Finite Group Databasebeta

← #28377819

Subgroups — #28377819

Order 13952.

1412 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1412 conjugacy classes of subgroups — too many to draw. Only its 545 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#20295862C211yesmaximal, characteristic, complemented
2#20295841C218yesmaximal, complemented
3#20295834C218yesmaximal, complemented
4#20295824C216yesmaximal, complemented
5C2×#16812730C216yesmaximal, complemented
6#20295793C211yesmaximal, characteristic, complemented
7C2×2−1+4×C109C211yesmaximal, characteristic, complemented
8#16812747—21no—
9#16812746C2214yescomplemented
10#16812745C2214yescomplemented
11#16812744—23no—
12C2×#13182029C2216yes—
13#41×D109—24no—
14#16812730C22112yescomplemented
15#16812730C22124yescomplemented
16#16812729C22112yescomplemented
17#16812725C2214yescomplemented
18#16812724C22112yescomplemented
19#16812719C22124yescomplemented
20C2×#13182019C22112yescomplemented
21C2×#13182016C22112yescomplemented
22#16812706C2218yescomplemented
23C2×#13182013C2216yescomplemented
242−1+4×C109—24no—
252−1+4×C109C2218yescomplemented
26C2×#41×C109—23no—
27C2×#41×C109C2211yescharacteristic
28C2×#41×C109C2213yescomplemented
29C22×Q8×C109—21no—
30C22×Q8×C109C2213yescomplemented
31#13182030—24no—
32#13182029—26no—
33#13182029—26no—
34#13182029C23112yes—
35#13182029C2314yes—
36#13182028—212no—
37C2×Dic218C2316yes—
38Q8×D109—24no—
39C2×D436C2316yes—
40C2×C22⋊D109—26no—
41D4×D109—212no—
42C2×C4×D109—26no—
43#13182019C23124yes—
44#13182017C2318yes—
45#13182016C23124yes—
46#13182014C2318yes—
47#13182013C2314yes—
48#13182013C23112yes—
49C2×#11724784C23112yes—
50#41×C109—224no—
51#41×C109—26no—
52#41×C109D414yes—
53#41×C109C23112yes—
54#41×C109C2314yes—
55C2×Q8×C109—28no—
56C2×Q8×C109—23no—
57C2×Q8×C109—26no—
58C2×Q8×C109D411yescharacteristic
59C2×Q8×C109C2313yes—
60C2×Q8×C109C23112yes—
61C2×D4×C109—23no—
62C2×D4×C109D413yes—
63C2×D4×C109C2311yescharacteristic
64C22×C4×C109—23no—
65C22×C4×C109—23no—
66C22×C4×C109C2313yes—
67C23×C109—21no—
68C23×C109C2×D413yes—
69C22×D109—26no—
70Q8×C109—212no—
71Q8×C109—23no—
72Q8×C109C2×D414yes—
73Q8×C109C2416yes—
74D4×C109—212no—
75D4×C109—21no—
76D4×C109C2×D4112yes—
77D4×C109C2412yes—
78C2×C4×C109—23no—
79C2×C4×C109—212no—
80C2×C4×C109—212no—
81C2×C4×C109C2×D413yes—
82C2×C4×C109C2×D416yes—
83C2×C4×C109C2411yescharacteristic
84C2×C4×C109C2416yes—
85C22⋊D109—212no—
86C22⋊D109—24no—
87C2×Dic109—26no—
88D436—26no—
89D436C2414yes—
90C4×D109—24no—
91C4×D109—212no—
92Dic218—26no—
93Dic218C2414yes—
94#11724784C2418yes—
95C22×C109—22no—
96C22×C109—43no—
97C22×C109—42no—
98C22×C109C22×D416yes—
99C22×C109C22×D411yescharacteristic
100C2×D109—24no—
101C2×D109—86no—
102C4×C109—26no—
103C4×C109C22×D416yes—
104C4×C109C22×D411yescharacteristic
105C4×C109C2511yescharacteristic
106Dic109—24no—
107C2×C109—26no—
108C2×C109—21no—
109C2×C109—42no—
110C2×C109C23×D411yescharacteristic
111D109—44no—
112#3586—1091nomaximal, Sylow
113C109#358611yescharacteristic, complemented, Sylow
114#585—1091no—
115C2×2−1+4D10911yescharacteristic, complemented
116#578—1098no—
117#577—1098no—
118#575—1096no—
