Finite Group Databasebeta

← #28374767

Subgroups — #28374767

Order 13952.

482 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 131 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×Q16×D109C211yesmaximal, characteristic, complemented
2#20295683C212yesmaximal, complemented
3#20295409C211yesmaximal, characteristic, complemented
4#20295267C211yesmaximal, characteristic, complemented
5#20295259C211yesmaximal, characteristic, complemented
6#20295246C212yesmaximal, complemented
7#20295188C212yesmaximal, complemented
8#20295163C212yesmaximal, complemented
9Q16×Dic109C211yesmaximal, characteristic, complemented
10C8×C22⋊D109C211yesmaximal, characteristic, complemented
11C2×#135×C109C211yesmaximal, characteristic, complemented
12C2×Q8×D109C2212yes—
13C2×Dic436C2211yescharacteristic
14C2×#13182019C2212yes—
15Q16×D109—24no—
16C2×#13182017C2212yes—
17C2×C8×D109C2211yescharacteristic
18#16812686—22no—
19#16812686C2212yes—
20#16812678—22no—
21#16812678C2212yes—
22#16812670—22no—
23#16812670C2212yes—
24Q8×Dic109C2212yes—
25C4×C22⋊D109—22no—
26C4×C22⋊D109C2211yescharacteristic
27#16812633C2211yescharacteristic
28#16812628C2211yescharacteristic
29#16812624C2212yes—
30#16812623C2212yes—
31#16812620C2212yes—
32C22⋊#11724784C2211yescharacteristic
33#16812611C2211yescharacteristic
34C8×Dic109C2211yescharacteristic
35C2×#41×C109C2212yes—
36#135×C109—24no—
37C2×Q16×C109C2211yescharacteristic
38C2×SD16×C109C2212yes—
39C2×D8×C109C2211yescharacteristic
40C22×C8×C109C2211yescharacteristic
41C2×Dic218—22no—
42C2×Dic218C2312yes—
43Q8×D109—24no—
44Q8×D109—42no—
45C2×C22⋊D109—22no—
46C2×C22⋊D109D411yescharacteristic
47C2×C4×D109—22no—
48C2×C4×D109C2311yescharacteristic
49Dic436—22no—
50#13182019—24no—
51#13182017—42no—
52C2×#11724784C2311yescharacteristic
53C2×#11724784C2311yescharacteristic
54C8×D109—22no—
55C4⋊Dic109—22no—
56C4⋊Dic109C2312yes—
57#13182008—22no—
58#13182008D411yescharacteristic
59C22⋊Dic109—22no—
60C22⋊Dic109—22no—
61C22⋊Dic109—22no—
62C22⋊Dic109—22no—
63C22⋊Dic109D411yescharacteristic
64#13182006D411yescharacteristic
65C4×Dic109—22no—
66C4×Dic109C2311yescharacteristic
67#41×C109—24no—
68#41×C109—42no—
69C2×Q8×C109C2312yes—
70C2×D4×C109—22no—
71C2×D4×C109C2312yes—
72C22×C4×C109—22no—
73C22×C4×C109C2311yescharacteristic
74Q16×C109D414yescomplemented
75SD16×C109—24no—
76D8×C109—22no—
77C2×C8×C109—22no—
78C2×C8×C109C2311yescharacteristic
79C2×C8×C109C2311yescharacteristic
80C23×C109—22no—
81C23×C109C2×D411yescharacteristic
82C22×D109C2×D411yescharacteristic
83Q8×C109—22no—
84Q8×C109C2×D414yes—
85D4×C109—22no—
86D4×C109—22no—
87D4×C109—44no—
88C2×C4×C109—22no—
89C2×C4×C109—21no—
90C2×C4×C109—21no—
91C2×C4×C109—44no—
92C2×C4×C109C2×D411yescharacteristic
93C2×C4×C109C2411yescharacteristic
94C22⋊D109—22no—
95C22⋊D109—44no—
96C2×Dic109—22no—
97C2×Dic109—22no—
98C2×Dic109C2×D411yescharacteristic
99C2×Dic109C2×D411yescharacteristic
100C2×Dic109C2×D411yescharacteristic
101C4×D109—21no—
102C4×D109—21no—
103C4×D109—44no—
104Dic218—22no—
105Dic218—22no—
106Dic218—44no—
107C8×C109—21no—
108C8×C109C2×D412yes—
109#11724784—21no—
110#11724784—21no—
111C22×C109—21no—
112C22×C109—21no—
113C22×C109—21no—
114C22×C109—42no—
115C22×C109—42no—
116C22×C109—42no—
117C22×C109C22×D411yescharacteristic
118C2×D109—21no—
119C2×D109—21no—
120C2×D109#13512yescomplemented
121C4×C109—24no—
122C4×C109—21no—
123C4×C109C22×D411yescharacteristic
124C4×C109H(5,𝔽2)11yescharacteristic
125Dic109—21no—
126Dic109—21no—
127Dic109—21no—
