Finite Group Databasebeta

← #28349519

Subgroups — #28349519

Order 13696.

362 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293363C211yesmaximal, characteristic
2#20293352C211yesmaximal, characteristic, complemented
3#20293018C212yesmaximal, complemented
4#20292847C212yesmaximal, complemented
5#20292834C211yesmaximal, characteristic, complemented
6#20292830C212yesmaximal, complemented
7#20292824C211yesmaximal, characteristic
8#20292757C211yesmaximal, characteristic, complemented
9#20292728C212yesmaximal, complemented
10C4×Dic428C211yesmaximal, characteristic, complemented
11#492×C107C211yesmaximal, characteristic
12C2×Dic428C2211yescharacteristic
13C2×#13133808C2212yes—
14C2×#13133806C2212yes—
15#16812107C2212yes—
16#16812100—21no—
17#16812099—22no—
18#16812096—21no—
19#16812096C2212yes—
20#16812090—22no—
21Q8×Dic107C2212yes—
22C4×Dic214C2211yescharacteristic
23C4×Dic214C2211yescharacteristic
24C4×Dic214C2211yescharacteristic
25#16812074—21no—
26D4×Dic107—21no—
27D4×Dic107C2212yes—
28#16812043C2211yescharacteristic
29C4⋊(C107⋊C8)C2211yescharacteristic
30Dic107⋊C8C2212yes—
31#16812034C2211yescharacteristic
32#16812034C2211yescharacteristic
33#16812033C2212yes—
34#16812030C2212yes—
35#16812019C2212yes—
36C2×SD16×C107C2212yes—
37C2×SD16×C107C2212yes—
38C4⋊Q8×C107C2211yescharacteristic
39#127×C107C2211yescharacteristic
40C4×C8×C107C2211yescharacteristic
41C2×Dic214D411yescharacteristic
42C2×Dic214C2311yescharacteristic
43C2×Dic214C2311yescharacteristic
44C22×Dic107—22no—
45Dic428—22no—
46#13133808—42no—
47#13133806—42no—
48C2×C107⋊C8C2312yes—
49C4⋊Dic107—22no—
50C4⋊Dic107D411yescharacteristic
51C4⋊Dic107C2311yescharacteristic
52C4⋊Dic107C2311yescharacteristic
53Dic107⋊C4—21no—
54Dic107⋊C4—21no—
55Dic107⋊C4—22no—
56Dic107⋊C4—22no—
57Dic107⋊C4D412yes—
58C22⋊Dic107—22no—
59C22⋊Dic107—22no—
60C22⋊Dic107—22no—
61C4×Dic107—21no—
62C4×Dic107—21no—
63C4×Dic107—22no—
64C4×Dic107C2312yes—
65C2×Q8×C107C2312yes—
66C2×D4×C107—21no—
67C2×D4×C107—21no—
68C2×D4×C107C2312yes—
69SD16×C107—24no—
70SD16×C107—42no—
71C2×C8×C107C2311yescharacteristic
72C2×C8×C107C2311yescharacteristic
73C4⋊C4×C107—21no—
74C4⋊C4×C107—21no—
75C42×C107C2311yescharacteristic
76C23×C107—22no—
77Q8×C107—22no—
78Q8×C107—22no—
79D4×C107—22no—
80D4×C107—22no—
81D4×C107—42no—
82D4×C107—81no—
83C2×C4×C107—22no—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2411yescharacteristic
87C2×Dic107—21no—
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107—21no—
91C2×Dic107—82no—
92C2×Dic107C2×D412yes—
93C2×Dic107C2×D412yes—
94Dic214—22no—
95Dic214—21no—
96Dic214—21no—
97Dic214—21no—
98Dic214—21no—
99C8×C107—21no—
100C8×C107#4112yes—
101C107⋊C8—22no—
102C22×C107—42no—
103C22×C107—42no—
104C22×C107—42no—
105C22×C107C22×D411yescharacteristic
106C4×C107—21no—
107C4×C107—42no—
108C4×C107#13712yes—
109C4×C107C2×#4111yescharacteristic
110C4×C107H(5,𝔽2)11yescharacteristic
111Dic107—22no—
112Dic107—22no—
113Dic107—41no—
114Dic107—41no—
115Dic107—41no—
116Dic107—41no—
117C2×C107—82no—
118C2×C107#55011yescharacteristic
