Finite Group Databasebeta

← #28336826

Subgroups — #28336826

Order 13696.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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#20293393C211yesmaximal, characteristic, complemented
3#20292644C211yesmaximal, characteristic, complemented
4#20292624C211yesmaximal, characteristic, complemented
5#20292398C211yesmaximal, characteristic, complemented
6#20292392C211yesmaximal, characteristic, complemented
7#408×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812140C412yescomplemented
10#16812139—21no—
11#16812135C412yescomplemented
12#16812134—21no—
13#16812129—22no—
14#16812117—22no—
15#16812115—22no—
16C2×#13133802C2211yescharacteristic
17C2×#13133802C2211yescharacteristic
18M16×D107—22no—
19#16812068C2211yescharacteristic
20#16812036—22no—
21#16812034—21no—
22#16812031—21no—
23C2×#41×C107C2211yescharacteristic
24C2×M16×C107C2211yescharacteristic
25C22×C8×C107C2211yescharacteristic
26C22⋊C8×C107—22no—
27#13133818—21no—
28#13133818—21no—
29#13133818C2×C412yes—
30#13133818D412yes—
31C2×Dic214D411yescharacteristic
32C2×D428D411yescharacteristic
33C2×C22⋊D107—21no—
34C2×C4×D107—21no—
35#13133808—22no—
36#13133806—22no—
37#13133805—22no—
38#13133803—22no—
39#13133802—21no—
40#13133802—21no—
41#13133802C2×C412yes—
42#13133802D412yes—
43#13133801—24no—
44C2×C107⋊C8—21no—
45C2×C107⋊C8—21no—
46C8×D107—24no—
47C4⋊Dic107—21no—
48C22⋊Dic107—21no—
49C4×Dic107—21no—
50#41×C107—21no—
51#41×C107—21no—
52#41×C107—21no—
53#41×C107—21no—
54C2×Q8×C107C2×C411yescharacteristic
55C2×D4×C107—21no—
56C2×D4×C107C2×C411yescharacteristic
57C22×C4×C107—21no—
58C22×C4×C107C2311yescharacteristic
59M16×C107—21no—
60M16×C107—21no—
61C2×C8×C107—21no—
62C2×C8×C107—21no—
63C2×C8×C107—21no—
64C2×C8×C107D412yes—
65C23×C107—21no—
66C23×C107#4111yescharacteristic
67C22×D107—21no—
68Q8×C107—21no—
69Q8×C107—21no—
70D4×C107—21no—
71D4×C107—21no—
72D4×C107—41no—
73D4×C107—41no—
74C2×C4×C107—21no—
75C2×C4×C107—41no—
76C2×C4×C107—41no—
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—22no—
84C22⋊D107—22no—
85C2×Dic107—21no—
86C2×Dic107—41no—
87D428—21no—
88D428C22⋊C412yes—
89C4×D107—22no—
90C4×D107—22no—
91Dic214—21no—
92Dic214C22⋊C412yes—
93C8×C107—22no—
94C8×C107—41no—
95C107⋊C8—22no—
96C107⋊C8—22no—
97C22×C107—41no—
98C22×C107—42no—
99C22×C107—41no—
100C22×C107C4×D411yescharacteristic
101C22×C107#12111yescharacteristic
102C22×C107#12311yescharacteristic
103C2×D107—22no—
104C2×D107—81no—
105C4×C107—41no—
106C4×C107C2×C22⋊C411yescharacteristic
107C4×C107C4×D411yescharacteristic
108C4×C107#12011yescharacteristic
109C4×C107#12111yescharacteristic
110Dic107—22no—
111Dic107—81no—
112C2×C107—21no—
113C2×C107—21no—
114C2×C107—21no—
115C2×C107—81no—
116C2×C107#38611yescharacteristic
117D107—42no—
118#1881—1071nomaximal, Sylow
119C107#188111yescharacteristic, complemented, Sylow
120#575—1071no—
121#568—1071no—
122#421—1071no—
123#416—1071no—
124#408D10711yescharacteristic, complemented
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—2142no—
135#131—2142no—
136C2×M16—1071no—
137C2×M16—1071no—
138C2×M16—2142no—
139C2×M16C2×D10711yescharacteristic
140C22×C8C2×D10711yescharacteristic
141#117—1071no—
142#104—2142no—
143#103—2141no—
144#102—2141no—
145C22⋊C8—22no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—1072no—
151#41—1072no—
152#41—2141no—
153#41—2141no—
154C2×Q8—1071no—
155C2×Q8C4×D10711yescharacteristic
156C2×D4—21no—
157C2×D4—1071no—
158C2×D4—2141no—
159C2×D4C4×D10711yescharacteristic
160C22×C4—21no—
161C22×C4—2141no—
162C22×C4C22×D10711yescharacteristic
163Q16—2142no—
164SD16—2142no—
165SD16—2142no—
166D8—2142no—
167M16—21no—
168M16—21no—
169M16—1072no—
170M16—1072no—
171M16—2144no—
172M16—2141no—
173M16—2141no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—2144no—
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—21no—
189Q8—1072no—
190Q8—2141no—
191D4—21no—
192D4—21no—
193D4—41no—
194D4—41no—
195D4—1072no—
196D4—2142no—
197D4—2142no—
198D4—2141no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—4281no—
203C2×C4—2141no—
204C2×C4—2142no—
205C2×C4—2142no—
206C2×C4C2×C4×D10711yescharacteristic
207C2×C4D4×D10711yescharacteristic
208C2×C4D4×D10711yescharacteristic
209C2×C4D4×D10711yescharacteristic
210C2×C4C2×C22⋊D10711yescharacteristic
211C2×C4#1313381711yescharacteristic
212C8—22no—
213C8—41no—
214C8—2142no—
215C8—2142no—
216C22—41no—
217C22—42no—
218C22—41no—
219C22—8561no—
220C22—2142no—
221C22C4×C22⋊D10711yescharacteristic
222C22#1681208211yescharacteristic
223C22#1681209111yescharacteristic
224C4—41no—
225C4—8561no—
226C4—2142no—
227C4C22⋊C4×D10711yescharacteristic
228C4(C22⋊D107)⋊C411yescharacteristic
229C4#1681207711yescharacteristic
230C4#1681208011yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—4282no—
236C2#2029250311yescharacteristic
subgroupG/H
centerC4#16812077
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#408×C107C2
socleC2×C107#386
prime radical#20293425C2
cosocleC22×C4×C107C23
hypercenter#408D107
hypocenterC107#1881
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #20293393C2semidirect
3. #20292644C2semidirect
4. #20292624C2semidirect
5. #20292398C2semidirect
6. #20292392C2semidirect
7. #408×C107C2semidirect
9. #16812140C4semidirect
11. #16812135C4semidirect
119. C107#1881semidirect
124. #408D107semidirect

Maximal subgroups

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

subgroupindex
#202934252
#202933932
#202926442
#202926242
#202923982
#202923922
#408×C1072
#1881107