128Dic109—42no—
129Dic109—42no—
130C2×C109—41no—
131C2×C109—82no—
132C2×C109D4211yescharacteristic
133C2×C109C2×#13511yescharacteristic
134C2×C109#57811yescharacteristic
135D109—22no—
136#3285—1091nomaximal, Sylow
137C109#328511yescharacteristic, complemented, Sylow
138C2×#135D10911yescharacteristic, complemented
139C22×Q16—1091no—
140#548—1092no—
141#495—1091no—
142#467—1091no—
143#466—1091no—
144#464—1092no—
145#452—1092no—
146#448—1092no—
147C4×Q16—1091no—
148C8×D4—1091no—
149C2×#41C2×D10912yes—
150C22×Q8—1092no—
151#135—24no—
152C2×Q16—1092no—
153C2×Q16—1091no—
154C2×Q16—2184no—
155C2×Q16C2×D10911yescharacteristic
156C2×SD16—1092no—
157C2×SD16C2×D10912yes—
158C2×D8C2×D10911yescharacteristic
159C22×C8—1091no—
160C22×C8C2×D10911yescharacteristic
161#124—1092no—
162#124—2182no—
163C22⋊Q8—1092no—
164C22⋊Q8—2182no—
165#121—1092no—
166#121—2182no—
167C4×Q8—1092no—
168C4×D4—1091no—
169C4×D4—2182no—
170#107—1091no—
171C4⋊C8—1091no—
172#103—1092no—
173#103—1092no—
174#102—1092no—
175C22⋊C8—1091no—
176C22⋊C8—1091no—
177C4×C8—1091no—
178#41—24no—
179#41—42no—
180C2×Q8—1092no—
181C2×Q8—4362no—
182C2×Q8—2182no—
183C2×Q8—2184no—
184C2×Q8C22×D10912yes—
185C2×D4—22no—
186C2×D4—1091no—
187C2×D4—2182no—
188C2×D4C22×D10912yes—
189C22×C4—22no—
190C22×C4—1091no—
191C22×C4—2182no—
192C22×C4C22×D10911yescharacteristic
193Q16—2184no—
194Q16—2182no—
195Q16C22⋊D10914yescomplemented
196SD16—24no—
197SD16—4362no—
198D8—22no—
199C2×C8—22no—
200C2×C8—1091no—
201C2×C8—1091no—
202C2×C8—2182no—
203C2×C8C22×D10911yescharacteristic
204C2×C8C22×D10911yescharacteristic
205C4⋊C4—1091no—
206C4⋊C4—1092no—
207C4⋊C4—2182no—
208C4⋊C4—2182no—
209C22⋊C4—1091no—
210C22⋊C4—1091no—
211C22⋊C4—2182no—
212C22⋊C4—2182no—
213C22⋊C4—2182no—
214C22⋊C4—2182no—
215C42—1091no—
216C42—2182no—
217C23—22no—
218C23—1091no—
219C23D4×D10911yescharacteristic
220Q8—22no—
221Q8—4364no—
222Q8—2182no—
223Q8—2182no—
224Q8C2×C22⋊D10914yes—
225D4—22no—
226D4—22no—
227D4—44no—
228D4—4364no—
229D4—2182no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—44no—
234C2×C4—1091no—
235C2×C4—1091no—
236C2×C4—1091no—
237C2×C4—4364no—
238C2×C4—2182no—
239C2×C4—2182no—
240C2×C4—2181no—
241C2×C4—2181no—
242C2×C4D4×D10911yescharacteristic
243C2×C4C23×D10911yescharacteristic
244C8—21no—
245C8—2181no—
246C8—2181no—
247C8C2×C22⋊D10912yes—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—1092no—
255C22—2181no—
256C22—2181no—
257C22C2×D4×D10911yescharacteristic
258C4—21no—
259C4—24no—
260C4—4362no—
261C4—4362no—
262C4—2181no—
263C4—2181no—
264C4—2181no—
265C4C22×C22⋊D10911yescharacteristic
266C4#1681274411yescharacteristic
267C2—41no—
268C2—82no—
269C2—2182no—
270C2D4×C22⋊D10911yescharacteristic
271C2#135×D10911yescharacteristic
272C2#2029584011yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×#135×C109C2
socleC22×C109C22×D4
hypercenterC2×#135D109
hypocenterC109#3285
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D109C2semidirect
2. #20295683C2semidirect
3. #20295409C2semidirect
4. #20295267C2semidirect
5. #20295259C2semidirect
6. #20295246C2semidirect
7. #20295188C2semidirect
8. #20295163C2semidirect
9. Q16×Dic109C2semidirect
10. C8×C22⋊D109C2semidirect
11. C2×#135×C109C2semidirect
74. Q16×C109D4semidirect
120. C2×D109#135semidirect
137. C109#3285semidirect
138. C2×#135D109semidirect
195. Q16C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1092
#202956832
#202954092
#202952672
#202952592
#202952462
#202951882
#202951632
Q16×Dic1092
C8×C22⋊D1092
C2×#135×C1092
#3285109