Finite Group Databasebeta

← #28345988

Subgroups — #28345988

Order 13696.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#16812140C211yesmaximal, characteristic
2#20293201C211yesmaximal, characteristic
3#20293105C211yesmaximal, characteristic, complemented
4#20292851C214yesmaximal, complemented
5#20292835C214yesmaximal, complemented
6#20292653C211yesmaximal, characteristic, complemented
7#20292636C211yesmaximal, characteristic, complemented
8#20292624C211yesmaximal, characteristic
9C22×Q16×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11#16812140—24no—
12C2×#13133808C2212yes—
13C2×#13133805C2212yes—
14C2×#13133802C2211yescharacteristic
15#16812107C2212yes—
16#16812098C2212yes—
17#16812092—22no—
18#16812085—22no—
19Q8×Dic107—22no—
20Q8×Dic107C2214yes—
21#16812068—22no—
22#16812068C2211yescharacteristic
23#16812068C2211yescharacteristic
24Dic107⋊C8C2214yes—
25#16812034C2212yes—
26#16812033C2214yes—
27#16812031C2212yes—
28C22×Q8×C107C2211yescharacteristic
29C22×Q8×C107C2211yescharacteristic
30C2×Q16×C107—24no—
31C2×Q16×C107C2214yes—
32C22×C8×C107C2211yescharacteristic
33#13133818—22no—
34#13133818—22no—
35C2×Dic214C2311yescharacteristic
36C2×D428C2311yescharacteristic
37C2×C22⋊D107—21no—
38C2×C4×D107—21no—
39#13133808—24no—
40#13133805—24no—
41#13133802—22no—
42C2×C107⋊C8C2312yes—
43C4⋊Dic107—24no—
44C4⋊Dic107D412yes—
45C4⋊Dic107C2312yes—
46Dic107⋊C4—24no—
47C22⋊Dic107—21no—
48C22⋊Dic107D412yes—
49(C2×D107)⋊C4—24no—
50C4×Dic107—21no—
51C4×Dic107—24no—
52C4×Dic107C2312yes—
53C2×Q8×C107—21no—
54C2×Q8×C107—21no—
55C2×Q8×C107—22no—
56C2×Q8×C107—22no—
57C2×Q8×C107—22no—
58C2×Q8×C107D414yes—
59C2×Q8×C107C2312yes—
60C2×Q8×C107C2312yes—
61C22×C4×C107—21no—
62C22×C4×C107—21no—
63C22×C4×C107C2311yescharacteristic
64Q16×C107—28no—
65C2×C8×C107—22no—
66C2×C8×C107C2312yes—
67C23×C107C2×D411yescharacteristic
68C22×D107—21no—
69Q8×C107—24no—
70Q8×C107—22no—
71Q8×C107—22no—
72Q8×C107—22no—
73Q8×C107—42no—
74Q8×C107C2×D414yes—
75C2×C4×C107—24no—
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—42no—
84C2×Dic107—22no—
85C2×Dic107—21no—
86C2×Dic107C2×D414yes—
87D428—21no—
88D428—21no—
89C4×D107—42no—
90Dic214—21no—
91Dic214—21no—
92C8×C107—22no—
93C107⋊C8—22no—
94C22×C107—22no—
95C22×C107#12112yes—
96C22×C107C22×D411yescharacteristic
97C2×D107—41no—
98C2×D107—42no—
99C4×C107—24no—
100C4×C107—42no—
101C4×C107#12112yes—
102C4×C107C22×D411yescharacteristic
103C4×C107C2×#4111yescharacteristic
104Dic107—24no—
105Dic107—41no—
106Dic107—42no—
107C2×C107—22no—
108C2×C107C2×#12111yescharacteristic
109C2×C107#57812yes—
110D107—81no—
111#3044—1071nomaximal, Sylow
112C107#304411yescharacteristic, complemented, Sylow
113C2×#137—1071no—
114C22×Q16D10711yescharacteristic, complemented
115#536—1071no—
116#519—1071no—
117#464—1074no—
118#462—1074no—
119#424—1071no—
120#419—1071no—
121#416—1071no—
122C2×#41—1071no—
123C22×Q8C2×D10711yescharacteristic
124C22×Q8C2×D10711yescharacteristic
125#137—2144no—
126C2×Q16—24no—
127C2×Q16—1072no—
128C2×Q16C2×D10714yes—
129C2×SD16—1072no—
130C2×M16—1071no—
131C22×C8C2×D10711yescharacteristic
132C4⋊Q8—1072no—
133#124—1072no—
134#123—2142no—
135C22⋊Q8—2142no—
136C4×Q8—1074no—
137C4×Q8—2142no—
138#117—1071no—
139#117—1071no—
140#117—2142no—
141C4⋊C8—1074no—
142#103—1074no—
143#103—1072no—
144#102—1072no—
145#41—2142no—
146#41—2142no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—1071no—
153C2×Q8C22⋊D10714yes—
154C2×Q8C22×D10712yes—
155C2×Q8C22×D10712yes—
156C2×D4—1071no—
157C2×D4—2141no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2141no—
161C22×C4C22×D10711yescharacteristic
162Q16—28no—
163Q16—2144no—
164SD16—2144no—
165M16—2142no—
166C2×C8—22no—
167C2×C8—1072no—
168C2×C8C22×D10712yes—
169C4⋊C4—1072no—
170C4⋊C4—1072no—
171C4⋊C4—2144no—
172C4⋊C4—2144no—
173C22⋊C4—1072no—
174C22⋊C4—2144no—
175C22⋊C4—2141no—
176C42—1072no—
177C42—2144no—
178C42—2141no—
179C23—2141no—
180C23D4×D10711yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—22no—
184Q8—22no—
185Q8—42no—
186Q8—2141no—
187Q8—2141no—
188Q8C2×C22⋊D10714yes—
189D4—4282no—
190D4—2141no—
191D4—2141no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—1074no—
197C2×C4—4282no—
198C2×C4—2142no—
199C2×C4—2141no—
200C2×C4D4×D10711yescharacteristic
201C2×C4C2×C22⋊D10712yes—
202C2×C4#1313381712yes—
203C2×C4C23×D10711yescharacteristic
204C8—22no—
205C8—2142no—
206C22—22no—
207C22—4282no—
208C22—4281no—
209C22#1681208112yes—
210C22C2×D4×D10711yescharacteristic
211C4—24no—
212C4—42no—
213C4—4282no—
214C4—4281no—
215C4—2144no—
216C4#1681208112yes—
217C4C22×C22⋊D10711yescharacteristic
218C4C2×#1313381711yescharacteristic
219C2—22no—
220C2—8561no—
221C2C2×#1681208111yescharacteristic
222C2#2029344012yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×Q16×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenterC22×Q16D107
hypocenterC107#3044
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
3. #20293105C2semidirect
4. #20292851C2semidirect
5. #20292835C2semidirect
6. #20292653C2semidirect
7. #20292636C2semidirect
9. C22×Q16×C107C2semidirect
112. C107#3044semidirect
114. C22×Q16D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
#202932012
#202931052
#202928512
#202928352
#202926532
#202926362
#202926242
C22×Q16×C1072
#3044107