Finite Group Databasebeta

← #28371513

Subgroups — #28371513

Order 13952.

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

normal subgroups only

This group has 704 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#20295792C212yesmaximal, complemented
3#20295010C211yesmaximal, characteristic, complemented
4#20295007C214yesmaximal, complemented
5C2×#16812617C212yesmaximal, complemented
6#101×D109C214yesmaximal, complemented
7#413×C109C211yesmaximal, characteristic, complemented
8#16812746—24no—
9#16812746—24no—
10C2×#13182029C412yes—
11C2×#13182029C2211yescharacteristic
12C2×#13182028C414yes—
13C2×#13182028C2212yes—
14C22×Dic218C412yes—
15C22×Dic218C2211yescharacteristic
16C2×Q8×D109C2212yes—
17#16812707—24no—
18#16812705—24no—
19C2×#13182013C2212yes—
20M16×D109C2218yes—
21#16812618C2212yes—
22#16812617C2218yes—
23#16812616C2212yes—
24C2×#41×C109C2211yescharacteristic
25C2×M16×C109C2212yes—
26#101×C109C2212yes—
27#100×C109C2212yes—
28#13182029—24no—
29#13182029—24no—
30#13182029D414yes—
31#13182029D414yes—
32#13182028—216no—
33#13182028—28no—
34C2×Dic218—22no—
35C2×Dic218—22no—
36C2×Dic218—24no—
37C2×Dic218—24no—
38C2×Dic218C2×C414yes—
39C2×Dic218C2×C412yes—
40C2×Dic218C2×C411yescharacteristic
41C2×Dic218C2312yes—
42Q8×D109—28no—
43C2×D436C2×C411yescharacteristic
44C2×C22⋊D109C2×C414yes—
45C2×C22⋊D109C2×C412yes—
46C22×Dic109C2×C414yes—
47C22×Dic109C2×C412yes—
48C2×C4×D109C2×C414yes—
49C2×C4×D109C2312yes—
50#13182013—22no—
51#13182013C2314yes—
52#13182012—28no—
53C2×#11724784—22no—
54C8×D109—28no—
55#41×C109—22no—
56#41×C109—22no—
57C2×Q8×C109C2311yescharacteristic
58C2×D4×C109C2×C412yes—
59C2×D4×C109C2311yescharacteristic
60C22×C4×C109C2×C412yes—
61C22×C4×C109C2311yescharacteristic
62M16×C109—22no—
63M16×C109C2314yes—
64C2×C8×C109—22no—
65C23×C109C22×C412yes—
66C23×C109C22×C411yescharacteristic
67C22×D109C22×C412yes—
68Q8×C109—22no—
69D4×C109—24no—
70D4×C109—22no—
71C2×C4×C109—22no—
72C2×C4×C109—22no—
73C2×C4×C109C22×C412yes—
74C2×C4×C109C22×C411yescharacteristic
75C2×C4×C109C2×D412yes—
76C2×C4×C109C2×D412yes—
77C2×C4×C109C2411yescharacteristic
78C22⋊D109—28no—
79C22⋊D109C22⋊C418yes—
80C2×Dic109—24no—
81C2×Dic109—28no—
82C2×Dic109C22×C414yes—
83C2×Dic109C22×C412yes—
84D436C22⋊C412yes—
85D436C22⋊C412yes—
86C4×D109—28no—
87C4×D109C2×D418yes—
88Dic218—22no—
89Dic218—22no—
90Dic218—24no—
91Dic218—28no—
92Dic218C22⋊C412yes—
93Dic218C22⋊C412yes—
94C8×C109—24no—
95#11724784—24no—
96C22×C109—22no—
97C22×C109—41no—
98C22×C109—42no—
99C22×C109C2×C22⋊C412yes—
100C22×C109C23×C411yescharacteristic
101C2×D109—42no—
102C2×D109C2×C22⋊C414yes—
103C4×C109—22no—
104C4×C109C2×C22⋊C411yescharacteristic
105C4×C109C2×C22⋊C411yescharacteristic
106C4×C109C22×D412yes—
107Dic109—24no—
108Dic109C2×C22⋊C414yes—
109C2×C109—21no—
110C2×C109—22no—
111C2×C109—42no—
112C2×C109C22×C22⋊C411yescharacteristic
