Finite Group Databasebeta

← #28362357

Subgroups — #28362357

Order 13952.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#20295770C211yesmaximal, characteristic
2#20295395C211yesmaximal, characteristic, complemented
3#20295386C211yesmaximal, characteristic, complemented
4#20295385C211yesmaximal, characteristic, complemented
5#20294767C211yesmaximal, characteristic, complemented
6#20294765C211yesmaximal, characteristic, complemented
7#329×C109C211yesmaximal, characteristic, complemented
8#16812699C2211yescharacteristic
9#16812691C2211yescharacteristic
10#16812687C2211yescharacteristic
11#16812683—21no—
12#16812681—21no—
13#16812676—21no—
14#16812675—21no—
15#16812672—21no—
16#16812671—21no—
17#16812624—21no—
18#16812624—21no—
19#16812624—21no—
20#16812623—21no—
21#16812621—21no—
22#16812620—21no—
23#16812610C2211yescharacteristic
24#124×C109C2211yescharacteristic
25#97×C109C2211yescharacteristic
26#97×C109C2211yescharacteristic
27C2×Dic218—21no—
28C2×Dic218D411yescharacteristic
29C2×D436D411yescharacteristic
30C2×C22⋊D109—21no—
31C2×C22⋊D109—21no—
32C22×Dic109—21no—
33C2×C4×D109—21no—
34C2×#11724784—21no—
35C4⋊Dic109—21no—
36C4⋊Dic109—21no—
37C4⋊Dic109—21no—
38C4⋊Dic109D411yescharacteristic
39C4⋊Dic109D411yescharacteristic
40#13182008—21no—
41#13182008—21no—
42#13182008—21no—
43C22⋊Dic109—21no—
44#13182006—21no—
45#13182006—21no—
46#13182006—21no—
47C2×Q8×C109D411yescharacteristic
48C2×D4×C109D411yescharacteristic
49C2×C8×C109—21no—
50C2×C8×C109—21no—
51C22⋊C4×C109—21no—
52C22⋊C4×C109—21no—
53C42×C109C2311yescharacteristic
54C23×C109—21no—
55C22×D109—21no—
56Q8×C109—41no—
57D4×C109—41no—
58C2×C4×C109—21no—
59C2×C4×C109C2×D411yescharacteristic
60C2×C4×C109C2×D411yescharacteristic
61C2×C4×C109C2×D411yescharacteristic
62C22⋊D109—81no—
63C22⋊D109—81no—
64C2×Dic109—21no—
65C2×Dic109—21no—
66C2×Dic109—21no—
67C2×Dic109—41no—
68C2×Dic109—81no—
69D436—41no—
70C4×D109—81no—
71Dic218—41no—
72Dic218—42no—
73C8×C109—41no—
74C8×C109—41no—
75#11724784—41no—
76C22×C109—41no—
77C22×C109—41no—
78C22×C109—41no—
79C22×C109#12011yescharacteristic
80C2×D109—41no—
81C2×D109—41no—
82C2×D109—41no—
83C4×C109—21no—
84C4×C109—21no—
85C4×C109—21no—
86C4×C109—41no—
87Dic109—41no—
88Dic109—41no—
89Dic109—41no—
90Dic109—81no—
91C2×C109—81no—
92C2×C109#45411yescharacteristic
93C2×C109#45511yescharacteristic
94C2×C109#45511yescharacteristic
95D109—81no—
96#1662—1091nomaximal, Sylow
97C109#166211yescharacteristic, complemented, Sylow
98#562—1091no—
99#491—1091no—
100#489—1091no—
101#489—1091no—
102#330—1091no—
103#329—1091no—
104#329D10911yescharacteristic, complemented
105C4⋊Q8—1091no—
106#125—1091no—
107#124—1091no—
108#124C2×D10911yescharacteristic
109#123—2181no—
110#123—2181no—
111C22⋊Q8—2181no—
112C22⋊Q8—2181no—
113#121—2181no—
114#121—2181no—
115#103—2181no—
116#103—2181no—
117#103—2181no—
118#103—2181no—
119#102—2181no—
120#102—2181no—
121#97—1091no—
122#97C2×D10911yescharacteristic
123#97C2×D10911yescharacteristic
124C2×Q8—1091no—
125C2×Q8—2181no—
126C2×Q8D43611yescharacteristic
127C2×D4—1091no—
128C2×D4—2181no—
129C2×D4—2181no—
130C2×D4D43611yescharacteristic
131C22×C4—2181no—
132C22×C4—2181no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—2181no—
136C4⋊C4—1091no—
137C4⋊C4—1091no—
138C4⋊C4—2181no—
139C4⋊C4—2181no—
140C4⋊C4—2181no—
141C4⋊C4—2181no—
142C4⋊C4—2181no—
143C4⋊C4—2181no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—2181no—
147C22⋊C4—2181no—
148C22⋊C4—2181no—
149C22⋊C4—2181no—
150C42C22×D10911yescharacteristic
151C23—21no—
152C23—2181no—
153Q8—41no—
154Q8—4362no—
155Q8—4361no—
156D4—41no—
157D4—8721no—
158D4—8721no—
159D4—4361no—
160C2×C4—21no—
161C2×C4—8721no—
162C2×C4—8721no—
163C2×C4—4361no—
164C2×C4—2181no—
165C2×C4—2181no—
166C2×C4—2181no—
167C2×C4D4×D10911yescharacteristic
168C2×C4D4×D10911yescharacteristic
169C2×C4C2×D43611yescharacteristic
170C8—41no—
171C8—41no—
172C8—4361no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—4361no—
177C22—4361no—
178C22—4361no—
179C22#1681266911yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—41no—
184C4—8721no—
185C4—4361no—
186C4—4361no—
187C4—4361no—
188C2—81no—
189C2—8721no—
190C2#2029520011yescharacteristic
191C2#2029520511yescharacteristic
192C2#2029520611yescharacteristic
subgroupG/H
centerC22#16812669
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#329×C109C2
socleC22×C109#120
prime radical#20295770C2
hypercenter#329D109
hypocenterC109#1662
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20295395C2semidirect
3. #20295386C2semidirect
4. #20295385C2semidirect
5. #20294767C2semidirect
6. #20294765C2semidirect
7. #329×C109C2semidirect
97. C109#1662semidirect
104. #329D109semidirect

Maximal subgroups

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

subgroupindex
#202957702
#202953952
#202953862
#202953852
#202947672
#202947652
#329×C1092
#1662109