Finite Group Databasebeta

← #28336823

Subgroups — #28336823

Order 13696.

310 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 89 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#20293425C211yesmaximal, characteristic
2#20292643C211yesmaximal, characteristic, complemented
3#20292623C211yesmaximal, characteristic, complemented
4#20292595C211yesmaximal, characteristic, complemented
5#20292398C211yesmaximal, characteristic, complemented
6#20292391C211yesmaximal, characteristic, complemented
7#568×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812140—21no—
10#16812139C412yescomplemented
11#16812135—21no—
12#16812134C412yescomplemented
13#16812129—22no—
14C2×#13133802C2211yescharacteristic
15C22×C107⋊C8C2211yescharacteristic
16#16812068C2211yescharacteristic
17#16812035—22no—
18#16812033—21no—
19#16812030—21no—
20(C2×D107)⋊C8—22no—
21C2×#41×C107C2211yescharacteristic
22#131×C107—22no—
23#131×C107—22no—
24C2×M16×C107—22no—
25C2×M16×C107C2211yescharacteristic
26C2×M16×C107C2211yescharacteristic
27#13133818—21no—
28#13133818—21no—
29#13133818—21no—
30#13133818—21no—
31C2×Dic214C2×C411yescharacteristic
32C2×D428C2×C411yescharacteristic
33C2×C22⋊D107—21no—
34C2×C4×D107—21no—
35#13133808—22no—
36#13133806—22no—
37#13133805—22no—
38#13133803—22no—
39#13133802—21no—
40#13133802C2×C412yes—
41C2×C107⋊C8—21no—
42C2×C107⋊C8—21no—
43C2×C107⋊C8—21no—
44C2×C107⋊C8—21no—
45C4⋊Dic107—21no—
46C22⋊Dic107—21no—
47C4×Dic107—21no—
48#41×C107—21no—
49#41×C107—21no—
50#41×C107C2×C412yes—
51#41×C107D412yes—
52C2×Q8×C107D411yescharacteristic
53C2×D4×C107—21no—
54C2×D4×C107D411yescharacteristic
55C22×C4×C107—21no—
56C22×C4×C107C2311yescharacteristic
57M16×C107—24no—
58M16×C107—21no—
59M16×C107—21no—
60M16×C107—21no—
61M16×C107D412yes—
62C2×C8×C107—24no—
63C2×C8×C107—21no—
64C2×C8×C107D412yes—
65C23×C107—21no—
66C23×C107#4111yescharacteristic
67C22×D107—21no—
68Q8×C107—21no—
69Q8×C107C22⋊C412yes—
70D4×C107—22no—
71D4×C107—22no—
72D4×C107—21no—
73D4×C107C22⋊C412yes—
74C2×C4×C107—21no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107C22×C411yescharacteristic
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107#4111yescharacteristic
83C22⋊D107—41no—
84C22⋊D107—41no—
85C2×Dic107—21no—
86C2×Dic107—41no—
87D428—21no—
88D428—21no—
89C4×D107—41no—
90C4×D107—41no—
91Dic214—21no—
92Dic214—21no—
93C8×C107—22no—
94C8×C107—22no—
95C107⋊C8—22no—
96C107⋊C8—41no—
97C22×C107—22no—
98C22×C107—41no—
99C22×C107—81no—
100C22×C107C4×D411yescharacteristic
101C22×C107#12111yescharacteristic
102C22×C107#12311yescharacteristic
103C2×D107—42no—
104C2×D107—41no—
105C4×C107—22no—
106C4×C107C2×C22⋊C411yescharacteristic
107C4×C107C4×D411yescharacteristic
108C4×C107#12011yescharacteristic
109C4×C107#12111yescharacteristic
110Dic107—41no—
111Dic107—81no—
112C2×C107—21no—
113C2×C107—21no—
114C2×C107—21no—
115C2×C107—42no—
116C2×C107#38611yescharacteristic
117D107—81no—
118#1881—1071nomaximal, Sylow
119C107#188111yescharacteristic, complemented, Sylow
