Finite Group Databasebeta

← #28379295

Subgroups — #28379295

Order 13952.

234 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 52 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#20296013C211yesmaximal, characteristic
2#20296006C211yesmaximal, characteristic
3#20295929C212yesmaximal, complemented
4#20295928C212yesmaximal, complemented
5#20295598C211yesmaximal, characteristic
6C2×#13182038—21no—
7#16812778—21no—
8#16812760—21no—
9#16812759C2212yes—
10#16812757—21no—
11#16812757C2212yes—
12C2×#13182028C2211yescharacteristic
13C22×Dic218C411yescharacteristic, complemented
14#16812698—21no—
15#16812685—21no—
16#16812683C2211yescharacteristic
17#16812682C2211yescharacteristic
18#16812680C411yescharacteristic, complemented
19#16812677—21no—
20#16812676—21no—
21#16812655C411yescharacteristic, complemented
22#123×C109C411yescharacteristic, complemented
23#13182038—22no—
24#13182038—41no—
25C2×#11724786—41no—
26C4⋊#89137—41no—
27C22⋊#89137—22no—
28C4×#89137—41no—
29#13182028—41no—
30#13182028—41no—
31C2×Dic218—21no—
32C2×Dic218—21no—
33C2×Dic218—41no—
34C2×Dic218—41no—
35C2×Dic218D411yescharacteristic
36C2×C22⋊D109D411yescharacteristic
37C2×C22⋊D109C2311yescharacteristic
38C22×Dic109C2×C411yescharacteristic
39C22×Dic109C2×C411yescharacteristic
40C22×Dic109D411yescharacteristic
41C2×C4×D109D411yescharacteristic
42C4⋊Dic109—21no—
43#13182008—21no—
44#13182008—21no—
45#13182008—21no—
46C22⋊Dic109—21no—
47C22⋊Dic109C2×C411yescharacteristic
48#13182006—21no—
49#13182006—21no—
50C4×Dic109—21no—
51C2×D4×C109C2×C411yescharacteristic
52C22×C4×C109C2×C411yescharacteristic
53C4⋊C4×C109—21no—
54C22⋊C4×C109—21no—
55C22⋊C4×C109C2×C411yescharacteristic
56C23×C109C22⋊C411yescharacteristic
57C23×C109C22×C411yescharacteristic
58C22×D109C2×D411yescharacteristic
59C2×#89137—42no—
60D4×C109—41no—
61C2×C4×C109—21no—
62C2×C4×C109—21no—
63C2×C4×C109—41no—
64C2×C4×C109C22⋊C411yescharacteristic
65C22⋊D109—41no—
66C22⋊D109—41no—
67C2×Dic109—21no—
68C2×Dic109—21no—
69C2×Dic109—21no—
70C2×Dic109—21no—
71C2×Dic109—21no—
72C2×Dic109—21no—
73C2×Dic109—41no—
74C2×Dic109C22⋊C411yescharacteristic
75C2×Dic109C22⋊C411yescharacteristic
76C2×Dic109C2×D411yescharacteristic
77C4×D109—21no—
78C4×D109—21no—
79Dic218—41no—
80Dic218—42no—
81Dic218—41no—
82Dic218—41no—
83#11724786—42no—
84C22×C109—21no—
85C22×C109—21no—
86C22×C109—41no—
87C22×C109—41no—
88C22×C109C2×C22⋊C411yescharacteristic
89C2×D109—21no—
90C2×D109—41no—
91#89137—82no—
92C4×C109—21no—
93C4×C109—41no—
94C4×C109—41no—
95Dic109—21no—
96Dic109—21no—
97Dic109—41no—
98Dic109—41no—
99Dic109#9911yescharacteristic
100Dic109#9911yescharacteristic
101C2×C109—21no—
102C2×C109—41no—
103C2×C109—41no—
104C2×C109C2×#9911yescharacteristic
105D109—41no—
106#2123—1091nomaximal, Sylow
107C109#212311yescharacteristic, complemented, Sylow
108#536—1091no—
109#413—1091no—
110#410—1091no—
111#352—1092no—
112#352—1092no—
113C2×#41—1091no—
114C22×Q8—1091no—
115C2×M16—2181no—
116C4⋊Q8—2181no—
117#124—2181no—
118#123—1091no—
119#123—1091no—
120#123—1091no—
121#123#8913711yescharacteristic, complemented
122C22⋊Q8—2181no—
123C22⋊Q8—2181no—
124#117—1091no—
125#117—2181no—
126#101—2181no—
127#100—1092no—
128#99—1092no—
129#99—2181no—
130#41—4361no—
131#41—4361no—
132C2×Q8—1091no—
133C2×Q8—4361no—
134C2×Q8—4361no—
135C2×Q8—2181no—
136C2×Q8—2181no—
137C2×D4—1091no—
138C2×D4—1091no—
139C2×D4C2×#8913711yescharacteristic
140C22×C4—1091no—
141C22×C4—1091no—
142C22×C4—1091no—
143C22×C4—1091no—
144C22×C4C2×#8913711yescharacteristic
145M16—4361no—
146M16—2182no—
147C2×C8—4361no—
148C4⋊C4—21no—
149C4⋊C4—4361no—
150C4⋊C4—2181no—
151C4⋊C4—2181no—
152C4⋊C4—2181no—
153C4⋊C4—2181no—
154C22⋊C4—21no—
155C22⋊C4—1091no—
156C22⋊C4—2181no—
157C22⋊C4—2181no—
158C22⋊C4—2182no—
159C22⋊C4—2181no—
160C22⋊C4C2×#8913711yescharacteristic
161C42—4361no—
162C42—2181no—
163C23—1091no—
164C23C22⋊#8913711yescharacteristic
165C23C22×#8913711yescharacteristic
166Q8—4361no—
167Q8—4361no—
168Q8—4361no—
169Q8—4362no—
170D4—41no—
171D4—4361no—
172D4—4361no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—1091no—
177C2×C4—1091no—
178C2×C4—1091no—
179C2×C4—4362no—
180C2×C4—4361no—
181C2×C4—2181no—
182C2×C4—2181no—
183C2×C4—2181no—
184C2×C4—2181no—
185C2×C4—2181no—
186C2×C4—2181no—
187C2×C4—2181no—
188C2×C4—2181no—
189C2×C4C22⋊#8913711yescharacteristic
190C8—4362no—
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—4361no—
196C22—2181no—
197C22C2×C22⋊#8913711yescharacteristic
198C4—21no—
199C4—41no—
200C4—41no—
201C4—8722no—
202C4—1091no—
203C4—1091no—
204C4—4361no—
205C4—4361no—
206C4—2181no—
207C4—2181no—
208C2—21no—
209C2—41no—
210C2—41no—
211C2—4361no—
212C2D109⋊#9911yescharacteristic
subgroupG/H
centerC2D109⋊#99
commutatorC23×C109C22×C4
FrattiniC23C22×#89137
Fitting#123×C109C4
socleC2×C109C2×#99
prime radicalC2×#13182028C22
cosocleC2×C22⋊D109C23
hypercenter#123#89137
hypocenterC109#2123
LCSG ▷ C23×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #20295929C2semidirect
4. #20295928C2semidirect
13. C22×Dic218C4semidirect
18. #16812680C4semidirect
21. #16812655C4semidirect
22. #123×C109C4semidirect
107. C109#2123semidirect
121. #123#89137semidirect

Maximal subgroups

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

subgroupindex
#202960132
#202960062
#202959292
#202959282
#202955982
#2123109