119C2×#137—1096no—
120#568—1091no—
1212−1+4—24no—
1222−1+4—1094no—
1232−1+4—2181no—
1242−1+4C2×D10918yescomplemented
125H(5,𝔽2)—1094no—
126H(5,𝔽2)—2183no—
127C2×#41—23no—
128C2×#41—1096no—
129C2×#41—2184no—
130C2×#41C2×D10911yescharacteristic
131C2×#41C2×D10913yescomplemented
132C22×Q8—21no—
133C22×Q8C2×D10913yescomplemented
134#137—10912no—
135#137—10912no—
136#137—10924no—
137AGL(1,ℤ/8)—10912no—
138AGL(1,ℤ/8)—1094no—
139#135—10924no—
140C2×Q16—10912no—
141C2×SD16—10912no—
142#131—1098no—
143C2×M16—1096no—
144#41—224no—
145#41—26no—
146#41—10912no—
147#41—1094no—
148#41—2184no—
149#41—21812no—
150#41—2186no—
151#41—2186no—
152#41C22⋊D10914yes—
153#41C22×D109112yes—
154#41C22×D10914yes—
155C2×Q8—28no—
156C2×Q8—23no—
157C2×Q8—26no—
158C2×Q8—1096no—
159C2×Q8—2184no—
160C2×Q8C22⋊D10911yescharacteristic
161C2×Q8C22×D109112yes—
162C2×Q8C22×D10913yes—
163C2×D4—23no—
164C2×D4—1096no—
165C2×D4—2186no—
166C2×D4—21812no—
167C2×D4C22⋊D10913yes—
168C2×D4C22×D10911yescharacteristic
169C22×C4—23no—
170C22×C4—23no—
171C22×C4—2186no—
172C22×C4C22×D10913yes—
173Q16—10924no—
174SD16—10924no—
175SD16—1098no—
176D8—1098no—
177M16—1094no—
178M16—10912no—
179C2×C8—10912no—
180C23—21no—
181C23—2186no—
182C23C2×C22⋊D10913yes—
183Q8—212no—
184Q8—23no—
185Q8—1094no—
186Q8—2186no—
187Q8C2×C22⋊D10914yes—
188Q8C23×D10916yes—
189D4—212no—
190D4—21no—
191D4—1094no—
192D4—2184no—
193D4—21812no—
194D4—2186no—
195D4C2×C22⋊D109112yes—
196D4C23×D10912yes—
197C2×C4—23no—
198C2×C4—212no—
199C2×C4—212no—
200C2×C4—2186no—
201C2×C4—2184no—
202C2×C4—21812no—
203C2×C4C2×C22⋊D10913yes—
204C2×C4C2×C22⋊D10916yes—
205C2×C4C23×D10916yes—
206C2×C4C23×D10911yescharacteristic
207C8—1098no—
208C22—22no—
209C22—43no—
210C22—42no—
211C22—8726no—
212C22—2184no—
213C22C22×C22⋊D10911yescharacteristic
214C22C22×C22⋊D10916yes—
215C4—26no—
216C4—2184no—
217C4C22×C22⋊D10916yes—
218C4C22×C22⋊D10911yescharacteristic
219C4C24×D10911yescharacteristic
220C2—21no—
221C2—26no—
222C2—42no—
223C2—4364no—
224C2C23×C22⋊D10911yescharacteristic
subgroupG/H
centerC2C23×C22⋊D109
commutator, cosocleC4×C109C25
FrattiniC4C24×D109
FittingC2×2−1+4×C109C2
socleC2×C109C23×D4
hypercenterC2×2−1+4D109
hypocenterC109#3586
LCSG ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #20295862C2semidirect
2. #20295841C2semidirect
3. #20295834C2semidirect
4. #20295824C2semidirect
5. C2×#16812730C2semidirect
6. #20295793C2semidirect
7. C2×2−1+4×C109C2semidirect
9. #16812746C22semidirect
10. #16812745C22semidirect
14. #16812730C22semidirect
15. #16812730C22semidirect
16. #16812729C22semidirect
17. #16812725C22semidirect
18. #16812724C22semidirect
19. #16812719C22semidirect
20. C2×#13182019C22semidirect
21. C2×#13182016C22semidirect
22. #16812706C22semidirect
23. C2×#13182013C22semidirect
25. 2−1+4×C109C22semidirect
28. C2×#41×C109C22semidirect
30. C22×Q8×C109C22semidirect
113. C109#3586semidirect
115. C2×2−1+4D109semidirect
124. 2−1+4C2×D109semidirect
131. C2×#41C2×D109semidirect
133. C22×Q8C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958622
#202958412
#202958342
#202958242
C2×#168127302
#202957932
C2×2−1+4×C1092
#3586109