113D109—24no—
114#2889—1091nomaximal, Sylow
115C109#288911yescharacteristic, complemented, Sylow
116C2×2−1+4—1091no—
117#568—1092no—
118#413—1091no—
119#413—1094no—
120#413D10911yescharacteristic, complemented
121C2×#101—1092no—
122C2×#101—1094no—
1232−1+4—2184no—
1242−1+4—2184no—
125C2×#41—1094no—
126C2×#41—1092no—
127C2×#41—1092no—
128C2×#41—1091no—
129C2×#41C2×D10911yescharacteristic
130C22×Q8—1092no—
131C22×Q8—1092no—
132C22×Q8—1091no—
133#131—2184no—
134#131—2184no—
135C2×M16—1092no—
136C2×M16—1098no—
137C2×M16C2×D10912yes—
138#101—1092no—
139#101—1098no—
140#101C2×D10912yes—
141#100—1092no—
142#100C2×D10912yes—
143#41—22no—
144#41—22no—
145#41—1094no—
146#41—1094no—
147#41—2184no—
148#41—2184no—
149#41—2188no—
150#41—21816no—
151C2×Q8—1091no—
152C2×Q8—1094no—
153C2×Q8—1092no—
154C2×Q8—1092no—
155C2×Q8—2184no—
156C2×Q8—2184no—
157C2×Q8—2188no—
158C2×Q8—2182no—
159C2×Q8—2182no—
160C2×Q8C22×D10911yescharacteristic
161C2×D4—1092no—
162C2×D4—1094no—
163C2×D4—1091no—
164C2×D4C4×D10912yes—
165C2×D4C22×D10911yescharacteristic
166C22×C4—1092no—
167C22×C4—1094no—
168C22×C4—1092no—
169C22×C4—1094no—
170C22×C4C4×D10912yes—
171C22×C4C22×D10911yescharacteristic
172M16—22no—
173M16—1094no—
174M16—2188no—
175M16—2182no—
176M16C22×D10914yes—
177C2×C8—22no—
178C2×C8—2188no—
179C2×C8—2182no—
180C23—1092no—
181C23C2×C4×D10912yes—
182C23C2×C4×D10911yescharacteristic
183Q8—22no—
184Q8—1092no—
185Q8—1092no—
186Q8—2188no—
187Q8—2182no—
188Q8—2182no—
189Q8—2184no—
190D4—24no—
191D4—22no—
192D4—1098no—
193D4—1092no—
194D4—1092no—
195D4—2188no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—1092no—
199C2×C4—1098no—
200C2×C4—1094no—
201C2×C4—2188no—
202C2×C4—2188no—
203C2×C4—2184no—
204C2×C4C2×C4×D10912yes—
205C2×C4C2×C4×D10911yescharacteristic
206C2×C4D4×D10912yes—
207C2×C4D4×D10912yes—
208C2×C4C23×D10911yescharacteristic
209C8—24no—
210C8—2184no—
211C22—22no—
212C22—41no—
213C22—42no—
214C22—1094no—
215C22—4362no—
216C22C22⋊C4×D10912yes—
217C22C22×C4×D10911yescharacteristic
218C4—22no—
219C4—1094no—
220C4—2184no—
221C4C22⋊C4×D10911yescharacteristic
222C4C22⋊C4×D10911yescharacteristic
223C4C2×D4×D10912yes—
224C2—21no—
225C2—22no—
226C2—42no—
227C2—2184no—
228C2C2×C22⋊C4×D10911yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D109
commutatorC22×C109C23×C4
FrattiniC2×C4C23×D109
Fitting#413×C109C2
socleC2×C109C22×C22⋊C4
prime radicalC2×#16812746C2
cosocleC2×C4×C109C24
hypercenter#413D109
hypocenterC109#2889
LCSG ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #20295792C2semidirect
3. #20295010C2semidirect
4. #20295007C2semidirect
5. C2×#16812617C2semidirect
6. #101×D109C2semidirect
7. #413×C109C2semidirect
115. C109#2889semidirect
120. #413D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127462
#202957922
#202950102
#202950072
C2×#168126172
#101×D1092
#413×C1092
#2889109