Finite Group Databasebeta

← #28346115

Subgroups — #28346115

Order 13696.

460 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C22×#13133808C211yesmaximal, characteristic, complemented
2C2×#16812117C211yesmaximal, characteristic
3#20293203C211yesmaximal, characteristic
4#20293201C211yesmaximal, characteristic
5#20292884C214yesmaximal, complemented
6#20292867C214yesmaximal, complemented
7#20292663C211yesmaximal, characteristic, complemented
8#20292637C211yesmaximal, characteristic, complemented
9#419×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×Dic214C2211yescharacteristic
12C2×#13133808—24no—
13C2×#13133808C2212yes—
14C2×#13133808C2212yes—
15#16812117—24no—
16C2×#13133801C2212yes—
17C22×C107⋊C8C2211yescharacteristic
18C2×C8×D107C2212yes—
19#16812109—21no—
20#16812107—21no—
21#16812098—21no—
22#16812097—21no—
23#16812093—22no—
24#16812092—22no—
25#16812089C2214yes—
26#16812085C2214yes—
27#16812068C2211yescharacteristic
28#16812042C2212yes—
29#16812040C2212yes—
30#16812034C2214yes—
31C22×Q8×C107C2211yescharacteristic
32C2×M16×C107C2211yescharacteristic
33#117×C107C2211yescharacteristic
34#103×C107C2214yes—
35#13133818—22no—
36#13133818—22no—
37C2×Dic214—22no—
38C2×Dic214—22no—
39C2×Dic214—21no—
40C2×Dic214D411yescharacteristic
41C2×Dic214C2312yes—
42C2×D428D411yescharacteristic
43C2×C22⋊D107D412yes—
44C22×Dic107—21no—
45C2×C4×D107C2312yes—
46#13133808—24no—
47#13133808—24no—
48#13133801—24no—
49C2×C107⋊C8D414yes—
50C2×C107⋊C8C2311yescharacteristic
51C2×C107⋊C8C2311yescharacteristic
52C8×D107—24no—
53C4⋊Dic107—24no—
54C4⋊Dic107C2312yes—
55Dic107⋊C4—24no—
56C22⋊Dic107—21no—
57(C2×D107)⋊C4—24no—
58(C2×D107)⋊C4—24no—
59C4×Dic107—21no—
60C2×Q8×C107—21no—
61C2×Q8×C107—22no—
62C2×Q8×C107—22no—
63C2×Q8×C107C2312yes—
64C22×C4×C107—21no—
65C22×C4×C107C2311yescharacteristic
66M16×C107—22no—
67C2×C8×C107C2312yes—
68C4⋊C4×C107C2312yes—
69C22⋊C4×C107—21no—
70C42×C107—21no—
71C23×C107C2×D411yescharacteristic
72C22×D107C2×D412yes—
73Q8×C107—22no—
74Q8×C107—22no—
75Q8×C107—42no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—42no—
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107C2×D412yes—
81C2×C4×C107#4112yes—
82C2×C4×C107C2411yescharacteristic
83C22⋊D107—24no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—42no—
87C2×Dic107C2×D412yes—
88D428—22no—
89C4×D107—22no—
90C4×D107—22no—
91Dic214—22no—
92Dic214—22no—
93Dic214—22no—
94Dic214—42no—
95C8×C107—22no—
96C107⋊C8C2×D412yes—
97C107⋊C8C2×D412yes—
98C22×C107—22no—
99C22×C107#12112yes—
100C22×C107C22×D411yescharacteristic
101C2×D107—22no—
102C2×D107—24no—
103C4×C107—42no—
104C4×C107—42no—
105C4×C107#12112yes—
106C4×C107C22×D411yescharacteristic
107C4×C107C2×#4111yescharacteristic
108Dic107—22no—
109Dic107—42no—
110Dic107—42no—
111C2×C107—22no—
112C2×C107C2×#12111yescharacteristic
113C2×C107#57812yes—
114D107—42no—
115#3059—1071nomaximal, Sylow
116C107#305911yescharacteristic, complemented, Sylow
117C22×Q16—1071no—
118C2×#131—1071no—
119#536—1071no—
120#536—1071no—
121#470—1074no—
122#467—1074no—
123#427—1071no—
124#419—1071no—
125#419D10711yescharacteristic, complemented
126C2×#41—1071no—
127C22×Q8—1071no—
128C22×Q8C2×D10711yescharacteristic
129C2×Q16—1072no—
130C2×Q16—1072no—
131C2×Q16—2144no—
132#131—2144no—
133C2×M16—1072no—
134C2×M16C2×D10711yescharacteristic
135C22×C8—1071no—
136C22×C8—1072no—
137C4⋊Q8—2141no—
138C4⋊Q8—2141no—
139#124—2141no—
140#124—2141no—
141#123—2142no—
142#123—2142no—
143C22⋊Q8—1074no—
144C22⋊Q8—1074no—
145#117—1071no—
146#117C2×D10711yescharacteristic
147#107—1072no—
148#106—1072no—
149#103—1074no—
150#103C2×D10714yes—
151#41—2142no—
152#41—2142no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1071no—
157C2×Q8—1072no—
158C2×Q8—2142no—
159C2×Q8—2142no—
160C2×Q8—2141no—
161C2×Q8C22×D10712yes—
162C2×D4—1072no—
163C2×D4—1071no—
164C22×C4—21no—
165C22×C4—1072no—
166C22×C4—2141no—
167C22×C4C22×D10711yescharacteristic
168Q16—2144no—
169Q16—2144no—
170M16—22no—
171M16—2144no—
172C2×C8—1074no—
173C2×C8—1071no—
174C2×C8—1071no—
175C2×C8—2144no—
176C2×C8C22×D10712yes—
177C4⋊C4—1072no—
178C4⋊C4—2144no—
179C4⋊C4—2144no—
180C4⋊C4C22×D10712yes—
181C22⋊C4—21no—
182C22⋊C4—2144no—
183C22⋊C4—2144no—
184C22⋊C4—2141no—
185C42—21no—
186C42—2141no—
187C23—1072no—
188C23D4×D10711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—4282no—
193Q8—2142no—
194Q8—2142no—
195Q8—2142no—
196D4—2144no—
197D4—2142no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—1072no—
202C2×C4—4282no—
203C2×C4—2142no—
204C2×C4—2142no—
205C2×C4—2142no—
206C2×C4—2142no—
207C2×C4D4×D10711yescharacteristic
208C2×C4D4×D10712yes—
209C2×C4#1313381812yes—
210C2×C4C23×D10711yescharacteristic
211C8—22no—
212C8—1072no—
213C8—1072no—
214C22—22no—
215C22—2144no—
216C22—2142no—
217C22#1681208212yes—
218C22C2×D4×D10711yescharacteristic
219C4—42no—
220C4—42no—
221C4—4282no—
222C4—4282no—
223C4—2142no—
224C4#1681208212yes—
225C4C2×D4×D10711yescharacteristic
226C4C2×#1313381811yescharacteristic
227C2—22no—
228C2—4282no—
229C2C2×#1681208211yescharacteristic
230C2#2029344312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#419×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenter#419D107
hypocenterC107#3059
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#13133808C2semidirect
5. #20292884C2semidirect
6. #20292867C2semidirect
7. #20292663C2semidirect
8. #20292637C2semidirect
9. #419×C107C2semidirect
116. C107#3059semidirect
125. #419D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338082
C2×#168121172
#202932032
#202932012
#202928842
#202928672
#202926632
#202926372
#419×C1072
#3059107