Finite Group Databasebeta

← #28349613

Subgroups — #28349613

Order 13696.

350 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 113 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#20293441C211yesmaximal, characteristic, complemented
2#20293440C211yesmaximal, characteristic, complemented
3#20293070C211yesmaximal, characteristic, complemented
4#20293069C212yesmaximal, complemented
5#20293066C212yesmaximal, complemented
6#20293058C212yesmaximal, complemented
7#20293057C211yesmaximal, characteristic, complemented
8C2×#16812055C211yesmaximal, characteristic, complemented
9Q32×D107C212yesmaximal, complemented
10#20293045C211yesmaximal, characteristic, complemented
11#510×C107C211yesmaximal, characteristic, complemented
12#16812157—21no—
13#16812156—21no—
14#16812141—22no—
15#16812140—21no—
16#16812139—21no—
17#16812138—22no—
18#16812132C2212yes—
19#16812131C2211yescharacteristic
20#16812128C2212yes—
21C2×Dic428C2211yescharacteristic
22Q16×D107C2212yes—
23Q16×D107C2212yes—
24#16812115C2211yescharacteristic
25Dic856C2212yes—
26#16812055C2211yescharacteristic
27#16812055C2212yes—
28#16812055C2212yes—
29#16812054C2212yes—
30#16812053C2211yescharacteristic
31#16812052C2211yescharacteristic
32#16812050C2211yescharacteristic
33#16812048C2212yes—
34C2×C107⋊C16C2211yescharacteristic
35C16×D107C2212yes—
36#135×C107C2211yescharacteristic
37C2×Q16×C107C2211yescharacteristic
38Q32×C107C2212yes—
39SD32×C107C2212yes—
40M32×C107C2211yescharacteristic
41#13133819—22no—
42#13133819—41no—
43#13133818—21no—
44#13133818—42no—
45#13133818—41no—
46#13133818D411yescharacteristic
47#13133817—22no—
48#13133817—42no—
49C2×Dic214—21no—
50C2×Dic214—41no—
51Q8×D107—22no—
52Q8×D107—22no—
53Q8×D107—41no—
54Dic428—21no—
55Dic428C2311yescharacteristic
56Dic428C2312yes—
57#13133808—22no—
58#13133808—22no—
59#13133807—21no—
60#13133806—22no—
61#13133805—22no—
62D856C2311yescharacteristic
63#13133802D411yescharacteristic
64#13133801D412yes—
65C8×D107C2312yes—
66C107⋊C16C2311yescharacteristic
67C107⋊C16C2311yescharacteristic
68#41×C107—21no—
69C2×Q8×C107—21no—
70Q16×C107—21no—
71Q16×C107C2311yescharacteristic
72Q16×C107C2312yes—
73SD16×C107—21no—
74D8×C107C2311yescharacteristic
75C2×C8×C107C2311yescharacteristic
76C16×C107C2312yes—
77Q8×C107—21no—
78Q8×C107—22no—
79Q8×C107—41no—
80D4×C107—21no—
81D4×C107—41no—
82C2×C4×C107—41no—
83C2×C4×C107—41no—
84C2×C4×C107C2×D411yescharacteristic
85C22⋊D107—41no—
86C22⋊D107—42no—
87C22⋊D107D812yes—
88C2×Dic107—41no—
89C2×Dic107—42no—
90D428—21no—
91D428—42no—
92D428D811yescharacteristic
93C4×D107—42no—
94C4×D107—42no—
95C4×D107—41no—
96C4×D107—42no—
97C4×D107C2×D412yes—
98Dic214—22no—
99Dic214—21no—
100Dic214—41no—
101Dic214—42no—
102Dic214—42no—
103Dic214D811yescharacteristic
104C8×C107C2×D411yescharacteristic
105C8×C107C2411yescharacteristic
106C107⋊C8C2×D412yes—
107C22×C107—41no—
108C22×C107C2×D811yescharacteristic
109C2×D107—41no—
110C2×D107C2×D812yes—
111C4×C107—41no—
112C4×C107—42no—
113C4×C107C2×D811yescharacteristic
114C4×C107C22×D411yescharacteristic
115Dic107—41no—
116Dic107—42no—
117Dic107C2×D812yes—
118C2×C107—21no—