120#575—1071no—
121#568D10711yescharacteristic, complemented
122#421—1071no—
123#416—1071no—
124#408—1071no—
125#344—1071no—
126#341—1071no—
127C2×#41—1071no—
128C2×#41C2×D10711yescharacteristic
129#137—1072no—
130#137—2141no—
131AGL(1,ℤ/8)—1072no—
132AGL(1,ℤ/8)—2141no—
133#135—2142no—
134#131—22no—
135#131—22no—
136C2×M16—22no—
137C2×M16—1071no—
138C2×M16C2×D10711yescharacteristic
139C2×M16C2×D10711yescharacteristic
140C22×C8—1071no—
141#117—1071no—
142#104—2142no—
143#103—2141no—
144#102—2141no—
145C22⋊C8—2142no—
146#41—21no—
147#41—21no—
148#41—2141no—
149#41—2141no—
150#41—2141no—
151#41—2141no—
152#41C4×D10712yes—
153#41C22⋊D10712yes—
154C2×Q8—1071no—
155C2×Q8D42811yescharacteristic
156C2×D4—21no—
157C2×D4—1071no—
158C2×D4—2141no—
159C2×D4D42811yescharacteristic
160C22×C4—21no—
161C22×C4—2141no—
162C22×C4C22×D10711yescharacteristic
163Q16—2142no—
164SD16—2142no—
165SD16—2142no—
166D8—2142no—
167M16—24no—
168M16—21no—
169M16—21no—
170M16—21no—
171M16—1072no—
172M16—2141no—
173M16C22⋊D10712yes—
174C2×C8—24no—
175C2×C8—21no—
176C2×C8—2141no—
177C2×C8—2141no—
178C2×C8—2141no—
179C2×C8—2141no—
180C2×C8C22⋊D10712yes—
181C4⋊C4—2141no—
182C22⋊C4—2141no—
183C42—2141no—
184C23—21no—
185C23—2141no—
186C23#1313381811yescharacteristic
187Q8—21no—
188Q8—2141no—
189Q8—2141no—
190Q8(C2×D107)⋊C412yes—
191D4—22no—
192D4—22no—
193D4—21no—
194D4—4281no—
195D4—4281no—
196D4—2141no—
197D4—2141no—
198D4(C2×D107)⋊C412yes—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—4281no—
203C2×C4—4281no—
204C2×C4—4281no—
205C2×C4—2141no—
206C2×C4C2×C4×D10711yescharacteristic
207C2×C4C2×C22⋊D10711yescharacteristic
208C2×C4C2×C22⋊D10711yescharacteristic
209C2×C4C2×C22⋊D10711yescharacteristic
210C2×C4C2×D42811yescharacteristic
211C2×C4#1313381811yescharacteristic
212C8—22no—
213C8—22no—
214C8—4281no—
215C8—2142no—
216C22—22no—
217C22—41no—
218C22—81no—
219C22—4282no—
220C22—4281no—
221C22C4×C22⋊D10711yescharacteristic
222C22#1681208311yescharacteristic
223C22#1681209211yescharacteristic
224C4—22no—
225C4—8561no—
226C4—4281no—
227C4C2×(C2×D107)⋊C411yescharacteristic
228C4C4×C22⋊D10711yescharacteristic
229C4#1681207811yescharacteristic
230C4#1681208311yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—42no—
235C2—8561no—
236C2#2029250611yescharacteristic
subgroupG/H
centerC4#16812078
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#568×C107C2
socleC2×C107#386
prime radical#20293425C2
cosocleC22×C4×C107C23
hypercenter#568D107
hypocenterC107#1881
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20292643C2semidirect
3. #20292623C2semidirect
4. #20292595C2semidirect
5. #20292398C2semidirect
6. #20292391C2semidirect
7. #568×C107C2semidirect
10. #16812139C4semidirect
12. #16812134C4semidirect
119. C107#1881semidirect
121. #568D107semidirect

Maximal subgroups

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

subgroupindex
#202934252
#202926432
#202926232
#202925952
#202923982
#202923912
#568×C1072
#1881107