Finite Group Databasebeta

← #28362017

Subgroups — #28362017

Order 13952.

280 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 53 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#20295588C211yesmaximal, characteristic
2#20295339C212yesmaximal, complemented
3#20294759C212yesmaximal, complemented
4#20294750C211yesmaximal, characteristic, complemented
5#397×C109C211yesmaximal, characteristic, complemented
6C22×C22⋊D109—21no—
7#16812683—22no—
8#16812673—21no—
9#16812673C2212yes—
10#16812669—22no—
11#16812632—22no—
12#16812619—22no—
13C22⋊#11724784C2212yescomplemented
14C2×C4⋊C4×C109C2211yescharacteristic, complemented
15C2×C22⋊C4×C109—21no—
16C2×C22⋊C4×C109C2211yescharacteristic
17#95×C109—21no—
18#95×C109C2211yescharacteristic, complemented
19C23×D109—21no—
20C2×D436—21no—
21C2×D436D412yes—
22C2×C22⋊D109—22no—
23C2×C22⋊D109—22no—
24C2×C22⋊D109—42no—
25C2×C22⋊D109—42no—
26C22×Dic109—21no—
27C2×#11724784—22no—
28C4⋊Dic109—21no—
29C4⋊Dic109D412yes—
30#13182006—22no—
31#13182006—22no—
32C24×C109D411yescharacteristic
33C22×C4×C109—21no—
34C22×C4×C109—21no—
35C22×C4×C109D411yescharacteristic
36C22×C4×C109C2311yescharacteristic
37C4⋊C4×C109—22no—
38C22⋊C4×C109—22no—
39C22⋊C4×C109—42no—
40C23×C109—21no—
41C23×C109—21no—
42C23×C109—41no—
43C23×C109—41no—
44C23×C109D812yes—
45C23×C109C2×D411yescharacteristic
46C22×D109—22no—
47C22×D109—42no—
48C22×D109—81no—
49C22×D109—81no—
50C2×C4×C109—21no—
51C2×C4×C109—41no—
52C2×C4×C109—41no—
53C2×C4×C109—41no—
54C2×C4×C109—41no—
55C2×C4×C109—41no—
56C2×C4×C109—41no—
57C2×C4×C109—41no—
58C2×C4×C109—41no—
59C2×C4×C109C2×D412yes—
60C22⋊D109—44no—
61C22⋊D109—42no—
62C22⋊D109—42no—
63C2×Dic109—22no—
64C2×Dic109—42no—
65D436—42no—
66D436—81no—
67#11724784—42no—
68C22×C109—21no—
69C22×C109—21no—
70C22×C109—22no—
71C22×C109—42no—
72C22×C109—41no—
73C22×C109—41no—
74C22×C109—81no—
75C22×C109#12011yescharacteristic
76C22×C109C2×D811yescharacteristic
77C22×C109AGL(1,ℤ/8)11yescharacteristic
78C2×D109—42no—
79C2×D109—42no—
80C2×D109—42no—
81C2×D109—84no—
82C4×C109—22no—
83C4×C109—41no—
84C4×C109—81no—
85C4×C109—81no—
86Dic109—42no—
87C2×C109—21no—
88C2×C109—21no—
89C2×C109—42no—
90C2×C109#44711yescharacteristic
91C2×C109#45511yescharacteristic
92C2×C109#45811yescharacteristic
93D109—82no—
94#1607—1091nomaximal, Sylow
95C109#160711yescharacteristic, complemented, Sylow
96#534—1091no—
97#480—1092no—
98#397D10911yescharacteristic, complemented
99#327—1092no—
100#323—1091no—
101C22×D4—2181no—
102#123—2182no—
103#121—1092no—
104#121—2181no—
105#120—2182no—
106C2×C4⋊C4C2×D10911yescharacteristic, complemented
107C2×C22⋊C4—21no—
108C2×C22⋊C4C2×D10911yescharacteristic
109#107—2182no—
110#102—2182no—
111C22⋊C8—1092no—
112#95—21no—
113#95C2×D10911yescharacteristic, complemented
114C24—2181no—
115C24C22⋊D10911yescharacteristic
116C2×D4—1092no—
117C2×D4—4362no—
118C2×D4—4362no—
119C2×D4—2181no—
120C2×D4—2182no—
121C2×D4—2182no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—2181no—
125C22×C4C22⋊D10911yescharacteristic
126C22×C4C22×D10911yescharacteristic
127C2×C8—2182no—
128C4⋊C4—22no—
129C4⋊C4—1092no—
130C4⋊C4—2181no—
131C22⋊C4—22no—
132C22⋊C4—42no—
133C22⋊C4—2182no—
134C22⋊C4—2182no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—8721no—
140C23—8721no—
141C23—4362no—
142C23—2182no—
143C23#1318201412yes—
144C23C2×C22⋊D10911yescharacteristic
145D4—8721no—
146D4—4362no—
147D4—4364no—
148D4—4362no—
149D4—4362no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—4362no—
160C2×C4—2182no—
161C2×C4D4×D10912yes—
162C8—4362no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—8724no—
171C22—4362no—
172C22—4362no—
173C22—4362no—
174C22#1681266711yescharacteristic
175C22C2×#1318201411yescharacteristic
176C22#1681272411yescharacteristic
177C4—22no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—4362no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—8722no—
186C2#2029515911yescharacteristic
187C2#2029520311yescharacteristic
188C2#2029521511yescharacteristic
subgroupG/H
centerC22#16812667
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
Fitting#397×C109C2
socleC22×C109#120
prime radical#20295588C2
hypercenter#397D109
hypocenterC109#1607
LCSG ▷ C22×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #20295339C2semidirect
3. #20294759C2semidirect
4. #20294750C2semidirect
5. #397×C109C2semidirect
13. C22⋊#11724784C22semidirect
14. C2×C4⋊C4×C109C22semidirect
18. #95×C109C22semidirect
95. C109#1607semidirect
98. #397D109semidirect
106. C2×C4⋊C4C2×D109semidirect
113. #95C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202955882
#202953392
#202947592
#202947502
#397×C1092
#1607109