119C2×C107—81no—
120C2×C107C22×D811yescharacteristic
121D107—22no—
122D107—81no—
123#3417—1071nomaximal, Sylow
124C107#341711yescharacteristic, complemented, Sylow
125#578—1071no—
126#578—1071no—
127#510—1072no—
128#510—1072no—
129#510—1071no—
130#510D10711yescharacteristic, complemented
131#508—1071no—
132#508—1072no—
133C2×Q32—1071no—
134C2×Q32—1072no—
135#504—1071no—
1362−1+4—2141no—
1372−1+4—2141no—
138#137—2142no—
139#137—2142no—
140#137—2141no—
141#137—2141no—
142#135—1072no—
143#135—1072no—
144#135—1071no—
145#135C2×D10711yescharacteristic
146C2×Q16—1072no—
147C2×Q16—1072no—
148C2×Q16—1071no—
149C2×Q16C2×D10711yescharacteristic
150#131—1071no—
151Q32—1071no—
152Q32—1072no—
153Q32—1072no—
154Q32—1072no—
155Q32C2×D10712yes—
156SD32—1071no—
157SD32—1071no—
158SD32—1072no—
159SD32C2×D10712yes—
160D16—1071no—
161M32—1072no—
162M32C2×D10711yescharacteristic
163C2×C16—1071no—
164C2×C16—1072no—
165#41—21no—
166#41—1071no—
167#41—4282no—
168#41—4281no—
169#41—4282no—
170#41—4281no—
171#41—2142no—
172#41—2142no—
173#41—2141no—
174C2×Q8—21no—
175C2×Q8—4281no—
176C2×Q8—4281no—
177C2×Q8—2142no—
178C2×Q8—2142no—
179C2×Q8—2141no—
180Q16—21no—
181Q16—1071no—
182Q16—1072no—
183Q16—2142no—
184Q16—2142no—
185Q16—2141no—
186Q16C22×D10711yescharacteristic
187Q16C22×D10712yes—
188SD16—21no—
189SD16—2142no—
190SD16—2142no—
191SD16—2141no—
192D8—1071no—
193D8C22×D10711yescharacteristic
194M16—1071no—
195M16—1072no—
196C2×C8—1072no—
197C2×C8C22×D10711yescharacteristic
198C16—1071no—
199C16—1071no—
200C16C22×D10712yes—
201Q8—21no—
202Q8—22no—
203Q8—41no—
204Q8—1071no—
205Q8—4282no—
206Q8—4282no—
207Q8—4281no—
208Q8—2142no—
209Q8—2141no—
210D4—21no—
211D4—41no—
212D4—1072no—
213D4—1071no—
214D4—4282no—
215D4—4282no—
216D4—4281no—
217D4—2141no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—1072no—
221C2×C4—4282no—
222C2×C4—4282no—
223C2×C4—4282no—
224C2×C4—4282no—
225C2×C4—4281no—
226C2×C4—4281no—
227C2×C4D4×D10711yescharacteristic
228C8—1072no—
229C8D4×D10711yescharacteristic
230C8C23×D10711yescharacteristic
231C22—41no—
232C22—1072no—
233C22—4281no—
234C22D8×D10711yescharacteristic
235C4—41no—
236C4—42no—
237C4—1072no—
238C4—4281no—
239C4—4282no—
240C4D8×D10711yescharacteristic
241C4C2×D4×D10711yescharacteristic
242C2—21no—
243C2—81no—
244C2—8561no—
245C2—2142no—
246C2C2×D8×D10711yescharacteristic
subgroupG/H
centerC2C2×D8×D107
commutator, cosocleC8×C107C24
FrattiniC8C23×D107
Fitting#510×C107C2
socleC2×C107C22×D8
hypercenter#510D107
hypocenterC107#3417
LCSG ▷ C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #20293441C2semidirect
2. #20293440C2semidirect
3. #20293070C2semidirect
4. #20293069C2semidirect
5. #20293066C2semidirect
6. #20293058C2semidirect
7. #20293057C2semidirect
8. C2×#16812055C2semidirect
9. Q32×D107C2semidirect
10. #20293045C2semidirect
11. #510×C107C2semidirect
124. C107#3417semidirect
130. #510D107semidirect

Maximal subgroups

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

subgroupindex
#202934412
#202934402
#202930702
#202930692
#202930662
#202930582
#202930572
C2×#168120552
Q32×D1072
#202930452
#510×C1072
#3417107