Finite Group Databasebeta

← #28365750

Subgroups — #28365750

Order 13952.

516 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 197 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×#16812697C211yesmaximal, characteristic
2C2×Q8×Dic109C211yesmaximal, characteristic
3C2×#16812664C211yesmaximal, characteristic
4#20294932C211yesmaximal, characteristic
5#20294913C214yesmaximal
6#20294908C214yesmaximal
7#20294891C211yesmaximal, characteristic
8#20294872C211yesmaximal, characteristic
9#391×C109C211yesmaximal, characteristic
10C22×Dic218C2211yescharacteristic
11#16812697C418yescomplemented
12Q8×Dic109—24no—
13#16812664—24no—
14C2×C4⋊Dic109C2212yes—
15C2×C4⋊Dic109C2211yescharacteristic
16C2×#13182008C2212yes—
17C2×#13182008C2214yes—
18C2×C4×Dic109C2212yes—
19C2×C4×Dic109C2212yes—
20C2×C4×Dic109C2211yescharacteristic
21C2×C4×Dic109C2211yescharacteristic
22#16812606C2212yes—
23#16812606C2214yes—
24#16812606C2214yes—
25#16812606C2212yes—
26C22×Q8×C109C2211yescharacteristic
27C2×C4⋊C4×C109C2211yescharacteristic
28C2×C42×C109C2211yescharacteristic
29#95×C109C2214yes—
30C2×Dic218—22no—
31C2×Dic218—22no—
32C2×Dic218C2×C414yes—
33C22×Dic109C2312yes—
34C22×Dic109C2312yes—
35C22×Dic109C2312yes—
36C22×Dic109C2311yescharacteristic
37C22×Dic109C2311yescharacteristic
38C4⋊Dic109—22no—
39C4⋊Dic109—24no—
40#13182008—24no—
41#13182008C2×C4116yes—
42C4×Dic109—24no—
43C4×Dic109—24no—
44C4×Dic109C2×C414yes—
45C4×Dic109D414yes—
46C2×Q8×C109—22no—
47C2×Q8×C109—22no—
48C2×Q8×C109C2×C414yes—
49C22×C4×C109C2312yes—
50C22×C4×C109C2312yes—
51C22×C4×C109C2312yes—
52C22×C4×C109C2311yescharacteristic
53C4⋊C4×C109—22no—
54C42×C109—22no—
55C23×C109C2411yescharacteristic
56Q8×C109—28no—
57C2×C4×C109—22no—
58C2×C4×C109—22no—
59C2×C4×C109—22no—
60C2×C4×C109—22no—
61C2×C4×C109—24no—
62C2×C4×C109—22no—
63C2×C4×C109—22no—
64C2×C4×C109C22×C414yes—
65C2×C4×C109C22×C412yes—
66C2×C4×C109C2×D412yes—
67C2×C4×C109#4112yes—
68C2×Dic109—22no—
69C2×Dic109—22no—
70C2×Dic109—22no—
71C2×Dic109—22no—
72C2×Dic109—22no—
73C2×Dic109—22no—
74C2×Dic109—22no—
75C2×Dic109—24no—
76C2×Dic109C22×C414yes—
77C2×Dic109C22×C414yes—
78C2×Dic109C2×D412yes—
79C2×Dic109C2×D412yes—
80Dic218—28no—
81C22×C109C23×C411yescharacteristic
82C22×C109C22×D411yescharacteristic
83C22×C109C2×#4111yescharacteristic
84C22×C1092−1+412yes—
85C22×C1092−1+412yes—
86C4×C109—24no—
87C4×C109—42no—
88C4×C109—42no—
89C4×C109C4×D414yes—
90Dic109—24no—
91Dic109—24no—
92Dic109—22no—
93Dic109—22no—
94Dic109—42no—
95C2×C109C2×C4×D411yescharacteristic
96C2×C109#51912yes—
97C2×C109#53611yescharacteristic
98C2×C109#53611yescharacteristic
99C2×C109#54112yes—
100#2327—1091nomaximal, Sylow
101C109#232711yescharacteristic, complemented, Sylow
102C2×C4⋊Q8—1091no—
103C2×C4×Q8—1091no—
104C2×C4×Q8—1091no—
105#391—1091no—
106#391D10911yescharacteristic
107#387—1094no—
108#387—1094no—
109#382—1091no—
110#376—1091no—
111C22×Q8—1091no—
112C22×Q8C2×D10911yescharacteristic
113C4⋊Q8—1098no—
114C4×Q8—2184no—
115C4×Q8—2184no—
116C2×C4⋊C4—1094no—
117C2×C4⋊C4—1092no—
118C2×C4⋊C4—1091no—
119C2×C4⋊C4—1092no—
120C2×C4⋊C4C2×D10911yescharacteristic
121C2×C42—1091no—
122C2×C42—1091no—
123C2×C42—1092no—
124C2×C42—1092no—
125C2×C42C2×D10911yescharacteristic
126#95—1092no—
127#95—1092no—
128#95—1094no—
129#95—1094no—
130#95C2×D10914yes—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—1094no—
134C2×Q8—2182no—
135C2×Q8—2182no—
136C2×Q8C4×D10914yes—
137C22×C4—1092no—
138C22×C4—1091no—
139C22×C4—1091no—
140C22×C4—1092no—
141C22×C4—1092no—
142C22×C4C22×D10912yes—
143C22×C4C22×D10912yes—
144C22×C4C22×D10912yes—
145C22×C4C22×D10911yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—10916no—
148C4⋊C4—2184no—
149C4⋊C4—2184no—
150C4⋊C4—2182no—
151C42—22no—
152C42—1094no—
153C42—1094no—
154C42—2184no—
155C42—2184no—
156C23C23×D10911yescharacteristic
157Q8—28no—
158Q8—2188no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—1094no—
167C2×C4—1094no—
168C2×C4—1092no—
169C2×C4—1092no—
170C2×C4—2182no—
171C2×C4—2184no—
172C2×C4—2182no—
173C2×C4—2182no—
174C2×C4—2182no—
175C2×C4—2182no—
176C2×C4—2182no—
177C2×C4—2182no—
178C2×C4C2×C4×D10914yes—
179C2×C4C2×C4×D10912yes—
180C2×C4D4×D10912yes—
181C2×C4#1318202812yes—
182C22C22×C4×D10911yescharacteristic
183C22C2×D4×D10911yescharacteristic
184C22C2×#1318202811yescharacteristic
185C22#1681274612yes—
186C22#1681274712yes—
187C4—24no—
188C4—42no—
189C4—42no—
190C4—4362no—
191C4—2184no—
192C4—2184no—
193C4—2182no—
194C4—2182no—
195C4#1681265914yes—
196C2C2×#1681265911yescharacteristic
197C2#2029550212yes—
198C2#2029559611yescharacteristic
199C2#2029560111yescharacteristic
200C2#2029563612yes—
subgroupG/H
center, FrattiniC23C23×D109
commutatorC22×C109C23×C4
Fitting#391×C109C2
socle, prime radical, cosocleC23×C109C24
hypercenter#391D109
hypocenterC109#2327
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #16812697C4semidirect
101. C109#2327semidirect

Maximal subgroups

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

subgroupindex
C2×#168126972
C2×Q8×Dic1092
C2×#168126642
#202949322
#202949132
#202949082
#202948912
#202948722
#391×C1092
#2327109