Finite Group Databasebeta

← #28371494

Subgroups — #28371494

Order 13952.

716 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16812746C211yesmaximal, characteristic
2#20295503C212yesmaximal, complemented
3#20295000C211yesmaximal, characteristic, complemented
4#20294999C212yesmaximal, complemented
5#20294998C218yesmaximal, complemented
6#410×C109C211yesmaximal, characteristic, complemented
7#16812746—24no—
8#16812746—24no—
9C2×#13182029C412yescomplemented
10C2×#13182029C2211yescharacteristic
11C2×#13182028C412yescomplemented
12C2×#13182028C2214yes—
13C22×Dic218C412yescomplemented
14C22×Dic218C2211yescharacteristic
15C2×Q8×D109C412yescomplemented
16C4×Dic218—24no—
17#16812664—24no—
18#16812658C2212yes—
19#16812655C2218yes—
20#16812614C2214yes—
21#16812613C2218yes—
22C2×#41×C109C2211yescharacteristic
23#117×C109C2212yes—
24#99×C109C2214yes—
25#13182029—22no—
26#13182029—22no—
27#13182029—24no—
28#13182029—24no—
29#13182028—216no—
30#13182028—28no—
31C2×Dic218—24no—
32C2×Dic218—24no—
33C2×Dic218C2×C414yes—
34C2×Dic218C2×C411yescharacteristic
35C2×Dic218C2×C412yes—
36C2×Dic218C2×C412yes—
37C2×Dic218D414yes—
38C2×Dic218D414yes—
39Q8×D109—28no—
40C2×D436C2×C411yescharacteristic
41C2×C22⋊D109C2×C414yes—
42C2×C22⋊D109C2312yes—
43C22×Dic109C2×C414yes—
44C22×Dic109C2312yes—
45C2×C4×D109C2×C412yes—
46C2×C4×D109C2×C414yes—
47C4⋊Dic109—22no—
48#13182008—28no—
49C22⋊Dic109C2314yes—
50C4×Dic109—22no—
51C4×Dic109—28no—
52#41×C109—22no—
53#41×C109—22no—
54C2×Q8×C109C2×C411yescharacteristic
55C2×D4×C109C2×C411yescharacteristic
56C2×D4×C109C2312yes—
57C22×C4×C109C2×C412yes—
58C22×C4×C109C2311yescharacteristic
59C4⋊C4×C109—22no—
60C22⋊C4×C109C2314yes—
61C42×C109—22no—
62C23×C109C22×C412yes—
63C23×C109C2411yescharacteristic
64C22×D109C22×C412yes—
65Q8×C109—22no—
66D4×C109—24no—
67D4×C109—22no—
68C2×C4×C109—22no—
69C2×C4×C109—22no—
70C2×C4×C109—24no—
71C2×C4×C109C22×C412yes—
72C2×C4×C109C22×C411yescharacteristic
73C2×C4×C109C22×C411yescharacteristic
74C2×C4×C109C2×D412yes—
75C2×C4×C109C2×D412yes—
76C22⋊D109—24no—
77C22⋊D109—28no—
78C2×Dic109—28no—
79C2×Dic109—24no—
80C2×Dic109C22×C412yes—
81C2×Dic109C22×C414yes—
82C2×Dic109C2×D418yes—
83D436—21no—
84D436—21no—
85C4×D109—24no—
86C4×D109—28no—
87Dic218—21no—
88Dic218—21no—
89Dic218—28no—
90Dic218C22⋊C418yes—
91Dic218C22⋊C414yes—
92Dic218C22⋊C414yes—
93C22×C109—22no—
94C22×C109—41no—
95C22×C109—42no—
96C22×C109C23×C411yescharacteristic
97C22×C109C22×D412yes—
98C2×D109—22no—
99C2×D109—42no—
100C4×C109—22no—
101C4×C109—44no—
102C4×C109C2×C22⋊C411yescharacteristic
103C4×C109C2×C22⋊C411yescharacteristic
104C4×C109C2×C22⋊C412yes—
105Dic109—22no—
106Dic109—44no—
107Dic109C2×C22⋊C418yes—
108C2×C109—22no—
109C2×C109—21no—
110C2×C109—42no—
111C2×C109C22×C22⋊C411yescharacteristic
112D109—42no—
113#2884—1091nomaximal, Sylow
