic
67C2×C4×C109C2×D412yes—
68C2×C4×C109#4112yes—
69C2×C4×C109C2411yescharacteristic
70C22⋊D109—44no—
71C2×Dic109—22no—
72C2×Dic109—22no—
73C2×Dic109C2×D412yes—
74D436—22no—
75D436—22no—
76D436—48no—
77C4×D109—24no—
78C4×D109—22no—
79C4×D109—22no—
80C4×D109—44no—
81Dic218—22no—
82Dic218—22no—
83C8×C109C2×D412yes—
84C8×C109C2×D412yes—
85#11724784—22no—
86C22×C109—22no—
87C22×C109#12112yes—
88C22×C109C22×D411yescharacteristic
89C2×D109—28no—
90C2×D109—22no—
91C2×D109—22no—
92C2×D109—42no—
93C2×D109—44no—
94C2×D109#13514yescomplemented
95C4×C109—44no—
96C4×C109#12112yes—
97C4×C109C22×D411yescharacteristic
98C4×C109C2×#4111yescharacteristic
99Dic109—22no—
100Dic109—42no—
101C2×C109—22no—
102C2×C109C2×#12111yescharacteristic
103C2×C109C2×#13512yes—
104D109—24no—
105D109—82no—
106#3050—1091nomaximal, Sylow
107C109#305011yescharacteristic, complemented, Sylow
108C2×#135—1091no—
109C23×C8—1091no—
110#525—1092no—
111#467—1092no—
112#466—1092no—
113#465—1094no—
114#427D10911yescharacteristic, complemented
115#416—1092no—
116C2×#41—1092no—
117C23×C4—1091no—
118#135—2184no—
119C2×Q16—1091no—
120C2×SD16—1092no—
121C2×D8—1091no—
122C22×C8—1091no—
123C22×C8—1092no—
124C22×C8—1092no—
125C22×C8—2184no—
126C22×C8C2×D10911yescharacteristic
127#123—2182no—
128C22⋊Q8—1094no—
129#121—1094no—
130#120—2182no—
131C4×D4—2184no—
132#117C2×D10912yes—
133#107C2×D10912yescomplemented
134#106C2×D10912yescomplemented
135#103—1094no—
136#102—1094no—
137C24—1091no—
138#41—2184no—
139#41—2184no—
140C2×Q8—1092no—
141C2×D4—1092no—
142C2×D4—2184no—
143C2×D4—2182no—
144C22×C4—1091no—
145C22×C4—1092no—
146C22×C4—1092no—
147C22×C4—2182no—
148C22×C4—2182no—
149C22×C4—2182no—
150C22×C4C22×D10911yescharacteristic
151Q16—2182no—
152SD16—2184no—
153D8—2182no—
154C2×C8—1092no—
155C2×C8—2182no—
156C2×C8—2184no—
157C2×C8—2184no—
158C2×C8D43614yescomplemented
159C2×C8C22×D10911yescharacteristic
160C2×C8C22×D10911yescharacteristic
161C4⋊C4—2184no—
162C4⋊C4C22×D10914yes—
163C22⋊C4—22no—
164C22⋊C4—2184no—
165C22⋊C4—2184no—
166C42—22no—
167C23—1092no—
168C23—2184no—
169C23—2182no—
170C23—2182no—
171C23D4×D10911yescharacteristic
172Q8—2182no—
173Q8—2182no—
174D4—4368no—
175D4—4364no—
176D4—2182no—
177D4—2182no—
178C2×C4—24no—
179C2×C4—1092no—
180C2×C4—4364no—
181C2×C4—2182no—
182C2×C4—2182no—
183C2×C4—2184no—
184C2×C4—2182no—
185C2×C4—2182no—
186C2×C4D4×D10911yescharacteristic
187C2×C4C2×D43612yes—
188C2×C4#1318203012yes—
189C2×C4C23×D10911yescharacteristic
190C8—2182no—
191C8C2×D43612yes—
192C8C2×D43612yes—
193C22—22no—
194C22—1094no—
195C22—4362no—
196C22—4364no—
197C22—2188no—
198C22—2182no—
199C22—2182no—
200C22#1681267412yes—
201C22C2×D4×D10911yescharacteristic
202C4—44no—
203C4—4362no—
204C4—2182no—
205C4#1681267412yes—
206C4C22×D43611yescharacteristic
207C4C2×#1318203011yescharacteristic
208C2—22no—
209C2—8722no—
210C2—2184no—
211C2C2×#1681267411yescharacteristic
212C2#135×D10912yes—
subgroupG/H
centerC2×C4D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#427×C109C2
socleC22×C109C22×D4
hypercenter#427D109
hypocenterC109#3050
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. C2×#16812721C2semidirect
2. C22×C8×D109C2semidirect
3. #20295538C2semidirect
4. #20295268C2semidirect
5. #20295263C2semidirect
6. #20295256C2semidirect
7. #20295020C2semidirect
8. #427×C109C2semidirect
12. C2×Dic436C22semidirect
13. C2×#13182018C22semidirect
14. C2×D872C22semidirect
24. #16812622C22semidirect
25. #16812619C22semidirect
28. #107×C109C22semidirect
29. #106×C109C22semidirect
54. C2×C8×C109D4semidirect
94. C2×D109#135semidirect
107. C109#3050semidirect
114. #427D109semidirect
133. #107C2×D109semidirect
134. #106C2×D109semidirect
158. C2×C8D436semidirect

Maximal subgroups

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

subgroupindex
C2×#168127212
C22×C8×D1092
#202955382
#202952682
#202952632
#202952562
#202950202
#427×C1092
#3050109