119C2×C107C2×#13711yescharacteristic
120C2×C107#57711yescharacteristic
121#3400—1071nomaximal, Sylow
122C107#340011yescharacteristic, complemented, Sylow
123#559—1071no—
124#557—1071no—
125#497—1072no—
126#492D10711yescharacteristic
127#464—1072no—
128#462—1071no—
129#461—1072no—
130#460—1071no—
131#446—1071no—
132#440—1072no—
133C4×Q16—1071no—
134C2×Q16—1072no—
135C2×Q16—1071no—
136C2×SD16—1072no—
137C2×SD16C2×D10712yes—
138C2×SD16C2×D10712yes—
139C4⋊Q8—1072no—
140C4⋊Q8C2×D10711yescharacteristic
141#127C2×D10711yescharacteristic
142#125—2142no—
143#125—2141no—
144#124—1072no—
145#124—2141no—
146#123—2142no—
147C4×Q8—1071no—
148C4×Q8—1072no—
149C4×Q8—1071no—
150C4×Q8—1071no—
151C4×Q8—2141no—
152C4×D4—1072no—
153C4×D4—2141no—
154#107—1071no—
155C4⋊C8—1072no—
156C4⋊C8—1071no—
157#103—1072no—
158#103—1071no—
159#103—1071no—
160#102—1072no—
161#97—1072no—
162C4×C8C2×D10711yescharacteristic
163C2×Q8—1071no—
164C2×Q8—1071no—
165C2×Q8—1071no—
166C2×Q8C22×D10712yes—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4C22×D10712yes—
170C22×C4—2142no—
171Q16—4282no—
172Q16—2142no—
173SD16—24no—
174SD16—42no—
175SD16—4282no—
176C2×C8—1072no—
177C2×C8C22×D10711yescharacteristic
178C2×C8C22×D10711yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—1071no—
182C4⋊C4—1072no—
183C4⋊C4—1071no—
184C4⋊C4—1071no—
185C4⋊C4—2142no—
186C4⋊C4—2142no—
187C4⋊C4—2142no—
188C4⋊C4—2141no—
189C4⋊C4—2141no—
190C22⋊C4—2142no—
191C22⋊C4—2142no—
192C22⋊C4—2142no—
193C42—1072no—
194C42—2142no—
195C42—2141no—
196C42—2141no—
197C42C22×D10711yescharacteristic
198C23—22no—
199Q8—22no—
200Q8—22no—
201Q8—2141no—
202Q8—2141no—
203Q8—2141no—
204Q8—2141no—
205Q8—2142no—
206D4—22no—
207D4—22no—
208D4—42no—
209D4—81no—
210C2×C4—22no—
211C2×C4—8562no—
212C2×C4—1072no—
213C2×C4—1072no—
214C2×C4—2141no—
215C2×C4—2141no—
216C2×C4—2141no—
217C2×C4—2141no—
218C2×C4D4×D10711yescharacteristic
219C2×C4D4×D10711yescharacteristic
220C2×C4C23×D10711yescharacteristic
221C8—21no—
222C8—2142no—
223C8#1313381712yes—
224C22—42no—
225C22—42no—
226C22—42no—
227C22C2×D4×D10711yescharacteristic
228C4—21no—
229C4—42no—
230C4—4281no—
231C4—4281no—
232C4—4281no—
233C4—4281no—
234C4—2142no—
235C4—2142no—
236C4#1681213812yes—
237C4C2×#1313381711yescharacteristic
238C4#1681215411yescharacteristic
239C2—82no—
240C2#2029329711yescharacteristic
241C2C2×#1681213811yescharacteristic
242C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#492×C107C2
socleC22×C107C22×D4
prime radical#127×C107C22
hypercenter#492D107
hypocenterC107#3400
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #20293352C2semidirect
3. #20293018C2semidirect
4. #20292847C2semidirect
5. #20292834C2semidirect
6. #20292830C2semidirect
8. #20292757C2semidirect
9. #20292728C2semidirect
10. C4×Dic428C2semidirect
122. C107#3400semidirect

Maximal subgroups

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

subgroupindex
#202933632
#202933522
#202930182
#202928472
#202928342
#202928302
#202928242
#202927572
#202927282
C4×Dic4282
#492×C1072
#3400107