Finite Group Databasebeta

← #28378459

Subgroups — #28378459

Order 13952.

284 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 58 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
1D109⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
2#20295969C211yesmaximal, characteristic, complemented
3#20295880C212yesmaximal, complemented
4#20295878C212yesmaximal, complemented
5#20295539C211yesmaximal, characteristic
6C2×D109⋊C8—21no—
7C2×C22⋊#89137—21no—
8#16812756C2212yes—
9C22⋊#11724786—21no—
10(C2×#89137)⋊C4—21no—
11(C2×#89137)⋊C4C2212yes—
12C2×#13182030C411yescharacteristic, complemented
13C22×C4×D109C2211yescharacteristic
14#16812684—21no—
15#16812675C2211yescharacteristic
16#16812674C2211yescharacteristic
17#16812673C411yescharacteristic, complemented
18#16812669—21no—
19#16812661—21no—
20#16812659—21no—
21#16812657C411yescharacteristic, complemented
22C22⋊Q8×C109C411yescharacteristic, complemented
23C22×#89137—22no—
24D109⋊C8—42no—
25C2×#11724786—22no—
26C22⋊#89137—42no—
27C23×D109D411yescharacteristic
28#13182030—42no—
29C2×D436—21no—
30C2×D436—21no—
31C2×D436C2×C411yescharacteristic
32C2×C22⋊D109—21no—
33C22×Dic109D411yescharacteristic
34C2×C4×D109—21no—
35C2×C4×D109—42no—
36C2×C4×D109C2×C411yescharacteristic
37C2×C4×D109D411yescharacteristic
38C2×C4×D109D411yescharacteristic
39C2×C4×D109C2311yescharacteristic
40C4⋊Dic109C2×C411yescharacteristic
41#13182008—21no—
42#13182006—21no—
43#13182006—21no—
44#13182006—21no—
45#13182006—21no—
46C4×Dic109—21no—
47C2×Q8×C109C2×C411yescharacteristic
48C22×C4×C109C2×C411yescharacteristic
49C4⋊C4×C109—21no—
50C4⋊C4×C109C2×C411yescharacteristic
51C22⋊C4×C109—21no—
52C23×C109C22⋊C411yescharacteristic
53C22×D109—22no—
54C22×D109—22no—
55C22×D109—21no—
56C22×D109—41no—
57C22×D109C22⋊C411yescharacteristic
58C22×D109C2×D411yescharacteristic
59C2×#89137—42no—
60C2×#89137—42no—
61C2×#89137—42no—
62Q8×C109—41no—
63C2×C4×C109—21no—
64C2×C4×C109—21no—
65C2×C4×C109—41no—
66C2×C4×C109C22⋊C411yescharacteristic
67C2×C4×C109C22×C411yescharacteristic
68C22⋊D109—42no—
69C2×Dic109—21no—
70C2×Dic109—41no—
71C2×Dic109C22⋊C411yescharacteristic
72C2×Dic109C2×D411yescharacteristic
73D436—41no—
74D436—42no—
75D436—42no—
76C4×D109—22no—
77C4×D109—22no—
78C4×D109—41no—
79C4×D109—41no—
80C4×D109—42no—
81#11724786—42no—
82C22×C109—21no—
83C22×C109—21no—
84C22×C109—21no—
85C22×C109C2×C22⋊C411yescharacteristic
86C2×D109—21no—
87C2×D109—21no—
88C2×D109—21no—
89C2×D109—21no—
90C2×D109—44no—
91C2×D109—41no—
92C2×D109—41no—
93C2×D109—41no—
94C2×D109#9912yes—
95C2×D109#13512yes—
96#89137—82no—
97C4×C109—21no—
98C4×C109—21no—
99C4×C109—41no—
100C4×C109—41no—
101Dic109—21no—
102Dic109—21no—
103Dic109—41no—
104C2×C109—41no—
105C2×C109C2×#9911yescharacteristic
106C2×C109#41611yescharacteristic
