Finite Group Databasebeta

← #28364074

Subgroups — #28364074

Order 13952.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16812724C212yesmaximal, complemented
2C2×#16812626C212yesmaximal, complemented
3#20294927C211yesmaximal, characteristic, complemented
4#20294795C211yesmaximal, characteristic, complemented
5#537×C109C211yesmaximal, characteristic, complemented
6C22×D436C2211yescharacteristic
7#16812724—24no—
8#16812724—24no—
9C2×#13182016—22no—
10C2×#13182014—22no—
11C2×#13182013C2212yes—
12C2×#13182006—22no—
13C2×C4×Dic109C2211yescharacteristic
14#16812626—24no—
15C2×#41×C109C2212yes—
16C22⋊Q8×C109—22no—
17#121×C109—22no—
18#121×C109—22no—
19C2×C4⋊C4×C109C2211yescharacteristic
20C23×D109—21no—
21C2×D436—22no—
22C2×D436—22no—
23C2×D436—21no—
24C2×D436D412yescomplemented
25C22×Dic109—21no—
26#13182016—28no—
27#13182014—28no—
28#13182013—22no—
29#13182013D414yescomplemented
30C2×#11724784—22no—
31#13182006—44no—
32C4×Dic109—22no—
33#41×C109—22no—
34#41×C109—22no—
35#41×C109—22no—
36#41×C109—22no—
37C2×Q8×C109D412yes—
38C2×D4×C109—22no—
39C2×D4×C109—22no—
40C2×D4×C109D412yes—
41C22×C4×C109—22no—
42C22×C4×C109D412yes—
43C22×C4×C109C2311yescharacteristic
44C4⋊C4×C109—22no—
45C4⋊C4×C109—22no—
46C22⋊C4×C109—24no—
47C23×C109—22no—
48C23×C109C2×D411yescharacteristic
49C22×D109—22no—
50C22×D109—42no—
51C22×D109—42no—
52C22×D109—41no—
53C22×D109—41no—
54Q8×C109—22no—
55Q8×C109—22no—
56D4×C109—22no—
57D4×C109—22no—
58D4×C109—42no—
59D4×C109—42no—
60D4×C109—82no—
61C2×C4×C109—22no—
62C2×C4×C109—22no—
63C2×C4×C109—42no—
64C2×C4×C109—42no—
65C2×C4×C109—42no—
66C2×C4×C109C2×D411yescharacteristic
67C2×C4×C109C2×D412yes—
68C2×C4×C109C2×D412yes—
69C2×C4×C109#4111yescharacteristic
70C2×Dic109—22no—
71C2×Dic109—41no—
72C2×Dic109—41no—
73D436—24no—
74D436—42no—
75#11724784—24no—
76C22×C109—21no—
77C22×C109—21no—
78C22×C109—42no—
79C22×C109—42no—
80C22×C109—42no—
81C22×C109#12011yescharacteristic
82C22×C109#12111yescharacteristic
83C22×C109#12711yescharacteristic
84C2×D109—44no—
85C2×D109—42no—
86C2×D109—82no—
87C2×D109—81no—
88C2×D109—81no—
89C4×C109—42no—
90C4×C109—42no—
91C4×C109#12012yes—
92C4×C109#12112yes—
93Dic109—42no—
94C2×C109—21no—
95C2×C109—21no—
96C2×C109—82no—
97C2×C109#39211yescharacteristic
98C2×C109#45311yescharacteristic
99C2×C109#45411yescharacteristic
100D109—82no—
101#2006—1091nomaximal, Sylow
102C109#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1092no—
104#537D10911yescharacteristic, complemented
105C2×#104—1092no—
106#390—1091no—
107#343—1091no—
108C2×#41C2×D10912yes—
109C22×D4—1091no—
110AGL(1,ℤ/8)—2184no—
111AGL(1,ℤ/8)—2184no—
112C2×SD16—2182no—
113C2×D8—2182no—
114C2×M16—1092no—
115C22⋊Q8—22no—
116#121—22no—
117#121—22no—
118C2×C4⋊C4C2×D10911yescharacteristic
119C2×C22⋊C4—2182no—
120C2×C42—1091no—
121#104—2184no—
122C24—2181no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—22no—
127C2×Q8C22⋊D10912yes—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—1092no—
131C2×D4—2182no—
132C2×D4—2182no—
133C2×D4—2181no—
134C2×D4C22⋊D10912yes—
135C22×C4—22no—
136C22×C4—2181no—
137C22×C4C22⋊D10912yes—
138C22×C4C22×D10911yescharacteristic
139SD16—2188no—
140D8—2188no—
141M16—1094no—
142M16—2182no—
143C2×C8—2182no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C22⋊C4—24no—
147C22⋊C4—4364no—
148C42—2182no—
149C23—22no—
150C23—4362no—
151C23—4362no—
152C23—4361no—
153C23—4361no—
154C23—2182no—
155C23C2×C22⋊D10911yescharacteristic
156Q8—22no—
157Q8—22no—
158D4—22no—
159D4—22no—
160D4—42no—
161D4—42no—
162D4—82no—
163D4—4362no—
164D4—2184no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—4361no—
171C2×C4—4361no—
172C2×C4—2182no—
173C2×C4D4×D10911yescharacteristic
174C2×C4D4×D10912yes—
175C2×C4C2×C22⋊D10912yes—
176C2×C4#1318202811yescharacteristic
177C8—2184no—
178C22—21no—
179C22—21no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—8722no—
184C22—8721no—
185C22—8721no—
186C22—4364no—
187C22—4362no—
188C22#1681266811yescharacteristic
189C22#1681267011yescharacteristic
190C22#1681269511yescharacteristic
191C4—42no—
192C4—42no—
193C4—4362no—
194C4#1681266712yes—
195C4#1681267112yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—8722no—
200C2#2029493411yescharacteristic
201C2#2029519411yescharacteristic
202C2#2029519911yescharacteristic
subgroupG/H
centerC22#16812668
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
Fitting#537×C109C2
socleC22×C109#120
hypercenter#537D109
hypocenterC109#2006
LCSG ▷ C22×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#16812724C2semidirect
2. C2×#16812626C2semidirect
3. #20294927C2semidirect
4. #20294795C2semidirect
5. #537×C109C2semidirect
24. C2×D436D4semidirect
29. #13182013D4semidirect
102. C109#2006semidirect
104. #537D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127242
C2×#168126262
#202949272
#202947952
#537×C1092
#2006109