Finite Group Databasebeta

← #28379167

Subgroups — #28379167

Order 13952.

360 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 74 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#20296031C211yesmaximal, characteristic, complemented
2C2×#16812764C212yesmaximal, complemented
3C2×#16812763C212yesmaximal, complemented
4C42×#89137C211yesmaximal, characteristic, complemented
5#20295575C211yesmaximal, characteristic
6C2×#13182039C2212yes—
7C2×C4×#89137—22no—
8C2×C4×#89137C2212yes—
9#16812766—22no—
10#16812764—24no—
11#16812763—24no—
12C2×#13182030C2211yescharacteristic
13C2×#13182028C2211yescharacteristic
14#16812697C411yescharacteristic, complemented
15#16812695C411yescharacteristic, complemented
16#16812687C411yescharacteristic, complemented
17#16812672—22no—
18#16812659—22no—
19C42×D109C2211yescharacteristic
20#124×C109C411yescharacteristic, complemented
21C22×#89137—22no—
22#13182039—22no—
23#13182039—22no—
24C2×#11724786—22no—
25C4×#89137—28no—
26C4×#89137—22no—
27C4×#89137D414yescomplemented
28#13182030—22no—
29#13182030—22no—
30#13182028—22no—
31#13182028—22no—
32C2×Dic218C2×C411yescharacteristic
33C2×D436—21no—
34C2×D436C2×C411yescharacteristic
35C2×C22⋊D109—21no—
36C2×C22⋊D109—22no—
37C22×Dic109—21no—
38C2×C4×D109—21no—
39C2×C4×D109D412yes—
40C2×C4×D109C2311yescharacteristic
41#13182008—22no—
42#13182006—22no—
43C4×Dic109C2×C411yescharacteristic
44C4×Dic109D412yes—
45C2×Q8×C109C2×C411yescharacteristic
46C2×D4×C109C2×C411yescharacteristic
47C22⋊C4×C109—22no—
48C42×C109C2×C411yescharacteristic
49C23×C109—21no—
50C22×D109—21no—
51C22×D109C2×D411yescharacteristic
52C2×#89137—28no—
53C2×#89137—24no—
54Q8×C109—22no—
55D4×C109—22no—
56C2×C4×C109—21no—
57C2×C4×C109C22⋊C412yes—
58C2×C4×C109C22×C411yescharacteristic
59C22⋊D109—42no—
60C22⋊D109—44no—
61C2×Dic109—21no—
62C2×Dic109—41no—
63C2×Dic109—41no—
64C2×Dic109C22⋊C412yes—
65C2×Dic109C2×D411yescharacteristic
66D436—22no—
67D436—42no—
68C4×D109—24no—
69C4×D109—41no—
70C4×D109—41no—
71C4×D109C2×D412yes—
72C4×D109#4112yes—
73Dic218—22no—
74#11724786—42no—
75C22×C109—41no—
76C22×C109—42no—
77C22×C109C2×C22⋊C411yescharacteristic
78C2×D109—22no—
79C2×D109—41no—
80C2×D109—42no—
81C2×D109#12112yes—
82#89137—28no—
83C4×C109—22no—
84C4×C109—41no—
85C4×C109C4×D412yes—
86Dic109—41no—
87Dic109#10414yescomplemented
88Dic109#12411yescharacteristic
89Dic109#12711yescharacteristic
90C2×C109—81no—
91C2×C109#39011yescharacteristic
92C2×C109C2×#10412yes—
93D109—22no—
94D109—81no—
95#1962—1091nomaximal, Sylow
96C109#196211yescharacteristic, complemented, Sylow
97#532—1091no—
98#423—1091no—
99C2×#104—1092no—
100C2×#104—1092no—
101C43—1091no—
102C2×#41—1091no—
103C2×#41—1091no—
104C2×M16—1092no—
105C4⋊Q8—1091no—
106#127—1091no—
107#124—1091no—
108#124#8913711yescharacteristic, complemented
109#121—2182no—
110C4×D4—2182no—
111C2×C42—1091no—
112C2×C42—1092no—
113C2×C42—2182no—
114C4⋊C8—2182no—
115#104—2184no—
116#104—2184no—
117#41—2182no—
118#41—2182no—
119#41—2182no—
120#41—2182no—
121C2×Q8—1091no—
122C2×Q8C2×#8913711yescharacteristic
123C2×D4—1091no—
124C2×D4—2181no—
125C2×D4—2182no—
126C2×D4—2181no—
127C2×D4C2×#8913711yescharacteristic
128C22×C4—1091no—
129C22×C4—1092no—
130C22×C4—2181no—
131C22×C4—2181no—
132C22×C4—2182no—
133M16—2182no—
134M16—2182no—
135C2×C8—2182no—
136C4⋊C4—2182no—
137C22⋊C4—22no—
138C22⋊C4—2182no—
139C42—1092no—
140C42—1091no—
141C42—1094no—
142C42—2182no—
143C42—2188no—
144C42C2×#8913711yescharacteristic
145C23—21no—
146C23—1091no—
147C23—2181no—
148Q8—22no—
149Q8—2182no—
150D4—22no—
151D4—4362no—
152D4—4364no—
153D4—4362no—
154D4—2182no—
155C2×C4—21no—
156C2×C4—1092no—
157C2×C4—1091no—
158C2×C4—1092no—
159C2×C4—1092no—
160C2×C4—4361no—
161C2×C4—4361no—
162C2×C4—4361no—
163C2×C4—4361no—
164C2×C4—2184no—
165C2×C4—2181no—
166C2×C4—2184no—
167C2×C4—2188no—
168C2×C4C22⋊#8913712yes—
169C2×C4C22×#8913711yescharacteristic
170C8—4362no—
171C22—41no—
172C22—42no—
173C22—1092no—
174C22—4361no—
175C22—4362no—
176C22—2182no—
177C22C2×C22⋊#8913711yescharacteristic
178C4—22no—
179C4—41no—
180C4—1091no—
181C4—1091no—
182C4—1094no—
183C4—4361no—
184C4—2188no—
185C4D4×#8913712yes—
186C2—81no—
187C2—8721no—
188C2—2182no—
189C2#2029597811yescharacteristic
190C2D109⋊#10412yes—
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
Fitting#124×C109C4
socleC22×C109C2×C22⋊C4
prime radical#20295575C2
cosocleC2×C4×D109C23
hypercenter#124#89137
hypocenterC109#1962
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20296031C2semidirect
2. C2×#16812764C2semidirect
3. C2×#16812763C2semidirect
4. C42×#89137C2semidirect
14. #16812697C4semidirect
15. #16812695C4semidirect
16. #16812687C4semidirect
20. #124×C109C4semidirect
27. C4×#89137D4semidirect
87. Dic109#104semidirect
96. C109#1962semidirect
108. #124#89137semidirect

Maximal subgroups

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

subgroupindex
#202960312
C2×#168127642
C2×#168127632
C42×#891372
#202955752
#1962109