107C2×C109#42111yescharacteristic
108D109—22no—
109D109—22no—
110D109—81no—
111#1516—1091nomaximal, Sylow
112C109#151611yescharacteristic, complemented, Sylow
113#525—1091no—
114C2×C22⋊C8—1091no—
115#381—1091no—
116#328—1092no—
117#327—1092no—
118C2×#41—1091no—
119C23×C4—1091no—
120C22×C8—2181no—
121#123—2181no—
122C22⋊Q8—1091no—
123C22⋊Q8#8913711yescharacteristic, complemented
124#121—1091no—
125#121—1091no—
126#120—2181no—
127C4×D4—2181no—
128C4×D4—2181no—
129#117—1091no—
130C2×C22⋊C4—2181no—
131C22⋊C8—1092no—
132C22⋊C8—2181no—
133#95—1092no—
134#95—2181no—
135C24—1091no—
136#41—4362no—
137C2×Q8C2×#8913711yescharacteristic
138C2×D4—1091no—
139C2×D4—2181no—
140C2×D4—2181no—
141C2×D4—2181no—
142C22×C4—1091no—
143C22×C4—1091no—
144C22×C4—1091no—
145C22×C4—1091no—
146C22×C4—1091no—
147C22×C4—4362no—
148C22×C4—2181no—
149C22×C4—2182no—
150C22×C4C2×#8913711yescharacteristic
151C2×C8—4362no—
152C2×C8—2182no—
153C4⋊C4—21no—
154C4⋊C4—1091no—
155C4⋊C4—2181no—
156C4⋊C4C2×#8913711yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—4362no—
159C22⋊C4—2181no—
160C22⋊C4—2181no—
161C22⋊C4—2181no—
162C22⋊C4—2181no—
163C42—2181no—
164C23—1091no—
165C23—1091no—
166C23—4361no—
167C23—2182no—
168C23—2182no—
169C23—2181no—
170C23C22⋊#8913711yescharacteristic
171Q8—41no—
172D4—4361no—
173D4—4362no—
174D4—4362no—
175D4—4362no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—1091no—
180C2×C4—1091no—
181C2×C4—4361no—
182C2×C4—4361no—
183C2×C4—4362no—
184C2×C4—4362no—
185C2×C4—4362no—
186C2×C4—4362no—
187C2×C4—4361no—
188C2×C4—2182no—
189C2×C4—2182no—
190C2×C4—2181no—
191C2×C4C22⋊#8913711yescharacteristic
192C2×C4C22×#8913711yescharacteristic
193C8—4362no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—1092no—
198C22—1092no—
199C22—4364no—
200C22—4361no—
201C22—4361no—
202C22—4361no—
203C22—2181no—
204C22—2181no—
205C22—2181no—
206C22—2181no—
207C22C2×C22⋊#8913711yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—8722no—
213C4—4361no—
214C4—2181no—
215C4—2181no—
216C2—41no—
217C2—8721no—
218C2—2182no—
219C2—2182no—
220C2D109⋊#9911yescharacteristic
221C2#2029601611yescharacteristic
222C2#2029602511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC22⋊Q8×C109C4
socleC22×C109C2×C22⋊C4
prime radical#20295539C2
cosocleC2×C4×D109C23
hypercenterC22⋊Q8#89137
hypocenterC109#1516
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. D109⋊(C22⋊C8)C2semidirect
2. #20295969C2semidirect
3. #20295880C2semidirect
4. #20295878C2semidirect
12. C2×#13182030C4semidirect
17. #16812673C4semidirect
21. #16812657C4semidirect
22. C22⋊Q8×C109C4semidirect
112. C109#1516semidirect
123. C22⋊Q8#89137semidirect

Maximal subgroups

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

subgroupindex
D109⋊(C22⋊C8)2
#202959692
#202958802
#202958782
#202955392
#1516109