114C109#288411yescharacteristic, complemented, Sylow
115C2×2−1+4—1091no—
116#519—1092no—
117#410—1091no—
118#410—1092no—
119#410—1098no—
120#410D10911yescharacteristic, complemented
1212−1+4—2184no—
1222−1+4—2184no—
123C2×#41—1092no—
124C2×#41—1094no—
125C2×#41—1092no—
126C2×#41—1091no—
127C2×#41C2×D10911yescharacteristic
128C22×Q8—1092no—
129C22×Q8—1092no—
130C22×Q8—1091no—
131C4×Q8—2184no—
132C4×Q8—2184no—
133#117—1092no—
134#117—1098no—
135#117C2×D10912yes—
136#99—1094no—
137#99—1098no—
138#99C2×D10914yes—
139#41—22no—
140#41—22no—
141#41—2184no—
142#41—2184no—
143#41—21816no—
144#41—2188no—
145#41—2182no—
146#41—2182no—
147C2×Q8—1092no—
148C2×Q8—1092no—
149C2×Q8—1094no—
150C2×Q8—1091no—
151C2×Q8—1094no—
152C2×Q8—1094no—
153C2×Q8—2184no—
154C2×Q8—2184no—
155C2×Q8—2188no—
156C2×Q8C4×D10911yescharacteristic
157C2×D4—1094no—
158C2×D4—1092no—
159C2×D4—1091no—
160C2×D4C4×D10911yescharacteristic
161C2×D4C22×D10912yes—
162C22×C4—1092no—
163C22×C4—1094no—
164C22×C4—1094no—
165C22×C4—1092no—
166C22×C4C4×D10912yes—
167C22×C4C22×D10911yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—2188no—
170C4⋊C4—2182no—
171C22⋊C4—1094no—
172C22⋊C4C22×D10914yes—
173C42—22no—
174C42—2188no—
175C42—2182no—
176C23—1092no—
177C23C2×C4×D10912yes—
178C23C23×D10911yescharacteristic
179Q8—22no—
180Q8—1094no—
181Q8—1094no—
182Q8—1098no—
183Q8—2188no—
184Q8—2181no—
185Q8—2181no—
186D4—24no—
187D4—22no—
188D4—2188no—
189D4—2184no—
190D4—2181no—
191D4—2181no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—1094no—
196C2×C4—1098no—
197C2×C4—1092no—
198C2×C4—2188no—
199C2×C4—2188no—
200C2×C4—2184no—
201C2×C4—2184no—
202C2×C4C2×C4×D10912yes—
203C2×C4C2×C4×D10911yescharacteristic
204C2×C4C2×C4×D10911yescharacteristic
205C2×C4D4×D10912yes—
206C2×C4D4×D10912yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—4362no—
211C22—2182no—
212C22C22×C4×D10911yescharacteristic
213C22C2×D4×D10912yes—
214C4—22no—
215C4—44no—
216C4—1098no—
217C4—4364no—
218C4—2182no—
219C4C22⋊C4×D10911yescharacteristic
220C4C22⋊C4×D10911yescharacteristic
221C4C22⋊C4×D10912yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—4362no—
226C2C2×C22⋊C4×D10911yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D109
commutatorC22×C109C23×C4
FrattiniC23C23×D109
Fitting#410×C109C2
socleC2×C109C22×C22⋊C4
prime radicalC2×#16812746C2
cosocleC23×C109C24
hypercenter#410D109
hypocenterC109#2884
LCSG ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20295503C2semidirect
3. #20295000C2semidirect
4. #20294999C2semidirect
5. #20294998C2semidirect
6. #410×C109C2semidirect
9. C2×#13182029C4semidirect
11. C2×#13182028C4semidirect
13. C22×Dic218C4semidirect
15. C2×Q8×D109C4semidirect
114. C109#2884semidirect
120. #410D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127462
#202955032
#202950002
#202949992
#202949982
#410×C1092
#2884109